Competitive Strategy: Options and Games 0262015994, 9780262015998

Citation preview

Competitive Strategy

Competitive Strategy Options and Games

Benoît Chevalier-Roignant Lenos Trigeorgis

The MIT Press Cambridge, Massachusetts London, England

© 2011 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please email [email protected] .edu This book was set in Times Roman by Toppan Best-set Premedia Limited. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Chevalier-Roignant, Benoit, 1983– Competitive strategy: options and games/Benoit Chevalier-Roignant and Lenos Trigeorgis; foreword by Avinash K. Dixit. p. cm. Includes bibliographical references and index. ISBN 978-0-262-01599-8 (hardcover: alk. paper) 1. Options (Finance) 2. Investment analysis. 3. Game theory. I. Trigeorgis, Lenos. II. Title. HG6024.A3C48535 2011 332.6—dc22 2010053619 10

9

8

7

6

5

4

3

2

1

Contents

Glossary xi Symbols xix Foreword by Avinash Dixit Preface xxv 1

The Strategy Challenge

xxiii

1

1.1 1.2 1.3

The Changing Corporate Environment 3 What Is Strategy? 10 Two Complementary Perspectives on Strategy 1.3.1 Corporate Finance and Strategy 15 1.3.2 Game Theory and Strategy 20 1.4 An Integrative Approach to Strategy 35 1.5 Overview and Organization of the Book 41 Conclusion 43 Selected References 43 I

STRATEGY, GAMES, AND OPTIONS

2

Strategic Management and Competitive Advantage 2.1

2.2

15

45 47

Strategic Management Paradigms 48 2.1.1 External View of the Firm 48 2.1.2 Internal View of the Firm 50 Industry and Competitive Analysis 50 2.2.1 Macroeconomic Analysis 57 2.2.2 Industry Analysis: Structure–Conduct–Performance Paradigm 58 2.2.3 Porter’s Industry and Competitive (Five-Forces) Analysis 59

vi

Contents

2.3

Creating and Sustaining Competitive Advantage 2.3.1 Value Creation 66 2.3.2 Generic Competitive Strategies 70 2.3.3 Sustaining Competitive Advantage 72 Conclusion 73 Selected References 74 3

Market Structure Games: Static Approaches

66

75

3.1 3.2

Monopoly 76 Duopoly 81 3.2.1 Bertrand Price Competition 86 3.2.2 Cournot Quantity Competition 92 3.2.3 Strategic Substitutes versus Complements 97 3.3 Oligopoly and Perfect Quantity Competition 99 3.4 Market Structure under Incomplete Information 102 Conclusion 107 Selected References 107 4

Market Structure Games: Dynamic Approaches

109

4.1

Commitment Strategy 110 4.1.1 Concept of Commitment 111 4.1.2 Taxonomy of Commitment Strategies 118 4.1.3 Sequential Stackelberg Game 131 4.2 Bargaining and Cooperation 135 4.2.1 Bargaining 135 4.2.2 Cooperation between Cournot Duopolists in Repeated Games 138 4.2.3 Co-opetition: Sometimes Compete and Sometimes Cooperate? 147 Conclusion 151 Selected References 151 5

Uncertainty, Flexibility, and Real Options 5.1

5.2

153

Strategic Investment under Uncertainty—The Electricity Sector 154 5.1.1 Need for New Investment in Europe 154 5.1.2 Sources of Uncertainty 156 5.1.3 Generation Technologies and Business Risk Exposure 159 Common Real Options 162

Contents

vii

5.3

Basic Option Valuation 169 5.3.1 Discrete-Time Option Valuation 176 5.3.2 Continuous-Time Options Analysis 184 Conclusion 189 Selected References 189 Appendix 5A Multistep Cox–Ross–Rubinstein (CRR) Option Pricing 190 II

OPTION GAMES: DISCRETE-TIME ANALYSIS

6

An Integrative Approach to Strategy: Option Games

193

6.1

195

Key Managerial Issues: Optimal Timing and Flexibility versus Commitment 196 6.1.1 Optimal Investment Timing under Uncertainty 196 6.1.2 The Trade-off between Flexibility and Commitment 197 6.2 An Illustration of Option Games 197 6.3 Patent-Fight Strategies 206 6.4 An Application in the Mining/Chemicals Industry 209 Conclusion 217 Selected References 217 7

Option to Invest

219

7.1 7.2

Deferral Option of a Monopolist 219 Quantity Competition under Uncertainty 224 7.2.1 Cournot Duopoly 224 7.2.2 Asymmetric Cournot Oligopoly 235 7.3 Differentiated Bertrand Price Competition 238 Conclusion 240 Selected References 242 8

Innovation Investment in Two-Stage Games 8.1 8.2

243

Innovation and Spillover Effects 243 Innovation and Patent Licensing 253 8.2.1 Patent Licensing: Deterministic Case 253 8.2.2 Patent Licensing under Uncertainty 261 8.3 Goodwill/Advertising Strategies 264 Conclusion 271 Selected References 272

viii

Contents

III

OPTION GAMES: CONTINUOUS-TIME MODELS

9

Monopoly: Investment and Expansion Options

275

277

9.1

Option to Invest (Defer) by a Monopolist 278 9.1.1 Deterministic Case 281 9.1.2 Stochastic Case 284 9.2 Option to Expand Capacity 298 9.2.1 Additional (Lumpy) Capacity Investment 299 9.2.2 Incremental Capacity Investment 303 Conclusion 306 Selected References 307 Appendix 9A Contingent-Claims Analysis of the Option to Invest in Monopoly 308 10

Oligopoly: Simultaneous Investment

311

10.1 Oligopoly: Additional Capacity Investment 312 10.1.1 Existing Market Model: Expansion Option 312 10.1.2 New Market Model: Investment (Defer) Option 314 10.2 Oligopoly: Incremental Capacity Investment 317 10.3 Perfect Competition and Social Optimality 322 Conclusion 325 Selected References 325 Appendix 10A Derivation Based on Dynamic Programming 326 11

Leadership and Early-Mover Advantage

331

11.1 A Basic Framework for Sequential Investment in a Duopoly 331 11.2 Duopoly with Sequential Investment under Uncertainty 339 11.3 Oligopoly with Sequential Investment under Uncertainty 348 11.4 Option to Expand Capacity 352 Conclusion 357 Selected References 357 12

Preemption versus Collaboration in a Duopoly

359

12.1 Preemption versus Cooperation 360 12.1.1 Preemption 363 12.1.2 Cooperation in an Existing Market 370 12.2 Option to Invest in a New Market under Uncertainty

372

Contents

ix

12.2.1 Symmetric Case 373 12.2.2 Asymmetric Case 379 12.2.3 Size of Competitive (Cost) Advantage 382 12.3 Option to Expand an Existing Market 389 12.3.1 Symmetric Case 389 12.3.2 Asymmetric Case 393 Conclusion 395 Selected References 396 Appendix 12A Strategy Space and Solution Concept 397 Appendix 12B Perfect Equilibrium in Deterministic Setting 398 Appendix 12C Perfect Equilibrium in Stochastic Setting 400 13

Extensions and Other Applications

403

13.1 Exogenous Competition and Random Entry 404 13.2 Real-Estate Development 405 13.3 R&D and Patenting Applications 407 13.4 Investment with Information Asymmetry 411 13.5 Exit Strategies 415 13.6 Optimal Capacity Utilization 417 13.7 Lumpy Capacity Expansion (Repeated) 419 13.8 Other Extensions and Applications 421 Conclusion 423 Selected References 423 Appendix: Basics of Stochastic Processes

425

A.1 Continuous-Time Stochastic Processes 426 A.1.1 Brownian Motion 427 A.1.2 Mean-Reversion Process 438 A.1.3 General Itô Processes 440 A.2 Forward Net Present Value 441 A.3 First-Hitting Time and Expected Discount Factor 447 A.3.1 Exercise Timing and First-Hitting Time 447 A.3.2 Expected Discount Factor 448 A.3.3 Profit-Flow Stream with Stochastic Termination 452 A.4 Optimal Stopping 453 Conclusion 458 Selected References 458 References 461 Index 473

Glossary

Accommodated entry Entry is accommodated if structural entry barriers are low and entry-deterring strategies are ineffective or too costly. An incumbent attempts to adopt strategies and build competitive advantage early on to cushion the future negative effects of accommodated competitive entry. Action An action or move by one of the players in a simultaneous game is a choice or decision she can make that affects the other player(s). Barriers to entry Barriers or factors that allow an incumbent firm to earn positive economic profits or excess rents by making it unprofitable for newcomers to enter the industry. Competitive markets are characterized by low entry barriers. Blockaded entry A condition where the incumbent need not undertake any entry-deterring strategies to enjoy monopoly rents in the marketplace. Structural or administrative barriers are sufficient conditions for blockaded market entry. Call option A contract or situation that gives its holder the right, but not the obligation, to buy or acquire an underlying asset at a predetermined price over a specified period (maturity). Chicken game See “war of attrition.” Closed-form solution A solution that gives an analytical answer to a mathematical formulation. Closed-loop strategies In dynamic games, closed-loop strategies allow players to condition their play on both the previous moves by the players and on calendar time. All past actions of all players is common knowledge at the beginning of each stage. Closed-loop equilibrium The equilibrium forms a perfect equilibrium in closed-loop strategies; that is, the closed-loop strategies form a Nash

xii

Glossary

equilibrium in each and every subgame (on and off the equilibrium path). Commitment

See “strategic commitment.”

Commitment value The incremental value (positive or negative) accruing to a firm from making a strategic commitment. Cost advantage One of the main strategies to achieve a competitive advantage is to seek to attain lower costs, while maintaining a perceived benefit comparable to that of competitors. A relative cost advantage influences the optimal investment timing of competing firms. Cost of capital The rate of return expected or required by an equally risky asset or investment to induce investors to provide capital to the firm. The cost of capital reflects the systematic (or market) risk of a traded asset perfectly correlated with the investment or asset to be valued. Deterred entry A situation occurring when an incumbent can keep an entrant out by employing an entry-deterring strategy. Dominant strategy A strategy that is the best decision for a player regardless of the action or strategy chosen by its opponent. Early-mover advantage An advantage from moving early that enables a firm to make a higher economic profit than its rivals. An early-mover advantage can stem from the uncontested market presence of a leader that enjoys monopoly rents for some time. Sources of sustainable earlymover advantages include the learning-curve effect (economies of cumulative production and learning), brand-name reputation—especially in situations where buyers are uncertain about product quality (“experience goods”)—, or high customer switching costs. Economic profit A profit concept that represents the difference between the profits earned by investing resources in a particular activity and the profits that would have been earned by investing the same resources in the best alternative activity in the market. Opportunity costs are subsumed in economic profits. Economies of scale Cost savings achieved when the unit production cost of a product decreases with the number of units produced. Economies of scope These involve cost savings externalities among product lines or production activities. Elasticity The elasticity of a variable with respect to a given parameter is the percentage change in the value of the variable resulting from a one

Glossary

xiii

percent change in the value of the parameter, all other factors remaining constant. European option An option that can only be exercised at maturity, not earlier. Extended (or expanded) net present value (E-NPV) The total value of a project including the option or flexibility value and the impact of any strategic commitment or interaction effects. Fixed costs Costs that are independent of the scale of production and are locked in for a given period of time. Some distinguish among fixed and sunk costs though the difference is subtle: “fixed” often refers to short-term commitments, while “sunk” or nonrecoverable costs generally involve a longer planning horizon. Focal-point argument In the event of multiple equilibria, the focalpoint argument suggests that an equilibrium that appears “natural” or logically compelling is the one more likely to arise. Games of incomplete information Games where players do not know some relevant characteristics of their opponents (e.g., their payoffs, available action sets or their beliefs). Games of perfect information Games where all players possess all relevant information. Extensive-form games of perfect information have the property that there is exactly one node or decision point in each information set. Game theory A branch of mathematics and economic sciences concerned with the analysis of optimal decision-making in multiplayer settings. Chess is a well-known example. Solution concepts meant to provide predictions on the likely outcomes impose certain behavioral restrictions on players. Information set The information a decision maker has at the moment she makes a decision. Isolating mechanisms Economic forces put in place by a firm to limit the extent to which its competitive advantage can be duplicated, eroded, or neutralized through the resource-creation activities of competitors. Learning-curve effect Learning-based cost advantage that results from accumulating experience and know-how in productive activities over time. Market structure Characteristic of a market in terms of number and size (power) distribution of firms. Monopoly, duopoly, and perfect competition are classic examples of market structure.

xiv

Glossary

Markov property This property asserts that for certain stochastic processes or dynamic problems, all the past relevant information is summarized into the latest value of a variable or price. One cannot therefore use past information to predict the future state of the variable. Efficient markets and Itô processes have this Markov property. Maturity The period or last moment at which an option can be exercised. If the option can be exercised before this date, it is called an American option; if only at maturity, a European option. Myopic strategy A strategy that does not take into account or is independent of the investment decisions of rivals. Nash equilibrium A classic equilibrium solution concept used in the analysis of multiplayer games: each player pursues its individual optimizing actions given the best actions of the other players. In equilibrium, no player has an incentive to unilaterally deviate. Nature Acts of nature are treated as the actions of a quasi-player that makes random choices at specified points in a game. Net present value (NPV) The NPV paradigm is well established in corporate finance. The net present value of a project is the present value of the expected cash flows minus the (present value of) required investment costs. In the absence of managerial flexibility, a stand-alone investment is deemed acceptable if its net present value is positive. Node A decision point in a game at which a player (or nature) can take an action. Open-loop strategies Strategies where players cannot observe the previous play of the opponents and therefore condition their play on calendar time only. Such strategies are also called “precommitment strategies.” Open-loop equilibrium A Nash equilibrium solution that is obtained when firms adopt open-loop strategies ignoring their rivals’ actions over the history of the game. Option A contract or situation that gives its holder the right but not the obligation to buy or acquire (if a call) or sell (if a put) a specified asset (e.g., common stock or project) by paying a specified cost (the exercise or strike price) on or before a specified date (the expiration or maturity date). If the option can be exercised before maturity, it is an American option; if only at maturity, a European option. Option to invest or option to defer An American-type call option embedded in projects where management has the right (but no obliga-

Glossary

xv

tion) to delay the project start for a certain time period. The exercise price is the cost needed to initiate the project. Option valuation Valuation process by which the total value or “expanded net present value” (E-NPV) of an investment opportunity is determined. The valuation approach is meant to capture management’s flexibility to adapt its decisions to the evolving uncertain circumstances. Path dependence A situation occurring when past circumstances or history condition the current outcome and can preclude or favor certain path evolutions in the future. Payoff The utility, reward, or value a player receives when the game is played out. In the option games context, the payoff can be the real option value (e.g., the value of a plant with the option to expand production). Perfect Nash equilibrium

See “subgame perfect Nash equilibrium.”

Players The individuals, firms, or actors that make decisions in a game situation. Each player’s goal is to maximize her payoff or value by choosing the best action or sequence of actions (strategy). Precommitment strategies See “open-loop strategies.” Preemption A situation whereby a firm invests ahead of its rivals to hinder their entry or profitable operation. Such a situation is often related to the presence of some first-mover advantage. Proprietary option An option held by a firm that entitles it to the full exclusive benefits resulting from exploiting the option. A monopolist firm has a proprietary investment option. Real option The flexibility arising when a decision maker has the opportunity to adapt or tailor a future decision to information and developments that will be revealed in the future. A real option conveys the right, but not the obligation, to take an action (e.g., defer, expand, contract, or abandon a project) at a specified cost (the exercise price) for a certain period of time, contingent on the resolution of some exogenous (e.g., demand) uncertainty. Real options analysis The field of application of option-pricing theory to valuing real investment decisions. Risk-neutral valuation A valuation method underlying option pricing analysis that adjusts for risk in the expectation of cash flows (certaintyequivalent), enabling discounting of future values at the risk-free interest rate. This contrasts to the standard valuation approach (NPV) consisting

xvi

Glossary

in discounting the (actual) expected cash flows at a (higher) risk-adjusted discount rate. Risk neutral Situations where investors are indifferent between a sure payoff (certainty-equivalent) and a risky outcome of equal expected value. By extension, the same description is used for the corresponding valuation method. Shared option An option simultaneously held or shared by several firms in the industry. Shared options characterize competitive industries where incumbent firms share the same investment opportunities. Shared options are more involved to analyze as they must account for rival reactions or interactions. Option holders’ investment behavior and project values are affected by the proprietary or shared nature of real options. Soft commitment (or accommodating stance) A strategic investment commitment that makes the rival firm better off in the (later) competition stage. In Cournot quantity competition, a soft commitment leads the committing firm to produce relatively less, while in Bertrand price competition a soft commitment induces the firm to maintain a higher price. Stochastic processes A set or collection of random variables such that the value of the process at any future time t is random though specified by a given probability distribution. Strategic commitment A strategic decision or move intended to alter the competitors’ behavior or beliefs about future market competition. Such commitments are generally difficult or costly to reverse. Strategic complements (or reciprocating actions) These characterize actions in situations where firms react (in equilibrium) in a reciprocating or complementary manner (i.e., reaction functions are upward sloping). For example, I will be nice to you if you are nice to me. In case of price competition, price-setting actions are strategic complements. Strategic substitutes (or contrarian actions) These characterize actions in situations where firms react (in equilibrium) in a contrarian or opposite manner (i.e., reaction functions are downward sloping). For example, you take advantage of me if I am nice to you. In case of quantity competition, capacity-setting actions are strategic substitutes. Strategy A behavioral rule adopted by a player that prescribes which contingent action(s) to choose at each stage in a game. A strategy specifies for each decision node or information set which action to pursue. Open-loop and closed-loop strategies refer to different strategy types:

Glossary

xvii

open-loop strategies make more restrictive assumptions on the information the players possess over the game play. Subgame perfect Nash equilibrium Solution concept used in dynamic games under complete information: (continuation) strategies form a Nash equilibrium at each and every stage of the game—even those stages that will not be actually played in equilibrium (“off the equilibrium path”). Sunk costs Investment costs that cannot be recouped. Once incurred, sunk costs are irrelevant for future decision-making. Tough commitment (or aggressive stance) A strategic commitment intended to hurt the rival. In Cournot quantity competition, tough commitment induces the committing firm to produce more, while in Bertrand price competition the firm will cut its price and enter a price war. Variable costs Costs, such as direct labor or commissions to sales people, that vary as the output level rises. War of attrition (or chicken game) In a duopoly involving a secondmover (follower) advantage, both firms have an incentive to be a follower or wait to be the last to move, leading to a “war of attrition.”

Symbols

i, j

Competing firms (in a duopoly)

−i

All other firms except firm i (in oligopoly)

n

Number of firms (in a competitive industry or market)

π i (⋅) πɶ i (⋅)

Firm i ’s certain (deterministic) profit function

p (⋅)

Deterministic (inverse) demand function

pɶ (⋅)

Uncertain (inverse) demand function

a, b

Known constant parameters in the (inverse) demand function

qi

Quantity produced by firm i

Q

Total quantity produced by all firms in the industry

ci

Firm i ’s variable (unit) cost

Ki

Up-front strategic investment outlay by firm i (in commitment games)

s

Degree of substitution (in differentiated Bertrand price competition)

x γ

R&D effort (in R&D investment games) Degree of R&D spillover (sharing) effects

k

Risk-adjusted discount rate (cost of capital)

r

(Instantaneous) risk-free interest rate

δ

Asset cash flow or dividend yield, convenience yield (for commodities), competitive erosion, or opportunity cost of waiting

g

Actual growth rate (drift parameter) for geometric Brownian motion

gˆ

Risk-neutral growth rate (for geometric Brownian motion)

Firm i ’s uncertain (stochastic) profit function

xx

Symbols

α αˆ

Actual growth rate for arithmetic Brownian motion

σ

Volatility (for the arithmetic or geometric Brownian motion)

h

Small time interval

dt

Infinitesimal time interval (h → dt )

q

Actual (empirically observed) probability (of up move)

p

Risk-neutral probability (of up move)

u

Up-move multiplicative factor in a binomial tree (process)

d

Down-move multiplicative factor in a binomial tree (process)

Ii

Investment outlay (cost) for firm i

e

Expansion factor (base value V expanded by e percent upon paying investment cost I , giving eV − I)

λ

Poisson jump arrival rate (frequency)

ρ

Correlation coefficient

E0 [⋅]

Expectation based on actual probabilities (conditional on time- 0 information)

Eˆ 0 [⋅]

Risk-adjusted expectation based on risk-neutral probabilities (conditional on time-0 information)

var ( )

Variance (or σ 2 )

t0

Starting time

t

Current time

T Tɶ

Time (years) to maturity of the option (in discrete-time models)

Tɶ *

Optimal random first-hitting time (i.e., the random time the optimal investment trigger X * is first reached)

Bt (Tɶ )

Risk-neutral growth rate for arithmetic Brownian motion

random first-hitting time (i.e., the random time at which the pre-set investment trigger XT is first reached)

Value (at time t ) of a bond paying 1 euro at random future time Tɶ . It also represents the expected discount factor.

zt

A standard Brownian motion or Wiener process

εp

Price elasticity of demand (ε p (Q) ≡ −[∂Q ∂p] × ( p Q))

β (βˆ )

Dummy variable of the fundamental quadratic function (^ for the risk-neutral version)

β1 (βˆ 1) Positive root of the fundamental quadratic function (elasticity of the option to invest)

Symbols

xxi

β 2 (βˆ 2 ) Negative root of the fundamental quadratic function (elasticity of the option to exit) x0

Time-0 value of the stochastic process in the case of arithmetic Brownian motion

xɶ t

Stochastic process value at time t in the case of arithmetic Brownian motion ( xɶ t ≡ ln ( Xɶ t ))

X0

Time-0 value of the stochastic process in the case of geometric Brownian motion

Xɶ t

Time-t value of the process in the case of geometric Brownian motion

XT

Value of the stochastic process at maturity T in discrete-time option games, or the preset investment trigger—not necessarily the optimal one—in continuous-time option games

X*

Optimal investment trigger (*)

M

Monopolist firm’s total value (with investment timing or expansion option)

S

Total firm value (with investment timing or expansion option) in an oligopoly characterized by simultaneous investment

L

Leader’s value (with investment timing or expansion option) in sequential investment option games

F

Follower’s value (with investment or expansion option) in sequential investment option games

*

(Superscript) denotes optimal or equilibrium value

^

(Superscript) denotes risk-neutral or risk-adjusted expectation, variable or parameter Euro

M

Million

Bn

Billion

≡

Means “is defined to be”

Foreword

President Truman once said: “Give me a one-handed economist. All of my economic advisers say ‘On the one hand this, on the other hand that.’” Economists do indeed recognize that there are multiple forces at work in most situations, and it takes quite subtle analysis to understand their interaction and balance. This book is an admirable effort at such an economic analysis. When facing an uncertain future, remaining flexible until more information arrives has value, because one can cherry-pick to make investments only when the prospects are relatively favorable. This is the starting intuition of real option theory. But in game theory, the strategy of making irreversible commitments to seize first-mover advantage and present rival players with a fait accompli to which they must adapt can have its own value. So what does one do when facing an uncertain future in the company of rivals? This is a difficult problem, conceptually as well as mathematically. The last two decades have brought a trickle of research contributions that address some aspect of this dilemma. In this book, Benoît Chevalier-Roignant and Lenos Trigeorgis synthesize and consolidate much of this literature and skillfully extend it. Generations of students and researchers over the next decade or two will find the book an invaluable starting point for their own work. The authors also deserve congratulations for their excellent exposition. They begin with very simple and clear overviews of the issues, concepts and models from the separate areas of real options and game theory. They focus on applications to industrial organization and business. For students in any of these areas, this book can serve as a hypermarket for one-stop intellectual shopping. I will endeavor a personal remark. I was involved in some of the early research on real option theory. I also contributed to popularizing game theory. But I seem to suffer from a kind of attention-deficit disorder in

xxiv

Foreword

research; therefore I got interested in, and diverted to, other fields like political economy and governance institutions. I am happy that my contributions to real option theory are still remembered, and feel honored to be asked to write a preface to this book. But, returning to the real options literature after many years to read this book, I also feel like Rip Van Winkle, amazed at how much has changed in the intervening years. Although I have enjoyed my excursions into other fields, perhaps I should have stayed and contributed to these equally exciting developments that have combined options and games to produce better twohanded economists. Avinash Dixit Princeton, NJ June 2010

Preface

In real life most situations corporate managers face are characterized by both strategic and market uncertainties with respect to the economic environment. Following the liberalization and deregulation of Western economies, very few industries remain protected, whereas most companies face fierce competition in their respective economic sectors. Certain European governments used to favor high administrative barriers shielding certain “natural monopolies” from competitive entry. Such protected monopolies included the telecommunications, electricity, and gas sectors. European governments recently had to enforce deregulation schemes, opening up many economic activities to new, potentially foreign market participants. At the same time sectors traditionally populated by many firms have undergone significant consolidation, yielding oligopolistic situations with a reduced number of players. These two ongoing concurrent phenomena—liberalization and continuing consolidation— highlight the emphasis corporate managers increasingly put on strategic uncertainty and market structure. Besides strategic uncertainty, managers face increasing market uncertainty. With a reduced life cycle for many products, firms can no longer rely on a given offering but have to renew their product portfolio frequently to sustain or enhance their revenue stream in light of competitive pressures. The IT industry has evolved most rapidly, putting companies unable to respond to market developments and technological uncertainty at a severe disadvantage. At the core of this dilemma lies a classic trade-off between commitment and flexibility. Managers can stake a claim by making large capital investments today influencing their rivals’ behavior or take a “wait-andsee” or step-by-step approach to avoid possible adverse market consequences tomorrow. The assessment and optimal management of strategic options is critical for firms to succeed in today´s constantly changing

xxvi

Preface

business environment. Whether to invest in a new product or enter a new market is a critical decision management should address with proper analytical tools in assessing options and deciding whether to make this or that strategic move. This book aims to make accessible to a wide audience recent results on how to achieve and quantify balance between flexibility and commitment through the new approach of “option games.” We believe that this approach can play an important role in managing modern business in a changing global marketplace. Option games help model situations where a firm that has a real option to (dis)invest in specific projects faces competition. Such situations lead to two, sometimes conflicting, main sources of value for the firm. First, there is a value of waiting or flexibility related to the real option that the firm holds to make future better-informed decisions. Second, there is a value of commitment in light of the strategic interaction with competitors. The trade-off between these two forces calls for a careful balancing act. From a modeling perspective, the examination of such a trade-off calls for a combination of the real options and game theory approaches to decision-making. In the first part of the book, we discuss prerequisite concepts and tools concerning basic game theory, industrial organization, and real options analysis. We are then in a position to amalgamate these diverse fields into a unified framework for option games. This makes the task at hand very ambitious because these fields of research normally require different tool kits and needed results are scattered around in disparate parts of the literature. We aim to fill this gap here by synthesizing the existing literatures to provide a consistent and accessible account of options and games. We combine some of the best materials, tools, and ideas found in diverse books and research works on game theory (e.g., Fudenberg and Tirole 1991; Osborne 2004), industrial organization (Tirole 1988), and real options analysis (Dixit and Pindyck 1994; Trigeorgis 1996) and go beyond current knowledge to chart new territory. The current book brings important materials and ideas together in a unified framework, which makes it much easier for managers and academics to enter and understand this new field. The second part of the book presents the new approach in discrete time, while the last part on continuous-time option games is not covered in any systematic way anywhere and is an important addition. For pedagogical convenience and for the sake of simpler and clearer exposition and buildup of the basic ideas, we first present each of the building blocks step by step to form the supporting foundation and columns of the structure, with the richer theory coming later

Preface

xxvii

(culminating toward chapter 12) as the keystone that will complete the arch.1 The book should be especially appealing to the academic community, particularly in the areas of strategy, economics, and finance. It is the first book that combines the aforementioned fields in such a way that it is accessible to an audience that is not necessarily expert in all fields involved. The book should also be of interest to (academically trained) practicing managers who are interested in applying these ideas. It provides many strategic insights and a ready-to-use tool kit offering quantitative guidance for important competitive trade-offs faced by the modern firm. We attempted to strike a balance between making the book accessible to a wider audience while simultaneously making it challenging and rigorous, subjecting the art of strategy to a scientific inquiry. The book provides a very pragmatic and intuitive, yet rigorous approach to strategy formulation. We owe an intellectual debt to the many scholars who made significant contributions to the related literatures and to the many individuals who provided us with generous comments, criticism, ideas, and suggestions. We thank Avinash Dixit for his feedback, prologue, and encouragement. Marcel Boyer, Marco Días, Kuno Huisman, Grzegorz Pawlina, Sigbjørn Sødal, and Jacco Thijssen made invaluable detailed comments and provided suggestions for improvement on the entire manuscript. Several other colleagues offered valuable comments on select parts of the book or on specific chapters: Christoph Flath, John Khawam, Bart Lambrecht, Richard Ruble, and Bruno Versaevel. We also thank Stefan Hirth, Helena Pinto, Artur Rodrigues, and Han Smit for useful comments. We thank Robert J. Aumann, Rainer Brosch, Peter Damisch, Marco Días, Avinash Dixit, Eric Lamarre, Scott Mathews, Robert C. Merton, Reinhard Selten, and Jean Tirole for enhancing the relevance of this book with their valuable interviews and comments. The authors would like to thank Arnd Huchzermeier for his support. Jane MacDonald from the MIT Press has been most enthusiastic, efficient, and supportive. Last but not least, we would like to thank our families, alive or departed, for their love and support. 1. Chapter 12 can be thought of as the climax of the book, but we need to introduce various key notions (e.g., investment trigger, open-loop vs. closed-loop equilibrium) in a step-bystep fashion to facilitate and smooth out the exposition. Several of the issues left unresolved in the earlier parts of the book are addressed in chapter 12.

1

The Strategy Challenge

At a time when national monopolies have been losing their secular wellprotected positions owing to market liberalization in the European Union and elsewhere across the globe, strategic interdependencies and interactions have become a key challenge for managers in many corporations. Strategic questions abound: How should a firm sustain or gain market share? How to differentiate oneself from others in the grueling global marketplace? When precisely should a firm enter or exit an industry when it faces uncertainty and significant entry and exit costs? Recent developments in economics, finance, and strategy equip management facing such challenges with a concrete framework and tool kit on how to behave strategically in such a complex and changing business environment. Corporate finance and game theory provide complementary perspectives and insights regarding strategic decision-making in business and daily life. Box 1.1 motivates the relevance of game theory to the understanding of daily life situations. The option games approach followed in this book paves the way for a more rigorous approach to strategy formulation in many contexts. It helps integrate in a common, consistent framework the recent advances made in these diverse disciplines, providing powerful insights into how firms should behave in a dynamic, competitive and uncertain marketplace. We first highlight in section 1.1 several environmental factors that justify why firms should be careful when formulating their corporate strategies in an uncertain, competitive business environment. In section 1.2 we discuss how an understanding of competitive strategy in terms of sound economic principles is useful to managers. Two complementary but separate perspectives on strategy (corporate finance and game theory) are discussed in section 1.3. We address the need for an integrative approach to corporate strategy in section 1.4. We provide an overview of the book organization in section 1.5.

2

Chapter 1

Box 1.1 Game theory in daily life

All Is Fair in Love, War, and Poker Tim Harford, BBC News Online What Do Love, War, and Poker Have in Common? High stakes, perhaps. Certainly, in all three you spend a lot of effort trying to work out what the other side is really thinking. There is another similarity: economists think they understand all three of them, using a method called “game theory.” Threats and Counterthreats Game theory has been used by world champion poker players and by military strategists during the cold war. Real enthusiasts think it can be used to understand dating, too. The theory was developed during the Second World War by John von Neumann, a mathematician, and Oskar Morgenstern, an economist. Mr. von Neumann was renowned as the smartest man on the planet—no small feat, given that he shared a campus with Albert Einstein—and he believed that the theory could be used to understand cold war problems such as deterrence. His followers tried to understand how a nuclear war would work without having to fight one, and what sort of threats and counterthreats would prevent the US and the Soviets bombing us all into oblivion. Since the cold war ended without a nuclear exchange, they can claim some success. Understand the World Another success for game theory came in 2000, when a keen game theorist called Chris “Jesus” Ferguson combined modern computing power with Mr. von Neumann’s ideas on how to play poker. Mr. Ferguson worked out strategies for every occasion on the table. He beat the best players in the world and walked away with the title of world champion, and has since become one of the most successful players in the game’s history. Game theory is a versatile tool. It can be used to analyze any situation where more than one person is involved, and where each side’s actions influence and are influenced by the other side’s actions. Politics, finding a job, negotiating rent, or deciding to go on strike are all situations that economists try to understand using game theory. So, too, are corporate takeovers, auctions, and pricing strategies on the high street. Financial Commitment But of all human interactions, what could be more important than love? The economist using game theory cannot pretend to hand out advice on

The Strategy Challenge

3

Box 1.1 (continued)

snappy dressing or how to satisfy your lover in the bedroom, but he can fill some important gaps in many people’s love lives: how to signal confidence on a date, or how to persuade someone that you are serious about them, and just as importantly, how to work out whether someone is serious about you. The custom of giving engagement rings, for instance, arose in the United States in the 1930s when men were having trouble proving they could be trusted. It was not uncommon even then for couples to sleep together after they became engaged but before marriage, but that was a big risk for the woman. If her fiancé broke off the engagement she could be left without prospects of another marriage. For a long time the courts used to allow women to sue for “breach of promise” and that gave them some security, but when the courts stopped doing so, both men and women had a problem. They did not want to wait until they got married, but unless the man could reassure his future wife, then sleeping together was a no-no. The solution was the engagement ring, which the girl kept if the engagement was broken off. An expensive engagement ring was a strong incentive for the man to stick around—and financial compensation if he did not. Not Committed Modern lovers might think the idea of engagement ring as guarantee is a thing of the past, but they can still use game theory to size up their partners. When a couple with separate homes move in together, selling the second home is an important signal of commitment. That second home is an escape route—valuable only if the relationship is shaky. If your partner wants to hang on to his bachelor pad, do not let him tell you it is merely a financial investment. Game theory tells you that he is up to something. Reprinted with permission of BBC News, bbc.co.uk/news. Publication date: August 17, 2006.

1.1 The Changing Corporate Environment The competitive environment around the globe is becoming increasingly challenging for managers as modern economies have witnessed tremendous changes over the last three decades. In this constantly evolving environment, where firms must often make quick decisions that have long-term impact, it is anybody’s guess what might happen in the future— market developments often prove expectations wrong. Firms must carefully commit to specific strategies while developing adaptive capabilities

4

Chapter 1

in an ever-changing marketplace. Globalization, deregulation, and the emergence of new economies (e.g., Brazil, Russia, India, and China) have created both threats and opportunities for incumbent firms who now have to adapt more effectively to the rapidly changing global environment or suffer damage by new entrants and risk extinction. Following the liberalization and deregulation of European economies, only a limited number of industries have remained secure, while most companies across the board face serious competitive pressures. At the same time other economic sectors traditionally characterized by a large number of companies have undergone significant consolidation, resulting in oligopoly structures with a reduced number of players. The recent economic crisis has amplified these consolidation pressures. The mining giant Rio Tinto has recently merged with BHP Billiton, forming a virtual duopoly together with Brazilian mining giant Vale. M&A deals have similarly reshaped the automotive sector, with the recent acquisitions of Chrysler by Fiat, of Porsche and Suzuki by Volkswagen, and of Mitsubishi by PSA (Peugeot, Citroën). British Airways together with Iberia claim the top two position in the fiercely competitive European airline business. In the United States, the merger between United Airlines and Continental Airlines created one of the world’s biggest airlines. A dramatic concentration has also taken place in the banking sector: out of the top five investment banks worldwide, only two (Goldman Sachs and Morgan Stanley) have remained independent. Notable banking deals include the acquisition of Washington Mutual by JP Morgan Chase, of Countrywide by Bank of America, and of the Belgian bank Fortis by BNP Paribas. These two concurrent phenomena—liberalization and consolidation— have put higher on the corporate agenda the assessment of strategic uncertainty. Italy’s dominant state electricity authority, Enel, is a good case in point. Just a decade ago Enel was in a very comfortable position, enjoying an established natural monopoly in the Italian electricity market with the benediction of the national government. The main concern for Enel during this period was to minimize or keep under control its mix of input or production costs, as the output electricity price was regulated. Several years later Enel lost its preferential monopoly position due to the liberalization of the European markets, and the competitive environment facing Enel changed dramatically. Enel was forced to sell half of its generating assets to half a dozen smaller local rivals, creating more competition in its home base. Electricity price deregulation accompanied by oil and other fuel price fluctuations have added considerable pressure

The Strategy Challenge

5

and uncertainty for Enel. With further deregulation other European competitors (e.g., the dominant electricity producer of France, EDF) also entered the Italian electricity market. Today Enel has to consider the actions of local as well as international competitors on the national soil, as well as contemplate investing itself in new or emerging markets, such as Russia, to sustain or leverage its once dominant position in the area. Just as its European and global counterparts, Enel now faces a broader range of uncertainties and challenges: How to cope with increased energy (input) and electricity demand (output) uncertainties? How to compete with local and global rivals in an ever-changing local and global competitive landscape? How to assure and diversify its energy portfolio mix in a globalized marketplace? How to formulate and dynamically adjust its strategy, knowing when to compete, threaten, bargain, or cooperate with its rivals? These are the kind of questions we will be addressing in this book. Many situations corporate managers face today are characterized by both market and strategic uncertainty with respect to the economic environment. When one desires to address such complex issues, some simplification of reality is useful to focus on the fundamental trade-offs. Some simple models in management are being revised as they offer rather simplistic approaches that no longer describe current economic reality. The field of investment under uncertainty falls in this category. Prevailing management approaches often lead to investment decisions detrimental to the overall firm’s long-term well-being. In an increasingly uncertain and competitive environment, corporate managers need appropriate management tools that can provide long-term guidance. This book describes a novel approach aimed to enable managers make rational decisions in a competitive environment under uncertainty. It allows managers to quantify and balance the conflicting impacts of managerial flexibility and the strategic value of early investment commitment in influencing rivals’ strategic behavior. Real options analysis is widely considered to be more reflective of reality than traditional financial methods (e.g., net present value) in that it takes managerial flexibility into account.1 To avoid dealing with complex models, however, standard real options analysis often ignores 1. Investment under uncertainty is part of the mainstream literature on finance, economics, and strategic management. In the last decades financial theory has been supplemented with real options analysis. Use of the financial options analogue can be insightful in assessing flexibility embedded in real asset situations. The real options approach to investment has reached a corporate finance textbook status and is currently applied in leading corporations for guiding real-world strategic investment decisions.

6

Chapter 1

the strategic interactions among option holders, analyzing investment decisions as if the option holder has a proprietary right to exercise. This simplifying perspective is far from being realistic in many situations because firms generally compete with rivals. Several firms may share an option in the industry and hence option exercise strategies cannot be formulated in isolation. Rather, optimal investment behaviors must be determined as part of an industry equilibrium. As a consequence of this more pragmatic view of the nature of the competitive environment, a new theory called “option games” has emerged. This theory combines the concepts and tools offered by traditional real options analysis with game theory principles designed to help figure out how players behave in strategic conflict situations. The option games approach we elaborate on herein provides powerful insights into understanding strategic interactions and challenges traditional thinking that presumes that firms pursue strategies in isolation. Game theory is for the most part deterministic. Option games help management better intuit how uncertainty can be modeled in a strategic setting. This approach helps improve prediction and understanding of industry dynamics in highly uncertain industries. It enhances previous industrial organization literature on strategic investment in a deterministic setting to better explain the strategic investment behavior of firms under changing conditions. Box 1.2 provides an overview of the challenges commercial airframe manufacturers Box 1.2 Evolving strategy in commercial aviation

Boeing Bets the House on Its 787 Dreamliner Leslie Wayne In recent years Boeing has stumbled badly, ceding its decades-long dominance in commercial aviation to Airbus and becoming mired in a string of scandals over Pentagon contracts. The terrorist attacks of 2001 depressed demand at a time when the company’s product line paled against appealing new planes from Airbus. In one year alone, from 2001 to 2002, Boeing’s profits dropped 80 percent. But the view from Seattle, the headquarters of Boeing’s commercial jet operations, has more of that Chinese pep-rally spirit than such gloomy talk might indicate . . .. With revenue having grown for the second consecutive year, to $54.8 billion in 2005, and a record number of orders on its books, Boeing has had a huge gain in its stock price—to more than $80 a share, more than three times its nadir of $25 in 2003. Boeing’s 1,002 orders last

The Strategy Challenge

Box 1.2 (continued)

year fell short of Airbus’s 1,055. But Boeing’s orders included more widebody planes, which analysts valued at $10 billion to $15 billion more than Airbus’s. But what is really driving the high spirits at Boeing—and the high stock price—is a plane that has not yet taken to the skies: the 787. It is Boeing’s first new commercial airplane in a decade. Even though it will not go into service until 2008, its first three years of production are already sold out— with 60 of the 345 planes on order going to China, a $7.2 billion deal. Other big orders have come from Qantas Airways, All Nippon Airways, Japan Airlines, and Northwest Airlines. Big orders mean big money, of course—and that is good, because analysts estimate that Boeing and its partners will invest $8 billion to develop the 787. Boeing is also risking a new way of doing business and a new way of building airplanes: farming out production of most major components to other companies, many outside the United States, and using a carbonfiber composite material in place of aluminum for about half of each plane. If it works, Boeing could vault back in front of Airbus, perhaps decisively. If it fails, Boeing could be relegated to the status of a permanent also-ran, having badly miscalculated the future of commercial aviation and unable to meet the changing needs of its customers. “The entire company is riding on the wings of the 787 Dreamliner,” said Loren B. Thompson, an aviation expert at the Lexington Institute, a research and lobbying group in Arlington, Virginia, that focuses on the aerospace and military industries. “It’s the most complicated plane ever.” Boeing calls the 787 Dreamliner a “game changer,” with a radically different approach to aircraft design that it says will transform aviation. A lightweight one-piece carbon-fiber fuselage, for instance, replaces 1,200 sheets of aluminum and 40,000 rivets, and is about 15 percent lighter. The extensive use of composites, already used to a lesser extent in many other jets, helps improve fuel efficiency. To convince potential customers of the benefits of composite—similar to the material used to make golf clubs and tennis rackets—Boeing gives them hammers to bang against an aluminum panel, which dents, and against a composite one, which does not. At the same time, the 787 has new engines with bigger fans that are expected to let the plane sip 20 percent less fuel per mile than similarly sized twin-engine planes, like Boeing’s own 767 and many from Airbus. This is no small sales point, with oil fetching around $70 a barrel and many airlines struggling to make a profit even as they pack more passengers into their planes. “The 787 is the most successful new launch of a plane—ever,” said Howard A. Rubel, an aerospace analyst at Jeffries & Company, an investment bank that has advised a Boeing subsidiary . . .. The 787 is designed to carry 220 to 300 people on routes from North America to Europe and Asia. Boeing is counting on it to replace the workhorse 767, which is being

7

8

Chapter 1

Box 1.2 (continued)

phased out, and, it hopes, a few Airbus models as well. Its advantages go beyond fuel efficiency: Boeing designed the 787 to fly long distances while keeping passengers relatively comfortable. That approach grows out of another gamble by Boeing—that the future of the airline business will be in point-to-point nonstop flights with medium-size planes rather than the current hub-and-spoke model favored by Airbus, which is developing the 550-seat A380 superjumbo as its premier long-haul jetliner. Flying point to point eliminates the need for most passengers to change planes, a competitive advantage so long as the Dreamliner is as comfortable and as fast as a bigger aircraft. And after talking with passengers around the world, Boeing designed the 787 to have higher humidity and more headroom than other airplanes, and to provide the largest windows of any commercial plane flying today. “We are trying to reconnect passengers to the flying experience,” said Kenneth G. Price, a Boeing fleet revenue analyst. With airlines squeezing every last cent and cutting back service, “flying is not enjoyable,” Mr. Price said. “Every culture fantasizes about flying,” he added. “All superheroes fly. But we were taking a magical experience and beating the magic out.” Even more innovative for Boeing is the way it makes the 787. Most of the design and construction, along with up to 40 percent of the estimated $8 billion in development costs, is being outsourced to subcontractors in six other countries and hundreds of suppliers around the world. Mitsubishi of Japan, for example, is making the wings, a particularly complex task that Boeing always reserved for itself. Messier-Dowty of France is making the landing gear and Latecoere the doors. Alenia Aeronautica of Italy was given parts of the fuselage and tail. Nor are these foreign suppliers simply building to Boeing specifications. Instead, they are being given the freedom, and the responsibility, to design the components and to raise billions of dollars in development costs that are usually shouldered by Boeing. This transformation did not come overnight, of course, nor did it begin spontaneously. Boeing changed because it had to, analysts said. “Starting in 2000, Airbus was doing well,” said Richard L. Aboulafia, an aerospace analyst with the Teal Group, an aviation research firm in Fairfax, Va. “Boeing had to reconsider how it did business. That led to the framework for the 787—getting the development risk off the books of Boeing and coming up with a killer application.” Boeing plans to bring the 787 to market in four and a half years, which is 16 to 18 months faster than most other models. All of that is good, Mr. Aboulafia added, if it works. It is a tall order for a wholly new plane being built with new materials, many from new suppliers and assembled in a new way. “The 787 is operating on an aggressive timetable and with aggressive performance goals,” he said. “It leaves no margin for error.”

The Strategy Challenge

Box 1.2 (continued)

Never before has Boeing farmed out so much work to so many partners—and in so many countries. The outsourcing is so extensive that Boeing acknowledges it has no idea how many people around the world are working on the 787 project. Airbus, Boeing’s sole rival in making big commercial airliners, is also making a big bet on the future, but in a different direction. The companies agree that in 20 years, the commercial aviation market may double, with today’s big orders from China, India, and the Middle East to be followed by increased sales to American and European carriers as they reorganize and reduce costs. By 2024, Boeing estimates, 35,000 commercial planes will be flying, more than twice the number now aloft, and 26,000 new planes will be needed to satisfy additional demand and replace aging ones. But how passengers will get from place to place, and in what planes, will depend on whether Boeing or Airbus has correctly forecast the future. Boeing believes that passengers will want more frequent nonstop flights between major destinations—what the industry calls “city pairs.” That is what led to the big bet on the Dreamliner, a midsize wide-body plane that can fly nonstop between almost any two global cities—say, Boston to Athens, or Seattle to Osaka—and go such long distances at a lower cost than other aircraft. Airbus believes that airplane size is more important than frequent nonstop flights and that passengers will stick with a hub-and-spoke system in which a passenger in, say, Seattle, will fly to Los Angeles and transfer to an Airbus 380 to go to Tokyo before catching a smaller plane to Osaka. That view has led it to spend $12 billion to develop the double-deck A380, the largest passenger jet ever—a bet that is as crucial to its future as the 787 is to Boeing’s. “We have a fundamental difference with Airbus on how airlines will accommodate growth,” said Randolph S. Baseler, Boeing’s vice president for marketing. “They are predicting flat growth in city pairs. We are saying that people want more frequent nonstop flights. They believe airplane size will increase, and we believe that airplanes will not increase in size that much. Those two different market scenarios lead to two different product strategies.” The market, of course, will determine the winner, but given the industry’s long lead times, that may not be clear for 10 to 20 years. For now, airlines have ordered 159 copies of the A380—which has a list price of $295 million and is scheduled to enter service this year—and more than twice as many 787s, which list for $130 million and are scheduled to enter service in two years. Publication date: May 7, 2006.

9

10

Chapter 1

have faced over the recent years, focusing on the changes in corporate strategy of Boeing compared with Airbus. A manager from Boeing discusses in box 1.3 the use of real options to capture and assess the diverse sources of uncertainty in his business; an analysis of the strategic interplay vis-à-vis Airbus is highlighted. 1.2 What Is Strategy? Corporate strategy is high on the agenda of every major corporation.2 The strategy a firm formulates and how it implements that strategy will determine to a large extent whether it will survive and be successful in the marketplace or become extinct. Formulating the right strategy in the right place at the right time is not an easy task. It requires deep analysis and ready-to-implement, adaptable solution programs. Our approach to strategy is based on the premise that strategic management is a structured, rational discipline relying on rigorous market and competitive analysis. One can understand why firms succeed or fail by analyzing their decision processes in terms of consistent principles of market economics and rational strategic actions. This is the reason why a large literature in strategic management relies on economic theory as it provides a reliable, rigorous foundation to understanding specific developments and reactions taking place in the market place. Good firm performance is considered the result of soundly formulated and well-implemented strategies. Grant (2005) identifies the following elements as key to a successful strategy: (1) simple, agreed-upon, longterm objectives; (2) deep understanding of the competitive landscape; (3) objective appraisal of the firm’s internal resources and capabilities; and (4) effective implementation. There are no easy “recipes for success” applicable to each firm in every industry.3 The pursuit of one-size-fits-all 2. Although it is commonly agreed that strategy is critical in today’s changing corporate environment, there is no universally agreed-upon definition of business strategy. According to Chandler (1962), strategy is “the determination of the basic long-term goals and objectives of an enterprise and the adoption of courses of action and the allocation of resources necessary for carrying out these goals.” According to Mintzberg et al. (2002), strategy is “the pattern or plan that integrates an organization’s goals, policies and action sequences into a cohesive whole.” 3. There are two main approaches to strategy formulation. The first approach looks at specific firms or case studies examining why these firms are successful and tries to deduce success factors that might be applicable to other firms. This is the “best-practices” approach. Herein we take a different approach. We formulate a conceptual framework for strategic management and assess if it provides prescriptive insights into real-world managerial problems.

The Strategy Challenge

Box 1.3 Interview with Scott Matthews, Boeing

1. Do you believe real options is more suitable than other capital budgeting approaches to provide managerial guidance? Where and to what extent is real options analysis used at Boeing? Real options provides a more informed decision for our strategic projects. Of special significance are the scenarios that we build around the real options analyses that help us understand both the risks and the opportunities of any venture. To date, real options analysis has been used mostly on large-scale projects. Because of the higher investment amounts, these large projects pose particular risks that require more careful analyses including the use of real options techniques. 2. What are the sources of uncertainty you face at Boeing, and how do you manage them with options, physical or contractual? Boeing projects have many sources of uncertainty. We build models that attempt to integrate technology development, design, manufacture, supply chain, and market forces, including possible actions of our competitors. These models have dozens, even hundreds, of variables modeled using various Monte Carlo and discrete event simulation capabilities. Usually there are just a handful of principal uncertainty drivers which are determined using sensitivity analysis. We then apply a series of targeted investments to investigate and better understand the true scale of these uncertainties. Since the uncertainty landscape is in constant evolution, due to both our risk mitigation efforts as well as exogenous events, we continue to update the model and modify our investments as appropriate. These are

11

12

Chapter 1

Box 1.3 (continued)

often modeled as real option investments with a type of varying volatility, as we are attempting to both reduce uncertainty while at the same time increase the value of the subsequent project stage. 3. Do you see a usefulness for game theory and option games in Boeing’s strategic thinking, for example, vis-à-vis Airbus? We find that game-theoretical approaches provide additional insight to a solution set as long as the number of actors is limited to just a few players. At a certain point market considerations dominate and provide a better approach to modeling the scenarios. We have managed to execute a few plays against our competitors, the origin of which could be traced back to strategic gaming scenarios and market timing. When we are successful, these plays are often highly leveraged, and therefore take on the characteristics of well-placed option investments. However, like other companies in dynamic markets, we find our competitive response limited by timing or technology and product availability considerations.

success factors has failed to provide a coherent direction to guide the actions and decisions of firms. Ghemawat (1991) criticizes the success factors approach and identifies commitment as a main driver of corporate success or failure. He sees commitment as a well-thought-out plan of action affecting the firm in the long term. A successful strategy should exhibit consistent but adaptive behavior over time and involve certain strategic commitments that might sometimes hinder managerial flexibility. This feature of strategy implies that, in an uncertain environment, resolving the investment or commitment timing issue is critical to a firm’s success. A firm should not always invest or commit immediately, but should be prepared to decide at the right moment to reap the benefits of a developing opportunity. As suggested by Dixit and Pindyck (1994), investment situations where decisions are costly or impossible to reverse compel corporate managers to be cautious and careful to make decisions at the right time. Strategy should also be dynamic in that it should be adaptable to changing market circumstances or competitive dynamics. Following Rumelt (1984, p. 569), the essence of competitive strategy is being in “constant search for ways in which the firm’s unique resources can be redeployed in changing circumstances.” The increasing cone of market and strategic (competitive) uncertainty makes the dynamic formulation of strategy key to survival and success in a changing marketplace. Box 1.4

The Strategy Challenge

Box 1.4 Flexible strategy and real options

Stay Loose: By Breaking Decisions into Stages, Executives Can Build Flexibility into Their Plans Lenos Trigeorgis, Rainer Brosch, and Han Smit, Wall Street Journal In turbulent times adaptability is critical. That’s why today flexibility is more valuable than ever in business strategy. Markets, technologies, and competition are becoming more dynamic by the day. To succeed in this environment, companies need to position themselves to capitalize on opportunities as they emerge, while limiting the damage if adversity hits. This requires a whole new level of flexibility. Good managers have always been able to think on their feet. But many widely applied tools of strategy development were designed for relatively stable environments. As a result business strategy may too often lock managers into decisions that turn out to be flawed because something outside their control doesn’t go as planned. What is needed is a systematic translation of managers’ flexibility into strategy—a plan that lays out a series of options for managers to pursue or decline as developments warrant. That is the essence of what is known as “real options” analysis, an approach that borrows from the workings of the financial markets. Just as stock options, for instance, give the holder the right, but not the obligation, to buy or sell shares at a given price at some time in the future, real options give executives the right, but not the obligation, to pursue certain business initiatives. Start Small Instead of making rock-hard plans and irreversible long-term commitments, the idea is to create flexibility by breaking decisions down into stages. When building a new plant, for example, it may be tempting to realize the full economies of scale by building the biggest facility the company can manage. But it may be wiser to first build a smaller plant that can be easily expanded later on. That way, if the market for the products the plant produces should decline, a smaller investment has been put at risk. At that point managers have the option to scale down or abandon operations. On the other hand, if things turn out well, they have the option to expand the plant. As a mind-set this approach encourages managers to be flexible in their planning. In more concrete terms it allows them to value investment decisions and business initiatives in a new way. Instead of making a decision based on a rigid financial analysis of a given project as a whole, managers can analyze, from the start, the financial implications of each step along the way and every potential variation—without committing to anything

13

14

Chapter 1

Box 1.4 (continued)

before they must. Once the project is under way, they also can account for the changing value of each option as events unfold. All that information gives them a clearer framework for decisions on whether to launch a project and whether to proceed, hold back, or retreat at each stage. What does this look like in practice? A leading European automaker was considering two investment alternatives for the production of a new vehicle. Under one alternative, production would be based entirely in one country. Under the other, the company would set up plants around the world, allowing it to switch production from site to site to take advantage of fluctuations in exchange rates or labor costs. The cost of the flexible system would be higher. But the company decided that the value of that flexibility, with its promise of cost savings and increased profits, exceeded the difference in cost between the two alternatives. So it chose the multinational plan. Competitive Edge Real options analysis can also be useful in helping strategic planners address the challenges of competition. Many managers already incorporate game theory into their planning to help predict how competition will play out. But with competition emerging and evolving more rapidly than ever, supplementing game theory with real options analysis can help companies be more flexible in how they react. Consider, for example, the question of whether a company should aim to preempt competition or choose to cooperate with other players in a way that could expand the market. This is a question of growing relevance as sometimes competing technologies are at the heart of more products. In deciding whether to fight or cooperate, companies can use real options analysis to better quantify the value of each contingency, including the value of the options that would be lost or gained depending on what competitive course is chosen. What this all adds up to is a portfolio of corporate real options, each with a value that will change along with the company’s developing markets. Those who manage that portfolio most effectively will be in the best position to realize their company’s growth potential. Reprinted with permission of The Wall Street Journal, Copyright © 2007 Dow Jones & Company, Inc. Publication date: September 15, 2007.

The Strategy Challenge

15

discusses the need for strategic plans to be flexible and adaptable in a changing environment. Since strategic decisions have long-term consequences, one should look not only at today’s advantages or drawbacks but also at the long-term consequences and value of such decisions. The trade-off between the benefits of commitment (as part of a consistent strategy over time) and remaining flexible and adaptive to changing circumstances calls for an integrative approach weighing the merits of flexibility against commitment. 1.3 Two Complementary Perspectives on Strategy Two approaches to strategy are of particular interest as they provide insights that help management deal with the flexibility or commitment trade-off: corporate finance and game theory. These disciplines are generally considered separate but are in fact complementary. We discuss each one next. 1.3.1

Corporate Finance and Strategy

At first sight the link between corporate finance and strategy may not be that clear. Within corporations, finance is in charge of raising firm resources, while the strategy department is concerned with how to allocate these resources strategically. The two departments deal, however, with two sides of the same coin. Financial managers are concerned with how to finance a project at a reasonable cost. They are aware that resource providers (e.g., shareholders or banks) will carefully scrutinize what the firm plans to do with the resources they are asked to provide, carefully assessing the firm’s strategic plans and the quality of its management. The formed opinion of the resource providers will influence the cost of the resources the firm has access to. A good financial manager cannot therefore ignore the firm’s strategy. Understanding and communicating the firm’s strategy should be one of her primary tasks. Following a finance theory approach, the objective of the firm is to maximize the wealth (utility) of shareholders. According to the Fisher separation theorem, this objective is achieved when maximizing the firm’s market value. A fundamental question in corporate finance is how to attain this objective. As part of corporate finance, capital budgeting considers this problem from an investment perspective, being concerned with the optimal allocation of scarce resources among alternative

16

Chapter 1

projects.4 A key issue is how to address the intertemporal trade-off faced by a firm between paying more dividends or cash distributions now and investing in growth projects meant to generate future cash flows. The established criterion in capital budgeting is the discounted cashflow (DCF) or net present value (NPV) method. The approach involves a relatively easy-to-understand logic and methodology that consists in assessing the current value of a project based on the expected future cash flows it will generate, net of the related costs. Management estimates the stream of future expected cash flows over the project’s life and discounts them back to the present using a risk-adjusted discount rate, obtaining the project’s present value V . It then subtracts the (present value of) investment outlays, I , obtaining the current (t = 0) net present value:5 NPV = V − I .

(1.1)

Alternatively, the present value represents the discounted sum of economic profits.6 The economic profit in a given period represents the firm’s total revenue earned in that period minus all relevant opportunity costs, including the cost of capital. Following the DCF or NPV paradigm, the firm creates shareholder value by following the NPV rule, prescribing to immediately undertake projects with positive NPV, meaning NPV > 0 or V > I . In the absence of managerial flexibility, net present value is the main valuation measure consistent with the firm’s objective to maximize shareholders’ wealth.7 Other valuation measures, such as payback period, accounting rate of return, or internal rate of return are considered inferior to NPV and sometimes even inconsistent. The above finance theory often appears rather technical and not so relevant for strategic management practice. Already in Myers (1984), a 4. Corporate finance provides a useful frame to help managers make investment and financing decisions. Two subfields of corporate finance are particularly relevant for corporate managers: capital budgeting, or how to make investment decisions, and financing, or how to finance projects at the lowest cost available. It is commonly agreed that real investments are more important for creating shareholder value than financial engineering. 5. Consider a project generating over its lifetime (T years) expected cash inflows E [ Rt ] in each future year t. Launching the project is costly, involving expected cash outflows t t E [Ct ]. Let V ≡ ∑Tt =0 E [Rt ] (1 + k ) and I ≡ ∑Tt =0 E [Ct ] (1 + k ) denote the present value of the stream of cash inflows and outflows, respectively. k denotes the appropriate riskadjusted discount rate. The necessary cash outflow I might be a single investment outlay incurred at the outset or the present value of a series of outflows. 6. The economic value added (EVA) approach is based on this notion. 7. Throughout the book, we ignore agency problems inside a firm that may invalidate the NPV rule. Myers (1977) discusses the problem of “underinvestment.” Managers acting in the shareholders’ interest may reject projects with positive NPV when the firm is close to bankruptcy since investing in these projects would only benefit debt-holders.

The Strategy Challenge

17

gap between finance and strategy was identified. Myers offers three main explanations for this gap: NPV is often mistakenly applied Firms in practice often pursue financial objectives that are inconsistent with basic financial theory. They may focus on short-term results rather than long-term value creation. For instance, firms may worry about the impact of their strategic decisions on today’s P&L and on today’s balance sheet.8 Financial theory in fact stresses the importance of taking a long-term perspective to enhancing firm value over short-term creative accounting.9 The balance sheet or income statement are accounting instruments presenting snapshots of the moment or period and do not necessarily mirror real long-term value creation. Another pitfall is that some managers may pursue corporate diversification to reduce total risk for their own benefit.10 In addition managers often treat available divisional resources as being limited. This internally imposed constraint is in sharp contrast with the basic finance assumption that firms have ready access to capital markets at the prevailing cost of capital. Even if acquiring new financial resources may be more costly, the project should be adopted if the project brings more value than it costs to undertake it. •

Finance and strategy mind-sets differ They represent two cultures looking at the same problem. In perfect competition the firm presumably makes no excess economic profit. Strategists are thus looking for deviations from perfect competition to generate excess profits. Such deviations result from distinctive sustainable competitive advantages. Given the linkage between competitive advantage and excess economic profits, strategists often find it superfluous to determine the net present value (as the discounted sum of economic profits) once they have identified the source of competitive advantages. •

NPV has limited applicability The DCF approach involves the estimation of a risk-adjusted discount rate, a forecast of expected cash flows, and an assessment of potential side effects (e.g., erosion or synergies between projects) or time-series links between projects. The last aspect is most difficult to handle with traditional techniques because •

8. Other common mistakes include the inconsistent treatment of inflation (deflated cash flows discounted back at a discount rate assuming inflation) and unrealistic hurdle rates (use of discount rates that take into account both systematic and diversifiable risk). 9. Short-term orientation is allegedly rampant in countries relying heavily on the capital markets. 10. Risk reduction through portfolio diversification had better be undertaken by investors directly in the capital markets; corporate diversification undertaken by managers is a less efficient means to diversify risk.

18

Chapter 1

today’s investment decisions may constrain or open up new future opportunities. NPV has other drawbacks. First, the NPV paradigm views investment opportunities as now-or-never decisions under passive management. This precludes the possibility to adjust future decisions to unexpected future developments in industry cycles, demand, or prices. Firms need to position themselves to capitalize on opportunities as they emerge while limiting the damage arising from adverse circumstances. If market developments deviate significantly from the expected future scenario, managers can generally revise their future decisions to protect themselves from adverse downward movements or tap on favorable developments and further growth potential. Applying the NPV rule strictly is ill-advised when managers can adjust their planned investment programs or delay and stage their investment decisions. Managers following the prescriptions offered by NPV may find themselves locked into decisions that are flawed when something outside their control does not go as planned. Second, NPV typically assumes a constant discount rate for each future time and state scenario regardless of whether the situation is favorable. Table 1.1 summarizes situations where NPV might give a good approximation of reality and when it might be misleading. Finally, NPV typically overlooks the consequences of competitive actions. Strategy is in need of a quantitative tool that allows for dynamic consideration of changing circumstances. Academics have attempted early on to use alternative approaches to overcome the problems inherent in NPV, particularly to deal with uncertainty and the dynamic nature of investment decision-making. Such methods include sensitivity analysis, Table 1.1 Use of NPV for financial and corporate real assets

Appropriate

Inappropriate

Financial assets

Corporate real assets

Valuation of bonds, preferred stocks, and fixed-income securities Valuation of relatively safe stocks paying regular dividends

Valuation of flows from financial leases

Valuation of companies with significant growth opportunities Valuation of call and put options

Source: Myers (1984, p. 135).

Valuation of “cash cows” Valuation of projects with substantial growth opportunities Valuation of R&D projects

The Strategy Challenge

19

simulation, and decision-tree analysis. Each has had, however, known drawbacks. Sensitivity analysis considers each variable in isolation, thereby ignoring correlations among them. Simulation (e.g., Monte Carlo) faces the same risk-adjustment (discount rate) problem as NPV and generally requires additional adjustment to handle certain recursive problems (e.g., American options) because it is a forward-looking process. Sensitivity analysis and standard simulation are static approaches in that they assume that management is precommitted to a previously agreedupon course of action. In real life this hardly holds. Managers have valuable flexibility and can adapt to the actual market developments once uncertainty gets resolved. They may, for example, abandon a onceundertaken project if the prospects prove gloomy. Managers may also have other options to alter project features in view of the actual market development. Decision-tree analysis (DTA) can be seen as a refined version of NPV aimed to take into consideration the dynamic nature and across-time linkages of decision-making. DTA attributes probabilities to different states of the world and determines in each case the strategy management should optimally formulate (e.g., increase the production scale, switch off the plant, exit the market). In this respect DTA considers the options management has and provides better insights on the dynamic structure of the problem. Trigeorgis and Mason (1987) point out that DTA fails, however, to be economically sound. The discount rate might not be constant over time or across states as typically assumed. Determining the real probabilities of occurrence of each state is often quite involved. Real options analysis (ROA) is an enhanced version of decision-tree analysis that provides improvement in terms of risk-adjustment and determination of probabilities.11 This enhancement is the result of using insights from option-pricing theory.12 Just as stock options give the holder the right, but not the obligation, to buy or sell shares at a given price at some time in the future, real options gives executives the right, but not the obligation, to pursue certain business initiatives. ROA is operationally similar to DTA, with the key difference that the probabilities are riskadjusted, which allows the use of the risk-free discount rate. Real options 11. Several corporate-finance textbooks subsume DTA into real options analysis. We disagree. Real options is the application of option-pricing theory (contingent claims analysis) and risk-neutral pricing to real investment situations (Myers 1977). 12. An alternative to NPV for risk-adjusting risky future cash flows is to consider the certainty equivalents of the uncertain future cash flows and discount them at the risk-free rate. This alternative approach for risk adjustment is a cornerstone of risk-neutral pricing and real options analysis.

20

Chapter 1

analysis is an innovative capital-budgeting tool suitable for the analysis of dynamic decision making under exogenous uncertainty. It enables quantifying strategic considerations that justify sometimes undertaking projects with negative (static) net present value or delaying projects with positive NPV. Box 1.5 highlights the challenges managers face under uncertainty, the inability of NPV to cope with them, and the usefulness of real options in practice. 1.3.2

Game Theory and Strategy

Corporate life would be rather comfortable if a single firm were the only one operating in the marketplace. As a monopolist the firm could Box 1.5 Uncertainty, NPV, and real options in practice

Getting Real: Want to Take More Uncertainty out of Capital Investment Decisions? Try Real Options S. L. Mintz, CFO Magazine “The Edsel is here to stay.” That’s what Ford Motor Co. chairman Henry Ford II told Ford dealers in 1957. “There is no reason why anyone would want a computer in their home.” Thus intoned Digital Equipment Corp. founder Kenneth Olsen in 1977. Even for business leaders with vision, the future is difficult to predict. So where does that leave less-than-legendary executives come budget-planning season? Stuck, largely, with the same venerable tools that guided their predecessors and their predecessors: net present value and gut instinct. Short of denigrating tools that account for many great successes (along with memorable flubs), many executives are wondering if that’s all there is. “There is definitely room for improvement,” concedes Rens Buchwaldt, CFO of Bell & Howell Publishing Services, in Cleveland. Large capitalinvestment decisions—whether it’s launching a new automobile, or building a chip-fabrication plant, or installing an ERP system, or making any number of other very pricey investments—hurl companies toward uncertain outcomes. Huge sums are at risk, in a competitive climate that demands ever-faster decisions. Is there a better way to evaluate capital investments? A growing and vocal cadre of academics, consultants, and CFOs say there is one: real options. Fans insist that real options analysis extends quantitative rigor beyond discount rates and expected cash flows. “Everybody knew there was some kind of embedded value” in strategic options, says an oil industry finance

The Strategy Challenge

Box 1.5 (continued)

executive. Real options analysis, he says, brings that embedded value to light. By quantifying the fuzzy realm of strategic judgment, where leaps of faith govern decisions, real options analysis fosters the union of finance and strategy. “It’s a way to be a little more precise about intuitive feelings,” says Tom Unterman, CFO of $3 billion Times Mirror Co., the Los Angeles based news and information company. A real options analysis recently bolstered the company’s decision to back away from an acquisition, says Unterman, and wider use of the approach is foreseeable. “We are quite actively looking for ways to apply it,” he says. Casting investment opportunities as real options increased both the top and bottom lines at Cadence Design Systems Inc., a San Jose, California, based provider of electronic design products and services. “We have closed a number of transactions that we would not have closed before,” CEO Ray Bingham declares . . .. The Value of Flexibility Unlike net present value measurements, real options analysis recognizes the flexibility inherent in most capital projects—and the value of that flexibility. To executives familiar with stock options, real options should look familiar. A stock option captures the value of an investor’s opportunity to purchase stock at a later date at a set price. Similarly a real option captures the value of a company’s opportunity to start, expand, constrain, defer, or scrap a capital investment depending on the investment’s prospects. When the outcome of an investment is least certain, real options analysis has the highest value. As time goes by and prospects for an underlying investment become clearer, the value of an option adjusts. Sweep away the rocket science, and real options analysis presents a more realistic view of an uncertain world beset by constant shifts in prices, interest rates, consumer tastes, and technology. To focus strictly on numerical value misses the depth and complexity of real options discipline, observes Nalin Kulatilaka, a professor of finance at Boston University School of Management. Kulatilaka is an evangelist for a methodology that obliges managers to weigh equally all imaginable alternatives, good and bad. Real options analysis liberates managers from notions of accountability that mete out blame when plans don’t go as expected. That’s not a healthy environment for workers or companies that need to be nimble all the time, if not right all the time. “The best decision may lead to a bad outcome,” says Soussan Faiz, manager of global valuation services at oil giant Texaco Inc. “If you are judged on a bad outcome, guess what? People will say,

21

22

Chapter 1

Box 1.5 (continued)

‘Why go through that?’” To succeed today, companies must create new options. But unless managers are rewarded for creating them, Faiz warns, “it ain’t gonna happen.” Certainty Is a Narrow Path By taking uncertainty into account, real options analysis fosters a more dynamic view of the world than net present value does. Net present value ultimately boils down to one of two decisions: go or no-go. When the net present value of expected cash flows is positive, companies usually proceed. As a practical consequence managers concentrate on prospects for favorable outcomes. Prospects for unfavorable outcomes get short shrift. In this analysis certainty enjoys a premium—and that’s a narrow path. Even without gaming the numbers to justify projects, this upside bias invites unpleasant surprises. “Unfortunately, discounted cash flow collapses to a single path,” says Texaco’s Faiz. Management and measurement are intertwined, she explains, yet companies manage with an eye to options, but measure performance as if options don’t exist. In the oil business, oil prices don’t remain low for the life of a project; they bounce back. “The likelihood of prices being low for the rest of the project is zero or nearly zero,” says Faiz. But even if prices do remain stagnant, defying the odds, managers don’t snooze the whole time. They wake up and react. Net present value, however, treats investments as if outcomes are cast in stone. This, needless to say, is not realistic. “Net present value makes a lot of heroic assumptions,” warns Tom Copeland, chief corporate finance officer of Monitor, a strategy consultancy in Cambridge, Massachusetts. Typically a multiyear project is plotted along a single trajectory worth pursuing only if the net present value exceeds zero or some hurdle rate. This type of reasoning may satisfy requirements for a midterm exam, says Copeland, but situations in the real world change constantly as new information surfaces. Most managers realize that flexibility ought to be included in valuations, Copeland says. “The bridge they have to cross is understanding the methodology to capture the value of flexibility.” Out of the Ivory Tower Experts have touted the merits of real options for at least a decade, but the sophisticated mathematics required to explain them has penned up those merits in ivory towers. That’s changing, as proponents tout the virtues of real options as a mind-set for decision-making . . . “The kinds of businesses companies go into today are difficult to go into with NPV,” says John Vaughan, vice president for business development

The Strategy Challenge

23

Box 1.5 (continued)

at M/A-COM, the Lowell, Massachusetts, based wireless products group of AMP Inc . . .. Net present value would have derailed this project long ago, Vaughan insists. “It would have been difficult to sell this business case, because of the high level of uncertainty,” he says. Real options analysis assembles diverse risks in a coherent fashion, Vaughan says, layer upon layer, like a papier-mâché creation. “It very much mimics the venture capitalist approach,” he says, “by timing expenditures to the maturity of the opportunity.” Handle with Care Real options “add richness and perspective I can’t get elsewhere,” says the oil industry executive. But like any metric that relies on judgment, he warns, real options must be used carefully. They are not tamper-proof. “Given enough volatility and time,” he says, “I can make an option a very big number.” Without solid, accurate measures of volatility, real options can lead companies astray. For evaluating an offshore oil lease, look at the history of oil-futures prices; for a petrochemical plant, look at historical futures and options contracts on margins . . . “I don’t think the value of great judgment or intuition is any less in using a more sophisticated model,” Bingham says. To the extent that real options analysis sheds more light on uncertainty, in his view, it provides a critical link between strategy and finance. Says Bingham: “Getting hold of real options will make a CFO more and more relevant and a valuable part of leadership.” In an uncertain world, that’s the sort of vision CFOs rely on. Reprinted with permission from CFO Magazine, website www.cfo.com © CFO Publishing LLC. Publication date: November 1999.

sometimes make wrong decisions with limited adverse consequences. In contrast, when several firms are active in the market, managers are under constant competitive pressure to make the right decisions all the time. Otherwise, the firm might go belly-up. In perfect competition the decisions of a single firm do not have a significant impact on others, and strategic interactions are again inconsequential. In reality, however, industries are rarely either purely monopolistic or perfectly competitive. In the real marketplace that is closer to oligopoly, firms typically respond to their rivals’ actions. This calls for an appropriate methodology, namely game theory.

24

Chapter 1

Over the last half century game theory has developed into a rigorous framework for assessing strategic alternatives.13 It helps managers formulate the right strategies and make the right decisions under competition. A recent article in CFO magazine epitomizes a renewed interest by companies in using game theory to aid decision-making (see box 1.6). The origins of game theory trace back to the 1900s when mathematicians got interested in studying various interactive games, such as chess and poker. The first comprehensive formulation of the concept of optimal Box 1.6 Game theory in business practice

More Companies Are Using Game Theory to Aid Decision-Making: How Well Does It Work in the Real World? Alan Rappeport, CFO Magazine When Microsoft announced its intention to acquire Yahoo last February, the software giant knew the struggling search firm would not come easily into the fold. But Microsoft had anticipated the eventual minuet of offer and counteroffer five months before its announcement, thanks to the powers of game theory. A mathematical method of analyzing game-playing strategies, game theory is catching on with corporate planners, enabling them to test their moves against the possible responses of their competitors. Its origins trace as far back as The Art of War, the unlikely management best seller penned 2,500 years ago by the Chinese general Sun Tzu. Mathematicians John von Neumann and Oskar Morgenstern adapted the method for economics in the 1940s, and game theory entered the academic mainstream in the 1970s, when economists like Thomas Schelling and Robert Aumann used it to study adverse selection and problems of asymmetric information. (Schelling and Aumann won Nobel Prizes in 2005 for their work.) Game theory can take many forms, but most companies use a simplified version that focuses executives on the mind-set of the competition. “The formal stuff quickly becomes very technical and less useful,” says Louis Thomas, a professor at the Wharton School of Business who teaches game theory. “It’s a matter of peeling it back to its bare essentials.” One popular way to teach the theory hinges on a situation called the “prisoner’s 13. Game theory is concerned with the actions of decision makers conscious that their actions affect those of rivals and that the actions of competitors, in turn, impact their own decisions. When many players can disregard strategic interactions as being inconsequential (perfect competition) or when a firm can reasonably ignore other parties’ actions (monopoly), standard optimization techniques suffice. Under imperfect competition such as in oligopoly, a limited number of firms with conflicting interests interact such that the actions of each can materially influence firm individual profits and values.

The Strategy Challenge

Box 1.6 (continued)

dilemma,” where the fate of two detainees depends on whether each snitches or stays silent about an alleged crime. Many companies are reluctant to talk about the specifics of how they use game theory, or even to admit whether they use it at all. But oil giant Chevron makes no bones about it. “Game theory is our secret strategic weapon,” says Frank Koch, a Chevron decision analyst. Koch has publicly discussed Chevron’s use of game theory to predict how foreign governments and competitors will react when the company embarks on international projects. “It reveals the win-win and gives you the ability to more easily play out where things might lead,” he says. Enter the Matrix Microsoft’s interest in game theory was piqued by the disclosure that IBM was using the method to better understand the motivations of its competitors—including Microsoft—when Linux, the open-source computer operating system, began to catch on. (Consultants note that companies often bone up on game theory when they find out that competitors are already using it.) For its Yahoo bid, Microsoft hired Open Options, a consultancy, to model the merger and plot a possible course for the transaction. Yahoo’s trepidation became clear from the outset. “We knew that they would not be particularly interested in the acquisition,” says Ken Headrick, product and marketing director of Microsoft’s Canadian online division, MSN. And indeed they weren’t; the bid ultimately failed and a subsequent partial acquisition offer was abandoned in June. Open Options wouldn’t disclose specifics of its work for Microsoft, but in client workshops it asks attendees to answer detailed questions about their goals for a project—for example, “Should we enter this market?” “Will we need to eat costs to establish market share?” “Will a price war ensue?” Then assumptions about the motives of other players, such as competitors and government regulators, are ranked and different scenarios developed. The goals of all players are given numerical values and charted on a matrix. The exercise is intended to show that there are more outcomes to a situation than most minds can comprehend, and to get managers thinking about competition and customers differently. “If you have four or five players, with four actions each might or might not take, that could lead to a million outcomes,” comments Tom Mitchell, CEO of Open Options. “And that’s a simple situation.” To simplify complex playing fields, Open Options uses algorithms to model what action a company should take—considering the likely actions of others—to attain its goals. The result replicates the so-called Nash

25

26

Chapter 1

Box 1.6 (continued)

equilibrium, first proposed by John Forbes Nash, the Nobel Prize winning mathematician portrayed in the movie A Beautiful Mind. In this optimal state, the theory goes, a player no longer has an incentive to change his position. As a tool, game theory can be useful in many areas of finance, particularly when decisions require both economic and strategic considerations. “CFOs welcome this because it takes into account financial inputs and blends them with nonfinancial inputs,” says Mitchell. Rational to a Fault? Some experts, however, question game theory’s usefulness in the real world. They say the theory is at odds with human nature because it assumes that all participants in a game will behave rationally. But as research in behavioral finance and economics has shown, common psychological biases can easily produce irrational decisions. Similarly John Horn, a consultant at McKinsey, argues that game theory gives people too much credit. “Game theory assumes rationally maximizing competitors who understand everything that you’re doing and what they can do,” says Horn. “That’s not how people actually behave.” (Activist investor Carl Icahn said Yahoo’s board “acted irrationally” in rejecting Microsoft’s bid.) McKinsey’s latest survey on competitive behavior found that companies tend to neglect upcoming moves by competitors, relying passively on sources such as the news and annual reports. And when they learn of new threats, they tend to react in the most obvious way, focusing on near-term metrics such as earnings and market share. Moreover finance executives have their own sets of metrics, and when favored indicators such as net present value clash with game theory models, choices become more complicated. “Sometimes [game theory] tells you things you don’t like,” says Koch. Game theory is still finding its place as a tool for companies, and its ultimate usefulness may depend on how quickly it moves from novelty to accepted practice. Practice in fact may be key. McKinsey takes that to heart with its “war game” scenarios, in which a company’s top managers play the roles of different parties in a simulation. In effect this boils game theory down to the schoolyard lesson that perfection comes through repetition. “Discipline is not a dirty word,” as basketball coach Pat Riley once said. Game theory is one way that companies can assess their options with more discipline. Reprinted with permission from CFO Magazine, website www.cfo.com © CFO Publishing LLC. Publication date: July 15, 2008.

The Strategy Challenge

27

strategies in a multiple-player setup came with The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern (1944). More critical advancements were made in the early 1950s when John F. Nash Jr. provided a broad mathematical basis for the study of equilibria in strategic conflict situations (Nash 1950a, b, 1951).14 More recently game theory has been applied in many fields, including political science, international relations, military strategy, law, sociology, psychology, and biology. Game theory has revolutionized microeconomics and given a strong analytical basis for studying real market structures. As firm competitiveness involves interactions among many players (firms, suppliers, buyers, etc.), it also brought appealing insights into strategic management. Box 1.7 provides an overview of the basics of game theory by Avinash Dixit. An interview with Professor Dixit concerning the interconnection between real options and game theory and his pedagogical approach is given in box 1.8. To perform strategic analysis, one needs to reduce a complex multiplayer problem into a simpler analytical structure that captures the essence of the conflict situation. As discussed later, conducting such an analysis involves a clear depiction of the “rules of the game,” namely (1) identifying the players, (2) describing their available alternative choices, (3) specifying the information structure of the game, (4) determining the payoff values attached to each possible strategy choice, and (5) specifying the order or sequence of the play.15 From a specified game structure, one may derive useful predictions on how rivals are likely to react (equilibrium strategies) in a given environment. A key question commonly arises: how should managers view these models? Should they interpret them in a literal or in a metaphorical sense? There is no clear-cut answer, but the metaphorical interpretation is generally accepted as more appropriate. One of the underlying premises of standard game theory is the presumed rationality of economic agents.16 This assumption is not always consistent with real-world behavioral phenomena, so excessive mathematical rigor may limit the 14. Before John F. Nash’s work, the focus of game theory was mostly on zero-sum games. Nash’s (1950b) equilibrium concept applies to a large set of problems beyond zero-sum games. Myerson (1999) provides a comprehensive analysis of how the Nash equilibrium concept shaped economic theory. 15. The order of play may affect both the possible actions players can select from and the information they possess at the time they make their decisions. 16. Defining rationality and optimality in a given strategic setting is one of the core objectives of game theory. This has an impact on the choice of the game-theoretic solution concept. Nash equilibrium makes behavioral assumptions beyond common knowledge of rationality.

28

Chapter 1

Box 1.7 Overview of game theory basics

Avinash K. Dixit, John J. K. Sherrerd ’52 University Professor of Economics, Princeton University Game theory studies interactive decision-making, where the outcome for each participant or “player” depends on the actions of all. If you are a player in such a game, when choosing your course of action or “strategy” you must take into account the choices of others. But in thinking about their choices, you must recognize that they are thinking about yours, and in turn trying to take into account your thinking about their thinking, and so on. It would seem that such thinking about thinking must be so complex and subtle that its successful practice must remain an arcane art. Indeed some aspects such as figuring out the true motives of rivals and recognizing complex patterns do often resist logical analysis. But many aspects of strategy can be studied and systematized into a science—game theory. The Nash Equilibrium The theory constructs a notion of “equilibrium” to which the complex chain of thinking about thinking could converge. Then the strategies of all players would be mutually consistent in the sense that each would be choosing his or her best response to the choices of the others. For such a theory to be useful, the equilibrium it posits should exist. Nash used novel mathematical techniques to prove the existence of equilibrium in a very general class of games. This paved the way for applications. Biologists have even used the notion of Nash equilibrium to formulate the idea of evolutionary stability. Here are a few examples to convey some ideas of game theory and the breadth of its scope. The Prisoner’s Dilemma In Joseph Heller’s novel Catch-22, allied victory in World War II is a forgone conclusion, and Yossarian does not want to be among the last ones to die. His commanding officer points out, “But suppose everyone on our side felt that way?” Yossarian replies, “Then I’d certainly be a dammed fool to feel any other way, wouldn’t I?” Every general reader has heard of the prisoner’s dilemma. The police interrogate two suspects separately, and suggest to each that he or she should fink on the other and turn state’s evidence. “If the other does not fink, then you can cut a good deal for yourself by giving evidence against the other: if the other finks and you hold out, the court will treat you especially harshly. Thus no matter what the other does, it is better for you to fink than not to fink—finking is your uniformly best or ‘dominant’ strategy.” This is the case whether the two are actually guilty, as in some

The Strategy Challenge

Box 1.7 (continued)

episodes of NYPD Blue, or innocent, as in the firm L.A. Confidential. Of course, when both fink, they both fare worse than they would have if both had held out; but that outcome, though jointly desirable for them, collapses in the face of their separate temptations to fink. Yossarian’s dilemma is just a multi-person version of this. His death is not going to make any significant difference to the prospects of victory, and he is personally better off alive than dead. So avoiding death is his dominant strategy. John Nash played an important role in interpreting the first experimental study of the prisoner’s dilemma, which was conducted at the Rand Corporation in 1950. Real-World Dilemmas Once you recognize the general idea, you will see such dilemmas everywhere. Competing stores who undercut each other’s prices when both would have done better if both had kept their prices high are victims of the dilemma. (But in this instance consumers benefit from the lower prices when sellers fink on each other.) The same concept explains why it is difficult to raise voluntary contributions, or to get people to volunteer enough time for worthwhile public causes. How might such dilemmas be resolved? If the relationship of the player is repeated over a long time horizon, then the prospect of future cooperation may keep them from finking; this is the well-known tit-for-tat strategy. A “large” player who suffers disproportionately more from complete finking may act cooperatively even when the small is finking. Thus Saudi Arabia acts as a swing producer in OPEC, cutting its output to keep prices high when others produce more, and the United States bears a disproportionate share of the costs of its military alliances. Finally, if the group as a whole will do better in its external relations if it enjoys internal cooperation, then the process of biological or social selection may generate instincts or social norms that support cooperation and punish cheating. The innate sense of fairness and justice that is observed among human subjects in many laboratory experiments on game theory may have such an origin. Mixing Moves In football, when an offense faces a third down with a yard to go, a run up the middle is the usual or “percentage” play. But an occasional long pass in such a situation is important to keep the defense honest. Similarly a penalty kicker in soccer who kicks exclusively to the goalie’s right, or a server in tennis who goes exclusively to the receiver’s forehand, will fare poorly because the opponent will anticipate and counter the action. In such situations it is essential to mix one’s moves randomly so that on any one occasion the action is unpredictable.

29

30

Chapter 1

Box 1.7 (continued)

Mixing is most important in games where the players’ interests are strictly opposed, and this happens most frequently in sports. Indeed recent empirical studies of serving in tennis grand slam finals, and penalty kicks in European soccer leagues, have found the behavior consistent with the theory. Commitments Greater freedom of action seems obviously desirable. But in games of bargaining, that need not be true because freedom to act can simply become freedom to concede to the other’s demands. Committing yourself to a firm final offer leaves the other party the last chance to avoid a mutually disastrous breakdown, and this can get you a better deal. But a mere verbal declaration of firmness may not be credible. Devising actions to make one’s commitment credible is one of the finer acts in the realm of strategic games. Members of a labor union send their leaders into wagebargaining with firm instructions or mandates that tie their hands, thereby making it credible that they will not accept a lower offer. The executive branch of the US government engaged in international negotiations on trade or related matters can credibly take a firm stance by pointing out that the Congress would not ratify anything less. And a child is more likely to get the sweet or toy it wants if it is crying too loudly to hear your reasoned explanations of why it should not have it. Thomas Schelling pioneered the study of credible commitments, and other more complex “strategic moves” like threats and promises. This has found many applications in diplomacy and war, which, as military strategist Karl von Clausewitz told us long ago, are two sides of the same strategic coin. Information and Incentives Suppose that you have just graduated with a major in computer science and have an idea for a totally new “killer app” that will integrate PCs, call phones, and TV sets to create a new medium. The profit potential is immense. You go to venture capitalists for finance to develop and market your idea. How do they know that the potential is as high as you claim it to be? The idea is too new for them to judge it independently. You have no track record, and you might be a complete charlatan who will use the money to live high for a few years and then disappear. One way for them to test your own belief in your idea is to see how much of your own money you are willing to risk in the project. Anyone can talk a good game; if you are willing to put enough of your money where your mouth is, that is a credible signal of your own true valuation of your idea. This is a game where the players have different information; you know the true potential of your idea much better than does your prospective

The Strategy Challenge

Box 1.7 (continued)

financier. In such games, actions that reveal or conceal information play crucial roles. The field of “information economics” has clarified many previously puzzling features of corporate governance and industrial organization, and has proved equally useful in political science, studies of contract and tort law, and even biology. The award of the Nobel Memorial Prize in 2001 to its pioneers, George Akerlof, Michael Spence, and Joseph Stiglitz, testifies to its importance. What has enabled information economics to burgeon in the last twenty years is the parallel development of concepts and techniques in game theory. Aligning Interests, Avoiding Enrons A related application in business economics is the design of incentive schemes. Modern corporations are owned by numerous shareholders who do not personally supervise the operations of the companies. How can they make sure that the workers and managers will make the appropriate efforts to maximize shareholder value? They can hire supervisors to watch over workers, and managers to watch over supervisors. But all such monitoring is imperfect: the time on the job is easily monitored, but the quality of effort is very difficult to observe and judge. And there remains the problem of who will watch over the upper-level management. Hence the importance of compensation schemes that align the interests of the workers and managers with those of the shareholders. Game theory and information economics have given us valuable insights into these issues. Of course, we do not have perfect solutions; for example, we are just discovering how top management can manipulate and distort the performance measures to increase their own compensation while hurting shareholders and workers alike. This is a game where shareholders and the government need to find and use better counterstrategies. From Intuition to Prediction While reading these examples, you probably thought that many of the lessons of game theory are obvious. If you have had some experience of playing similar games, you have probably intuited good strategies for them. What game theory does is to unify and systemize such intuitions. Then the general principles extend the intuitions across many related situations, and the calculation of good strategies for new games is simplified. It is no bad thing if an idea seems obvious when it is properly formulated and explained; on the contrary, a science or theory that takes simple ideas and brings out their full power and scope is all the more valuable for that.

31

32

Chapter 1

Box 1.8 Interview with Avinash K. Dixit

1. You have helped establish and popularize both real options and game theory as separate disciplines. How do you see the interconnection or interplay among the two? The real options concept emphasizes the value of flexibility, whereas an irreversible commitment has value in many situations of strategic competition. Analyzing the two together enables us to understand when one prevails over the other, or more generally, the trade-off between the two. This is clearly an important research program. 2. In your teaching of game theory at Princeton you rely a lot on stories, movies, literature, sports, games, and other engaging tools to motivate your students. Can you elaborate on your view or approach? Undergraduate students are rightly skeptical of abstract theory, and demand evidence of its relevance before they will spend their time and effort on studying theory. Game theory is fortunate in having so many compelling and entertaining examples readily available. Using examples to bring out theoretical concepts is similar to the case method used in most business schools. But MBA students are narrowly focused on business and moneymaking. Undergraduates have richer and more varied lives; therefore examples from sports, games, literature, and movies appeal to them. 3. What other current or future areas of research in economic sciences do you find interesting? Connections between economics and other social sciences are enriching them all. Economists are becoming aware of aspects of human behavior

The Strategy Challenge

33

Box 1.8 (continued)

that differ from selfish rationality assumed in most traditional economic theory; sociologists, political scientists, and even psychologists and anthropologists are learning the value of economists’ conceptual framework of choice and equilibrium and of the issues of endogeneity and identification in empirical work. I find this confluence of the social sciences interesting and exciting. I don’t expect complete reintegration of fields that separated more than a century ago; the benefits of specialization remain. But I do expect much closer communication and collaboration that will benefit research in all these fields.

applicability of game theory in certain real-world situations. A metaphorical interpretation of game-theoretic models can nonetheless hold significant value. Game theory can generally be used to deduce principles and insights from simplified models of reality. Simpler models are generally more prescriptive. Oftentimes, when game theory is applied in more complex situations, it results in no outcome or in multiple equilibria. In such cases complex modeling may lose its predictive ability.17 One objective of microeconomic modeling is to simulate qualitatively the type of environment being studied. Useful insights into an issue or behavior of practical relevance can be then deduced. A simple game-theoretic framework has several advantages for strategic management.18 First, it provides an audit track that enables researchers and practitioners to go back to basic premises. Management should formulate the basic assumptions explicitly and consider whether they are of practical relevance. Second, it offers a methodology that is conducive to rigorous analysis and can help derive novel insights, which might be counterintuitive in some cases. Such insights may hardly stem from a “boxes-and-arrows” conceptual framework.19 Finally, it helps bring discipline by enforcing a common language that enables researchers and managers to compare results and refine earlier models.20 In view of such potential extensions, game theory holds out considerable promise for 17. See Grant (2005, p. 111). 18. This discussion follows Saloner (1991, pp. 120–25, 127). 19. Saloner (1991) includes Porter’s (1980) five-forces framework among these conceptual frameworks for strategic management. A game-theoretic refinement of Porter’s (1980) framework is offered by Brandenburg and Nalebuff (1996). 20. Some microeconomic models are quite tractable for studying new problems. Cournot and Bertrand duopoly models discussed in chapter 3 form the basis for many industrial organization setups which are often much more complex. For instance, the commitment theory addressed in chapter 4 can be understood in light of these two pillar models.

34

Chapter 1

studying human interactions. This explains the success of game theory particularly in economics and strategic management. The following three fields have greatly benefited from game-theoretic modeling:21 Industrial organization This field focuses on strategic interactions arising in the external environment of the firm, addressing issues such as competitive interactions in oligopolies, first- and second-mover advantages, firm entry and exit decisions, strategic commitment, reputation, signaling and information asymmetries among different players in an industry. •

• Organization theory This field focuses on the firm’s internal or organizational aspects such as vertical and horizontal scope and conflicting incentives inside an organization (e.g., optimal compensation schemes).22

Interaction between the internal and external environment of the firm This area addresses issues at their interface like optimal incentive schemes in oligopolistic market structures,23 organizational design to achieve competitive advantage, cooperation versus competition in R&D.24 •

A good understanding of the competitive environment will thus enable managers to ascertain the strategic implications of their actions in the marketplace and determine how they should behave. A normative role for the strategic management literature is to provide a broad qualitative understanding of such strategic interactions and give qualitative prescriptions for managerial action.25 Although game theory has witnessed rapid developments since the 1950s, its use for practical strategic management purposes has remained limited so far.26 Nevertheless, some managers already incorporate game-theoretic thinking into their planning. 21. This discussion follows Saloner (1991, pp. 119–20). 22. Theories of the firm attempt to explain why firms exist and operate as they do. They include property-rights theory, incentive-system theory, rent-seeking theory, adaptation theory, and contract theory. 23. Ferschtman and Judd (1987), for example, argue that in case of product market competition, a firm has an incentive to design incentive schemes in a way that would not be optimal for a stand-alone organization. In their model, managers have an incentive to maximize output in a quantity competition setting, which makes the firm better off since the rival interprets the incentive contract as a commitment. This exemplifies the possible use of industrial organization thinking to better the understanding of firms’ organizational design. 24. D’Apremont and Jacquemin (1988) analyze the effect of R&D spillovers on the incentive to cooperate or compete during the R&D stage and the market competition stage. 25. See Saloner (1991, pp. 107–31). 26. In politics, game theory has been used to analyze the 1962 Cuban missile crisis, President Reagan’s 1982 tax cut, and certain public auctions.

The Strategy Challenge

35

1.4 An Integrative Approach to Strategy Both corporate finance and game theory provide useful insights for strategic management. As discussed by Jean Tirole in box 1.9, the interface between finance and game theory enables attaining a better understanding on a number of firm- or market-related issues. Viewed separately, however, these approaches have limited applicability. Integrating these approaches in a consistent manner is at the core of the option games approach. Standard real options analysis overcomes many of the drawbacks of the NPV approach but neglects other aspects. When management assesses its real options, it must determine whether the benefits resulting from the exercise of its options are fully appropriable. Kester (1984) distinguishes two categories of real options depending on whether the benefits are proprietary or shared. If management has an exclusive exercise right, retaining all potential benefits for itself, the investment opportunity is a proprietary option.27 When the firm is not in a position to appropriate all of the project’s benefits for itself but rivals share the same opportunity, it is a shared option.28 In this case, the presence of market contenders introduces strategic externalities (positive or negative) that can significantly affect the value and optimal exercise strategy of the firms’ real options. The value loss resulting from strategic interactions is seen as a competitive value erosion. Standard or naïve real options analysis typically assumes a proprietary or monopolistic mind-set, ignoring such shared options.29 A firm here formulates its investment decisions in isolation, disregarding interactive competition. 27. Rivals’ investment decisions have no material impact on project values or optimal strategies. For instance, a monopolist protected by significant entry barriers faces such a situation. Proprietary options are also encountered when a firm is granted an infinitely lived patent on a product that has no close substitutes or when it has unique know-how of a technological process. 28. Shared options include the opportunity to launch a new product that is unprotected from the entry of close substitutes, or the opportunity to penetrate a newly deregulated market. 29. One reason why strategic interactions among option holders are not typically considered in standard real options analysis is that early on real options were seen as an extension of standard option-pricing theory to real investment situations. In capital markets, except in special cases like valuation of warrants, strategic interactions among option holders rarely affect the asset or the option values. Certain continuous-time real option models attempt to account for market and competitive uncertainty in an exogenous manner, such as through a higher dividend yield or a jump process. These models represent an improvement over standard models developed with a monopolistic mind-set, but fall short of adequately accounting for the endogenous nature of strategic interactions in an oligopolistic setting. Trigeorgis (1991) discusses continuous-time real options models involving strategic uncertainty exogenously.

36

Chapter 1

Box 1.9 Interview with Jean Tirole

1. You have contributed greatly to extending game theory for the analysis of economic problems. What are the merits of this mathematical discipline for economic analysis? Which other social sciences do you believe can benefit from the use of game theory? Game theory aims at describing and predicting behaviors in environments in which actors are interdependent and have potentially conflicting objectives. It deepens our understanding of when the quest for specific goals may lead to inefficiencies and of how players choose actions with an eye on changing other actors’ incentives. As such, game theory applies to all social sciences and beyond. The most obvious applications, besides economics, include political sciences, sociology, law, and psychology. Psychology might look like an outlier as it usually focuses on the individual, but it is not. Experimental evidence confirms the old notion that we “play games with ourselves.” These can be represented as games among successive incarnations of the self. Biologists use game theory to understand mutualism between species, inefficient signals, or fights. Computer scientists also take a keen interest in game theory. Part of the appeal of game theory is that it accommodates diverse objective functions, which enables us to conceptualize the behavior of different actors, from consumers to politicians, from firms to suborganisms. Game theory more and more integrates behavioral approaches. Mainstream game theory focuses on optimal strategies given the strategic interdependences and the actors’ limited information and various constraints. This rational choice approach has served social sciences well by identifying the key strategic features of conflict situations. At the same time, limited cognition and various

The Strategy Challenge

Box 1.9 (continued)

behavioral biases are being increasingly incorporated into our thinking about strategic interactions, extending the reach of game theory beyond purely rational choice. Finally, experimental economists have been testing our equilibrium concepts and behavioral predictions, and empiricists use game theory to put more structure on their estimation strategies. 2. Your work on industrial organization has helped popularize this important discipline and extend its areas of application. Do you think industrial organization will become increasingly more useful for managerial practice and understanding or predicting of market developments? Yes. Game theory and its applications to industrial organization have made their way into business books and have affected managerial practice. For example, concepts developed in industrial organization are used in deliberations on how to design new platforms and get all sides on board. Game theory is taught to MBAs and strategy textbooks now incorporate game-theoretic thinking. Game-theoretic analyses have become a language for antitrust practitioners to conceptualize impacts of behaviors on market outcomes. Empirical work on estimating demand and strategic choices of price and non–price competition also make substantial use of game-theoretic industrial organization. 3. Do you see a connection between industrial organization and corporate finance? In what ways? Can game theory help reshape corporate finance as it has reshaped standard microeconomics? There is indeed a strong connection between industrial organization and finance. I am really happy that your book “cross-breeds” options theory and industrial organization and connects it to business, as this is an area of very fruitful cross-fertilization. Not only do finance and industrial organization share common tools (e.g., economics of incentives, game theory), they also interface in many areas. For example, it is hard to fully understand predation or entry into industries without understanding financial constraints and therefore corporate finance. Conversely, the industrial organization of finance and banking is a hot topic on our research agenda. Game theory has already made its way into various subfields of corporate finance: takeover strategies, liquidity hoarding, expectations of refinancing and bailouts, bank runs, issues of securities under asymmetric information, and conglomerate strategies are just a few examples. Many areas of corporate finance have benefited significantly from importing ideas coming from game theory.

37

38

Chapter 1

Table 1.2 Comparison of main advantages and drawbacks of standard stand-alone approaches Approach

Advantages

Drawbacks

Standard NPV

Easy to use; convincing logic; widely used; easy to communicate

Assumes precommitment to a given plan of action, often treating investment as a one-time decision (“invest now or never”); ignores flexibility to adapt to unexpected market developments or strategic interactions

Real options

Incorporates market uncertainty and managerial flexibility; recognizes that investment decisions can be delayed, staged, or adjusted under certain future contingencies

Typically applied to the valuation of a monopolist or proprietary option; ignores (endogenous) competitive interactions

Game theory

Incorporates competitive reactions endogenously; considers different player payoffs

Typically disregards market uncertainty involving stochastic variables

Game theory has been generally applied to studying strategic interactions in settings involving steady or deterministically changing states, where players could accurately predict the evolution of the external environment. Standard game theory falls short of explaining the firm’s incentive to stay flexible to react to unexpected developments. Under uncertainty this prescription of standard game theory is inadequate.30 The main advantages and drawbacks of each stand-alone approach (standard NPV, standard real options analysis, and game theory) are summarized in table 1.2. Many real-world problems, however, require a simultaneous assessment of both market (exogenous) as well as competitive (endogenous) uncertainty. Stand-alone NPV, real options analysis, and game theory alone fail in providing the necessary tool kit. The NPV paradigm deals with static situations where firms make now-or-never decisions or precommit to a certain plan of action. Real options analysis allows for 30. Different forms of uncertainty are considered in game theory. For instance, some solution concepts (e.g., Bayesian and perfect Bayesian equilibrium) are designed to address problems involving information uncertainty. This form of uncertainty is not equivalent to what we consider here. In stochastic environments, payoffs may be affected by exogenous factors or shocks, whose future values are not known with certainty but follow a known probability law. The appendix of the book provides a discussion of such stochastic processes.

The Strategy Challenge

39

Table 1.3 Classification of decision situations and relevant theories Decision theory (no strategic interaction)

Game theory (strategic interaction)

Net present value (DCF) Chapter 1

Static industrial organization Chapter 3

Deterministic

Resource extraction/ forest economics Chapter 9

Dynamic industrial organization Chapters 4, 11, 12

Stochastic

Real options analysis Chapters 5, 9, appendix

Option games Chapter 6 onward

Static

Dynamic

dynamic decision-making in situations where firms face exogenous stochastic uncertainty.31 Static industrial organization (IO) has limited applicability to situations involving simultaneous games where firms are ignorant of both past and future actions and payoffs.32 Dynamic industrial organization analysis permits a long-term perspective but assumes steady state or a deterministic evolution of the market environment.33 We here discuss an integrated approach employed to help overcome the shortcomings of stand-alone approaches, analyzing key value drivers concurrently. Option games are meant to capture dynamic strategic interactions in stochastic environments. Table 1.3 positions the option games approach within the traditional decision and game-theory paradigms.34 We indicate in which chapters each approach is most relevant in the book. The importance of incorporating options analysis and game theory is confirmed by the number of Nobel Prizes awarded in related fields. A 31. Real options analysis provides many applications for dynamic programming. The term “dynamic programming” was originally used by Bellman (1957) to describe a recursive process for solving dynamic problems. This method can be extended to stochastic environments. Stochastic dynamic programming or stochastic control is discussed by Harrison (1985), Dixit (1993), and Stokey (2008), with applications in economics and finance. 32. Static industrial organization rests on static game theory with the related notion of, for example, Nash or Bayesian Nash equilibria. See Osborne (2004) for an introduction. 33. Fudenberg and Tirole (1986) present a number of dynamic models of oligopoly. This field rests on dynamic game theory. It involves the use of “dynamic” solution concepts such as subgame perfect Nash, perfect Bayesian, or sequential equilibrium. A subfield is differential games that study dynamic strategic interactions in settings where an industry state evolves according to a differential equation. See Dockner et al. (2000) for an introduction to differential games. 34. Option games share some aspects with stochastic timing and differential games.

40

Chapter 1

Table 1.4 Selected Nobel Prizes awarded in Economic Sciences Year

Nobel Prize winner(s)

Noted contribution

2007

L. Hurwicz, E. S. Maskin, R. B. Myerson

For laying the foundations of mechanism design and contract theory

2005

R. J. Aumann, T. C. Schelling

For enhancing our understanding of conflict and cooperation through game theory analysis

2001

G. A. Akerlof, A. M. Spence, J. E. Stiglitz

For analyzing markets with asymmetric information

1997

R. C. Merton, M. S. Scholes

For developing the option pricing method to value derivatives (and thereby real options)

1994

J. C. Harsanyi, J. F. Nash Jr., R. Selten

For their analysis of equilibria in the theory of noncooperative games

1990

H. M. Markowitz, M. H. Miller, W. F. Sharpe

For their pioneering work in the theory of financial economics

1982

G. J. Stigler

For his studies of industrial structures, functioning of markets, and causes and effects of public regulation

1981

J. Tobin

For his analysis of financial markets, expenditure decisions, employment, production, and prices

1978

H. A. Simon

For his research into the decisionmaking process within economic organizations

1975

L. V. Kantorovich, T. C. Koopmans

For their contributions to the theory of optimum allocation of resources

1970

P. A. Samuelson

For developing dynamic economic theory and raising the rigor of analysis in economics

Source: Nobel Prize committee website.

selected list of Nobel Prize awards in economic sciences is shown in table 1.4. Real options analysis is a natural extension of major breakthrough developments in financial economics to real investments. It builds on the seminal works of Paul Samuelson, Robert C. Merton, Fischer Black, and Myron Scholes. Concurrently the analysis of industrial organization has been greatly facilitated by developments in game theory. Von Neumann, Morgenstern, and Nash have made significant early contributions to game theory. Selten, Harsanyi, Schelling, and Aumann also earned Nobel

The Strategy Challenge

41

Prizes for refinements to the theory.35 Option games, being at the intersection of option and game theories, benefited from the cumulative developments in these subfields of economic sciences. Today option games represent a powerful strategic management tool that can guide practical managerial decisions in a competitive context, as discussed by Ferreira, Kar, and Trigeorgis (2009). It enables a more complete quantification of market opportunities while assessing the sensitivity of strategic decisions to exogenous variables (e.g., demand volatility, costs) and competitive interactions. 1.5

Overview and Organization of the Book

The book is organized in three parts. Part I, “Strategy, Games, and Options,” presents the three building blocks or prerequisite fields for the option games approach. Chapter 2, “Strategic Management and Competitive Advantage,” reviews the main strategic management paradigms used to analyze or explain a firm’s performance in creating value for shareholders. We describe industry and competitive analysis and discuss how to create sustainable competitive advantage in an industry utilizing generic competitive strategies. Chapters 3 and 4 on “Market Structure Games” introduce game theory principles and industrial organization concepts providing economic foundations for strategic management. Chapter 3, focusing on “Static Approaches,” discusses benchmark cases where firms interact in one-time situations. Quantity and price competition are discussed in detail. Chapter 4, focusing on “Dynamic Approaches,” supplements the previous analysis by allowing firms to interact in the marketplace over many periods, attempting in the long term to shape the market in their own advantage or collaborating with rivals for mutual benefit. Chapter 5 on “Uncertainty, Flexibility, and Real Options” discusses the strategyformulation challenges facing the firm when the underlying market is uncertain. Motivated by various sources of uncertainty electricity utilities face today, we discuss how real options analysis can be used to analyze such situations, value strategic options, and optimally chose among them. We also briefly discuss discrete-time and continuous-time tools for the pricing of embedded real options. We ignore here the fact 35. Akerlof, Spence, Stiglitz, Hurwicz, Maskin, and Myerson won the Nobel Prize for insightful applications of game theory for the understanding of incentives in social groups (e.g., industrial organization, contract theory).

42

Chapter 1

that many firms may face counteracting business opportunities affected by rival behavior. Part II, “Option Games: Discrete-Time Analysis,” fleshes out in more detail in discrete time the integration of real options with game theory and industrial organization and explains how to capture the flexibility and strategic-interaction aspects of real investment situations. Chapter 6 presents core issues in option games analysis, namely optimal investment timing under uncertainty and competition and the trade-off between commitment and flexibility. We provide a number of examples to illustrate the discrete-time option games approach. Chapter 7, “Option to Invest,” rigorously sets the premise for analyzing option games in discrete time, building upon models developed in chapter 3. Chapter 8, “Innovation Investment in Two-Stage Games,” discusses at length the trade-off between commitment and flexibility in sequential investment settings. The focus, here, is on two-stage competition models where real options analysis tools are combined with industry organization insights. The two-stage analysis provides guidance into how and when strategic investments enhance value creation or are detrimental to the firm. We examine appropriate investment strategy applications in different settings, such as R&D and advertising. Part III, entitled “Option Games: Continuous-Time Models,” extends the analysis of option games by use of continuous-time modeling techniques. Chapter 9, “Investment and Expansion Option: Monopoly,” introduces the methodology employed throughout part III and sets the benchmark case of a monopolist firm. Two categories of options are discussed, the option to invest in a new market and the option to expand an existing market. Chapter 10, “Oligopoly: Simultaneous Investment,” extends the analysis to simultaneous investment oligopoly markets. Chapter 11, “Leadership and Early-Mover Advantage,” discusses the appeal of having a competitive advantage to turn the investment-timing game into one’s own advantage. Chapter 12, “Preemption versus Cooperation in a Duopoly,” deals with preemptive investments and the possibility of tacit collusion among firms, delaying investment until a later date. Chapter 13, “Extensions and Other Applications,” provides a short overview of important contributions on various subjects and other applications discussed in the literature. The appendix that follows discusses the basics of stochastic processes and provides a compendium of tools in stochastic calculus and control for the more analytically minded reader. To smooth out the exposition in the text, part III occasionally refers to this appendix when a derivation is more involved.

The Strategy Challenge

43

Conclusion In this introductory chapter we discussed key changes firms have faced over the past decades. We discussed the development and changes in a challenging business environment and the evolution of strategy, and examined to which extent the two pillar approaches underlying this book, namely game theory and real options analysis, can be rigorous and relevant for analyzing and understanding business strategies. We concluded with the need to combine both into an integrative option games approach. Selected References Grant (2005) discusses the role of strategy in corporate performance, stressing the need for sound strategy formulation based on consistent principles. Myers (1984) discusses the gap between financial theory and its practical implementation in corporate strategy, highlighting the differential insights corporate finance can add to strategy. Saloner (1991) and Rasmussen (2005) discuss the usefulness of game theory for strategic management, emphasizing its prescriptive value in competitive settings involving strategic interactions. Grant, Robert M. 2005. Contemporary Strategy Analysis, 5th ed. Malden, MA: Blackwell. Myers, Stewart C. 1984. Finance theory and financial strategy. Interfaces 14 (1): 126–37. Rasmussen, Eric. 2005. Games and Information: An Introduction to Game Theory. Oxford: Blackwell. Saloner, Garth. 1991. Modeling, game theory, and strategic management. Strategic Management Journal 12 (Winter): 119–36.

I

STRATEGY, GAMES, AND OPTIONS

2

Strategic Management and Competitive Advantage

Managing an enterprise in an uncertain competitive environment is not an easy task. Strategic management attempts to explain why some firms are more successful than others in the marketplace. At the core of strategy is a dilemma between flexibility and commitment. Flexibility to adapt strategy and operations is clearly valuable when the environment changes unexpectedly. An early investment commitment may yet have strategic value because it can influence the behavior of rivals in equilibrium, potentially creating a future competitive advantage for the firm. The flexibility perspective partly draws on the resource-based view of the firm and core-competence arguments: a firm should invest in resources and competencies that give it a distinctive ability to pursue a set of market opportunities. During the 1990s this view became a dominant paradigm in strategic management. It has also helped spark further developments in what is referred to as the “knowledge-based” view of the firm. The commitment view has been firmly anchored in industrial organization. In recent years competitive advantage has become rather temporary because industries and competition within industries change continually. Firms make tremendous efforts to sustain competitive advantage and protect their future growth options from duplication efforts by rivals. Whether competitive advantage can be sustained in the long term partly depends on mechanisms put in place by the firm to renew and protect its resource position and capabilities. In section 2.1 we provide an overview of some of the prevailing strategic management concepts and theories. We discuss in section 2.2 several approaches used in practice that help provide a deeper understanding of a given industry. We then examine, in section 2.3, the sources of competitive advantage and their sustainability as an industry evolves.

48

2.1

Chapter 2

Strategic Management Paradigms

Various paradigms in strategic management approach the underlying sources of firm value creation from distinct viewpoints. Categorized in figure 2.1 are the main strategic management approaches. Many of these managerial perspectives trace their roots to primary economic disciplines (microeconomics, organization theory, finance). Behind many of them lies the notion that competitive advantage is a function of the firm’s position, resources, and capabilities compared to those of its rivals, the opportunities these resources and capabilities create in a dynamic environment, and the firms’ adaptive capability to respond to market changes. It is commonplace to distinguish between approaches with an external perspective from those with an internal firm view. The former consider competitive advantage and value creation through the eyes of an external observer, focusing on external forces, opportunities, and threats in the external environment. The latter focus on internal forces within the firm, viewing value creation from in-house combinations of expertise, resources, and capabilities. These perspectives consider competitive advantage as dependent on unique, firm-specific resources and capabilities that enable the firm to create and exploit advantaged opportunities. Real options and option games lie at their intersection and can bring together these complementary perspectives. Real options are created by internal adaptive capabilities developed over time but may be affected by the actions of external parties. 2.1.1

External View of the Firm

Outside forces, such as firm’s rivals, suppliers, and customers, continually shape the external environment. External approaches view the sources of value creation as lying in market imperfections, synergies, or economies of scale. The industry and competitive analysis framework popularized by Michael Porter (1980) views strategy in terms of industry structure, entry deterrence, and strategic positioning. Key competitive variables include entry and exit barriers, as well as product differentiation and availability of information. Porter proposes to look at rivalry situations among suppliers of a product and analyze the impact of additional forces, such as the threat from substitute products and technology disruption, new market entry, and the bargaining power of suppliers and buyers. He emphasizes actions the firm should take to protect itself from these market forces and threats. Another external approach is that of strategic

Rumelt (1984) Wernerfelt (1984) Collins and Montgomery (1995)

Schelling (1960) Shapiro (1989) Ghemawat (1991) Dixit and Nalebuff (1991)

Dixit and Pindyck (1994) Trigeorgis (1996)

Real options

Dynamic capabilities/ core competencies

Prahalad and Hamel (1990) Teece, Pisano, and Shuen (1997)

Transaction costs

Property rights

Organization theory

Internal view of the firm

Resourcebased view

Agency theory

Strategic conflict

Game theory/ industrial organization

External view of the firm

Porter (1980)

Industry and competitive analysis

Structure conduct performance

Microeconomic theory

Grenadier (2000) Huisman (2001) Smit and Trigeorgis (2004)

Option games

Option pricing

Financial economics

Figure 2.1 Strategic management frameworks Rows 2 and 3 are related: economic theory (row 2) supports notions developed in strategic management (row 3) via formal models.

Representative authors

Strategic management

Economic theory

Broad field

Strategic Management and Competitive Advantage 49

50

Chapter 2

conflict (e.g., Shapiro 1989), which focuses on conflict behavior and views value creation as the result of strategic moves in a competitive environment. Game theory ideas are used to help management understand better the strategic interactions among rivals, predict rivals’ reactions, and determine the optimal competitive strategy. 2.1.2

Internal View of the Firm

The internal view of the firm traces its roots to The Theory of the Growth of the Firm by Edith Penrose (1959). This approach became known subsequently as the resource-based view of the firm in articles by Wernerfelt (1984), Rumelt (1984), Teece (1984), and others.1 Its distinguishing characteristic is that competitive advantage arises from within the firm. A firm becomes profitable not so much because it undertakes strategic investments that may deter entry or raise prices above long-run cost, but rather because it manages to achieve significantly lower costs or obtain markedly higher quality or product performance through internal mechanisms. Excess economic profits stem from imperfect markets for firmspecific intangible assets like distinctive competences, know-how, and capabilities. A more recent variant rests on corporate capabilities to adapt in environments of rapid technological change. The theory of “dynamic capabilities” proposed by Teece, Pisano, and Shuen (1997) views capabilities to adapt in a changing environment as resting on distinctive processes, shaped by the firm’s asset position and the evolutionary path it has adopted or inherited. Box 2.1 discusses the concept and process of strategy and their evolution in a changing and rapidly transformed competitive landscape. 2.2

Industry and Competitive Analysis

A key objective of strategic management is to help decision makers understand what leads to a firm’s success. Strategic management provides both explanation and direction. A useful strategy framework should address the following types of questions: Why are some firms more successful than others? Are there different paths to success? Can some firms be successful by adopting product innovation, and others by enhancing operational efficiency via process innovation? 1. According to Wernerfelt (1984), the resource-based view aims to understand the relationship between profitability and the different ways of managing resources over time. Resources are “those tangible and intangible assets which are tied semi-permanently to the firm” (p. 172).

Strategic Management and Competitive Advantage

Box 2.1 Strategy in a changing competitive environment

Changes in the Competitive Battlefield C. K. Prahalad, Financial Times Many of the concepts used in strategy were developed during the late 1970s and 1980s when underlying competitive conditions evolved within a well-understood model . . .. While the canvas available to today’s strategists is large and new, companies will need to understand global forces, react quickly, and innovate when defining their business models . . .. It is hardly surprising that the conceptual models and administrative processes used by managers often outlast their usefulness. It takes researchers time, after all, to identify new problems and emerging solutions before they can produce theories about them. Then there is the time lag between the development of these theories and their conversion into common business practice. Where management concepts are concerned, this time lag—often a decade—brings with it an interesting conundrum. In an era of rapid and disruptive change in the economic, political, social, regulatory, and technological environment do managers have to discard established and tested analytical tools equally as fast? How can they identify the ongoing relevance of concepts and tools in a changing environment? The Heritage of Strategy Concepts The most prevalent and widely used tools of strategy analysis are: strength weakness, opportunities, and threats (SWOT) analysis, industry structure analysis (five forces), value chain analysis, generic strategies, strategic group analysis, barriers to entry, and others of the genre . . .. The concepts and tools—many of them the staples of economists—were adopted and simplified for the use of managers. The formalization of these concepts was instrumental in pushing strategy development from the realm of “the intuitive genius of the founder or a top manager” to that of logical process. However, most of these concepts were developed during the late 1970s and the decade of the 1980s. During this period, underlying competitive conditions evolved but within a well-understood paradigm. A major competitive disruption during this period, certainly for US and European companies, was the spectacular success of Japanese manufacturing in industries as diverse as steel, consumer electronics, autos, and semiconductors. The sources of competitive advantage, during this decade, accrued to those who could wrest major efficiencies in operations through focus on quality, cycle time, reengineering, and team work. Operational efficiencies, within a relatively stable industry structure paradigm, became the focus. In fact this focus on wresting competitive advantage through operational efficiencies led some managers to believe that strategy was unimportant and management was all about implementation.

51

52

Chapter 2

Box 2.1 (continued)

The Emerging Competitive Landscape The decade of the 1990s has witnessed significant and discontinuous change in the competitive environment and an accelerating global trend to deregulate and privatize. Large and key industries like telecoms, power, water, health care, and financial services are being deregulated. Countries as diverse as India, Russia, Brazil, and China are at various stages of privatizing their public sectors. Technological convergence—such as that between chemical and electronic companies; computing, communications, components, and consumer electronics; food and pharmaceuticals; and cosmetics and pharmaceuticals—is disrupting traditional industry structures . . .. Ecological sensitivities and the emergence of nongovernmental organizations such as the green movement are also new dimensions of the competitive landscape. Are these discontinuities changing the very nature of the industry structure—the relationships between consumers, competitors, collaborators, and investors? Are they challenging the established competitive positions of incumbents and allowing new types of competitors and new bases for competition to emerge? We can identify a long list of discontinuities and examples to illustrate each one of them . . .. We need to acknowledge the signals (weak as they may be) of the emergence of a new competitive landscape where the rules of engagement may not be the same as they were during the decade of the 1980s. Strategists have to make the transition from asking the question: How do I position my company and gain advantage in a known game (a known industry structure)? Increasingly the relevant question is: How do I divine the contours of an evolving and changing industry structure and therefore the rules of engagement in a new and evolving game? Industries represent such a diversity of new, emerging, and evolving games. The rules of engagement are written as companies and managers experiment and adjust their approaches to competition. Strategy in a Discontinuous Competitive Landscape Strategists must start with a new mind-set. Traditional strategic planning processes emphasized resource allocation—which plants, what locations, what products, and sometimes what businesses—within an implicit business model. Disruptive changes challenge the business models. Four transformations will influence the business models and the work of strategists in the decades ahead: The strategic space available to companies will expand Consider, for example, the highly regulated power industry. All utilities once looked alike and their scope of operations was constrained by public utility commissions and government regulators. Because of deregulation, utilities can now determine their own strategic space. Today utilities have

•

Strategic Management and Competitive Advantage

Box 2.1 (continued)

a choice regarding the level of vertical integration: Do I need to be in power generation? Do I need to be in power transmission? Companies can unbundle assets and can also segment their businesses: Should we focus more on industrial or domestic consumers? They can decide their geographical scope: Should I become global, regional, national, or just remain local? And finally, they can change their business portfolio: Should I invest in water, telecoms, gas lines, services? The forces of change—deregulation, the emergence of large developing countries such as India, China, and Brazil as major business opportunities—provide a new playing field. Simultaneously forces of digitalization, the emergence of the Internet, and the convergence of technologies provide untold new opportunities for strategists. The canvas available to the strategist is large and new. One can paint the picture one wants. Business will be global Increasingly the distinction between local and global business will be narrowed. All businesses will have to be locally responsive and all businesses will be subject to the influences and standards of global players. Consider, for example, McDonald’s and Coca-Cola—held up as examples of truly global players unconstrained by local customers and national differences. In India, McDonald’s had to change its recipe to serve lamb (instead of beef) and vegetarian patties (a radical departure from its normal western fare). Coke had to recognize the power of “Thums Up,” a local cola (which Coke purchased) and promote that product. The need for local responsiveness, especially when global companies want to penetrate markets with different levels of consumer purchasing power, is very clear. On the other hand, Nirula’s, a local fast-food chain in India, was, in its own restaurants, forced to respond to the cleanliness and ambience of McDonald’s. This is a case of global standards being imposed on a local player. Global and local distinctions will remain in products and services. Globalization may have as much to do with standards—quality, service levels, safety, environmental concerns, protection of intellectual property, and talent management. Needless to say, globalization will force strategists to come to terms with multiple geographical locations, new standards, capacity for adaptation to local needs, multiple cultures, and collaboration across national and regional boundaries in everything from manufacturing, product development, global account management, and logistics.

•

Speed will be a critical element Given the nature of competitive changes, speed of reaction will be a critical element of strategy. At a minimum it will challenge the yearly planning cycle. For example, consider the traditional strategic planning process in a

•

53

54

Chapter 2

Box 2.1 (continued)

large company. The process of strategy discussion and commitments typically starts in October. It identifies the strategic issues for the next calendar year and three to four years hence. What is the use of such a process in an Internet-based start up? Speed of reaction, not tied to a rigid corporate calendar is of the essence. Strategy must be a topic of discussion and debate all the time not just during the planning sessions—strategy making and thinking cannot be a “corporate rain dance” during October! Speed is also an element in how fast a company learns new technologies and integrates them with the old. As all traditional companies are confronted with disruptive changes, the capacity to learn and act fast is increasingly a major source of competitive advantage. Innovation is the new source of competitive advantage Innovation was always a source of competitive advantage. However, the concept of innovation was tied to product and process innovations. In many large companies, the innovation process is still called the “product creation process.” Reducing cycle time, increasing modularity, tracking sales from new products introduced during the last two years as a percentage of total sales, and global product launches were the hallmarks of an innovative company. Increasingly the focus of innovation has to shift toward innovation in business models. For example, how does an auctionbased pricing market (e.g., airlines, hotels) in an industry with excess capacity change the business model? How do you think about resources available to the company for product development when customers become co-developers of the product or service? Should we have an expanded notion of resource availability? What impact does mass customization or, more importantly, personalization of products and services have on the total logistics chain? Business innovations are crucial in a competitive landscape subject to disruptive changes.

•

Strategy in the Next Millennium Given the dramatic changes taking place in the competitive landscape, I believe that both the concept of strategy and the process of strategy making will change. Older approaches will not suffice. Managers will have to start with two clear premises. First, they can influence the competitive environment. Strategy is not about positioning the company in a given industry space but increasingly one of influencing, shaping and creating it. What managers do matters in how industries evolve. This is not just about the reactions of large, well-endowed companies. Smaller companies can also have an impact on industry evolution . . .. Second, it is not possible to influence the evolving industry environment if one does not start with a point of view about how the world can be, not how to improve what is available but how radically to alter it. Imagining a new competitive space

Strategic Management and Competitive Advantage

Box 2.1 (continued)

and acting to influence the migration toward that future is critical. Strategy is therefore not an extrapolation of the current situation but an exercise in “imagining and then folding the future in.” This process needs a different starting point. This is about providing a strategic direction—a point of view—and identifying, at best, the major milestones on the way. There is no attempt to be precise on product plans, or budgets. Knowing the broad contours of the future is not as difficult as people normally assume. For example, we know with great uncertainty the demographic composition of every country. We can recognize the trends—the desire for mobility, access to information, the spread of the web, and the increasing dependence of all countries on global trade. The problem is not information about the future but insights about how these trends will transform industries and what new opportunities will emerge. While a broad strategic direction (or strategic intent and strategic architecture) is critical to the process, it is equally important to recognize that dramatic changes in the environment suggest managers must act and be tactical about navigating their way around new obstacles and unforeseen circumstances. Tactical changes are difficult if there is no overarching point of view. The need constantly to adjust resource configuration as competitive conditions change is becoming recognized. A critical part of being strategic is the ability quickly to adjust and adapt within a given strategic direction. This may be described as “inventing new games within a sandbox,” the sandbox being the broad strategic direction. The most dramatic change in the process of strategy making is the breakdown in the traditional strategy hierarchy—top managers develop strategy and middle managers implement it. By its very nature discontinuous change in the competitive environment is creating a whole new dynamic. People who are close to the new technologies, competitors, and customers appear as managers in the middle. They have the information, urgency, and motivation to act. They are also the ones who have direct control over people and physical resources. Top managers, in an era of discontinuous change, are rather removed from the new and emerging competitive reality. . . . Middle managers must take more responsibility for developing a strategic direction and, more important, in making decentralized decisions consistent with the broad direction of the company. The involvement of middle managers is a critical element of the strategy process. Finally, creating the future is a task that involves more than the traditional stand-alone company. Managers have to make alliances and collaborate with suppliers, partners, and often competitors to develop new standards, infrastructure, or new operating systems. Alliances and networks are an integral part of the total process. This requirement is so well understood that it is hardly worth elaborating here. Resources available

55

56

Chapter 2

Box 2.1 (continued)

to the company are dramatically enhanced through alliances and networks. The New View of Strategy The emerging view of strategy contrasts dramatically with the traditional view . . .. The shift in emphasis in the concept of strategy and the process of strategy making is dramatic. It is clear the disruptive forces that have wrought this change are accelerating. It is time for managers to abandon the comfort of the traditional, and tried and tested, tools and concepts and embrace the new. Disruptive competitive changes will challenge the status quo. Those who take up the challenge and proactively change will create the future. The markets will decide the drivers, passengers, and the rate of “roadkill” soon enough. Reprinted from Financial Times. Publication date: October 4, 1999.

Industry and market attractiveness

Revenue position compared to rivals

Cost position compared to rivals

Firm profitability Firm’s relative competitive positioning in the industry

Figure 2.2 Drivers of firm profitability From Besanko et al. (2004, p. 360)

Figure 2.2 summarizes the drivers of firm profitability according to Besanko et al. (2004). The first driver determines whether a firm operates in an attractive market or industry, as can be ascertained via detailed industry analysis. Profitability does not only vary across industries but also across firms within a given industry, requiring a closer look at the revenue or cost position of the firm relative to its competitors. Analyzing the firm’s environment involves taking a close look at a range of factors, some of which may be beyond the firm’s control (e.g., macroeconomic factors) as well as at industry-specific factors. We examine below how to analyze the macroeconomic environment and subsequently examine industry-specific factors.

Strategic Management and Competitive Advantage

2.2.1

57

Macroeconomic Analysis

The firm’s macroeconomic environment can have a critical impact on industry attractiveness. One way suggested to analyze the external environment is the PEST framework (standing for politics, economics, social culture, and technology). Politics (e.g., environmental legislation, antitrust rulings, tax policies, employment laws, trade restrictions, tariffs) may influence the profitability and viability of the overall industry, as well as specific firms within it. For multinational corporations (MNCs), assessing the impact of such political factors can be involved as MNCs have to constantly monitor governmental stability, monitor import and export regulations, and be abreast of local business legislations.

•

Economics can have a serious impact on the profitability of an industry or firm. The unemployment level, the relative value of the domestic versus foreign currencies, inflation, growth, and productivity can influence the health of the economy, by way of consumer demand or confidence. These economic factors determine the growth potential. A poor economic situation, such as the recent financial crisis, may lead to reduced spending by consumers. A firm must carefully monitor these factors and devise appropriate strategies to ride out an unfavorable economic environment or take advantage of opportunities arising from an expanding economy. Multinationality, in particular, enables a firm to shift operations among several countries, taking advantage of opportunities or reducing risks as relative labor costs, exchange rates, tax policies, and the like, change. •

Social-cultural factors, such as the proportion of women in the workforce, health and fitness trends, or the growth in discretionary spending by the young, can also have a significant impact. A firm should monitor these trends to ensure that its products and services meet changing consumer needs. Moreover, as the firm expands into new markets or countries, differences in purchasing behavior due to social and cultural idiosyncrasies must be taken into account. •

• Technological changes have been unprecedented in recent decades in terms of incremental improvements and technological disruptions. Today one can travel faster, communicate instantly globally, and produce in more efficient ways than ever before. New industries have emerged (e.g., cellular phones and dot-coms), while others have entirely disappeared (e.g., typewriters or argentic photo cameras). Businesses

58

Chapter 2

that thrived in the past have often taken advantage of emerging technologies in other industries to serve their customers better or cheaper. 2.2.2

Industry Analysis: Structure–Conduct–Performance Paradigm

A thorough industry and microeconomic analysis must complement the analysis of the macroeconomic environment. Two countries with similar macroeconomic environments may be quite different with respect to their microeconomic profiles. For example, France and Germany are roughly similar in terms of macroeconomic features, but many industries are organized quite differently in the two countries (e.g., the banking system or the waste management industry). Microeconomic analysis enables a better understanding of the overall attractiveness of a particular market and the achievable profitability of the typical firm within that industry. A well-known industry analysis framework is the “structure–conduct– performance” (SCP) paradigm.2 At its core this paradigm assumes a causal link from market structure to firm conduct and performance: market structure determines firm conduct, which in turn determines industry and firm performance. This also works in the reverse direction, since firms may pursue strategies that can alter the market structure (e.g., M&A activities). Government intervention and basic demand and supply conditions may also influence the components discussed above. Market Structure In perfect competition, many firms (facing no entry or exit barriers) populate the market. Moreover clients have a perfect flow of information. Firms offer homogeneous products and behave as price takers. In the long run a competitive industry will supply a good at a price that reflects the marginal cost of the resources required to manufacture that product. On the other extreme, in an industry characterized by a monopoly structure, there are high entry and exit barriers (e.g., large fixed costs), and the customer base is relatively large and homogeneous. A monopolist can earn excess profits stemming from market power as reflected in its ability to set a price higher than the marginal cost of production. Market structure, as proxied by the number and market power of firms in the industry, is an important driver of industry 2. The SCP paradigm was developed by the “Harvard School,” including Joe Bain and Edward Mason, based on empirical observations.

Strategic Management and Competitive Advantage

59

profitability. The fewer the number of producers in the market, the greater their ability to wield market power and earn excess profits. The structure and bargaining power of suppliers and customers may also influence whether a firm exercises market power. This “extended rivalry” can put significant pressure on the firm. Firm Conduct in the Industry Conduct refers to whether a firm actually exercises its market power to earn excess profits, whereas market structure refers to whether market power is structurally possible in the industry. In addition to individual exercise of market power, firms may collude (explicitly or tacitly) in an oligopolistic industry to sustain a higher price and earn excess profits.3 The ability to exercise market power is influenced by strategic investments, such as engaging in advertisement campaigns or investing heavily in research and development, preventing potential rivals from market entry. Industry Performance The performance of an industry is based on its profitability as well as its static and dynamic efficiency. Excess profits earned by incumbent firms imply that the given industry is not perfectly competitive as some firms wield market power. A secure incumbent may have little incentive to improve its production processes.4 Although a consolidated industry can attain higher static efficiency resulting from larger production scale, the technology employed may not be the most efficient, with firms investing less in R&D compared to a highly competitive industry. In this sense there may be a “cost” to society from consolidated industries since they tend to be less dynamically efficient than a highly competitive industry. This issue has raised a big debate among economists with no clear consensus emerging on whether monopoly and the existence of patents is beneficial or harmful in terms of encouraging innovation.5 2.2.3

Porter’s Industry and Competitive (Five-Forces) Analysis

The industry analysis above provides a benchmark for the profitability of the “average” firm in an industry. A complementary approach to the industry analysis above is Michael Porter’s (1980) “five-forces” 3. Usually a cartel attempts to set a price equal to what a monopolist would charge as cartel members are attempting to maximize joint profits. The stability of a cartel is another matter. 4. A widely known remark on monopolistic industries is that of Hicks (1935, p. 8): “the best of all monopoly profits is a quiet life.” 5. See Tirole (1988, ch. 10) for a comprehensive economic analysis of the incentives to conduct R&D in oligopolistic industry structures.

60

Chapter 2

Potential entrants Threat of new entrants

Competitors Bargaining power of customers

Bargaining power of suppliers

Suppliers

Customers Rivalry among existing firms Threat of substitute products or services

Substitutes

Figure 2.3 Porter’s “five-forces” industry and competitive analysis From Porter (1980)

framework. Since the early 1980s, it has become a well-known paradigm used both in academe and in practice for analyzing industry performance. According to Porter (1980), the attractiveness of an industry depends on two kinds of strategic interactions: Interactions along the “value chain” These relationships, depicted along the horizontal axis in figure 2.3, involve the relative bargaining power of different parties along the value chain, from suppliers to manufacturers and distributors, to end customers. Extended rivalry This includes the internal rivalry within a particular industry (nature and degree of intra-industry competition) as well as the threat posed to the firm from external forces, such as potential entrants and the threat of substitute products or services. These “extended-rivalry” interactions are illustrated along the vertical axis in the figure. Porter categorizes the relevant economic factors into five groups or forces (hence, the “five-forces” analysis). He proposes a thorough assessment of (1) the impact of these factors on the intensity of competition within the industry or internal rivalry, (2) the threat of substitute products or services, (3) the threat of new firms entering the market, (4)

Strategic Management and Competitive Advantage

61

the bargaining power of suppliers, and (5) customers. These forces are in turn affected by industry structure and other relevant variables, like entry and exit barriers, product differentiation and availability of information.6 We look at these forces in detail next. Internal Rivalry Internal rivalry is a central piece of Porter’s analysis. It refers to competitive actions taken by each firm to gain market share within the industry. A means to assess the intensity of competition within an industry is to use a concentration ratio. Industrial economists and antitrust authorities have long attempted to find a quantitative measure of the intensity of rivalry using various concentration ratios. A common feature is that they are based on the market shares of the firms already active in the industry.7 Two concentration indexes are most commonly used to measure internal rivalry within an industry: (1) the cumulative market share of the k biggest firms, Ck, and (2) the Herfindhal–Hirschman index, developed by Herfindhal (1950) and Hirschman (1945, 1964), commonly referred to as HHI.8 Cumulative Market Share of the k Biggest Firms (Ck ) Let si denote the market share of firm i in an industry consisting of n active (incumbent) firms. A simple proxy of the concentration in a industry with n active firms is the cumulative market share of the k biggest firms (k ≤ n), that is, k

Ck ≡ ∑ si .

(2.1)

i =1

The market shares of the few “big” players are generally available in market reports. This concentration ratio has a major drawback: it fails to capture heterogeneity among the largest firms. For this reason industrial economists and antitrust authorities generally prefer alternative concentration measures, such as the Herfindhal–Hirschman index. This is discussed in example 2.1 below. 6. Although conceptually very useful, Porter’s (1980) framework does not give a clear prescription as to whether to invest in a given industry. Its primary benefit is that it helps frame an industry and identify key forces affecting market attractiveness. 7. These concentration ratios fail to capture whether external parties can easily enter these markets. 8. Tirole (1988) provides an alternative to these concentration indexes, namely the entropy index that involves a logarithmic function of the market shares. Encoua and Jacquemin (1980) describe desirable properties a concentration index should satisfy and assert that the HHI and entropy indexes are better suited to analyze industry concentration. Here we only discuss the Ck and HHI indexes due to their relative simplicity.

62

Chapter 2

Herfindhal–Hirschman Index (HHI) The HHI index has the merit of combining information about all firms in the market, not just the five (or k ) biggest firms. The Herfindhal– Hirschman index is given by n

HHI ≡ ∑ si ².

(2.2)

i =1

From a practical viewpoint, it is more difficult to gather information (market shares) on all firms operating in a given industry. Example 2.1 Concentration Indexes Consider the market share distribution of five firms in the three different industry situations described in table 2.1. The last two columns give the values of C 4 and HHI in each case. The industry in case 3 is clearly more concentrated than in case 2, which in turn is more concentrated than in case 1, as the market shares are less uniformly distributed. Using the C4 index, one cannot distinguish the difference in concentration within the industry in case 3 or 2, as the index is equal to 88 percent in both cases. The C4 index does not capture the fact that in case 3 a single firm is dominant with a 50 percent market share, compared to case 2 where the three biggest firms have comparable power. The HHI does a better job since it properly captures this difference, giving a higher value in case 3 relative to case 2. As summarized in figure 2.4, concentration indexes are used by economists and antitrust authorities to identify and differentiate among different market structures. A monopoly is generally characterized by only one seller and very high barriers to entry and exit. An example, from the diamond business, is DeBeers, which has exclusive access to scarce Table 2.1 Concentration indexes based on the first three (C3 ), four (C4 ), or all firms in an industry Case

Market share distribution

1 2 3

s1 1 /5 ¼ ½

s2 /5 ¼ ⅛

1

s3 1 /5 ¼ ⅛

Concentration index s4 1 /5 ⅛ ⅛

s5 1 /5 ⅛ ⅛

C4 80% 88% 88%

HHI 2,000 2,188 3,125

Note: By convention, the HHI is often given as 10,000 times the HHI number given by the definition in equation (2.2).

Strategic Management and Competitive Advantage

Ck(%) HHI

63

Perfect competition

Monopolistic competition

Oligopoly

Dominant firm

Monopoly

0

0 < C4 < 40

40 ≤ C4 < 60

60 ≤ C1 < 90

C1 ≥ 90

HHI < 1,000

1,000 ≤ HHI < 1,800

HHI ≥ 1,800

Figure 2.4 Classification of market structures based on different concentration index ranges

resources giving it a (quasi-) monopoly position due to high barriers to market entry. A monopoly is characterized by C1 ≥ 90 percent. In an industry with a dominant firm and some fringe competitors, the dominant firm can typically ignore strategic interactions with the smaller rivals. Barriers to entry and exit are fairly high. Apple and its iPod® have a dominant position in the portable music industry: despite smaller competitors, Apple has built an uncontestable entry barrier thanks to its unique brand name. Firm dominance is characterized by 60% ≤ C1 < 90% (and HHI ≥ 1, 800). An oligopoly is characterized by few competing firms with 40% ≤ C 4 < 60%. A classic case is the duopoly involving Airbus and Boeing in the airframe industry. Barriers to entry due to technological expertise or huge sunk capital costs are generally very high. National quality standards may also raise entry barriers for foreign competitors and explain oligopoly situations.9 Monopolistic/differentiated competition is characterized by many suppliers with heterogeneous products appealing to different customer or market segments. A good example is the luxury goods industry where distinct brands lure different customer groups (youth, fashion followers, etc.). Product differentiation makes it possible for firms to price products above the marginal cost even if many firms compete in the marketplace.10 Differentiated competition is characterized by 0 ≤ C4 < 40 percent. Perfect competition involves a homogeneous product (commodity) with no single firm having significant market power to influence the price-setting process. The agricultural sector, for example corn production, is considered a good approximation to a perfectly competitive industry. Typically HHI is below 1, 000. 9. A case in point is the French pipeline and valve industry. In France, valves for public water management pipes turn counterclockwise, whereas in most other European countries valves turn clockwise. For this reason only manufacturers that meet French standards are active in France, mostly French companies. 10. In monopolistic competition free entry of new brands leads to zero economic profit with gains from pricing over marginal costs exactly offsetting fixed cost.

64

Chapter 2

One explanation for the existence of imperfect competition lies with the large fixed costs incurred by incumbent firms: if the market is not sufficiently large, only a limited number of firms can afford incurring these high fixed costs. Even when the market-clearing price exceeds marginal production costs, no external party enters since it cannot recoup its large fixed costs. Threat of Substitutes In assessing the threat of substitute products, a key prerequisite is to determine what the relevant product market is and define the relevant substitute products. Are train and plane substitutes in the transportation market? To travel from France to Cyprus, the train is certainly not an option. However, from Paris to London the train is a valuable option thanks to the Eurotunnel.11 Many economists and antitrust authorities define the relevant market as the smallest set of products for which a firm, should it become dominant, would find it profitable to raise prices significantly (and permanently).12 Once the relevant market is defined, the threat of close substitutes can be assessed. The extent to which a company might be affected by substitute products depends on several factors, including the propensity of buyers to substitute, switching costs, the value added by the company’s product or service in the clients’ perception, as well as the price-performance characteristics of substitute offerings. Potential Entrants Another key force affecting firm performance is the threat of entry. New entrants pose a threat to incumbent firms as higher rivalry generally leads to lower profit margins (negative externality). How easy or difficult it is for a potential entrant to penetrate a market depends on entry barriers.13 Entry barriers may be structural (exogenous) or strategic (endogenous). An example of structural entry barriers is when the incumbent is protected by a favorable government policy or other administrative barriers (e.g., property rights,14 state, market or environmental regulation). 11. This travel option has become more attractive especially after September 11 since it reduces substantially the time to check in and out. 12. To determine the relevant product market, economists utilize various measurement tools, such as cross-price elasticity of demand, price correlation analysis, and shock analysis. 13. Bain (1956) has extensively analyzed entry barriers. A barrier to entry is a mechanism that allows incumbents to make economic profits without threat of entry by competition. 14. Examples include patents, copyrights, and licenses. The property right problem for a social planner hinges on the trade-off between erecting entry barriers (which is not socially optimal) and giving firms incentives to invest in R&D.

Strategic Management and Competitive Advantage

65

Structural barriers may also stem from other economic phenomena, such as fixed entry costs, economies of scale, scope, or learning, access to scarce resources,15 reputation, network effects,16 product differentiation advantages, process innovation, and vertical integration. Strategic entry barriers involve actions taken by incumbents to deter the entrance of new competitors into the market, for example, by manipulating prices before (limit pricing) or after entrance (predatory pricing), by building excess capacity, or by launching aggressive advertising campaigns to create brand loyalty. They may also create a new brand to capture a previously unsatisfied customer segment (product proliferation). We elaborate on corporate strategies to deter entry later in chapter 4. Bain (1956) identifies three kinds of behavior by incumbents in the face of an entry threat. Entry is blockaded if the incumbent is well protected by insurmountable structural barriers to entry and exit so that it continues to enjoy incumbency profits. The incumbent can ignore the threat of entry so that strategic interactions play a minor role in this case. Entry is deterred if the incumbent is not exogenously protected and behaves strategically to make it unprofitable for new competitors to enter. For instance, by massively investing in automated production processes and reducing its production costs (a strategic first-stage investment), a firm may ensure that the rival’s post-entry profits are driven close to zero. This kind of deterrence strategy may be extremely expensive to pursue, but nonetheless strategically justified. Entry is accommodated when the incumbent finds it preferable to allow entry than to erect costly barriers. A strategy of entry accommodation is not tantamount to a passive stance toward one’s rivals: the incumbent can still build up an early competitive advantage (e.g., via excess capacity or strong brand image). Bargaining Power of Suppliers and Customers Both suppliers and customers can extract profits from manufacturers depending on who influences the price-setting process when the manufacturer buys or sells. The firm’s ability to influence or set prices depends on a number of factors: The concentration and market power of customers and suppliers. If the customer base is small (e.g., B-to-B clients) but suppliers are numerous, •

15. A case in point is Nutella. The firm has an exclusive right to acquire a given species of nuts, so no entrant can imitate Nutella’s unique taste. 16. The combination “PC + MS DOS” won the computer war against “Apple Mac OS” partly because IBM had a less restrictive licensing policy than Apple.

66

Chapter 2

customers will wield more bargaining power. L’Oréal, one of the major cosmetic companies worldwide, puts significant pressure on its packaging services suppliers, which have to toe the line. •

The price elasticity of demand for the product (inputs or outputs).

•

Customer and supplier switching costs (purchase from another party).

The ability of customers to backward integrate or of suppliers to forward integrate. •

The relation specificity of the firms’ assets (specific to clients or suppliers). •

2.3

Creating and Sustaining Competitive Advantage

The profitability of a firm depends both on industrywide characteristics (e.g., competitive pressures, market power, bargaining power) as well as on firm-specific attributes. We already discussed the main strategic management frameworks for assessing industry attractiveness. We next focus on firm-specific factors that might help explain why certain firms in a given industry are more successful than the “average” firm considered in industry analysis. We review the “generic strategies” proposed by Michael Porter (1980), namely the incentive to be a cost leader, a differentiation leader, or focused on a specific customer segment. To understand these concepts from a microeconomic viewpoint, it is necessary to first address the notion of value creation and competitive advantage.17 2.3.1 Value Creation To earn profits in excess of the industry average, a firm must strive to be superior to its competitors in the industry. A key to superior performance is creating more value for customers than one’s rivals by exploiting some competitive advantage. Firms’ strategies aim to create such comparative advantages. To assess the value created by a firm, we need to go back to basic microeconomics. For each product there is a maximum value the consumer is willing to pay. This maximal value (“willingness to pay” or “consumer value”) determines whether the customer purchases the product. If the price asked for the good exceeds the customer’s willingness to pay, the customer will reject the product. If the price is lower, the customer will purchase it, “earning” the difference (between the 17. For a more detailed analysis of how firms can create competitive advantage, refer to Besanko et al. (2004, ch. 11) and Grant (2005, chs. 8 and 9).

Strategic Management and Competitive Advantage

67

customer value and the product price) as consumer surplus or utility. If p denotes the price and u the consumer’s willingness to pay (utility), the consumer surplus is represented by u − p if p ≤ u, and 0 otherwise. If the product is homogeneous, meaning the products offered by rivals are considered perfectly identical, a rational consumer will purchase from the lowest price supplier. For instance, if a consumer sees no significant difference among airline carriers and does not care much about small differences in the time schedules, she would choose to fly with a low-cost airline like Ryanair. We must also consider the incentive of a given firm to sell in the marketplace. This incentive is captured by the producer surplus or economic profit, and represents the bottom-line criterion for many managerial decisions. The value created by the producer is the difference between the price at which the firm sells the good, p, and its unit cost of producing it, c.18 The total economic value created through such an exchange in the marketplace is the difference between the utility the product brings to the consumer and the amount it costs to be produced, namely u − c. A portion of this value is captured by the consumer in the form of consumer surplus, u − p, and the remaining part is earned by the producer as economic profit, p − c. That is, Total value created = consumer surplus + producer surplus u−c = + (u − p) ( p − c)

(2.3)

The relation above is depicted in figure 2.5. A necessary condition for market exchange to take place is that the value (utility) the product represents for the consumer (u) exceeds the cost of producing it (c). In this case both the consumer and the producer gain. In equilibrium the producer does not set the price arbitrarily; this choice is closely related to the consumer price elasticity of demand, the production cost, the nature and degree of competition, as well as the size of the customer base. In a perfectly competitive market all economic value created by the product offering is captured by the consumer and competing firms make no economic profit. In order to be economically profitable, firms need to find ways to deviate from perfect competition. A firm has a competitive advantage when it is able to deliver more economic value than its rivals. 18. Included in c are the labor costs, input prices, and cost of capital per unit of the product. The profit and value measures we discuss here are net of the opportunity cost of capital. The term used for profits in this case is “economic profit.” A financial translation of this economic concept is economic value added (EVA).

68

Chapter 2

Consumer value u Consumer surplus (u−p) Total economic value u− c

Price p Producer surplus (p− c) Cost c

Figure 2.5 Total value created consisting of consumer surplus plus producer surplus

In reality, if consumers are not identical, a given product may represent a higher utility for one consumer than for another. Distinct market segments can be identified in which consumers (with uniform preferences) are more willing to accept one particular offering versus another, namely the one for which their consumer surplus (u − p) is the highest. This basic economic principle underlies market segmentation. To create higher economic value, firms pay considerable attention to what drives customer value and production costs. The following factors typically drive the customer’s willingness to pay: The appeal of physical characteristics and the perceived fitness of the product (e.g., the size of furniture may be an important criterion if one’s flat is only 20 square meters or 200 square feet). •

The quality of services associated with the product (the price premium one may be willing to pay for a night at Sofitel in NYC rather than the Best Western Hotel is linked to the perceived services one receives rather than the quality of the bed). •

Brand name (Apple®’s brand name creates a unique competitive advantage for the iPhone® in the now-commoditized mobile telephony market). •

•

Perception by customers.

Strategic Management and Competitive Advantage

69

Firm infrastructure (e.g., finance, accounting, legal) Human resource management

Support activities

Technology development Procurement

Primary activities

Inbound logistics

Production

Outbound logistics

Marketing and sales

Services

Main activities

Figure 2.6 Porter’s value chain From Porter (1980)

The following factors typically drive production costs: Drivers related to the size of the firm (economies of scale and scope, capacity utilization and bargaining power). •

•

Experience of the firm (learning-curve effects).

Factors related to organizational structure (organization of transactions, agency efficiency, vertical chain). •

•

Other costs (lower input prices, production in a low labor cost country).19

To understand how value is created along the vertical chain, Porter (1980) proposes a conceptual framework (shown in figure 2.6) involving the organization of value-creating activities referred to as the value chain. Porter’s value chain essentially divides the organization into a series of value-creating activities that can be broadly classified as supportive or primary. Regardless of whether an activity belongs in the first or the second category, it can potentially contribute to creating more economic value either by increasing the value perceived by the consumer or by reducing the cost of producing it. Primary activities are involved with the physical creation of the product or development of the service until it reaches the end customer. Inbound logistics, production or operations, outbound logistics, marketing and sales, and services are typical primary activities. Additional activities support these primary activities by providing procurement and technology services, human resources management, and various firmwide functions such as finance, accounting, and 19. For more details on benefit and cost drivers, see Besanko et al. (2004, pp. 411–18).

70

Chapter 2

legal services. Increased value added may also be a result of these firmwide functions. The organization as a whole is embedded in a larger stream of activities that include suppliers, distributors, and buyers’ value chains, which together with the firm’s value chain form the so-called value system. These activities are all interrelated, so events in one of them may affect the company’s overall strategy basis. The notions of value creation and value chain are essential for understanding the “generic competitive strategies” proposed by Porter (1980). 2.3.2

Generic Competitive Strategies

A generic competitive strategy refers to how a firm positions itself to compete in the marketplace. Porter (1980) identifies three generic strategies—cost leadership, differentiation, and focus strategy. The choice of one strategy over another partly depends on the size of the market the firm wants to cover. When covering a broad market, firms can mainly pursue two generic strategies: cost leadership or differentiation. In homogeneous goods markets, cost leadership can play a central role since customers primarily decide based on the price. In a market where customers prefer customized products, a benefit leader or differentiated firm that offers a perceived superior product may obtain a stronger competitive advantage. A third strategic alternative is to specialize in a narrow customer segment, namely develop a niche, following a focus strategy. Cost Leadership In homogeneous goods markets, cost leadership consists in offering similar products as rivals but at lower price. This strategy typically involves producing larger quantities and appealing to a broader market (broad scope). It is more likely to succeed when customers are sensitive to prices, as measured by the price elasticity of demand. A costleadership strategy is advisable when products are commodity-like or customer services are hard to differentiate. Cost advantage can result from economies of scale, scope, or learning-curve effects. Economies of scale offer unit-cost savings that increase with the level of output: the average production cost of a single product decreases with the number of units produced. Fixed costs here play a critical role: when volume rises, the fixed-cost component per unit declines. Such savings may have a material impact on the equilibrium market structure. If fixed costs are large, only a limited number of firms can profitably operate in the market: the price markup over the marginal cost needs to be sufficiently large to justify spending the fixed costs by the incumbents. Economies of scope

Strategic Management and Competitive Advantage

71

correspond to cost savings gained by operating several product lines or businesses: the production of good A reduces the production cost of product B. Learning-curve effects are related to the experience and know-how firms cumulate over time when they produce a given product or provide a certain service. The classic BCG matrix originated from work related to the learning-curve effect.20 Cost leadership can also result from improved process efficiencies, increased bargaining power with suppliers, higher capacity utilization (especially when fixed costs are high), lean organizational structure, better compensation systems, or location advantages (manufacturing in low-cost countries). Firms employ several approaches to achieve cost leadership: cost leadership under benefit parity, cost leadership under benefit proximity, and product redesign. In case of benefit parity, the cost leader offers a product identical to the products offered by competitors but the cost of producing it is significantly lower. In case of benefit proximity, the cost of producing the good is lower but the consumer perceives the product to be of lower quality; here the increase in consumer surplus due to the lower price offsets the quality decrease. Alternatively, a firm may offer a product to consumers that is qualitatively different from competitors’ offerings (product redesign). Whether achieving cost leadership enhances firm value also depends on the size of the initial investment. Differentiation Strategy Consumer preferences in the marketplace may not be homogeneous. Within the same market some customers may have preferences for specific product features and others for other features. A differentiation strategy enables a firm to create more economic value than its rivals by offering higher consumer value than other products supplied in the marketplace. In exchange for this higher consumer surplus, a consumer may be willing to pay a price premium. A differentiation strategy may encompass the whole value chain. This strategy is more appealing when price elasticity is low or when cost advantages are limited. Traditionally one distinguishes two kinds of differentiation. In case of horizontal differentiation, product differentiation can be achieved thanks to new combinations of product characteristics (e.g., MP3-player for mobile telephone handsets), new distribution or sales outlets (e.g., online shopping for people living on the fast lane), better marketing differentiation (lenovo® laptop, mainly focused on corporate customers), 20. The BCG matrix paradigm asserts that a conglomerate should finance promising “stars” with excess cash generated by “cash cows.” By so doing, the “stars” commence production early on and accumulate experience ahead of competitors, leveraging on learning-curve effects.

72

Chapter 2

or better complementarities owing to network effects (e.g., Blu-ray players on Sony’s PlayStation3®).21 Vertical differentiation is achieved when all perceive the product as being of higher quality. Customers without budget constraints would be willing to pay a premium for it. Again, three cases may be distinguished here. In case of cost parity, the cost of producing the good remains unchanged, but the differentiated product is perceived as being of higher quality by a given customer segment. In case of cost proximity, the differentiated product implies higher production costs (than the standard product), but consumers appreciate the higher quality and willingly pay a price premium that compensates for these higher costs. Product redesign consists in offering a product with new characteristics that appeal to new or existing customers. Focus Strategy Alternatively, a firm can devise a focus strategy and tailor its products and services to the requirements of a specific customer group. In doing so, the firm creates higher economic value for this specific segment. One type of focused strategy is to be a local player or appeal to a given population group (e.g., Mecca Cola® as an alternative to CocaCola® and Pepsi® for Muslim consumers). A key advantage of a focused strategy is that the targeted market may not be large enough to accommodate many producers, enabling the focused firm to enjoy a quasi-monopolistic position within its segment. 2.3.3

Sustaining Competitive Advantage

It is said that history repeats itself in cycles. Many once-leading firms have been dethroned abruptly for not being able to protect and sustain their once-formidable competitive advantage. Imitators and new entrants with superior technologies can quickly erode competitive advantages that have taken years to build. Firms that persistently outperform their peers are those able to sustain or renew their competitive advantage over the long run. Sustainability of competitive advantage invariably depends on the dynamics of the market. In a market with low entry and exit barriers, it is more likely that a firm will enter when it sees a window of opportunity. To avoid this, an incumbent can erect entry barriers. Just as there are various mechanisms to protect the profitability of an industry (e.g., erection of entry and exit barriers), there exist mechanisms within an industry to protect a firm’s competitive advantage from rivals’ 21. See Besanko et al. (2004, p. 415).

Strategic Management and Competitive Advantage

73

imitation. A firm’s ability to create and sustain a competitive advantage depends on its firm-specific resources and the distinctive capabilities arising from these resources. To keep competitors from duplicating its competitive advantage, the firm should constantly strive to create or enhance asymmetries by acquiring and managing distinctive resources and capabilities that cannot be readily duplicated by would-be imitators. Heterogeneity among firms is the cornerstone of the resource-based view of the firm, emphasizing firm specificity and uniqueness. Imitation can only be precluded if there is heterogeneity and imperfect transferability of key resources. Besides the scarcity and immobility of unique resources, firms can erect isolating mechanisms to prevent rivals from internally developing similar resources. Isolating mechanisms fall into two main categories: (1) barriers to imitation (e.g., property rights or exclusive access to scarce resources) and (2) early-mover advantages, such as learning-curve effects and brand-name building. When a firm cannot acquire distinctive key resources, it has to build them up over time. In its history of investment and commitment, a firm accumulates skills, assets, and resources involving path dependencies. The experience, relationships, and reputation a firm has built over time enable it to leverage its resources and capabilities and represent sources of uniqueness. It is the exploitation of firm-specific resources and capabilities in a dynamic way that enables a firm to enjoy higher (excess) profit flows. Apple®, frequently ranked as one of the most famous brands in the world, has successfully leveraged its brand image to sustain its profitability over time. Conclusion In this chapter we provided a brief overview of the main paradigms and issues in strategic management, focusing on how prevalent frameworks view the key drivers of market attractiveness and firm profitability. We elaborated on market structure and discussed how limited access to a market and the existence of entry barriers can enhance a firm’s profitability. We described Porter’s five-forces paradigm and how it can help structure market analysis. We also discussed how, within a given industry, a firm can build competitive advantage through production efficiency (cost leadership), branding (differentiation leadership), or a focus strategy. Finally, we discussed the importance of sustaining competitive advantage dynamically.

74

Chapter 2

Selected References Bain (1956) examines barriers to market entrance in detail. Porter’s (1980) seminal work on industry and competitive analysis is a cornerstone for understanding the art of strategy and competitive advantage. Besanko et al. (2004) provide an easy-to-read economic perspective on strategy. Bain, Joe S. 1956. Barriers to New Competition. Cambridge: Harvard University Press. Besanko, David, David Dranove, Mark Shanley, and Scott Schaefer. 2004. Economics of Strategy, 3rd ed. New York: Wiley. Porter, Michael E. 1980. Competitive Strategy. London: Macmillan.

3

Market Structure Games: Static Approaches

Studying industrial organization is useful to deduce managerial insights to help explain how firms should behave strategically when faced with competition. In this chapter and the following one, we review some basic principles and models in this area, discuss industry structures, and examine information asymmetry, commitment, and collaboration. Basic ideas developed in these chapters serve as building blocks in subsequent discussions. In this chapter we deal mainly with static models that help explain the modus vivendi in competitive situations when firms focus on the short-run impact of their decisions. We discuss simple economic models that characterize optimal firm behavior under different industry structures. Chapter 4 extends the discussion to include long-term, dynamic strategy formulation, helping explain phenomena such as restraining one’s own freedom (commitment) or collaboration among rival firms. This chapter is organized as follows. Section 3.1 discusses the benchmark case of a monopolist, developing the basic building blocks for subsequent models. We then turn to oligopolistic situations where firms have enough market power to influence the decisions of rivals. Such analysis requires the use of game-theoretic tools. Section 3.2 introduces static models of duopoly, presenting first the case of Bertrand price competition and then discussing the classic Cournot quantity duopoly model. Section 3.3 extends the standard Cournot quantity competition analysis to accommodate a larger number of firms in oligopoly, obtaining the market equilibrium in perfect competition as a polar case. Going beyond earlier models that involved complete information, section 3.4 explores the impact of incomplete information on market structure in a duopoly.

76

3.1

Chapter 3

Monopoly

In monopoly entry barriers are typically sufficiently high to shield the incumbent firm from competitive entry (blockaded entry). Customers take price as given and cannot influence the price-setting process. Monopoly may naturally arise when demand is not sufficient or fixed costs are too high to accommodate other incumbents (natural monopoly) or when regulation prohibits further market entry (e.g., patents).1 In western Europe the electricity market used to be considered a national natural monopoly because of the prohibitive fixed costs. Price setting in a competitive market leads to lower prices, making it difficult or impossible for more electricity utilities to make a profit once large fixed costs are considered. The equilibrium market structure critically depends on such fixed costs. Economies of scale and scope are thus one of the main drivers of industry structures (Viner 1932). When large-scale production plants are needed (e.g., nuclear power plants), a monopoly is economically justified and may be socially preferable.2 In this chapter we simplify and ignore fixed production costs, focusing instead on the relationship between the market-clearing price and marginal production costs.3 In perfect competition a firm takes the market price as given and produces until the marginal cost of an additional unit is just below the market price. In contrast, a monopolist is a price setter. Consider a monopolist facing (inverse) demand function p (⋅) and variable production cost C (Q), as a function of the quantity Q supplied in the marketplace. Given a downward-sloping demand (∂p ∂Q < 0), the monopolist faces a trade-off between price and quantity. Setting a higher price p implies earning a higher profit margin but selling a lower quantity. Inversely, if the monopolist sells an additional unit of output, it moves down the (inverse) market demand curve, suffering a (marginal) price reduction. This trade-off is linked to marginal revenues. The monopolist’s profit function is concave in its own quantity, Q,

π (Q) = R (Q) − C (Q), 1. A natural monopoly may exist when a firm can make excess economic profits when operating alone, while a duopolist cannot. If π (i) indicates the equilibrium profit of a firm in an industry with i incumbent(s), π (2) < 0 < π (1) holds for natural monopoly. 2. Today, following deregulation, the electricity market is sufficiently mature to accommodate more suppliers. 3. Under certain conditions we show that firms can make excess economic profits (gross of fixed costs). Considering fixed costs might invalidate some of these results.

Market Structure Games: Static

77

where total revenues are given by R (Q) = p (Q) × Q. To earn maximum profit, the firm must choose to produce that output, Q *, that equates its marginal cost, MC (Q) ≡ C ′ (Q), to its marginal revenue, MR (Q) ≡ R ′ (Q).4 This principle, MR (Q *) = MC (Q *), holds whatever the industry structure and the assumed profit function. In perfect competition, the marginal revenue equals the price (since the output decision of an individual firm does not affect the price-setting process, p = MC in equilibrium). In monopoly, since the firm’s output decision influences the market-clearing price and ∂p ∂Q < 0, the firm loses revenue on already produced output when it decides to produce more. Box 3.1 and figure 3.1 describe common types of demand functions. Consider the linear (inverse) demand function p (Q) = a − bQ

(3.1)

and a linear cost function C (Q) = cQ with c < a, b > 0. In this case the marginal revenue is MR (Q) = R ′ (Q) = a − 2bQ and the marginal cost is MC (Q) = C ′ (Q) = c. In monopoly we thus have a − 2bQM = c. The equilibrium quantity produced by the monopolist is QM =

a−c . 2b

(3.2)

The equilibrium price, corresponding to point E′ in figure 3.2 (determined by substituting equation 3.2 for QM into the inverse demand function 3.1) is pM =

a+c a−c = c+ 2 2

(> c ) .

(3.3)

From the equilibrium quantity and price, equations (3.2) and (3.3), the equilibrium profit for the monopolist is 4. The marginal benefit (revenue) is MR (Q) ≡ R′ (Q) = p (Q) +

∂p (Q) × Q. ∂Q

The profit function is concave so the first-order condition is both sufficient and necessary for a maximum output (Q *) to obtain. The first-order derivative of the profit function is ∂π ∂p (Q) = p(Q) − C ′(Q) + (Q) × Q . ∂Q ∂Q The first two terms yield the profitability of an extra unit of output (i.e., the difference between the price and marginal cost), while the third term recognizes that an increase in output (implying a decrease in ∂p ∂Q < 0 ) affects the profitability of units already produced.

78

Chapter 3

Box 3.1 Common demand functions

Industry structure models rely on certain assumptions. One key assumption relates to the (inverse) demand function used to determine marketclearing prices and firm profits. We assume that firms can observe customer demand. There are several deterministic inverse demand functions. The most widely used are the following: The linear demand function, p(Q) = a − bQ, where p (Q) is the marketclearing price, Q the total quantity produced by all firms in the industry, b > 0 , and a is the demand intercept. Figure 3.1a depicts this linear demand function. Here p(·) is downward sloping (with slope − b < 0 ), indicating that for an increased industry output ( Q) the market-clearing price ( p (Q)) declines. This property holds for most markets (with notable exception the luxury market, where a high price may signal a more exclusive product). One limitation, however, is that for the price to be (or remain) positive, total industry output must be limited to a certain range (if Q > a b, the price is negative). • The isoelastic demand function, p(Q) = 1 (Q + W ), where W is a positive constant. This curvi-linear demand is depicted in figure 3.1b. Here •

Price (p)

Price (p)

a

p(Q) =

p(Q) = a−bQ

1 Q+W

Slope − b 0

a /b

0

Quantity (Q)

Quantity (Q)

(a)

(b)

Price (p)

Price (p)

p(Q) = Q−1 ⁄ ep p(Q) = a × exp(−e p Q)

0

(c)

Quantity (Q)

0

Quantity (Q)

(d)

Figure 3.1 Different (inverse) demand functions (a) Linear demand; (b) isoelastic demand; (c) exponential demand; (d) constantelasticity demand

Market Structure Games: Static

79

Box 3.1 (continued)

whatever the total industry output Q, the market-clearing price remains positive. • The exponential demand function, p(Q) = a exp ( − ε p × Q), where ε p is the price elasticity of demand. This is illustrated in figure 3.1c. This steeply declining demand function has the property of excluding negative prices even for very high quantity levels and allowing a limited price for low output. −1 • The constant-elasticity demand function, p (Q) = Q ε p . This demand function is often used because of its relative simplicity. It is depicted in figure 3.1d. A condition on the price elasticity parameter ε p is often imposed to obtain nice mathematical properties. Since the linear (inverse) demand function is the simplest and most widely used, many industrial organization models rely on it to obtain simple expressions for equilibrium quantities, prices, and profits.

Price (p)

a Equilibrium price

pM =

E'

a+c 2

Marginal cost (c)

E

c

Demand function

p(Q) = a−bQ Marginal revenue

MR(Q) = a−2bQ −2b

0

Equilibrium quantity

QM =

a− c 2b

Figure 3.2 Equilibrium price and quantity in a monopoly

−b

Quantity (Q)

80

Chapter 3

πM =

(a − c ) ² 4b

.

(3.4)

As confirmed in figure 3.2, the monopolist firm maximizes its profit by selecting the output that makes its marginal revenue, MR (Q), just equal to its marginal cost, c, as represented by the intersection point E. Inverse Elasticity Markup Rule or Lerner Index Lerner (1934) complements the discussion on price setting by a monopolist by adopting an approach based on the price elasticity of demand, ε p. This elasticity measure corresponds to the percentage decrease in quantity sold due to a 1 percent increase in price, all other things equal. It is given by

H p (Q ) ≡ −

pQ ∂Q Q . =− ∂p p ∂p ∂Q

(3.5)

From the profit-maximizing condition, a markup formula measuring the firm profit margin in equilibrium obtains as5 L≡

pM − c 1 . = εp pM

(3.6)

This markup rule, commonly known as the Lerner index, provides a practical rule of thumb for pricing a product in a monopoly. It asserts that the profit margin received by a monopolist in equilibrium, L, is inversely proportional to the price elasticity of demand, ε p ≡ ε p(QM ). If customers are highly sensitive to a price increase (i.e., ε p is high) and would refrain from purchasing the product in case of a price increase, the firm’s profit margin L will be low, since ∂L ∂ε p < 0. In contrast, if the price elasticity of demand is low such that customers are hardly sensitive to the price increase, the monopolist can set a very high price ( pM ) and still enjoy a high profit margin (L). The price distortion is larger when customers reduce their demand only slightly in response to an increased price. Table 3.1 gives an indication of price-elasticity ranges. 5. We can rewrite the marginal revenue in note 4 based on the price-elasticity formula (3.5), obtaining MR (Q) = p +

∂p (Q) × Q = ∂Q

⎛ 1 ⎞ Q ∂p ⎛ ⎞ p⎜1 + (Q)⎟ = p ⎜ 1 − . ⎝ ⎠ ⎝ ε p (Q) ⎟⎠ p ∂Q

Expression (3.6) obtains from the first-order condition in the special case of a linear cost function. If demand is linear as in (3.1), it obtains from (3.3) that L = (a − c ) (a + c ) and ε p = (a + c ) (a − c ), giving L ≡ 1 e p.

Market Structure Games: Static

81

Table 3.1 Range of price elasticity of demand Perfectly elastic

εp → ∞

Elastic

1 < εp < ∞

Unit elastic

εp = 1

Inelastic

0 < εp < 1

Perfectly inelastic

εp = 0

Example 3.1 Market Equilibrium in Monopoly Assume a linear demand as in equation (3.1) with parameters a = 10 and b = 1; that is, p (Q) = 10 − Q. The marginal cost is constant, c = 1, and the profit function is π (Q) = ( p (Q) − 1) × Q. In this case equations (3.2) to (3.6) give a − c 10 − 1 a + c 10 + 1 (a − c ) = = 5.5 , π M = = 20.25, = = 4.5 , pM = 4b 2 2 2b 2 2

QM =

e p (QM ) = −

5.5 pM / Q M = − ( −1) = 1.22 , ∂p / ∂Q 4.5

and L=

pM − c 5.5 − 1 4.5 ⎛ 1 ⎞ . = = = pM 5.5 5.5 ⎜⎝ ε p ⎟⎠

Figure 3.3 illustrates the price-setting trade-off (higher profit margin vs. lower sales) that occurs in this situation and the optimal price markup using the Lerner index. 3.2

Duopoly

A duopoly is characterized by competition among two firms in an industry. The analysis of duopoly or oligopoly situations requires a clear depiction of the influence of each firm on the other firms in the industry. An appropriate tool kit is offered by game theory that gives prediction about the likely outcome of such strategic interactions. A game has a certain structure, rules, and characteristics as discussed in box 3.2. The tools borrowed from game theory, used throughout the book, are sometimes criticized for being based on the strong assumption of rationality by all agents. In box 3.3 Reinhard Selten addresses

82

Chapter 3

Demand function, marginal revenue

Profit 25

12

5.5−1 1 L≡ ≈ 5.5 1.22

a =10 10

πM = 20.25 20

8 Equilibrium price

pM = 5.5

Demand function E'

6

15

Profit function

p(Q)=10−Q 10

4 Marginal revenue

5

MR(Q)=10−2Q

2 E

c =1

−1

0 0

2

4 Equilibrium quantity

6

8

QM =4.5

0 10

Quantity (Q)

Figure 3.3 Trade-off and optimal price markup for a monopolist The demand function is p (Q) = 10 − Q and marginal cost is constant ( c = 1).

Box 3.2 “Rules of the game”

The players These are the individuals, firms, entities, or actors who make decisions. In games under exogenous uncertainty, a player’s actions depend not only on the strategies played but also on external events, called “states of the world.”a Alternative actions and strategies Each time players are called upon to “play” (possibly only once), they face different alternative actions they can choose from. The collection of alternative actions at a given stage is called the action set. It is either discrete (e.g., involving choices whether or not to enter a market) or continuous (e.g., a decision on the output to supply or the size of the production capacity expansion). A strategy is a contingent a. Occasionally so-called pseudoplayers, such as nature, are used to account for or explain outside exogenous factors, such as R&D success or demand realization. Nature selects the state of the world at random regardless of the players’ actions. The probability distribution of nature’s moves is a key element of certain games, especially in option games modeling demand uncertainty.

Market Structure Games: Static

Box 3.2 (continued)

plan of actions indicating which action to take at each and every stage or state.b Information set Players condition their actions on the information they possess. An information set consists of all relevant information available to a player at the time of a decision.c Over time players generally collect more information. A key question is whether players can react to information revealed over time or if they stay committed.d Payoff structure Each strategy combination, also called strategy profile, results in a specified payoff value for each player. In equilibrium, each player pursues a strategy that maximizes its expected well-being. In a business context, ignoring agency problems, this corresponds to managers acting to maximize shareholder value. Order (sequence) of decisions In a simultaneous game, all players make their decisions at the same time so that no player observes other players’ actions before making its own decision.e If one player makes its decision after the other, having observed the earlier actions, we face a sequential or dynamic game. In such games, “time” is interpreted in terms of decision time, not necessarily real time.f b. At the start of the game, each player determines, as part of a contingent rule, what to do at each subsequent decision node, conditional on information available then. A strategy prescribes action choices at decision nodes that might not be reached during the actual (equilibrium) evolution of the game (i.e., decision nodes off the equilibrium path). A strategy profile encompasses the strategies pursued by all the players. The distinction between actions in one-stage problems and strategies as a contingent plan of actions in dynamic problems is essential to the understanding of dynamic games. c. One often distinguishes various types of information structure, such as perfect, incomplete, or imperfect information. Under perfect information, decision makers know the previous moves over the play of the game. Under incomplete information, players do not know the exact payoffs received by rivals. Such games are typically transformed into games of imperfect information by assuming that players have probabilistic information (beliefs) about some characteristics (types) of their rivals (Harsanyi transformation). d. In dynamic games, strategies that depend on calendar time but not on previous plays are called open-loop. Strategies that take account of previous plays as well are closed-loop. This distinction can substantially affect the resulting equilibrium outcome. We come back to this distinction later when discussing commitment and games of timing. e. When competing firms face such a problem under perfect information, each is aware that there is another player that is similarly aware of the potential choices of the former and the impact of its decisions on their payoffs (common knowledge). f. A two-period game where the second player faces information asymmetry concerning the move of the first player is tantamount to a simultaneous game.

83

84

Chapter 3

this issue and suggests how to refine the standard approach by incorporating behavioral considerations. There are two archetypical economic models of duopoly competition, Cournot quantity competition and Bertrand price competition. These two basic models serve to illustrate how competitive advantage can result from a better cost position or from product differentiation. They thus provide an economic foundation to Porter’s (1980) generic business strategies and enable an economic study of strategic management issues via option games. Cournot (1838) first introduced the quantity competition model. The original treatise did not explicitly rely on game theory, but Cournot (1838) roughly anticipated the equilibrium concept developed by Nash (1950) over a century later. A half century after Cournot presented his analysis of quantity competition, Bertrand (1883) criticized the model’s fundamental premise, arguing that in the real world firms do not compete in quantity but in prices. The basic Cournot model essentially assumes that each individual firm sets the quantity it wishes to provide to the market, and once all delivered quantities are aggregated, the market (acting as a “representative auctioneer”) determines the market-clearing price given the total quantity supplied by producers and the demand sought by consumers. In contrast, rather than assuming that the market price is exogenously determined, Bertrand posited that individual incumbent firms directly set their prices. The theory of industrial organization has since largely relied upon these two cornerstone models. Apart from the early criticisms of these models, economists have interpreted (for reasons explained later) Bertrand price competition as a short-run, tactical approach, while Cournot quantity competition is thought to involve long-term capacity commitments. For this reason we first present the basic Bertrand model where identical firms compete in price over homogeneous products. This model leads to what is called the Bertrand paradox, describing a situation where, due to competitive pressures, duopolists make no excess profits at all (a fortiori if fixed costs are material). Subsequently, we discuss a refinement allowing firms to produce differentiated products (differentiated Bertrand model). This extended model gives interesting insights for understanding the incentive of firms to differentiate their product offerings from those of rivals, avoiding cutthroat competition. We next consider the basic Cournot model of duopoly where firms compete in quantity

Market Structure Games: Static

Box 3.3 Interview with Reinhard Selten, Nobel Laureate in Economics (1994)

1. Your work on game theory refined existing solution concepts to select equilibria more in line with intuitive prediction. Do you think there is room for heuristics in economic analysis? My work on equilibrium refinement was guided by rational game theory and intuitions based on this approach. I think that rational game theory has to be complemented by a descriptive approach. This leaves room for heuristics but of a different kind. These heuristics involve plausible assumptions about nonoptimizing decision procedures that are within the boundaries of human cognitive abilities. 2. You are a pioneer of experimental economics. Do you believe that experimental economics has revealed limited usefulness or relevance of mathematical modeling in economic analysis? Experimental economics is very important for the development of a behavioral approach of decision and game theory. Mathematical methods are also needed for that purpose, but they are different from those used in the mainstream of economic theory. They might involve the analysis of dynamic systems, for example, learning processes and stationary concepts replacing game equilibrium. 3. Game theory generally hinges on the assumption of rationality by all players. What if people do not react rationally? Do you believe that the use of rational game theory in social sciences is necessary or appropriate? Rational game theory assumes common knowledge of the rationality of all players. When rationality is defined in terms of Bayesian decision

85

86

Chapter 3

Box 3.3 (continued)

theory, human behavior is not rational in this sense but it is not irrational. Therefore it is necessary to build a theory of bounded rationality. However, this task is far from being completed, and as long as we do not have a good substitute, rational game analysis may still be valuable because it reveals the strategic structure of a problem and may serve as a benchmark for experimental economics. We must be very skeptical, however, about the descriptive validity of rational decision and game theory.

facing the same production cost and subsequently extend the analysis to allow for (variable) cost asymmetry. 3.2.1

Bertrand Price Competition

Here, to analyze strategic interactions based on the Bertrand model of price competition, we distinguish between two cases, depending on whether firms offer identical products or not. Standard Bertrand Model Consider first a market where two symmetric firms compete over a homogeneous product, have no capacity constraint and can supply whatever quantity is required. Firms decide what price to set, and customers decide to purchase the product if the set price is lower than the value they attribute to the product (their willingness to pay). As the products offered by the rival firms are not distinguishable (they are perfect substitutes), customers have an incentive to purchase from the firm with the lowest price. Suppose that firm i sets a price pi. The best response for firm j is to set a price pj just lower than its rival’s, namely pj = pi − ε, where ε (> 0) is small, as long as its profit margin is nonnegative. Similarly firm i will then consider selling its product at a somewhat lower price pi − 2ε (< pj ) if this price is still higher than (or equal to) its marginal cost of production. Firm j will likely follow suit, and so on and on. Eventually, the rival duopolists will reduce the price they set all the way down to the marginal cost of production (assumed identical for both firms at a constant level c), at which point they stop further pointless price reduction. Consequently the duopolists will make no excess economic profit, a result analogous to the outcome found in perfectly competitive markets.

Market Structure Games: Static

87

Thus, according to the standard Bertrand model of price competition, there is no need for a large number of players to be active in an industry to create the economic conditions most beneficial to customers (i.e., perfect competition). Two firms competing in price over an undifferentiated product suffice to create a situation involving no excess economic profit for any firm and the highest possible consumer surplus. This outcome has been coined the “Bertrand paradox” in industrial organization since it challenges common business sense. One may wonder why firms bother to enter the market if they make no economic profit once they enter. A firm typically incurs a fixed cost upon entry. If a firm already operates, the other firm will not invest, however small the fixed entry cost; the market is thus likely to yield a monopoly. This standard Bertrand model rests on strong assumptions that can be relaxed but derives its appeal by illustrating cutthroat competition among a few firms. Porter’s business strategies are essentially meant to circumvent such undesirable outcome, allowing for firms to earn excess profits. To achieve this, firms compete over dimensions other than price. Differentiated Bertrand Model One noted business strategy is for firms to differentiate their product offerings. This is the differentiation strategy proposed by Porter (1980). Economists can justify this strategy thanks to the differentiated Bertrand model. Following Gibbons (1992), suppose that two firms, i and j, produce a differentiated product while facing symmetric (linear) cost6 Ci (qi ) = cqi, where qi is the quantity produced by firm i, with constant marginal cost of production c. The (modified) linear inverse market demand function for each differentiated good (allowing to charge a different price by firm i, pi) is given by pi (qi , q j ) = a − b ( qi + sq j ),

(3.1′)

where a > c ≥ 0, b > 0, and parameter s ∈ [ 0, 1) represents the substitution effect between the two differentiated product offerings.7 Products are unrelated if s = 0; here each firm can disregard the rival’s price decision since it has no impact on its own, with demand function (3.1′) reducing to the linear demand in (3.1) faced by a monopolist. If s approaches 1, 6. We formulate the demand function somewhat differently than Gibbons (1992). 7. The substitution effect captures the price change of one product for a unit change in the supply of its substitute product.

88

Chapter 3

products are undifferentiated or perfect substitutes; equation (3.1′) then reduces to the traditional linear demand faced by duopolists in the homogeneous-product case. Note that here products are horizontally differentiated, whereby the characteristics of a differentiated product appeal to a certain customer base with particular taste.8 In the case above, by inverting (3.1′), one gets the demand function9 qi ( pi , pj ) =

a (1 − s ) − pi + spj . b (1 − s 2 )

(3.7)

Firms select their prices pi , pj (≥ 0 ) simultaneously not knowing the rival’s price decision. The Nash equilibrium prices under Bertrand competition ( piB , pBj ) are such that each firm’s price choice is the best response to the other’s optimal price decision (see related box 3.4): ⎧ π i ( piB , pBj ) ≥ π i ( pi , pjB ) , ∀pi ≥ 0, ⎨ B B B ⎩π j ( pi , pj ) ≥ π j ( pi , pj ) , ∀pj ≥ 0. In the continuous case the first-order profit maximizing condition for a Nash equilibrium is a(1 − s) − 2 pi + spj + c ∂π i = 0. ( pi , pj ) = b (1 − s 2 ) ∂pi Firm i’s best-reply or reaction function is piB ( pj ) =

a (1 − s ) + c s + pj. 2 2

(3.8)

Solving the system of two equations (first-order conditions for firms i and j) with two unknowns ( pi and pj ) results in the following equilibrium prices and quantities: 1− s ⎧ B B B ⎪⎪ p = pi = pj = c + 2 − s (a − c ) ( > c ) , ⎨ a−c ⎪ q B = qiB = q Bj = . b(1 + s)(2 − s) ⎪⎩

(3.9)

8. In this case there are no commonly agreed best features for the product, just individual preferences. Products are not qualitatively ranked on a scale where customers desire the top-quality products but cannot afford them due to budget constraints. This category of differentiated products (Ferrari vs. a standard car) relates to vertical differentiation. Horizontal differentiation refers to situations where customers have taste preferences among a pool of products with comparable quality standards. 9. The resulting profit function is (twice) differentiable and concave, as can be seen from ∂ 2π i 2 =− < 0. b (1 − s 2 ) ∂pi 2

Box 3.4 Solution concepts and Nash equilibrium

Once model assumptions are laid out, one can solve a game-theoretic model using a so-called solution concept.a A solution concept is a methodology for predicting players’ behavior intended to determine the decisions (actions or strategies) that maximize each player’s payoff.b The rationality of each player is typically accepted as a common knowledge; namely each player is aware of the rationality of the other players and acts accordingly.c To obtain stronger predictions about a game outcome, one may impose assumptions beyond common knowledge of rationality.d Here equilibrium strategy profiles must form a Nash equilibrium. Consider n players. Let a i (∈ Ai ) denote firm i ’s pure strategic action (and Ai its strategy set). By convention, a −i represents the strategies played by all other players except i . When firm i chooses strategy a i and her rivals a −i , firm i receives payoff p i (a i , a − i ). A Nash equilibrium is a set of decisions—strategy profile—such that no player can do better by unilaterally changing their decision. A strategy profile (a i*,a − i*) forms a Nash equilibrium if, for any player i, i = 1, …, n,e p i (a i*, a − i*) ≥ p i (a i , a − i*)

∀a i ∈ Ai .

Equivalently, each firm i, i = 1, . . ., n, is faced with the following profitoptimization problem: max p (a i , a − i*). a i ∈Ai

We can interpret the above in terms of best-reply or reaction functions. Let R(a − i ) denote firm i ’s best response to her rivals’ strategies a −i . In Nash equilibrium each player formulates her best response to the other players’ optimal decision; that is, a i * ∈ R(a − i*). Depending on the game specification, such as the order of the play or the available information, the solution concept used may be more stringent as discussed in chapter 4. a. We do not intend here to provide a mathematical account of game theory. Readers should refer to the readings suggested at the end of this chapter for a more formal discussion of game theory. b. We are being vague on the optimality notion. While in problems involving a single decision maker optimality has an unambiguous meaning, in multiplayer decision settings “optimality” is not a well-defined concept. Its understanding is closely related to the solution concept chosen. c. The rationality assumption imposes an important limitation in situations where pride and irrationality play a role. However, building up a reputation as being irrational or unpredictable can also make strategic sense: One can turn such reputation to its advantage by altering opponents’ beliefs. Reagan’s foreign policy toward the Soviet Union might be interpreted in this light. Similarly a firm that has a reputation to wage war with any contestant imposes a credible threat on potential entrants who might think twice before entering. d. Rationalizability, introduced by Bernheim (1984) and Pearce (1984), is considered a fundamental solution concept. Based on the weak assumption that payoffs and players’ rationality are common knowledge, it predicts that a player will not play a strategy that is not a best response to some beliefs about her opponents’ strategies. By iteration, we can narrow down the set of strategies that could be reasonably played (so-called rationalizable strategies). By design, rationalizability makes very weak predictions: the set of rationalizable strategies can be large since it contains all strategy profiles that cannot be excluded based on the assumption of common knowledge of rationality. Nash’s (1950b) equilibrium concept is usually preferred for being more restrictive. e. We consider, here, pure strategies. An equivalent definition exists for mixed strategies, namely convex combinations of pure strategies. By employing mixed strategies, one ensures that (at least) one Nash equilibrium exists (Nash 1950b).

90

Chapter 3

Thus, in differentiated Bertrand–Nash equilibrium firms set a price, pB, strictly higher than the marginal cost of production, c, earning excess profits. This contrasts to the previous standard Bertrand case involving perfect substitutes (homogeneous products). Moreover, in equilibrium, prices for both differentiated products are equal ( piB = pBj ); this results from the fact that products are horizontally differentiated and that there is (by assumption) no quality premium for one of the products over the other. Finally, the equilibrium price decreases with the degree of substitutability s (∂pB ∂s = − (a − c ) ( 2 − s ) < 0). From (3.9) firm profits are

π B = π iB = π Bj =

1− s ⎛ a − c⎞ ⎟ ⎜ b (1 + s ) ⎝ 2 − s ⎠

2

(> 0) .

(3.10)

Thus, for products that are close to being perfect substitutes (s → 1), the excess profits for the duopolists become zero, a result equivalent to the outcome obtained in the standard Bertrand price competition case. Clearly, firms have an incentive to differentiate their product offerings to soften price competition, departing from the undesirable Bertrand paradox where no firm makes excess profits. This is in line with most marketing and strategic management practices.10 Example 3.2 Differentiated Price Competition Assume the linear market demand function of equation (3.1′) with parameters a = 24, b = 2 3, s = 1 2 and constant variable production cost c = 3. From (3.7) the demand functions are given by ⎧ qi ( pi , pj ) = 24 − 2 pi + pj , ⎨ ⎩q j ( pi , pj ) = 24 − 2 pj + pi . From equation (3.8), firm i’s reaction function is piB ( pj ) = 7.5 +

1 p j. 4

Firm j’s reaction function is obtained symmetrically. In (differentiated) Bertrand price competition, reaction functions are upward-sloping. In Nash equilibrium, equilibrium prices, quantities, and profits, obtained from equations (3.9) and (3.10), are pB = 10, q B = 14, and π B = 98, respectively. This situation is illustrated in figure 3.4. 10. For an economic discussion of the advisable degree of differentiation, refer to Tirole (1988, pp. 286–87).

Market Structure Games: Static

91

Price by firm j (pj ) 20

Firm i’s reaction function 15

piB( pj )

10

E

Firm j’s reaction function

pjB( pi)

5

0 0

5

10

15

20

Price by firm i (pi) Figure 3.4 Upward-sloping reaction functions in differentiated Bertrand competition a = 24, b = 2 3, s = 1 2 , and c = 3

The differentiated Bertrand model above is fairly descriptive of many real-world rivalry situations in the short term. This model underscores that competition is not just about price but that additional parameters may come into play, such as heterogeneous tastes among customers. The main alternative to price competition is quantity competition represented by the Cournot model.11 Although Bertrand price competition seems more descriptive of certain real-world situations in the short run, Kreps and Scheinkman (1983) suggest that the implicit assumption of a “market auctioneer” in the Cournot model is not necessary. Cournot 11. In Cournot quantity competition, producers simultaneously and independently make production quantity decisions. The output is brought to the market and a “market auctioneer” sets the market-clearing price. In (differentiated) Bertrand price competition, producers simultaneously and independently set prices and there is no need for a price-setting party.

92

Chapter 3

outcomes may also result from a two-stage competition game where duopolists first make a capacity choice, and once these capacity choices become common information, firms set prices as in Bertrand price competition under capacity constraints.12 The two-stage model of Kreps and Scheinkman (1983) results in Cournot-like outcomes and may be more realistic than strict Bertrand price competition in situations involving long-term investments and constrained capacities. For this reason capacity (quantity) is often thought of as a long-term strategic variable, while price is a short-run, tactical action variable. Most of the models discussed in subsequent chapters deal with long-horizon investment problems and therefore build upon the Cournot quantity setting. 3.2.2

Cournot Quantity Competition

Cournot’s (1838) classic quantity competition model characterizes industries in which firms set production schedules in advance and cannot alter them in the short run. The price-setting process is driven by the capacity or quantity chosen by the active firms in the industry.13 Firms produce a homogeneous good. We discuss next two variants of the Cournot duopoly model. We consider first the situation where firms have symmetric costs, and then relax the symmetric cost assumption. A comparison of these two models helps illustrate why firms have an incentive to be cost leader in quantity competition involving a homogeneous good. Cost Symmetry Consider two identical firms (i and j) that face the same, constant unit variable production cost c (≥ 0 ). The linear (inverse) demand function, driven by the total quantity supplied in the market Q ( = qi + q j ), is given by p(Q) = a − bQ,

(3.1)

with a > c and b > 0. Firms’ strategies consist in selecting an output that maximizes profit.14 Equivalently, one can assume that they maximize net 12. From this perspective we can consider the (equilibrium) profit functions in Cournot competition as reduced-form profit functions in which later price competition has been subsumed. Kreps and Scheinkman (1983) show this property in the case where firms face a concave demand function and cannot satisfy all demand according to the “efficientrationing rule.” Investment in new capacity units must be costly for this result to hold. 13. The ex ante chosen production schedules are hard to reverse, so firms cannot influence the market-clearing price-setting process in the short-run. 14. An important assumption of the Cournot model is that firms invest simultaneously and therefore do not observe the strategy chosen by their rivals. This is a case of complete information in a simultaneous game. The appropriate solution concept is the “simple” Nash equilibrium.

Market Structure Games: Static

93

present value (NPV) where value is given by the perpetuity of profits (π k , where k is the risk-adjusted discount rate).15 Firm i’s profit, resulting from the action profile ( qi , q j ), is given by16 p i ( qi , q j ) = [ p (Q) − c ] qi.

(3.11)

As in the monopoly case there is a trade-off between increasing the quantity supplied (qi) and receiving a lower price (and profit margin) because of the upsurge in total industry quantity. This time, however, the firm’s own output choice is not the only factor influencing the pricesetting process; the rival influences it as well. The Nash equilibrium outputs ( qiC , qCj ) are such that each firm’s quantity choice is the best response to the other’s optimal quantity decision. Taking the rival’s quantity choice as given, the first-order condition for firm i’s profit maximization (∂π i ∂qi = 0) yields a − 2bqiC − bq j = c,

(3.12)

or qiC (q j ) =

1⎛a−c − q j ⎞⎟ . ⎜ ⎠ 2⎝ b

(3.13)

Firm j’s first-order profit maximizing condition is obtained symmetrically. If one of the firms does not produce (i.e., if q j = 0), the sole active firm i will supply in equilibrium QM = (a − c ) 2b, as in monopoly. In quantity competition, the reaction functions of duopolists given by (3.13) are downward sloping (i.e., decreasing in rival’s capacity-setting action), as depicted in figure 3.5. The Nash equilibrium outputs can be determined by solving the system of the two equations (3.12) (for firms i and j) with two unknowns (qiC and qCj ) or by substituting firm i’s reaction function from (3.13) into firm j’s (and reciprocally). The equilibrium is found at the intersection (point E) of the two reaction functions: qiC ≡ qiC (qCj ( qiC )) =

1⎛a−c 1⎛a−c − ⎜ − qiC ⎞⎟ ⎞⎟ . ⎜ ⎠⎠ 2⎝ b 2⎝ b

Since symmetric firms presumably produce the same output, the Cournot–Nash equilibrium output by each firm is 15. We assume that firms settle in steady state and that there is no (expected) growth for the underlying profit flow π . 16. The profit function π i (⋅, ⋅) is concave and twice continuously differentiable in qi.

94

Chapter 3

Quantity supplied by firm j (qj)

( 0; (a−c )/ b)

Firm i’s reaction function

( 0; ( a−c ) / 2b) (qiC ; qjC ) E Firm j’s reaction function

(( a−c ) / 2b; 0 )

(( a−c )/ b; 0 ) Quantity supplied by firm i (qi)

Figure 3.5 Downward-sloping reaction functions in (symmetric) Cournot quantity competition

qC = qiC = qCj =

a−c . 3b

(3.14)

The total industry output then is QC = qiC + qCj =

2 ⎛ a − c⎞ ⎟. ⎜ 3⎝ b ⎠

(3.15)

Two observations are of particular interest. First, in Cournot duopoly each individual firm produces less than does a monopolist firm: qC ≤ QM

⎛ 1 a − c ≤ 1 a − c⎞. ⎟ ⎜⎝ 3 b 2 b ⎠

Second, firms collectively produce more than does a monopolist: QC ≥ QM

⎛ 2 a − c ≥ 1 a − c⎞ . ⎟ ⎜⎝ 3 b 2 b ⎠

Consequently, given the downward-sloping demand (∂p ∂Q < 0), the market-clearing price in Cournot duopoly is lower than in monopoly.

Market Structure Games: Static

95

One can readily deduce the equilibrium price and profit. The equilibrium market-clearing price in Cournot duopoly, pC, obtained by substituting the equilibrium quantity of equation (3.15) into demand equation (3.1), is pC = c +

a−c 3

(> c ) .

(3.16)

The equilibrium profit for firm i ( j) is

π C = π iC = π Cj =

(a − c )2 9b

.

(3.17)

Due to the lower individual output and lower market-clearing price, a Cournot duopolist earns lower profits than a monopolist (π iC ≤ π iM ). From a social-welfare viewpoint, Cournot duopoly is better than monopoly as the total quantity supplied is higher, the equilibrium marketclearing price lower, and the profit or producer surplus lower.17 What would happen if the duopolists could collaborate to improve their joint profits? Suppose that firms could (tacitly) collude, choosing to maximize their joint profit instead of their individual profits. In this case the symmetric duopolists would select the optimal cumulated quantity, Q, and then divide up the higher joint-profit pie. The optimal total industry output selected by the cartel would be the monopoly quantity, QM. Each firm could, for example, produce half of it and earn half of the monopoly rent. Since π M 2 ≥ π iC, a collusive agreement seems preferable at first sight. This strategy profile is not stable, however. The contract ruling the cartel is not self-enforceable because each firm has an incentive to cheat, increasing the quantity it supplies to benefit from a higher price.18 Firms would increase their output until the Cournot–Nash equilibrium output ((a − c ) 3b ; (a − c ) 3b) is reached. This is the only stable equilibrium in this simultaneous quantity game. Collusion is not likely 17. This efficiency result is based on the premise that fixed costs are immaterial. The presence of material fixed costs might substantially alter this result, since a natural monopoly might turn out to be socially optimal. 18. To demonstrate the instability of the collaborative outcome, consider first the optimal reaction of the deviating party (e.g., firm i). Substituting half the monopoly quantity given in (3.2) into firm i’s reaction function in (3.13), the best reply obtains qiC (Q M 2 ) = 3 (a − c ) 8b . The resulting market-clearing price is p = c + 3 (a − c ) 8 . The profit for the deviating party is higher than half the monopoly profit, creating an incentive to deviate from the cartel agreement:

πD =

9 πM ⎛ πM ⎞ ⎟. ⎜> 8 2 ⎝ 2 ⎠

96

Chapter 3

to occur since each firm has an incentive to deviate, being pulled as in a prisoner’s dilemma to the inferior Nash equilibrium outcome. Collusion will thus not occur in such a static one-shot duopoly game. This ultimately benefits consumers who can purchase the product at a lower price.19 Cost Asymmetry Suppose now that the two firms have different marginal costs ci ≠ c j, with firm i’s cost function given by Ci (qi ) = ci qi , ci ≥ 0. If ci < c j , firm i produces the (homogeneous) good more efficiently than firm j, enjoying a cost-leader advantage. Assuming the same (inverse) demand function as before, firm i’s profit function is p i (Q) = ( p (Q) − ci ) qi. Due to strategic interaction, firm i’s quantity choice ultimately depends on its own cost ci as well as on its rival’s cost c j. The cost differential, c j − ci (≠ 0 ), does matter. From the first-order profit-maximizing condition, firm i’s reaction function is qiC (q j ) =

1 ⎛ a − ci − q j ⎞⎟ . ⎜ ⎠ 2⎝ b

(3.18)

Substituting the reaction functions (into each other), firm i’s equilibrium quantity is20 qiC =

a − 2ci + c j . 3b

(3.19)

In equilibrium, firm i produces more when its rival is cost disadvantaged or the latter’s marginal production cost is increased (∂qiC ∂c j > 0). The resulting industry equilibrium price is pC =

a + ci + c j . 3

(3.20)

Firm i’s resulting equilibrium profit is

π = C i

(a − 2ci + c j )2 9b

.

(3.21)

Useful insights can be deduced from the latter expression. The cost leader (here firm i) earns higher profits in the market than its less efficient rival. This provides an economic rationale for the cost leadership strategy proposed by Porter (1980). As the firm’s profit decreases in its 19. We discuss later how (tacit) collusion can sustain itself as stable industry equilibrium in certain repeated games. 20. The formulas for firm j are analogous, meaning one can obtain firm j’s quantity by substituting j for i and i for j in the expression above.

Market Structure Games: Static

97

own cost (∂π iC ∂ci < 0), the cost leader would further benefit from an improved cost position. An alternative, indirect tactic to achieve cost leadership is to increase the rival’s cost (∂π iC ∂c j > 0), for example, by limiting its access to scarce resources it needs. 3.2.3

Strategic Substitutes versus Complements

As noted earlier, reaction functions may be downward sloping (as in the Cournot quantity competition model) or upward sloping (as in differentiated Bertrand price competition). This distinction between these two kinds of reactions is at the core of the notion of strategic substitutes versus strategic complements introduced by Bulow, Geanakoplos, and Klemperer (1985). The notions of strategic substitutes and complements are meant to capture how competitors are expected to react when a firm changes its strategic decision variables, such as price or quantity. This distinction is informative on whether a firm reacts to its rivals’ strategic actions in a reciprocating or in a contrarian way. The notion of commitment—analyzed extensively in industrial organization—rests on these types of reaction functions. When reaction functions are downward sloping, the actions are strategic substitutes. Let α i ∈ Ai be firm i’s strategic action variable (with Ai its action set) and Ri (α j ) its best reply to rival action αj.21 The general form of the equilibrium in case of strategic substitutes is depicted in figure 3.6. In figure 3.6a, a particular case of interest is the reaction function in Cournot quantity competition. Assuming a linear (inverse) demand function as in equation (3.1) and a linear cost function, Ci (qi ) = cqi with c constant, firm i’s reaction function is obtained in (3.13) as qiC (q j ) =

a−c 1 − qj. 2b 2

Since ∂qiC ∂q j < 0, if a Cournot duopolist increases its output, its rival will do the opposite by reducing it, a contrarian reaction. Cournot quantity competition is a game where actions are strategic substitutes. In case of strategic complements, competitive reactions are similar or reciprocating. The distinction again relates to the shape (slope) of the reaction function. When reaction functions are upward sloping, as in figure 3.6b, the firms’ actions are strategic complements or reciprocal. In the (differentiated) Bertrand model, we deal with strategic complements 21. We assume that the firms’ reaction functions are characterized by slopes of the same sign, meaning both are downward sloping or both upward sloping.

98

Chapter 3

aj

R i (a j)

a j*

E

R j (a i )

a i*

ai

(a)

aj

R i (a j)

a j*

E'

R j (a i )

a i*

ai

(b) Figure 3.6 Downward and upward-sloping reaction functions compared: strategic substitutes versus complements (a) Strategic substitutes or contrarian reactions ( ∂Ri ∂α j < 0 ); (b) strategic complements or reciprocating reactions ( ∂Ri ∂α j > 0 )

since a reduction in price by a firm is an optimal response to its competitor’s price cut. If the inverse demand function is linear as in equation (3.1′), firm i’s reaction function is given by (3.8) above, namely piB ( pj ) =

a (1 − s ) + c s + pj . 2 2

Since ∂piB ∂pj ≥ 0, when a Bertrand duopolist decreases its price, its rival will follow suit and decrease its price as well, a reciprocating reaction. For perfect substitutes (s → 1), such reciprocating reactions can lead

Market Structure Games: Static

99

to the Bertrand paradox, where firms end up waging a fierce price war and make zero economic profits (as in perfect competition). Equilibrium outcomes in case of strategic substitutes vs. complements are illustrated as outcomes E and E ′ in figure 3.6. The slope of the reaction function matters. Whether actions are strategic substitutes or complements can greatly affect the optimal decisionmaking in such settings. Following Gal-Or (1985), when reaction functions are downward sloping (involving strategic substitutes or contrarian reactions), each firm has an incentive to take the lead, acting as a Stackelberg leader and enjoying a first-mover advantage. This can lead, however, to situations involving preemption as firms strive to seize this advantage ahead of competitors. By contrast, in case of strategic complements or reciprocating actions, the Stackelberg follower is better off, benefiting from a second-mover advantage. For instance, under price competition no one has an incentive to set its price first if the follower can undercut it later.22 This kind of strategic interaction could result in a war of attrition where firms compete to be the last to make a move, each firm doing its utmost not to be considered a coward (chicken game). 3.3

Oligopoly and Perfect Quantity Competition

In the previous sections we analyzed industry structure where only two firms are incumbent. Although duopoly situations exist (e.g., the competition between Boeing and Airbus in the aircraft manufacturing industry), there is often more than two players operating in a given industry. Numerous examples of such industries abound. A case in point is the automotive sector, with giants such as Volkswagen (including Audi, Porsche, and Seat), Toyota, Ford, BMW, Daimler, and GM. The economic analysis of such oligopolistic structures can provide interesting insights as well. The basic quantity duopoly setup analyzed by Cournot (1838) may be refined to consider more than two competitors. At first we analyze below an oligopoly with cost asymmetry. Subsequently we discuss the simpler case of oligopoly under cost symmetry (as a special case). We confirm that as the number of competitors substantially increases, the Cournot oligopoly outcome tends toward the perfect competition benchmark. This analysis is useful later in the analysis of option games in both discrete and continuous time. 22. A firm is better off to defer its decision and select its own price until after having full knowledge of the price charged by the leader.

100

Chapter 3

Assume n oligopolist firms competing in quantity over a homogeneous product. Each firm i faces a linear demand function as in equation (3.1) and marginal production cost ci. Without loss of generality, firms are ranked in cost advantage with firm 1 having the largest cost advantage or lowest cost c1. Firm i’s profit function is given by

π i (qi , Q− i ) = [ p (qi , Q− i ) − ci ] qi ,

i = 1, . . ., n,

where Q− i stands for the quantity produced collectively by all other suppliers except firm i, with Q = qi + Q− i . To obtain the Cournot–Nash equilibrium, one has to deduce the optimal production strategy profile (q1C , . . ., qnC ) such that each firm i, i = 1, . . ., n, maximizes profit p i ( qi , Q−Ci ) , considering other rivals’ choices as given. This leads to the first-order condition: MRi ( qiC , Q−Ci ) = qiC ×

∂p C C (qi , Q− i ) + p (qiC , Q−Ci ) = ci , ∂qi

i = 1, . . ., n. (3.22)

In the linear demand case, marginal revenue is MRi (qi , Q− i ) = a − 2bqi − bQ− i . In the general Cournot oligopoly with n asymmetric firms, we must solve the following system of n equations with n unknowns: ⎧ a − 2bq1C − b (QC − q1C ) = c1 , ⎪ ⋮ ⎪⎪ C C C ⎨ a − 2bqi − b (Q − qi ) = ci , ⎪ ⋮ ⎪ C ⎪⎩a − 2bqn − b (QC − qnC ) = cn . Summation leads to a total quantity for the entire market of n ⎞⎛a−c⎞ QC (n) = ⎛⎜ , ⎝ n + 1⎟⎠ ⎜⎝ b ⎟⎠ n

(3.23)

where c ≡ ∑ j = 1 c j n is the average (variable) production cost in the industry. Denote by c− i ≡ ∑ j ≠ i c j (n − 1) the average production cost for all other firms except i. Individual firm quantities are obtained by substituting the total industry quantity of (3.23) into each equation of the system above:

Market Structure Games: Static

qiC (n) =

101

1 ⎛ a − nci + ( n − 1) c− i ⎞ ⎜ ⎟⎠ , n + 1⎝ b

i = 1, . . ., n .

(3.24)

The equilibrium price obtains from (3.1) and (3.23) as pC ( n) = c +

a−c n+1

(> c1 ).

(3.25)

Regardless of how many firms operate in the market, the most costadvantaged firm (absolute cost leader) always enjoys a price strictly higher than its variable cost c1.23 The resulting profit for firm i is p iC ( n) =

1

(a − nci + (n − 1) c− i )2

(n + 1)2

b

,

i = 1, . . ., n .

(3.26)

It can be seen that profits are increasing in the cost advantage. Moreover, a firm’s profitability depends on its production cost, its market share, and the price elasticity of demand.24 The special case of cost symmetry (with ci = c) has clear-cut insights. The equilibrium quantity, obtained from equation (3.24), is the same for each symmetric firm: qC ( n) =

1 ⎛ a − c⎞ ⎟, ⎜ n + 1⎝ b ⎠

i = 1, . . ., n.

(3.27)

The equilibrium price, resulting from equation (3.25), is given by pC (n) = c +

a−c n+1

( > c ).

(3.28)

The expression above suggests that even for a large oligopoly (n large), the equilibrium price margin, pC − c, remains positive. From (3.26), the profit for each identical oligopolist firm is

π C (n) =

1

( a − c )2

(n + 1)2

b

,

i = 1, . . ., n.

(3.29)

23. This holds as long as the demand intercept exceeds the average unit cost in the industry. 24. Let si ≡ qiC QC denote firm i’s (equilibrium) market share and recall the price elasticity of demand e p in equation (3.5). The first-order condition in (3.22) yields firm i’s Lerner index: Li ≡ ( pC − ci ) pC = − si ε p . This measure proxies for firm i’s profit margin and depends on ci , si , and ε p .

102

Chapter 3

In the special case when there is only one firm operating (n = 1), the previous monopoly results in equations (3.2) to (3.4) obtain. In duopoly (n = 2), we obtain the symmetric Cournot duopoly results of equations (3.14) to (3.17). As the number of active firms is greatly increased (n → ∞), the quantity produced by any one firm becomes negligible, the market-clearing price approaches marginal production cost, c, and the economic profit of an individual firm approaches zero. These outcomes confirm the well-known results in perfect competition: firms make no excess profits and set prices at marginal cost. The presence of fixed production or investment costs may affect the number of firms operating in the sector. 3.4

Market Structure under Incomplete Information

The standard Cournot model assumes that duopolists know perfectly the cost structure of their rivals; their profit-maximizing quantities are derived as a function of the known variable costs. This assumption might not hold in certain industry settings characterized by information asymmetry. Suppose that firm j does not know for sure firm i’s cost but that “nature” will reveal it with certain probability. For example, firm i may have invested in a new innovative process that could alter its cost structure if its R&D efforts succeed (with probability P). The rival’s cost can be either low (cL) with probability P or high (cH ) with probability 1 − P (cL < cH).25 That is, firm j has probabilistic beliefs about its rival’s type (cost). Suppose also that firm i can perfectly observe what “nature” decided concerning its own cost (e.g., the outcome of its own R&D efforts), as well as its rival’s cost (e.g., the rival uses the prevalent production technology). Firm j is aware of the information asymmetry in favor of firm i. Once firm i knows its own cost, the two firms choose simultaneously their capacity or output. Figure 3.7 depicts this situation. In selecting its optimal quantity, firm j no longer maximizes its deterministic profit function but instead maximizes its expected profit function, taking into account the uncertainty it faces concerning firm i’s type or cost. Firm j knows that if the low-cost outcome (cL) occurs (with probability P), firm i’s reaction function would be similar to equation (3.18) in the Cournot game: qiC ( q j , cL ) =

1 ⎛ a − cL − q j ⎞⎟ . ⎜⎝ ⎠ 2 b

25. Subscript L stands for low- and H for high-cost type.

(3.30)

Market Structure Games: Static

103

Nature

cL (low cost)

Firm i qi (cL ) = qL

cH (high cost)

Firm i qi (cH) = qH

Firm j

qj

Firm j

qj

Figure 3.7 Extensive form of Cournot competition under asymmetric information The dashed box indicates that firm j has two nodes in its information set and is unable to anticipate its rival’s quantity decision.

However, in case the high-cost outcome (cH ) occurs (with probability 1 − P), firm i’s reaction function is instead qiC ( q j , cH ) =

1 ⎛ a − cH − q j ⎞⎟ . ⎜ ⎠ 2⎝ b

(3.31)

Firm j’s expected profit reflects its belief (probabilistic distribution) about firm i’s possible production costs. It is given by26

π j ( q j , qL , qH ) ≡ E [π j (qi , qL , qH )] = P × π j ( q j , qL ) + (1 − P ) × π j ( q j , qH ) ,

(3.32)

where qL and qH stand for firm i’s quantity choice as a function of its low or high marginal cost. Firm j’s (Bayesian Nash) equilibrium strategy consists in selecting its output q j so as to maximize its above expected profit in (3.32), considering its rival’s optimal decision as given. From the first-order profit optimization condition (and substitution of equations 26. Firm j ’s profit function is differentiable and concave in its own strategic action q j ( ∂ 2π j ∂q j 2 < 0 since b > 0). The first-order condition is both necessary and sufficient for a maximum to obtain.

104

Chapter 3

(3.30) and (3.31) into firm j’s reaction function) firm j’s equilibrium quantity is27 qj * ≡

a − 2c j + ci , 3b

(3.33)

where ci = PcL + (1 − P ) cH is the mean (expected) value of firm i’s cost. This result is similar to the equilibrium outcome in Cournot quantity competition under complete information, except that here we utilize firm i’s expected cost (ci). Firm i faces no information asymmetry and may adapt the quantity it produces to its actual production cost realization (cL or cH ). But firm j cannot. Firm i’s equilibrium quantities, depending on nature’s move, are obtained by substituting firm j’s equilibrium quantity from (3.33) into the reaction functions (3.30) and (3.31). This gives 1− P ⎧q C C ⎪ L* = qi ( q j *, cL ) = qL − 6b (cH − cL ) ⎨ ⎪qH * = qiC ( q j *, cH ) = qLC + P (cH − cL ) ⎩ 6b

(≤ qLC ) , ( ≥ qHC ) ,

(3.34)

where qLC = ( a − 2cL + c j ) 3b and qCH = ( a − 2cH + c j ) 3b are the asymmetric complete-information Cournot quantities for firm i depending on whether it has low or high cost. The equilibrium quantities for firm i are not the same as the ones derived for the standard Cournot model (under complete information), even though firm i has the same information as before; that is, it knows both its cost and its rival’s. The difference arises from the fact that firm j has an informational disadvantage and that firm i knows it. In the favorable case where firm i has lower cost cL, the total quantity supplied in the marketplace, Q (cL ), is higher under asymmetric information than under complete information. Indeed, from equations (3.33) and (3.34), we have Q (cL ) = q j * + qL* = QC (cL ) +

1− P ( c H − cL ) 6b

( ≥ QC (cL )),

(3.35)

27. The first-order derivative of firm j’s profit with respect to its own quantity (holding firm i’s quantity choices qL and qH fixed) is ∂p j ∂q j = a − 2bq j − bqi − c j , with qi ≡ PqL + (1 − P ) qH . From the first-order condition, qCj (qi ) =

1 ⎛ a − cj ⎞ − qi ⎟ . ⎜ ⎠ 2⎝ b

Equation (3.33) obtains from substituting firm i’s reaction functions (3.30) and (3.31) in the expression above.

Market Structure Games: Static

105

where QC (cL ) = qCj + qLC = ( 2a − c j − cL ) 3b is the total industry output in the complete-information case obtained by summing the quantities for firms i and j in equation (3.19). As the inverse demand function is downward sloping, the market-clearing price is lower in the asymmetricinformation case since Q(cL ) ≥ QC (cL ). Firm i would have been better off if its rival had known its cost.28 Due to the combined effect of lower price and lower quantity for firm i, the informed party (i) is worse off as a result of the rival’s information disadvantage in the favorable cost situation (cL).29 This rests on the fact that firm j forms a higher expectation on i’s cost (than the actual low cost cL) and consequently produces more compared to the situation where firm j actually knows the cost realization. Since rivals’ actions are strategic substitutes or contrarian, firm i will produce less as a best reply to firm j’s oversupply. In the opposite (unfavorable) case where firm i turns out to have a high production cost (cH ), the effects are reversed. Firm i produces more than under complete information since its rival, firm j, believes it has a lower average cost. The market-clearing price is higher, and the informed party makes larger excess profit.30 This underscores that an informed 28. Given the linear demand function of equation (3.1) and the total industry output in (3.35), one obtains the market-clearing price in the asymmetric-information case: p (Q (cL )) = pLC −

1− P ( c H − cL ) 6

(≤ pLC ),

where pLC = ( a + c j + cL ) 3 from equation (3.20). 29. Firm i’s equilibrium profit is p L = p i ( cL ) =

1 ⎡ ⎛ a + c j − 2 cL ⎞ 1 − P (cH − cL )⎤⎥ ⎜ ⎟⎠ − 3 6 b ⎢⎣⎝ ⎦

2

(≤ p LC ) .

The equilibrium profit in the complete-information Cournot case, π LC = (a + c j − 2cL ) 9b , comes from (3.21). 30. In this case total industry output is obtained from equations (3.33) and (3.34): 2

Q (cH ) = q j * + qH * = QC (cH ) −

P (cH − cL ) 6

(≤ QC (cH )) ,

where QC (cH ) = ( 2a − c j − cH ) 3b is the total industry output in the complete-information case. The equilibrium price in the asymmetric-information case is p (Q (cL )) = pHC +

P ( c H − cL ) 6

(≥ pHC )

with pLC = ( a + c j + cH ) 3 . Firm i’s equilibrium profit is obtained from the above and equation (3.34) as

π H = π i ( cH ) =

1 ⎡ a − 2cH + c j P ⎤ + ( c H − cL ) ⎥ 3 6 b ⎣⎢ ⎦

2

(≥ π CH )

where the equilibrium profit of the high-cost firm, π CH = (a − 2cH + c j ) 9b, is obtained from equation (3.21). 2

106

Chapter 3

Table 3.2 Equilibrium outcomes under various industry structures A: Cost symmetry Industry structure

Firm i ’s quantity ( qi*)

Industry quantity (Q*)

Firm i’s profit

Monopoly

1 ⎛ a − c⎞ ⎟ ⎜ 2⎝ b ⎠

Cournot duopoly

1 ⎛ a − c⎞ ⎟ ⎜ 3⎝ b ⎠

1 ⎛ a − c⎞ ⎟ ⎜ 2⎝ b ⎠ 2 ⎛ a − c⎞ ⎟ ⎜ 3⎝ b ⎠

1 (a − c ) 4 b 2 1 (a − c ) 9 b

Stackelberg leadera

1 ⎛ a − c⎞ ⎟ ⎜ 2⎝ b ⎠

Stackelberg followera

1 ⎛ a − c⎞ ⎟ ⎜ 4⎝ b ⎠ 1 ⎛ a − c⎞ ⎟ ⎜ n + 1⎝ b ⎠

Cournot oligopoly ( n players) Bertrand duopoly (differentiated)

1

2

1 (a − c ) 4 b

2

3 ⎛ a − c⎞ ⎟ ⎜ 4⎝ b ⎠

1 (a − c ) 16 b 2 1 ⎛ a − c⎞ ⎟⎠ ⎜⎝ b n+1 2

n ⎛ a − c⎞ ⎟ ⎜ n + 1⎝ b ⎠ ⎛ a − c⎞

⎟ ⎜ (1 + s ) (2 − s) ⎝ b ⎠

2

⎛ a − c⎞

⎟ ⎜ (1 + s ) (2 − s) ⎝ b ⎠

⎛ ⎞ ( a − c )2 1 − s2 b ⎝⎜ (1 + s )2 ( 2 − s )2 ⎠⎟

B: Cost asymmetry Industry structure

Firm i ’s quantity ( qi*)

Industry quantity (Q*)

Firm i’s profit

Monopoly

1 ⎛ a − ci ⎞ ⎟ ⎜ 2⎝ b ⎠

1 ⎛ a − ci ⎞ ⎟ ⎜ 2⎝ b ⎠

1 (a − ci ) 4 b

Cournot duopoly

1 ⎛ a − 2ci + c j ⎞ ⎜ ⎟⎠ 3⎝ b 1 ⎛ a − 2ci + c j ⎞ ⎜ ⎟⎠ 2⎝ b 1 ⎛ a − 2ci + c j ⎞ ⎜ ⎟⎠ 4⎝ b 1 a − nci + (n − 1)c− i n+1 b

2a − ( ci + c j ) 3b

1 (a − 2ci + c j ) b 9 2 1 (a − 2ci + c j ) b 4 2 1 (a − 2ci + c j ) b 16

Stackelberg leadera Stackelberg followera Cournot oligopoly ( n players)

3 ⎛ a − ci ⎞ ⎟ ⎜ 4⎝ b ⎠ n ⎛a−c⎞ ⎟ ⎜ n + 1⎝ b ⎠

2

2

1 ⎛ a − nci + (n − 1)c− i ⎞ ⎟⎠ ⎜ b⎝ n+1

2

Note: p (Q) = a − bQ with Q = qi + q j ; s is the substitutability parameter in p (Q) = a − b ( qi + sq j ) ; the cost function is Ci (qi ) = cqi (symmetry) or Ci (qi ) = ci qi (asymmetry). c = ∑ nj =1 c j n and c− i = ∑ j ≠i c j (n − 1). a. The outcomes in case of the Stackelberg game are derived in chapter 4.

Market Structure Games: Static

107

party cannot always benefit from its informational advantage when strategic interactions come into play. Rivals form beliefs about the informed party’s cost type and may over- or under-react as a result of their informational disadvantage. These inaccurate beliefs can sometimes prove detrimental for the better-informed firm. Conclusion In this chapter we provided an overview of static market structure games. We first discussed the monopoly benchmark. We then gave applications illustrating how noncooperative game theory can provide powerful insights about real-world problems. We considered several static models involving price and quantity competition, and examined how such models motivate certain generic competitive strategies of being a cost leader or differentiating from one another. We analyzed the impact of the number of players in a given industry on an individual firm’s profit, as well as potential effects of asymmetric information on the industry structure. Several equilibrium outcomes discussed in this chapter are summarized in table 3.2, panels A and B, for cost symmetric and cost asymmetric firms, respectively. These tables may be useful for later reference. The next chapter extends this analysis to dynamic models of oligopoly, highlighting that, in the long term, a competitive stance such as commitment or collaboration may be both value-enhancing for the individual firms and sustainable as industry equilibrium. Selected References Osborne (2004) is a good introductory textbook on game theory. Tirole’s (1988) appendix provides a short, concise manual on noncooperative game theory. Averted readers might prefer the rigorous and more advanced treatment of game theory offered by Fudenberg and Tirole (1991). Besanko et al. (2004) provide a good overview of industrial organization issues focusing on the models’ strategic insights. Tirole (1988) is considered a key reading on industrial organization.

108

Chapter 3

Besanko, David, David Dranove, Mark Shanley, and Scott Schaefer. 2004. Economics of Strategy, 3rd ed. New York: Wiley. Fudenberg, Drew, and Jean Tirole. 1991. Game Theory. Cambridge: MIT Press. Osborne, Martin J. 2004. An Introduction to Game Theory. New York: Oxford University Press. Tirole, Jean. 1988. The Theory of Industrial Organization, Cambridge: MIT Press.

4

Market Structure Games: Dynamic Approaches

In chapter 3 we dealt with benchmark models of market structure from a static perspective. These static games allow predicting the short-run impact of a firm’s actions on its rivals. The current chapter extends this discussion to dynamic approaches. These allow the inclusion of long-term strategy formulations and discussion of phenomena that require more dynamic thinking such as commitment and collaboration. Dynamic games make explicit how today’s decisions may affect or induce distinct future strategic situations. For example, if one adopts today an aggressive stance toward a rival, this behavior might induce the rival to act aggressively in the future. When assessing the benefit the firm could gain from an aggressive stance, it has to figure out what is the potential long-term impact of its decision today. One way to do this is to “look forward and reason backward” (Dixit and Nalebuff 1991). This dynamic way of thinking consists in figuring out what will be the likely future reaction in the marketplace to today’s firm actions. Once this is understood, one can assess alternative actions based on the impact they may have in the future—possibly rejecting certain initiatives that appear beneficial today but may have a negative long-term impact. This forms the underlying logic behind backward induction and subgame perfection used in dynamic games under perfect information. These notions require that for every history (or subgame) players act optimally as part of Nash equilibrium and that they make their early decisions knowing that all future actions will be optimal given the current set of information. Based on these notions, we discuss below two streams of models dealing with multiplayer decision-making in multistage settings. In section 4.1 we elaborate on commitment and discuss how limiting one’s own flexibility might create strategic value for the committing firm, thereby contradicting standard real options thinking that suggests that flexibility is always of value. Subsequently, in section 4.2, we discuss

110

Chapter 4

situations when firms may find it beneficial to cooperate with their rivals and have no long-term incentive to “cheat” on them. 4.1

Commitment Strategy

Industrial organization and game theory have evolved to help analyze situations and concepts encountered in everyday life. Commitment is a core topic in several books and movies. A case in point is Dr. Strangelove or How I Learned to Stop Worrying and Love the Bomb (1964) by Stanley Kubrick. The movie depicts an imaginary setting of the Cold War in the aftermath of an unauthorized launch of nuclear weapons on the Soviet Union. US General Ripper (who “went a little funny in the head”) utilizes flaws in the US defensive system, deciding on his own to attack selected targets with hydrogen bombs. He does this in the hope that the US president will react by ordering an all-out attack, destroying all Soviet military bases to circumvent any retaliation. The president, however, reacts differently. He summons his senior officials to the War Room to formulate alternatives to the unauthorized attack. He decides to warn the Russians of the imminent US attack, but learns that the USSR has installed a deterrence device—the “Doomsday Machine”— that would be triggered automatically in case of a sneak US nuclear attack, with no possibility to be reversed. This automatic retaliation would destroy all human life. Astonished by the threat of this Doomsday device, the president summons the director of Weapons Research and Development, Dr. Strangelove, for advice: “How is it possible for this thing to be triggered automatically and at the same time be impossible to un-trigger?” the president asks. Dr. Strangelove replies: It is not only possible, it is essential. That is the whole idea of this machine. Deterrence is the art of producing in the mind of the enemy the fear to attack. Because of this automated decision-making process—which rules out human meddling—the Doomsday Machine is terrifying and simple to understand. It is completely credible. . . . The whole point of this Doomsday Machine is [however] lost if you keep it a secret.

The only flaw in the Soviet deterrence strategy was that General Ripper, when deciding to attack the Russians, was ignorant of the very existence of this deterrence device.1 1. Dixit and Nalebuff (1991) discuss a simple setting to underline the effectiveness of strategic commitment to deter nuclear warfare (pp. 128–31), and explain how the Doomsday Machine should have deterred sneak attacks if effectively communicated (pp. 155–56).

Market Structure Games: Dynamic

111

(4, 3) c

a

Firm j d

(2, 4)

Firm i

Firm j c

d

a

(4, 3)

(2, 4)*

b

(3, 2)

(1, 1)

Firm i

c

(3, 2)*

b Firm j d

a. Simultaneous game

(1, 1)

b. Sequential game

Figure 4.1 Payoffs in a simultaneous versus sequential game for strategic commitment The first element in (·, ·) corresponds to firm i‘s payoff, while the second denotes the rival’s payoff.

4.1.1

Concept of Commitment

Commitment relates to strategic moves. A strategic move is intended to alter the beliefs or actions of rivals to one’s own benefit by purposely limiting one’s own freedom of action (e.g., killing one’s options). The raison d’être of strategic moves counteracts a common belief held in the real options literature that it is preferable under uncertainty to always keep options open (e.g., see Dixit and Pindyck 1994). As is well known in the industrial organization literature, lack of freedom can have strategic value if it can change the rival’s expectations about one’s future response, turning it to one’s own advantage. One approach to considering strategic moves is to look at how firms can be better off by altering the rules of the game, such as by transforming a simultaneous game into a sequential one. Consider the simultaneous game under complete information described in panel a of figure 4.1. Firm i may choose between action a and b, and firm j between action c and d . Firm i has a dominant strategy to take action a since whatever firm j decides (c or d ), the payoff for firm i is always higher when choosing action a (4 > 3 and 2 > 1). The best response to firm i’s choosing action a is for firm j to choose action d (receiving 4 rather than 3). Given this,

112

Chapter 4

the Nash equilibrium (*) is the right-top situation, with equilibrium payoffs of 2 for firm i and 4 for firm j.2 An alternative strategy for firm i is to move ahead of firm j, influencing its subsequent reaction. In this way, firm i turns a simultaneous game into a sequential one. To deal with the resulting sequential game shown in panel b of figure 4.1, we need to refine the Nash equilibrium solution concept to a multistage setting. Nash’s contribution to game theory and refinements of the Nash equilibrium concept are discussed in box 4.1. Figure 4.2 summarizes four basic game theory solution concepts. The appropriate solution concept here is the subgame perfect Nash equilibrium that rules out cheap talk and empty threats by the first mover (e.g., having the possibility to un-trigger the Doomsday Machine). In this finite-horizon problem (two stages) the solution is obtained by backward induction along the game tree of figure 4.1b. Once firm i chooses action a in the first stage and firm j observes this, the latter’s optimal action in stage two is to choose action d since its payoff for the strategy profile ( a, d ) is higher than for (a, c ) as 4 > 3. Thus, if firm i pursues action a, it will receive a payoff of 2 provided that firm j behaves optimally in the next stage (choosing d to receive 4 rather than c receiving 3). However, if firm i chooses action b in the first stage, firm j will select c (receiving 2 rather than 1), and so firm i will then receive 3. Firm i is thus better off choosing action b in the first stage, receiving 3 rather than 2. Firm i’s equilibrium payoff in the sequential game of figure 4.1b is higher than the equilibrium payoff it would have received in the simultaneous game depicted in figure 4.1a. To acquire this advantage obtained in the sequential game, firm i can announce early on it will pursue strategy b, whatever the decision of its rival (e.g., no wavering by the Soviet Union following a sneak attack by the United States). To behave strategically, firm i must commit not to follow the static optimal action of the simultaneous game (which was action a). Such strategic commitment makes firm i better off, receiving 3 rather than 2. The discussion above provides an example of how a player can act strategically, changing the ex post subgames with its ex ante strategic decision. If firm i declares upfront it will pursue action b regardless of its rival’s subsequent action, it can change the rules of the game in its favor. But for such a move to have real commitment value, it must be, among other things, credible and observable. To be credible, such strategic 2. The Nash equilibrium (a, d ) consists of strategies that are rationalizable since they survive iterated removal of never-best responses. This equilibrium selection relies on very weak assumptions, namely common knowledge of rationality.

Market Structure Games: Dynamic

113

Box 4.1 Development of game theory and refined solution concepts

As far back as the early nineteenth century, beginning with Auguste Cournot in 1838, economists have developed methods for studying strategic interaction. But these methods focused on specific situations, and for a long time no overall method existed. The game-theoretic approach now offers a general toolbox for analyzing strategic interaction. Game Theory Whereas mathematical probability theory ensued from the study of pure gambling without strategic interaction, games such as chess and cards became the basis of game theory. The latter are characterized by strategic interaction in the sense that the players are individuals who think rationally. In the early 1900s mathematicians such as Zermelo, Borel, and von Neumann had already begun to study mathematical formulations of games. It was not until the economist Oskar Morgenstern met the mathematician John von Neumann in 1939 that a plan originated to develop game theory so that it could be used in economic analysis. The most important ideas set forth by von Neumann and Morgenstern in the present context may be found in their analysis of two-person zerosum games. In a zero-sum game, the gains of one player are equal to the losses of the other player. As early as 1928 von Neumann introduced the minimax solution for a two-person zero-sum game. According to the minimax solution, each player tries to maximize his gain in the outcome that is most disadvantageous to him (where the worst outcome is determined by his opponent’s choice of strategy). By means of such a strategy each player can guarantee himself a minimum gain. Of course, it is not certain that the players’ choices of strategy will be consistent with each other. von Neumann was able to show, however, that there is always a minimax solution, namely a consistent solution, if so-called mixed strategies are introduced. A mixed strategy is a probability distribution of a player’s available strategies, whereby a player is assumed to choose a certain “pure” strategy with some probability. John F. Nash In his dissertation Nash introduced the distinction between cooperative and noncooperative games. His most important contribution to the theory of noncooperative games was to formulate a universal solution concept with an arbitrary number of players and arbitrary preferences, that is, not solely for two-person zero-sum games. This solution concept later came to be called Nash equilibrium. In a Nash equilibrium all of the players’ expectations are fulfilled and their chosen strategies are optimal. Nash proposed two interpretations of the equilibrium concept: one based on

114

Chapter 4

Box 4.1 (continued)

rationality and the other on statistical populations. According to the rationalistic interpretation the players are perceived as rational, and they have complete information about the structure of the game, including all of the players’ preferences regarding possible outcomes, where this information is common knowledge. Since all players have complete information about each others’ strategic alternatives and preferences, they can also compute each others’ optimal choice of strategy for each set of expectations. If all the players expect the same Nash equilibrium, then there are no incentives for anyone to change his strategy. Nash’s second interpretation—in terms of statistical populations—is useful in so-called evolutionary games. This type of game has also been developed in biology in order to understand how the principles of natural selection operate in strategic interaction within and among species. Moreover Nash showed that for every game with a finite number of players, there exists an equilibrium in mixed strategies. Despite its usefulness there are problems associated with the concept of Nash equilibrium. If a game has several Nash equilibria, the equilibrium criterion cannot be used immediately to predict the outcome of the game. This has brought about the development of so-called refinements of the Nash equilibrium concept. Another problem is that when interpreted in terms of rationality, the equilibrium concept presupposes that each player has complete information about the other players’ situation. It was precisely these two problems that Selten and Harsanyi undertook to solve in their contributions. Reinhard Selten The problem of numerous noncooperative equilibria has generated a research program aimed at eliminating “uninteresting” Nash equilibria. The principal idea has been to use stronger conditions not only to reduce the number of possible equilibria, but also to avoid equilibria that are unreasonable in economic terms. By introducing the concept of subgame perfection, Selten (1965, 1975) provided the foundation for a systematic endeavor. An example might help to explain this concept. Imagine a monopoly market where a potential competitor is deterred by threats of a price war. This may well be a Nash equilibrium—if the competitor takes the threat seriously, then it is optimal to stay out of the market—and the threat is of no cost to the monopolist because it is not carried out. But the threat is not credible if the monopolist faces high costs in a price war. A potential competitor who realizes this will establish himself on the market and the monopolist, confronted with fait accompli, will not start a price war. This is also a Nash equilibrium. In addition, however, it fulfills Selten’s requirement of subgame perfection, which thus implies systematic

Market Structure Games: Dynamic

115

Box 4.1 (continued)

formalization of the requirement that only credible threats should be taken into account. Selten’s subgame perfection has direct significance in discussions of credibility in economic policy, the analysis of oligopoly, the economics of information, and so forth. It is the most fundamental refinement of Nash equilibrium. Nevertheless, there are situations where not even the requirement of subgame perfection is sufficient. This prompted Selten to introduce a further refinement, usually called the “trembling-hand” equilibrium. The analysis assumes that each player presupposes a small probability that a mistake will occur, that someone’s hand will tremble. A Nash equilibrium in a game is “trembling-hand perfect” if it is robust with respect to small probabilities of such mistakes. This and closely related concepts, such as sequential equilibrium (Kreps and Wilson 1982), have turned out to be very fruitful in several areas, including the theory of industrial organization and macroeconomic theory for economic policy. John C. Harsanyi In games with complete information all of the players know the other players’ preferences, whereas they wholly or partially lack this knowledge in games with incomplete information. Since the rationalistic interpretation of Nash equilibrium is based on the assumption that the players know each others’ preferences, no methods had been available for analyzing games with incomplete information, despite the fact that such games best reflect many strategic interactions in the real world. This situation changed radically in 1967–68 when John Harsanyi published three articles entitled Games with Incomplete Information Played by “Bayesian” Players. Harsanyi’s approach to games with incomplete information may be viewed as the foundation for nearly all economic analysis involving information, regardless of whether it is asymmetric, completely private, or public. Harsanyi postulated that every player is one of several “types,” where each type corresponds to a set of possible preferences for the player and a (subjective) probability distribution over the other players’ types. Every player in a game with incomplete information chooses a strategy for each of his types. Under a consistency requirement on the players’ probability distributions, Harsanyi showed that for every game with incomplete information, there is an equivalent game with complete information. In the jargon of game theory, he transformed games with incomplete information into games with imperfect information. Such games can be handled with standard methods. Source: Nobel Prize Committee Website.

• Determine stable (fixed) points among possible strategy profiles

• Eliminate dominated strategies

• Select dominant strategies

• Complete information

• Static game

• Simultaneous moves

• Avoid profitable one-stage deviations

• Go back to the beginning (backward induction)

• Determine Nash equilibria at each subgame

At date t, players know all moves before t –1

• Almost perfect information

Each player knows all actions taken previously

• Perfect information

• Dynamic game

• Players move sequentially

Subgame perfect equilibrium

• Players form expectation about the outcome of the game and act accordingly

Players have probabilistic beliefs about other players’ type

• Imperfect

Players do not know other players’ payoffs

• Incomplete

• Static game

• Simultaneous moves

Bayesian equilibrium

Beliefs are obtained from strategies and observed actions using Bayes’s rule

• Actions are optimal given the beliefs

• Mixed form of perfect and Bayesian equilibrium

Players have probabilistic beliefs about other players’ type

• Imperfect

Players do not know other players’ payoffs

• Incomplete

• Dynamic game

• Players move sequentially

Perfect Bayesian equilibrium

Figure 4.2 Main game theory solution concepts The subgame perfect, Bayesian, and perfect Bayesian equilibria are essentially refinements of Nash equilibrium. Other solution concepts include correlated equilibrium and sequential equilibrium.

Model use

Information set

Order of moves

Nash equilibrium

116 Chapter 4

Market Structure Games: Dynamic

117

commitments involve early decisions with a long-term impact that are costly or difficult to reverse (e.g., they involve substantial fixed or sunk costs).3 This is the “whole idea” of the Doomsday Machine that cannot be un-triggered once initiated. The automatic Soviet retaliation is suboptimal from the second-stage perspective (annihilation of all human life on earth) but has commitment value in a dynamic setting provided that the attacking party is aware of the deterrence device (which was not the case in Dr. Strangelove). Such strategic commitments differ from tactical decisions that can be easily reversed. According to Schelling (1960), for a strategic move to have commitment value it must be: Observable Rivals must be aware of the committing party’s strategic move before making their own decision. The Doomsday Machine could not impact on General Ripper’s decision since he was not aware of its existence. Understandable The commitment should affect the incentives of the opponent. The strategic move must be able to change the ex post choices. Credible A strategic decision has little commitment value if an announcement is perceived as a bluff. Credibility influences the rival’s estimation whether the committing party is willing to carry through what it has stated regardless of circumstances. Costly to reverse For a move to be credible, it must be hard, costly, or impossible to reverse once set in motion. This typically involves substantial sunk costs for the committing party.4 Credibility is crucial. Dixit and Nalebuff (1991) enumerate eight devices for achieving credibility (coined the “eightfold path to credibility”).5 These are summarized in table 4.1. Ex ante commitments can significantly alter ex post incentives, altogether affecting the outcome of strategic interactions in dynamic games. A classic case is when an incumbent can deter rival entry to protect its own market by building overcapacity (e.g., see Spence 1977, 1979, or Dixit 1980).6 When firms commit, they do not only anticipate the direct 3. If firm i were to announce it will pursue action b but later it may choose to implement alternative action a —that is optimal from a simultaneous-game perspective—its announcement will be seen as “cheap talk” and will be of little strategic value. 4. Sunk costs are significant when the ex post value of the investment outlay is significantly lower. In the case of sunk investment with low or no salvage value, there is effectively no option to abandon. This may be the case with relationship- or industry-specific assets. 5. See Dixit and Nalebuff (1991, pp. 142–67). 6. In the first stage, the incumbent can choose excessively high capacity (higher than the second-stage optimal capacity) to deter entry by potential entrants.

118

Chapter 4

Table 4.1 Paths to credible commitment Basic concept

Credibility device

Make it costly to break your commitment

(1) Establish and use a reputation (bluffing may be costly in terms of reputation if revealed; in repeated games, reputation is crucial and should be carefully cultivated). (2) Make contracts (agree on punishment if the announcement is not followed through).

Limit the options to reverse actions

(3) Cut off communication (e.g., seal off the base as General Ripper did). (4) Burn bridges behind you (deny yourself the opportunity to retreat or to reverse your action). (5) Leave the outcome beyond control (i.e., provide automatic response to your rivals’ action). This was the “whole idea” behind the Doomsday Machine.

Engage in a repeated relationship

(6) Move in small steps (build up a reputation of carrying through with your announcement via repeated relationships rather than one-time agreements).

Build credibility on others

(7) Develop credibility through teamwork (individual weaknesses can be resolved by forming groups, peer pressure) (8) Employ negotiating agents (agents have the permission to negotiate up to a point).

Source: Adapted from Dixit and Nalebuff (1991).

effect of the commitment (e.g., direct cost savings from investment) but also what will be the ex post strategic impact on rivals, called the strategic effect. Under uncertainty there is a trade-off between this positive strategic value of early commitment and the flexibility to wait to invest. Strategic commitment kills one’s option to wait, but it can make the firm better off due to strategic interactions. It is thus of critical importance to closely examine the trade-off between flexibility and commitment in an integrated framework under uncertainty (considered in chapter 7). 4.1.2 Taxonomy of Commitment Strategies Fudenberg and Tirole (1984) extend the theory on strategic commitment, introducing a new classifying scheme for business strategies.7 They 7. The following taxonomy is based on Tirole (1988), who complements Fudenberg and Tirole (1984) with elements from Bulow, Geanakoplos, and Klemperer (1985). The terminology “strategic substitutes” and “strategic complements” was not used in the original paper (of 1984). The names for the four main business strategies are unchanged. We collectively refer to this work as Fudenberg and Tirole (1984) thereafter, although parts are more directly based on Tirole (1988, pp. 323–28).

Market Structure Games: Dynamic

119

identify four main business strategies: top dog strategy, puppy dog ploy, lean and hungry look, and fat cat strategy. These are closely linked to the type (and sign) of the strategic effect brought about by the strategic commitment. This strategic effect ultimately depends on whether the (ex post) actions are strategic complements or substitutes, and whether the commitment makes the firm tough or soft.8 We discuss next a firm’s incentive to be tough or soft and present Fudenberg and Tirole’s taxonomy. Entry Strategies Standard competition models (e.g., Cournot, Bertrand) assume that the number of firms in an industry is given exogenously. In reality the number of firms may be endogenous and driven, for example, by the magnitude of fixed entry costs. If there are low or no barriers to entry and exit and incumbents make excess profits, other firms will contemplate entry to take a slice of the pie. In such a contestable market, economic profits will be forced down to zero. Excess profits may be sustainable if there exist significant barriers to entry and exit. Bain (1956) initiated the study of entry barriers, distinguishing three entry types: blockaded, deterred, and accommodated. Fudenberg and Tirole (1984) build on this framework to examine commitment strategies. The authors disregard the case of blockaded entry, since it is an exogenous state that cannot be altered by incumbents. They focus instead on entry-barrier erection strategies, specifically on deterred and accommodated entry. For simplicity, consider two players: firm i is the incumbent and firm j a would-be entrant. In the first stage, firm i may commit by incurring a sunk investment outlay, K i. In the second stage, both firms compete (deciding simultaneously) over tactical or short-term variables α i and α j (e.g., prices).9 Firm i’s profit p i ( K i , a i , a j ) depends on the investment commitment Ki and later action choices α i and α j.10 Under perfect information, the relevant solution concept is the subgame perfect Nash equilibrium. In subgame perfect equilibrium, firms’ second-stage actions, α i* ( K i ) and α j* ( K i ), must form a Nash equilibrium for commitment choice Ki. The first-stage strategic investment thus affects ex post equilibrium choices; that is, the second-stage actions are functions of the first-stage investment.11 Firm i can deter entry (entry 8. Smit and Trigeorgis (2001) use the terms “aggressive” versus “accommodating” instead of “tough” versus “soft.” 9. If the two firms choose their actions simultaneously, the outcome will be, for example, that of Cournot quantity or Bertrand price competition. 10. π i(⋅, ⋅, ⋅) and π j (⋅, ⋅, ⋅) are twice continuously differentiable with respect to α i and α j , and concave in one’s action. π i(⋅, ⋅, ⋅) is concave in K i . 11. α i* (⋅) and α j * (⋅) are differentiable in K i .

120

Chapter 4

deterrence) if its investment Ki prevents firm j from making profits in the second stage:

π j ( Ki , α i* ( Ki ) , α j * ( Ki )) ≤ 0. Firm i accommodates entry (entry accommodation) if, despite committing, it allows its rival to make profits:

π j ( Ki , α i* ( Ki ) , α j * ( Ki )) > 0. To analyze the strategic impact, we need to consider the total derivative (total effect) of firm i’s equilibrium profit, π i, with respect to the firststage strategic commitment, Ki, namely dp i ⎛ ∂p i ⎞ ⎛ ∂p i da i * ⎞ ⎛ ∂p i da j * ⎞ . =⎜ ⎟ +⎜ ⎟+ dKi ⎝ ∂Ki ⎠ ⎝ ∂a i dKi ⎠ ⎜⎝ ∂a j dKi ⎟⎠

(4.1)

The total derivative for firm j’s profit is obtained symmetrically. Business strategies may differ depending on whether entry is deterred or accommodated. Deterred Entry As noted, firm i’s commitment strategy is driven by its rival’s profit (π j). We must thus consider the total derivative of π j with respect to Ki to determine firm i’s optimal first-stage investment policy. Since in the second stage firm j will select its action α j optimally (∂π j ∂α j = 0), the third term in the total derivative of firm j’s profit—the symmetric version of equation (4.1) above—drops out. This leads to dπ j = dKi (total effect)

⎛ ∂π j ⎞ ⎛ ∂π j dα i * ⎞ ⎜⎝ ⎟ +⎜ ⎟. ∂Ki ⎠ ⎝ ∂α i dKi ⎠ (direct (strategic effect) effect)

(4.2)

Firm i’s first-period strategic investment, Ki, has several value effects. First, it has a direct impact on firm j’s profit value, ∂π j ∂Ki ; this is the direct effect. For instance, if firm i acquires all scarce resources in the first stage, firm j cannot operate and make any profit; this is a positive direct effect. In many cases, however, this effect is negligible. This is the case when firm i invests in a process innovation that does not improve firm j’s cost position (no spillover). The second term, ∂π j ∂α i × dα i* dKi , represents the strategic effect of the commitment. It results from firm i’s ex post behavioral change due to its own commitment (dα i* dKi ) and

Market Structure Games: Dynamic

121

from the impact of this behavioral change on firm j’s profit (∂π j ∂α i ). The total effect of the strategic investment is the sum of the direct and the strategic effects.12 To determine what commitment strategy to follow, we further need to discuss tough as opposed to soft commitment. These two concepts are meant to capture whether a commitment by one firm places its rival at an advantage (benefits them) or at a disadvantage (hurts them). Firm i makes a tough commitment if the rival, firm j, is hurt, meaning dπ j dKi < 0. Such an announcement is bad news for the competitor. In this case firm i should overinvest in the first stage to hurt its rival and deter entry.13 Fudenberg and Tirole (1984) coin this approach “top dog” strategy because it consists in “being big or strong, to look tough or aggressive.” On the contrary, an early investment is a soft commitment if the rival benefits from it, meaning if dπ j dKi > 0. If firm i commits or invests too much in the first stage (Ki is high), entry will hardly be deterred since π j will be increased. Firm i should then underinvest to deter entry, as part of a “lean and hungry look” strategy. Whether one of the two approaches is advisable also depends on the trade-off between the sunk investment commitment cost and the value increment resulting from the direct and strategic effects in equation (4.2). A general recommendation cannot be readily formulated. 12. Fudenberg and Tirole (1991, pp. 132–33) propose an alternative, more technical interpretation of equation (4.2) based on the notion of open-loop and closed-loop strategies. Open-loop strategies depend only on calendar time but not on the actions previously played in the dynamic game. Dynamic games where firms can only adopt open-loop strategies are in a way static since firms cannot react optimally to previous plays by rivals. Closed-loop strategies allow this since they take account of previous moves as well. In this setting, the key difference between these two notions depends on whether the first-stage strategic move by firm i is effectively communicated to firm j when the latter decides to act. The direct effect in equation (4.2) refers to the first-order optimality condition in the open-loop strategy case (open-loop equilibrium), whereas the second term (strategic effect) only exists if firms observe their rival’s first-stage strategic moves, that is, if they can devise closed-loop strategies. In Dr. Strangelove, the existence of the Doomsday Machine has no strategic effect since General Ripper was not aware of it; that is, he could not formulate a closed-loop strategy and act optimally (i.e., not attack the Russians) given this information. We come back to the notions of open-loop versus closed-loop strategies (and the corresponding equilibrium concepts) in later chapters. 13. The notion of “overinvestment” (and later of “underinvestment”) is defined with respect to the benchmark case where firms cannot formulate closed-loop strategies, namely where the strategic move is of no commitment value since firm j does not take its (secondstage) action based on the information conveyed by the commitment. The difference between the subgame perfect equilibrium outcome (in closed-loop strategies) and the Nash equilibrium (in open-loop strategies) is captured by the strategic effect. This term fully reflects the incentive to under- or overinvest.

122

Chapter 4

Accommodated Entry If deterring entry is too costly or infeasible, firm i may accommodate entry instead. This friendlier stance still leaves room for strategic behavior because firm i can make an early move that enhances its position in the (ex post) product market competition stage. Contrary to the case of entry deterrence, firm i considers here its own profit function (not its competitor’s) when it devises its business strategy. We thus look at the total derivative of firm i’s own profit (π i) with respect to its investment Ki, dπ i ⎛ ∂π i ⎞ ⎛ ∂π i dα i * ⎞ ⎛ ∂π i dα j * ⎞ =⎜ . ⎟+ ⎟ +⎜ dKi ⎝ ∂Ki ⎠ ⎝ ∂α i dKi ⎠ ⎜⎝ ∂α j dKi ⎟⎠ Since firm i selects action α i as part of a Nash equilibrium in the second stage, ∂π i ∂α i = 0 so the expression above simplifies to ⎛ ∂π dα j * ⎞ dπ i ⎛ ∂π ⎞ . = ⎜ i⎟ + ⎜ i ⎝ ⎠ ⎝ ∂α j dKi ⎟⎠ dKi ∂K i (total (direcct (strategic effect) effect) effect)

(4.3)

The first (right-hand) term in equation (4.3) is the direct effect of the investment on firm i’s own profit, ∂π i ∂Ki . This effect occurs even if the rival does not observe the strategic move. Since sequential strategic interaction has no impact on the direct effect, we set it aside as irrelevant from a strategic viewpoint. The last term, ∂π i ∂α j × dα j * dKi , is the strategic effect. If this is positive, firm i should overinvest to increase its total (gross) value. If this strategic effect is negative, the firm should underinvest instead. Thus the strategic effect provides guidance whether to over- or underinvest. Its sign depends on:14 14. Note that ∂π i dα j ∗ ⎛ ∂π i ∂α j ∗ ⎞ dα i ∗ . = ∂α j dKi ⎜⎝ ∂α j ∂α i ⎟⎠ dKi If the two reaction functions are both downward or both upward sloping, ⎛ ∂ π dα j * ⎞ ⎛ ∂α j * ⎞ ⎛ ∂ π j dα i * ⎞ . × sign ⎜ = sign ⎜ sign ⎜ i ⎝ ∂α i ⎠⎟ ⎝ ∂α i dKi ⎟⎠ ⎝ ∂α j dKi ⎟⎠ The last term, ∂α j * ∂α i , captures the effect of a change in firm i’s action α i on firm j’s best reply. Its sign depends on whether the actions are strategic complements (positive sign) or substitutes (negative). The first right-hand term is the strategic effect in (4.2) for the entrydeterrence case. Ignoring the direct effect, its sign depends on whether the firm faces tough (negative sign) or soft commitment (positive sign).

Market Structure Games: Dynamic

Strategic substitutes

(∂a * j

Tough investment

( dp

j

dKi < 0)

Soft investment

( dp

j

dKi > 0)

∂a i < 0)

123

Strategic complements

(∂a * j

∂a i > 0)

Positive strategic effect

Negative strategic effect

Negative strategic effect

Positive strategic effect

Figure 4.3 Sign of the strategic effect

1. whether firm i’s strategic commitment (Ki) hurts (tough investment) or benefits (soft investment) its rival firm j (see the sign of dπ j dKi ); 2. whether firms react to each other in a reciprocating (strategic complements or ∂D j* ∂D i > 0) or a contrarian (strategic substitutes or ∂D j* ∂D i < 0) manner. This analysis can be represented by the two-by-two matrix of figure 4.3. Consider a different situation involving two incumbents where firm i would like firm j to exit the market. Firm i would design its commitment strategy to hurt firm j, considering the first-order (total) derivative of its rival’s profit, dπ j dKi . This case is equivalent to the entry-deterrence case discussed above. Choosing Commitment Strategies Following our previous discussion, we can differentiate among four main business strategies. These rest on (1) whether the investment is intended to be tough or soft, and (2) whether firms behave in a contrarian (substitutes) or reciprocating (complements) way. The objective of the firm in each of the four strategies is to induce the rival to behave less aggressively. The four main business strategies are summarized in figure 4.4 in case of deterred entry (figure 4.4a) and of accommodated entry (figure 4.4b). The four business strategies are discussed next: Top dog strategy Two cases are distinguished. In the entry-deterrence case (figure 4.4a) this means overinvesting when the investment is tough to ensure that the rival will renounce entering the market (earning zero profit). In the accommodated-entry case (figure 4.4b), it means

124

Chapter 4

Tough investment

Top dog strategy

Soft investment

Lean and hungry look

(a) Strategic substitutes (Cournot quantity competition)

Strategic complements (Bertrand price competition)

Tough investment

Top dog strategy

Puppy dog ploy

Soft investment

Lean and hungry look

Fat cat strategy

(b)

Figure 4.4 Four main business strategies (a) Deterred entry; (b) accommodated entry. Adapted from Fudenberg and Tirole (1984).

overinvesting for tough commitment (dπ j dKi < 0) when second-stage actions are strategic substitutes (∂D j* ∂D i < 0) inducing rivals to back down. Lean and hungry look In the deterred-entry case (figure 4.4a), this means underinvesting to be soft or flexible (dπ j dKi > 0), ensuring that the rival stays out. In the accommodated-entry case (figure 4.4b), this business strategy is advisable if second-stage actions are strategic substitutes (∂D j* ∂D i < 0) and the investment makes firm i soft, cushioning the potentially negative strategic effect (see figure 4.3). Puppy dog ploy In the accommodated-entry case (figure 4.4b), if the investment makes firm i tough, hurting its rival (dπ j dKi < 0), the rival will be more aggressive in the second stage as actions are strategic complements (∂D j* ∂D i > 0). Firm i should then underinvest (stay flexible). Fat cat strategy In the accommodated-entry case (figure 4.4b), if second-stage actions are reciprocating or strategic complements

Market Structure Games: Dynamic

125

Strategic substitutes

Strategic complements

Tough investment

Submissive underdog

Mad dog

Soft investment

Suicidal Siberian

Weak kitten

Figure 4.5 Suboptimal business strategies observed in practice (accommodated entry) Adapted from Besanko et al. (2004)

(∂D j* ∂D i > 0), firm i should avoid being aggressive (dπ j dKi < 0), behaving as a “fat cat.” In the real world some companies sometimes follow suboptimal strategies. Besanko et al. (2004) extend the taxonomy above to include additional business strategies that have a harmful effect on the committing firm.15 They identify the following strategies, summarized in figure 4.5: Submissive underdog The firm underinvests to accommodate entry when its commitment is tough (dS j dKi < 0) and actions are strategic substitutes (∂D j* ∂D i < 0). This strategy is not advisable; the firm should be a “top dog” instead (overinvest). Suicidal Siberian Here the firm overinvests in case of strategic substitutes (∂D j* ∂D i < 0) even though its investment makes it soft (dπ j dKi > 0), instead of underinvesting to keep a “lean and hungry look.” Mad dog The firm overinvests in case of tough commitment (dπ j dKi < 0) even though actions are strategic complements (∂D j* ∂D i > 0). It would be better to be a clever, harmless “puppy dog,” avoiding the negative effect depicted in figure 4.3 (underinvestment). Weak kitten Here the firm underinvests in case of soft commitment (dπ j dKi > 0) when actions are strategic complements (∂α j ∂α i > 0). The “weak kitten” is not mature enough to recognize that it should put on more weight (overinvest), becoming a “fat cat.” 15. See Besanko et al. (2004, pp. 246–47).

126

Chapter 4

Commitment in Differentiated Bertrand and Cournot Competition Bertrand and Cournot competition are examples of games where actions are strategic complements and strategic substitutes, respectively. We illustrate next application of the four main business strategies in the context of these benchmark models. Bertrand Price Competition Differentiated Bertrand price competition involves reciprocating actions (strategic complements). If firm i decreases its price, firm j will respond by following suit, and vice versa. Consider firm i’s reaction function from equation (3.8): piB ( pj ) =

a (1 − s ) + c s + pj , 2 2

where s ∈[0, 1) is the degree of substitutability. We thus have ∂piB ∂pj ( = s 2 ) ≥ 0, confirming that price choices are strategic complements. Should firm i invest early in a new technology to attain reduced future production costs? Firm i could then commit to a price cut, being more aggressive toward its rival in the later product market stage. This tough commitment by firm i is detrimental to rival firm j (as dπ j dKi < 0). If firm i decides to charge a lower price in the second stage, the marketclearing price will spiral downward since the reaction functions are upward sloping. For a given level of the rival’s price pj , firm i’s price pi will be lower, with the reaction curve shifting to the left as depicted in figure 4.6a.16 Firm j is worse off ex post because it is obliged to reduce its own price, following a reciprocating reaction. This strategy is not advisable because the strategic move by firm i will backfire as the strategic effect (represented in the top-right in figure 4.3) is negative. Firm i should instead underinvest, maintaining higher prices, to avoid entering an intensified price war. The optimal business strategy here is to be a nice “puppy dog.” In contrast, if investment makes firm i soft (dπ j dKi > 0), the rival will be less aggressive in the second stage. For a given level of competitor price pj , firm i optimally charges a higher price pi as a result of this 16. The case before (without) investment is indexed “before,” whereas the case after investment commitment as “after.” The case “before” corresponds to the open-loop equilibrium where product-market decisions are taken in ignorance of the rival’s strategic move, whereas the case “after” refers to closed-loop equilibrium where firms know that their commitment is recognized in their rivals’ strategy formulation and firms maximize profits accordingly (subgame perfection).

Market Structure Games: Dynamic

127

pj RiBEFORE(pj) RiAFTER(pj)

pjAFTER *

Rj (pi)

E

pjBEFORE * E'

piAFTER *

piBEFORE *

pi

(a)

pj RiBEFORE(pj) RiBEFORE(pj)

E'

pjAFTER * pjBEFORE *

Rj (pi)

E

piBEFORE *

piAFTER *

(b) Figure 4.6 Tough versus soft commitment in differentiated Bertrand competition (a) Tough commitment (puppy dog); (b) soft commitment (fat cat srategy)

pi

128

Chapter 4

commitment with the reaction function shifting to the right. The original price, pj , however, is not on the reaction curve. To reach the Nash equilibrium, firm j charges a higher price than before. The two firms are better off due to this accommodating strategy. In this case firm i should overinvest becoming fatter (fat cat), amplifying the positive strategic effect (bottom-right in figure 4.3) as much as possible. This is illustrated in figure 4.6b. An application of “soft” advertising strategy to the German telecom market is discussed in box 4.2. Cournot Quantity Competition To illustrate the importance of tough versus soft commitment in case of strategic substitutes or contrarian reactions, consider a tough investment commitment such as an R&D investment in process innovation. This would enhance firm i’s cost advantage in the Cournot duopoly setting (section 3.2.2). By contrast, a soft commitment would worsen the cost differential in favor of the rival, firm j; this would be the case, for instance, if the firm announces the building up of new, fancy headquarters. The (inverse) market demand function is linear as per equation (3.1). The cost functions are also linear. We allow for distinct marginal production costs, ci and c j (ci , c j < a). As seen from equation (3.13), firm i’s reaction function in Cournot duopoly is qiC (q j ) =

1 ⎛ a − ci − q j ⎞⎟ . ⎜ 2⎝ b ⎠

(4.4)

Firm i’s reaction function qiC (⋅) is differentiable and decreasing in its own cost (∂qiC ∂ci = − 1 2b < 0), implying that for a cost increase (soft commitment) the reaction curve shifts to the left, whereas for a cost decrease (tough commitment) it shifts to the right. As seen in figure 4.7a, if firm i makes a tough commitment (curve shifts to the right), then no matter what output its rival produces it will produce more output (than it would without the commitment). Since quantities are strategic substitutes, firm j behaves in a “contrarian” way, producing less in equilibrium than it would have if this investment did not occur. Ex post profit is higher for firm i if it makes a tough commitment ex ante (without accounting for the cost involved in the first-stage investment). This investment has a positive strategic effect and firm i should overinvest as in a “top dog” strategy. By contrast, a soft commitment would lead firm i to produce less than it would otherwise. The reaction curve now shifts to the left and firm i produces less after the commitment, while firm j produces more. Firm

Market Structure Games: Dynamic

129

Box 4.2 European liberalization, “soft advertising” and the persistence of high prices in the German telecom market

Fudenberg and Tirole’s (1984) framework can provide powerful insights. Their framework is particularly useful to help explain how an industry evolves once liberalization of the market is enforced. In most European countries certain sectors (e.g., railway, electricity, post and telecommunications) have long been considered as natural monopolies. Governments have erected high entry barriers (blockaded entry), believing that competitive entry would be socially detrimental. Presumably there were not enough room for several firms in these markets to make positive profits or at least high enough profits to pay for the large fixed infrastructure costs. In the last two decades a revolution has occurred in Europe. Policy makers decided to enforce liberalization of these markets, believing that enhanced competition would result in lower prices and be more beneficial to European consumers. A case in point is the telephone industry in Germany where Deutsche Telekom, once a formidable monopolist, faces competitive entry from rival companies such as Vodafone, Telecom Italia (Alice), and other regional companies. Deutsche Telekom has been one of the world’s leading telecommunication and information-technology service providers, with an annual turnover of )60 bn a year. Although liberalization of the telecommunication sector has taken place many years ago, the German Federal Network Agency (“Bundesnetzagentur”) must still wonder why no significant price decreases and market share shifts have occurred in Germany up to now. This may be partly due to Deutsche Telekom’s heavy investment in advertising campaigns. Every year Deutsche Telekom channels nearly 15 percent of its revenues into ads. The telecommunication industry is typical of Bertrand price competition. Firms face no serious capacity constraints and the key purchasing criterion for many customers is the price. As it is difficult to differentiate among offerings in the telecommunication industry (with the exception of bundling for the iPhone), the advertising campaign of Deutsche Telekom was not meant to build a differentiation advantage. It rather promoted telecommunication services generally. As such, its ad campaigns have been beneficial to rivals as well. The massive general advertising investment made by Deutsche Telekom is soft (benefits its rivals) and rivals have been less aggressive in second-stage competition. Consequently, despite liberalization, prices for telecommunication services have not decreased significantly in Germany. Market shares have not declined dramatically either since customers do not find it worthwhile to incur switching costs if alternative offerings are not sufficiently better or cheaper. Still, even though competition is not yet that fierce, Deutsche Telekom has not taken full benefit since its advertising campaigns have been quite expensive.

130

Chapter 4

qj

RiBEFORE(qj) RiAFTER(qj)

qjBEFORE *

E

E'

qjAFTER *

Rj (qi)

q iBEFORE *

qiAFTER *

qi

(a)

qj RiAFTER(qj) RiBEFORE(qj)

qjAFTER *

E'

E

qjBEFORE *

Rj (qi)

qiAFTER *

qiBEFORE *

qi

(b) Figure 4.7 Tough versus soft commitment in Cournot quantity competition (a) Tough commitment (top dog strategy); (b) soft commitment (lean and hungry look strategy)

Market Structure Games: Dynamic

131

i’s investment is beneficial to firm j who is more aggressive in the product market stage. Firm i should refrain from committing and should thus underinvest. This “lean and hungry look” strategy is depicted in figure 4.7b. This is what the strategic effect in the case of tough versus soft commitment for strategic substitutes is about. There is a positive strategic effect when its strategic investment makes firm i tough. This may dominate even though the direct effect might be negative (e.g., due to sunk costs). When its investment makes the first-mover soft, there is a negative strategic effect and firm i should refrain from making this kind of investment. Box 4.3 describes entry accommodation in the Italian electricity market, previously dominated by state monopoly Enel. Overinvestment in capacity may also result in an alternative model, also famous in industrial organization. It is the duopoly model developed by Stackelberg (1934) where one of the firms makes an early move selecting its quantity (i.e., capacity) first, while the second firm selects its own quantity in view of the leader’s quantity choice. The following section develops the sequential model by Stackelberg and discusses in which circumstances it might be useful. 4.1.3

Sequential Stackelberg Game

Heinrich von Stackelberg (1934) proposed a dynamic duopoly model in which a firm (the leader L) moves first (for whatever reason) and its rival (the follower F ) follows suit. A key assumption is that the follower observes the leader’s choice before selecting its own output. The profit and cost functions are the same as used in Cournot quantity competition (section 3.2.2). The (inverse) demand function is linear as given by equation (3.1). Cost functions are also linear, allowing for distinct marginal costs cL and cF . In the second period, the follower chooses output qF (≥ 0) to maximize its own profit, given the leader’s observed quantity choice qL (≥ 0). The leader has no possibility to revise its production plan and installed capacity ex post (i.e., it is irrevocably committed to its announced capacity decision, qL). The subgame perfect Nash equilibrium of this sequential, finite-horizon game can be obtained by backward induction. The follower’s optimization problem is to select its output qF (≥ 0) to maximize its profit:17

π F ( qL , qF ) = [ p (Q) − cF ] qF . 17. The follower’s profit function is differentiable and concave in its own strategic variable, qF .

132

Chapter 4

Box 4.3 Entry accommodation in the Italian electricity market

As electric utilities are subject to serious capacity constraints in offering a homogeneous, nonstorable product, they effectively compete over capacity as in Cournot competition. Firms first decide on their installed capacity base (a long-term decision) and then compete in the short run in prices under capacity constraints. The Italian electric utility Enel is recently facing new challenges, having been a protected monopolist in the Italian marketplace for decades. Since July 2007 the Italian electricity market has opened up to foreign utilities in the wake of the European Commission enforcing liberalization across European electricity markets. In the current environment, traditional frameworks (e.g., SWOT, five-forces analysis, or generic strategy framework) offer little guidance for Enel executives about how to deal with the issues they face under uncertainty because they lack the dynamic perspective necessary to understand market-entry problems. Fudenberg and Tirole’s (1984) framework of business strategies can be helpful. The Italian electricity industry was previously blockaded due to high administrative and other entry barriers. Now Enel cannot rely on the Italian government to protect its home market from competitive forces since the government has to toe the line dictated by the European Commission. Even if Enel manages to deter competitive entry (deterred entry), DG Competition, the European body responsible for implementing and monitoring adherence to European antitrust law, may intervene. Enel has one main choice left, to accommodate entry. Of course, Enel may accommodate entry but still behave in a way that makes entry less profitable for potential entrants. In this industry, capacities are strategic substitutes in that if one firm adds capacity, the optimal reaction of rivals is not to raise capacities or revise their construction plans to accommodate less capacity. Being the sole electricity provider in the Italian market (until July 2007), Enel had the opportunity to make a “first-stage” strategic move to alter the “second-stage” competition game. Given that investing in new capacity was perceived as a tough initiative in this setting, Enel had the opportunity to “over-invest” in capacity over the years to discourage potential entrants from building new production units. Accommodation was likely a preferred strategy.

Market Structure Games: Dynamic

133

The follower’s reaction function (where superscript S stands for Stackelberg) is 1 a − cF qFS ( qL ) = ⎛ − qL ⎞ . ⎠ 2⎝ b The leader L knows that the follower will choose its output once it observes its own and will infer the follower’s reaction before determining its own optimal output. The leader will thus choose its quantity qL so as to maximize its profit:

π L(qL , qFS ( qL )) = [ p (qL + qFS (qL )) − cL ] qL. The resulting Stackelberg equilibrium quantities for the leader (L) and the follower (F ) are given by ⎧qS = a − 2cL + cF , ⎪ L 2b ⎨ ⎪qFS = a + 2cL − 3cF . ⎩ 4b

(4.5)

The Stackelberg equilibrium price, obtained by substituting the equilibrium quantity above in (3.1), is pS =

a + 2 c L + cF . 4

(4.6)

The profits for the leader (L) and the follower (F ) in subgame perfect equilibrium are ⎧ S ( a − 2cL + cF )2 , ⎪⎪π L = 8b ⎨ 2 ⎪π S = ( a + 2cL − 3cF ) . F ⎪⎩ 16b

(4.7)

In case of cost symmetry between the leader and the follower (cL = cF = c), the above equilibrium profit expressions simplify to ⎧ S ( a − c )2 ⎪⎪S L = 8b ⎨ 2 a 1 ⎪ S ( − c) = = S LS S ⎪⎩ F 16b 2

(> S C ), (< S ),

(4.8)

C

where π C = ( a − c ) 9b is the Cournot profit under cost symmetry given in equation (3.17). 2

134

Chapter 4

The leader in the above sequential Stackelberg game is better off than a symmetric Cournot duopolist (S LS > S C). The Stackelberg follower receives a lower profit than the single-period Cournot duopolist (S FS < S C ). Moreover the aggregate (total) quantity produced in the sequential symmetric Stackelberg game (3 4 × (a − c ) b) is higher than the industry output in symmetric Cournot duopoly (2 3 × (a − c ) b). Therefore, given the downward-sloping demand, the market-clearing price is lower in the sequential Stackelberg game. The leader could defer its output decision to the second period but is better off not to do so. The profit value of being a leader in the sequential Stackelberg game is higher than in the simultaneous wait-and-see case, even if the price is lower. Here the leader is not acting optimally from the second-stage static (Nash-equilibrium) perspective since its output is not a best response to the rival’s quantity in the second stage. From a dynamic perspective, the leader is actually better off, thanks to its commitment to a certain output. By contrast, the Stackelberg follower produces less than under symmetric Cournot competition. Since the market price is lower, the Stackelberg follower makes lower profits than in the Cournot case. The sequential Stackelberg model provides an example of first-mover advantage. In the sequential Stackelberg setting, the follower might have been better off ignoring the quantity chosen by the leader. Although the leader is better off communicating its output decision to the follower, it should be careful not to lose credibility because if the follower suspects “cheap talk,” it will choose its quantity as if no communication occurred, namely à la Cournot. The sequential Stackelberg game should thus be interpreted in light of commitment theory: the leader should follow a “top dog” strategy and overinvest in capacity, with investment costs being fully sunk. The sequential Stackelberg model is also useful to help assess the importance of information in multiperson games. In Stackelberg competition, the firm that knows its rival’s quantity decision (the follower) is worse off than a firm that ignores this information (Cournot case). Conversely, when one gains a competitive advantage by deciding first (e.g., regarding capacity choice or market entry), the less-informed party (i.e., the leader) is not necessarily worse off. This may justify an early commitment as a sound strategy for a firm even if not all information concerning the market development is known or predictable. But why should one firm have the possibility to commit and not the other? We discuss in chapter 12 the timely interplay that occurs when firms compete over early commitment (e.g., in the case of market entry).

Market Structure Games: Dynamic

4.2

135

Bargaining and Cooperation

So far we have discussed models where firms take a noncooperative stance toward their rivals, ignoring the possibility of more complex or repeated relationships based on both competition and collaboration. In reality, however, fierce competition is not the only modus vivendi adopted by firms in the marketplace. Firms often bargain or cooperate with other parties, including competitors. Antitrust authorities generally prohibit formal collaboration (explicit collusion) but may tolerate other forms (e.g., tacit collusion). Such cooperation between rival firms is not necessarily detrimental to customers. For example, customers may find it advantageous to be faced with only one technology standard owing to product externalities. An agreement on common technological norms (e.g., Blue ray) may be beneficial both to the cooperating firms and to the end consumers. Real-world competition calls for analytic models that can better explain this kind of complex strategic interactions observed in the marketplace. We discuss in turn bargaining and (tacit) collusion in repeated games.18 4.2.1

Bargaining

Bargaining problems are treated as dynamic games under complete or incomplete information. They are solved using solution concepts suited to a multistage setting (e.g., subgame perfection or perfect Bayesian equilibrium). Here we use subgame perfect equilibrium strategy profiles under perfect information. When reasoning backward, one needs to consider the time value of money (discounting) and the trade-off between reaching an agreement today or continuing bargaining. The “patience” of the bargaining parties is thus a key factor in negotiations. The bargaining process can be limited in time (finite horizon) or continue indefinitely until parties reach an agreement (infinite horizon). Consider first a simple example of bargaining to help identify the key drivers. A typical problem is how to split a pie among players. Ståhl (1972) discusses such a model for finite horizon, and Rubinstein (1982) extends it to infinite horizon.19 Suppose that two firms “bargain” over market shares in the marketplace. One of the two firms (the bargaining 18. We discuss bargaining and tacit collusion in sequence owing to some similar economic interpretations. We do not mean to suggest that the theory of repeated games is built upon bargaining games. In fact sustainability of cooperative behavior in infinitely repeated games can be established directly by using so-called grim-trigger strategies. 19. We here follow the treatment by Shaked and Sutton (1984) and Gibbons (1992).

136

Chapter 4

Nature

Leader

Follower

Leader

Follower

leader L) is allowed first to make a proposal to the follower (e.g., due to its technological edge or market power). Subsequently the follower (F ) may make a proposal as roles are reversed. The slice (as a percentage of the total market value) that accrues to the leader (L) is denoted by si where subscript i stands for firm i that makes the proposal; the follower (F ) receives the remaining slice, 1 − si. In the first period, the leader (L) proposes the following deal to the rival: the leader takes sL (%) of the pie (0 ≤ sL ≤ 1), with the follower (F ) receiving 1 − sL. The follower may either accept the proposed deal or refuse it. If F rejects the proposal, it may in turn make a new offer: the leader takes sF and the follower 1 − sF . The leader may accept it or not. If no agreement is reached after two negotiation rounds, an exogenously given settlement is applied: the leader receives s (%) and the follower 1 − s (0 ≤ s ≤ 1). The extensive form of this sequential bargaining game is depicted in figure 4.8. Over the bargaining process, firms have a certain degree of “impatience” captured by a common discounting factor δ (0 ≤ δ < 1). Note that δ ≡ 1 (1 + k ), where k (k > 0) is the appropriate discount rate. Discounting future payoffs at a higher discount rate gives the players an incentive to reach an agreement earlier. We can conjecture that the equilibrium slicing is

t jec

( s, 1 − s )

re

sF

t

Decision maker

(sF , 1 − sF )

accep

sL

pt acce

t ec rej

( sL , 1 − s L )

Leader

Follower

Follower

Leader

(Nature)

Decision time

1

2

3

4

5

Real time

1

2

3

Figure 4.8 Extensive form of the bargaining game under complete information Here ( si , 1 − si ) are the payoffs to the leader and the follower, respectively. Two time references are considered: the order of the play (decision time) and the real time at which agreements (or settlement) may occur. Discounting relates to the second dimension.

Market Structure Games: Dynamic

137

affected by the magnitude of the discount rate since a later ( 1) settlement has lower present value. In such a sequential game with perfect information and a finite horizon, the subgame perfect equilibrium is found backward. At decision time 4, the leader may either accept the follower’s proposal, receiving sF today, or reject it, receiving the exogenous amount s in the last period. This latter settlement is worth δ s today (at real time 2). The leader will accept the offer sF if sF ≥ δ s.20 What is the follower’s best continuation strategy from decision time 3 onward, when the leader acts optimally in the subsequent period? The follower (F ) faces this decision if it rejects the offer made by the (bargaining) leader previously. The follower may construct the offer (at decision time 3) in two alternative ways: Deal A The leader accepts the offer. In this case, the follower is better off offering the minimum acceptable level for the leader, namely sF * = δ s, instead of any higher share. The follower would then receive a slice of 1 − sF * = 1 − δ s (at real time 2). Deal B The leader rejects the offer. In this case, the follower receives 1 − s at the last stage, worth δ (1 − s ) today (at real time 2). The leader receives δ s (at real time 2). When selecting between the two alternative deals, the follower considers the relative value of 1 − δ s (deal A) versus δ (1 − s ) (deal B). For 0 ≤ δ < 1, we have 1 − δ s > δ (1 − s ). The optimal deal at decision time 3 is deal A, where follower F proposes δ s to its rival who accepts it. The follower would then earn 1 − δ s. One step earlier (at decision time 1), the leader may design its proposal in a similar manner: Deal C The follower accepts the offer. F then receives 1 − sL. Deal D The follower refuses the offer. The follower then proposes a new settlement. As determined above, L would then receive δ s at real time 2, which is worth δ 2 s at real time 1. F receives the present (real time 1) value of 1 − δ s, meaning δ(1 − δ s ). The follower will not accept the offer unless it receives a higher value than otherwise. In other words, firm F accepts if 1 − sL ≥ δ (1 − δ s ). The leader would optimally retain for itself the maximum acceptable amount, 20. We assume that if a firm is indifferent between the present values of the two different deals, it will select the proposal made first. This assumption enhances the model readability but it can be relaxed.

138

Chapter 4

which is sL * ≡ 1 − δ (1 − δ s ) = 1 − δ + δ 2 s (≥ δ 2 s ). Both parties will thus accept the equilibrium pie sharing ( sL*, 1 − sL*) at the outset. The three-period model above is rather simplistic and not descriptive of many real-world bargaining situations. In reality, although decision times are discrete, firms sometimes bargain over a longer period until an agreement is reached. In such a setting one can truncate the infinitehorizon problem, leveraging on the earlier three-period setup. Previously the final settlement ( s, 1 − s ) required the presence of a third party or arbiter in the bargaining process. Instead of ( s, 1 − s ) being exogenously given, suppose that this pie-sharing rule results from another similar three-period bargaining game. As a result in perfect equilibrium the bargaining leader’s payoff satisfies s* = 1 − δ + δ 2 s *; hence s* = 1 (1 + δ ). In the subgame perfect Nash equilibrium for the infinitely repeated game, the split is

( s*, 1 − s *) = ⎛⎝

δ ⎞ 1 , . 1+ δ 1+ δ ⎠

(4.9)

This equilibrium result is related to the firms’ “patience” via the discount factor δ . The bargaining leader is better off receiving a higher share than its rival.21 A case in point is negotiating a short-sale agreement over a real-estate property. Suppose that newcomers have decided to acquire a cozy apartment in the city center but stay in a hotel until they find the appropriate opportunity. Staying in a hotel is costly and increases their opportunity cost of waiting, resulting in a lower δ . They naturally prefer to find their ideal apartment early on and avoid paying out rents for a prolonged time. In bargaining for the apartment, they will likely accept to pay a premium due to their higher opportunity cost and impatience. The owner of the apartment may take advantage of the situation and sell the property at a higher price. 4.2.2

Cooperation between Cournot Duopolists in Repeated Games

We analyzed in section 3.2.2 the classic case where Cournot duopolists choose their quantity simultaneously in a noncooperative manner. The simultaneous Cournot competition model rests on the premise that firms maximize their short-term profit. This leads to a situation where even 21. If the follower suffers from delay, it is more likely to accept the deal offered by the leader earlier as it is costly to wait longer for a more lucrative deal. The leader can take advantage of this, proposing a less attractive deal. This first-mover advantage increases with higher risk (higher discount rate or lower discount factor). Moreover the values obtained in equilibrium by the parties decrease with the degree of impatience δ .

Market Structure Games: Dynamic

139

though the monopoly outcome would have been preferable for both firms considered individually, it is never reached as a stable industry equilibrium (prisoner’s dilemma). This result is based on the debatable assumption that firms formulate their strategies at the same decision time regardless of past history and ignoring the long-term impact of today’s decisions. In real life, firms frequently compete with one another over an extended period and are faced with recurring competitive situations over multiple stages. This kind of strategy setting may be captured by repeated games or supergames.22 Box 4.4 provides a real-world example of collusion in the German retailing market. Here we sketch a refinement of the Cournot model to examine how repeated strategic interactions may alter the equilibrium play.23 We again consider duopolist firms facing linear demand as in equation (3.1) and symmetric production cost, c. The Nash equilibrium for a duopolist in the standard symmetric Cournot setup, obtained in equation (3.14), is qC = ( a − c ) 3b, with total industry output being QC = 2 (a − c ) 3b. This industry output in Cournot duopoly is higher than in monopoly (QC ≥ QM). The price, however, is lower. Clearly, the two firms would be better off setting jointly their aggregate production equal to the monopoly quantity (QM). However, in the standard Cournot model, each player has an incentive to deviate from the collusive quantities (QM 2 , QM 2). Eventually Nash equilibrium quantities (qC , qC ) that are not jointly optimal are chosen. In infinitely repeated games, however, the desirable cooperative equilibrium (QM 2 , QM 2) may be sustainable under certain conditions. Friedman (1971) considers such a duopoly game where the market is assumed in a steady state.24 In each period, each duopolist makes a tactical decision on the quantity to supply to the market. In this repeated game, duopolists end up being better off if they behave as follows: 22. In simultaneous games, firm actions maximize short-term profits. This mechanism explains why the collectively optimal outcome is not necessarily reached as a (stable) Nash equilibrium. In multistage games, a firm may have an incentive to cooperate if a (tacit) agreement may create a favorable platform for future higher profits. Even though a firm would earn a higher short-term profit by deviating from the collusive agreement, it will not be enticed to do so if it values future collaborative profits more. The long-term benefits gained from a tacit agreement may offset the short-term gains from behaving selfishly. Infinitely repeated games can result in collaborative behaviors being sustainable in situations where sufficiently patient firms act in their own interest. 23. We do not intend to give an advanced account of repeated games here. For a comprehensive treatment of repeated games, reputations, and long-run relationships, see Maileth and Samuelson (2006). A good treatment of repeated games is also given in Fudenberg and Tirole (1991, ch. 5). 24. Growth is here assumed zero, so there is no need to account for the interplay between growth and discounting.

140

Chapter 4

Box 4.4 Suspected collusion in the German retailing market

German Antitrust Authorities Investigate Mass Retailers S. Amann and F. Ott, Spiegel Online German antitrust officials are currently probing some of the country’s largest supermarkets and retailers over suspected cases of price fixing on chocolate—both at the production and resale levels. Investigators believe as many as 24 companies set illegal minimum prices on products. On Thursday morning, German antitrust investigators launched a surprise search of the offices of some of the biggest names in the country’s retail sector. A total of 56 German Cartel Office employees and 62 police officers visited the offices of retail giant Metro, Germany’s largest supermarket chains, including Edeka, Rewe, and Lidl, as well as the drug store chain Rossmann and pet supply store Fressnapf. Authorities also visited the offices of chocolate bar-maker Mars. Investigators have evidence that a total of 24 companies may have illegally agreed to price-fixing deals on coffee, sweets, and pet food. They believe millions of consumers may have paid more for these goods than they should have. For industry observers, however, the raids were less surprising. Late last year, the Federal Cartel Office imposed )160 million ($233 million) in fines against coffee producers Melitta, Dallmayr, and Tchibo for entering into a secret agreement to coordinate price increases. American food conglomerate Kraft was also involved but avoided penalty because it turned itself in and cooperated with the competition authority during the investigation. Both Tchibo and Melitta have contested the fines, and the case is expected to go before the higher regional court in Düsseldorf. . . . But it is also clear that the agency has been investigating some of the larger players in the field over other products for some time now. Industry expert Roeb says it’s not surprising that investigations are now focused on makers of sweets following the coffee investigations. “Certain product groups are particularly prone to price fixing,” he says. There are high fluctuations in commodity prices, there’s a relatively small number of countries that can produce the commodities and the number of producers is limited. “That makes the market clear and manageable,” he says. Attractive for All Involved What’s new in the current investigation, however, is the suspicion that both producers and retailers may have agreed to price fixing. “The vertical price fixing is unique in this case,” Cartel Office spokesman Kay Weidner said. Companies are allowed to set recommended prices, but it is illegal to agree

Market Structure Games: Dynamic

141

Box 4.4 (continued)

to concrete prices. Antitrust investigators believe the 24 companies came to a deal on a minimum price for products. If the allegations are true, it would mean that both traders and companies adhered to that agreement, thus violating German competition law. Such agreements are, of course, attractive for all involved as they guarantee both producers and retailers reliable prices and higher profits. Such arrangements are additionally attractive for retailers in that they can boost the sales of their own brands. “Merchants can price their own brands below the agreed-upon prices for name brands without the fear that the name brand prices will follow,” said one branch insider . . .. Those familiar with the market are convinced that the Cartel Office wanted to set an example. “With today’s investigation, we wanted to make it clear that vertical price fixings are also in violation of antitrust law,” said Cartel Office spokesman Weidner. A 2000 law allowing for those involved in price fixing to turn state’s evidence and thus avoid penalty—a regulation recently taken advantage of by Kraft Foods—is giving antitrust regulators a boost. One measure of the Cartel Office’s recent success is the value of fines it has levied from year to year. In 2006, the antitrust authorities collected just )2.5 million in fines. In 2007, however, that number exploded to )114 million before almost tripling in 2008 to )317 million. In 2009, following the penalties handed down to the coffee producers, the Cartel Office brought in over )400 million. Such statistics are good news for those companies not involved in price fixing. And for consumers. According to German consumer protection groups, coffee drinkers in Germany paid a total of )4.8 billion too much for their morning jolt of caffeine from 2000, when the coffee producers began colluding, until the three companies were first searched in July 2008. Price Fixing Probe: German Antitrust Authorities Investigate Mass Retailers by Susanne Amann and Friederike Ott reprinted with permission from SPIEGEL Online. Publication date: January 15, 2010.

1. Agree to produce half the monopoly quantity if the other player also produces half.25 2. Deviate from the (tacit) agreement and produce the Cournot–Nash quantity forever if the other player deviates. This aggressive stance is the (harsh) punishment from having deviated in the first place. 25. This assumes that (QM 2 , QM 2 ) is already the industry state at the outset.

142

Chapter 4

Such behavior relates to tit-for-tat or trigger strategies. Such strategies are also observed in nature, as discussed in box 4.5. This strategic stance is a translation of the biblical saying “an eye for an eye, a tooth for a tooth.” It supposes that fair behavior by one player induces fairness by the other, and vice versa. Under monopoly, the equilibrium quantity 2 is QM = ( a − c ) 2b and profit equals π M = (a − c ) 4b. In the standard 2 Cournot duopoly game, firms earn a profit π C = ( a − c ) 9 b in equilibrium. If firms behave cooperatively, such as by following a tit-for-tat 2 strategy, they each produce QM 2 = (a − c ) 4b, earning π M 2 = (a − c ) 8 b at each stage. If firm i sticks to the tacit agreement and produces QM 2 while firm j deviates, the optimal quantity for firm j obtains by solving QM ⎞ ⎡ ⎛ QM ⎤ ⎛ ⎞ max ⎢ p ⎜ + q j ⎟ − c ⎥ q j = max ⎜ a − bq j − b − c⎟ q j . q q j ≥0 ⎣ ⎝ 2 ≥ 0 ⎠ ⎝ ⎠ j 2 ⎦ When deviating, firm j would optimally produce q D = q j* = 3 (a − c ) 8b, 2 earning profit π D = 9 (a − c ) 64b. In the aftermath, however, firm j would compete forever à la Cournot, receiving Cournot duopoly profit π C each period. Since π D > π M 2 > π C , firm j has an incentive to deviate in the short run. However, it must also assess the long-term negative effect of this deviation. We next analyze this trade-off. The value of cooperating forever is ∞ ⎛ πM ⎞ πM C ≡ ∑δ t ⎜ = ⎝ 2 ⎟⎠ 2 t =0

∞

∑ ⎛⎝ 1 + k ⎞⎠ 1

t =0

t

=

1 ⎛ πM ⎞ . 1 − δ ⎜⎝ 2 ⎟⎠

Alternatively, if the firm deviates this period and operates in Cournot duopoly forever thereafter, it receives the present value: ∞

∞

t

δ ⎞ C 1 ⎞ = πD + ⎛ D ≡ π D + ∑ δ tπ C = π D + π C ∑ ⎛ π . ⎝ ⎝ ⎠ − k δ⎠ + 1 1 t =1 t =1 For tacit cooperation to sustain itself in each period, the following condition must hold: C > D, or M ⎛ 1 ⎞ π > π D + ⎛ δ ⎞ πC. ⎝ 1−δ ⎠ ⎝ 1−δ ⎠ 2

Considering the given profit values π C , π M, and π D, the condition above holds if δ > 9 17 or k < 8 9. So once again we confirm that the degree of impatience or the opportunity cost of waiting (which is exogenous to the firms’ decisions) affects critically the sustainability of cooperative behavior. In industries characterized by high riskiness (high k

Market Structure Games: Dynamic

143

Box 4.5 Repeated prisoner’s dilemma and tit for tat in nature

Tit for Tat Chris Meredith, Australian Broadcasting Co. Long before humans started playing games, natural selection discovered the fundamentals of game theory and shaped animal societies according to its rules. Within species, individuals adopt alternative competing strategies with frequencies that reflect the success of each strategy . . .. However, cooperation within and between species has generated only one strategy, tit for tat . . . You Scratch My Back . . . Evolutionary biologists have had considerable trouble explaining the evolution of cooperative behavior. The problem is that cooperation can always be exploited by selfish individuals who cheat. It seems that natural selection should always favor the cheats over the cooperators. Co-operation involves doing and receiving favors, and this means that the opportunity to cheat and not return a favor is a very real possibility. Trivers (1971) tackled this problem and developed the theory of reciprocal altruism based on the idea that cooperation could evolve in species clever enough to discriminate between cooperators and cheats. The concept is summarized in the saying “you scratch my back and I’ll scratch yours.” Trivers’s theory of reciprocal altruism is particularly successful in explaining human behavior because reciprocal altruism is a major part of all human activities . . .. [it] was an important advance in our understanding of the evolution of cooperation, but it was a “special theory” rather than a “general theory.” The discovery of how cooperative behavior could evolve in species far less intelligent than humans came in a surprising way—from a detailed study of the well-known paradox “the prisoner’s dilemma.” The Prisoner’s Dilemma The prisoner’s dilemma refers to an imaginary situation in which two individuals are imprisoned and are accused of having cooperated to perform some crime. The two prisoners are held separately, and attempts are made to induce each one to implicate the other. If neither one does, both are set free. This is the cooperative strategy available to both prisoners. In order to tempt one or both to defect, each is told that a confession implicating the other will lead to his or her release and, as an added incentive, to a small reward. If both confess, each one is imprisoned. But if one individual implicated the other, and not vice versa, then the implicated partner receives a harsher sentence than if each had implicated the other. The prisoner’s dilemma is that if they both think rationally, then each one will decide that the best course of action is to implicate the other although they would both be better off trusting each other. Consider how

144

Chapter 4

Box 4.5 (continued)

one prisoner thinks. If his partner fails to implicate him, then he should implicate his partner and get the best possible payoff. If his partner has implicated him, he should still “cheat”—since he suffers less than if he trusts his partner. However, the situation is more complicated than this analysis suggests. It is fairly obvious that the players’ strategic decisions will also depend on their likelihood of future encounters. If they know that they are destined never to meet again, defection is the only rational choice. Both individuals will cheat and both will end up relatively badly off. But if the prisoner’s dilemma is repeated a number of times, then it may be advantageous to cooperate on the early moves and cheat only toward the end of the game. When people know the total number of games of prisoner’s dilemma, they do indeed cheat more often in the final games. Robert Axelrod was interested in finding a winning strategy for repeated prisoner’s dilemma games. He conducted a computer tournament. The result of the tournament was that the simplest of all strategies submitted attained the highest average score. This strategy, called tit for tat, had only two rules. On the first move cooperate. On each succeeding move do what your opponent did the previous move. Thus tit for tat was a strategy of cooperation based on reciprocity . . .. Tit for Tat These results provide a model for the evolution of cooperative behavior. At first sight it might seem that the model is relevant only to higher animals that can distinguish between their various opponents. If so, tit for tat would simply be Trivers’s theory of reciprocal altruism restated. But tit for tat is more than this and can be applied to animals that cannot recognize each other—as long as each individual starts cooperative encounters with very minor, low-cost moves and gradually escalates as reciprocation occurs . . .. Four features of tit for tat emerged: 1. Never be the first to defect. 2. Retaliate only after your partner has defected. 3. Be prepared to forgive after carrying out just one act of retaliation. 4. Adopt this strategy only if the probability of meeting the same player again exceeds 2/3. According to Axelrod, tit for tat is successful because it is “nice,” “provokable,” and “forgiving.” A nice strategy is one that is never first to defect. In a match between two nice strategies, both do well. A provokable strategy responds by defecting at once in response to defection. A forgiving strategy is one which readily returns to cooperation if its opponent does so; unforgiving strategies are likely to produce isolation and end cooperative encounters . . . Source: Excerpts from Tit for Tat by Chris Meredith, [The Slab] Australian Broadcasting Co., 1998.

Market Structure Games: Dynamic

145

or low δ ), tacit collusion between duopolists may not be individually preferable and sustainable as an industry equilibrium. In such situations firms may prefer higher short-run profit π D (π D > π M / 2) even at the cost of sacrificing cooperative profits later on. A small value of δ (or equivalently a higher value of k ) makes a punishment meant to begin next period and apply thereafter (where punishment is tantamount to going back to Cournot behavior) less effective in deterring a profitable deviation this period (π D > π M / 2). If the market is more stable or less volatile (high δ or low k ), tacit collusion may become self-enforceable as firms have no incentive to cheat or deviate from the agreed-upon behavior. Friedman’s (1971) model of tacit collusion among Cournot duopolists is a special case of what is referred to as the folk theorem (e.g., see Friedman 1977; Fudenberg and Maskin 1986). This theorem basically asserts that for infinitely repeated games, many strategy profiles, including the tacit collusion profile, that are not optimal from a short-run, static perspective can sustain as a perfect industry equilibrium provided that players are sufficiently patient or discount factors are large (discount rates are low).26 For more on the economic intuition behind repeated games, see our interview with Robert Aumann in box 4.6 where he also discusses the underlying notion of rationality in game theory. Chamberlin (1933) recognizes other factors that influence the sustainability of collusive behaviors, such as detection lags and firm heterogeneity. Tacit collusion is enforced by the threat of retaliation, for example, as part of a trigger strategy. If firms cannot immediately react when rivals deviate due to detection lags, retaliation is delayed. This makes deviation 26. The folk theorem formalized by Fudenberg and Maskin (1986) is more general than suggested here. It asserts that if the players are sufficiently patient then any “feasible,” “individually rational” payoffs can be enforced by a perfect equilibrium in infinitely repeated games under complete information. In itself, the folk theorem has little to do with collaborative behavior. Many strategy profiles are perfect equilibria for sufficiently patient players in such infinitely repeated games. To improve the models’ prediction power, economists often look for an argument that allows selecting one equilibrium among many as being more likely to arise. A selection process often used rests on the assumptions that (1) symmetric players are likely to coordinate on an equilibrium that allows for symmetric payoffs, and (2) the “focal” equilibrium should be Pareto optimal from the players’ perspective. This suggests that duopolists would coordinate on half the monopoly output. The “correctness” of this selection is more a belief than a result relying on solid game-theoretic foundations. One should ideally resort to more appropriate selection techniques, such as risk-dominance, to select among equilibria. Harsanyi and Selten (1988) examine such approaches. Fudenberg and Tirole (1986) suggest that infinitely repeated games are too “successful” to provide a solid structure for analyzing oligopoly dynamics; they instead propose several alternative approaches.

146

Chapter 4

Box 4.6 Interview with Robert J. Aumann, Nobel Laureate in Economics (2005)

1. Economic sciences have been criticized for the presumed rationality of economic agents. Do you envision a role for a Center for the Study of Irrationality? I belong to the Center for Rationality at Hebrew University. We study both, rational as well as psychological or behavioral approaches. Recent experiments, such as the ultimatum game, probability matching, or Tversky and Kahneman’s behavioral work, confirm that our acts are not always rational, i.e., they do not follow self-interested optimizing behavior in each situation. Rather, we often follow certain rules or norms of behavior that usually, or on average, lead to broad desirable outcomes for survival or success over many similar decision situations over time. Thus optimization is over rules that provide repeated guidance over time rather than over individual situation actions (I refer to this as rule- vs. act-rationality). Rules may lead to optimal behavior in general, but not necessarily in every case. Such rules may be evolutionary or unconsciously adopted. For example, we eat when we feel hungry. 2. Your work on repeated games is particularly interesting. Do you believe that cooperation is more descriptive of real-world firm interactions than classic competition? If so, what is the intuitive logic for this?

Market Structure Games: Dynamic

147

Box 4.6 (continued)

In repeated games players encounter the same situation over and over again. In such games a player has to take into account the impact of his current action on the future actions of other players, i.e., reputation or relationship effects. In such situations cooperation is more likely to be sustainable under certain conditions. Intuitively, the presence of a cooperative equilibrium arises because the threat of retaliation is real, since one will play the game again with the same person. On many occasions the optimal way of playing a repeated game is not to repeat a Nash strategy of the constituent game (e.g., repeated prisoner’s dilemma), but to cooperate and play a socially optimum strategy or follow the “social norm.” In the repeated prisoner’s dilemma, cooperation can be sustained if the discount rate is not too high, i.e., if the players are interested enough in future outcomes of the game. Cooperation is more likely when interaction over time occurs among a few players. The time element/repetition is important. Cooperation may be harder to attain when there are many players.

less costly in present-value terms and tacit collusion harder to sustain.27 In case of asymmetry among firms (e.g., involving different fixed or variable production costs), it is difficult to determine a “focal” choice (e.g., production quantities) on which firms are likely to cooperate. This further hinders the sustainability of tacit collusion. 4.2.3

Co-opetition: Sometimes Compete and Sometimes Cooperate?

It is by now well accepted by corporate executives that partnerships with suppliers, customers, and sometimes even with competitors, may bring about significant competitive advantages. As noted, some game-theoretic models can provide economic rationale for this notion. However, until recently strategic management frameworks have not adequately addressed this issue. Porter’s five-forces framework, for instance, tends to view other parties (competitors, suppliers, or buyers) as threats to a firm’s profitability. The various relationships along the value chain, however, do not necessarily present a threat. Firms may create win-win relationships with 27. In the extreme case of infinite detection lags, firms cannot at all react to their rivals’ actions over the play of the game. This case collapses to a situation where firms formulate open-loop strategies that are adapted to calendar time only. This is equivalent to a setting where firms make their decisions simultaneously. In quantity competition, firms would produce Cournot–Nash equilibrium quantities forever.

148

Chapter 4

their suppliers and even their competitors. There are various situations where collaboration with external parties may create value: •

Joint R&D ventures with rivals.

Partnership with suppliers to enhance the benefit to end consumers or improve productive efficiency (e.g., via just-in-time production). •

Co-development with a customer’s R&D team to better tailor endproducts to customer requirements. •

Agreement among competitors to set industry standards (e.g., Blu-ray technology). •

•

Coordinated lobbying among competitors in the same industry.

As a result of opportunities created by these broader collaborations, the entire value chain, including activities performed by outside parties, should be re-considered. Porter’s (1980) value chain is based on the premise that everything is done in-house. Today it is possible to redesign the value chain in many industries to better cope with uncertainty and interactive relationships, including outsourcing. Strategists have called for a more collaborative model of the value chain based on partnerships. With an increasing number of business processes becoming commoditized within and across industries, executives need to rethink the very basis for competition in their businesses. “Cooperative” strategies nowadays appear on top of corporate agendas. To provide a better framework that takes account of collaborative relationships, Brandenburger and Nalebuff (1995) propose the concept of value net, founded in microeconomics and game theory. They term this kind of strategies co-opetition, as they involve a mix of cooperation and competition. Firms need to decide which of their business processes are core or crucial to make their strategies succeed, and which can be performed in a relatively generic and low-cost fashion, potentially outsourced to external parties. Interactions among firms are structured along the value net depicted in figure 4.9. The value net consists of four key players: suppliers, customers, substitutors, and complementors. Substitutors are external parties who act in a contrarian fashion, whereas complementors help create value since their actions benefit both firms. The value net includes relationships with external parties presenting threats or opportunities. A firm can increase economic profits and value either through value creation (increasing the size of the pie) or through value redistribution (sharing the pie in more

Market Structure Games: Dynamic

Reduce prices

Customers

Dividing up markets Substitutors

Increase prices

Making markets Company

Price competition

Complementors Creating new value

COMPETITION Reduce margin

149

COOPERATION Suppliers

Increase margin

Figure 4.9 Value net and co-opetition Adapted from Brandenburger and Nalebuff (1995)

favorable terms). Value redistribution can be accomplished, for instance, through more skillful bargaining with buyers and suppliers. Such a focus is risky as firms rarely outperform their rivals solely through value redistribution. Competition in redistribution is fiercer than competition to create new value. Yet firms can cooperate in different ways to increase the size of the pie. Once this is accomplished (in cooperation with others), value redistribution can be easier.28 The value net is helpful to better account for cooperation opportunities that might be beneficial to the firm. Such situations where firms can either compete or cooperate can be analyzed with tools offered by option games. Trigeorgis and Baldi (2010) assess the value of optimal patent leveraging strategies under both demand uncertainty and competitive rivalry using a methodology allowing for a mix between competition and cooperation. They 28. Brandenburger and Stuart (2007) subsequently develop a hybrid approach involving both noncooperative and cooperative game theory called biform games to better account for such complex strategic interactions. Their approach is characterized by two stages. In the first stage, players behave cooperatively to increase the value of the total market “pie.” Afterward they behave noncooperatively to share the market pie based on their market power. The bargaining power wielded in the second stage may stem from competitive advantages, such as technological leadership, created in the first stage. Biform games focus on shaping the competitive environment, namely first creating or reshaping a market and then building up competitive advantage to capture a larger slice of it. Such approaches may gain in acceptance over the coming decades. A practical approach providing management with state-of-the-art conceptual frameworks would be helpful.

150

Chapter 4

7

No (zero) (symmetric)

5

6

8

9

Small

High

Medium

Low

(monopoly/ abandon)

ate per

Coo

Bracketing (asymmetric)

L

4

3

t

h Fig

Patent wall

Cross-licensing

M

2

Sleep/ abandon

1

H

Licensing

Bracketing (symmetric)

Demand

Large (asymmetric)

Competitive advantage (patent innovation)

Figure 4.10 Compete versus cooperate as part of a patenting strategy Adapted from Trigeorgis and Baldi (2010)

illustrate how optimal patent strategy depends on the level and volatility of demand and on the size of competitive advantage arising from the patented innovation. The key question they consider is, when business conditions are uncertain, under what circumstances should rivals fight and when should they collaborate in using their intellectual property (IP) assets, namely when should two competitors follow a fighting or a cooperating patent strategy? They show that the circumstances under which firms should fight or cooperate are not trivial. Under demand uncertainty rivals may sometimes find it preferable to compete (e.g., defending themselves via raising a patent wall around their core patent or fighting fiercely by attacking each other via patent bracketing) and at other times to collaborate (e.g., via cross-licensing to each other or even licensing one’s patented innovation to its rival). The menu of different fight or cooperate strategies is summarized in figure 4.10. The option games approach enables to capture certain features of an adaptive business strategy. The increasing cone of market and strategic uncertainty makes the value of a dynamic strategy that enables switching among a broader menu of competing or cooperating alternatives key to survival and success in a changing marketplace.

Market Structure Games: Dynamic

151

Conclusion This chapter and the previous one discussed how industrial organization can be helpful in providing insights about certain competitive situations. Chapter 3 discussed some static benchmark models involving price and quantity competition. Chapter 4 considered the time dimension, discussing several dynamic settings. Dynamic models of complete or incomplete information are more challenging than their simultaneous-move counterparts and give richer insights on how firms should consider the long-term consequence of their current actions. In such multistage games, strategic decisions that appear suboptimal from a short-term, static perspective may indeed be optimal in the long term. Dynamic games are useful to understand that committing and killing one’s options may sometimes be advisable as a devise to create competitive advantage in subsequent stages. They also provide economic foundations for collaborative behavior in certain markets involving repeated relationships. The results discussed in chapters 3 and 4 serve as building blocks for the option games methodology we deploy later.29 Selected References Osborne (2004) offers an accessible account of game theory. Averted readers might prefer the more advanced treatment offered by Fudenberg and Tirole (1991). Besanko et al. (2004) provide a good overview of industrial organization focusing on the strategic insights of the models. Tirole (1988) is a key reference in this area. Fudenberg and Tirole (1984) discuss the taxonomy of commitment strategies. Rubinstein (1982) examines bargaining over an infinite horizon. Friedman (1971) elaborates on the sustainability of tacit collusion in infinitely repeated Cournot games. Besanko, David, David Dranove, Mark Shanley, and Scott Schaefer. 2004. Economics of Strategy, 3rd ed. New York: Wiley. Friedman, James W. 1971. A non-cooperative equilibrium for supergames. Review of Economic Studies 38 (1): 1–12. 29. Most dynamic models in industrial organization assume either steady state or a deterministic evolution of the underlying parameters. Mainstream industrial organization does not explicitly consider the impact of stochastic market uncertainty on the equilibrium results. Merging dynamic games with real options analysis helps bridge this gap, better explaining optimal decision-making in the face of uncertainty.

152

Chapter 4

Fudenberg, Drew, and Jean Tirole. 1984. The fat cat effect, the puppy dog ploy and the lean and hungry look. American Economic Review 74 (2): 361–66. Fudenberg, Drew, and Jean Tirole. 1991. Game Theory. Cambridge: MIT Press. Osborne, Martin J. 2004. An Introduction to Game Theory. New York: Oxford University Press. Rubinstein, Ariel. 1982. Perfect equilibrium in a bargaining model. Econometrica 50 (1): 97–109. Tirole, Jean. 1988. The Theory of Industrial Organization. Cambridge: MIT Press.

5

Uncertainty, Flexibility, and Real Options

Up to now we have mainly focused on industrial organization models in a certain or deterministic world. Before discussing further how to combine game theory with real options under uncertainty, we review in this chapter the basics and insights of real options analysis to improve our understanding of flexible real investment decisions.1 The real options approach to analyzing investment under uncertainty has by now gained standard corporate finance textbook status. Real options analysis involves the application of methods utilized to price financial options and other derivatives to real assets; that is, it is an extension of option-pricing theory or contingent-claim analysis to real investment situations. Financial options are financial assets (generally traded on the capital market) that give their holders the right—but not the obligation—to buy or sell an underlying asset at a specified price for a given time period. Option-pricing theory was developed to price this asset class. Real options draw on an analogy between financial options (calls or puts) and real-world contingent cash-flow streams. And yet, even when an analogy can be drawn, real options are quite different. For example, they may not be traded on the capital market and may simultaneously be held by more than one investors. Still, the analogy between real investment projects and certain traded derivative assets holds conceptual value. For instance, having the option to abandon a project for a salvage value should the market evolve worse than expected is similar to holding a put option giving the right—but not the obligation—to “sell” the underlying asset at a strike price equal to the salvage value. In such a case, the analogy applies well. Real options are now commonly 1. Readers familiar with real options analysis may skip this chapter and go directly to the description of option games provided in the subsequent chapter.

154

Chapter 5

accepted as a powerful real asset valuation and management approach. Several economics and finance books have been dedicated to this approach, more prominently Dixit and Pindyck (1994) and Trigeorgis (1996). To put our examination of real options in a concrete, applied context, we discuss in section 5.1 the investment challenges faced by European electricity suppliers. Section 5.2 discusses how managers of an electric utility can address such challenges utilizing real options analysis. Section 5.3 describes option-pricing theory and illustrates its application in situations involving concrete investment problems from the energy sector. 5.1

Strategic Investment under Uncertainty—The Electricity Sector

Over the past few decades many firms have experienced new developments in their competitive landscape, including regulatory, technological breakthrough and demand pattern changes. Few of these developments could have been predicted. Increasing uncertainty casts doubt on current predictions about market developments, technological trends or rivals’ behavior. In the midst of such uncertainty, adaptability to market developments can be of significant strategic impact. The European energy sector exemplifies this challenge quite well. National markets have opened up to new entrants and energy prices have become increasingly volatile. Even though the need to add generation capacities is well recognized in the electric utility sector (see International Energy Agency 2007), tackling complex issues such as when or where to invest in generation units is all the more difficult in this highly unpredictable environment. From this industry perspective we examine next the major sources of uncertainty and the business risk exposures implied by the new generation technology choices. 5.1.1

Need for New Investment in Europe

A key factor monitored by bodies regulating electric utilities is reserve margins. This indicator measures the power adequacy in terms of installed power generation capacity over peak demand in a given geography. Energy suppliers have to build up sufficient generation capacity to cover the base load as well as excess capacity to ensure that even in peak demand periods electricity output is sufficient. In

Uncertainty, Flexibility, and Real Options

155

Europe (EU12)

France

Italy

CAGR%

CAGR%

CAGR%

4.5

4.5

4.5

4.0

4.0

4.0

3.5

3.5

3.5

3.0

3.0

3.0

2.5

2.5

2.5

2.0

2.0

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

0.0

0.0 86–91 91–96 96–01 01–06

86–91 91–96 96–01 01–06

Electricity consumption

86–91 91–96 96–01 01–06

Generation capacity

Figure 5.1 Growth trends in the European electricity markets From Energy Information Administration Database. CAGR stands for “compound annual growth rate.”

other words, reserve margins must remain positive. If they fall to low levels, consumers face generation inadequacy potentially leading to blackouts. In the last decades most European countries have experienced substantial energy demand increases. As shown in figure 5.1, these increases were hardly met by capacity additions, resulting in shrinking reserve margins. In addition, most power plants built in the 1970s in the aftermath of the oil crisis are expected to be decommissioned or refurbished in the next decade across Europe as they become obsolete or no longer abide by new environmental standards. Reserve margins in some European regions (e.g., Brittany in France) are already approaching critical levels, indicating dire investment need in the coming years. Another means to improve reserve margins is to curb electricity demand, especially in peak load, by implementing energy-efficiency

156

Chapter 5

measures or other market mechanisms (e.g., customer participation) to induce consumers shift demand from peak load to lower load periods. Such initiatives are, however, not always as effective, so the need for additional generation capacity remains. Electric utilities across Europe are expected to channel huge financial resources into new power generation capacity to ensure system adequacy and reliability going forward. These investments will take place under growing uncertainty and increased environmental constraints with the aim to promote green technologies. 5.1.2

Sources of Uncertainty

As illustrated in table 5.1, utilities face a number of uncertainties, be they idiosyncratic (e.g., changing demand patterns, electricity demand growth), technological, or firm specific. Rapid changes occurring in recent years challenge would-be investors in determining when to invest (optimal investment timing), where to add new generation facilities (network Table 5.1 Uncertainty factors for electric utilities Idiosyncratic risks

Technology-specific risks

Firm-specific risks

Demand uncertainty • Demand growth • Changing demand patterns • Country-specific prices • Price developments • Effectiveness of energy-efficiency measures Regulatory risks • Plant licensing and approval • Regulation of transmission lines Competitive risks • Market power of incumbents • Market entrance

Construction risks • Investment cost overruns • Unforeseeable lead time Fuel-related uncertainty • Unsecured supply (e.g., wind, gas)a • Input price volatility (e.g., gas, coal, uranium) Environmental uncertainties • Prices on CO2 and greenhouse gas emissions • Support on green technologies • Waste management and phaseouts (nuclear power) • New environmental standards

Capital structure • Bankruptcy risks • Interest payments Cost structure • Economies of scale and scope • Learning-curve effects Company’s portfolio of corporate activities • Internal hedging

a. The 2008 dispute on gas between Russia and Ukraine led to serious supply disruptions in eastern and central Europe (e.g., Ukraine, Poland, Germany). Source: Adapted from International Energy Agency (2007).

Uncertainty, Flexibility, and Real Options

157

Electricity price (monthly averages in €/MWh) 100

80

60

40

20

0 2002

2003

2004

2005

2006

2007

2008

Figure 5.2 Electricity price fluctuations in Europe from 2002 to 2008 From Powernext

design), or what mix of technologies to invest in (technology choice). We next look at the business risks each company in the sector faces and the risks involved in the generation technology choice.2 Idiosyncratic Business Risks Since electricity cannot be effectively stored, electricity prices are very sensitive to shocks (e.g., demand pattern fluctuations, plant disruption) and are therefore extremely volatile. Figure 5.2 shows electricity price fluctuations during the period 2002 to 2008. Wholesale electricity prices in many European countries started to skyrocket in 2004 and have remained at a substantial high average level since. This overall price increase can be attributed to three underlying factors. First, most fuel prices (e.g., gas, oil, coal) have trended upward. Second, utilities now have to pay for CO2 emissions. Finally, the tightening of reserve margins has resulted in utilizing more expensive generation technologies, leading to higher average generation costs. Energy suppliers for the most part managed to pass this cost through to their customers (given the low price elasticity of demand). 2. We do not intend to elaborate on firm-specific risk factors but instead adopt the viewpoint of an “average firm” in the electricity sector.

158

Chapter 5

Besides, demand patterns can vary markedly across regions, some being more often subject to huge demand peaks (e.g., due to intense use of air-conditioning in southern European geographies). Other country-specific factors might also play a role. For instance, temperatures are different across northern and southern Europe and have differential effects. Some countries have a high share of industrial consumption (e.g., Germany), while others have more tertiary activities (e.g., the United Kingdom). The national production mix may also spill over into the electricity prices. When demand increases, utilities need to operate more generation units, including costly ones, to ensure power adequacy. The national average market price is likely to be higher owing to the costly national production mix if transmission lines are ineffective. Local resistance in the form of “not-in-my-backyard” attitudes may also act as a hindrance to ensuring adequate capacity and investment. Various European governments contemplate initiating energy-efficiency measures to curb future electricity demand, besides achieving environmental objectives. Smart meters have been introduced in some European countries (e.g., in Italy) both for industrial customers and households, enabling customers to be better informed of current electricity prices and alter their consumption patterns accordingly. Although governments and regulatory bodies have shown interest at pushing energy-efficiency measures, it is not clear whether such initiatives will be enforced and will achieve their goals. Governmental intervention may take many forms: subsidies or fiscal incentives toward new technological options, lengthy plant licensing processes for nuclear power plants, antitrust regulation or administrative market-entry barriers to protect incumbents. Technology-Related Business Risks Capital expenditures may greatly vary from technology to technology and might follow different development paths. The steel price is an important variable in any generation plant investment. Other construction-related costs as well might fluctuate depending on general realestate market conditions. New technologies may be characterized by decreasing investment costs over time due to economies of learning. Through plant design standardization, R&D costs can be spread out to a larger number of projects. This is critical for nuclear plants, especially when lack of process optimization due to limited experience results in substantial cost overruns.

Uncertainty, Flexibility, and Real Options

159

Operating and fuel costs are a major source of uncertainty. Coal-fired plants are more sensitive to the underlying fuel price volatility than nuclear power, but much less than combined-cycle gas turbines (CCGTs). Operating and maintenance costs may differ depending on the technology considered and may undergo distinct development. At the end of the day, the relative competitiveness of generation technologies varies over time. Investment decision-making must be dynamic, with the up and down potential of each underlying factor being seriously considered in adjusting the generation mix. Policy makers’ reluctance toward a steady, long-term environmental policy is another source of uncertainty. From an individual value-maximizing viewpoint, utilities tend to select the “cheapest” technology regardless of total societal costs. A practical means to avoid this problem is to internalize the indirect costs (e.g., damage on the environment) resulting from power generation. This is what the European Union intended when introducing the EU Emission Trading Scheme (EU ETS) in 2005. 5.1.3

Generation Technologies and Business Risk Exposure

A key question is not simply whether to invest but also which available generation technologies to choose. The technology choice ultimately determines the project’s sensitivity to certain underlying risk factors and its overall business risk exposure. When selecting among available technological options, one considers the variable and fixed costs involved in each case. The fixed cost components are the upfront investment cost, the decommissioning cost, and the fixed operating and maintenance costs. The up-front investment cost is very high for a nuclear or hydro plant and less so for CCGTs. The major variable cost component is the input fuel cost. Completion delays, cost overruns, and the plant’s lifetime are also important factors. Supply security may also come into play. Certain technologies may possibly put the system at risk as they might not be always available or reliable (e.g., wind). Cost factors may vary, which drives cost uncertainty for utilities. There is no superior power generation technology among the group of mature technological options. Each has certain features that make it better suited in certain situations. No technology turns out to be cheaper, cleaner, and more reliable than the others. Depending on whether a firm is willing to pay large fixed costs (e.g., nuclear power) or large variable costs (e.g., coal fired), one might prefer one technology over another.

160

Chapter 5

Among the alternative technological options some are considered more conventional, such as coal fired, combined-cycle gas turbines (CCGT), and nuclear power. The coal-fired technology involves the lowest (variable) costs under most circumstances. This technology, however, has a high carbon footprint and is significantly more expensive if environmental costs are factored in. The CCGT technology has several advantages, including lower capital investment costs and short construction times. It is also one of the technologies offering the greatest operational flexibility. This is offset by current high fuel costs. This technology is highly sensitive to gas prices since fuel costs represent a significant share of total costs. CCGT technology, however, has a poor environmental footprint in terms of CO2 emissions. Hydro power generation also involves very large (sunk) investment costs. Reservoirs are very flexible, however, since they can be closed and opened at very little cost.3 Available sites though are limited and plant standardization is hardly possible, which may lead to substantial delays and investment cost overruns. Recent high natural gas prices and CO2 emission costs have lead certain countries (mainly France) to boost investment in nuclear power plants, whose efficiency has been significantly improved relative to other technological options. However, civil nuclear power has a bad track record of delays, cost overruns, and catastrophic disruptions (e.g., Fukushima). Many countries (e.g., Italy, Germany) are reluctant to embrace civil nuclear power, while a few still invest in this technology with France taking the lead. Certain alternative technological options are gradually emerging as mainstream technologies. Onshore wind power has nearly reached a 10 percent capacity share in Europe (EU12). Other renewable technologies such as biomass, small hydro, and solar, have not yet gained enough acceptance but pave the way for more environmental-friendly power generation. Over the past two decades investments in various new generation units have significantly altered the generation mix in many European countries. A noticeable trend is the increasing share of natural gas-fired capacity (especially CCGTs) and of renewables technologies (especially onshore wind-power turbines). The offshore wind power technology has also gained in acceptance, with Germany commissioning a 400-MW wind farm (BARD offshore 1) in 2011 and the United Kingdom building up a 500-MW offshore wind farm on the eastern coast (Greater Gabbard). EU member countries generally give subsidies to 3. Other technologies such as nuclear, coal or fuel, and gas (with the exception of CCGT) power typically require a significant time lag between the decision to fire up the plant or turbine and the production of electricity itself.

Uncertainty, Flexibility, and Real Options

161

increase investment in renewables, primarily wind power and hydro. Despite being environmentally friendly, the wind power technology is often challenging for utilities as it can only operate with sufficient wind. Another alternative for electric utilities is to invest in transmission lines to enable improved integration of electricity systems with neighboring European countries. Transmission lines enable an electric utility to compensate for low reserve margins in one region through imports from another neighboring country. The presence of interconnection lines between regions is beneficial not only to utilities, which have the option to supply the market under better economic conditions, but also to the general public. Transmission lines pave the way for “shared” options or resources, enabling better power supply management when reserve margins get tighter at the national level with higher regional demand compensated by higher generation in areas that were not connected before. Business risk exposure depending on generation technology is summarized in table 5.2. Current market trends are not solely due to the emergence of superior technologies but also to the prevailing market and regulatory conditions favoring these technologies. Figure 5.3 shows the development in terms of generation capacity of various technological options over the recent two decades in Europe. Changes in fuel costs, construction costs and CO2 emission prices significantly affect the relative attractiveness of each technology. Since Table 5.2 Business risk exposure of conventional generation technologies Technology

Investment cost

Time-tobuild

Operating cost

Fuel cost

Security of supply

CO2 emissions

Coal-fired CCGT Hydro Nuclear Wind Low exposure Medium exposure High exposure Note: The table excludes idiosyncratic uncertainty (which is the same whatever the technology) as well as firm-specific risk.

162

Chapter 5

CAGR% (1987–2007)

Generation capacity by technology type in EU12 (in million kW) 600

Total

1.5

Renewables 17.7

500 Nuclear

1.2

Hydro

0.2

Thermal

1.2

400

300

200

2007

2006

2005

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

1993

1992

1991

1990

1989

1988

0

1987

100

Figure 5.3 Generation capacity by technology type in Europe From Energy Information Administration Database

investment in generation capacity is largely sunk, firms face high business risks when deciding at the outset which technology to select. Unexpected events may make other technological alternatives more suitable under specific future market conditions. A key parameter an electric utility should take into consideration when committing to a given power plant is exposure to fuel price (variable cost) volatility. Wind power needs no fuel as such. For nuclear plants, fuel costs (mainly uranium) represent a small share of total operating and investment costs (most being dedicated to enrichment, conversion, and fabrication, being therefore quasifixed). In contrast, CCGTs are very sensitive to fuel costs. Because natural gas prices are very volatile, this brings substantial fuel-sensitivity for CCGTs. 5.2

Common Real Options

Electric utilities must take into account the aforementioned risk factors and market dynamics when they work out their investment strategies. They can benefit from an improved understanding and guidance concerning the risk factors, particularly when formulating investment decisions to enhance their generation capacity and future economic profits.

Uncertainty, Flexibility, and Real Options

163

Table 5.3 Common real options Real option type

Description

Relevant industries

Deferral or waiting option

Management can wait before making the investment to see how the market unfolds.

Resource extraction industries, real-estate development, capitalintensive industries.

Staging or time-to-build option

When a managerial decision takes time or is done in stages, management can default if market prospects prove worse than expected.

Technology-based firms (R&D), longdevelopment capitalintensive industries (e.g., electric utilities), startup ventures.

Expand or extend option

If the project turns out better than expected, management can spend more to expand the project scale or it can extend the project’s useful life.

Natural-resource industries (e.g., mining), real-estate development.

Contract or abandon option

If the market prospects are worse than expected, managers can contract or abandon it for salvage.

Capital-intensive industries (e.g., airplane manufacturers), new product introductions.

Switching option

Management can select among the best of several alternatives, e.g., inputs, outputs or locations, under the prevalent market conditions.

Multinational firms with production facilities in different currencies, platform strategy in the automotive sector.

Compound option

If investment takes place in stages, the first project can be valued in view of the future growth options it creates.

High-tech, R&D, industries with multiple product generations, strategic acquisitions.

As summarized in table 5.3, management can benefit from different types of real options. We here discuss common types of real options in the context of an electric utility. The necessary tools to quantify the values of such real options are discussed in the subsequent section. Deferral or timing option Management is not always confronted with a now-or-never investment decision. Often it might have the flexibility to time its investment decision after observing how events unfold. Suppose that French utility EDF has identified that in the Brittany region reserve margins are falling to such low levels that they may jeopardize power supply security. EDF, being authorized to open up nuclear power plants in France, resolves to operate such a plant in Brittany if it is deemed worthwhile. Since the involved capital investment cost I is large and cannot be recouped, management wishes to decide on solid ground when more sure that the need for new generation capacity in this

164

Chapter 5

region is here to stay. EDF in effect holds an investment timing or “waitand-see” option to benefit from the resolution of uncertainty about electricity prices (reflective of the regional imbalance between demand and supply). Suppose that EDF may exercise its option to invest only once, at maturity (year T ). Let VT indicate the time- T value of the investment’s expected operating cash flows. Just before expiration the option will pay off the greater of the net value created, VT − I , or 0, meaning at maturity the option is worth max {VT − I ; 0}. The option to invest or defer is thus analogous to a call option on the gross project value VT with exercise price equal to the required capital investment outlay I . Waiting, though it may come at the cost of forgoing early cash flows, may pay off under certain conditions and help avoid costly premature investment. Option to stage or default during construction (time-to-build option) Cash flows are not generated overnight once managers make an investment decision. Typically large construction projects are staged. During power plant construction, if market conditions deteriorate, management can choose to forego any future planned capital outlays. The actual investment staging consists of a series of capital outlays, offering at any given stage the opportunity to revise the initial go-decision and stop incurring further costs should future prospects turn out worse than expected. In such a case each stage t ( ≤ T ) can be viewed as an option written on the value of subsequent stages by incurring the (time-t) investment cost outlay I t required to proceed to the next stage. The investment opportunity can be valued similarly to compound options (options on options). Option to expand or extend Given the raised public awareness of environmental challenges ahead and the enforcement of more stringent environmental legislation, electric utilities are increasingly investing in renewables technologies, such as wind power. A possible plan design for the utility is to start up small, scalable wind farms with a limited number of turbines, with the possibility to expand. The value of the initial wind farm is Vt at time t ( ≤ T ). If future electricity prices or governmental subsidies are higher than expected, management can expand the number of turbines or the scale of production (by e percent) by making an additional cash outlay I e. At time t ( ≤ T ), the entire project can be viewed as the base-scale wind farm, Vt, plus a call option on future (expanded) investment, namely Vt + max {eVt − I e ; 0} = max {(1 + e ) Vt − I e ; Vt }. The initial wind farm thus enables the firm to capitalize on future expansion opportunities. This

Uncertainty, Flexibility, and Real Options

165

expansion option, which will be exercised only if future electricity prices or subsidies turn out more favorable, can make the basic small, scalable wind farm project worth undertaking. Option to scale down or contract If market conditions turn out weaker than originally expected, management can operate below total power generation capacity or even reduce the scale of operations (by c percent), thereby saving part of the planned cash outflows or fixed costs (Ic). This flexibility to mitigate loss is analogous to a European put option on part (c percent) of the base-scale project, with exercise price equal to the potential cost savings (I c), paying off max {I c − cVT ; 0} at maturity. It may sometimes be preferable to build a plant with lower initial construction costs and higher maintenance expenditures in order to acquire the flexibility to contract operations by cutting down on maintenance if market conditions turn out unfavorable. Option to shut down (and re-start) operations A power plant does not necessarily have to operate in each and every period. It may be profitable to operate flexible (e.g., CCGT peak-load power) plants with higher marginal costs if they can be shut down when electricity prices are low and not sufficient to cover variable operating costs. In this case the utility might be better off not operating temporarily, especially if the costs of switching between the operating and idle modes are relatively small.4 In peak-demand periods, as market electricity prices rise to peak levels, operation can start up again and be quite profitable. Enron allegedly made huge profits in California operating such peak power plants only a few days or weeks in the year when prices peaked, exploiting the tremendous volatility in local electricity prices. Operating such a peak-load plant is analogous to a call option enabling management to receive in year t revenues Pt by incurring variable costs Ct as exercise price. The contingent payoff at time t (≤ T ) is π t = max {Pt − Ct ; 0}, with present (option) value π 0(t ) ≡ e − rt Eˆ 0[π t ]. Because the plant embeds such shutdown and restart options at each (discrete) time period t until maturity T T , the total plant value is V0 = ∑ t =0 π 0( t ) provided switching costs are small or negligible.5 Option to switch use (e.g., inputs or outputs) Instead of committing to a certain input an electric utility may select the best of several fuel alternatives should future conditions vary. We discuss below two types 4. For simplicity, we assume henceforth that switching costs are negligible, allowing decision-making to be path independent. 5. This case with no switching cost has been analyzed by McDonald and Siegel (1985) along the Black–Scholes lines.

166

Chapter 5

of switching options, relating to inputs or outputs. Over the past years natural gas has been under increasing competitive pressure owing to the introduction of substitute products, especially liquefied natural gas (LNG). The prices for natural gas and LNG are positively correlated, but they undergo different market development trends that can make one input more affordable than the other (for an equivalent output efficiency) in a given period. An effective strategy is to invest in flexible gas-fired generation technologies (especially CCGT) that can use either of these fuels. The utility would be willing to pay a certain premium to acquire such a flexible technology, compared to the (best of the) rigid alternatives that confer no or less choice (either natural gas or LNG).6 Naturally the correlation between natural gas and LNG is fairly high since the products are close substitutes. If the correlation between the two fuels is lower, the switching option will be even more valuable. Generally, process flexibility can be achieved not only through an adequate technology choice (e.g., building a flexible generation facility that can switch among alternative fuel inputs) but also by maintaining relationships with a variety of suppliers, changing the mix as their relative prices change. Another form of switching option relates to the optionality to design operations such as to produce alternative outputs, selecting the most profitable output once price uncertainty gets resolved. Recently there has been a consolidation trend in the energy industry toward the merging of electric and gas utilities. A case in point is the merger of the French gas utility GDF with Suez. This move partly resulted from the synergies arising from the increasingly growing share of gas-fired power generation (CCGTs) in Europe. From this viewpoint the gas utility (GDF) may either sell the natural gas to the end consumers or supply natural gas to gas-fired power generation units in view of the actual energy prices (gas vs. electricity). Another example of the option to switch outputs relates to the integration of European electricity markets due to the introduction of merchant transmission lines. Such interconnection lines enable the Italian electric utility Enel buy cheap nuclear power from France, or to 6. Let At be the value (as of time t) of utilizing natural gas and Bt of using LNG as input. The cost of switching from one input to the other may differ depending on which fuel was previously in use, exhibiting some form of path dependency. Denote by I ( At −1 → Bt ) the switching cost when swapping from natural gas to LNG and by I ( Bt −1 → At ) the cost of switching from LNG to natural gas. Should market conditions prove currently better for LNG over natural gas, the firm may switch from natural gas to LNG by incurring the switching cost I ( At −1 → Bt ) . The utility can select the best of the two operating modes, max { At ; Bt − I ( At −1 → Bt )} . Inversely, if the LNG fuel was initially used, the payoff at the following node will be max {Bt ; At − I ( At −1 → Bt )} .

Uncertainty, Flexibility, and Real Options

167

sell electricity in France should the price there be higher than the Italian market price. Option to abandon for salvage value If electricity prices suffer a longstanding decline, management does not have to sustain operations forever. It may find it better to shut down operations, decommissioning the less performing generation units (e.g., plants with higher variable costs). Management has a valuable option to abandon a project permanently and stop paying the production costs. This option is analogous to an American put option on current project value, Vt, with exercise price the “salvage value,” St, entitling management to receive Vt + max {St − Vt ; 0} or max {Vt ; St } before option expiration in year T . The option to close down the plant provides downside risk protection if the firm is not committed to go on generating electricity when market prospects worsen. Since most assets in the electricity sector are dedicated to generate power with no possibility to produce other goods, the resale price of production facilities is limited. The prospect to forgo negative cash flows when shutting down a plant can provide a sufficient incentive and make the abandonment option valuable. The salvage value for which the plant can be sold or exchanged, St, may fluctuate over time as does the project’s value Vt.7 Corporate (compound) growth options An early investment may set the path for future opportunities to follow. Such growth options are particularly valuable when learning-curve effects are involved. Because of more stringent environmental constraints and demand for higher energy efficiency, several new generation technologies are being developed (while older ones are being substantially researched on and improved). A case in point is the evolutionary pressurized reactor (EPR) technology developed by Areva, EDF, and Siemens as the new generation of pressurized water nuclear reactors. Installing a single nuclear plant with the EPR reactor may appear unattractive to developers owing to high R&D costs and lack of scale economies. It could be of interest, however, to build one as an operating prototype. Future nuclear power plants could leverage on the experience gained from developing, designing, and building the first EPR plant. The investment in the first EPR plant is a prerequisite in a chain of interrelated projects. The prototype derives its value not so much from its expected project-specific cash flows but rather from unlocking future growth opportunities in the form 7. In this case the abandonment option can be viewed and valued as a switching option to select between two stochastic assets (with no switching cost). Myers and Majd (1990) examine a similar problem.

168

Chapter 5

of a chain of new, better performing nuclear power plants. From an options perspective the opportunity to invest in an innovative generation technology is analogous to an option written on options (construction of other new-generation plants); that is, it is an interproject compound option. Even if the initial stand-alone project does not generate positive net value on its own, the experience generated during the development of the first-generation EPR plant may serve as a springboard for developing future enhanced-efficiency nuclear power generation units. Unless the firm decides to commit and initiate the first EPR investment, subsequent generations would not even be feasible. Such growth options are found in many other industries especially in industries which are infrastructure-based, R&D-intensive, or involving multiple product generations or applications (e.g., in semiconductors, computers, pharmaceuticals) as well as in industries where having a global footprint is key. Portfolios of real options A well-diversified power generation portfolio includes several affordable technologies that hedge exposure to various technology-specific business risks while achieving power adequacy. Fluctuations in the key cost factors (e.g., investment costs, fuel costs, CO2 emissions prices) can alter the relative attractiveness of alternative generation technologies. A well-diversified portfolio of generation capacities allows utilizing technologies better suited to changing market conditions. Portfolios of options may also involve negative interactions or positive synergies.8 Even though the installed generation capacity is well-diversified on the overall European level, it is not necessarily the case at the national level. There are noticeable “unbalanced” mixes in France (large share of nuclear power) or the Netherlands (almost 70 percent gas fired). The technology portfolios in the European Union compared to individual member countries are shown in figure 5.4. Diversification in terms of technologies is a sensible strategy for an electric utility to help manage risks. As part of their risk management strategies, utilities can consider investing in other technology types or setting up transmission lines. Transmission lines make it possible to leverage on the Europe-wide welldiversified generation mix and reduce exposures in a given geography to certain technology-specific business risks. Plants involving high fixed costs (e.g., nuclear plants) should be committed to base-load demand in 8. For general option interactions, see Trigeorgis (1996, ch. 7). In the electricity market, coupling wind farm and hydro with pump storage creates a portfolio of power generation technologies whose value is higher than the sum of the parts. In such coupled systems, overproduction from wind farm can be used to pump water back in the hydro system using this energy source (in the hydro system) at a later point in time.

Uncertainty, Flexibility, and Real Options

169

Total capacity 100% Renewables Nuclear

80%

Hydro

60%

40% Thermal

20%

0%

BE

DK

FR

DE

GR

IE

IT

LU

NL

PT

ES

UK EU12

15

13

112

127

13

6

82

1

23

14

77

80

562

Total capacity (in million kW)

Figure 5.4 European generation capacity in 2007 by technology type—Europe (EU12) and member countries From Energy Information Administration Database

order to spread these costs over as many hours as possible. Flexible power plants (e.g., CCGTs) characterized by large variable costs should operate during peak hours when load is higher and electricity prices sufficiently high to cover these costs. A precise rigid recommendation for the generation mix is unwarranted since the optimal mix changes continuously over time. Figure 5.5 shows a static version of the recommended generation mix as a function of electricity demand and marginal costs. The marginal costs are hardly stable over time, so the graph changes dynamically. Box 5.1 discusses how to think about managing portfolios of options via a gardening metaphor. Marco A. G. Días from Petrobras discusses some real options applications in Brazil in box 5.2. 5.3

Basic Option Valuation

Should electricity prices or other market factors evolve differently than initially expected, the utility’s management has several options to adapt to enhance the future cash flow stream or limit losses. The aforementioned real options, however, cannot be properly valued using the classical discounted cash-flow (DCF) approach. Real options analysis is a more useful tool to analyze managerial flexibility since it

170

Chapter 5

Marginal costs (in €/MWh)

Demand-off peak

Demand peak

Extreme demand

Old plants

Gas-fired

Coal-fired

Nuclear, hydro Must run (e.g., wind)

Electricity demand (in MW)

Figure 5.5 Technology use as a function of demand level Adapted from International Energy Agency (2007, p. 127)

Box 5.1 Managing portfolios of options: A gardening metaphor

Strategy as a Portfolio of Real Options T. Luehrman, Harvard Business Review Managing a portfolio of strategic options is like growing a garden of tomatoes in an unpredictable climate. Walk into the garden on a given day in August, and you will find that some tomatoes are ripe and perfect. Any gardener would know to pick and eat those immediately. Other tomatoes are rotten; no gardener would ever bother to pick them. These cases at the extremes—now or never—are easy decisions for the gardener to make. In between are tomatoes with varying prospects. Some are edible and could be picked now but would benefit from more time on the vine. The experienced gardener picks them early only if squirrels or other competitors are likely to get them. Other tomatoes are not yet edible, and there’s no point in picking them now, even if squirrels do get them. However, they are sufficiently far along, and there is enough time left in the season, that

Uncertainty, Flexibility, and Real Options

171

Box 5.1 (continued)

many will ripen unharmed and eventually be picked. Still others look less promising and may not ripen before the season ends. But with more sun or water, fewer weeds, or just good luck, even some of these tomatoes may make it. Finally, there are small green tomatoes and late blossoms that have little likelihood of growing and ripening before the season ends. There is no value in picking them, and they might just as well be left on the vine. Most experienced gardeners are able to classify the tomatoes in their garden at any given time. Beyond that, however, good gardeners also understand how the garden changes over time. Early in the season, none of the fruit falls into the “now” or “never” categories. By the last day, all of it falls into one or the other because time has run out. The interesting question is: What can the gardener do during the season, while things are changing week to week? A purely passive gardener visits the garden the last day of the season, picks the ripe tomatoes, and goes home. The weekend gardener visits frequently and picks ripe fruit before it rots or the squirrels get it. Active gardeners do much more. Not only do they watch the garden, but based on what they see, they also cultivate it: watering, fertilizing, and weeding, trying to get more of those in-between tomatoes to grow and ripen before time runs out. Of course, the weather is always a question, and not all the tomatoes will make it. Still, we would expect the active gardener to enjoy a higher yield in most years than the passive gardener. In option terminology, active gardeners are doing more than merely making exercise decisions (pick or not to pick). They are monitoring the options and looking for ways to influence the underlying variables that determine option value and, ultimately, outcomes. Option pricing can help us become more effective, active gardeners in several ways. It allows us to estimate the value of the entire year’s crop (or even the value of a single tomato) before the season actually ends. It also helps us assess each tomato’s prospects as the season progresses and tells us along the way which to pick and which to leave on the vine. Finally, it can suggest what to do to help those in-between tomatoes ripen before the season ends. Source: Reprinted with permission of Harvard Business Review from “Strategy as a Portfolio of Real Options,” by T. Luehrmann, September– October 1998, pp. 89–99. Copyright © 1998 by the Harvard Business Review School Publishing Corporation; all rights reserved.

172

Chapter 5

Box 5.2 Interview with Marco A. G. Días, Petrobras

1. Besides the United States, Brazil is likely the most popular user of real options analysis in the world. Why you think is that? What is the role of instability and uncertainty historically in Brazil? Can you comment on the adaptability of the Brazilian people and businesses? Real options (RO) is more popular in Brazil than in most developed countries. The history of instability and uncertainty obligates both people and businesses to be more flexible to adapt to the frequently changing environment. We say that in Brazil even the past is uncertain! So flexibility is more valuable here than elsewhere. One example is the flex-fuel car. Here RO started as the villain and ended up as a hero! Due to the oil price shocks in the 1970s, Brazil initiated the ethanol-fuel automobile production in the 1980s. But with low petroleum prices the owners of sugar mills preferred to make sugar instead of ethanol (switch output option), leaving the service stations without ethanol for customers. Ethanol car production practically disappeared with the fall of consumer confidence in the ethanol automobile. But in the 2000s a new technology appeared in Brazil: the flex-fuel car using either gasoline or ethanol. The flex-fuel car provided switch-input options for the consumer, offsetting the fear of the producer switch-output (sugar-ethanol) option. The flex-car was an immediate success. In the last years almost all automobiles sold in the Brazilian market are flex-fuel. The best antidote to the producers’ switch output option was the consumers’ switch input option! The flex-fuel technology increased consumer confidence and boosted the automobile market demand, leaving everybody better off: ethanol producers, automobile producers, and consumers. A real options success story! 2. How, and to what extent, is real options thinking used at Petrobras? Can you give specific examples of its use and its importance in influencing key decisions? There are many examples of application of real options at Petrobras. Some of these applications are listed below, including several public cases where

Uncertainty, Flexibility, and Real Options

173

Box 5.2 (continued)

the decision of the petroleum regulatory agency (ANP) was often favorable to Petrobras. i. The petroleum sector was opened up in Brazil in the late 1990s. In 1998, ANP presented for public debate a proposal for the duration of the exploratory phase, suggesting only 5 years for deepwater blocks exploration. We conducted analysis using real options, suggesting between 8 and 10 years. Subsequently ANP, once it became aware of the results, enlarged the duration to 9 years. Petrobras thus had more time to discover and appraise new oilfields; the recent large pre–salt discoveries needed more than 6 years to be discovered after the auction. RO thus contributed in the public debate with a lot of success! ii. In 2000 to 2001, the Brazil–Bolivia gas pipeline became a subject of litigation between TBG (a pipeline company controlled by Petrobras) and British Gas (BG) and Enersil for the pipeline free access. BG and Enersil wanted free access to the pipeline with flexibility to use it or not, while paying the same tariff as other companies without this flexibility (“takeor-pay” contracts). We demonstrated to ANP that this flexibility has value so that the tariff must be higher for BG compared with “take-or-pay” tariffs. The decision of ANP was to permit free access to the pipeline but require paying a higher price for the tariff. iii. In 2005, a biodiesel project was analyzed with real options because of the “flex” technology for the inputs, namely flexibility to use vegetable oils from soybean, cotton, castorbean, etc. The RO value was decisive for the project to get approved by the board of directors. iv. In 2003, Petrobras International solicited another real options project named “Strategic Valuation of E&P International at West Africa Offshore.” Using RO, we showed the importance to stay in Africa offshore. Petrobras kept its business there, and nowadays the African production is very significant for Petrobras’s international operations. v. We studied GTL (gas-to-liquid) technology using RO (switch-input and switch-output options) in 2006 to 2008. The study recommended the project (though the 2008 crisis and other priorities, such as large discoveries in pre-salt, put the project in wait mode). 3. Do you see a role for using game theory in conjunction with real options analysis? Yes, definitely. At Petrobras game theory is being taught since 2007, but separate from courses on real options. This course, although recent, motivated three real-life applications already. In the future, I think option games courses will be offered at Petrobras also and related applications will follow.

174

Chapter 5

can allow future decisions to be adapted to future market conditions. Static NPV represents an extreme case where the management commits at the outset (at t = 0) to a stringent plan of action, as if the utility could not alter its operational or investment decisions over time. Here we review basic option valuation principles to help equip those readers in need of a basic tool kit for valuing the real options an electric utility or any other firm faces. The two most known option-pricing models are attributed to Black and Scholes (1973) and Cox, Ross, and Rubinstein (1979).9 The Black–Scholes (BS) or Black–Scholes–Merton model involves advanced mathematics and notions of financial theory in continuous time, while the discrete-time multiplicative binomial model of Cox–Ross–Rubinstein (CRR) offers a more intuitive introduction to option pricing.10 Originally these models were designed to price financial options but have since been extended to valuing real options. Robert C. Merton discusses the development and impact of continuous-time finance, option-pricing theory, and the relevance of real options in box 5.3. The valuation approaches underlying the net-present-value (NPV) paradigm and real options analysis involve similar assumptions but alternative versions of risk adjustment. The traditional approach based on net present value essentially involves discounting expected cash flows at a discount rate k that reflects the nondiversifiable risk of the project. An alternative approach, that is core to option pricing, is to make the adjustment for risk to the expected cash flow rather than to the discount rate: the resulting certainty-equivalent cash flow can thus be discounted at the risk-free rate of interest r (instead of k ). Discrete-time and continuoustime models offer alternative ways to characterize these certainty equivalents. Discrete-time models are generally better suited when one needs to handle practical or complex valuation problems (e.g., portfolios of real options). The continuous-time approach assumes instantaneous decisionmaking. Discrete-time models are easier to implement, though continuous-time models have an appeal as they help better identify the theoretical value drivers and examine the underlying trade-offs. We first present the basic valuation idea in discrete time and later extend it briefly to the continuous-time context. 9. The Black–Scholes model is also often referred to as the Black–Scholes–Merton model to pay tribute to Robert C. Merton for developing much of the foundation work in Merton (1973). 10. The two setups are closely related. The discrete-time CRR model converges to the continuous-time Black–Scholes model for small time increments or large number of inbetween steps. In both cases the basic underlying idea is that in a complete capital market one can replicate over the next time period an asset’s price dynamics by creating a portfolio that provides the same payoffs in each state of the world.

Uncertainty, Flexibility, and Real Options

175

Box 5.3 Interview with Robert C. Merton, Nobel Laureate in Economics (1997)

1. Did you envision the widespread use or application of options thinking and tools when you first worked on options and stochastic-calculus applications? How did you come up with the idea? I recognized early on that option pricing had a much wider application than to stock options narrowly, such as using it to develop a unified theory for pricing the capital structure of the firm. I traded warrants, convertibles, and OTC options before I ever studied economics. At MIT I worked with Paul Samuelson in 1968 on warrant pricing, and separately attacked the intertemporal optimal lifetime portfolio problem with continuous trading using the Itô calculus as a tool to describe the dynamics of actual sample paths. When I discussed with Myron Scholes what he and Fischer Black were doing in discrete-time intervals between trades to find a dynamic strategy for hedging out the beta risk of an options/stock portfolio, it was natural to apply this same technique and show in the limit of continuous trading that one could hedge all (not just beta) risk and thus could replicate the payoffs to any derivative via dynamic portfolio trading in the underlying asset. 2. Paul A. Samuelson recently passed away. His seminal work spans many economic fields, including finance. Can you comment on his early contribution to continuous-time finance? In his 1965 rational theory of warrant pricing, Paul Samuelson introduced geometric Brownian motion for underlying stock prices; he generated the Kolmogorov equations for warrant pricing and with H. P. McKean derived

176

Chapter 5

Box 5.3 (continued)

pricing for the early exercise provision of American-style warrants. In an attached appendix to Samuelson’s published 12th von Neumann Lecture, I provided the connection between his warrant pricing model and the Black–Scholes model.a This was the first time that the Black–Scholes model appeared in published print. 3. To what extent has continuous-time finance reshaped the theory of pricing and valuation? The replication idea has been employed for decades in just about every venue of financial security pricing; my Nobel Lecture (Merton 1998) includes a list of various applications of the model methodology, including real options. 4. What are your views on the usefulness of real options analysis? Is real options analysis handicapped if the underlying asset (project) is not traded or portfolio replication of the embedded option is not readily available? I believe real options analysis is extremely useful. Real options valuation is no more handicapped than any of the other tools of valuation, such as NPV, where you need data on an equally risky asset to estimate beta, etc. Conceptually the model can be adapted to nontrading of the underlying asset and even to nonobservable interim prices (see Merton 1998, pp. 326–36). Since in all equilibrium asset-pricing models (e.g., the CAPM or arbitrage pricing theory) assets that have only nonsystematic or diversifiable risk are priced to yield an expected return equal to the riskless interest rate, and given that the portfolio tracking error (following a portfolio strategy that minimizes that error) is uncorrelated with all traded asset returns, the option pricing formula would apply even in those applications in which the underlying asset is not traded. a. Paul Samuelson was John von Neumann Lecturer at the SIAM Annual Meeting in 1971. This lecture resulted in the article by Samuelson (1973).

5.3.1

Discrete-Time Option Valuation

Assume that a new power plant would generate a stream of cash flows having present value V today (t = 0). In one period, the plant value can take one of two possible values: it will move up to V + or down to V − with real probabilities q and 1 − q, respectively. Figure 5.6 depicts the underlying asset’s value dynamics in a one-period binomial lattice. Assume further that this asset is traded in capital markets or that there

Uncertainty, Flexibility, and Real Options

177

V+ V V− t=0

1

Figure 5.6 Binomial lattice for the underlying asset (project) value

is a twin security traded in the market that has the same risk profile as the asset under consideration. Under traditional NPV analysis, the future values of the power plant (V + and V − ) are related to its present value (V ) as a discounted expectation: V=

E [V1 ] qV + + (1 − q) V − . = 1+ k 1+ k

(5.1)

Suppose that investing in the plant involves capital cost I . Investing immediately would result in project NPV = V − I . The standard (NPV) approach, however, is unable to properly value projects involving operational flexibility, such as the option to delay the investment for a year.11 Real options involve cash flows that are asymmetric on the downside versus the upside and are contingent on future uncertain events. Risk, and therefore discount rates, may vary in a complex way over time and across various future states. Optionality can be properly valued using option-pricing theory. The basic idea behind option valuation is that one can replicate an option by constructing a portfolio consisting of a (long or short) position in the underlying asset and a (short or long) position in a risk-free bond. This portfolio can be constructed so as to exactly replicate the future cash flows or returns of the option in each state of the world.12 Since the option and its equivalent portfolio would provide the same future returns in all states, they must sell for the same current value to avoid risk-free arbitrage profit opportunities. Equivalently, we could create a riskless replicating portfolio and discount the payoffs from this portfolio at the 11. When we consider a project with differing starting dates as mutually exclusive alternatives, one can capture a form of timing flexibility and act optimally by choosing the alternative with the highest NPV. This approach based on mutually exclusive alternatives extends the standard “NPV rule” asserting to invest in positive NPV projects. 12. We assume that the market is complete so that the replication argument holds.

178

Chapter 5

risk-free rate, r.13 To value the project as if in a risk-neutral world, one needs to utilize the so-called risk-neutral probabilities.14 Alternatively, the present value of any contingent claim can be obtained from its expected certainty-equivalent value discounted at the risk-free rate r. Through the ability to construct a riskless portfolio, investors’ risk attitudes do not matter in valuing the option and need not be explicitly considered. Therefore, we can equivalently—and more conveniently— obtain the option value as if we were in a risk-neutral world. In such a world all assets would earn the risk-free return (r), and so risk-neutral expected cash flows can be appropriately discounted at the risk-free rate. Let p denote the risk-neutral probability of an upward move (and 1 − p the probability of a downward move). The present value of the underlying asset, such as the power plant, in a risk-neutral world is V=

Eˆ [V1 ] pV + + (1 − p) V − , = 1+ r 1+ r

(5.2)

where Eˆ [⋅] denotes the risk-neutral expectation. The risk-neutral probability of an up move is then p≡

(1 + r ) V − V − V+ −V−

.

(5.3)

13. Consider a portfolio made of N shares of the underlying asset Vt and B risk-free bond(s) that pay €1 next period. Since the bond is risk-free, it yields in each state of the world a return equal to 1 plus the risk-free return r. Holding B bonds at time t is worth B (1 + r ) in one period. Consider a portfolio that replicates the payoffs of the option in each state. To avoid arbitrage opportunities in the capital market, this portfolio must sell for the same price as the option it replicates, that is, C + = N × V + − (1 + r ) B and C − = N × V − − (1 + r ) B. This system of two equations with two unknowns gives the following values for N and B N=

C+ − C− V+ −V−

and

B=

C (1 − H ) , 1+ r

with ⎛ C+ −C− ⎞⎛ V ⎞ H =⎜ + ⎟ − ⎟⎜ ⎝ V − V ⎠⎝ C ⎠ being a discrete measure of the elasticity of the option with respect to the underlying asset value. The position, N , invested in the asset to replicate the option payoff is the option’s hedge ratio (or option’s delta). In discrete time, it is obtained as the difference (spread) of the option prices divided by the spread of asset prices. 14. The NPV approach factors investors’ risk-aversion in the risk-adjusted discount rate k . The ability to construct a replicating portfolio enables the current value of the option claim to be independent of the actual probabilities or investors’ risk preferences. Option-pricing theory bypasses risk-aversion by valuing options as if investors were in a risk-neutral world.

Uncertainty, Flexibility, and Real Options

179

If the underlying asset evolves according to the multiplicative binomial process with up factor u (where V + = uV ) and down factor d (where V − = dV), the risk-neutral probability above is given by15 p≡

(1 + r ) − d u−d

.

(5.4)

Consider now a deferral option on the underlying power plant’s value. Suppose that this real option has value C + or C −, as a function of V + or V − , respectively, in one period. In a risk-neutral world, the expected return of this option must also equal the risk-free rate r. This results in the following fundamental option-pricing formula:16 C=

Eˆ [C1 ] pC + + (1 − p)C − . = 1+ r 1+ r

(5.5)

Example 5.1 Option to Invest (or Defer) Suppose that EDF has the option to build a power plant in Brittany this year or wait one year until regional elections are settled. EDF has no fear of preemption and, depending on the party chosen, may gain political support through subsidies. By waiting, EDF will forgo one period of profit but can benefit from the resolution of political uncertainty. If the plant is built immediately, EDF will receive gross project value V = C100 million (m) ; net of the capital investment cost I = C80 m, this results in an NPV of C20 m. Suppose that the plant’s value in one period will move up by 80 percent (or u = 1.8) or down by 40 percent (d = 0.6) depending on the outcome of next year’s elections, with each outcome equally likely (q = 0.5). A year later, the plant will have an expected value (from subsequent cash flows) of V + = C180 m if the 15. Note that d < 1 < 1 + r < u for the no-arbitrage argument to hold. Alternatively, formula (5.4) can be obtained by considering returns. In the up case, the return is R + ≡ V + V − 1 = u − 1. In the down case, it is R − ≡ V − V − 1 = d − 1. The risk-neutral probability, p, is obtained from the equilibrium condition that the expected return on the asset in a risk-neutral world must equal the risk-free rate r: pR + + (1 − p) R − = r, or

p≡

(1 + r ) − d u−d

.

Probability p is lower than the real probability of an upward move, q, since p accounts for investors’ risk-aversion, giving lower weight to upside events. p is the value probability q would have in equilibrium if investors were risk neutral. 16. Equation (5.5) provides a formula for the value of the option in terms of V , r and the asset volatility (u and d ). Effectively, u is the discrete-time equivalent of the continuoustime volatility parameter eσ h (which is a function of the volatility of the asset σ and the time step h). In the electricity case we can derive the risk-neutral probability p from the price dynamics of electricity prices using forward prices for electricity.

180

Chapter 5

outcome is favorable or V − = C60 m in the adverse case. Suppose that the expected rate of return or discount rate is k = 20 percent and the risk-free interest rate is r = 8 percent. The risk-neutral probability, obtained from equation (5.4), is p=

(1 + r ) − d u−d

=

1.08 − 0.6 = 0.4 ( < 0.5) . 1.8 − 0.6

If there are no options creating asymmetry in the payoff profile, the riskneutral valuation formula of equation (5.2) would yield the same value for the plant as traditional DCF valuation based on (5.1): 0.4 × 180 + (1 − 0.4 ) 60 1.08 0.5 × 180 + (1 − 0.5) 60 = 1.20 = C100 m.

V=

The option to defer the investment can be quite valuable, however, since the firm would invest only if prices and project value rise sufficiently, while it has no obligation to invest under unfavorable developments. The value of the plant launched one year from now is C + = max {180 − 80; 0} = 100 in the event of a positive election outcome, or C − = max {60 − 80; 0} = 0 in the event of a negative outcome. The value of the option to invest next year, based on equation (5.5), is C=

0.4 × 100 + 0.6 × 0 ≈ C 37 m. 1.08

EDF thus has a very valuable option to defer the investment for a year (37 > 20), waiting until further information about the outcome of political elections and the electricity prices in Brittany is revealed. Example 5.2 Option to Expand Alternatively, EDF may install new turbines in one of its wind farms in Brittany, expanding production scale by making a complementary investment if the newly elected regional government gives support to renewable energies. In year 1, EDF can choose to maintain base-scale operation (receiving project value V − or V + at no extra cost) or expand operations by 50 percent (e = 0.50) by adding capacity at extra cost I e = 40. The original investment opportunity can be seen as the base-scale project plus a call option on future growth, providing C + = V + + max {eV + − I e ; 0} = max {(1 + e ) V + − I e ; V + } after an up move,

Uncertainty, Flexibility, and Real Options

181

or C − = max {(1 + e ) V − − I e ; V − } in the adverse case. In the case at hand, EDF will choose to expand in the up case, obtaining C + = max {1.5 × 180 − 40; 180} = 230, but it will maintain the base scale in the adverse case earning C − = max {1.5 × 60 − 40; 60} = 60. The expanded NPV (including the value of the option to expand if the market grows) based on equation (5.5) is C=

0.4 × 230 + 0.6 × 60 ≈ C118.5 m . 1.08

Example 5.3 Interconnection Line The Italian electric utility Enel is considering installing an interconnection line with France that would enable it to benefit from electricity price differentials between the two countries. Suppose that the electricity price in France is constant at PF = 1 due to reliable nuclear power technology. The Italian electricity price, being PI = 1 today, will move up by 50 percent to PI+ = 1.5 (u = 1.5) or down by 33 percent to PI− = 0.67 (d = 2 3) with equal probability (q = 0.5) each year. If the Italian electricity price rises next year, Enel can take advantage of the transmission line importing cheaper electricity from France by buying at the fixed French price PF = 1 and receiving the higher Italian selling price PI+ = 1.5. This is analogous to the previous option to expand with payoff max {PI+ − PF ; 0} = 0.5. In the adverse case, if Italian prices drop to PI− = 0.67 next year, Enel can export electricity to France receiving the higher French price PF = 1; the option payoff is then max {PF − PI− ; 0} = 0.33. This option is analogous to a put that allows to receive the French price (strike price) of C1 by paying the lower Italian electricity price (underlying asset) of PI− = 0.67. The interconnection line thus provides Enel with two valuable options: one creating an upside potential to sell more electricity in Italy by importing cheaper nuclear power from France (a call option), and another limiting the downside risk to Enel by selling in France rather than in Italy if local prices drop (a put option). The price received by Enel will be P + = max {PI+ ; PF } or

P − = max {PI− ; PF }.

If local prices move up next year, Enel will charge P + = max {1.5; 1} = 1.5. If they move down, it will obtain the guaranteed (French nuclear) price P − = max {0.67; 1} = 1. Without the transmission line, Enel could only sell at local prices, being exposed to the full price fluctuation risk, receiving PI+ = 1.5 or PI− ≈ 0.67. The average (next-year) price in Italy would have been E [ PI ] = qPI+ + (1 − q) PI− = 0.5 (1.5 + 0.67 ) ≈ 1.1. With the transmission

182

Chapter 5

line, however, the average price is E [ PI ] = 0.5 (1.5 + 1) = 1.25 ( > 1.1). Thus the transmission line allows to sell at higher prices on average. The transmission line would be mispriced if managers ignore the embedded switching flexibility value and base their investment decision on the future average price, max {E [ PI ]; PF } = 1.1. Real options analysis, by considering the optimal switching decision at each period in each state, makes it possible to circumvent this “flaw of averages.”17 What is the correct “average” price discounting for the value of the transmission line? From equation (5.4), the risk-neutral probability of an up move (with risk-free rate r = 0.05) is p ≡ (1.05 − 0.67) (1.5 − 0.67) = 0.46. From equation (5.5) the value of the merchant interconnection line per unit of output is 0.46 × 1.5 + 0.54 × 1 ≈ 1.17 . 1.05 Outside Europe, Hydro-Québec, a state-owned corporation in Canada with large production capacity in hydro (95 percent), is profitably exploiting such transmission lines. Hydro reservoirs are very flexible resources, so that Hydro-Québec can increase or decrease output at very low cost almost overnight. In contrast, Ontario and the northeastern states of the United States rely mainly on nonflexible thermal power plants that virtually have a constant output, explaining why prices there are very sensitive to changes in demand patterns. At night, Hydro-Québec buys electricity from Ontario and the NE-US at very low prices (due to low local demand), downscaling its own production. During the day, it raises own production and sells them back electricity at high prices (high local demand).18 One can extend the above one-period binomial analysis to a larger number of steps though a repetitive process. This is discussed in appendix 5A and illustrated briefly in example 5.4 below for two steps. Example 5.4 Option to Invest or Defer (Two Steps) Consider a situation analogous to example 5.1, where Enel of Italy has the option to enter the Russian electricity market in two years’ time. Suppose again that the market value, currently C150 m (V = 150), follows a binomial process with an upward multiplicative factor u = 1.5 and 17. We can characterize this value difference by applying Jensen’s inequality to convex or concave functions. The payoff function f (⋅) of the call option is convex in the underlying factor X , such that E [ f ( X )] ≥ f ( E [ X ]). On the converse, a put has a concave payoff function, so that the opposite inequality holds. Savage (2009) applies Jensen’s inequality to challenge certain managerial practices based on the “flaw of averages.” 18. We thank Marcel Boyer for suggesting this example.

Uncertainty, Flexibility, and Real Options

183

downward factor d = 1 u ≈ 0.67 in each period. Enel must incur an infrastructure investment cost of I = C80 m to enter this geography at T = 2 years. The risk-free interest rate is r = 0.04. In the scenario following two consecutive up moves, the gross market value, V ++ = uuV = 1.5 × 1.5 × 150 ≈ C 338 m, exceeds the investment cost, I = C80 m. Enel will then enter the market, receiving a net (forward) investment opportunity value of C + + = max {338 − 80; 0} = C 258 m. In the intermediary state (that occurs after a downward move following an up move or inversely), the gross market value is V +− = udV = 1.5 × 0.67 × 150 = C150 m. If Enel invests, it receives V +− − I = 150 − 80 = C 70 m. It will thus again exercise its investment option at maturity, receiving C +− = C70 m. In the down state, after two consecutive down moves, the gross market value is V − − = ddV = 0.67 × 0.67 × 150 ≈ C 67 m. Enel will not invest I = C80 m for a negative NPV (67 − 80 = −13). It will instead abandon the investment option, receiving C − − = 0. Figure 5.7 shows the market value evolution and the investment option’s payoffs at maturity. The current investment option value is assessed by a backward process using risk-neutral valuation along the two-step binomial tree. The riskneutral probability of an up move, determined from equation (5.4), is p = 0.45. Going backward to t = 1, in the up state the investment option value is given by equation (5.5) as Binomial tree representing market value evolution

Option payoff

Option tree

338 C ++ = 338 − 80 = 258

C ++ = 258

C + = 148

225

150

p

p 150

C +– = 70

1−p

C = 80

C +– = 70 p

1−p

100

C – = 30 1−p

67

t=0

1

2

C

––

C –– = 0

=0

t=0

Figure 5.7 Binomial tree evolution and payoff for the option to defer (2 steps)

1

2

184

Chapter 5

C+ =

pC ++ + (1 − p)C +− 0.45 × 258 + 0.55 × 70 = ≈ C148 m. 1+ r 1.04

In the down state, the option has (time-1) value C− =

0.45 × 70 + 0.55 × 0 = C 30 m. 1.04

Finally, one step earlier, the present value (t = 0) of the investment option for Enel is pC + + (1 − p)C − 0.45 × 148 + 0.55 × 30 = ≈ C 80 m. 1+ r 1.04

C=

Alternatively, using the t = 2 expectation, based on applying the equation from appendix 5A for the binomial distribution with n = 2 steps: p2C + + + 2 p (1 − p) C + − + (1 − p) C −− 2

C=

(1 + r )2

0.45 × 258 + 2 × 0.45 × 0.55 × 70 + 0.552 × 0 1.04 2 ≈ C 80 m. =

2

That is, the current value of the investment opportunity for Enel to enter the Russian market in two years is &80 m, which is more that the value to enter it today. The net present value of entering the market immediately (at t = 0) is NPV = V − I = 150 − 80 = C 70 m. 5.3.2

Continuous-Time Options Analysis

Real options analysis can be either dealt with in discrete or in continuous time. Each approach has its own merits. The continuous-time approach is more helpful to derive basic investment principles.19 The CRR discrete-time analysis rested on the assumption that the underlying asset, Vt, follows a multiplicative binomial process. In continuous time this corresponds to assuming that the underlying asset follows the geometric Brownian motion (GBM) dV = ( gV ) dt + σ V dz,

(5.6)

where g denotes the actual growth trend of the process, σ 2 its variance and z is a standard Brownian motion.20 19. Continuous-time models are useful when model assumptions enable the derivation of analytical solutions. 20. See the appendix of the book for detail on the GBM.

Uncertainty, Flexibility, and Real Options

185

Black and Scholes (1973) use this process to derive their famous formula for pricing European call options. At maturity T , a European call option pays off the greater of the net value VT − I or zero, meaning CT = max {VT − I ; 0}. Under risk-neutral valuation, the call option at time t = 0 is worth C = e − rT Eˆ [CT ], where r is the continuously compounded risk-free interest rate.21 The Black–Scholes (BS) formula for the value of a European call option (on a nondividend-paying asset) is C = V N( d1 ) − Ie − rT N(d2 ),

(5.7)

where N (⋅) is the cumulative standard normal distribution.22 Parameters d1 and d2 are given by d1 =

ln (V I ) + [ r + (σ 2 2)]T

σ T d2 = d1 − σ T .

,

(5.8)

This formula is a cornerstone in option-pricing theory. One can obtain it based on the replicating portfolio argument as in Black and Scholes (1973) and Merton (1973).23 The discrete-time and continuous-time approaches are not really competing paradigms; they view the same problem from different mathematical angles. The multistep Cox–Ross–Rubinstein (CRR) option-pricing formula is derived in appendix 5A. For a given maturity, T , as the time 21. Under the risk-neutral probability measure, the asset price does not exactly follow the GBM of equation (5.6), but an adjusted GBM involving a “risk-neutral drift,” gˆ , equal to r in the Black and Scholes model. 22. Let p (·) denote the probability density under the risk-neutral measure. By decomposing C , we obtain ∞

C = e − rT ∫ max {VT − I ; 0} p(VT ) dVT 0

∞

= e − rT ∫ (VT − I )p(VT ) dVT I

= e − rT × ξ − e − rT I × θ , where θ ≡ Pr (VT ≥ I ) and ξ ≡ Eˆ [VT | VT ≥ I ] are, respectively, the probability that the option ends up “in the money” and gets exercised at maturity, and the expected value of the asset when it does. From equations (A.16) and (A.19) in the appendix of the book, θ = N ( d2 ) and ξ = Ve rT N ( d1 ) . We employ the risk-neutral drift term r, here, instead of g. 23. Our derivation of BS in note 21 rests on probabilistic properties of the European call option and of geometric Brownian motion. The replicating-portfolio approach generalizes to a larger family of processes (Itô processes) and can accommodate other payoff functions. As noted, under mild conditions, it is possible to create a portfolio replicating the payoff to a call option. This portfolio consists of N shares of the underlying asset and a short position, B, in a risk-free bond. In continuous time, N = CV (V ) and B = e − rtC (1 − ε ) , with ε ≡ CV (V ) × V C . In the BS case involving geometric Brownian motion, the hedge ratio (position in the underlying asset) used in the replicating portfolio is N = CV (Vt ) = N( d1 ) .

186

Chapter 5

step, h ≡ T n, becomes increasingly smaller (as h approaches 0 or the partition of the time interval becomes finer with n going to infinity), continuous replication is approximated and the multiplicative binomial process converges to the geometric Brownian motion. In the binomial model the parameters u, d , and p are consistent with their continuoustime counterparts if 24 u = eσ

h

1 and d = . u

(5.9)

Under these conditions the multistep CRR binomial option-pricing formula, discussed in appendix 5A, converges to the BS formula in (5.7).25 Example 5.5 Option to Invest (Black–Scholes) Let us revisit example 5.4 above. Enel may delay investment in the Russian electricity market for two years until the true market development is revealed. Underlying project value is V = 150. Suppose that this value fluctuates continuously over time (driven by global energy supply and demand factors) with volatility σ = 40%.26 The risk-free interest rate is r = 4%. The investment outlay for the power plant is I = 80. From equation (5.8), d1 =

ln (150 80 ) + [ 0.04 + ( 0.4 2 2)] 2 0.4 2

≈ 1.54,

and d2 ≈ 1.54 − 0.4 2 ≈ 0.97. From the cumulative standard normal distribution, N (d1 ) ≈ 0.94 and N (d2 ) ≈ 0.84. From the Black–Scholes formula of equation (5.7), the investment option value is C = 150 × 0.94 − 80 × e −0.04 × 2 × 0.84 ≈ C 79 m. This is close to the value obtained in example 5.4 above using the CRR binomial model (C80 m) with only two steps (n = 2). 24. The probability of an up move is given by p=

1 2

+

1 αˆ ′ 2 σ

h,

where αˆ ′ ≡ ln r − (σ 2 2) . Cox, Ross, and Rubinstein (1979) show that, in the limit, the complementary binomial distribution function B used in appendix 5A converges to the cumulative standard normal distribution function N(⋅). 25. The CRR formula converges to the BS formula if h ≤ σ 2 αˆ 2 , with αˆ ≡ r − (σ 2 2). 26. This volatility value, σ = 40 percent, is close to u = 1.5 in example 5.4, as given by u = eσ h .

Uncertainty, Flexibility, and Real Options

187

The Black–Scholes formula of equation (5.7) deals with a relatively simple situation where a single risk factor evolves stochastically. In reality, several risk factors may affect firm value. Margrabe (1978) examines a more general situation where a firm can exchange one (nondividend-paying) stochastic asset (I) for another (V). For example, the capital investment cost as well as the project value may evolve stochastically. Suppose that both project value, Vt, and investment cost, I t, follow correlated geometric Brownian motions dV = ( gV V ) dt + σ V Vdz, dI = ( g I I ) dt + σ I I dz′.

(5.6′)

These factors are assumed correlated with correlation coefficient ρ. We want to value a European call option giving the right to receive at maturity uncertain project value VT by paying uncertain investment cost IT , i.e., C (V , I ) = e − rT Eˆ [max {VT − IT ; 0}]. One way to simplify the problem is to define project value, V , in units of investment outlay, I , thereby reducing the problem dimensionality by one dimension (from two to one). Let X ≡ V I be this composite (relative) stochastic factor, with relative (instantaneous) variance given by

σ X 2 = σ V 2 + σ I 2 − 2 ρσ V σ I .

(5.10)

The value of the investment outlay I in terms of itself is 1 and the interest rate on a riskless loan (paying no dividend) in units of I becomes 0. The Black–Scholes formula still applies, with two needed adjustments: (1) r = 0 and (2) σ 2 becomes σ X 2 as given by equation (5.10).27 This results in Margrabe’s (1978) extension for the European option to exchange stochastic cost I for V at maturity T : C(V , I ) = V N(d1 ′ ) − I N( d2 ′ ),

(5.11)

where 1 ln (V I ) + σ X 2T 2 d1 ′ = , σX T d2 ′ = d1 ′ − σ X T .

(5.12)

27. Given these adjustments and the homogeneity of the option value function, we obtain from Black–Scholes formula (5.7) that C (V , I ) = X N ( d1′ ) − 1e −0 T N ( d2 ′ ) . I

188

Chapter 5

Example 5.6 Option to Temporarily Shut Down Operations28 The Italian electric utility is equipped with CCGT technology. The generation cost, Ct , is driven by two factors: the input gas price, PtG, which evolves stochastically according to dPtG = (rPtG ) dt + σ G PtG dzt, and technological efficiency, measured by a constant heat rate H.29 The output electricity price, Pt E, follows the geometric Brownian motion, dPt E = (rPt E ) dt + σ E Pt E dzt. Electricity and gas prices are correlated with coefficient ρ . The electric utility is not compelled to operate the CCGT plant at any time t if variable costs are not covered. The contingent profit stream at time t is thus π t ≡ max {Pt E − Ct ; 0}, where the generation cost, Ct = H × PtG, where H is the efficiency coefficient, is the exercise price of this European call option. The present value of the time-t contingent profit π 0(t ) can be valued (as of time 0) using Margrabe’s formula from (5.11) with maturity t. From equation (5.11) the expected present value of π 0( t ) = C( P0E , P0G ) is C( P0E , P0G ) = P0E N( d1′ ) − H × P0G N( d2 ′ ),

(5.11′)

where H × P0G is the present value of the (time-t) exercise price. The parameters d1’ and d2’, obtained from equation (5.12), are given by 1

d1′ =

ln ( P0E ( H × P0G )) + σ X t 2

σX t

(5.12′)

and d2 ′ = d1′ − σ X t with σ X2 = σ E2 + H 2 × σ G2 − 2ρHσ E σ G following (5.10).30 Suppose that the current prices of electricity and gas are P0E = C12 and P0G = C10, with efficiency coefficient H = 1. Electricity and gas prices have volatility σ E = 30 percent and σ G = 20 percent, with correlation coefficient ρ = 0.40. From equation (5.10), σ X2 = 0.32 + 12 × 0.2 2 − 2 × 0.4 × 1 × 0.3 × 0.2 ≈ 0.08. For a two-year horizon (t = 2) we have 28. The present example is adapted from McDonald (2006, ch. 17). 29. The heat rate H corresponds to the number of British thermal units necessary for the generation of one MWh of electricity. 30. McDonald and Siegel (1985) derive the closed-form solution for this problem. Assuming that the two assets pay no dividends, the general solution in McDonald and Siegel (1985) obtains as Margrabe’s (1978) formula for the option to trade one risky asset for another.

Uncertainty, Flexibility, and Real Options

d1′ =

ln (12 (1 × 10)) + (0.08 2 2) × 2 0.08 2

189

≈ 1.67 ,

d2 ′ = 1.67 − 0.08 2 ≈ 1.55, so N (d1′ ) ≈ 0.95 and N (d2 ′ ) ≈ 0.94. Having flexible operations with the possibility to shut down operations in two years results in a value, based on equation (5.11), of π 0 ( 2 ) = C(12, 10 ) = 12 × 0.95 − 1 × 10 × 0.94 ≈ 2.0. Assuming the CCGT plant will be decommissioned in T years, the power plant that embeds such operational flexibility in each year t has total T value ∑ t = 0 π 0( t ). Conclusion We discussed multiple sources of risk affecting companies in the energy sector and the breakthrough innovation of real options analysis. Optionbased valuation is a useful tool to corporate managers and strategists, providing a consistent and unified approach toward incorporating the value of real options associated with the investment decisions of the firm. We discussed how to quantify in principle the value of various types of operating options embedded in capital investments with applications in the energy sector. Some of these options enhance the upside potential (e.g., options to defer or expand), while others reduce downside risk (e.g., options to contract, switch use, interconnect or default on staged planned outlays). This valuation approach serves as foundation for the option games approach developed later in the book. Selected References The International Energy Agency (2007) identifies various challenges ahead for companies in the energy sector and suggests possible ways to tackle the business risks involved. Trigeorgis (1996) offers a broad overview on real options. Dixit and Pindyck (1994) discuss various real options in continuous time. Cox, Ross, and Rubinstein’s (1979) binomial model is widely used for pricing options in discrete time. Black and Scholes (1973) and Merton (1973) derive the key properties and formula for valuing European call options. Black, Fischer, and Myron S. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81 (3): 637–54.

190

Chapter 5

Cox, John C., Stephen A. Ross, and Mark E. Rubinstein. 1979. Option pricing: A simplified approach. Journal of Financial Economics 7 (3): 229–63. Dixit, Avinash K., and Robert S. Pindyck. 1994. Investment under Uncertainty. Princeton: Princeton University Press. International Energy Agency. 2007. Tackling Investment Challenges in Power Generation. Paris: IEA Publications. Merton, Robert C. 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4 (1): 141–83. Trigeorgis, Lenos. 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. Cambridge: MIT Press.

Appendix 5A: Multistep Cox–Ross–Rubinstein (CRR) Option Pricing Consider a European call option that gives the right—but not the obligation—to receive at maturity the underlying asset value VT by paying the exercise price or investment cost I . This call option pays off at maturity (T ) the greatest of VT − I and 0, or CT = max {VT − I ; 0}. Suppose that the time to maturity T is divided into n time steps, each of equal length h ≡ T / n, and let j be the number of up moves in n time steps. At maturity in the last period n, after j up moves, the call option pays off C n = max { u j d n− jV − I ; 0}. The risk-neutral probability of an up move, p, over an interval is given in equation (5.4).31 We can generalize equation (5.5) to obtain a binomial option pricing formula for n periods C=

Eˆ [CT ]

(1 + r )T

=

⎤ ⎡ n ⎛ n⎞ j n− j p (1 − p) max {u j d n − j V − I ; 0}⎥, T ⎢∑ ⎜ ⎟ (1 + r ) ⎣ j = 0 ⎝ j ⎠ ⎦ 1

(5A.1)

where n! ⎛ n⎞ ⎛ n⎞ j n− j ⎜⎝ j ⎟⎠ ≡ (n − j )! j ! and ⎜⎝ j ⎟⎠ p (1 − p) denotes the binomial distribution giving the probability that the asset will take j upward jumps in n steps, each jump occurring with risk-neutral 31. As discussed elsewhere, in equation (5.9), for consistency the value of parameters u and d should be related to the time increment, h.

Uncertainty, Flexibility, and Real Options

191

probability p. The summation of all possible (from j = 0 to n) option values at expiration, multiplied by the state probability that each scenario will occur, gives the expected option value at maturity T ( n) . This expected option value, Eˆ [C ], is discounted at the risk-free rate r over the n time periods. If we let m be the minimum number of up moves (over n periods) “triggering” call option exercise (above which V > I ), the binomial option-pricing formula in (5A.1) becomes C = VB ( m, n, p′ ) −

I

(1 + r )n

B ( m, n, p),

(5A.2)

where B( ) is the complementary binomial distribution function that gives the probability of at least m up moves in n periods n ⎛ n⎞ n− j B ( m, n, p) ≡ ∑ ⎜ ⎟ p j (1 − p) , ⎝ ⎠ j j=m

with p′ ≡

u p. 1+ r

II

OPTION GAMES: DISCRETE-TIME ANALYSIS

In the first part of the book, we introduced basic principles of strategic management and real options, and discussed how an industrial organization perspective may provide useful insights to managers about how to behave vis-à-vis rivals under certain business conditions. Our belief is that these separate approaches should be combined, leveraging the strengths of each discipline to provide enhanced managerial guidance, dealing concomitantly with both market and strategic uncertainties. In part II of the book, we put these perspectives together, discussing some useful models at the interface between game theory and real options as part of a unified, discrete-time analysis. Discrete-time option games are more suitable to help explain intuitively the basic concepts and logic behind the timing and interactions of real investment decisions and determining optimal investment strategies to guide the behavior of rational option holders. In what follows we introduce concepts and tools gradually in terms of increasing complexity, starting first with a simple “equilibrium selection procedure” and building up in terms of complexity over subsequent chapters. We illustrate the simple idea behind option games via illustrative examples in chapter 6. In subsequent chapters we analyze how strategic interactions among rivals may alter the firms’ behavior in cases where the investment opportunity is analogous to a shared European call option, circumventing the issue of optimal timing under rivalry.1 We base much of this analysis on Smit and Trigeorgis’s (2004) extension 1. Investment opportunities are often viewed as being analogous to American call options. To simplify, we start with the simpler case of European options that give the option holder only one investment exercise possibility, at maturity. Using European options has two main advantages. First, it allows circumventing the problem of optimal timing in multiplayer settings, thereby precluding preemption or war of attrition effects. Second, for discrete-time analysis with finite horizons, it enables us to have investment triggers that are independent of the time or stage considered. Here the investment trigger is fixed and prescribes the investment decision at the end period (maturity).

194

Part II

of Fudenberg and Tirole’s (1984) framework for analyzing business strategies under uncertainty in light of whether firms react in a reciprocating or contrarian manner. Exogenous demand uncertainty is modeled via discrete-time multiplicative binomial models, as in Cox, Ross, and Rubinstein (1979). Uncertainty arising from strategic interactions at each end demand node is modeled via two-by-two matrices (in strategic form), whose equilibrium payoff replaces the node payoffs in the standard binomial tree. The cases of Cournot quantity and Bertrand price competition are analyzed in discrete time, both in the context of a proprietary R&D investment and when there are spillover effects (shared benefits). An extension and application in case of two-stage R&D investment games is also provided. In chapter 7 we discuss how the above principles can be applied in the context of Cournot quantity and Bertrand price competition. In chapter 8 we describe how commitment and early investment in two-stage games (e.g., in R&D or advertising campaigns) may alter the strategic value of an investment opportunity. Whether the firststage commitment investment increases or decreases the option value depends on the effect of the commitment (tough or soft) and on the reactions of competitors (strategic substitutes or complements). The analysis is extended to derive optimal investment strategies under uncertainty, providing new insights about R&D investment, spillover effects, goodwill building, and patent strategies.

6

An Integrative Approach to Strategy: Option Games

In part I we discussed basic approaches providing insights about how to behave in an uncertain competitive environment. We reviewed strategic management, industrial organization, and real options as separate, standalone disciplines. Each approach can separately bring about useful insights in analyzing business situations. Nonetheless, each discipline, taken separately, also has drawbacks. Standard game theory has not dealt so far with stochastic dynamics; static games by their very nature do not involve such a problem, while dynamic games are mostly cast in a steady or deterministically evolving environment. Real options analysis, while suitable to account for stochastic uncertainty, often makes the simplifying assumption that strategic interactions do not materially affect the investment decisions or project values of firms. Strategic management has a lot to benefit by more explicitly incorporating the trade-off between the value of flexibility and commitment. The best way to overcome these drawbacks is to follow an integrative approach using the combined tools and concomitant insights of game theory and real options. This is what “option games” is all about, standing at the intersection between the theory of investment under uncertainty (real option analysis) and the game-theoretic analysis of strategic investments. In section 6.1 we discuss how option games can provide useful insights to management faced with conflicting strategic choices. We discuss two prominent issues here: optimal investment timing, and the trade-off between flexibility and commitment.1 We elaborate on these issues further in upcoming chapters. Section 6.2 provides a simple illustration to help figure out the logic behind option games. Sections 6.3 and 6.4 go deeper and discuss applications to R&D and the mining/chemicals industry, respectively. 1. These two issues are not really that distinct. Optimal timing is concerned with the choice of the right moment at which to incur a sunk investment cost, which is a form of commitment. Thereby the firm kills the option to delay further, losing valuable flexibility.

196

Chapter 6

6.1 Key Managerial Issues: Optimal Timing and Flexibility versus Commitment 6.1.1

Optimal Investment Timing under Uncertainty

Consider the example of a multinational financial institution, such as Deutsche Bank, contemplating investing in China where profits have been growing and prices escalating. Deutsche Bank ponders whether to acquire a stake in a local bank at the earliest opportunity or wait to see if prices subside as the hype diminishes. Many international organizations face such a dilemma. The market in China is not fully liberalized and administrative barriers have long blockaded entry by foreign financial institutions. These foreign institutions are faced with a trade-off between investing now incurring a large investment outlay or waiting for the liberalization of the market in the wake of WTO negotiations.2 A good understanding of investment timing requires a simultaneous assessment of both market uncertainty and the anticipated reactions of competitors eager to seize their slice of the pie. Should the firm invest now? Invest later? Abandon the project altogether? If the market develops favorably, investing can prove quite valuable. But if the market flops, investing prematurely may prove a regrettable mistake, especially when investment costs are sunk. To protect itself from adverse market developments, a firm that has the flexibility to wait will require a delay investment premium until the market is sufficiently mature to compensate for the risk. Real options analysis can provide valuable insights concerning the optimal timing decisions of firms under uncertainty. Investing when the project NPV is positive is inferior to selecting optimally the time at which the project value is maximal, with due accounting for the time value of money. Optimal investment timing under uncertainty is a key challenge. In competitive settings, the optimal timing policy may differ substantially from that of a monopolist. Rivals may preempt, eroding the incentive to defer the decision. In other situations, rivals may prefer to wage a war of attrition in the hope that their rival will retreat sooner. In these settings the optimal timing decisions, as the result of a multiplayer problem, 2. Deutsche Bank actually decided not to wait, acquiring several local banks such as Harrest Fund Management and Huaxia Bank. In their decision-making, Deutsche Bank took into account the risk of international rival banks preempting it.

An Integrative Approach to Strategy

197

must form an industry equilibrium. In such investment problems the presence of competition generally leads firms to invest earlier than a monopolist and erodes the deferral option value of the firm. 6.1.2 The Trade-off between Flexibility and Commitment In an uncertain competitive environment, any company considering a capital-intensive decision, like whether to invest in a new technological process to develop a new product, faces a trade-off between investing early to build up competitive advantage over rivals versus delaying investment to acquire more information and mitigate the potentially unfavorable consequences of market uncertainty. At the heart of this trade-off between commitment and flexibility are two conflicting value drivers: (1) exogenous uncertainty in the form of fluctuating demand, prices, or input costs may restrain a firm from committing early, and (2) in the face of competitive threat or pressure, a firm might be better off to build first-mover competitive advantage. The value of commitment as discussed in industrial organization challenges the belief that flexibility is always valuable.3 One thus needs to weight the relative merits of commitment versus flexibility, providing a balancing act between the two. This trade-off can be addressed by combining real options analysis and game theory. In boxes 6.1 and 6.2 senior consultants from BCG and McKinsey & Company discuss the usefulness of real options analysis for management practice and the prospects offered by taking a game-theoretic perspective. 6.2 An Illustration of Option Games Viewed in discrete time, an option game is an overlay of a binomial tree onto a payoff matrix.4 A binomial tree, as the one shown in figure 6.1, is 3. The reader may recall that chapter 4 dealt with the value of early commitment (industrial organization), while chapter 5 focused on the flexibility value via real options analysis. 4. In real options analysis, two approaches are commonly used: (1) the discrete-time approach, using binomial trees depicting the evolution of the demand state, and (2) the continuous-time approach, using stochastic processes to model the underlying variables. In chapter 5 we described how project value dynamics can be modeled based on a binomial tree lattice, how risk-neutral probabilities are derived, and how the current value of an investment option can be determined using the backward risk-neutral valuation principle. Smit and Ankum (1993), Smit and Trigeorgis (2001, 2004), and Ferreira, Kar, and Trigeorgis (2009) discuss option games in discrete time. This approach enables deriving key insights by use of simple numerical examples.

198

Chapter 6

Box 6.1 Interview with Rainer Brosch and Peter Damisch, Boston Consulting Group

1. In what ways do you believe real options thinking can contribute to strategy development? How useful have you found real options thinking in your consulting practice? We believe that real options thinking can contribute to strategy development in a very powerful way: it helps to analytically support strategic decision-making, especially in environments that are complex and exposed to many sources of uncertainty. The approach provides structural elements and a “vocabulary” that can be very valuable in strategy discussions that, by definition, center around managerial flexibility, options, and contingent decisions. A question like “how would the exercise price of a strategic option be affected by a specific managerial action?” could very well uncover a key issue. In many contexts a detailed valuation of business cases or strategies is an important element of our work; this typically can be achieved with much precision by an in-depth DCF and scenario analysis. However, in settings where significant uncertainty and managerial flexibility is involved, such as flexibility to change trajectory, abandon, or accelerate, the value of such optionality must be considered explicitly: this flexibility possibly allows capitalizing on positive market developments or reducing the impact of negative market developments, and thus impacts strategic choices. In every practical setting it is important to rigorously assess possible sources of optionality and decide how to best capture them. 2. Would you envision extending BCG matrix thinking using options and games for competitive analysis? Portfolio thinking is inseparably connected to strategy and is among the most challenging—and rewarding—aspects of management. Looking at a company as a portfolio of businesses opens up an important perspective, enabling to actively realize value which is above and beyond the sheer

An Integrative Approach to Strategy

199

Box 6.1 (continued)

sum of the parts. At the same time, this lens requires to actively address trade-offs within the portfolio, namely to tackle portfolio effects and synergies that are the basis of the portfolio perspective. From its first days BCG has pioneered strategy through portfolio analysis. The BCG Portfolio Matrix is one well-known example, capable of providing awareness and tremendous insight. Over the years we have continued to develop strategy and portfolio analysis in many directions, such as by integrating additional parameters, including among these strategic fit, value creation potential, role-based views, dynamic paths and trajectories, options, and competitive interaction. Today’s portfolio manager can rely on very advanced approaches, consolidating various views which are inseparably connected to options thinking and competitive analysis. Admittedly, it is always a challenge to highlight competitive interaction and even more so when options are involved. This is exactly what option games is about. Option games thus make a very important and directional contribution to strategy, tying together game theory and options thinking into a joint, analytically rigorous framework.

Box 6.2 Interview with Eric Lamarre, McKinsey & Company

1. In your corporate finance and risk management practice with clients at McKinsey & Company, how, and to what extent do you believe real options analysis offers better guidance to your clients than traditional capital budgeting techniques?

200

Chapter 6

Box 6.2 (continued)

There is no doubt that real options analysis generates a more insightful dialogue among senior managers. It helps uncover additional value not fully recognized by traditional NPV techniques and, more important, creates a structured conversation around the sources of risk and how a company plans to respond to these risks. While a typical investment business case merely lists potential risks, RO techniques place much greater emphasis on these risks, as well as on the managerial flexibility to react. 2. How did you first come up with or run into the option games concept? We first came across this concept in the mining industry. Option games are particularly helpful for a special class of strategy problems where uncertainty is high and a company’s actions or those of competitors can shape industry dynamics. This is sometimes the case in mining when major capacity expansions can materially impact the supply/demand balance. The first mover in adding capacity will hold an advantage, but must also take on more risk because future demand is uncertain. Managers intuitively understand this dynamics, but they may find that quantifying the value is difficult with standard DCF methodologies. 3. How useful have you found real options analysis in your consulting practice? Have your clients shown interest in this approach? Despite its compelling benefits, clients have been slow to adopt the RO approach for three reasons. First, often too much focus is placed on analytics and not enough on the organizational changes required to generate broad adoption inside a corporation. Second, having received little training on RO techniques, senior managers often revert to familiar methods despite the shortcomings these methods present in certain situations. Finally, poor execution during first experimentations often leads to disappointing results. There is sometimes too much focus on computer simulations and not enough on structuring the risk problem properly. If properly used, this new methodology can be a real source of competitive advantage, even more so when integrated within the organizational and strategy processes.

An Integrative Approach to Strategy

State value

201

Cumulative probability

p p

pp

V+ 1− p

V

p 1− p

V ++

V +− = V −+ 2 p (1 − p )

V− 1− p

V −−

2

(1 − p )

Figure 6.1 Binomial tree representing evolution of market uncertainty and associated probabilities

used to model the stochastic evolution of project value V , while two-bytwo matrices are used to capture the competitive interactions among players. Suppose that p is the risk-neutral probability of an up move per period. To each scenario at the end node in the binomial tree of figure 6.1 corresponds the cumulative risk-neutral probability shown in the right column after two steps: pp for the high demand state, 2 p (1 − p) 2 for the intermediary demand state, and (1 − p) for the low demand state. Consider a duopoly consisting of firms i and j sharing a European option to invest in an emerging market within two years. The annual risk-free interest rate is 4 percent (r = 0.04). Firms i and j can both invest now, wait and invest later (at maturity in year 2), or let the option expire. If none invests now, at the end node in year 2 the firms’ strategic choices (represented in two-by-two payoff matrices) are: invest or do not invest (abandon). At maturity both firms can invest, both can abandon, or only one invests (potentially involving a coordination problem). The basic structure of this option game in discrete-time is depicted in figure 6.2. Once the binomial tree charts the evolution of potential demand scenarios until maturity (year 2), in each end node a two-by-two payoff matrix depicts the resulting competitive interaction. The resulting equilibrium outcome (*) and corresponding player payoffs can be anticipated for each of the three payoff matrices. Once the equilibrium (*) strategic option values are obtained in each end state (C*++, C*+−, C*−−), working the tree backward enables the firm to assess the value each strategy creates under rivalry. The analysis reveals the benefits to each player from

202

Chapter 6

Binomial tree representing market uncertainty

Payoff matrix for Strategic option strategic uncertainty value Firm j

V

Invest

V+ V +−

V−

Invest

Invest Abandon

C*+ −

Abandon

V

C*++

Abandon

High demand ++

Firm i

Invest Abandon

t=0

1

2

Abandon

V −− Low demand

Invest

Invest Abandon

C*−−

Figure 6.2 Structure of an option game involving both market (demand) and strategic (rival) uncertainty

pursuing a given strategy and enables management to determine how these benefits might change if certain key variables, such as growth or volatility, would change. Let us put this in a concrete context from the electricity sector (see example 5.4). Suppose that the Italian utility Enel and EDF of France both possess the option to enter the Russian electricity market in two years’ time.5 This investment option is analogous to a shared European call option with maturity two years, with the underlying market value following a multiplicative binomial tree process. Suppose that the market is currently worth C150 m (V = 150) and evolves with an upward multiplicative factor u = 1.5 and downward factor d = 1 u ≈ 0.67 in each period. If only one firm invests when the market is mature in year 2, that firm would seize the entire market pie (in effect will be a monopolist). If both utilities invest at t = 2, we assume that the market will be divided among the two firms relative to their market power. Suppose that the Italian 5. The intent here is to value, for each firm, the option to invest in this market as incorporated in the firm market value. A different issue might be of interest as well. Suppose the Russian government is selling two options to invest, with Enel and EDF as bidders. What will they bid? This analysis should include the possibility that one of the utilities may find it optimal to buy up both options and then exercise only one (if the rules and regulations made by the Russian government permit this). Such a problem might have a different answer.

An Integrative Approach to Strategy

203

utility (Enel) has a comparative advantage enabling it to acquire a higher slice of the market pie (s = 0.60), with its French rival obtaining the remainder (40 percent). Each firm must incur an infrastructure investment outlay of I = C80 m to enter the Russian market. At maturity in year 2, each firm will either invest (receiving the net project value at that time) or abandon the investment (receiving nothing). The values obtained by each firm in each competitive scenario are presented in two-by-two matrices; each of the tree end nodes of demand has a corresponding two-by-two matrix. Consider first the upper demand state in figure 6.3 that occurs after two consecutive demand up moves. In this case, the gross market value is V ++ = uuV = 1.5 × 1.5 × 150 ≈ C 338 m. In the event where only one firm (Enel or EDF) enters the market, it gets the full gross market value V ++ incurring the entry cost of I = C80 m, receiving a net (forward) value of V ++ − I = 338 − 80 = C 258 m. If the firm does not enter the market, letting its investment option expire, it receives nothing (0). If both firms invest at maturity (invest, invest), they split the market as follows: Enel obtains 60 percent of the gross market value (338) at the cost of I = C80 m, Binomial tree representing market uncertainty

Payoff matrix for Strategic option strategic uncertainty value Firm j

Firm i

338

Abandon

High demand

Invest

Invest Abandon

225

C*++ = (123, 55) (0, 258)

(0, 0)

150

100

Invest

Invest Abandon (10, –20)

Abandon

150

(123, 55) (258, 0)

(0, 70)

(70, 0)

C*+− = (70, 0) (0, 0)

Low demand

t=0

1

2

Abandon

67

Invest

Invest Abandon (–40, –53) (–13, 0)

C*−− = (0, 0) (0, –13)

(0, 0)

Figure 6.3 Binomial tree and equilibrium end-node payoff values in the Enel versus EDF rivalry over the Russian market The first element in (·, ·) represents firm i ’s (Enel’s) net value or payoff, while the second entry denotes the net value or payoff to firm j (EDF).

204

Chapter 6

Invest

(123, 55) (258, 0)

Abandon

Firm i (Enel)

Firm j (EDF) Invest Abandon

(0, 258)

(0, 0)

Figure 6.4 Two-by-two matrix in the upper demand node The first element in (·, ·) represents firm i’s (Enel’s) net value or payoff, while the second entry denotes the net value or payoff to firm j (EDF).

receiving net (forward) value: s × V ++ − I = 0.6 × 338 − 80 = C123 m; EDF gets 0.4 × 338 − 80 = C 55 m. These competitive situations are summarized in the two-by-two matrix of figure 6.4. The first number in each pair of payoff values (in parenthesis) represents the payoff to Enel (firm i), and the second to EDF (firm j). We next determine which competitive outcome is more likely to occur as a stable solution, applying the Nash equilibrium concept. Whatever EDF chooses, Enel earns a higher payoff by investing than by abandoning the project: if EDF invests, Enel receives C123 m when investing and 0 otherwise; if EDF abandons the option, Enel is better off investing obtaining net value C258 m rather than 0. Enel thus has a dominant strategy to invest. EDF also has a dominant strategy to invest, earning !55 m if Enel invests or C258 m if Enel lets its option expire. The resulting Nash equilibrium in the upper end node is for both firms to invest, with Enel receiving C123 m and EDF !55 m. The equilibrium payoffs, C*++ = (123, 55), are shaded or shown with *. The case arising in the medium demand state (occurring after a downward move following an up move or inversely) can be analyzed similarly. Here the gross market value is V − + = udV = 1.5 × 0.67 × 150 = C150 m. In the intermediate demand state there is room for only one firm to operate profitably in the market (150 < 2 × 80). If Enel or EDF ends up in a monopoly, it receives V − + − I = 150 − 80 = C 70 m. If the firm abandons, it receives nothing (0). If both firms invest, Enel receives s × V − + − I = 0.6 × 150 − 80 = C10 m, while EDF obtains 0.4 × 150 − 80 = − C 20 m. Again, Enel has a dominant strategy to invest, receiving a positive value in both situations. Given that Enel has a dominant strategy to invest, the best response of EDF is to abandon the market, receiving 0 rather than a negative payoff (−C20 m). Given that

An Integrative Approach to Strategy

205

each firm plays its best response to its rival’s actions, the Nash equilibrium payoff in the intermediate demand state is C*+− = ( 70, 0 ). In the down state (after two consecutive down moves), the gross market value is V − − = ddV = 0.67 × 0.67 × 150 ≈ C 67 m, which is lower than the needed investment outlay of I = C80 m. Neither firm therefore has an incentive to invest, since whatever the resulting competitive outcome, the firm would incur losses. Both firms thus have a dominant strategy to abandon the project, receiving 0. The resulting Nash equilibrium payoff is C*−− = ( 0, 0 ). The binomial tree and resulting Nash equilibria (highlighted) in each of the three demand states are shown in figure 6.3. The next step is to substitute these equilibrium payoff or strategic option values in the end nodes of the binomial tree, and determine for each firm the investment option value by backward valuation along the binomial tree. For this, we need to assess the risk-neutral probabilities of each end node that will allow determining the present value of the strategic option. Following the discussion in chapter 5, the risk-neutral probability of an up move is given by equation (5.4). In the present case with r = 0.04, u = 1.5, and d ≈ 0.67, the risk-neutral probability of an up move is p=

(1 + 0.04) − 0.67 1.5 − 0.67

= 0.45.

Going backward along the binomial tree at t = 1, in the up state the investment option for Enel is worth C+ =

pC ++ + (1 − p)C +− 0.45 × 123 + (1 − 0.45) × 70 = ≈ C 90 m. 1+ r 1.04

The (time-1) value for Enel in the down state is C− =

0.45 × 70 + 0.55 × 0 ≈ C 30 m. 1.04

Finally, one step earlier, the present value (t = 0) of the strategic investment option for Enel is C=

pC + + (1 − p)C − 0.45 × 90 + 0.55 × 30 = ≈ C 55 m. 1+ r 1.04

That is, the current value of the investment opportunity for Enel to enter the Russian market in two years, accounting for strategic interactions with European rival EDF in each possible state of market demand evolution, is C55 m.

206

Chapter 6

The value of the investment opportunity for EDF is similarly obtained to be 10 m.6 This basic valuation procedure is a cornerstone for the discrete-time analysis of option games. It will be used again in the examples in the rest of this chapter and in subsequent chapters. The next section provides an illustration of option games in the case of patentleveraging strategies. Section 6.4 provides a case application in the context of mine investments. 6.3

Patent-Fight Strategies

As another illustration of the usefulness of simple option games, Trigeorgis and Baldi (2010) consider different patent-leveraging strategies among two patent-holding firms in a duopoly. The setup of their problem together with the payoff matrices at maturity are depicted in figure 6.5.

Binomial tree representing market uncertainty

Payoff matrix for Strategic option strategic uncertainty value Firm j Sleep

Firm i

324

(220, 8) (92, 31)

(9, 3) (–29, 33)

180

60

36 Low demand

t= 0

1 Stage I

Patent acquisition for firm i (old technology for firm j )

2

Sleep

Invest

108

Invest

100

Invest

(98, 33) (68, 152)

Invest

High demand

Sleep

Sleep

Sleeping

Wall

Wall

Bracketing

Sleep

Invest

Sleeping

Wall

C *++ = (92, 31)

C *+– = (34, 0)

(34, 0) (3, –17) Wall

Bracketing

Sleep

Invest

(9, 0) (–14, –1) Sleeping

Wall

C *–– = (9, 0)

(–5, 0) (–20, –7) Wall

Bracketing

Stage II Choice of patent fighting strategy

Figure 6.5 Binomial tree and equilibrium end-node payoffs for patent fighting strategies where firm i has a strong patent advantage 6. At time t = 1 in the up node, EDF will receive 55 m in the up state or 0 in down state, worth C + = (0.45 × 55 + 0.55 × 0 ) 1.04 = C 24 m. At t = 1 in the down state, EDF’s investment opportunity is worth zero. Going back one step earlier, the current ( t = 0 ) value of EDF is C = (0.45 × 24 + 0.55 × 0 ) 1.04 = C10 m.

An Integrative Approach to Strategy

207

At time t = 0 (or at the beginning of stage I), firm i innovates and acquires a new core patent that is superior to the old patented technology used by rival firm j. At time t = 2 (or at the beginning of stage II), each firm decides on its best patent-leveraging strategy, whether to fight, cooperate, or wait, depending on firm i’s cost advantage and the state of demand (high, intermediate, or low). The weaker rival, firm j, believing it has a fighting chance, may go on the offensive to identify and exploit gaps around firm i’s core patent. Firm i may pursue a similar offensive strategy, resulting in a patent-bracketing war. When both firms attack each other via patent bracketing, they share a reduced market value reflective of their respective market power. Alternatively, firm i can solidify its large advantage building a defensive patent wall around its core patent in the hope of driving the rival out. When one firm invests while the rival waits, it captures full monopoly NPV.7 Last, each firm can “wait and see,” virtually putting its patent in a “sleep mode.” When both firms sleep (wait) and postpone their fighting decision, firm i has an advantage (capturing more of the option value according to its higher market power). The type of competition depends in part on the size of the cost advantage resulting from the patented innovation. Absence of such advantage is likely to induce symmetric rivals to cooperate, while a large cost advantage by firm i will favor a fight mode instead. In case of a large cost advantage, different types of fight mode may result, depending on realized demand. Figure 6.5 focuses on the case where firm i has a strong advantage resulting from the innovation. This situation will result in patent fighting. In the two-by-two matrices under high, intermediate, or low demand, different types of fighting equilibria may result: Under high demand, each firm should invest regardless of the opponent’s decision (for firm i, 220 > 98 and 92 > 68; for firm j, 152 > 33 and 31 > 8). Invest is a dominant strategy for both. The resulting equilibrium strategies are for both firms to invest, ending up in a bracketing war with bottom-right payoffs C*++ = ( 92, 31). Under asymmetric reciprocating competition with high demand, both firms feel induced to fight via •

7. Firm i has an incentive to fortify and exploit its large cost advantage to drive the rival out of the market. The rival will be inclined to fight (even attack) if demand or volatility is high. Entering a reciprocating fight is costly and erodes profit margins for both firms, reducing total market pie (to 70 percent)—except when one firm ignores its fighting rival and lets its patent sleep.

208

Chapter 6

reciprocal patent bracketing—even though they would be better off to let their patents sleep, obtaining ( 98, 33).8 Under intermediate demand, firm i should invest, regardless of its rival’s decision (34 > 9 and 3 > −29), a dominant strategy. Knowing that firm i will invest and fight, firm j prefers to wait (sleep) rather than engage in a costly bracketing fight (0 > −17), resulting in patent walls with Nash equilibrium payoffs C*+− = ( 34, 0). Firm i will invest building a defensive patent wall around it, with firm j waiting (sleeping).

•

Under low demand, each firm has a strictly dominant strategy to let its patent sleep (for firm i, 9 > −5 and −14 > −20; for firm j, 0 > −1 and 0 > −7). Given the low level of demand, both firms would actually lose value by fighting each other. The equilibrium strategy is for both firms to “sleep.” Firm i lets its superior patent sleep, maintaining its option to become a monopolist should the market recover in the future (with continuation value 9). This amounts to the disadvantaged firm abandoning the market. The Nash equilibrium for the upper-left game payoff is C*−− = ( 9, 0 ). •

In the problem above the risk-free interest rate is r = 0.08 and the riskneutral probabilities for an up or down move are p = 0.4 and 1 − p = 0.6 . Given that the equilibrium investment option values at maturity (t = 2) for firm i are C*++ = 92, C*+− = 34, and C*−− = 9, in the high, medium, and low demand states, with associated probabilities pp = 0.16, 2 2 p (1 − p) = 0.48, and (1 − p) = 0.36, respectively, the current (t = 0) value of the strategic investment opportunity for firm i is p2C*++ + 2 p (1 − p)C*+− + (1 − p) C*−− ( 1 + r )2 0.16 × 92 + 0.48 × 34 + 0.36 × 9 = = 29. (1.08)2 2

C=

This yields an expanded net present value for the patent fight strategy of firm i of 29 m in case of a large cost advantage. Firm j’s value of the investment opportunity obtains similarly as C4 m.9 8. The fear of the rival investing in a patent wall and strengthening their position if they let their own patent sleep puts pressure on both firms to invest aggressively bracketing each other’s patent, a situation analogous to the prisoner’s dilemma. 9. Firm j obtains in each demand state C*++ = 31, C*+− = 0 , and C*−− = 0 , respectively. Firm j’s investment opportunity is worth C=

0.16 × 31 + 0.48 × 0 + 0.36 × 0 = C 4 m. (1.08)2

An Integrative Approach to Strategy

209

6.4 An Application in the Mining/Chemicals Industry Ferreira, Kar, and Trigeorgis (2009) present an actual application of option games in the mining/ chemicals industry.10 Deciding when to add capacity in the face of rivals is a challenging problem for any industry. This approach involves American-type real options and games and is solved backward by use of numerical analysis. Between 1995 and 2001 annual revenues for the US commodity chemicals industry fell from $20 bn to $12 bn, while companies’ operating profits fell on average by 26 percent a year. The collapse was in large measure caused by a tight economic environment and a rising dollar. But outside forces were only part of the story—industry players also made some very poor decisions. . . . It’s a story that regularly plays out in many industries. Indeed any company making big-budget investment decisions faces the same basic dilemma. On the one hand, it must make timely, strategic investments to prevent rivals from gaining ground. On the other, it must avoid tying up too much cash in risky projects, especially during times of market uncertainty. Using game theory models, managers can incorporate the collective effect on market-clearing prices of other companies that are expanding their capacity at the same time. Typically the way to do this is to create a payoff matrix, which compares your payoffs with those of your competitor under different scenarios. Unfortunately, the standard calculation of payoffs does not allow managers to factor in uncertainty for key market variables such as prices and demand, nor does it assign any value to flexible investment strategy. To get around this problem, we use a hybrid model that overlays real options binomial trees onto game theory payoff matrices. To Mine or Not to Mine? MineCo is planning to open a new mine to expand its capacity to produce minerals for its regional market. In this market, if demand exceeds local supply, customers will import from foreign sources, which effectively sets a cap on prices. From MineCo’s perspective there are two key sources of uncertainty: the growth rate of local demand, which has varied in recent years with shifts in the country’s political economy, and the risk that CompCo, its largest competitor, will invest in a similar project first. 10. This section is an excerpt from “Option games: The key to competing in capitalintensive industries” by Nelson Ferreira, Jayanti Kar, and Lenos Trigeorgis, Harvard Business Review, March 2009. It is reprinted by permission of Harvard Business School Publishing.

210

Chapter 6

The current demand is 2,200,000 tons and the current price (set by imports) is $1,000 per ton. The MineCo project involves adding 250,000 tons of capacity at a cash operating cost of $687 per ton (incurred each year the project is up and running) and a capital expenditure of $250 per ton, spread over three years. The Comp Co project faces cash operating costs of $740 per ton annually, projected capacity of 320,000 tons, and a capex of $150 per ton, also spread over three years. The investments take three years to complete, and both new mines have a lifespan of 17 years. For the purposes of simplicity, we assume that each firm can decide to invest in Y0 (with capex in Y0, Y1, and Y2 and production starting in Y3) or in Y3 (with capex to be invested in Y3, Y4, and Y5 and production starting in Y6). We begin by calculating the inputs that will serve as the basis for determining payoff values for each of the scenarios: demand evolution and the probabilities of upward and downward shifts in demand. We assume that demand will go up or down by a fixed multiple in each period (in this case the period is a year). Using historical data and surveys of the company’s managers, we predict demand will move up or down by about 5 percent in each period. We estimate the risk-adjusted probability of an upward shift in each period at 30 percent (therefore, a 70 percent probability of a downward shift in each period). Next we input these data into a binomial tree that tracks the evolution of demand over the next six years and overlay it with a tree that tracks the cumulative probabilities at each node in the demand tree (see figure 6.6). We will refer to this tree throughout the analysis in this section. Now let’s calculate the payoffs for MineCo and CompCo for each of the four scenarios arising from their decisions to invest now or wait until year three to decide. Scenario 1: Both Companies Invest Now If both firms decide to invest now, they will incur capital expenditures in Y0, Y1, and Y2, and both projects will start producing in Y3. Given this, we can model how evolution in demand and capacity will affect prices and thereby revenues and profits for each of the two companies. First, we create a binomial tree showing how market prices might evolve (see figure 6.7). The price at each node is determined by demand and supply, driven by the cash operating cost of the marginal producer (the producer just barely able to remain profitable at current levels of price and demand). If demand rises and MineCo or its competitor adds capacity at a higher marginal operating cost, local prices will rise.

An Integrative Approach to Strategy

Year 0

2200 100%

Year 1

2313 30% 2093 70%

Year 2

2431 9% 2200 42% 1991 49%

211

Year 3

2556 3% 2313 19% 2093 44% 1894 34%

Year 4

Year 5

Year 6

2687 1% 2431 8% 2200 26% 1991 41% 1801 24%

2825 0.2% 2556 2.8% 2313 13.2% 2093 30.9% 1894 36.0% 1713 16.8%

2970 0.1% 2687 1.0% 2431 6.0% 2200 18.5% 1991 32.4% 1801 30.3% 1630 11.8%

Figure 6.6 Demand evolution and probability tree

To calculate the annual operating profits at each node for each firm, we subtract that firm’s estimated annual cash operating costs per ton from the prices at each node for each operating year and multiply that number by the demand filled by the added capacity, estimated over the remaining life of the project. To illustrate, at the upper demand node in Y5, MineCo gets a margin of 313 (the 1,000 price less its cost of 687) per ton, which for 250, 000 tons of added capacity represents $78,250,000. In Y6 nodes, we have to add in the terminal value, which is an estimated present value of cash flows for the remaining 14 years of the mine’s useful life. To calculate this, we assume that price and demand remain constant subsequently and apply the standard discounting formula, which gives us a terminal value of $774,569,000. We add that to the Y6 annual operating profit (again, $78,250,000), and get a total value for the upper node in Y6 of $852,819,000. The resulting tree for MineCo (with the added capacity of 250,000 tons) is shown in the lower panel of figure 6.7. The tree for CompCo is similar (but not shown here)—the numbers are a little higher on the upside and more negative on the downside. Our final step is to weight the numbers at each node by the corresponding risk-adjusted probability (from the demand tree) and discount those expected payoff values by 5 percent per year (the risk-free interest rate) back from the position of the node to the present. We then sum up these numbers—the weighted, discounted annual operating profits at each node plus the terminal value—and subtract from that sum the

212

Market-clearing prices (in US$/ton)

Chapter 6

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

1,000 1,000 1,000 1,000 1,000 1,000 1,000

740 700

1,000 1,000

1,000 1000 740 700

700 700

1,000

700 700

687 685

687 685

685

685 685 680

Payoffs for MineCo (in US$ thousands)

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6 852,819

78,250 78,250 78,250 (20,833) (20,833) (20,833)

13,250 3,250

(20,833) (20,833)

852,819 78,250 144,407 3,250

3,250 3,250

(20,833)

35,421 3,250

0 (500)

0 (500)

(500)

(5,449) (500) (19,073)

Figure 6.7 Scenario 1: Both companies invest now

present value of the annual capex investments made by each company. This gives us the net current payoff value, or final payoff value, for each company under scenario 1: For MineCo the expected payoff in Y0 is −$36 m; for CompCo, −$195 m. If both firms invest now, both lose money. Scenario 2: MineCo Invests Now, While CompCo Waits In this scenario MineCo invests first, giving it the advantage of being the sole producer from Y3 to Y6, while CompCo waits until Y3 to decide whether to invest. If demand evolves favorably, CompCo enters in Y3; if not, it abandons the project. We begin the valuation by calculating the market-clearing prices from Y0 through Y3, using the demand tree and given the fact that MineCo has invested in extra capacity and CompCo so far has not. Next, we calculate how prices will evolve from Y3 through Y6. There are four possible Y3 scenarios, each with an associated probability of occurrence (see demand tree in figure 6.8). At each of the nodes, we determine the market-clearing prices, operating profits, and terminal value for each firm

An Integrative Approach to Strategy

Evolution of demand till year 3 Y0

Y1

Y2

Probability of reaching node (Y3)

213

Competitor’s decision (Y3)

Y3 3%

• CompCo

• MineCo = 328 • CompCo = 71

wn Do

Up

Up

decides to invest

19%

• MineCo = 263 • CompCo = ∅

Up

wn Do

Up

Expected value of payoffs (in US$ million)

wn Do

wn Do wn Do

Up

44%

CompCo abandons project

• MineCo = -6 • CompCo = ∅

wn Do 34%

• MineCo = -64 • CompCo = ∅

Figure 6.8 Scenarios 2 and 3: One company invests, the other waits

assuming that CompCo does invest in Y3. In other words, for each of the four scenarios, we create a three-year binomial tree (Y4, Y5, and Y6) showing what the annual operating profits plus terminal value would look like at each node if CompCo were to invest then. Next for each Y3 scenario we weight the node values by the demand probabilities for Y4, Y5, and Y6 and discount the values back to Y0, taking into account the NPV of CompCo’s investment costs (Y3 through Y5) and MineCo’s (Y0 through Y3). The result is four pairs of expected Y0 net payoff values: $71 m for the upper demand node in Y3, and –$114 m, –$169 m, and –$185 m for the other three nodes. As a rational investor, CompCo will not invest in Y3 unless its payoff value is positive, which is only the case in the top node where demand evolution from Y3 is high enough to accommodate a second entrant. At all the other demand nodes, CompCo will abandon the project, preferring a payoff of zero to losing money. We thus recalculate the operating profits plus terminal value for both companies, based on the assumption that CompCo will not invest in any but the top demand node. These expected net Y0 payoffs for MineCo and CompCo (taking into account the investment costs incurred in Y0

214

Chapter 6

through Y2 for MineCo in each subscenario and in Y3 through Y5 for CompCo in the uppermost subscenario) are shown in the last column in the figure. We finally weight these four pairs of Y0 payoff values according to the probabilities associated with the Y3 demand nodes. We arrive at the final payoff for each company by summing up the four weighted, discounted payoff numbers. For MineCo, the expected final payoff at Y0 is ($328 m × 3%) + ($263 m × 19%) + (−$6 m × 44%) + (−$64 m × 34%), which yields $35 m. For CompCo, the expected final payoff at Y0 is ($71 m × 3%) + ($0 × 19%) + ($0 × 44%) + ($0 × 34%), which yields about $2 m. Scenario 3: CompCo Invests Now, While MineCo Waits This is estimated in the same way as scenario 2, but with MineCo as the follower. The final payoffs are $4 m for MineCo and −$83 m for CompCo. Scenario 4: Both Companies Wait In the last scenario, where both firms wait until Y3 to decide whether to invest, we start by looking at the four possible demand nodes in Y3 (see figure 6.9). For each, we need to consider four subscenarios: both firms

Y1

Y2

Y3

Up

Y0

MineCo

Evolution of demand till Y3

Abandon Invest

CompCo Invest

Abandon

(143, 71) (410, 0) (0, 405)

Probability of reaching node 3%

(0, 0)

Expected value of payoffs (in US$ million) Dominant strategy • MineCo = 143 • CompCo = 71

Abandon Invest

MineCo

wn Do MineCo

wn Do

wn Do

Abandon Invest

wn Do

Up

Abandon

(–2, –114) (87, 0) (0, 71)

19%

Mixed strategy (average) • MineCo = 43.5 • CompCo = 35.5

44%

Dominant strategy • MineCo = ∅ • CompCo = ∅

34%

Dominant strategy • MineCo = ∅ • CompCo = ∅

(0, 0)

CompCo Invest

Abandon

(–45, –169)

(–11, 0)

(0, –102)

(0, 0)

MineCo

CompCo

Figure 6.9 Both companies wait to decide

Abandon Invest

Up

wn Do

Up

wn Do

Up

CompCo Invest

Invest

Abandon

(–57, –185)

(–55, 0)

(0, –158)

(0, 0)

An Integrative Approach to Strategy

215

investing in Y3, only MineCo investing in Y3, only CompCo investing in Y3, and both abandoning. We thus have 16 subscenarios, each with its own three-year market-clearing price evolution tree. The price at each node, as ever, is based on the demand evolution (captured by the demand tree) and on total industry capacity, which varies depending on the Y3 investment decisions of MineCo and CompCo. Let’s take the upper demand node in Y3 as an example. In the first subscenario, both firms invest from Y3 to Y5 and enter in Y6. We calculate the expected net Y0 payoffs in the same way we did in scenario 1 but with a three-year tree, weighting annual operating profit plus terminal value, discounting back to Y0, and subtracting net present capex costs. For the upper demand node, this results in Y0 net expected payoffs of $143 m for MineCo and $71 m for Comp Co. We perform similar exercises to calculate the expected net Y0 payoffs in the remaining three subscenarios, with the firm not investing receiving a zero payoff and the firm investing receiving payoffs determined by demand evolution and industry capacity. This exercise is then repeated for the remaining three sets of subscenarios. We present all expected net Y0 payoffs in a series of two-by-two game matrices, one for each demand node in Y3, which is when decisions are made. We then identify the Nash equilibria—outcomes from which neither player has an incentive to deviate. In the top demand node, for example, we see that both MineCo and CompCo will find it optimal to invest in that year (receiving $143 m and $71 m, respectively). MineCo cannot do better since the alternative (abandoning) would entail a lower (zero) payoff whatever CompCo does; CompCo reaches the same conclusion. The remaining three two-by-two matrices are similarly analyzed to find Nash equilibria. At three nodes (the top and the lower two) there is a single (pure) equilibrium. In one (the second), we have two. There are theories about how to determine which of two equilibria one should favor, but suppose here that the companies are roughly symmetrical such that there is an equal chance that either equilibrium will prevail; in other words, each player will choose one strategy 50 percent of the time and the other the remaining 50 percent. The resulting expected payoffs from the two mixed equilibria therefore are simply the average of the payoffs associated with each equilibrium for each player. For MineCo, the expected net Y0 payoff for the node is (0.5 × $87 m) + (0.5 × $0), which yields $43.5 m. Finally, we weight these four pairs of net Y0 payoff values according to the probabilities associated with the Y3 demand nodes. We arrive at

216

Chapter 6

the final payoff for each company by summing up the four weighted, discounted payoff numbers. This yields an expected net Y0 payoff value for MineCo of $12 m [($143 m × 3%) + ($43.5 m × 19%) + ($0 × 44%) + ($0 × 34%)]. For CompCo the payoff is $8 m. How Do the Results Stack Up? Having analyzed the four different strategic scenarios one at a time, we now put them together into a time-zero payoff matrix for a final decision, as shown in figure 6.10. We see that scenario 2 (MineCo invests now and CompCo waits) is a Nash equilibrium scenario, as no player has an incentive to deviate from the associated strategy choices. MineCo cannot do better (if it decides to wait as well, moving to scenario 4, it will get $12 m instead of $35 m). The optimal decision for MineCo therefore is to invest at once. How does this recommendation compare with the traditional valuation methods? Given the data, a standard NPV analysis (assuming MineCo invests now and the competition never enters) would have indicated values for the project of $41 m for MineCo and $13 m for CompCo. This would suggest that both companies should invest immediately, with disastrous results. A conventional real options calculation using the same data would have indicated that delaying the project would add $8.5 m in flexibility value to the NPV number for MineCo

Wait

MineCo

Invest

CompCo Invest

Wait

Scenario 1

Scenario 2

(–36, –195)

(35, 2)

Scenario 3

Scenario 4

(4, –83)

(12, 8)

Note: Current value of payoffs in each strategic scenario (in US$ million)

Figure 6.10 Comparing strategic scenario payoffs for a final time-0 decision

An Integrative Approach to Strategy

217

and $5 m for CompCo. This would suggest that both should delay, which, although not disastrous, would still misrepresent value for both players. With the benefit of an option games analysis, each player can see how the flexibility and commitment tradeoff works out for it. In MineCo’s case, the flexibility value from delaying is more than outweighed by the commitment value created by investing now, whereas CompCo is better off waiting. Conclusion In this chapter we illustrated via simple numerical examples the benefits of an integrative approach to strategy, deriving fruitful insights into when and how firms should invest in uncertain environments when they also face strategic interaction. Option games give valuable guidance as to when to pursue different investment strategies and how to assess the trade-off between flexibility and strategic commitment. In the next two chapters we elaborate further on option games in discrete-time settings. We first consider the option to invest in a new market in different competitive situations. Chapter 7 discusses this investment option, considering models of quantity or price competition. In Chapter 8, we extend our tool kit to quantify the trade-off between flexibility and commitment, examining when early commitment might be worthwhile in an uncertain environment in the context of R&D and advertising strategies. Selected References Smit and Ankum (1993) and Trigeorgis (1996) discuss related issues of market structure under uncertainty. Smit and Trigeorgis (2004) develop the option games approach in discrete time and give a number of case applications. Ferreira, Kar, and Trigeorgis (2009) present a concrete, actual application and discuss the insights offered by option games, highlighting their relevance for strategic management practice. Trigeorgis and Baldi (2010) discuss an application of the aforementioned methodology in the context of patent games. Ferreira, Nelson, Jayanti Kar, and Lenos Trigeorgis. 2009. Option games: The key to competing in capital-intensive industries. Harvard Business Review 87 (3): 101–107.

218

Chapter 6

Smit, Han T. J., and L. A. Ankum. 1993. A real options and game-theoretic approach to corporate investment strategy under competition. Financial Management 22 (3): 241–50. Smit, Han T. J., and Lenos Trigeorgis. 2004. Strategic Investment: Real Options and Games. Princeton: Princeton University Press. Trigeorgis, Lenos. 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. Cambridge: MIT Press. Trigeorgis, Lenos, and Francesco Baldi. 2010. Patent leveraging strategies: Fight or cooperate? Working paper. University of Cyprus.

7

Option to Invest

This chapter describes a simple framework—based on the combined insights from real options and game theory—that enables managers to analyze strategic investment and quantify flexibility in a competitive setting in relatively simple terms. Encompassing more aspects of reality would involve more complications to the modeling of option games. The simple models presented herein offer pedagogical value and make easier the understanding of subsequent chapters. The chapter is organized as follows. In section 7.1, we present a benchmark model for later analysis, focusing on a monopolist’s option to invest. In section 7.2, we discuss quantity competition models and examine the importance of having a cost advantage from a dynamic perspective. In section 7.3, we analyze investment settings in which, upon market entry, firms compete in price. Chapter 8 then builds upon these two benchmark duopoly models and puts them in dynamic perspective in light of earlier-stage commitment decisions. 7.1

Deferral Option of a Monopolist

Suppose that in the discrete-time model of Smit and Trigeorgis (2004) the demand intercept in the linear market demand function of equation (3.1) follows a multiplicative binomial process. That is, the inverse (stochastic) demand function is given by pɶ ( Xɶ t , Q) = aXɶ t − bQ,

(7.1)

where Xɶ t follows a multiplicative binomial process, a and b are constant parameters, and Q is the total output (capacity) offered in the marketplace.1 The binomial stochastic process followed by Xɶ t is shown in 1. Throughout part II we assume a linear market demand function. Most of the results and insights from the models developed here would carry over to other types of inverse demand curves.

220

Chapter 7

~

X2++ ~

X1+

X0

~

X2+ − ~

X1− ~

X2−− Figure 7.1 Multiplicative binomial process followed by demand (intercept) Xɶ t is the underlying asset (stochastic market demand) at time t.

figure 7.1 for two periods: after each up move, Xɶ t is multiplied by u, while after each down move by d (d = 1 u).2 Consider a European option held by a monopolist to invest at maturity t = T ( = 2 ).3 At each end state or node in the binomial tree, the monopolist can choose at maturity between investing or abandoning the investment. When the monopolist decides to invest, it chooses its strategic variable (output) to maximize its payoff; that is, it selects the monopoly profit-maximizing output QM ( Xɶ T ). Under market demand uncertainty the firm has an option to invest but can wait until new information is revealed, deferring the investment decision until maturity. Suppose that the monopolist’s cost function is linear and given by C ( q) = cq (where variable cost c is lower than the demand intercept). As seen in chapter 3, equation (3.4), the equilibrium profit for a monopolist in the deterministic case (if Xɶ t were constant over time and equal to 1) is

πM =

(a − c )2 4b

.

2. The binomial lattice, developed by Cox, Ross, and Rubinstein (1979), is meant as a discrete-time equivalent of the geometric Brownian motion. Many of the models used in the continuous-time part assume that Xɶ t follows a geometric Brownian motion. These assumptions used in the discrete-time and continuous-time parts are compatible. 3. We here focus on European options to simplify the problem structure and derive better intuition concerning the strategic interactions taking place at maturity among the option holders. We could also consider American-type options, but these models are more demanding mathematically. To solve this problem, we would need to resort to numerical analysis, for example, as in Smit and Trigeorgis (2004). Part II focuses on European options in discrete time due to their relative simplicity, while part III discusses perpetual American options in continuous time that may admit closed-form solutions.

Option to Invest

221

Given the stochastic uncertainty about the demand parameter Xɶ t , the equilibrium profit for the monopolist at maturity T now is

π M ( Xɶ T ) =

(aXɶ

T

− c)

4b

2

,

(7.2)

provided that the firm invests, and π M ( Xɶ T ) = 0 otherwise.4 Suppose the monopolist firm invests amount I at time t = T , receiving at the end of the year the equilibrium profit π M ( Xɶ T ) that grows thereafter in perpetuity at an average annual growth rate g . The appropriate risk-adjusted discount rate is k (k > g ). The monopolist’s net project value at the end node (at maturity T ) is given by

π NPV M ( Xɶ T ) =

M

( Xɶ ) − I, T

δ

(7.3)

where δ ≡ k − g ( > 0 ) represents some form of dividend yield or opportunity cost of waiting.5 Assuming after maturity T the project enters steady state (with g = 0 subsequently), the expression above (with δ = k) simplifies to NPV M ( Xɶ T ) =

π M ( Xɶ T ) k

− I.

The firm cannot delay the investment decision beyond maturity. At maturity it must decide either to invest or abandon. If the monopolist firm invests, it becomes committed to the project and receives the NPV of the investment as given in equation (7.3). In this sense the classical NPV rule holds at maturity because the monopolist is then faced with a now-or-never decision. The firm will decide to invest at maturity T if and only if NPV M ( Xɶ T ) ≥ 0. This occurs when the random demand reaches or exceeds a specified trigger X M , or Xɶ T ≥ X M , provided that aXɶ T ≥ c. The monopolist’s investment trigger, X M , is given by 4. The firm will not invest if aXɶ T < c since demand would not cover marginal production costs. 5. The perpetuity formula above (used as the present value) is often referred to as the Gordon formula. It equals the infinite sum of subsequent cash flows beginning with π M ( Xɶ T ) starting in one period (end of the year) and growing in perpetuity at a rate g per year, namely M t 1 ⎡ ∞ M ɶ ⎛ 1 + g ⎞ ⎤ π ( Xɶ T ) π ( XT ) ⎜ , = ⎟ ∑ ⎥ ⎢ ⎝ 1+ k⎠ ⎦ δ 1 + k ⎣ t =0

provided that δ ≡ k − g > 0.

222

Chapter 7

Binomial tree evolution

Threshold

Investment decision

Payoff value

Invest

~ NPVM XT ≥ 0

High

~

XT ≥ X M

( )

~

X0

XM

XT ~

XT < X M

Do not invest (abandon)

0

Low

Figure 7.2 Critical threshold, investment decision and payoff for the monopolist at maturity X M = (2 bδ I + c) a is the monopolist’s fixed investment trigger (critical threshold).

XM ≡

2 bδ I + c . a

(7.4)

Since all parameters (k , g , a, b, c, and I ) determining the investment trigger of the monopolist X M are constant and known at the outset, the optimal investment decision of the monopolist depends solely on whether the value of Xɶ T at maturity exceeds the fixed investment trigger X M . Figure 7.2 illustrates the payoff and investment decision of the monopolist at maturity, T , depending on the random value of Xɶ T exceeding or being below the specified fixed critical threshold X M . Example 7.1 Investment Trigger for a Monopolist A decade ago the Italian public electric utility Enel had an option to invest in a new power plant involving a capital expenditure of I = 250. At the time the unit variable cost for the monopolist was c = 15. The appropriate discount rate for the firm was 10 percent or k = 0.10. The demand parameter Xɶ t evolved stochastically with a = 5 and b constant and equal to 1. The market growth ( g ) was 0 percent p.a. Note that δ = k − g = 0.10 − 0 = 0.10. The investment trigger for the monopolist, based on equation (7.4), is XM =

2 1 × 0.10 × 250 + 15 = 5. 5

Option to Invest

223

NPVM at maturity (in thousands)

30

20

10

0

0

5

XM

10

15

20

25 ~

End-node value XT

Figure 7.3 Payoff value ( NPV M) at maturity for the monopolist’s option to invest We assume linear demand with a = 5 and b = 1. Unit variable cost is c = 15. I = 250, k = 0.10 , and g = 0 .

The payoff function for the monopolist option holder at maturity, NPV M ( Xɶ T ), is shown in figure 7.3. The payoff at maturity (as a function of end-node value Xɶ T ) increases at an increasing pace beyond the trigger point X M = 5.6 Once the optimal investment trigger level X M has been determined as per equation (7.4), the resulting payoff values at maturity T can be easily derived based on equations (7.2) and (7.3). The maximal value of the investment must be determined simultaneously with the monopolist’s optimal investment behavior or exercise strategy. Once the end-node project values are determined, we can work out the present value of the investment by backward induction using risk-neutral valuation along the binomial tree. When moving back along the binomial tree, we use the risk-neutral probability from equation (5.4), p=

(1 + r ) − d u−d

,

to calculate the expected option values and discount the resulting certainty-equivalent values at the risk-free rate r. 6. The monopolist’s profit function in equation (7.2) is a quadratic function of the demand shock Xɶ T .

224

Chapter 7

If the time to maturity T is subdivided into n equal subintervals of length h = T n as in appendix 5A, the expanded NPV of the investment for the monopolist firm is obtained recursively as

E−NPV =

∑

n j =0

⎛ n⎞ j n− j M j n− j ⎜⎝ j ⎟⎠ p (1 − p) NPV (u d X 0 )

( 1 + r )n

,

(7.5)

where n! ⎛ n⎞ ⎜⎝ j ⎟⎠ ≡ j !(n − j )! and ⎛ n⎞ j n− j ⎜⎝ j ⎟⎠ p (1 − p) is the binomial distribution giving the probability that the market demand parameter Xɶ t will take j upward jumps in n time steps, each with (riskneutral) probability p. 7.2

Quantity Competition under Uncertainty

In the previous section we developed an investment option model characterizing the optimal behavior of a monopolist firm having an exclusive right to enter the market. We next look at the situation when several firms may enter the market, involving uncertainty about the future industry structure as a result of both market demand uncertainty and (endogenous) multiplayer strategic interactions. 7.2.1

Cournot Duopoly

To value the option to invest under both demand and strategic uncertainties in the case of a duopoly, we consider at each end node at maturity a simultaneous game where each player does not know the strategic action chosen by the rival firm. We assume complete information concerning the cost structure of the players and the level of demand reached at maturity (time T ). One way to illustrate the time elements of the problem is to look at decision time versus real time. Game theory models strategic interactions in terms of decision time, whereas in real-world decisions real time matters in the evolution of exogenous demand

firm j

225

firm i

Option to Invest

qi Decision time 1

qj 2

Strategic uncertainty (2 decisions) Real time 1

2

3

T–2 T–1 T

Demand uncertainty (T periods)

Figure 7.4 Cournot model involving demand (real-time) and strategic (decision-time) uncertainties

uncertainty. Real time is essential in financial theory. It drives the evolution of the underlying asset in the binomial tree. Figure 7.4 illustrates the binomial tree approach to time and uncertainty evolution. We subsequently analyze Cournot quantity competition under uncertainty, considering in turn the case of (1) cost symmetry and (2) asymmetry. Suppose two firms compete over a homogeneous product but differ in their cost structure. The variable (marginal) cost of firm i is ci and of firm j is c j, with ci , c j ≥ 0.7 Firm i’s (linear) cost function is given by Ci (qi ) = ci qi, and its profit by π i ( qi , q j , Xɶ t ) = ⎡⎣ p (Q, Xɶ t ) − ci ⎤⎦ qi. Firm j’s cost and profit functions are given similarly. The equilibrium profit value in a Cournot duopoly under certainty (with Xɶ t constant and equal to 1) was shown in equation (3.21) to be8

π iC =

(a − 2ci + c j ) 2 . 9b

Under market uncertainty (with Xɶ T being random), this becomes

(aXɶ T − 2ci + c j ) 2 . π iC ( Xɶ T ) = 9b

(7.6)

To be entitled to these profits, each firm must incur investment cost I i (I j, respectively). Let k be the appropriate risk-adjusted discount rate. 7. The fixed costs are not explicitly considered or alternatively are included into the market-entry cost. 8. The superscript C stands for “Cournot.”

226

Chapter 7

Table 7.1 Comparison of project value payoffs in monopoly and Cournot (quantity) competition Monopoly (only firm i) Cournot duopoly

NPVi M ( Xɶ T ) = Vi M ( Xɶ T ) − I i NPViC ( Xɶ T ) = ViC ( Xɶ T ) − I i

2 2 Note: Vi M ( Xɶ T ) = ( aXɶ T − ci ) 4bδ , ViC ( Xɶ T ) = (aXɶ T − 2ci + c j ) 9bδ .

The Cournot–Nash equilibrium value of this investment generating the above annual profits in perpetuity (from the end of year T ) is given by9

π iC ( Xɶ T ) NPViC ( Xɶ T ) = − I i. δ

(7.7)

If both firms decide to invest at maturity, the (asymmetric) Cournot duopoly game payoffs obtain. If the cost asymmetry among the players is sufficiently large, we assume that only one firm, the advantaged one, enters at an intermediary demand level, the industry structure then becoming a monopoly.10 If demand is low, no one enters (both firms make zero profits). Table 7.1 compares the value functions in monopoly and in Cournot duopoly. We here consider pure strategies, namely firms chose whether or not to enter the market. This decision is driven by the firms’ respective optimal exercise or trigger policies. When a specified threshold demand level is reached, firm i (firm j) will invest. These optimal investment triggers depend on the (exogenously given) values of k , g , a, b, ci, c j, and on the level of demand reached at maturity, Xɶ T . Analogous to the monopoly case, the optimal investment strategies in Cournot duopoly are based on whether the demand parameter Xɶ T at maturity T exceeds or is below certain threshold levels. However, the equilibrium strategies in the 9. The first term is again the value of a perpetuity starting at the end of year T or equivalently in year T + 1. 10. This statement has common-sense appeal. As stated here, it is, however, based on weak game-theoretic foundations. In chapter 12 we analyze the coordination problem arising in the intermediary region in a context involving perpetual American investment options. We show there that depending on the magnitude of the cost asymmetry, two situations may emerge. For “small” cost difference, the disadvantaged firm may try to enter first as well, leading to preemption effects. For large cost asymmetry, the disadvantaged firm may accept its role as follower, whereby the advantaged firm enters peacefully as leader with no risk of preemption. At this early stage of discussion, we prefer to rely on the assumption that the advantaged firm enters first (becoming monopolist in the intermediary region) for pedagogical reasons since the technicalities involved to demonstrate the latter result are fairly involved and hence are deferred to chapter 12.

Option to Invest

227

duopoly case differ depending on whether the industry consists of symmetric or asymmetric-cost firms. The discrete-time analysis of the investment dynamics under quantity competition that follows reveals more intuitively the role of these investment thresholds. The optimal investment policy of each firm depends on the actual level reached by the underlying random variable compared to the investment thresholds. These thresholds are crucially important in the analysis of investment strategies under endogenous competition as they help induce firms’ optimal investment strategies. Cournot under Cost Symmetry Suppose that firms i and j share a European investment option in a duopoly. Firms compete over quantity (capacity) when they both operate. Costs are identical between the two firms, each facing variable unit cost c. Investment outlays are the same, I (≥ 0). For symmetric firms we might assume symmetric investment (exercise) policies; the industry structure will be either “both invest” or “neither invests.” Consequently investment thresholds are identical and, provided they are reached at maturity ( Xɶ T ≥ X C ), both firms will invest. In the case of the investment opportunity being analogous to a European call option the firms decide to invest only at the end nodes.11 At maturity, each firm decides whether to enter (à la Cournot) or not. Firm i will invest at the end node at maturity if realized demand Xɶ T is high enough such that the forward investment NPV is positive, meaning if NPV C ( Xɶ T ) ≥ 0.12 The investment condition is therefore13 Xɶ T ≥ X C (provided that Xɶ T ≥ c a). The Cournot investment trigger for both symmetric firms in a duopoly is XC ≡

3 bδ I + c , a

(7.8)

where δ ≡ k − g. The Cournot investment trigger strategies discussed above are shown in figure 7.5. Both firms have the same investment 11. The case of sequential investment with one firm investing as leader in the expanding phase of a market and the rival as follower when the market is mature is excluded by assumption. 12. At maturity there is no further option to defer the investment, so the NPV rule holds. 13. When Xɶ T ≥ X C , condition Xɶ T ≥ c a is satisfied. Therefore the condition Xɶ T ≥ c a becomes redundant. If Xɶ T < X C, then the project end node value is zero since the option is not exercised. If Xɶ T ≥ X C , the option is exercised since the project has positive NPV.

228

Chapter 7

Binomial tree evolution

Threshold

Investment decision

High

~

XT ≥ X C

X0

~

Invest

NPV C (XT) NPV

C

~

(X ) T

~

XC

XT X T < X C Low

t=0

Payoff (firm i, j)

Do not invest (abandon)

0 0

T

Figure 7.5 Investment decisions and payoffs at maturity for symmetric Cournot duopolists X C = (3 bδ I + c) a is the investment trigger of symmetric Cournot duopolists.

trigger. We can derive the present value of the investment by backward induction working back through the binomial tree. Example 7.2 Investment Trigger for Symmetric Cournot Duopolists Two symmetric firms compete in quantity, facing symmetric variable costs of c = 15. Firms are identical, facing the same risk-adjusted discount rate k = 10 percent and investment outlay I = 250 to enter the market. After maturity T steady state is reached. The demand parameter Xɶ t evolves stochastically, with a = 5 and b = 1. Given these parameters, the (common) investment trigger for the Cournot duopolists obtained from equation (7.8) is X C = (3 1 × 0.10 × 250 + 15) 5 = 6. Thus each duopolist will invest at maturity provided the demand parameter Xɶ T exceeds X C = 6. Once the investment decisions at maturity are determined, if demand is high and both firms enter, we utilize the Cournot equilibrium profit of equation (7.6) to obtain the net project value from equation (7.7). Figure 7.6 depicts the Cournot duopoly end-node payoffs (NPV) as a function of the stochastic demand level at maturity, Xɶ T . The payoff function in the symmetric cost case looks like the payoff function of a call option with exercise price X C = 6. One difference compared with the monopoly case lies in the relevant profit function once investment is made. Since monopoly profits are higher than individual profits of Cournot duopolists the payoff (NPV) values at maturity in the monopoly case (see in figure 7.3) are higher.

Option to Invest

229

NPV C at maturity (in thousands)

25

20

15

10

5

0

0

5 XC

10

15

20

25 ~

End-node value XT

Figure 7.6 Payoff ( NPV C ) at maturity for option to invest in (symmetric) Cournot duopoly

Cost Asymmetry In case of cost asymmetry, the investment triggers for Cournot quantity competition are determined similarly but require taking a closer look at the strategic interactions, since firms are less likely to follow symmetric strategies. In the case of cost asymmetry, we assume a cost leader with comparative cost advantage (subscript L) and a high-cost firm (subscript H) as follower. At the end nodes (at maturity), each firm (i = L, H ) will decide to invest (enter) or not. Under cost asymmetry an investment game results with industry structure (monopoly or duopoly) depending on the specific trigger policies of the two firms. Once the firms enter, they choose the appropriate output given knowledge of the prevalent number of active firms. To help determine the investment strategies, consider the strategic (normal) form of the simultaneous game shown in table 7.2. If both firms invest (e.g., under high demand), the resulting industry structure is an asymmetric Cournot duopoly. If only one firm invests, it results in a monopoly, and if none invests no one turns a profit (0). Once they have invested, firms select their output optimally. We next look closely at the resulting payoffs in the strategic form to deduce under which conditions each firm invests.

230

Chapter 7

Table 7.2 Strategic form of end-node investment game in asymmetric Cournot duopoly Firm j (high cost) Invest Firm i (low cost)

Invest Do not invest (abandon)

⎛ NPViC ( Xɶ T )⎞ ⎜ ⎟ C ⎝ NPVj ( Xɶ T )⎠ 0 ⎛ ⎞ ⎜ NPV M Xɶ ⎟ ⎝ j ( T )⎠

Do not invest (abandon) ⎛ NPVi M ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 ⎛ 0⎞ ⎝⎜ 0⎠⎟

Note: NPVi M ( Xɶ T ) and NPViC ( Xɶ T ) are given in equations (7.3) and (7.7).

If NPViC ( Xɶ T ) ≥ 0, meaning if Xɶ T ≥ X iC , where X iC ≡

3 bδ I i + ( 2ci − c j ) , a

(7.9)

firm i has a dominant strategy to invest at maturity regardless of the decision of its rival. The firm is always better off (its investment has positive NPV) because its value as a monopolist exceeds the value as Cournot duopolist (at least for the low-cost firm). If NPVi M ( Xɶ T ) < 0, meaning if Xɶ T < X iM with X iM ≡ (2 bδ I i + ci ) a, firm i has a dominant strategy not to invest at maturity. If realized demand Xɶ T is such that NPViC ( Xɶ T ) < 0 and NPVi M ( Xɶ T ) ≥ 0, meaning if X iM ≤ Xɶ T < X iC , firm i has no dominant strategy. The same arguments apply for firm j. Deriving the outcome of strategic interactions is more involved in the absence of dominant strategies. Two cases can be distinguished to facilitate further analysis: 1. Firm i has no dominant strategy (if X iM ≤ Xɶ T < X iC ), but firm j has a dominant strategy ( Xɶ T ≥ X Cj or Xɶ T < X jM ). If Xɶ T ≥ X Cj , firm j has a dominant strategy to invest whereas firm i should not invest since NPViC ( Xɶ T ) < 0. If Xɶ T < X jM , firm j has a dominant strategy not to invest, but firm i should invest since NPVi M ( Xɶ T ) ≥ 0. 2. Neither firm i (if X iM ≤ Xɶ T < X iC ) nor firm j (if X jM ≤ Xɶ T < X Cj ) has a dominant strategy. In this case there are two pure-strategy Nash equilibria. If firm j invests, the optimal (re)action of firm i is not to invest. If firm j does not invest, the optimal response of firm i is to invest. Symmetrically, if firm i invests, the best reply for firm j is not to invest. If firm i chooses not to invest, firm j should invest. The two pure-strategy Nash equilibria are (Invest, Abandon) and

Option to Invest

231

(Abandon, Invest).14 To solve this problem, we employ a focal-point argument.15 The optimal investment strategies again depend on the level of demand Xɶ T reached at maturity T. The end-node equilibrium payoffs (NPVT) can again be determined as a function of the demand parameter Xɶ T. Three regions or demand zones can be distinguished for firm i: Xɶ T < X iM , X iM ≤ Xɶ T < X iC , and Xɶ T ≥ X iC . Similarly there are three demand regions for firm j: Xɶ T < X jM , X jM ≤ Xɶ T < X Cj , and Xɶ T ≥ X Cj . The discussion so far applies for the general case. Let’s now consider explicitly that firm i is the low-cost firm (L) and firm j the high-cost firm (H), denoting the cost subscript accordingly (cL < cH ). When this cost asymmetry is taken into account, some cases sort themselves out, being mutually exclusive given that X LM < X HM and X LC < X HC. If Xɶ T < X LM , the case where Xɶ T ≥ X HM is not possible. Similarly, if Xɶ T < X LC , Xɶ T ≥ X HC does not hold. These explicit investment trigger restrictions enable deducing the optimal investment policies for the duopolist option holders. Once the investment policies are determined, we can deduce the equilibrium value payoffs at maturity depending on the level of demand reached at time T . The above are summarized in table 7.3. Depending on the future evolution of the underlying demand, three industry structures may result at maturity (T ): •

No one invests if Xɶ T < X LM (< X HM ).

Monopoly Only one firm (the cost leader) invests, while the rival has no incentive to invest and compete with the low-cost firm. The low-cost firm (L) becomes a monopolist if X LM ≤ Xɶ T < X HC.

•

Cournot competition Both firms invest simultaneously, choosing their individually rational output à la Cournot. This case occurs when Xɶ T ≥ X HC ( > X LC > X LM ).

•

Consequently the low-cost firm would choose not to invest, become a monopolist, or become a cost-advantaged Cournot duopolist, depending 14. A third Nash equilibrium in mixed strategies also exists, as discussed later. 15. Schelling (1960) introduced the notion of “focal point” to support the use of Nash equilibrium as a solution concept. Some use this approach to select among several purestrategy Nash equilibria. The focal-point argument suggests that among multiple equilibria some are more likely to occur due to common sense or psychological reasons. The “focal point” here is that the low-cost firm invests and the high-cost firm does not. This outcome may be the most likely since it Pareto dominates other strategy profiles. In other words, a social planner would give incentives to the low-cost firm to invest and to the high-cost firm to stay out if it wants to achieve the socially optimal equilibrium. The focal-point argument has questionable mathematical foundations so that one might prefer to rely on alternative equilibrium selection procedures. Such an alternative is alluded to later and discussed in detail in chapter 12.

(

)

No investment

Xɶ T < X LM

Low demand

)

Monopoly

X LM ≤ Xɶ T < X LC

Intermediate demand

(

Monopoly

Xɶ T ≥ X LC

High demand

⎛ 0⎞ ⎜⎝ 0⎟⎠ 2

⎛ NPVLM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 ⎛ NPVLM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 Monopoly

Focal point

Monopoly

0 ⎛ ⎞ ⎜ NPV M Xɶ ⎟ ⎝ H ( T )⎠

⎛ NPVLM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 ⎛ NPVLM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0

2

Monopoly

Monopoly

Cournot

Xɶ T ≥ X HC

X HM ≤ Xɶ T < X HC

Xɶ T < X HM

⎛ NPVLC ( Xɶ T )⎞ ⎜ ⎟ C ⎝ NPVH ( Xɶ T )⎠ 0 ⎛ ⎞ ⎜ NPV M Xɶ ⎟ ⎝ H ( T )⎠ 0 ⎛ ⎞ ⎜ NPV M Xɶ ⎟ ⎝ H ( T )⎠

Note: X iM ≡ 2 bδ I i + ci a, X iC ≡ 3 bδ I i + 2ci − c j a, NPVi M ( Xɶ T ) ≡ ⎡(aXɶ T − ci ) 4bδ ⎤ − I i , NPViC ( Xɶ T ) ≡ ⎡( aXɶ T − 2ci + c j ) 9bδ ⎤ − I i , for i = L, H. ⎣ ⎦ ⎣ ⎦

Low-cost firm (L)

High demand

Intermediate demand

Low demand

High-cost firm (H)

Table 7.3 Investment triggers and equilibrium payoffs (NPV) at maturity for Cournot quantity competition

232 Chapter 7

Option to Invest

233

Binomial tree

High

Intermediate

X0

Low

t=0

Investment decision/ industry structure

Threshold

Both invest (asymmetric Cournot)

~

XT ≥ XHC

Payoff function

~

NPVLC (XT)

~

NPVHC (XT)

Low-cost firm invests (monopoly) High-cost firm does not (0)

~

XLM ≤ XT < XHC

~

XT < XLM

~

NPVLM (XT)

0

0 0

None invests (0)

T

Figure 7.7 Investment decisions, industry structure and payoffs (NPV) at maturity in asymmetric Cournot duopoly X iM = (2 bδ I i + ci ) a, X iC = (3 bδ I i + 2ci − c j ) a , for i = L, H .

on demand realization Xɶ T . The high-cost firm does not invest or becomes a Cournot duopolist (it cannot become a monopolist). The resulting industry structures are summarized in figure 7.7. Example 7.3 Investment Triggers in Asymmetric Cournot Duopoly To help understand industry dynamics under cost asymmetry, let us revisit the example used previously (in the case of cost symmetry), assuming now that there is a low-cost firm with cost cL = 10 and a high-cost firm with cH = 15. Both face an identical capital investment cost I = 250. The investment triggers under asymmetry are now given from (7.4) and (7.9) by X LM = 2 1 × 0.10 × 250 + 10 5 = 4 and X HC = 3 1 × 0.10 × 250 + 2 × 15 − 10 5 = 7. Figure 7.8 shows the firm payoffs at maturity (NPV) when costs are asymmetric. One can discern three demand regions: for realized demand Xɶ T lower than the lower investment trigger X LM, no one invests; in the intermediary demand zone, X LM ≤ Xɶ T < X HC, only the low-cost firm (L) invests; at high demand (i.e., when Xɶ T ≥ X HC = 7), the industry becomes an (asymmetric) Cournot duopoly. There is a noticeable discontinuity in the low-cost firm’s (L) payoff function as the low-cost firm suffers a sudden value drop upon entry of the rival at X HC. At intermediate levels of demand, given its cost advantage, the low-cost firm is the only one to enter the market and chooses its quantity accordingly to enjoy temporary monopoly profits. However,

(

)

(

)

234

Chapter 7

NPV at maturity (in thousands)

15

10

5 Low-cost firm (L) High-cost firm (H)

0

0

XLM

5

XHC

10

15 ~

End-node value XT Figure 7.8 Payoffs at maturity for option to invest in asymmetric Cournot duopoly

as demand ( Xɶ T ) rises, the low-cost firm recognizes that the high-cost firm is also able to enter and make a profit, and thus adjusts its optimal quantity à la Cournot. The discontinuity occurs at this point. The low-cost firm produces more than the high-cost firm due to its cost advantage, still earning higher profits. The situation above rests on the resolution of the coordination problem (who enters first) in the intermediate demand region by use of the focalpoint argument, namely that the low-cost firm (L) naturally invests first or is the only entrant. An alternative would be to assume that the Cournot duopolists decide to leave their market-entry decision to chance and use mixed actions at each end node. In this case firm i chooses its (equilibrium) investment probability qi* such as to make its competitor (firm j) indifferent between investing, receiving qi* NPVjC + (1 − qi* ) NPVjM, and abandoning, receiving zero. The equilibrium investment probability for firm i is thus16 16. This logic is close to notions used later in chapter 12, in particular the equilibrium investment “intensity” or probability used to solve the coordination problem by use of continuous-time mixed strategies. In the present context, however, the follower’s value is equal to zero since we consider the problem at finite maturity (T ).

Option to Invest

qi* =

NPVjM . NPVjM − NPVjC

235

(7.10)

For asymmetric firms, one specific pure-strategy Nash equilibrium yields a Pareto-optimal payoff allocation. We assume that this strategy profile is the most likely based on a focal-point argument. The mixed-strategy approach provides an alternative to resolve the coordination problem. In the intermediate region, it is socially optimal for only one firm to invest. In this region each firm will invest with a positive probability (nondegenerate mixed strategy) as per equation (7.10). Note, however, that a “coordination failure” where both firms invest in the market, although only one can enter profitably, may still emerge in equilibrium. This point is addressed in chapter 12. In the following discussion, we do not allow for mixed strategies (this approach is less analytically tractable). 7.2.2 Asymmetric Cournot Oligopoly In the previous section on Cournot duopoly, we have seen that in the case of two asymmetric firms, three market structures may result: (1) no investment, (2) monopoly, or (3) duopoly. The threshold from the noinvestment demand region to the monopoly zone depends on the NPV of the low-cost firm (L), once it has invested. The threshold from the “monopoly zone” to “duopoly” is a function of the NPV expression of the high-cost firm seen as a “natural-born” follower. We can recast the results of the previous section in a more general setting. The focal point argument used above was based on the idea that the low-cost firm invests first in case of a coordination problem. Our present approach consists in generalizing this argument considering that the high-cost firm (H) is a “born” follower due to its cost disadvantage and will only invest if its NPV as Cournot duopolist is positive. We also recognize that the low-cost firm (L) will enter the market even if the investment is unprofitable for the high-cost firm. We here generalize this approach to an asymmetric oligopolistic industry where n firms share an (European) investment option, with firms having different, firm-specific costs. Let the transcript i = 1, . . ., n indicate the rank in relative cost advantage: firm 1 is the absolute (lowest) cost leader, firm 2 has a relative cost advantage compared to firms 3, 4, . . ., n, and so on and on. T indicates the number of periods to maturity. Let πɶ i( n, Xɶ T ) and NPVi ( n, Xɶ T ) denote the profit and the net present value resulting from the investment by firm i, when there are n active

236

Chapter 7

firms in the market. In chapter 3, equation (3.26), we derived the profit for firm i in an oligopoly setup with n firms in case of certainty ( Xɶ t = 1) as

π iC ( n, 1) =

1 (a − nci + ( n − 1) c− i ) , b ( n + 1)2 2

i = 1, . . ., n,

n

where c ≡ ∑ j =1 c j n is the average variable production cost in the industry and c− i ≡ ∑ j ≠i c j (n − 1) the average production cost for all other firms except firm i. Alternatively, we have 2

n ⎤ ⎡ 1 ⎢ a + ∑ j =1 c j − ( n + 1) ci ⎥ π ( n, 1) = π ( n) = , i = 1, . . ., n. ⎥ n+1 b⎢ ⎦ ⎣ The equilibrium profit value for firm i in case of uncertain end-node profits is similarly C i

C i

2

n ⎤ ⎡ ɶ 1 ⎢ aXT + ∑ j =1 c j − ( n + 1) ci ⎥ ɶ , i = 1, . . ., n. πɶ i( n, XT ) = (7.11) ⎥ b⎢ n+1 ⎦ ⎣ Let δ ≡ k − g, where k is the risk-adjusted discount rate and g the rate at which the underlying uncertain factor grows. The end-node NPV for firm i in case of investment with uncertain profits is 2

n ⎤ ⎡ ɶ 1 ⎢ aXT + ∑ j =1 c j − ( n + 1) ci ⎥ ɶ − Ii , NPVi ( n, XT ) = i = 1, . . ., n. ⎥ bδ ⎢ n+1 ⎦ ⎣ Firm n has an absolute cost disadvantage and will invest (as the latest entrant) if and only if NPVn ( n, Xɶ T ) ≥ 0, namely if Xɶ T ≥ X nC , where17 n

C n

X ≡

( n + 1) bδ I n + ( n + 1) cn − ∑ j =1 c j a

.

Similarly firm n − 1 will invest if and only if NPVn−1 ( n − 1, Xɶ T ) ≥ 0, n−1 namely if Xɶ T ≥ X nC−1, where18 X nC−1 ≡ n bδ I n−1 + ncn−1 − ∑ j =1 c j a. This can be generalized to any firm i, i = 2, . . ., n. Firm i will invest if and only if NPVi (i, Xɶ T ) ≥ 0, namely if XT ≥ X iC , where

(

X iC ≡

i ⎤ 1⎡ ⎢( i + 1) bδ I i + (i + 1) ci − ∑ c j ⎥ , a⎣ j =1 ⎦

)

i = 2, . . ., n.

(7.12)

17. If n = 2 , this reduces to expression (7.9) for the asymmetric Cournot duopolist. This is subject to aXɶ T + ∑ nj =1 c j − (n + 1) cn ≥ 0. If this condition is not met, the profit for firm n is negative and so the nth investor will not invest. 18. This is subject to aXɶ T + ∑ nj =−11 c j − ncn−1 ≥ 0 .

Option to Invest

237

In the special case of the first investor (i.e., of firm 1) preempting all others, the investment trigger is that of a monopolist given in equation (7.4) previously: X iM ≡

2 bδ I i + ci . a

Equation (7.12) thus applies to the case where i = 1 is included in the set {1, . . ., n}. The first (lowest cost) investor adopts a myopic stance when determining its investment strategy disregarding all rivals’ investment policies since they affect only the overall value of the investment but not its own optimal exercise strategy. Since X 1M < X 2C < . . . < X iC < . . . < X nC, the equilibria and payoff functions for the asymmetric firms in a Cournot oligopoly can be ranked as shown in figure 7.9. To further benchmark this result with previous known results, consider the special case of a duopoly with a cost leader (i = L) and a cost follower ( j = H ). The investment trigger of the low-cost firm (L) is X LM =

2 b δ I L + cL , a

and for the high-cost firm (H),

Binomial tree evolution

Threshold

Investment decision

Payoff function ~ NPV1 (n, XT)

~

High

XT ≥ XnC

...

All firms invest

~ NPVi (n, XT)

...

~ NPVn (n, XT)

…

…

… ~ NPV1 (i, XT)

X0

Intermediate

~

C XiC ≤ XT < Xi+1

...

Only first i firms invest

~ NPVi (i, XT)

0

...

…

…

Low

~

XT < X1C t=0

T

None invests

0

… 0 ... 0

Figure 7.9 Investment decisions and payoffs (NPV) in an asymmetric Cournot oligopoly with n firms i X iC = ⎡(i + 1) bδ I i + (i + 1) ci − ∑ j =1 c j ⎤ a , i = 1, . . ., n. ⎣ ⎦

238

Chapter 7

Table 7.4 Project payoffs (end-node NPVs) under differentiated Bertrand price competition Monopoly (firm i ) Bertrand duopoly ( i = 1, 2)

NPVi M ( Xɶ T ) ≡ Vi M ( Xɶ T ) − I NPVi B ( Xɶ T ) ≡ Vi B ( Xɶ T ) − I

2 2 2 Note: Vi M ( Xɶ T ) ≡ (ai Xɶ T − c ) 4bδ , Vi B ( Xɶ T ) ≡ (1 − s) ( ai Xɶ T − c ) b(1 + s) ( 2 − s ) δ .

X HC =

3 b δ I H + ( 2 c H − cL ) . a

This confirms equations (7.4) and (7.9) derived in the preceding section on asymmetric Cournot duopoly under uncertainty. 7.3

Differentiated Bertrand Price Competition

The analysis in the previous section assumed that firm actions were strategic substitutes and that firms engaged in quantity competition. In this section we turn to the case of actions that are strategic complements characterizing price competition. For more realism we consider the general case where products are differentiated and firms may have different demand functions. The uncertain market (inverse) demand function for firm i, following equation (3.1′), is assumed to be of the form: pi (Q, Xɶ t ) = ai Xɶ t − b ( qi + sq j ),

(7.13)

where b > 0 characterizes a downward-sloping demand, s (0 ≤ s < 1) captures the degree of substitutability, and ai is a constant price parameter assumed specific to firm i (ai ≠ aj). Each firm must incur the same capital cost I upon entry. Costs are here assumed symmetric: firm i’s cost function is Ci (qi ) = cqi, with same variable cost c ( ≥ 0 ). Using the equilibrium profits for monopoly and for simultaneous differentiated Bertrand price competition derived in chapter 3, equations (3.4) and (3.10), we obtain the end-node NPVs. These are summarized in table 7.4 for each of the above two industry structures. The results, when presented in strategic form, help better illustrate under which conditions these outcomes may occur. This strategic form representation is depicted in table 7.5. If NPVi B ( Xɶ T ) ≥ 0, namely if Xɶ T ≥ X iB, where19 2 − s ⎞ bδ I (1 − s 2 ) + c X iB ≡ ⎛ , ⎝ 1− s⎠ ai

(7.14)

19. This formula is an approximation. In the present context we assume asymmetric demand intercepts (ai ≠ aj) to be able to rely on a focal-point argument in favor of the demand-advantaged firm. For expositional simplicity we rely on equation (3.10), which uses identical demand intercepts.

Option to Invest

239

Table 7.5 Strategic form of end-node investment game under differentiated Bertrand price competition Firm j Invest Firm i

Invest Do not invest (abandon)

⎛ NPVi B ( Xɶ T )⎞ ⎜ ⎟ B ⎝ NPVj ( Xɶ T )⎠ 0 ⎛ ⎞ ⎜ NPV M Xɶ ⎟ ⎝ j ( T )⎠

Do not invest (abandon) ⎛ NPVi M ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 ⎛ 0⎞ ⎜⎝ 0⎟⎠

2 Note: NPVi M ( Xɶ T ) ≡ ⎡( ai Xɶ T − c ) 4bδ ⎤ − I, ⎣ ⎦ 2 2 NPVi B ( Xɶ T ) ≡ ⎡(1 − s )( ai Xɶ T − c ) b(1 + s) ( 2 − s ) δ ⎤ − I. ⎣ ⎦

firm i has a dominant strategy to invest since its value as a monopolist is higher than its value as a (differentiated) Bertrand duopolist. A similar argument holds for firm j. However, if NPVi M ( Xɶ T ) < 0, namely if Xɶ T < X iM , where, by extension of equation (7.4),20 X iM ≡

2 bδ I + c , ai

(7.15)

firm i will have a dominant strategy not to invest. If X iM ≤ Xɶ T < X iB, there is no dominant strategy for firm i. In this case, to identify the purestrategy Nash equilibria two cases need to be distinguished: 1. Firm i has no dominant strategy to invest while firm j has a dominant strategy. If Xɶ T ≥ X jB, firm j has a dominant strategy to invest, but firm i should not invest. If Xɶ T < X jM, firm j has a dominant strategy not to invest, but firm i should invest. 2. Neither firm i nor firm j has a dominant strategy to invest. In this case there are two Nash equilibria in pure strategies. If firm j invests, the optimal reaction for firm i is not to invest. If firm j does not invest, the optimal response for firm i is to invest. Similarly, if firm i invests, the best reply for firm j is not to invest. If firm i chooses not to invest, firm j should invest. The two Nash equilibria are (Invest, Abandon) and (Abandon, Invest). A focal point may be found, where the firm with the higher differentiated price parameter ai invests and the other does not. 20. Subject to Xɶ T ≥ c ai .

240

Chapter 7

Let firm i be the firm with the high-price parameter aH. Firm i is denoted H henceforth. Firm j is the firm with the low-price parameter aL. Some combinations are again mutually exclusive, for example, X HM < X LM and X HB < X LB. Based on these investment triggers, the optimal investment policies and equilibrium payoffs at maturity T can be deduced depending on the level of demand reached at time T (see table 7.6). Again, we identify three regions of demand. For low demand (i.e., Xɶ T < X HM ), no one invests; for high demand (i.e., Xɶ T ≥ X LB), both firms invest as (differentiated) Bertrand duopolists; and for an intermediary demand zone ( X HM ≤ Xɶ T < X LB), the resulting industry structure is a monopoly. The outcomes of the shared investment option game under differentiated Bertrand price competition at maturity are depicted in figure 7.10. As noted, the Cournot model is often used to describe industry structures where players first select the production capacity level for the long run and then compete in the short-term over prices (à la Bertrand). The Cournot model or its extensions may be more appropriate in case of option games involving long-term capacity investment decisions. For this reason most of the option games we consider subsequently are described in a Cournot quantity competition framework as strategic investment decisions typically have a long-term impact. Conclusion In this chapter we have built upon the materials developed earlier in chapter 3 to derive the value and optimal exercise formulas for benchmark option games under uncertainty. We first examined the monopolist’s deferral option to help identify the main factors influencing the investment policy of a firm. We later extended the analysis to quantity and price competition. We showed the importance of deriving the trigger strategies as part of the firm’s investment policy under uncertainty. Each firm should select a trigger level for the stochastic demand factor and decide to invest when the actual value of the process exceeds this investment trigger. Due to the existence of strategic interactions in the market, firms cannot simply set a trigger as a monopolist; they have to anticipate whether and when the rival will exercise its investment option in future states. The optimal investment option value reflects firms’ optimal future behavior in interaction with rivals and the resulting equilibrium industry structures.

No investment

Xɶ m < X HM *

Low demand

(

)

( )

Monopoly

X HM ≤ Xɶ T < X HB

Intermediary demand ⎛ 0⎞ ⎜⎝ 0⎟⎠

⎛ NPVHM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 ⎛ NPVHM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0

2

Focal point

Monopoly

NA

⎛ NPVHM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0 ⎛ NPVHM ( Xɶ T )⎞ ⎜ ⎟ ⎝ ⎠ 0

2

Bertrand

NA

⎛ NPVHB ( Xɶ T )⎞ ⎜ ⎟ B ⎝ NPVL ( Xɶ T )⎠ NA

2 bδ I (1 − s 2 ) + c ai ⎤⎦, NPVi M ( Xɶ T ) ≡ ⎡(ai Xɶ T − c ) 4bδ ⎤ − I, NPVi B ( Xɶ T ) ≡ ⎡(1 − s )( ai Xɶ T − c ) bδ (1 + s) ( 2 − s ) ⎤ − I, ⎣ ⎦ ⎣ ⎦

Monopoly

Xɶ T ≥ X HB

Xɶ T ≥ X LB

X LM ≤ Xɶ T < X LB

Xɶ T < X LM

High demand

2 − s⎞ ⎡ Note: X iM ≡ 2 bδ I + c ai, X iB ≡ ⎛⎜ ⎝ 1 − s ⎟⎠ ⎣ i = L, H. NA = nonapplicable.

Highdemand Firm (H)

High demand

Intermediary demand

Low demand

Low-demand firm (L)

Table 7.6 Investment triggers and equilibrium payoffs (NPV) under differentiated Bertrand price competition

Option to Invest 241

242

Chapter 7

Binomial tree evolution

Threshold

Investment decision

Payoff function (firm H, L)

~

High

X0

Intermediate

Low

t=0

~

XT ≥ XLB

~

XHM ≤ XT < XLB

~

XT < XHM

Both firms invest

High-demand firm invests (monopoly); Low-demand firm does not (0)

None invests (0)

NPVHB (XT)

~

NPVLB (XT)

~

NPVHM (XT)

0

0 0

T

Figure 7.10 Thresholds, investment decisions, and payoffs (NPV) in differentiated Bertrand price competition 2 NPVi M ( Xɶ T ) ≡ Vi M ( Xɶ T ) − I ; Vi M ( Xɶ T ) ≡ ( ai Xɶ T − c ) 4bδ ; X iM ≡ (2 bδ I + c) ai ; 2 2 NPVi B ( Xɶ T ) ≡ Vi B ( Xɶ T ) − I , Vi B ( Xɶ T ) ≡ (1 − s) ( ai Xɶ T − c ) bδ (1 + s) ( 2 − s ) ; X iB ≡ ⎡⎣((2 − s) (1 − s)) bδ I (1 − s 2 ) + c ⎤⎦ a , i = L, H .

Selected References Smit and Ankum (1993) extend binomial trees with embedded strategicform games. Smit and Trigeorgis (2004) develop a systematic approach to option games involving several option games models in discrete time meant to describe various industry settings. Dixit and Pindyck (1994) discuss some of these issues in continuous time. Dixit, Avinash K., and Robert S. Pindyck. 1994. Investment under Uncertainty. Princeton: Princeton University Press. Smit, Han T. J., and L. A. Ankum. 1993. A real options and gametheoretic approach to corporate investment strategy under competition. Financial Management 22 (3): 241–50. Smit, Han T. J., and Lenos Trigeorgis. 2004. Strategic Investment: Real Options and Games. Princeton: Princeton University Press.

8

Innovation Investment in Two-Stage Games

R&D investments are typically made under uncertainty. Firms cannot safely predict the state of demand when the resulting product offering will be launched in the marketplace, nor what the competitive situation will be. Box 8.1 discusses how real options analysis can provide intuitive insights regarding R&D investment. Here we complement such analysis with game-theoretic thinking. Strategic investments are often made under conditions of uncertainty about key market factors (e.g., demand, costs, or competitors’ strategies). When moves are hard or costly to reverse, the value of preserving flexibility must be explicitly assessed and traded off against the strategic benefits gained from early investment commitment. The trade-off between keeping one’s options open (the option to wait) and committing earlier to benefit from a positive strategic impact or rival behavior must be quantified. To address this issue, Smit and Trigeorgis (2001, 2004) extend Fudenberg and Tirole’s (1984) business strategy framework when one key underlying factor, namely demand, is uncertain. The uncertainty in demand is modeled as a multiplicative binomial process (e.g., see Cox, Ross, and Rubinstein 1979). In this chapter we propose a methodology to capture the trade-off between commitment and flexibility and present valuation expressions for quantifying the value of flexibility and commitment. The chapter is organized as follows. In section 8.1 we discuss the problem of investment in R&D when firms face spillover effects. Section 8.2 considers an application involving determination of optimal patenting strategies, while section 8.3 discusses the incentive to create goodwill in a context where firms compete in price. 8.1

Innovation and Spillover Effects

In chapter 7 we analyzed quantity competition involving asymmetric costs that were given exogenously. Now we consider two-stage games

244

Chapter 8

Box 8.1 Real options and gut feeling in R&D

In R&D, the Next Best Thing to a Gut Feeling Amal Kumar Naj, Wall Street Journal The tendency of Japanese companies to take advantage of innovations devised in the US is almost automatically attributed to Japan’s lower capital costs and its supportive government-industry alliances. But it’s often overlooked that from the very start US corporations put their emerging technologies at an enormous disadvantage because of the techniques they use to evaluate payoffs. The rigid equations and models currently used unfortunately have replaced the instinct and intuition that once guided US entrepreneurs. The fundamental test used in most US companies is: Does the return exceed the cost of capital over the life of the project? To find the answer companies subject the project to a variety of discounted-cash flow measures. Dozens of assumptions go into this methodological overkill. Many of the assumptions aren’t reliable. Some are tenuous at best. And if it’s a long-term project that would take, say, 10 years to commercialize (five years being the outer limits of American corporations’ horizon), the distant estimates are considered highly speculative and are heavily discounted. Even the “terminal value,” the value of the plant and other assets upon commercialization, is assumed to be zero. When all is said and done, the payoff estimates produced by the quantitative analyses are usually too low. They stand little chance of beating that ultimate yardstick—the cost of capital. Nine out of 10 projects, on average, fail the test. Perhaps the most fundamental problem with the current techniques is that they fail to take into account the consequences of not pursuing a technology. An auto maker may give up on ceramic engines, because technically they seem unfeasible. But a rival who makes prosaic ceramic parts with the aim of one day using the knowledge to build a ceramic engine—as the Japanese are doing—can quickly change the competitive balance. How to get American companies to act on their intuition again? There is a new technique that will go a long way toward that. It deserves wide attention, for it comes closest to simulating the old-fashioned gut feeling. Designed after well-developed stock-options theories, the technique works something like this: Suppose there are two emerging technologies, X and Y, that have some bearing on the company’s main business. The X may involve a well-understood technology and a market. The Y may involve a wide range of possible outcomes and new markets but with, say, 40 percent chance of technical failure. The benefits of X are easily quantifiable, and hence salable to top management for funding. The Y is unsalable because it’s unquantifiable, even though it seems more promising.

Innovation Investment in Two-Stage Games

245

Box 8.1 (continued)

Instead of ignoring Y, the company commits small amounts to develop the technology, in effect taking a “call option” on the underlying technology. The option allows R&D people to explore its technical possibilities, market opportunities, development costs and competitors’ strategies. If these don’t become clear within the time period of the option, the option is allowed to “expire.” The loss is limited to the small initial funding (the value of the R&D option), just as in the stock option. “The important thing to realize is that the initial expenditures are not directed so much as an investment as they are toward creating an option,” says William Hamilton of the University of Pennsylvania’s Wharton School, who has co-authored a paper on options valuations of R&D with Graham Mitchell, director of planning at GTE Laboratories in Waltham, Massachusetts. R&D options, fundamentally, help identify the unapparent outcomes that may be the most important reasons for undertaking the investment. Consider: In 1984, W.R. Grace decided to invest in a new technology for catalytic converters for the automobile aftermarket. The discounted cash flow analysis—which used a generous “terminal” value—showed that it was an “attractive” project. What looked like a sure-fire technology now can’t compete on price. “We didn’t accurately predict the price-performance requirement we needed in the aftermarket,” says Peter Boer, chief technical officer.

•

Grace would have dropped the project, except that other opportunities for that technology intervened. The recent clean-air legislation has created applications in cogeneration plants, in plants to reduce ozone emissions, and in utilities to control emissions. Suppose the automotive catalyst technology had been given a thumbs-down by the quantitative techniques, but the other downstream markets loomed on the horizon. Could Grace have overturned the negative signal and invested in the automotive catalyst anyway? “I don’t think I would have been able to communicate that to the finance people,” says Mr. Boer. In the early 1980s, General Electric initially ignored the emerging magnetic resonance imaging (MRI) technology for medical diagnosis. The MRI market was unclear. And most important, it would have cannibalized the market for GE’s existing CT diagnostic machine, which uses X rays. GE overruled the directive of the discounted cash flow technique, as it realized that if it didn’t cannibalize the CT market, someone else was going to. Walter Robb, director of GE’s R&D, says he doesn’t have much use for quantitative techniques now. “The challenge is to find 30 projects a year that will pay off not by NPV (net present value) but by the seat of our pants,” he says.

•

246

Chapter 8

Box 8.1 (continued)

It’s the downstream rewards that are allowing the pharmaceutical company Merck & Co. to plan extensive automation. When it considered the idea first for a drug packaging and distribution plant, the labor savings didn’t justify the investment. Moreover, with no prior experience with robots, Merck also wasn’t certain about the technical success.

•

“It was a tricky thing to convince the management. But options valuation allowed engineers to articulate a whole range of outcomes and their benefits,” says Judy Lewent, chief financial officer at Merck. The management agreed to take an “option” on automating the plant, and results from the pilot project since early this year have clarified the potential future benefits to the point that the company is now willing to expand automation to its diverse manufacturing operations. US West, a telecommunications concern in Denver, faced with fastchanging technologies in its industry, also has taken the options approach. It’s pursuing technologies—that it otherwise wouldn’t be able to justify— aimed at making its cellular phones and paging devices more user-friendly. “Our company doesn’t have the maturity and experience to act on gut feelings,” says David Sena, strategic technology planner of US West, a company spawned in 1984 by the AT&T breakup. •

The options technique needs refining (how to calculate the value of an R&D option is a subject of debate, for instance). But it is the first clear mechanism for R&D people to communicate with their finance men. Reprinted with permission of The Wall Street Journal, Copyright © 1990 Dow Jones & Company, Inc. Publication date: May 21, 1990.

where a firm may make a first-stage strategic investment (K) that can influence future variable costs. Suppose that the low-cost firm in a duopoly invests in an R&D process innovation to further reduce its variable cost cL.1 By reducing its second-stage variable cost, the low-cost firm faces a lower investment trigger and the project is 1. Having the low-cost firm ( L ) make the strategic investment is simpler. If the high-cost firm ( H ) is the one making such an R&D investment, two effects would result: (1) firm H ’s cost position is improved, affecting all the players’ investment triggers, and (2) for substantial change in the cost differential, the industry structure might change in the intermediate demand region from firm H staying out (without strategic commitment) to investing as a monopolist (with strategic commitment). This second effect creates a discontinuity in the value function. By focusing on the problem where the low-cost firm may improve its cost position, we are more in line with Fudenberg and Tirole’s (1984) analysis where the value function is differentiable in the first-stage investment. Considering the reverse problem would make the analysis more involved while being unable to isolate the pure strategic effect resulting from the commitment.

Innovation Investment in Two-Stage Games

247

more likely to be worthwhile (“in the money”), namely the opportunity to invest will be of higher value. At the same time this cost improvement reduces the rival’s incentives to enter in that it cannot extract as much value upon entering. That is, when the incumbent firm (L) lowers its future variable cost, the investment trigger for its rival X HC rises such that the high-cost firm is less likely to invest as a Cournot duopolist. Furthermore the firm investing in the lower cost production technology receives a higher temporary monopoly profit as well as higher duopoly rents compared to the case where it operates under the old technology. Four concurrent effects influence the investment decision as a result of this action: 1. The investment trigger of the low-cost firm ( X LM ) decreases, so it is more likely to invest (its option value increases). 2. The investment trigger of the rival ( X HC) increases, and consequently the low-cost firm is more likely to enjoy longer temporary monopoly profits. 3. At the intermediate demand zone, X LM ≤ Xɶ T < X HC, the profit made by the low-cost firm is higher (with process innovation leading to a decrease in its variable cost). 4. When demand is very high ( Xɶ T ≥ X HC ) and the high-cost firm also enters resulting in a Cournot duopoly structure, the low-cost firm may achieve higher profit (than without the upfront R&D investment). The firm therefore has an incentive to reduce variable costs not only because it enhances its chance of being NPV positive but also because of strategic interactions that may make the investing firm tougher, resulting in a (greater) cost disadvantage for its rival. We next discuss this problem at length. In analyzing strategic investment commitment, one has first to set a benchmark. At t = 0 firm i may decide to commit to an early strategic investment (e.g., R&D in process innovation) or not. If firm i does not commit now, it still has managerial flexibility to wait. This is the “base case.” By comparing the base case of no investment and the case involving strategic investment, we can determine the incremental value of commitment. Once the base case is set, we can assess whether early strategic investment commitment has a positive or a negative incremental value and compare the relative value of the investment strategy with versus without commitment. We first analyze the case when firms compete in quantity à la Cournot.

248

Chapter 8

The model developed previously makes it possible to value option games when one of the underlying factors is uncertain, such as when the demand shock parameter Xɶ t in the demand function p (Q, Xɶ t ) = aXɶ t − bQ

(8.1)

follows a multiplicative binomial process. To analyze strategic commitment, for example in R&D investment, it is useful to distinguish between “proprietary” and “shared” R&D investment. Firm L is contemplating investing ex ante in R&D (process innovation) that can reduce its unit production costs in the ex post competition phase. The base case involves the situation where firm L does not invest or reap the benefits of R&D investment. We assume that there is already a cost advantage for the low-cost firm even without the new commitment (cL < cH). Suppose that the R&D effort by firm L decreases ex post production cost, cL, by an amount ω (0 < ω < cL) to cL ′ ≡ cL ′(ω ) = cL − ω. The high-cost firm H does not itself invest in R&D, but due to R&D spillovers in case R&D is made and shared, it may partly or fully benefit from firm L’s investment in R&D as well. The unit production cost for the high-cost (noninvesting) firm reduces to cH ′ ≡ cH ′ (ω ) = cH − γω . The degree of spillover effect (shared R&D benefits) is reflected in the unit production cost savings of the high-cost firm via parameter γ ∈[ 0, 1]. If γ = 0, there are no R&D spillovers (the proprietary R&D case); if γ = 1, the high-cost firm simply free-rides from innovative firm L’s R&D investment (fully shared R&D).2 In some cases (e.g., in patent licensing) γ can be a decision variable. From chapter 7, equations (7.4) and (7.9), the investment triggers under the pre-R&D asymmetric cost structure are given by X LM ≡

2 b δ I + cL a

and 2. Spillovers measure to what extent the investment in R&D is “shared.” High spillovers mean that the competitor also benefits or free-rides on the investing firm. Low spillovers means that the investing firm can effectively protect its innovation and is the sole party benefiting from the R&D investment.

Innovation Investment in Two-Stage Games

X HC ≡

249

3 b δ I + ( 2 c H − cL ) . a

Let X LM ′ and X HC ′ be the investment triggers under the new, post-R&D cost structure. These are given by X LM ′ =

2 b δ I + cL − ω a

(8.2)

3 bδ I + ( 2cH − cL ) + (1 − 2γ ) ω . a

(8.3)

and X HC ′ =

Comparing these investment triggers, note first that X LM ′ < X LM (for all ω such that 0 < ω < cL). With process innovation, the cost and investment trigger of the investing firm decline and the likelihood that its option is in the money increases. In the American-type option framework we analyze later on (see later continuous-time analysis), this means that the low-cost firm will invest earlier in the production stage. A lower investment trigger resulting from innovation implies that the investment will be more attractive. Another insight is that the first-stage R&D investment of firm L may alter the investment trigger of its rival, depending on the size of the cost savings from innovation (ω ) and the degree of spillover (γ ). Since X HC ′ − X HC = (1 − 2γ ) ω a, X HC ′ ≥ X HC if γ ≤ 1 2 (a and ω are positive numbers). This has interesting implications on whether firm L should invest in R&D or not. For a low degree of spillover (γ ≤ 1 2), the investment threshold of the rival firm rises if the low-cost firm invests in R&D. For an American-type investment option this means that the rival invests later. This case with low spillover corresponds to the proprietary R&D investment case in Smit and Trigeorgis (2004). In patent licensing (see later section 8.2) it corresponds to the case of not licensing out, so the innovating firm keeps the innovation benefits to itself. By making the strategic R&D investment, firm L invests earlier in production (lower investment threshold for the low-cost firm); if the investment is proprietary or spillovers are low, this investment makes the rival firm less aggressive in the product market stage (i.e., it invests later on). This kind of positive strategic effect leads to the “top dog strategy” in Fudenberg and Tirole’s (1984) taxonomy. Investing in R&D makes innovative firm L tough, and the rival firm responds less aggressively in the competition

250

Chapter 8

stage (strategic substitutes), producing a positive strategic effect for the investing firm. Firm L should thus “overinvest.” In case of high spillover (γ > 1 2), however, the strategic investment in R&D by firm L reduces its own investment trigger as well as the investment trigger of its rival, so both firms benefit. Since firm L’s strategic move is also beneficial to the rival, it represents a soft commitment for firm L, who should refrain from investing. A third insight is obtained if we examine the difference between the investment thresholds of firm L as a monopolist in equation (7.4) and firm H as a duopolist in equation (8.3): X HC ′ − X LM =

bδ I + 2 ( cH − cL ) + (1 − 2γ ) ω , a

noting that ∂ ( X HC ′ − X LM ) ∂γ

=−

2ω (< 0 ). a

The higher the spillover effect (γ ), the lower the discrepancy between the investment thresholds of the two firms. If the spillover effect is high and firm L’s investment decision (to do R&D or license its technology to its competitor) also benefits the rival firm (the investment is “shared”), although the investment trigger of the low-cost firm is reduced the likelihood to become a monopolist is lower. In this case, firm L should refrain from making the strategic investment commitment. Firm L should wait or “underinvest” and keep a lean-and-hungry look. In this case “overinvesting” has a negative strategic effect.3 If the spillover effect (γ ) is low, however, the discrepancy between the two investment thresholds under the new cost structure widens and the attractiveness of firm L investing to reap proprietary benefits as a monopolist increases. The low-cost firm should overinvest (in R&D or licensing out its patent), acting as a top dog.4 So far we examined whether R&D investment has a positive or a negative strategic effect. To decide whether the firm should invest in R&D (license out its patent) or not, the net value of the investment under the two cost structures must be determined as discussed in the previous chapter. 3. Only a “suicidal Siberian” would overinvest under strategic substitutes even though investment would make it soft. 4. The low-cost firm would be a “submissive underdog” if it decided to underinvest to accommodate entry when its commitment makes it tough and actions are strategic substitutes.

Innovation Investment in Two-Stage Games

251

Example 8.1 Investment Triggers in a Duopoly Faced with Low R&D Spillover Let us revisit previous example 7.3, where cL = 10, cH = 15, b = 1, a = 5, k = 10 percent, g = 0 percent, I = 250 . Under the preR&D investment cost structure, the investment triggers given by equations (7.4) and (7.9) were X LM = 2 1 × 0.10 × 250 + 10 5 = 4 and X HC = 3 1 × 0.10 × 250 + 2 × 15 − 10 5 = 7. Now suppose that the lowcost firm L makes an upfront strategic investment in R&D of K = 200 to decrease its variable costs in the competition stage, achieving cost savings of ω = 4. Suppose that the spillover effect is low, with γ = 0.25 ( < 1 2 ). Firm L should then invest in R&D given that this commitment makes it tough and that firms compete in quantity (strategic substitutes). Under the new cost structure cL ′ = 10 − 4 = 6 and cH ′ = 15 − 0.25 × 4 = 14. The new investment triggers given by equations (8.2) and (8.3) are X LM ′ = 2 1 × 0.10 × 250 + 6 5 = 3.2 (lower than X LM = 4); and X HC ′ = 3 1 × 0.10 × 250 + 2 × 15 − 10 + 0.5 × 4 5 = 7.4 C (larger than X H = 7). As confirmed in figure 8.1, panel a, the investment trigger of the lowcost firm decreases ( X LM ′ < X LM ) such that the firm is more likely to invest. Furthermore the investment trigger of its rival increases so firm L is more likely to act as a monopolist in the marketplace. The project value is consequently higher for the low-cost firm and lower for its competitor. Depending on the size of the upfront strategic investment cost, firm L may be better off refraining from investing in R&D even if doing so has a positive strategic effect. It will do so when the cost of the investment necessary to obtain this strategic effect is higher than the incremental option value created by the strategic effect.5

(

)

(

(

(

)

)

)

Example 8.2 Investment Triggers in a Duopoly Faced with High R&D Spillover Consider the same situation but suppose now that the spillover effect is higher at γ = 0.75 ( > 1 2 ). The post-R&D costs now are cL ′ = 6 and cH ′ = 15 − 0.75 × 4 = 12. The new investment triggers are X LM ′ = 3.2 (same as with low spillover) and X HC ′= 6.6 (lower). Figure 8.1, panel b, confirms that as a result of the shared benefits of this R&D investment, both firms now face lower triggers and are more likely to invest in the product 5. Assuming a maturity of T = 5 years over 10 equally spaced time steps, volatility of σ = 30 percent, and risk-free rate r = 3 percent, the value of the option for the low-cost firm under the post-R&D cost structure is estimated to be 168 . The value of the option under the pre-R&D cost structure is 82 . If the additional cost advantage via R&D costs more than 168 − 82 = 86, it is not worthwhile creating a costly competitive advantage since the probability that this advantage will be useful does not offset today’s necessary cash outlay.

252

Chapter 8

NPV at maturity (in thousands)

3

2

Low-cost firm (L)

1 High-cost firm (H)

0 2.5

5.0

XLM'

XLM

7.5

XHC

10.0

XHC '

~

End-node value XT Old cost structure (pre-R&D) New cost structure (post-R&D)

(a)

NPV at maturity (in thousands)

3

2

Low-cost firm (L)

1 High-cost firm (H)

0 0

2.5

5.0

XLM'

XLM

7.5

XHC '

XHC

10.0 ~

End-node value XT Old cost structure (pre-R&D) New cost structure (post-R&D)

(b)

Figure 8.1 R&D investment (a) With low spillover (γ = 0.25); (b) with high spillover (γ = 0.75)

Innovation Investment in Two-Stage Games

253

market stage. Firm L, however, is less likely to enjoy monopoly profits than in the previous example since the rival’s entry trigger is now lower ( X HC ′ drops from 7.4 to 6.6) and will more likely enter. When spillover is high, the incentive to invest in R&D is greatly reduced and the value is lower, compared to the previous case with low spillover.6 The analysis above confirms that firm L should refrain from investing in process innovation if spillover is high and its rival will likely get a free-ride. This result, however, may get reversed if the innovation benefits are shared with the rival benefiting not for free but with adequate compensation, such as when firm L decides to license out its new innovation to firm H for a fixed cash payment or for a royalty fee. In this case the degree of spillover (γ ) in effect becomes a decision variable for firm L, namely whether to license out (γ = 1) or not (γ = 0). We examine this interesting application next. 8.2

Innovation and Patent Licensing

Previously we analyzed the impact of spillover effects on the investment decision made by two firms in a Cournot duopoly owning a shared investment option. Here we examine a strategic situation where whether there will be a “spillover” effect (sharing the benefits of the innovation) or not is an endogenous choice variable by the innovating firm through its outlicensing decision. In this context there is no a priori external pure spillover effect (γ = 0) unless the firm holding the patented process innovation or cost-reducing technology decides to license out its technology to the rival (in which case γ = 1) in exchange for some upfront payment or royalty fee. Option games analysis can extend previous gametheoretic literature on patent licensing and provide new insights about optimal decisions under uncertainty. In this section we review related literature on patent licensing and the trade-off between fixed fee and royalty fee in a Cournot duopoly setting, extending the analysis to incorporate uncertain demand along the product development phase. 8.2.1

Patent Licensing: Deterministic Case

Patents and Licensing Patents are meant to balance incentives for firms to innovate while ensuring dissemination of benefits for consumers. Patents presumably 6. We assume the same parameter values as in example 8.1. Under this post-R&D cost structure with high spillover (shared) benefits the value of the option for the low-cost firm is 134 . Here the threshold at which the R&D investment destroys value is lower (134 − 82 = 52) compared to the low-spillover case.

254

Chapter 8

encourage innovation by providing firms with temporary monopoly rights enforceable by law.7 Licensing is a means by which a patentholding firm can derive benefits from its intellectual property (IP) rights.8 Arrow (1962) analyzed cost-reducing innovations and the impact of drastic versus nondrastic innovation on competitive market equilibria. According to Arrow, a drastic innovation is one for which the postinnovation price of a monopolist is below the pre-invention competitive price. If the cost of producing under the old technology is constant and equal to c, the post-invention monopoly price must be less than c (the price set in perfect competition) for an innovation to be “drastic.” Licensing payments generally take one of the following three forms: (1) a fixed fee the licensee pays to the licensor, (2) a royalty rate as a function of the revenues or volumes the licensee produces, or (3) a mix of the above, namely an upfront fixed fee plus regular royalty payments. Only the first two forms are discussed here. In patent licensing, the game-theoretic interaction between the patentholding firm and would-be licensee(s) is typically as follows. At first, the patent-holding firm proposes a deal, choosing the type of licensing contract (fixed fee vs. royalty rate) and the terms of the contractual relationship (e.g., amount of the fee). Would-be licensees may either accept or refuse the proposal made by the patent holder. If they accept it, they can use the new (cost-reducing) technology conditional on the payment of the fees to the patent-holding firm. In the last stage simultaneous competition occurs between producing firms (potentially including the patent holding firm). They may compete either in quantity or in price. The game is played once and all relevant information is common knowledge to all the players. The licensor works out its licensing proposal as part of a subgame perfect Nash equilibrium, with the optimal decisions determined by backward induction. Below we discuss the case of a homogeneous-good duopoly where one of the rivals acquires a cost-reducing process innovation that it may license to a single would-be licensee.9 Firms i and j have a shared (European) option to launch the new product in the market. Firm i holds a patent with a technological edge and can decide to license its technology to its rival for a certain fixed fee or royalty rate. In the last stage the 7. This proprietary right is granted if the innovator provides public authorities with sufficient information concerning the innovation content so that society and researchers can benefit and further build on it. 8. For a general overview of the game-theoretic literature on patent licensing, see Kamien (1992). 9. This model follows Wang (1998) who models Cournot duopoly with symmetric quantity competition. Here the asymmetric case is considered.

Innovation Investment in Two-Stage Games

255

duopolists compete in quantity, setting their output independently and simultaneously (Cournot quantity competition). The patent-holding firm is concerned with maximizing its total profit when designing the offer (setting its fixed fee or royalty rate). For the patent holder the total profit consists of the profit derived from its own production plus any licensing revenues.10 The would-be licensee is also profit maximizing and may reject the offer if it does not make it better off. Suppose that the two firms face a linear (inverse) demand function as in equation (3.1), namely p (Q) = a − bQ, with Q = qi + q j. Prior to introducing the new process innovation (pre-innovation or base case), firms’ unit production cost are ci and c j (old technology) with ci ≤ c j. If the market is not sufficiently large for both firms to produce, the only active firm (say firm i) would earn monopoly profits, based on equation (3.4), equal to:

π iM ( ci , c j ) =

(a − ci )2 4b

.

(8.4)

If the market is large for both firms to be producing, firm i’s profit equals the (asymmetric) Cournot equilibrium profit of equation (3.21)

π iC ( ci , c j ) =

(a − 2ci + c j )2 . 9b

(8.5)

The equilibrium profits for firm j are analogous. Suppose that firm i develops a process innovation and obtains a superior technology that enables it to lower its marginal production cost by (savings) amount ω (> 0), resulting in a post-invention unit production cost ci’ ≡ ci − ω . Firm i’s innovation can be drastic (offering it the possibility to pre-empt and set monopoly prices) or nondrastic (letting room for competition). Drastic process innovations reduce costs by a sufficient amount such that the patent-holder can reap monopoly rents for some time. By contrast, nondrastic innovations are associated with a slight cost advantage over competitors, not sufficient to drive out rivals. Firm i will license its technology if doing so makes it better off. Four cases (six subcases) can be distinguished: (A) innovation is drastic and firm i prefers not to license out its process innovation; (B) innovation is nondrastic and firm j is given no access to the technology (no licensing); (C) innovation is drastic 10. The model by Wang (1998) differs from that of Kamien and Tauman (1986) who also analyze licensing in a Cournot oligopoly in that the patent holder in Wang’s (1998) model is one of the incumbents competing in the industry for production. Kamien and Tauman’s (1986) key result relating to the superiority of the fixed-fee licensing over royalty licensing stems from this differentiating feature.

256

Chapter 8

Table 8.1 Alternative cases for licensing out a patented technology and type of innovation Innovation

Innovator firm

i

No licensing out Licensing out

Fixed fee Royalty rate

Drastic

Nondrastic

Case A Case C1 Case C2

Case B Case D1 Case D2

and firm i chooses to license it; (D) innovation is nondrastic and firm i licenses it. In addition, when firm i licenses its technology, it may either select a fixed fee or a royalty payment. These cases are summarized in table 8.1. Consider first cases A and B in which licensing out does not occur (because it is not in the patent holder’s interest to do so). In both cases firm i produces with its new (lower cost) technology alone, while its competitor is forced to operate its business using the old technology (if it produces at all).11 Firm i’s marginal production cost with the new technology is ci′ ≡ ci − ω, whereas firm j’s cost remains c j. Drastic Innovation If innovation is drastic, firm j is driven out of the market; the (monopoly) price set by the patent holder is lower than (or equal to) the marginal production cost of its competitor (using the old production technology), that is, from equation (3.3): pM =

a + ci′ ≤ cj. 2

(8.6)

The monopoly price is lower than the marginal cost of the rival (based on the old technology) when the savings from innovation are greater than (or equal to) ω ≡ a + ci − 2c j (i.e., if ω ≥ ω ). If firm i decides not to license out its technology when the innovation is drastic (i.e., ω ≥ ω ), case A emerges. By contrast, if innovation is nondrastic (i.e., if ω < ω ) and firm i insists not to license, case B arises. Case A (ω ≥ ω ) If innovation is drastic, firm j can be driven out of the market while firm i acts as a monopolist earning monopoly profits π iM ( ci′, c j ). Equilibrium profits for firms i and j, based on equation (3.6), are 11. Under no licensing, it is assumed that no spillover effects occur (γ = 0 ).

Innovation Investment in Two-Stage Games

π iM ( ci′, c j ) =

( a − ci + ω )2 4b

,

257

πM j ( ci′, c j ) = 0.

(8.7)

Case B (ω < ω ) If the process innovation is nondrastic and the market is large enough for both firms to produce, both firms maximize their profits by optimally selecting their output (simultaneously), resulting in asymmetric Cournot-Nash equilibrium profits:

π iC ( ci′, c j ) =

(a − 2ci + 2ω + c j )2 , 9b

π Cj (ci′, c j ) =

(a − 2c j + ci − ω )2 . 9b

(8.8)

Alternatively, conditional on the monetary profit the patent holder will receive from licensing out, it may decide to license out its technology to its rival. Firm i selects the type of licensing contract (fixed-fee or royalty payment) and chooses the (optimal) price to be charged for licensing out the new technology. As the leader, it determines its optimal price by backward induction. Consider the cases where the patent holder decides to license out its technology. The innovation can be drastic (case C) or nondrastic (case D). The patent holder may license out its technology either for a fixed fee (cases C1 and D1) or for a royalty payment (cases C2 and D2). Fixed-Fee Licensing Consider first the fixed-fee licensing cases (C1 and D1). Suppose that the patent-holding firm licenses its process innovation for a fixed fee F. If innovation is nondrastic and licensing out occurs for a fixed fee (case D1), both firms will produce with the new technology at a marginal production cost ci′ ≡ ci − ω (respectively, c ′j ≡ c j − ω ). Equilibrium Cournot profits are

π iC ( ci′, c ′j ) =

(a − 2ci + ω + c j )2 , 9b

π Cj ( ci′, c ′j ) =

(a − 2c j + ω + ci )2 . 9b

(8.9)

The profit expression π iC (ci′, c ′j ) is taken into account (backward induction) when the patent-holding firm sets its optimal fixed fee (case D1). In addition to the profit firm i receives as a Cournot duopolist (if both firms produce), it will also receive the fixed-fee payment F from outlicensing. The maximum fee firm j would be willing to pay for the license is the difference between its profit as a privileged licensee (case D1) versus a nonlicensee under nondrastic innovation (case B). Firm i therefore sets the optimal fixed fee (F*) such that π Cj ( ci′, c ′j ) − F ≥ π Cj ( ci′, c j ). The maximum fixed fee the patent-holding firm i can charge licensee

258

Chapter 8

firm j is such that the licensee is indifferent between producing with the new technology (having lower production cost c ′j) versus the old technology (operating with cost c j). This yields the maximum licensing fixed fee the patent holder could charge: F * = 4ω

(a − 2c j + ci ). 9b

(8.10)

Firm i’s total profit is thus made up of last-stage production profits plus the fixed-fee licensing payment, namely π iC (ci′, c ′j ) + F *. Firm i will choose to license out its patent if π iC ( ci′, c ′j ) + F * > π iC ( ci′, c j ), with c ′j < ci. This occurs if the cost reduction obtained by the innovation ω is small (ω < ω , with ω ≡ 2 ( a + 4ci − 5c j ) 3). Equivalently, firm i will achieve a higher profit through licensing out its new nondrastic technology when the amount of cost reduction (ω ) is lower than ω . If the cost reduction is large (ω ≥ ω ), it will prefer not to license out under the fixed-fee licensing contract.12 If innovation is drastic (case C1 with ω ≥ ω ), the maximum fixed fee is thus given by F * = π Cj ( ci′, c ′j ) − π Cj ( ci′, ∞ ) = π Cj ( ci′, c ′j ).

(8.11)

Comparing its profit under no licensing (π iM ( c′i )) and its total profit under fixed-fee licensing (π iC (ci′, c ′j ) + F *) allows firm i’s management to decide whether it is justified licensing out its technology. Since when the innovation is drastic (case C1) π iM (ci′) > π iC ( ci′, c j′ ) + F *, the patent holder would prefer to keep its technology for itself (not licensing it out) and become a monopolist, driving the rival out of the market. Royalty Rate Licensing Consider next the royalty licensing case (cases C2 and D2). Here the patent holder considers licensing out its technology to a licensee at a set royalty rate based on the quantity the licensee (firm j) produces using the new licensed technology.13 The marginal cost of the patent-holding company (firm i) stays unchanged at the lower level ci′ ≡ ci − ω . Firm j benefits from the new technology but has to pay a proportional amount R per unit of output. Its marginal cost is now effectively higher, given by c ′j + R (with c ′j ≡ c j − ω ) if it licenses the technology, and c j otherwise. 12. The threshold for an innovation to be drastic (ω ) is strictly greater than the threshold for firm i selecting a fixed-fee licensing contract (ω > ω ). 13. In practice, royalty rates are often set as a percentage of revenues but for simplicity we adopt the assumption made in the literature of the royalty being a set amount per unit of quantity produced.

Innovation Investment in Two-Stage Games

259

The licensee would not accept to pay a royalty rate higher than the marginal benefit from producing with the new technology, namely 0 ≤ R ≤ ω. From the perspective of the patent-holding firm i, in addition to its own production profit it also receives an extra revenue from licensing (of R q j). Drastic Innovation (Case C2) When both firms are active in the market, they will each select (simultaneously) their quantity to maximize their profits. Firm j’s profit is given by

π j ( qi , q j ) = [ p (Q) − c ′j − R ] q j . Firm j considers firm i’s quantity as given when maximizing its own profit (by selecting q j ), leading to a best-reply (reaction) function analogous to (3.13) of 1 ⎛ a − c ′j − R ⎞ q j * ( qi ) = ⎜ − qi ⎟ . ⎠ 2⎝ b

(8.11)

Firm i’s total profit is made up of two components: the profit obtained in simultaneous (quantity) competition with firm j and the revenue it receives from royalty fees

π i ( qi , q j , R ) = [ p (Q) − ci′] qi + R q j . Firm i will select its output qi to maximize its total profit π i ( qi , q j , R). By differentiating the profit function with respect to qi, the right-hand term R q j drops out, so the best-reply function for firm i is 1 a − ci′ qi* ( q j ) = ⎛ − qj ⎞ . ⎝ ⎠ b 2

(8.12)

The equilibrium quantities, found at the intersection of the two bestreply curves, are qi* ( R ) =

a − 2ci + c j + ω + R , 3b

q j * (R ) =

a − 2c j + ci + ω − 2R . 3b

(8.13)

Equilibrium profits are therefore given by

π i* ( R ) = π iC ( ci′, c ′j + R) + R q j * ( R ),

π j * ( R) = π Cj ( ci′, c ′j + R).

(8.14)

The optimal royalty rate (R*) that maximizes the licensor’s total profit (in subgame perfect Nash equilibrium) is obtained from the first-order condition as

260

R* =

Chapter 8

5 ( a + ω ) − ( ci + 4c j ) . 10

(8.15)

Nondrastic Innovation (Case D2) In this case the maximum royalty rate the licensee would be willing to pay (in deciding whether to accept or reject the offer) is the one at which firm j is indifferent between being a licensee (having cost c ′j) and continuing operating with the old technology (at production cost c j). In the case of nondrastic innovation, the maximum royalty rate is R* = ω , where c ′j + R* = c j. Equilibrium profits for firms i and j are

π i* (ω ) = π iC ( ci′, c ′j + ω ) + ω q j * (ω ) = π iC ( ci′, c j ) + ω q j * (ω ), π j* (ω ) = π Cj (ci′, c ′j + ω ) = π iC (ci′, c j ). Firm j’s cost is the same whether firm i licenses out the technology or not. As ∂π j * (ci , c j ) ∂ci > 0 and ω > 0, firm j’s post-innovation profit is lower than its pre-innovation profit. Once the patent holder has determined the optimal royalty rate to set, it can decide whether to license out or not. When the innovation is nondrastic, since the total profit the patent holder receives from licensing under a royalty rate is higher than under no licensing, namely π iC ( ci′, c j ) + ω q j * (ω ) > π iC ( ci′, c j ), licensing out under a royalty rate fee makes the patent holder better off. Given the additional benefit ω q j * (ω ), the patent holder will prefer licensing out its patent when a royalty licensing method is used and process innovation is nondrastic. Comparison In case of drastic innovation (case C2), firm j will not produce.14 The would-be licensee is indifferent between being a licensee or not producing at all, in both cases earning zero profit.15 The patent-holding firm thus ends up earning monopoly profits when the innovation is drastic. Moreover, as the profit earned by the patent holder (firm i) under the optimal royalty contract, π i* ( R*), equals the monopoly profit π iM under drastic innovation and no licensing, the patent holder is indifferent between being monopolist and licensing for a high royalty rate, R*. Thus, under drastic innovation, licensing out via a royalty rate yields the same outcome as choosing not to license. Under the royalty-payment licensing

{(

}

2 ci − c j ) ; 0 = 0 . 5 15. In case ci < c j , firm j will refuse the licensing contract and earn zero profit (as in case A). 14. Since ci ≤ c j , max {q j* (ω*); 0} = max

Innovation Investment in Two-Stage Games

261

method the patent holder will license its innovation to its rival if the innovation is nondrastic (case D2) but will keep it for itself and become a monopolist if the process innovation is drastic (case C2). How do the fixed fee versus the royalty rate alternatives compare? Several cases result depending on (1) the nature of the innovation (drastic vs. nondrastic) and (2) the type of licensing payment (fixed-fee vs. royalty rate). When innovation is drastic (ω > ω ), the patent holder will choose to keep the new technology for itself and become a monopolist. No licensing contract makes the patent holder better off. For nondrastic innovation involving small cost savings due to process innovation (i.e., for ω < ω < ω ), the patent-holding firm earns more by licensing out using a fixed-fee contract than not licensing, receiving π iC ( ci′, c j′ ) + F *. If a royalty rate is selected instead, the patentee will earn π iC ( ci′, c j ) + ω q j * (ω ). The incremental profits under the two licensing methods (royalty vs. fixed fee) are

ω [π iC (ci′, c j ) + ω qj * (ω )] − [π iC (ci′, c j′ ) + F *] = 9b [a − c j − 5 (ci − c j )] (> 0). That is, for nondrastic innovation involving small cost savings (ω < ω < ω ), the patent holder is better off licensing out its technology via a royalty rate payment (q j R*) than via a fixed fee (F*). For somewhat larger cost savings (ω < ω < ω ), the patent holder is better off not to license than to license under the fixed-fee contract. By contrast, in this region the patentee is better off licensing under the royalty rate method than not licensing. In other words, in case of nondrastic innovation (ω < ω ) the patent-holding firm will prefer to license out its technology using the royalty rate contract. In this case equilibrium profits will be π i* (ω ) = π iC ( ci′, c j ) + ω q j * (ω ), and π j * (ω ) = π Cj ( ci′, c ′j + ω ). In case of drastic innovation, the patent holder will not license out, ending up as a monopolist.16 8.2.2

Patent Licensing under Uncertainty

The game-theoretic analysis above can be extended to incorporate product demand uncertainty. Depending on market development, different industry structures may occur once uncertainty is considered. 16. These results differ from Kamien and Tauman (1986) who show that licensing by means of a fixed fee makes a (nonoperating) patent holder better off. The difference lies in the fact that, in the model above, royalty rate licensing gives an increased cost advantage to the patent holder through the royalty payment acting as an additional marginal cost for the licensee. This extra marginal cost advantage does not exist under fixed-fee licensing and does not affect equilibrium quantities.

262

Chapter 8

However, first-stage innovation investment and second-stage licensing policy may alter industry structure altogether. Suppose that in each end state at maturity firms behave optimally, that is, set Cournot-Nash quantities and select the optimal licensing policy. If the innovation is nondrastic (i.e., if ω < ω ), the patent-holding firm will license its technology to its rival and earn a royalty payment (at an optimal rate R* = ω ). If innovation is drastic, the patent holder will drive its competitor out of the market and earn monopoly profits. Suppose that the low-cost firm (firm L) is the patent holder (having a pre-invention cost cL). Its process innovation allows it to reduce its production cost further by ω , facing a reduced post-invention cost of cL′ ≡ cL − ω . The patent holder may license its technology to the high-cost firm (firm H). The necessary investment outlay is identical for both firms (I ). From equations (7.4) and (7.9) the investment triggers in the standard Cournot option game for a monopolist and an asymmetric Cournot duopolist are given by X LM =

2 bδ I + cL , a

X HC =

3 bδ I + ( 2cH − cL ) . a

Compared to the previous option games, here a new threshold needs to be considered. This is the critical threshold which distinguishes drastic from nondrastic innovation, namely ω ≡ a + cL − 2cH. This threshold increases for higher levels of the demand intercept (a). The demand intercept a used in deterministic cases earlier is now substituted by aXɶ t in expression (3.1), where Xɶ t follows a stochastic process in discrete time. The demand threshold separating drastic from nondrastic innovation is determined as follows. Innovation is drastic if ω > ω ( ≡ aXɶ t + cL − 2cH ), or alternatively if demand is limited such that Xɶ t < X , with X≡

ω − cL + 2cH . a

(8.16)

Innovation is nondrastic if demand is sufficiently high for both firms to operate profitably, namely if Xɶ t > X . If X LM > X or ω < ω M ≡ cL − cH + bδ I , the patent holder will not be able to drive its competitor out of the market. If ω ≥ ω M, there exists a region of demand ( X LM < Xɶ t < X) where the patent holder will be a monopolist. For very low demand ( Xɶ t < X LM ), the low-cost firm does not enter the market. If innovation is nondrastic and the high-cost firm H also enters the market, firm L will license out its cost-reducing technology to its com-

(

)

Innovation Investment in Two-Stage Games

263

petitor. The unit production cost for firm H using the new technology is cH′ + R* = cH. The investment trigger (in case of licensing) for the highcost firm is X HC =

3 b δ I + ( 2 c H − cL ) . a

(8.17)

(

)

If X < X HC , that is, if ω < ω C ≡ 3 bδ I , the existence of drastic innovation does not impact the investment decision of the high-cost firm at maturity; firm H will invest if Xɶ t ≥ X HC > X . For ω > ω C and X > X HC , drastic innovation will affect the investment behavior of the high-cost firm. The firm will invest if Xɶ t ≥ X > X HC. Both in the base case and in the licensing case above the investment trigger of the high-cost firm is unchanged as the firm earns the same profit. Four possible cases are distinguished, resulting in different industry structures. These are summarized in figure 8.2. Pre-innovation

∞

Post-innovation

∞

∞

∞ Duopoly

Duopoly

XHC

XHC Case A

X

Monopoly

Monopoly

M L

X

Case B

X

X

M L

No investment No investment

0

0

0 ∞

∞

0 ∞ Duopoly

X

M L

Case C

X

X Mutually exclusive conditions for w

Case D

XHC

Monopoly

XLM X

C H

0 Figure 8.2 Possible outcomes for patent licensing game

No investment

0

264

Chapter 8

Case A If (ω < ω C ) ⇔ ( X < X HC ) and (ω < ω M ) ⇔ ( X < X LM )) (small cost savings).

•

Case B If (ω < ω C ) ⇔ ( X < X HC ) and (ω > ω M ) ⇔ ( X < X LM ) (intermediate cost savings). •

Case C If (ω > ω C ) ⇔ ( X > X HC ) and (ω < ω M ) ⇔ ( X < X LM ) (intermediate cost savings). •

Case D If (ω > ω C ) ⇔ ( X > X HC ) and (ω > ω M ) ⇔ ( X < X LM ) (large cost savings). •

In cases A and B the first-stage innovation investment only impacts the optimal investment strategies through small cost savings and the royalty payments. Firm L’s profit and value as a monopolist in the market is enhanced through these cost savings, resulting in a lower investment trigger. In addition firm L also receives the royalty payment paid by licensee firm H. For firm H the marginal cost savings it gains from using the new technology (ω ) are offset by the royalty rate fee payment— chosen optimally by firm L at R* = ω . That is, the patent-holding firm can set its optimal royalty rate (R*) such as to extract the full cost savings resulting from its new technology (ω ). Hence its profits under no licensing and under licensing based on the optimal royalty rate are the same. However, from the licensee’s perspective compared to the pre-innovation case, its profit is lower because its competitor (firm L) has reduced its marginal cost as a result of the process innovation. Thus firm H’s investment strategy is impacted adversely since its post-innovation profit and net present value decline and its investment trigger increases. The patent-holding firm L is more likely to exercise its investment option as a monopolist. Case C will never occur since cL < cH . The inequalities cL − cH + bδ I < bδ I < 3 bδ I hold and preclude case C. Case D is the one where the impact of innovation is most pronounced. Beyond the effect on profits and value from the cost savings, the first-stage innovation investment here changes the industry structure dramatically. Compared to the pre-innovation case (where the high-cost firm invests if Xɶ t > X HC ), post-innovation firm H is driven out of the market in the region X HC < Xɶ t < X . Its actual investment trigger is not X HC but rises to X given by equation (8.16). 8.3

Goodwill/Advertising Strategies

In this section we discuss advertising/goodwill in price competition situations where the entry decisions by firms depend on both the firm’s

Innovation Investment in Two-Stage Games

265

goodwill (endogenous factor) and on market development (exogenous). Suppose that two firms have an option to launch a new innovative product generation. One of the two firms (high-demand firm H) has the possibility to make a first-stage advertising investment to raise customer awareness by enhancing its brand image. The firm realizes that it can influence the rival’s second-stage behavior by committing to a strategic marketing campaign. Suppose that firms compete in prices and sell horizontally differentiated products (differentiated Bertrand). A differentiation parameter enters both firms’ profit function. It is given by s (0 ≤ s < 1), representing the substitution effect between the two differentiated product offerings. The (inverse) linear demand function for firm i, i = L, H (with aH > aL), analogous to equation (3.1′) and (7.13), is given by pi (Q, Xɶ t ) = ai Xɶ t − b ( qi + sq j ). Firm H considers spending a certain amount on advertising and goodwill building (K i) that can alter customers’ beliefs and the substitution between the two competing products. The rival, firm L, also has a shared option to expand and launch at maturity a substitute product competing with firm H’s innovative product. This scenario will likely occur in highdemand regions. Unlike its rival, however, firm L has no possibility to launch a strategic advertising campaign. Firm H can promote its brand image by stressing (1) its product’s distinctive features or (2) the fact that firm H’s and firm L’s products are competing in the same field. Apple® serves to illustrate the first kind of investment. Over the last decade the California-based company has developed a strong brand identity that it can leverage in many segments: having its roots in information technology, the firm has successfully diversified into legal music downloading (iTunes®), music players (iPod®), and mobile telecommunication (iPhone®). Customers of Apple’s products appreciate the unique image of Apple and prefer them against would-be competing products. Even if the iPod® is not of better quality than its MP3 player counterparts, these products are differentiated by the brand image of the product and the manufacturer. Through its strategic advertising and marketing campaigns, Apple has managed to lower the substitution effect with competing products setting a substitution effect (s) close to zero. This kind of marketing strategy also has a beneficial effect for rivals. By differentiating one’s product from others, these firms are behaving as in a monopoly-like segment and can set higher prices than in a fierce competitive environment.

266

Chapter 8

If products were perfect substitutes (s → 1), firms would face the risk of zero economic profit as into the Bertrand paradox. The advertising warfare PepsiCo and Coca Cola Company waged in the United States illustrates another kind of advertising investment. Each firm advertised that its product has a better taste than its competitor’s, creating in customers’ mind the feeling that the products of these firms are on an equal footing. Launching such kind of marketing campaigns increases the substitution effect between the competing products. Instead of dividing the market in two differentiated segments, each firm tried to capture a larger slice of the market by luring customers from the other side. Competing fiercely over close substitute products, firms end up earning lower prices. The Bertrand paradox, like the sword of Damocles, stands ready to serve punishment on both firms if they deviate in setting prices. Investment in such advertising campaigns represents a tough investment trying to hurt the competitor. As noted, strategic interactions in the second stage (in case entry is accommodated) may alter dramatically firm H’s optimal investment decision in the first stage. Here firms compete over prices (differentiated Bertrand) with their strategic actions being reciprocating (strategic complements). Firm L’s entry decision in the second stage may be endogenously influenced by firm H’s first-stage strategic commitment. In addition to such endogenous factors (strategic effects), exogenous factors (e.g., demand uncertainty) may also alter investment incentives. Firm L’s investment policy may be affected as well by actual market developments. Depending on the first-stage investment by firm H, its rival (firm L)’s investment triggers will be impacted accordingly. Several implications result: In case of advertising campaign of the first type (soft investment), investment by firm H eventually benefits the rival, making firm H soft and less aggressive in second-stage price competition (at the limit setting prices as if in a monopoly-like segment). The very fact that these firms are operating on distinct segments of the market gives them less incentive to set a low price in the second period. Firm H can be “soft” and less aggressive toward its rival in the second stage. The incumbent may actually increase prices to accommodate entry and soften second-stage competition. In this situation it might be advisable for the incumbent to (over-)invest in goodwill building/advertising to accommodate entry in the second period. Following such soft strategic commitment, firm H will •

Innovation Investment in Two-Stage Games

267

accommodate entry and raise its price. Its rival firm L (if it enters) will adjust its price according to its reaction curve (representing its best response to firm H’s actions). As a best reply to firm H’s price increase, firm L would increase its price in a reciprocating manner (strategic complements). This results in a Bertrand–Nash equilibrium where the second-stage market-clearing price is high. Firm H thus has an incentive to overinvest in soft advertising commitment, even though this is also beneficial to its rival (fat cat effect). In case of an advertising campaign of the second type (tough commitment), firm H will set a lower price (products are more substitutable); this entails a price decrease by its competitor, which will be eventually detrimental to both firms. Firm H therefore has an incentive not to invest or underinvest, playing the puppy dog ploy. The first-stage advertising campaign will eventually result in a lower equilibrium price in the second period as the firms’ reaction curves are upward sloping. This causes a negative strategic effect involving fiercer competition in the second stage. •

The results above apply when both firms invest in the second stage in a deterministic world. In an uncertain world, the insights concerning the effects of strategic commitment must be revisited in terms of optimal investment triggers as they also depend on market development. It is not always advisable to invest in the market when it is less attractive than previously expected. It may be that a tough first-stage advertising investment results in equilibrium in a higher investment trigger for firm L (hurting it) but decreases firm H’s investment option value. By contrast, a large investment in first-stage strategic commitment that makes firm H soft, even though it decreases the competitor’s investment trigger, may generate a higher option value. To illustrate the effect of market uncertainty, consider the following situation. Example 8.3 Investment Triggers and Value in Goodwill/Advertising The demand function parameter b is 2 3. Firm H has a slight competitive advantage over its competitor, reflected in firm-specific demand parameters aH = 12 and aL = 10. Unit marginal cost for both firms is c = 1. The risk-adjusted discount rate is k = 0.13 for both firms and there is no expected growth ( g = 0). The investment cost or option’s exercise price amounts to I = 100 for both firms. Figure 8.3 summarizes the option values obtained in the three cases of base case, soft, and tough commitment. Through soft commitment investment, firm H can increase its investment option value. To determine the net commitment effect, one

268

Chapter 8

Millions of euros 35 + 6.8

30.9

Total commitment effect

Soft commitment

30 25

(3.2)

24.1

Total commitment effect

Base case

21.0

20 15 10 5 0 Tough commitment

Incentive to invest in soft commitment

Figure 8.3 Investment values depending on the type of up-front strategic commitment

has to take into account the necessary outlay for the first-stage strategic commitment. Consider first the base case. Case A: Base case (no investment, s = 0.5) Firm H does not invest in the first-stage and the substitution effect parameter is s = 0.5. The maturity of the investment option T is 5 years. High-demand firm’s investment trigger, from equation (7.4), is X HM ≡ 2 Iδ b + c aH = 2 100 × 0.13 × 2 3 + 1 12 ≈ 0.57 and firm L’s, from equation (7.14), is X LB ≡ ⎡⎣((2 − s) (1 − s)) Iδ b (1 − s 2 ) + c ⎤⎦ aL = 3 13 × 1 2 + 1 10 ≈ 0.86 . To value the investment opportunity, a binomial tree made up of 20 steps, each of size h = 0.25, is used. Given an annual risk-free rate (r) of 5 percent, the discount factor used to value 1 received at time t + h as of time t is e − rh ≈ 0.987. Given a market volatility of σ = 30 percent, the up multiplicative parameter is u = eσ h ≈ 1.16, d = 1 u = 0.86, and the (risk neutral) up probability is 0.504.17 The demand process starts at

(

)

(

(

)

)

17. Here, we adjust equation (5.4) to consider that the risk-free rate over a period of length h is rh rather than the annual risk-free rate r, yielding p = (1 + rh − d ) (u − d ) ≈ 0.504 .

Innovation Investment in Two-Stage Games

269

X 0 = 0.40. Given these assumptions and trigger values, we can construct the binomial tree, determine the industry structure at the end nodes, and assess the resulting values for each firm. The value of the investment opportunity for high-demand firm H in the base case is 24.1 m.18 Consider next the case where firm H makes a first-stage commitment to alter second-stage market conditions for the better. There are two subcases: (1) The firm promotes unique branding, decreasing the substitution effect between the products (soft commitment), or (2) it harasses its competitor trying to lure customers away from its rival’s customer base, eventually leading to an increase in the substitution effect (tough commitment). Case B: Soft commitment (low, s = 0.3) Firm H invests in first-stage advertising emphazing brand uniqueness. Demand parameters aL and aH are unchanged, but the substitution effect is decreased from 0.5 to 0.3. The investment trigger of firm H as a monopolist is not affected (it remains X HM = 0.57). The low-demand firm’s trigger is slightly altered through the strategic effect of the first-stage investment, being reduced from X LB ≈ 0.86 to X LB ′ ≈ 0.78. Firm L’s investment trigger is driven by the profit the firm would make under differentiated Bertrand competition. Firm H’s upfront strategic investment affects second-stage profits in competitive equilibrium, altering market conditions and firm L’s investment trigger altogether. In the soft commitment case, firm H’s investment option value rises (from 24.1 m in the base case) to 30.9 m. Through its first-stage strategic commitment, firm H enhances its option value by nearly 6.8 m, even though its investment has benefited firm L as well (through larger Bertrand duopoly profit values, lower investment trigger and consequently higher option value). Firm H thus has an incentive to invest in this kind of advertising campaign. This confirms the fat cat business strategy to “overinvest” in soft commitment where actions are strategic complements or reciprocating. If, however, the required investment cost for the first-stage investment is more than 6.8 m, firm H should not make this investment as the cost exceeds the expected benefits. Case C: Tough commitment (high, s = 6.8) Now consider the second type of advertising investment. Firm H directly attacks its rival, convincing customers that the products are close substitutes. Such an advertising campaign represents a tough commitment, aiming at hurting the 18. The low-demand firm’s value is determined likewise.

270

Chapter 8

competitor in the second stage. In doing so, firm H increases the substitution effect (from 0.5 in the base case) to 0.8. Firm H’s investment trigger again remains unchanged ( X HM ≈ 0.57), whereas firm L’s rises (from X LB ≈ 0.86 in the base case) to X LB ′′ ≈ 1.16. The rival’s trigger is driven by strategic interactions in Cournot duopoly, affected by the first-stage strategic investment by firm H. In the tough commitment case the option value for firm H is reduced to 21 m. Regardless of the necessary upfront commitment investment cost, the tough advertising campaign destroys more than 3 m compared to the base case. Figure 8.4 illustrates the profit impact (not the option effect) from changing the substitution parameter s. A lower substitution effect is beneficial to both firms. In the extreme case (s = 0), firms earn monopoly profits. Yet increased substitution effect leads to lower (equilibrium) profits and hurts both firms. In the opposite extreme case (s → 1), firms sell perfect substitute products and make a zero profit (Bertrand paradox), a result analogous to perfect competition. Millions of euros 50

40

Firm H

30

Firm L

20

10

0 0.0 Monopoly rent

0.1

0.2

0.3

0.4

0.5

0.6

Substitution parameter (s)

0.7

0.8

0.9

1.0

Bertrand paradox (0 profits)

Figure 8.4 Equilibrium profits in differentiated Bertrand competition We assume a linear (inverse) demand function of the form pi (Q, Xɶ t ) = ai Xɶ t − b (qi + sq j ), with parameters: aH = 12, aL = 10, X 0 = 1, b = 2 3. Unit cost is c = 1.

Innovation Investment in Two-Stage Games

271

Conclusion Strategic interactions in two-stage games influence the incentive to invest and alter the value of future investment plans in a competitive context. Both the flexibility value and the strategic effects in the marketplace should be brought together to address both market and strategic uncertainty. The resulting expanded NPV should incorporate three concomitant effects: (1) the flexibility effect,19 (2) second-stage strategic interactions,20 and (3) the strategic commitment effect.21 An early strategic investment may have a high or low (even negative) net commitment value depending on the direct and strategic effects. The sign of the strategic effect itself may be positive or negative, depending on whether the benefits are proprietary or shared, and may be opposite for reciprocating competition (under strategic complements, e.g., price competition) than for contrarian competition (under strategic substitutes, e.g., quantity competition). Using option games, we have analyzed under conditions of market uncertainty various competitive strategies, depending on whether competitive actions are strategic complements or substitutes and whether the investment makes the firm tough (e.g., proprietary benefits) or soft (involving shared benefits). Several key results might be advisable to take into consideration: 19. This encompasses managerial flexibility to revise investment schedules over time, such as by delaying or staging investment. A strategic investment commitment signals competitors which kind of competition would emerge in the second stage. Although this involves sunk cost with a short-term negative effect on the P&L, it may nonetheless enhance option value by creating more favorable competitive conditions. From an options perspective, flexibility value increases with the level of interest rates, time to maturity, and market demand uncertainty (volatility). 20. In a multiplayer competitive environment strategic interactions can affect the shared option value by giving rise to different industry structures. For a monopolist option-holding firm, only two situations are possible: no investment or monopoly. In case of duopolist option holders with shared options, three industry structures are possible: no one invests, monopoly, or duopoly. 21. Early commitment may result in both a direct effect (additional cash flows) and a strategic effect. The strategic effect alters the competitor’s investment trigger favorably (under optimal business strategies) or detrimentally (if suboptimal strategies are pursued). The nature of equilibrium second-stage actions (reciprocal or contrarian) may critically influence the optimal choice of commitment strategy. In some cases an early commitment may create a competitive disadvantage if it limits the investor’s leeway and reduces its ability to be aggressive in later stages (under strategic substitutes) or if it induces the investor into being aggressive in the second stage and the competitor will follow suit (under strategic complements).

272

Chapter 8

When strategic actions are contrarian and the benefits of strategic commitment are proprietary, the firm should build up competitive advantage that makes it tough in later competition stages (e.g., invest in R&D under quantity competition). This strategy results in a higher investment trigger for the follower and increases the innovator’s likelihood of being a monopolist. •

When the innovation benefits are shared between duopolists and actions are contrarian, the firm should refrain from investing. When spillover effects are high, investing in R&D creates worse competitive conditions in the second stage for the investing firm. An alternative may be to license out the technology. •

When the investing firm benefits exclusively (or primarily) from its investment commitment and aggressive moves induce aggressive response by the rival (strategic complements), the firm should avoid investing to preserve its flexibility and avoid intensified competition.

•

When the strategic investment is beneficial to both firms (spillover effects) and strategic actions are reciprocating, there is an incentive to invest and commit. Even if the rival also benefits from the investment, the investing firm is eventually better off. •

The optimal competitive strategy thus depends not only on the nature of the investment commitment but also on the type of competitive reactions (strategic complements vs. substitutes). Option games enable simultaneously the determination of the equilibrium market structure under uncertainty (in a binomial-tree process) along with taking into account strategic interactions in multistage settings. Management may thus formulate appropriate dynamic competitive strategies that enable it to react effectively to changes in the market environment and competitive landscape. Selected References Smit and Trigeorgis (2001, 2004) analyze a number of option game applications in discrete time to illustrate the balanced effects of commitment versus flexibility. The authors present a number of case applications on R&D, infrastructure and goodwill-building investments. Wang (1998) discusses the licensing problem and the type of fee in a deterministic setting.

Innovation Investment in Two-Stage Games

273

Smit, Han T. J., and Lenos Trigeorgis. 2001. Flexibility and commitment in strategic investment. In Eduardo S. Schwartz and Lenos Trigeorgis, eds., Real Options and Investment under Uncertainty: Classical Readings and Recent Contributions. Cambridge: MIT Press, pp. 451–98. Smit, Han T. J., and Lenos Trigeorgis. 2004. Strategic Investment: Real Options and Games. Princeton: Princeton University Press. Wang, X. Henry. 1998. Fee versus royalty licensing in a Cournot duopoly model. Economic Letters 60 (1): 55–62.

III

OPTION GAMES: CONTINUOUS-TIME MODELS

We focused so far on discrete-time analysis of option games. The discrete-time approach is well suited for practical investment applications and is more accessible to corporate managers since it does not require knowledge of advanced mathematics. In part III we discuss continuoustime modeling of option games. Conditions for obtaining analytical solutions can be quite restrictive but continuous-time models that admit closed-form solutions can be useful to analyzing the main value drivers and depicting the trade-offs faced by firms. The continuous-time approach is often preferable for research purposes. We particularly focus here on models dealing with optimal investment timing. Even if an investment has a positive NPV if undertaken today, the firm may be better off to defer the investment and undertake the project in the future to limit potential undesirable effects of unexpected adverse market developments. The basic investment principle (i.e., the NPV rule) is now revised to: “undertake a project when its value exceeds the value of the deferral option.” Option game models can address the issue of investment timing in oligopolistic markets. They specifically deal with whether and when firms should exercise their shared investment options when strategic interactions among rivals are explicitly considered. A key benchmark is the seminal paper by McDonald and Siegel (1986) who study the optimal investment timing and option value of a monopolist when the underlying project value follows a stochastic process (geometric Brownian motion). The investment outlays are treated here as a sunk cost, highlighting the long-term impact of strategic investment. The investment timing issue is thus of critical importance.

9

Monopoly: Investment and Expansion Options

We argued previously that the firm’s ability to delay investment invalidates the common “NPV rule” that asserts firms should “invest when the value of a project exceeds the cost of investment.”1 This static investment rule effectively turns a blind eye to the opportunity cost of investing now when firms can wait for more accurate information. The real options paradigm explicitly takes into consideration this opportunity cost, revising the investment rule as follows: “invest when the project value exceeds the opportunity cost of waiting.” This raises the following implementation issues: When exactly should a firm invest? When does the value of a project exceed its opportunity cost? What is the current value of an investment opportunity, accounting for the embedded flexibility to defer investment? Before embarking on a study of these issues in an oligopolistic setting, it is useful to first dwell on the benchmark model of a monopolist, which does not involve strategic interactions. In doing so, we also lay down the foundations of continuous-time real options analysis, demonstrating the basic methodology to solve investment-timing games. The same methodology is amenable (with some adjustment) to situations involving strategic interactions, and thus helps highlight common characteristics of optimal investment strategies. We here adopt an approach based on determining investment trigger strategies that is not yet widespread in the real options literature. This approach provides the same results as more standard techniques but with greater ease. It is also more intuitive. Alternative approaches build on complex techniques originating from stochastic calculus and control theory. Our preferred approach levers on basic 1. A refinement of the NPV rule prescribes to select the project with the highest NPV among mutually exclusive projects. Viewing a project with differing starting dates as mutually exclusive alternatives, one can attempt to capture timing flexibility and provide insights about optimal investment timing. The problem of determining the appropriate discount rate, however, remains unresolved in such a setting. Real options analysis addresses this issue properly.

278

Chapter 9

principles of microeconomics instead.2 To smooth out the exposition, we first discuss the corresponding deterministic problem. In sections 9.1 and 9.2, respectively, we analyze two kinds of options faced by a monopolist firm: 1. The option when to invest (or wait) This option arises when a firm not currently producing considers the opportunity to enter or develop a new market (called the “new market” model), such as when to develop an oil reserve. 2. The option to expand This option emerges when a firm already active in the market has an option to increase or expand its production capacity should the market become more attractive than initially expected (called the “existing market” model), such as lump-sum capacity addition by an electric utility. Box 9.1 introduces two practical examples involving such options related to investment and expansion, namely the development of oil reserves and the trade-off between scale and flexibility in capacity addition. 9.1

Option to Invest (Defer) by a Monopolist

Consider a government-protected natural monopoly or a firm that has acquired a patent ensuring it an exclusive access to a new market. Suppose that there are insurmountable structural entry barriers or the patent has a very long (effectively infinite) life. This protection enables the firm to invest whenever it wants, with no fear of rival entry. The investment problem in this case boils down to deriving the optimal investment-timing rule for a single agent (the monopolist) considering that investment can occur anytime, today or in the future. Since the monopolist faces no potential rivals, it can form its optimal investment strategy in isolation, disregarding strategic interactions. In effect, the monopolist has a perpetual American call option to wait for new information to come.3 Below we discuss two deferral option cases: the case of investment with deterministic growth (certainty) and the case of investment with stochastic growth (uncertainty). The latter serves as the foundation of continuous-time real options analysis. 2. Our alternative approach has also been used by Dixit, Pindyck, and Sødal (1999) and Sødal (2006, 2012). The mathematical prerequisites we develop in the appendix to the book are a cornerstone for option games models applied in later chapters. 3. This problem was investigated by McDonald and Siegel (1986) in their seminal work on “the value of waiting to invest” and served as a building block for Dixit and Pindyck’s (1994) analysis of real options.

Monopoly: Investment and Expansion Options

279

Box 9.1 Investment and expansion options in business practice

The Options Approach to Capital Investment A. K. Dixit and R. S. Pindyck, Harvard Business Review As companies in a broad range of industries are learning, opportunities to apply option theory to investments are numerous. Below are a few examples to illustrate the kinds of insight that the options theory of investment can provide. Investment in Oil Reserves Nowhere is the idea of investments as options better illustrated than in the context of decisions to acquire and exploit deposits of natural resources. A company that buys deposits is buying an asset that it can develop immediately or later, depending on market conditions. The asset, then, is an option—an opportunity to choose the future development timetable of the deposit. A company can speed up production when the price is high, and it can slow it down or suspend it altogether when the price is low. Ignoring the option and valuing the entire reserve at today’s price (or at future prices following a preset rate of output) can lead to a significant underestimation of the value of the asset. The US government regularly auctions off leases for offshore tracts of land, and oil companies perform valuations as part of their bidding process. The sums involved are huge—an individual oil company can easily bid hundreds of millions of dollars. It should be not surprising, then, that unless a company understands how to value an underdeveloped oil reserve as an option, it may overpay or it may lose some very valuable tracts to rival bidders. Consider what would happen if an oil company manager tried to value an undeveloped oil reserve using the standard NPV approach. Depending on the current price of oil, the expected rate of change of the price, and the cost of developing the reserve, he might construct a scenario for the timing of development and hence the timing (and size) of the future cash flows from production. He would then value the reserve by discounting these numbers and adding them together. Because oil price uncertainty is not completely diversifiable, the greater the perceived volatility of oil prices, the higher the discount rate that he would use; the higher the discount rate, the lower the estimated value of the undeveloped reserve. But that would grossly underestimate the value of the reserve. It completely ignores the flexibility that the company has regarding when to develop the reserve—that is, when to exercise the reserve’s option value. And note that just as options are more valuable when there is more uncertainty about future contingencies, the oil reserve is more valuable when the price of oil is more volatile. The result would be just the opposite of what a standard NPV calculation would tell us: In contrast to the standard

280

Chapter 9

Box 9.1 (continued)

calculation, which says that greater uncertainty over oil prices should lead to less investment in undeveloped oil reserves, option theory tells us it should lead to more. By treating an undeveloped oil reserve as an option, we can value it correctly, and we can also determine when is the best time to invest in the development of the reserve. Developing the reserve is like exercising a call option, and the exercise price is the cost of development. The greater the uncertainty over oil prices, the longer an oil company should hold undeveloped reserves and keep alive its option to develop them. Scale versus Flexibility in Utility Planning The option view of investment can also help companies value flexibility in their capacity expansion plans. Should a company commit itself to a large amount of production capacity, or should it retain flexibility by investing slowly and keeping its options for growth open? Although many businesses confront the problem, it is particularly important for electric utilities, whose expansion plans must balance the advantages of building large-scale plants with the advantages of investing slowly and maintaining flexibility. Economies of scale can be an important source of cost savings for companies. By building one large plant instead of two or three smaller ones, companies might be able to reduce their average unit cost while increasing profitability. Perhaps companies should respond to growth opportunities by bunching their investments—that is, investing in new capacity only infrequently but adding large and efficient plants each time. But what should managers do when demand growth is uncertain, as it often is? If the company makes an irreversible investment in a large addition to capacity and then demand grows slowly or even shrinks, it will find itself burdened with capital it doesn’t need. When the growth of demand is uncertain, there is a trade-off between scale economies and the flexibility that is gained by investing more frequently in small additions to capacity as they are needed. Electric utilities typically find that it is much cheaper per unit of capacity to build large coal-fired power plants than it is to add capacity in small amounts. But at the same time, utilities face considerable uncertainty about how fast demand will grow and what the fuel to generate the electricity will cost. Adding capacity in small amounts gives the utility flexibility, but it is also more costly. As a result knowing how to value the flexibility becomes very important. The options approach is well suited to the purpose. For example, suppose a utility is choosing between a large coal-fired plant that will provide enough capacity for demand growth over the next 10 to 15 years or adding small oil-fired generators, each of which provides

Monopoly: Investment and Expansion Options

281

Box 9.1 (continued)

for about a year’s worth of demand growth as needed. The utility faces uncertainty over demand growth and over the relative prices of coal and oil in the future. Even if a straightforward NPV calculation favors the large coal-fired plant, that does not mean that it is the more economical alternative. The reason is that if it were to invest in the coal-fired plant, the utility would commit itself to a large amount of capacity and to a particular fuel. In so doing, it would give up its options to grow more slowly (should demand grow more slowly than expected) or to grow with at least some of the added capacity fueled by oil (should oil prices, at some future date, fall relative to coal prices). By valuing the options using option-pricing techniques, the utility can assess the importance of the flexibility that small oil-fired generators would provide. Utilities are finding that the value of flexibility can be large and that standard NPV methods that ignore flexibility can be extremely misleading. A number of utilities have begun to use option-pricing techniques for long-term capacity planning. The New England Electric System (NEES), for example, has been especially innovative in applying the approach to investment planning. Among other things, the company has used optionpricing techniques to show that an investment in the repowering of a hydroelectric plant should be delayed, even though the conventional NPV calculation for the project is positive. It has also used the approach to value contract provisions for the purchase of electric capacity and to determine when to retire a generating unit. Source: Reprinted with permission of Harvard Business Review from “The Options Approach to Capital Investment,” by A. K. Dixit and R. S. Pindyck, May–June 1995. Copyright © 1995 by the Harvard Business Review School Publishing Corporation; all rights reserved.

9.1.1

Deterministic Case

We consider first the deterministic case to help give intuition and guidance into how to solve investment-timing problems, paving the way for development of the case involving uncertainty. The opportunity to defer investment even in the deterministic case dramatically alters traditional investment principles (e.g., the NPV rule). Overlooking the basics of investment timing may lead to suboptimal choices. Suppose that the monopolist has a perpetual option to invest in a new market by incurring a necessary investment outlay I . Suppose that the underlying market (monopoly profit) grows compoundly at a rate g percent per year and that the current project value V0 is lower than the required investment

282

Chapter 9

cost I , resulting in a negative NPV. In this case the monopolist has an incentive to defer the investment and wait until the project becomes sufficiently profitable before investing. In the deterministic case with no volatility, project value is sure to increase at a rate of g percent each year (i.e., the actual growth exactly matches the expected one). Suppose that the discount rate is r ( > g ). The discount factor—that translates future time-t values in today’s (time t0 = 0) terms—is B0( t ) = e− rt .

(9.1)

The underlying profit starts at π 0 and by time t ( > t0 ) it grows with certainty to π t = π 0 e gt. The time-0 gross value of the project, obtained as a discounted perpetual stream of profits, is ∞

V0 = ∫ (π 0 e gt ) e − rt dt = 0

π0 , δ

(9.2)

where δ ≡ r − g ( > 0 ) is analogous to a dividend yield or opportunity cost of waiting. If the project is initiated at a future (nonrandom) time T and profits are received from that moment on, the value of the project at time T is VT =

πT = V0 e gT . δ

(9.3)

Note that both the profit and the project value grow at the rate g . From equation (9.3), we can express T as a function of the initial value V0 and the time-T value VT. From (9.1), an alternative expression for the discount factor is4 b

V  B0 (T ) = B (V0 ; VT ) =  0  ,  VT 

(9.4)

where b ≡ r / g . The discount factor, as expressed in (9.4), can be thought of as a discount factor over states (V0 and VT). In the deterministic case the firm’s strategy consists in selecting a prespecified investment time T at which to invest. Alternatively, the firm may select directly a critical threshold value VT and invest when this value is first reached (at time T ). These two strategy formulations are equivalent: the investor looks for the optimal timing decision or for the optimal target level at which to invest. For a given timing strategy (i.e., to invest at time T ), the time-0 value of the investment option equals 4. By inverting (9.3), we obtain gT = ln (VT V0 ) so that B0 (T ) = exp [ − ln (VT V0 ) b]. The result follows.

Monopoly: Investment and Expansion Options

283

the pre-specified forward value (net of the investment cost) VT − I , discounted back to time 0 using the discount factor in equation (9.1) or (9.4). Denote by M0 (T ) or M0 (VT ) the time-0 value of the monopolist’s investment option. It is then given by M0 (T ) ≡ M0 (VT ) = B0 (T ) × (VT − I ).

(9.5)

Pre-specifying a target strategy enables separating the value function into two separate multiplicative components: the forward NPVT ≡ VT − I and the discount factor B0 (T ). This version of the optimal-timing problem is known as Wicksell model in the theory of natural (e.g., forest) resources, where VT represents the forward (time-T ) value of the forest, g the growth rate of the trees, I the cost of cutting them and r the opportunity cost that captures the patience of the landowner. From expression (9.4), the discount factor B (V0 ; VT ) is a decreasing function of the target threshold VT. As VT − I is clearly increasing in the threshold, the investor faces a trade-off between obtaining a higher forward net present value, NPVT ≡ VT − I , and a lower present value due to discounting. The value function in equation (9.5) is concave in T (or equivalently in VT). The optimal timing strategy is found based on standard optimization techniques. The first-order condition (MV (VT ) = 0) leads to the following sufficient and necessary condition for the optimal (tree-size) threshold V*: V* = Π*, I

(9.6)

where Π* ≡

b r = ( ≥ 1) b−1 δ

(9.7)

is the profitability level that indicates whether the project should be undertaken (e.g., whether the trees should be cut). Note that b ≡ r / g can be seen as the (constant) elasticity of the discount factor with respect to the trigger value VT.5 Alternatively, the investment rule in equation (9.6) can be rewritten to provide guidance into the return on investment that should be attained at the time of optimal investment. This form of the investment rule reads 5. From equation (9.4), ∂B B(V0 , VT ) . (V0 , VT ) = −b ∂VT VT The elasticity expression obtains readily.

284

Chapter 9

π* = r, I

(9.8)

where π * satisfies V* = π * δ . The optimal investment rule suggests to invest whenever the profitability index (i.e., the ratio of the project value over the investment cost) V I exceeds the specified profitability target Π* given in equation (9.6), or equivalently to invest whenever the return on investment π* / I exceeds the investor’s opportunity cost of capital r as per equation (9.8). The first investment rule is analogous to Tobin’s q theory of investment. The second relates to the Jorgensonian rule of investment. From b ≡ r g , it obtains that the target profitability measure Π* strictly exceeds 1 (provided that r > g > 0 or δ > 0). It is 1 only when the project value remains unchanged over time (as the limit of Π* for g → 0), prescribing then to invest when the project value equals or exceeds the investment cost. If there is growth, however, the investment rule based on equation (9.7) brings out a contrast with the commonly taught NPV rule advising to invest when project value VT [ = π T δ ] exceeds the investment cost I . The static NPV rule, suggesting to invest now if NPV ≥ 0, is strictly correct in the case wherein the asset is not subject to growth. Even in the absence of uncertainty, when an investor is faced with an opportunity to delay investment in an asset that grows ( g > 0), she should require project value to strictly exceed the investment cost (Π* > 1) to account for the opportunity cost to kill her growth option. As the standard NPV paradigm does not explicitly take into consideration optimal timing or investment flexibility value, it potentially leads to suboptimal investment timing and early investment. When an investor has no timing flexibility (i.e., is required to invest immediately), the NPV rule will then hold. 9.1.2

Stochastic Case

The basic approach used previously can be extended and adapted to take account of market uncertainty. The stochastic process Xɶ t here describes a shock affecting firms’ profits. Our approach for the stochastic investment case is based on our proposed new methodology for valuing the option to invest, which consists of the following steps: 1. Specify the value of the investment as a function of the target strategy chosen. 2. Determine the optimal investment strategy (trigger) given the investment payoff function (derived in the previous step).

Monopoly: Investment and Expansion Options

285

3. Reassess the investment payoff given the derived optimal investment strategy. Before we do this, we need to define more precisely what is an investment strategy. Investment Strategy An investment strategy is a contingent plan of action that stipulates what investment action to take for each contingency (i.e., in each possible future state of the process). In case of a perpetual American investment option, the investment strategy consists in choosing for each possible value of the stochastic process Xɶ t the action to “invest” or to “wait” (the action set is discrete). In most cases the strategy can be simplified and implemented using the following principle: choose ex ante a specified future target (trigger) value XT to be reached by the stochastic process Xɶ t and invest in the project when this value is first reached.6 The strategy space is continuous: the option holder may adopt a continuum of different strategies (any value for XT ). However, only one investment strategy is optimal and should be followed by a rational option holder.7 An important metric is the first time, Tɶ, the threshold XT is reached

{

}

Tɶ ≡ inf t ≥ 0 Xɶ t ≥ XT . The first-hitting time Tɶ is determined by the chosen investment trigger and depends on the process value dynamics with (Tɶ being a random variable).8 In terms of investment strategy formulation (in the uncertain 6. Equivalently one could think of partitioning the state space. The strategy consists in waiting for Xɶ t located in the region ( −∞, XT ) and investing when Xɶ t is found for the first time in [ XT , ∞ ) . Since investment is irreversible, there is path-dependency for the industry structure. The current value of the process does not perfectly reflect whether the firm is operating. Xɶ t could be located in ( −∞, XT ), but the firm may be active if XT was previously exceeded. These alternative definitions of the investment strategy could be readily extended in case of multiple option holders. For multiple option holders the optimal investment strategy is part of an industry equilibrium. When a decision maker has to choose among a number of alternatives, the optimal investment strategy does not always take the form of a trigger strategy since the optimal investment region may be dichotomous. Días (2004) and Décamps, Mariotti, and Villeneuve (2006) discuss such situations. 7. The existence and uniqueness of the investment trigger is established in Dixit and Pindyck (1994, pp. 128–38). 8. In paradigms where the investment decisions are made based on expected profit flows, investment timing (e.g., capacity expansion) is deterministic. Given the growth of the market, one can easily determine when the (expected) market size will be sufficiently large for the firm to invest profitably. By contrast, real options theory considers stochastic investment timing, whereby the volatility of the underlying investment is explicitly taken into account. The real options approach makes sense because managers are rarely committed to an investment time schedule and can frequently revise it, for example, by deferring their decision if the market is less attractive than initially expected. Since actual profits evolve stochastically, the investment time cannot be selected ex ante.

286

Chapter 9

case), it is convenient to use XT as the reference point for strategy definition rather than Tɶ .9 In a stochastic framework, the time to invest cannot be determined ex ante. The use of the random variable Tɶ to define the investment strategy is therefore nontrivial.10 The monopolist will invest when the underlying stochastic (Itô) process Xɶ t first reaches XT , receiving the forward project value VT (as a function of XT ) and incurring the investment cost outlay I at time Tɶ . The option payoff at time Tɶ is the forward NPV (i.e., NPVT ≡ VT − I ). Let M0 ( XT ) be the present (time-0) value of the option to invest by the monopolist (new market model) who chooses to invest at random time Tɶ . For XT ≥ X 0, the expected present value of the investment opportunity (considering the deferral option) equals the forward NPV (NPVT) discounted back at the present time (t = 0), namely ɶ M0 ( XT ) = Eˆ  NPVT × e− rT  ,

where Eˆ [⋅] denotes the expected value under the risk-neutral probability measure.11 If the investment strategy chosen by the monopolist is to invest now (at time t0 and state X 0), the value of the investment opportunity is simply the (static) NPV of the project, NPV0 = V0 − I. Investing now, however, would kill the opportunity to delay the investment. Immediate investment therefore entails an opportunity cost. Thus investing now may not be the optimal decision. Since the forward value VT is determined ex ante and does not depend on the actual path of the process (being a deterministic function of the chosen trigger level XT ) and the investment outlay I is also deterministic, the forward net present value, NPVT, is independent of the path of the stochastic process Xɶ t and is solely conditional on the target level XT . Thus, in the monopolist investor’s value expression above, the expectation relates only to the discount factor term whose value is driven by the random timing parameter Tɶ . Therefore, for XT ≥ X 0, 9. In the certain case discussed above and in Reinganum’s (1981a, b) and Fudenberg and Tirole’s (1985) game-theoretic investment timing models, the investment time T defines the investment strategy. These models are, however, deterministic. The models by Reinganum (1981a) and Fudenberg and Tirole (1985) are explained in detail later in chapters 11 and 12 and are amended to allow for stochastic profit flows. 10. More advanced mathematical treatments of this problem generally optimize over the space of first-hitting (stopping) times (Snell envelope problem). We prefer a more intuitive definition of the strategy in terms of triggers as in the present case involving a simple partition of the state space. 11. In part III we generally follow the risk-neutral valuation perspective presented in chapter 5. This valuation approach applies for complete markets with no arbitrage opportunities. In the appendix at the end of the book, we also present an alternative exposition that would apply generally, even if the assumptions of risk-neutral valuation do not hold. This latter formulation consists in using an exogenous risk-adjusted discount rate k rather than the risk-free rate r.

Monopoly: Investment and Expansion Options

287

ɶ M ( XT ) = NPVT × Eˆ e − rT  = NPVT × B0 (Tɶ ),

where the expected discount factor is ɶ B0 (Tɶ ) ≡ Eˆ e− rT .

Alternatively, analogous to the certainty case, B0(Tɶ ) can be understood as an expected discount factor over states X 0 and XT . It can be formulated as B0 (Tɶ ) = B0 ( XT ) = B( X 0 ; XT ). If the target value XT selected by the firm is lower than (or equal to) the current level X 0 (or X 0 is in the investment region [ XT , ∞ )), the firm would invest immediately and receive the static net present value of the project, NPV0 = V0 − I. The time-0 value of the monopolist’s option to invest (defer) when the target value XT is first reached is thus given by  NPVT × B0 (Tɶ ) if M 0 ( XT ) =  if  NPV0

X 0 < XT , X 0 ≥ XT .

(9.9)

Consider now the case where Xɶ t follows a geometric Brownian motion (GBM) of the form dXɶ t = ( gXɶ t ) dt + σ Xɶ t dzt,

(9.10)

where g and σ are the constant drift and volatility parameters and zt is a standard Brownian motion.12 Under risk-neutral valuation, g gets replaced by gˆ = r − δ . As shown in equation (A.43) in the appendix at the end of the book, the expected discount factor in case of geometric Brownian motion is β

1 X  B0 (Tɶ ) =  0  ,  XT 

(9.11)

where (in the risk-neutral case) from equation (A2.4)

β1 = −

2 αˆ r  αˆ  +  2  + 2 2 σ2 σ σ

( > 1)

(9.12)

with αˆ = gˆ − (σ 2 2).13 The time-0 value for the monopolist investor that invests when XT is first reached is thus obtained from equations (9.9) and (9.11) as 12. More information on the geometric Brownian motion is provided in the appendix, section A.1. 13. In the appendix at the end of the book, we use two distinct fundamental quadratic functions, one with an exogenous given discount rate and the other under risk-neutral valuation, using β1 and βˆ 1 respectively. In part III we generally employ the risk-neutral valuation approach. For notational simplicity we denote here (in the main text) the positive root of the fundamental quadratic in the risk-neutral case by β1 , instead of βˆ 1 as in the appendix.

288

Chapter 9

β1 ⎧ ⎛ X0 ⎞ NPV ⎪ T ⎜ ⎝ XT ⎟⎠ M 0 ( XT ) = ⎨ ⎪ NPV ⎩ 0

(wait) (invest)

if if

X0 < XT ,

(9.13)

X0 ≥ XT .

For an investment target or trigger higher than the starting value ( X 0 < XT), the value of the strategy equals the forward NPV at future date Tɶ discounted back to the present time (time 0) using the expected discount factor of equation (9.11). The result given in equation (9.13) confirms the option value for a monopolist obtained by McDonald and Siegel (1986).14 Optimal Investment Strategy The investment value presented in equation (9.13) of the previous section has no prescriptive implications from a strategic-management viewpoint. It does not necessarily provide guidelines whether or when the monopolist firm should invest (optimal behavior). It simply helps management assess the value of an arbitrarily chosen investment strategy, but it does not prescribe which strategy is the optimal to pursue. In this (monopoly) case the value-maximizing investment strategy can be deduced from standard optimization techniques. In this section we determine the optimal investment strategy and derive the optimal investment trigger that allows obtaining the “expanded NPV” value for the monopolist investor, namely the overall value obtained given the optimal investment timing decision. We start with the general case of an Itô process. The investor may potentially choose XT among a continuous set (with infinite possibilities). What matters for the investor is to determine ex ante the optimal contingent investment rule. Let us denote by X* ɶ the random time when the optimal investment target (trigger) and by T* X* is first reached. The optimal trigger X* is such that it maximizes the given investment option value expression; that is, it is such that M0 ( X*) ≥ M0 ( XT ) for all XT, where M0 ( XT ) is in general given by equation (9.9) or by (9.13) for GBM. The expression for the expanded net present value (total investment value taking account of the value of managerial flexibility) is given by M0 ( X*) = max M0 ( XT ). XT

(9.14)

14. In McDonald and Siegel’s (1986) model, the stochastic process Xɶ t corresponds to the project value, so that NPVT = XT − I . We have not yet determined the optimal strategy to be followed by the monopolist, namely when exactly the monopolist should invest. This will be discussed next. The “expanded NPV” is the value of the investment option when the monopolist invests at the optimal time. Compared to formula (9.13), McDonald and Siegel (1986) embed the optimal timing behavior in their value function. They also consider a case where both the underlying project value and the investment cost follow correlated geometric Brownian motions.

Monopoly: Investment and Expansion Options

289

In the case of geometric Brownian motion, an analytical solution to the expression above exists if the dividend yield δ is (strictly) positive. It is of the form β1 ⎧ ⎛ X0 ⎞ ⎪ NPVT* ⎜ ⎟ M0 ( X *) = ⎨ ⎝ X *⎠ ⎪ NPV ⎩ 0

(wait) (invest)

if if

X 0 < X *,

(9.13′)

X 0 ≥ X *.

If this condition does not hold (δ ≤ 0), the optimal solution for the perpetual American investment option is to delay investment indefinitely (i.e., never exercise the option). Over time there is no opportunity cost or value loss due to missed “dividend payments” while the present value of the investment cost I declines by delaying. The parameter δ may represent some form of a payout rate or opportunity cost (below-equilibrium return).15 As in the certainty case the monopolist’s optimization problem here also presents the option-holding investor with a trade-off. First, to avoid adverse developments, the monopolist may delay deciding to invest when a higher forward net present value (NPVT ≡ VT − I ) is achieved, namely by selecting a higher value for the target (trigger) XT .16 Then again, a higher target leads to investment postponement further out in the future so that the present value of the investment reward is eroded due to a lower discount factor. This trade-off confirms that the value function is concave such that there exists an optimal value for the investment trigger. This situation is illustrated in figure 9.1. The trade-off above is analogous to the situation we discussed in chapter 3 concerning the price-setting dilemma faced by a monopolist when demand or the market-clearing price is downward sloping, namely when the price decreases with industry output (∂p ∂q < 0). We saw then that if the monopolist sets a higher price p, it can earn a higher profit margin ( p − c) on each unit sold but the reachable market (volume of sales or number of units sold) will be lower (∂q ∂p < 0). A higher price set for the product, although intended to achieve a higher profit margin, may actually backfire because it may deter potential customers from purchasing it. Although there are exceptions (e.g., when demand increases with price as in the case of luxury products), in most situations 15. Merton (1973) shows that the dividend yield δ must be (strictly) positive for a finitely lived American option to be exercised before maturity. Here we deal with perpetual American options, but the result is equivalent for a pre-specified constant exercise price. 16. Dixit, Pindyck, and Sødal (1999) explain this trade-off in terms of an option premium. The premium mirrors the difference between the value of the project at the time of investment and the exercise price of the option ( I ). The option premium is analogous to the forward net present value defined earlier.

290

Chapter 9

Discount factor ~

B0(T )

Forward NPV

6

1.2 1.0 Forward NPV NPVT ≡ VT − I

0.8

4

Discount factor ~

0.6

B0(T )

2

0.4 0.2 0.0 (0.2)

1

2

3

4

5

6

7

0

Investment trigger XT

(0.4)

–2

Figure 9.1 Trade-off between higher profitability and lower discount factor for optimal investment strategy Xɶ t follows a geometric Brownian motion driving the project value dynamics. The discount rate is r = 7 percent. The drift or growth parameter is ˆg = 6 percent. Volatility is σ = 20 percent. The starting value for the process is X 0 = 1. The investment outlay is I = 2.

monopolists face this classic trade-off in setting prices. To capture a larger market, a monopolist may lure more customers into buying by setting lower prices; the resulting revenues may be higher as the quantity sold in the marketplace is higher, even though the profit margin per unit is lower. This trade-off is governed by the rate at which demand declines as price is raised, namely by the price elasticity of demand. A similar approach is used here to deduce the optimal investment trigger chosen by the monopolist option holder. The optimal threshold X* is the target level XT that maximizes the monopolist’s value M ( XT ). The first-order condition gives BX ( X*) × V* + B0( X*) × VX ( X*) = BX ( X*) × I, where BX = ∂B ∂XT , VX = ∂V ∂XT , and V* ≡ V ( X *) is the optimal project value. The left-hand term is the discounted extra or marginal total reduction in gross project value V from raising the investment trigger XT by a small (infinitesimal) amount dXT . The right-hand term is the discounted marginal investment cost savings from delaying investment. At the optimum level, X*, they become equal. The expression above thereby leads to the following markup condition in equilibrium:

Monopoly: Investment and Expansion Options

Λ≡

V* − I ε ( X*) =− V , V* ε B ( X*)

291

(9.15)

where ε B(⋅) and ε V (⋅) denote, respectively, the elasticity of the discount factor and the elasticity of the forward net present value, given by

ε B ( XT ) ≡ − BX ( XT ) ×

XT , B0 ( XT )

XT . ε V ( XT ) ≡ −VX ( XT ) × VT ( XT )

(9.16)

The optimal investment trigger X* and the corresponding optimal discount factor B0 (Tɶ*) are found at the point where the markup is given as in equation (9.15). At the time of optimal investment, the project return Λ ≡ (V* − I ) / V* must be equal or exceed the specified level given in the right-hand side of equation (9.15). This term is affected by the rate at which the discount factor decreases for a higher target value (via ε B ( XT )) and the rate at which the forward value VT is increased by the choice of a higher target XT (via εV ( XT )). In case the underlying process is the gross project value (i.e., the stochastic factor is Vɶt) and the investment trigger is a given target project value VT, the elasticity of the forward NPV is constant and equal to εV (VT ) = −1. In this case the optimal markup rule of equation (9.15) reduces to Λ≡

V* − I 1 = . V* ε B (V*)

(9.17)

This confirms a main result given in Dixit, Pindyck, and Sødal (1999). In their model the optimal investment trigger is solely governed by the rate at which the discount factor declines as the target project value VT is increased. In our more general setting the elasticity of the forward (terminal) value with respect to the selected investment trigger also intercedes. It suggests that the optimal investment rule is governed as well by the rate at which the forward value increases when the target investment trigger XT is raised. In the setting of Dixit, Pindyck, and Sødal (1999), it is easy to see the analogy with the optimal price-setting problem of a monopolist. As noted, that pricing problem is solved based on the Lerner index L≡

p* − c 1 = , εp p*

292

Chapter 9

where p* is the equilibrium market-clearing price, c the constant marginal cost of production, and ε p the price elasticity of demand (ε p ≡ − ∂p ∂q × q p). In our general setting, due to the Markov property of the underlying stochastic (diffusion) process, the elasticity of the discount factor is independent of the starting value X 0.17 The elasticity of the discount factor is strictly positive (as BX (⋅) < 0). However, it may not always be constant. The elasticity of the forward value is negative and is also independent of the starting value.18 From the sign of these elasticities, the markup Λ in equation (9.15) is strictly positive. Consequently the value received from pursuing the optimal strategy V* strictly exceeds the cost required to undertake the project, requiring that the project’s NPV at the time of optimal investment be strictly positive. This confirms once again that the static NPV rule, advising to invest when the project’s gross present value equals the investment cost, is not strictly correct. A certain positive premium must be attained before the investment is undertaken (in optimum). This premium results from considering both the flexibility in delaying the investment and the underlying uncertainty. Rewriting equation (9.15) and setting19 Π ( XT ) ≡

ε B ( XT ) , ε B ( XT ) + ε V ( XT )

(9.18)

the premium (cushion) can be determined based on V* = Π*, I

(9.19)

where Π* ≡ Π ( X*) is obtained from equation (9.18) with XT = X*. From equations (9.19) and (9.9), the ‘expanded NPV’ given the optimal investment strategy is M0 ( X*) = [ Π* −1] IB0 (Tɶ*).

(9.20)

The metric Π* is the equilibrium gross profitability index and Π* − 1 is the excess profitability index. The gross profitability index is sometimes interpreted as Tobin’s q or as the market-to-book ratio attained at the 17. See also Dixit, Pindyck, and Sødal (1999). 18. Since XT and VT are either both negative or positive and VX (⋅) > 0, the elasticity of the forward value is strictly negative, that is, ε V ∈[ −1, 0 ). 19. Since I > 0 , Λ ≠ 1 and ε V ( X*) ≠ ε B( X*). The finiteness of Π ( XT ) at the optimal threshold X* is thus ensured.

Monopoly: Investment and Expansion Options

293

time of optimal investment.20 Dixit and Pindyck (1994) propose to interpret Π* = V* I as Tobin’s q, the ratio between the real value of an asset to the cost needed to produce it. Since the elasticities may not be constant, the profitability index Π ( XT ) may vary with the target investment trigger XT . Although the profitability index may change, what matters is its value Π* at the optimal investment trigger X*. The geometric Brownian motion has the convenient feature of admitting constant elasticities and profitability index. The elasticity of the monopolist’s deferral option equals the elasticity of the discount factor at equilibrium for timehomogeneous Itô processes.21 Example 9.1 Geometric Brownian Motion Suppose that project value Vɶt follows a geometric Brownian motion similar to (9.10) with constant drift g (under risk neutrality gˆ = r − δ ) and volatility σ . Since the project value is itself the underlying stochastic factor, εV = −1. The expected discount factor is given analogous to equation (9.11) as β1

V  B0 (VT ) =  0  ,  VT  where β1 is given in equation (9.12). The first-order derivative of the discount factor with respect to the investment trigger XT is BV (VT ) = −β1

B0(VT ) . VT

Hence the elasticity of the discount factor is constant and equal to

ε B = β1 ( > 1).

(9.21)

The parameter β1 refers, interchangeably, to the elasticity of the discount factor, B0 (Tɶ ), or the elasticity of the investment option. The option markup formula in equation (9.15) becomes Λ = 1 β1. The profitability index Π* of equation (9.18) is constant: 20. Dixit and Pindyck (1994) investigate this profitability index in the case of geometric Brownian motion. Here the underlying project is generally assumed to follow an Itô process. In traditional accounting, the balance sheet is used to record assets as the sum of the costs incurred to procure them. The market value, on the contrary, reflects the real value of the asset, namely the current value of the benefits the asset provides. The market-to-book ratio determines the excess return of the project over the necessary investment outlay. 21. From optimal-stopping theory the investment trigger is determined by two “boundary conditions”: the value-matching condition, M( X*) = V* − I ; and the smooth-pasting condition, M X ( X*) = VX ( X*). Both must hold at the point of optimal investment. The elasticity of the option is ε M ( XT ) ≡ − M X ( XT ) × XT M( XT ). From the markup formula in (9.15), equation (9.16), and the two conditions above, one obtains ε B( X*) = ε M ( X*). Section A.4 in the appendix summarizes briefly the relevant optimal-stopping theory.

294

Π* ≡

Chapter 9

β1 . β1 − 1

(9.22)

The expression above is similar to equation (9.7) in the certainty case discussed earlier. The difference lies in the fact that under deterministic growth, b is a function of r and g solely, whereas here β1 is the solution of a quadratic equation involving r, gˆ , and the additional volatility term σ capturing uncertainty about market developments. This version of the profitability index given in (9.22) has been used extensively by, for example, McDonald and Siegel (1986) and Dixit and Pindyck (1994).22 Their demonstration is based on the standard contingent-claims analysis approach (see appendix to this chapter).23 The optimal investment rule extended in case of uncertainty for the geometric Brownian motion is V* = Π* I

⇔

V* β1 = Π* ≡ . I β1 − 1

(9.23)

The investment rule derived in the deterministic case is a special case of this investment rule under uncertainty.24 If σ = 0, β1 = b = r g .25 Utilizing the standard rule (under certainty) for investment problems involving uncertainty, however, would lead to a suboptimal result where the chosen investment trigger is lower than the optimal, leading to 22. McKean (1965) first derived this closed-form formula for perpetual American call options. Karlin and Taylor (1975, pp. 364–65) and McDonald and Siegel (1986) developed close models. McKean and later Karlin and Taylor use an expression for the fundamental quadratic involving an exogenously given discount rate, while McDonald and Siegel adopt the risk-neutral pricing approach. See the appendix of the book, box A.2, for more details on the two quadratic equations. 23. The standard contingent-claims analysis or dynamic programming approaches (presented in Dixit and Pindyck 1994) consist in solving partial differential equations under appropriately specified boundary conditions. Given these conditions, it may be possible (if a solution exists) to derive a closed-form solution (e.g., in case of GBM). This methodology requires the use of involved mathematics (stochastic calculus and optimal control), which may be quite unintuitive for the nonseasoned reader. Our preferred approach is simpler and more intuitive, relying on an analogy with basic markup trade-off problems common in other fields of economics. Stochastic calculus is not absent from our proposed approach either. The calculation of terminal values and expected discount factors relies on such calculus. The techniques involved in these derivations are discussed in the appendix at the end of the book, rather than directly in each real option valuation. Our simplified derivation of optimal investment rules is done by using the expected discount factor in case of GBM. 24. In the deterministic growth case, the optimal trigger is given by (9.6) as V * = Π * I, with the profitability index in (9.7) given by Π* = r δ . This result is analogous to the one derived under uncertainty in (9.23), except that here Π* = β1 ( β1 − 1) . 25. As σ = 0, the fundamental quadratic simplifies to r − gβ (see box A.2 in the appendix at the end of the book), which admits only one root, b = r g.

Monopoly: Investment and Expansion Options

295

premature investment.26 From equations (9.13) and (9.23) the monopolist’s investment option value under uncertainty is27 β1   V0  (Π* − 1) I   M0 (V*) =   V*  V − I  0

(wait)

if V0 < V*,

(invest)

if V0 ≥ V* .

(9.24)

Modified Jorgensonian Rule of Investment In the case the uncertain profit πɶ t follows a geometric Brownian motion, the project value follows the diffusion process Vɶt ≡ πɶ t δ , where Vɶt and πɶ t are characterized by the same stochastic differential equation (but have different starting values). Let π* be the profit earned at optimal investment. Equation (9.23) can be reformulated as a modified version of the Jorgensionan rule prescribing to invest when πɶ t is such that

πɶ t π * 1 > = Π* δ = r + β1σ 2 I I 2

( > r ).

(9.25)

The modified Jorgensonian rule of investment suggests to invest when the project’s return on investment exceeds a certain hurdle compensation for both the interest rate (r) and the incentive (option) to delay investment in an uncertain world (β1σ 2 2). This modified rule under uncertainty is more stringent in terms of profitability (hurdle) 26. Suppose, by contraposition, that θ ≡ r (r − g ) − Π* ≤ 0 , with Π* as in (9.23). Since β1 > 1 and δ > 0, this implies that Θ ≡ δ (β1 − 1)θ ≤ 0. Since gˆ = αˆ + (σ 2 2), Θ = r − β1αˆ − (β1σ 2 2) . ˆ − (σ 2 β 2 2) (fundamental quadratic), it obtains that With β1 being a root of r − αβ Θ = β1 ( β1 − 1) σ 2 2 > 0 , which contradicts the assumption. The markup is thus larger in the uncertainty case. Moreover, as ∂Θ ∂σ > 0 , the higher the volatility, the higher is the discrepancy between the optimal investment trigger and the trigger obtained when ignoring the volatility. 27. The exit option (full closure) can be priced analogously. Suppose that a salvage value S is received upon exiting. The net present value received at that time is S − VT , where VT is forgone project value. The expected discount factor in case of exit (low barrier) involves the negative root of the (risk-neutral) fundamental quadratic,

β2 = −

2 r αˆ  αˆ  −  2 +2 2 , 2 σ  σ σ

with

1

αˆ = gˆ − σ 2 . 2

The optimal lower barrier is V*, such that V* S = Π*, with Π * = β 2 ( β 2 − 1). The exit option value is then worth β2   V0  (S − V*)   M0(V*) =   V*   S − V0

(wait)

if V0 > V*,

(divest)

if V0 ≤ V* .

296

Chapter 9

Value for monopolist 5

4

NPV = V0 − I

3

V* − I 2 Expanded NPV

M(V*)

1

0 0

1

2

I=2 Waiting region

3

4

V* = 4.46

5

6 7 Initial value V0

Investment region

Figure 9.2 Waiting versus investment regions for a monopolist under uncertainty Project value V follows a geometric Brownian motion with g = 5 percent and σ = 20 percent. The discount rate is k = 12 percent. Investment cost I = 2.

requirements than the standard version under certainty.28 Under certainty (σ = 0), the second (volatility) term in equation (9.25) vanishes and equation (9.25) reduces to the expression given in equation (9.8) for the certainty case. In a world of uncertainty, projects that do not meet the requirement using the standard Jorgensonian rule of investment (under certainty) a fortiori do not fulfill the requirements under uncertainty either since uncertainty imposes an extra hurdle (the second term). This property may be used by managers to readily identify (some) projects to be deferred, waiting longer if the standard Jorgensonian rule is not fulfilled. Figure 9.2 illustrates the value (waiting vs. investment) regions for the monopolist’s investment option. For V0 > V* = 4.46, it is optimal to invest 28. Building upon such optimal investment rules, McDonald (2000) examines whether the use of arbitrarily chosen hurdle rates and profitability indexes by management can roughly proxy for optimal investment decision making. He finds that (under mild conditions) such hurdle rate and profitability index recommendations can provide close-to-optimal investment rules.

Monopoly: Investment and Expansion Options

297

immediately. At this point the value of the opportunity to invest (E-NPV) just equals the value of the committed investment (NPV = V* − I ), which is strictly positive in this region. For V0 ≤ V*, there is an incentive to defer investment as investing immediately would kill the option to wait. At V*, the option holder is indifferent between keeping the option open or investing immediately (at this point the slopes of the two value functions are equalized).29 The NPV line (=V0−I) is tangent to the option value (E-NPV) function at X*. Since for any value of the elasticity of the waiting option (for σ ≠ 0), the profitability index (Π*) in equation (9.22) is strictly greater than one, from equation (9.23) the value of the project V* should exceed the necessary investment cost I by a certain positive premium. A number of variables affect the size of this premium through their impact on the profitability index: As ∂Π* ∂σ > 0, the greater the underlying uncertainty (σ ) the higher the markup Π* between the optimal investment trigger and the investment cost. Monopolists require a larger excess return before irreversibly investing when the market is riskier. If the project is extremely risky (σ very high), the monopolist will defer the investment indefinitely and never invest (with Π* becoming extremely large). •

As ∂Π* ∂δ < 0, the higher the dividend yield or opportunity cost of waiting, δ , the lower the investment trigger. The monopolist ends up investing earlier when δ is high. •

As ∂Π* ∂r > 0, when the risk-free rate is higher, the monopolist will defer the investment longer.

•

Example 9.2 Arithmetic Brownian Motion Suppose that project value vɶt is the underlying factor that follows an arithmetic Brownian motion of the form dvɶt = α dt + σ dzt.

(9.26)

The elasticity of terminal value is ε V = −1. As obtained in equation (A.41) in the appendix, the expected discount factor is exp ( β1v0 ) B0(Tɶ ) = , exp ( β1vT )

(9.27)

29. This property stems from the value-matching and smooth-pasting conditions, asserting that both the value function and its first-order derivative are continuous at the optimal trigger threshold X*.

298

Chapter 9

where β1 is given in (9.12).30 In this case, the elasticity of the discount factor is not constant. The first-order derivative of the discount factor with respect to the investment trigger is Bv (vT ) = −β1 B (v0 ; vT ). Hence from equation (9.16), ε B ( vT ) = −β1vT . The markup for the option to invest in equation (9.15) is Λ ≡ (v* − I ) v* = 1 β1v*. The profitability index (at equilibrium) in equation (9.18) becomes Π* = Π ( v *) =

β1v* . β1v* −1

Since the profitability index is not constant, the derivation of an analytical solution is cumbersome in this case. The preceding examples serve to confirm the extent to which the geometric Brownian motion is convenient for the derivation of closed-form solutions. Since (as explained in the appendix in the last chapter) this stochastic process is reasonably descriptive of many economic phenomena, it is widely used in economic and financial analysis. 9.2

Option to Expand Capacity

In the previous section we discussed how to analyze the deferral option or timing option to invest by a monopolist. Now we look at a slightly modified investment problem relating to the option to expand. Here we principally interpret this model in the context of adding capacity, though it may also apply in other contexts, for instance, in the adoption of new technology, expansion to a new country, or refurbishment of existing assets to improve efficiency. Although this model can accommodate many applications, for expositional simplicity we here follow one interpretation, that of capacity expansion. Suppose that the monopolist is already active in the market and holds a certain amount of production capacity. If the market becomes more profitable than expected, the monopolist may invest in added capacity to take advantage of the demand upsurge. The monopolist thus has an option to expand its production capacity. If the monopolist exercises 30. As noted, dynamic programming is considered the “general” approach as long as one can identify the correct discount rate (which may not be an exogenous constant, as Dixit and Pindyck 1994 assume). It applies to both incomplete markets (where there is a range of solutions) and complete markets (where risk-free arbitrage ensures a single unique solution, the risk-neutral version). The fundamental quadratic comes from a different ordinary differential equation (ODE) for ABM than for GBM. In the dynamic programming version of the ABM, the drift is α , whereas in the contingent-claims analysis version, it is replaced by αˆ vɶt with αˆ = r − δ .

Monopoly: Investment and Expansion Options

299

its real expansion option, its new (higher) profit flow will reflect both the production capacity owned before the new investment as well as the newly added capacity. As in the case of the deferral option, the problem for the monopolist is to decide when exactly to invest in added capacity. Once again, the monopolist’s strategy under uncertainty consists in choosing a priori an investment trigger XT and investing at the (random) time Tɶ ≡ inf {t ≥ 0 | Xɶ t ≥ XT } when XT is first hit. Expansion options may be modeled in two ways. The first approach, referred to as additional capacity models, involves a lumpy capacity expansion investment (∆Q) generating a profit flow increase by a given discrete amount, for example, 20 percent. The second approach involves incremental (very small) capacity expansion additions (dQ)—these models are referred to as incremental capacity investment models. Here the marginal effect on firm profit matters.31 9.2.1 Additional (Lumpy) Capacity Investment Consider first the case where the firm can increase its capacity and profit flow by a lumpy (large) amount. Once a project is undertaken, management may have the flexibility to alter it in various ways at different times during its life. Management may find it desirable, for example, to build additional capacity if it turns out that its product is more enthusiastically received in the marketplace that initially thought. In this sense the investment opportunity can be thought of as the initial scale project plus an (American) call option on the future expansion opportunity. The exercise price of the expansion option is I. In the additional capacity case two stages are usually distinguished. In the first stage, the monopolist is already active in the market and earns a profit flow (π 0) as a function of the (old) production capacity it already employs. In the second stage, when the monopolist expands its capacity, it earns a higher profit flow than previously (π 1) due to the added capacity. The additional capacity investment should occur at an optimal (random) time Tɶ . The firm does not immediately expand capacity since incurring the sunk investment cost may not be justified under the present 31. The second (incremental) approach seems less realistic, but it sometimes allows for powerful analytical results from a research viewpoint. The incremental capacity investment approach makes it possible to take the derivative of the profit function with respect to the capacity stock and obtain analytic formulas. Models of incremental capacity investment typically require the use of instantaneous control techniques (e.g., supercontact condition), whereas models involving investment in additional capacity rely on optimal stopping and/ or impulse control methods (e.g., smooth-pasting condition). We look at the additional capacity case first because it is more intuitive and realistic.

300

Chapter 9

Table 9.1 Firm’s profits in the two stages (regions) surrounding capacity expansion Industry structure

t

Stochastic profit

Certain profit

Before capacity investment

0 ≤ t ≤ Tɶ t ≥ Tɶ

πɶ 0 πɶ 1

π0 π1

After capacity expansion

economic conditions. The notation for the profit flows is summarized in table 9.1.32 The expected value for the monopolist (as of time t0 = 0) from investing in additional capacity at random time Tɶ equals (for X 0 ≤ XT) ∞ T ɶ M0 ( XT ) = Eˆ  ∫ πɶ 0 e− rt dt + ∫ ɶ πɶ 1 e− rt dt − I e− rT . (9.28)  0  T Using property (A.45) in the appendix at the end of the book, one obtains for the value of the strategy to invest in additional capacity when XT is first hit (at time Tɶ ): ɶ

M0 ( XT ) = V0 [πɶ 0 ] + B0 (Tɶ ) [VT [πɶ 1 − πɶ 0 ] − I ].

(9.29)

where Vt [πɶ ] is the forward perpetuity value of receiving profit πɶ from time t on. As before, the monopolist maximizes its value by selecting the optimal investment trigger X*. Using standard optimization techniques, the first-order derivative of the value function is obtained as M X ( XT ) = BX ( XT ) × [VT [πɶ 1 − πɶ 0 ] − I ] + B0 ( XT ) × VX [πɶ 1 − πɶ 0 ],

(9.30)

The first right-hand term, V0 [πɶ 0 ], in the value expression for M ( XT ) in equation (9.29) above is independent of the trigger and drops out (the perpetuity value of already-in-place capacity units does not depend on subsequent investments). Letting ∆VT ≡ VT [πɶ 1 − πɶ 0 ] and ∆V * ≡ ∆VTɶ *, the first-order condition becomes Λ′ ≡

∆V* − I ε ∆V ( X*) , =− ∆V* ε B ( X*)

(9.31)

where the elasticity of the discount factor (ε B) and of the additional terminal value (ε ∆V) are XT ε  B ( XT ) = − BX ( XT ) × B0( XT ) ,  ε ∆V ( XT ) = − ∆VX ( XT ) × XT , ∆V( XT ) 

32. Note that πɶ 1 ( Xɶ t ) > πɶ 0 ( Xɶ t ) for all Xɶ t , and π 1 > π 0 .

Monopoly: Investment and Expansion Options

301

with BX (⋅) = ∂B ∂XT and ∆VX (⋅) = ∂∆V / ∂XT. Equation (9.31) can be reformulated as ∆V* = Π*, I

(9.32)

where Π* ≡ Π ( X*) and Π ( XT ) ≡ ε B ( XT ) [ε B ( XT ) + ε ∆V ( XT )] as per equation (9.18). These expressions are analogous to the results obtained for the investment option in equations (9.15), (9.16), and (9.19), with ∆V replacing V . This equivalence makes sense since the perpetual option to invest in additional capacity is analogous to a perpetual American call option to invest involving the additional profit flows for a firm already active in the market. Let us be more specific concerning the uncertainty in the market. Suppose that the stochastic profit flow consists of two parts: A deterministic profit component that mirrors the capacity held by the firm (present or future). Let us define it generally as π (π 0 before capacity addition and π 1 afterward). This component results from optimal behavior at each stage by the monopolist. It is a reduced-form expression of the monopolist’s profit, which depends on demand, costs, and price.33 ɶ t , which follows an (time-homoge• A multiplicative stochastic shock, X •

neous) Itô process of the form dXɶ t = g ( Xɶ t ) dt + σ ( Xɶ t ) dzt,

where g ( Xɶ t ) is the drift, σ ( Xɶ t ) is the diffusion and zt is a standard Brownian motion. This shock multiplies the certain profit component π capturing exogenous uncertainty about the market development. This approach to viewing total uncertain profit as being made up of two components (including a multiplicative shock) is useful for analyzing problems of incremental capacity investment. In the following chapters this decomposition of the stochastic profit flow into two components will be used extensively.34 This modeling assumption is slightly different from what we used in the discrete-time part. Previously uncertainty was introduced in the 33. This notion of deterministic reduced-form profits can be readily extended to competitive settings. Several models to derive the certain profit in oligopolistic markets have been investigated previously (see chapter 3). 34. In the previous section on the option to invest (defer), we assumed a setting close to McDonald and Siegel’s (1986) model as summarized by Dixit and Pindyck (1994, p.142) and Trigeorgis (1996, p. 204), where project value follows a geometric Brownian motion with no decomposition of uncertain profits into two components.

302

Chapter 9

linear inverse demand function of equation (7.1), p ( Xɶ t , Q) = aXɶ t − bQ, via the demand intercept Xɶ t hypothesized to follow a specified stochastic process. This represented an additive shock affecting firms’ profits. Here the demand level is deterministic as captured by the reduced-form certain profits, while profits are affected by a stochastic multiplicative shock. The multiplicative shock may represent, for example, the exchange rate between a foreign currency and the domestic currency. Suppose that investment in added capacity enhances production made in the domestic country but that products are sold in a foreign country. Foreign currency must be repatriated to the parent firm at the prevailing currency exchange rate Xɶ t . The profit from the project’s production in domestic currency at time t is given by πɶ 1 ( Xɶ t ) = π 1 × Xɶ t if additional capacity investment has occurred (or πɶ 0 ( Xɶ t ) = π 0 × Xɶ t otherwise).35 The terminal value (forward NPV) function is linear, so M ( XT ) =

π0 (π − π 0 )  X 0 + B0 (Tɶ )  1 XT − I . δ δ  

(9.33)

If Xɶ t follows the geometric Brownian motion, the elasticity of the discount factor B0 (Tɶ ) is β1. With V0 ≡ π 0 δ , V1 ≡ π 1 δ , and ∆V ≡ V1 − V0 = (π 1 − π 0 ) δ defined, the elasticity of the terminal value obtained from equation (9.16) is ε ∆V ( XT ) = −1. From equations (9.32) and (9.33), the time-0 value of the monopolist firm with the option to expand production capacity (assuming it behaves optimally) is β1   X0  V0 X 0 + ( ∆V X* − I )  M( X*) =   X*   V1 X 0 − I

if

X 0 < X *,

if

X 0 ≥ X*.

(9.34)

Here X* is such that ∆VX* = Π*, I

(9.35)

with Π* ≡

β1 . β1 − 1

(9.36)

The above model of a monopolist with the option to invest in additional capacity investment can also be used for a firm that is not currently 35. The underlying (exchange-rate) stochastic process, Xɶ t , is assumed the same for both stochastic profit flows. There is no reason to believe that the (idiosyncratic) underlying factor evolves differently before versus after the capacity expansion investment.

Monopoly: Investment and Expansion Options

303

active, namely for a firm having the option to invest (defer) treated previously. If πɶ 0 (⋅) = 0, the present model reduces to the previous one. That is, the previous model of new investment can be thought of as a special case of the model of additional capacity investment (where prior investment is zero). 9.2.2

Incremental Capacity Investment

Consider now a slightly modified situation where a monopolist already active in the market contemplates investing in incremental capacity, namely to increase its production capacity by a very small amount. At the beginning, the monopolist firm already owns production capacity and earns a stochastic profit flow, πɶ ( Xɶ t , Q), which is a function of the capacity already installed, Q. The firm has an option to invest in incremental capacity should the project be more profitable than expected. If the firm exercises this real expansion option, it installs dQ of incremental capacity at an incremental cost i per unit of new capacity. For an additional capacity dQ, the firm must incur an additional investment cost of I ≡ i × dQ. At the (unknown) time Tɶ the monopolist invests in new capacity and receives afterward a stochastic profit flow πɶ Q × dQ (where πɶ Q ≡ ∂πɶ ( Xɶ t , Q) ∂Q), in addition to the profit flow stemming from the existing production capacity. Let vT (Q) ≡ VT [πɶ Q × dQ]. Again, the monopolist’s investment strategy consists in choosing upfront an investment trigger XT and investing at the (random) time Tɶ ≡ inf t ≥ 0 Xɶ t ≥ XT when XT is first reached. The time-0 value of the monopolist firm consists of the perpetuity profit flow resulting from the existing (old) production capacity plus the value of the option to expand capacity:

{

}

ɶ M0 ( XT , Q) = Eˆ  ∫ πɶ ( Xɶ t , Q) e− rt dt + (vT (Q) − I ) e− rT   0  ɶ = V0 πɶ ( Xɶ t , Q) + Eˆ {vT (Q) − I } e− rT  . ∞

(9.37)

Since the incremental net present value from the project, vT (Q) − I , does not depend on the actual path of the stochastic process, M0 ( XT , Q) = V0 πɶ ( Xɶ t , Q) + (vT (Q) − I ) B0 (Tɶ ) . The optimal investment trigger X* is such that M0 ( X *, Q) ≥ M0 ( XT , Q)

∀XT .

The first-order condition, BX ( X*) × vT (Q) + B0( X*) × vX (Q) = BX ( X*) × I ,

304

Chapter 9

results in the option markup formula v* (Q) − I ε ( X*) =− v , v* (Q) ε B ( X*)

(9.38)

where XT ε  B( XT ) = − BX ( XT ) × B0( XT ) ,  ε v ( XT ) = −vX (Q) × XT . vT (Q)  Alternatively, the optimal investment rule is v *(Q) = Π* I ,

(9.38′)

where, analogous to (9.18), Π* =

ε B ( X*) . ε B ( X*) + ε v ( X*)

Suppose again that stochastic project profit flow consists of two parts: A deterministic profit flow component corresponding to the capacity held by the firm (present Q or future Q + dQ), namely π (Q). This value results from market-clearing mechanisms. If the firm invests in incremental capacity, the extra profit flow for the monopolist is given by π Q × dQ, where π Q ≡ ∂π (Q) ∂Q.36 ɶ t , that follows geometric Brownian motion. • A multiplicative shock, X •

The (stochastic) profit resulting from the project at time t is thus given by

πɶ ( Xɶ t , Q) = π (Q) Xɶ t , while the marginal profit flow resulting from an incremental investment in capacity is ∂πɶ ɶ ( X t , Q) dQ = (π Q × dQ) Xɶ t. ∂Q Consider next the terminal value (forward NPV) and its elasticity. Given the above, the terminal value is 36. The deterministic profit function as a function of capacity is standard in industrial organization. For the sake of simplicity, we oftentimes equate capacities with quantities (assuming constant returns-to-scale production technologies). Given these parameters, the monopolist can optimally determine the production capacity needed.

Monopoly: Investment and Expansion Options

vT (Q) = (π Q(Q) × dQ)

305

XT δ

with δ ≡ k − g = r − gˆ . Letting v (Q) ≡ (π Q(Q) × dQ) δ , vT (Q) = XT v (Q).

(9.39)

For an initially specified (planned) amount of capacity (Q), the term v (Q) represents the perpetuity value of the deterministic incremental profits (π Q × dQ) stemming from the new capacity investment (growing at an annual risk-adjusted rate gˆ , with r being the interest rate). Alternatively, π Q × dQ is the contribution of the incremental capacity investment to the deterministic profit flow for the monopolist and v (Q) is the expected present value of this incremental profit flow contribution. The elasticity of vT is again constant and equal to ε v = −1. The optimal investment rule again is v* (Q) = Π* I , where Π* = β1 ( β1 − 1). Since from equation (9.39), v* (Q) = X* v(Q), the optimal investment trigger for the monopolist is given by37 v (Q) X* (Q) = Π* or I

π Q(Q) X*(Q) 1 = r + β1σ 2. 2 I

(9.40)

Suppose, for example, that the monopolist faces a constant-elasticity demand function of the form p (Q) = Q−1 ε p , where ε p is the constant price elasticity of demand. Assume, for simplicity, that the monopolist has no production cost. In this case the certain profit flow for the monopolist is π Q(Q) = Q(ε p − 1) ε p and therefore ∂ π ∂Q = [(ε p − 1) ε p ]Q−1 ε p . For this special case the monopolist’s investment trigger (as a function of the capacity in place) is  ε p  1 εp 1 X* (Q) =  r + β1σ 2   Q I.    ε p − 1 2

(9.41)

Figure 9.3 illustrates the above investment trigger for incremental capacity investment by a monopolist as a function of the level of capacity initially held (Q). It is shown that the higher the level of initial capacity, the higher the investment trigger X* (Q), implying that large firms are less sensitive to positive demand shocks than smaller firms in emerging markets. The additional (lumpy) capacity investment model in the previous section can be thought of as a discretized version of the continuous 37. This result is also found in Pindyck (1988), Bertola (1988, 1989), and Dixit and Pindyck (1994, p. 364).

306

Chapter 9

Optimal investment trigger value 15

10

Investment trigger

X*

5

0 1

2

3

4

5

6

Initial capacity (Q)

Figure 9.3 Investment trigger for incremental capacity investment by a monopolist (as function of the initial capacity held) The underlying process follows a geometric Brownian motion with drift g = 12 percent, volatility σ = 30 percent, and starting value X 0 = 1. Total return is k = 12 percent. The necessary investment cost is i = 20 . The elasticity of the demand function is ε p = 2 .

incremental capacity investment model of this section. In the previous model the optimal investment trigger was found in equation (9.35) at the point where X* was such that (∆VX *) I = Π* (with I being the investment outlay for additional capacity ∆Q). This reduces to the trigger expression of equation (9.40) as ∆Q → 0.38 Thus the incremental capacity investment model is the continuous equivalent of the additional (lumpy) capacity investment model discussed earlier.39 Conclusion In this chapter we discussed the situation faced by a firm having an exclusive access to a market or enjoying insurmountable structural 38. Note that lim

∆Q→0

( ∆V

ΠI δ ΠI = . = ΠI π Q(Q) dQ vQ(Q) ∆Q) × ∆Q

39. If in the additional capacity investment model the monopolist’s profits before and after the additional capacity investment are determined by equilibrium conditions (e.g., derived from profit-optimization techniques), a first-order derivative of the profit function (with respect to starting capacity level Q) might be difficult to find (discontinuity of the profit in Q). Then the equivalence between the discrete/lumpy and continuous/incremental problems may not hold.

Monopoly: Investment and Expansion Options

307

barriers enabling it to ignore strategic interactions by rivals when devising its investment strategy. The standard investment timing rule has to be revised under conditions of uncertainty to properly consider the volatility in the underlying market, with the monopolist waiting longer than in the deterministic growth case. We analyzed in turn the problem of entering a new market and the problem of expanding the firm’s scale of production. Both lump-sum and incremental capacity expansion models were considered. We derived additional intuition exploiting the analogy between the monopolist investor’s entry decision and the price markup set by a monopolist. Selected References McDonald and Siegel (1986) investigate the investment timing decision of a monopolist under uncertainty. Dixit and Pindyck (1994) extend this analysis to other settings.40 Dixit, Pindyck, and Sødal (1999) examine optimal investment timing and the (perpetual American call) investment option problem using classical techniques from microeconomics. This approach based on the elasticity of the expected discount factor is also amenable to the study of other real options settings. Sødal (2006) analyses hysteresis and the market entry and exit decisions along these lines.41 Dixit, Avinash K., and Robert S Pindyck. 1994. Investment under Uncertainty. Princeton: Princeton University Press. Dixit, Avinash K., Robert S. Pindyck, and Sigbjørn Sødal. 1999. A mark-up interpretation of optimal investment rules. Economic Journal 109 (455): 179–89. McDonald, Robert L., and Daniel Siegel. 1986. The value of waiting to invest. Quarterly Journal of Economics 101 (4): 707–28. 40. Dixit and Pindyck (1994) provide a solution based on dynamic programming using an exogenously given discount rate also applicable in case of incomplete markets. Since option pricing is typically used for real options problems, we discuss here the contingent-claims version (under the assumption of complete markets and no arbitrage). The two methodologies (contingent-claims analysis and dynamic programming) rely on somewhat different assumptions. For further details on these two approaches regarding the investment timing problem of a monopolist, see Dixit and Pindyck (1994, chs. 4 and 5). The dynamic programming approach can be used, in principle, even if the underlying asset is not spanned in the economy. 41. Dixit (1989) analyzes hysteresis in a real options setting. The preceding analysis of the expected discount factor focuses on the upper threshold with the firm investing when the threshold is first hit (investment option). If the firm may receive a scrap value upon exiting the market, the “hysteresis band” involves a lower bound at which the firm exits. The presence of sunk costs (as the difference between entry and exit costs) creates “hysteresis” or delay effects: firms are reluctant to enter and are likely to stay in the market longer in the hope that the market might recover.

308

Chapter 9

Sødal, Sigbjørn. 2006. Entry and exit decisions based on a discount factor approach. Journal of Economic Dynamics and Control 30 (11): 1963–86. Appendix 9A: Contingent-Claims Analysis of the Option to Invest in Monopoly Real investment opportunities can be seen analogous to perpetual American call options, giving the right but not the obligation to acquire an asset at a predetermined exercise price. The investment-timing problem is about when to invest, or equivalently when to acquire an asset given that the exercise price (which here equals the investment cost) is exogenously given and known. This problem is generally solved by means of dynamic programming or contingent-claims analysis (CCA). To apply the CCA approach, we assume that changes in Xɶ t are spanned by existing assets in the capital markets and that there exist no arbitrage opportunities. These assumptions generally hold true in complete capital markets , as is the case for most commodities. In this case a replicating portfolio can be constructed. Suppose that Vɶt follows the geometric Brownian motion as per equation (9.10) with g the expected percentage rate of change or growth parameter and σ the volatility parameter. The difference between total return k and the growth rate g represents some form of dividend yield, denoted δ ≡ k − g . Unless δ > 0, the option will never be exercised because the present value of the exercise price decreases for later investment. Let M be the value function of the monopolist firm’s option to invest.42 To determine M , we can (1) construct a replicating, risk-free portfolio and (2) determine its expected rate of return and equate that expected rate of return to the risk-free rate, r. Consider a portfolio that consists of a long position in M , and a short position (N units) in the underlying asset Vɶt. The portfolio is therefore worth M − N Vɶt. The short position pays, within each time interval dt , a dividend δ per unit investment in the underlying asset. Assuming that the short position N is held fixed for an infinitesimal time period dt , the portfolio has a total expected return over the short time interval dt of E [ dM ] − N E dVɶt  − δ NVɶt dt. 42. We drop the dependence of M (and its derivatives) on the asset value Vɶt for notational convenience.

Monopoly: Investment and Expansion Options

309

By Itô’s lemma given in equation (A2.1) in the appendix at the end of the book, 1 dM =  gMV Vɶt + σ 2 MVV Vɶt 2  dt + σ MV Vɶt dzt, 2  

so that 1 E [ dM ] =  gMV Vɶt + σ 2 MVV Vɶt 2  dt. 2  

From the two previous equations, the total expected return on the portfolio is 1

( gMV − gN − δ N ) Vɶt + σ 2 MVV Vɶt2. 2

If arbitrage is precluded, the portfolio is risk-free earning the risk-free return r so that N = MV . Therefore 1 2

σ 2 MVV Vɶt 2 − δ MV Vɶt = r  M − MV Vɶt  .

This results in the partial differential equation: 1 ˆ V Vɶt + σ 2 MVV Vɶt 2, rM = gM 2

with gˆ = r − δ . A solution to this equation (if it exists) is of the form M ≡ M(Vɶt ) = AVɶt β1 + BVɶtβ2 , where A and B are constants to be derived, β1 and β 2 are the positive and negative roots of the “fundamental quadratic” given in appendix equation (A2.4). Applying the boundary condition that limVɶt → 0 M (Vɶt ) = 0, it obtains that B = 0 (since β 2 < 0). From the value-matching and smoothpasting conditions (see section A.4 in the appendix to the book) we get A = (V* − I ) V*(− β1 ) and V* = Π*I, where Π* = β1 ( β1 − 1).

10

Oligopoly: Simultaneous Investment

In the previous chapter we discussed optimal investment timing under uncertainty for a monopolist. We considered two types of options: invest (defer) and expand. The models developed in the monopoly case help pave the way and set a benchmark for analyzing investmenttiming problems under uncertainty involving competition among two or more firms. This last extension requires the use of game theory. Here we will deal with simple cases of option games involving symmetric firms. Suppose, for simplicity and pedagogical usefulness, that simultaneous investment occurs at the same trigger value because firms agree to do so or tacitly collude.1 A social planner interested in maximizing joint firm profit would select the same investment trigger. In the following sections dealing with oligopolistic industry structures, we show how the presence of more rivals lowers the threshold that triggers investment, so investment occurs sooner. The value of the investment timing (or deferral) option deteriorates with more competition. The option to wait for new information before investing (new market model) can be seen as a special case of the expansion or growth option (existing market model), so we concentrate on the latter also obtaining analytical results for the option to defer investment as a special case. We consider, in turn, the following industry structures: oligopoly situations where firms add capacity in lump sums (section 10.1), oligopoly situations where firms can increase their capacity incrementally (section 10.2), and perfect competition, obtained in the limit as the number of incumbents becomes larger (section 10.3). 1. Whether such simultaneous investment can occur as part of a perfect Nash equilibrium will be discussed in the next chapters. This analysis is relevant in cases of collusive agreement between firms having shared investment options.

312

10.1

Chapter 10

Oligopoly: Additional Capacity Investment

We discuss next the existing market model (expansion option), and then obtain the new market model (investment deferral option) as a special case by setting an initial zero profit flow for each firm. 10.1.1 Existing Market Model: Expansion Option Consider two identical firms already active in the market that contemplate investing in additional capacity. We ignore firm-specific risk factors and concentrate on the industrywide demand shock, modeled as a stochastic process Xɶ t following the general diffusion (or time-homogeneous Itô process) of the form dXɶ t = g ( Xɶ t ) dt + σ ( Xɶ t ) dzt,

(10.1)

where g( Xɶ t ) is the drift, σ ( Xɶ t ) the diffusion of the underlying process, and zt a standard Brownian motion.2 This shock can capture exchangerate uncertainty or unexpected change in demand patterns. For simplicity, consider a low initial value X 0 for the demand process. Suppose also that firms share the option to expand capacity and decide to invest simultaneously. Suppose the duopolist firms can only add capacity by a given lump sum. At the outset, before adding investment in extra capacity, each firm receives profits based on their existing capacity (assets in place). We denote this initial profit flow by πɶ 0. After investment in added capacity units, each firm receives a higher profit flow, πɶ 1, the same for both firms (as investment among symmetric firms takes place simultaneously). As before, we decompose the stochastic total profit flow (πɶ 0 or πɶ 1) into a deterministic reduced-form profit component (π 0 or π 1) and a multiplicative stochastic shock, Xɶ t . Firms’ joint investment strategy consists in investing simultaneously the first time Tɶ the trigger XT (≥ X 0) is reached. Since the process Xɶ t evolves stochastically, the first-hitting time is a random variable, given by Tɶ ≡ inf t ≥ 0⏐ Xɶ t ≥ XT . Two demand regions are distinguished. If the process remains below the jointly selected investment trigger XT , that is, if Xɶ t < XT for all past t, firms stay put (wait). This set of values ( −∞, XT ) is the continuation or inaction region. At the first time the demand process hits the boundary, XT , both firms invest. The corresponding set of values [ XT , ∞ ) is the stopping or action region. As long as the process

{

}

2. Readers interested in the comparative effects of firm-specific or idiosyncratic shocks may refer to Caballero and Pindyck (1996) or Dixit and Pindyck (1994, pp. 277–80).

Oligopoly: Simultaneous Investment

313

remains in the inaction region ( −∞, XT ), each firm receives πɶ 0 = π 0 Xɶ t. As soon as the process enters the action region [ XT , ∞ ) at random time Tɶ , each firm invests and earns the higher profit amount πɶ 1 = π 1 Xɶ t onward.3 The assessment of the investment option proceeds in several steps: (1) assess the value induced by a given investment strategy XT , (2) determine the investment strategy that maximizes the firm’s value given the strategy choice of rivals, and (3) use the optimal investment trigger to obtain the expanded net present value (optimal investment value). If both firms follow the joint strategy to invest when XT is first reached, each has a time-0 value of ∞ T ɶ C0 ( XT ) ≡ C ( XT ; X 0 ) ≡ Eˆ 0 ⎡ ∫ π 0 Xɶ t e− rt dt + ∫ ɶ π 1 Xɶ t e− rt dt − I e− rT ⎤, T ⎣⎢ 0 ⎦⎥ ɶ

where I and r denote the capital investment cost and the risk-free interest rate, respectively. Set V0 ≡ π 0 δ and V1 ≡ π 1 δ , with δ = k − g = r − gˆ ( > 0 ) representing the opportunity cost of delaying or a “dividend yield.” From equation (A.45) in the appendix at the end of the book, we can decompose the value expression into C0 ( XT ) = V0 X 0 + B0(Tɶ ) ⎡⎣(V1 − V0 ) XT − I ⎤⎦.

(10.2)

The first right-hand term represents the perpetuity value of the firm if it stays put with its existing capacity forever, while the second term is the additional net forward value ⎣⎡(V1 − V0 ) XT − I ⎦⎤ discounted back to the present using the expected discount factor, B0(Tɶ ), giving the time-0 value of )1 received at random future time Tɶ . In the special case that the underlying stochastic factor Xɶ t follows the geometric Brownian motion, dXɶ t = ( gXɶ t ) dt + (σ Xɶ t ) dzt,

(10.3)

the expected discount factor is given by equation (A.43) in the appendix as ⎛X ⎞ B0 (Tɶ ) = ⎜ 0 ⎟ ⎝ XT ⎠

β1

(10.4)

with β1 (under risk neutrality) given in equation (A2.4) by

β1 = −

2 αˆ r ⎛ αˆ ⎞ + +2 2 ⎜ ⎟ 2 2⎠ ⎝ σ σ σ

( > 1)

3. The reduced-form certain profits π 0, π 1 capture the equilibrium profits at each time t considered. We do not specify the type of competition governing the time- t marketplace. In a context involving quantity competition it could be Cournot profits or the Paretooptimal outcome where both firms produce half the monopolist’s profit if it is enforceable over time (see chapter 4, section 4.2.2).

314

Chapter 10

with αˆ = gˆ − (σ 2 2). By the methodology developed in chapter 9, the optimal joint investment trigger X* must satisfy ΔV X* = Π*, or I

⎛ I X* = Π* ⎜ ⎝ ΔV

⎞ ⎟⎠ ,

(10.5)

where ΔV ≡ V1 − V0 is the deterministic value increment obtained upon investing and Π* is the level of profitability required at the time of optimal investment, given by Π* =

β1 . β1 − 1

(10.6)

If the two firms pursue the optimal joint investment strategy characterized by trigger value X*, based on equations (10.2) and (10.4), each will receive at the outset: β1 ⎧ ⎛ X0 ⎞ if ⎡(V1 − V0 ) X* − I ⎤⎦ (wait) ⎪V0 X 0 + ⎜ C0 ( X*) = ⎨ ⎝ X* ⎟⎠ ⎣ ⎪ (invest) if ⎩V1 X 0 − I

X 0 < X*, X 0 ≥ X*. (10.7)

Equation (10.7) can be interpreted as follows. In the waiting region, ( −∞, X*), the firm’s total value involves two components. In the first stage, each firm earns a reduced-form profit π 0 that reflects the initial capacity ɶ , both firms levels, receiving V0 X 0 as perpetuity value. At random time T* invest simultaneously, receiving net forward value (V1 − V0 ) X* − I . In effect each firm “exchanges” its original perpetuity value stemming from its initial installed capacity V0 for the new perpetuity value resulting from ɶ , the demand process has value expanded industry capacity V1. At time T* X*. This “exchange” involves a transaction cost I . To assess this net forward value in time-0 terms, we use the expected discount factor in equation (10.4). If the current demand value, X 0, is higher than the prescribed optimal investment threshold level X*, both firms will invest immediately receiving the higher net present value V1 X 0 − I. 10.1.2

New Market Model: Investment (Defer) Option

The value of the option to invest in a new market can be deduced as a special case of equation (10.7) for the capacity expansion problem. Setting the initial deterministic profit π 0 to zero with V0 = π 0 δ being zero, the value of the option to invest in a new market is thus given by

Oligopoly: Simultaneous Investment

β1 ⎧ ⎛ X0 ⎞ ⎪(V1 X* − I ) ⎜ C0 ( X*) = ⎨ ⎝ X* ⎟⎠ ⎪ ⎩V1 X 0 − I

315

if

X 0 < X*,

(invest) if

X 0 ≥ X*,

(wait)

(10.8)

where the optimal investment trigger X* is given as a special case of (10.5): ⎛ I ⎞ X* = Π* ⎜ ⎟ , ⎝ V1 ⎠ with the profitability index, Π*, as given by equation (10.6). The expression in equation (10.8) looks similar to that for the value of the option to invest by a monopolist obtained previously in equation (9.13′).4 Suppose that firms collude both on investment timing (investment stage) and on output levels (market stage), for example, by investing in a joint venture in which each pays half the investment cost, collectively paying the same investment cost as the monopolist. This situation is equivalent to the monopoly case. In this joint venture, π 1 = π M 2, where π M is the monopoly profit as given in equation (3.4). The perpetuity value for each firm is V1 = V M 2 with V M ≡ π M δ . Moreover, each incurs cost I 2. The optimal investment trigger in this collusive duopoly case is ⎛ I 2 ⎞ X* = Π* ⎜ , ⎝ VM 2 ⎟⎠ the same optimal investment trigger as in the monopoly case. That is, the joint venture or collusive duopolist investment trigger is identical to the monopolist’s optimal target.5 Example 10.1 Cournot Quantity Competition in Oligopoly Consider the oligopoly situation where, as in the Cournot model of section 3.3 of chapter 3, n symmetric firms face a linear demand p (Q) = a − bQ given by equation (3.1) and pay a constant marginal cost 4. Although the form of the solution is similar, the equilibrium reduced-form profits in duopoly and monopoly may differ (all other things being equal). Since the profit of a duopolist is generally lower than that of a monopolist, the investment trigger is higher in a duopoly (as ∂X* ∂V1 < 0) so that the duopolist firm invests later. In this setting the duopolist waits longer to ensure that the value it receives upon investment is sufficiently large to justify spending the investment cost I . This result rests on two key assumptions: (1) the investment cost incurred by one firm is the same whatever its market power (monopoly vs. duopoly) or duopolists collectively face twice the investment cost of a monopolist, and (2) firms collude or agree on the joint investment timing (investment stage) but compete “as usual” in the market after the investment (market stage). 5. The relationships we develop in this section are useful later (chapter 12) when we examine the option to expand when firms face a coordination problem.

316

Chapter 10

Table 10.1 Stage profits under Cournot quantity competition Industry structure

Number of firms

Equilibrium profit

Monopoly

1

Duopoly

2

1 (a − c) 4 b 2 1 (a − c ) 9 b

Oligopoly

n

2

1 (a − c ) (n + 1)2 b

2

c per unit output. Suppose that firms coordinate their actions in the investment stage; that is, they agree on a joint investment trigger, X n*, but do not necessarily collude in the market stage, competing à la Cournot once they have entered the market.6 The reduced-form stage profit flows, obtained in chapter 3, are summarized for convenience in table 10.1. The investment cost paid by each firm, I , is the same whatever the industry structure. Let π ( n) and V1 ( n) ≡ π ( n) δ be the deterministic profit and perpetuity value components obtained by each of the n (symmetric) oligopolist firms. The multiplicative stochastic shock follows geometric Brownian motion. In the inaction region, namely if X 0 < X n*, the value of the investment option for any of the n firms when all firms follow the joint optimal investment strategy is, by extension of equation (10.8), given by ⎛ X ⎞ C0 ( X n*) = (V1 ( n) X n* − I ) ⎜ 0 ⎟ ⎝ X n* ⎠

β1

β1

⎛ X ⎞ = (Π* − 1) I ⎜ 0 ⎟ , ⎝ X n* ⎠

(10.9)

where the investment trigger optimally chosen by the n colluding firms, X n*, is given by ⎛ I ⎞ X n* = Π* ⎜ ⎝ V1 ( n) ⎟⎠ with Π * given in equation (10.6). Since V1 ( n) decreases with the number of firms (n), the optimal investment trigger X n* increases. This result relates to the fact that investments are lumpy and that the investment cost, I , is constant, regardless of the industry structure. Figure 10.1 confirms that the investment option value dissipates fast when more firms are active in the marketplace (here, beyond n = 4 or 5 firms the option 6. We could assume instead that firms collude as well in the market stage, for example, producing half the monopoly output after making a joint simultaneous investment if it is more descriptive of the problem at hand.

Oligopoly: Simultaneous Investment

317

Option value (in millions of euros)

50

40

30 Aggregate industrywide option value

20

Option value of an individual firm

10

0 1

2

3

4

5

6

7

8

9

10

Number of firms (n)

Figure 10.1 Value of the option to invest in oligopoly as more firms are active in the market Here the individual investment cost is considered constant whatever the number of firms in the industry. It is possible to consider an investment cost decreasing with the number of firms. This would lead to higher values for the investment trigger and the option. p (Q) = 6 − Q; c = 2 ; I = 100 , k = 12 percent, g = 0.08 ; σ = 20 percent; X 0 = 1.

value becomes very small). The investment option value decreases as the profit value of an oligopolist firm declines with the number of firms, n. The option value does not vanish completely, however. Sustainability of such collusive agreement is doubtful, especially in an oligopoly with a large number of firms. Each firm may have an incentive to invest earlier to wield more market power and temporarily earn higher profits.7 10.2

Oligopoly: Incremental Capacity Investment

An interesting variation of the above investment problem has been developed by Grenadier (2002) to deal with situations where firms can 7. Simultaneous investment models in a dynamic setting are more challenging. In most cases the problem is formulated with open-loop strategies, and the resulting Nash equilibrium strategy profiles may fail to form a subgame perfect Nash equilibrium if firms have the possibility to observe and react to their rivals’ past actions, that is, if firms are permitted to devise closed-loop strategies. In lumpy investment cases—as discussed, for instance, in Huisman and Kort (1999)—tacit collusion can sustain as (Markov) perfect equilibrium for a certain range of parameters, but in most cases tacit collusion is not likely. For both openloop and closed-loop strategies the monopolist will wait more since there is no preemption threat. But with more firms following closed-loop strategies, preemption poses a real threat. Rivalry can hasten rather than delay investment. We discuss this issue at length in chapter 12.

318

Chapter 10

expand capacity by any small increment, rather than by a lump sum.8 This model is fairly general since it can accommodate various stochastic (Itô) processes and demand functions. In this section we develop a comparable model based on our approach for investing at an optimal target level.9 Suppose again that n firms already active in the market face Cournot quantity (or capacity) competition and contemplate investing in new capacity. Since firms can add capacity by any incremental amount, they are more likely to react to small, favorable shocks by adding capacity incrementally (by an amount dq), rather than waiting longer for a large favorable shock to occur to justify a larger, lump-sum investment. In this incremental capacity investment problem, we make the assumption that symmetric firms invest simultaneously and by the same capacity increment. Consider the general case first. The profit flow firm j receives depends on current total installed industry capacity and the current value of the demand shock. Suppose that the shock follows the general diffusion process in equation (10.1). Let Q be the current level of total industry capacity, where qj denotes firm j’s individual capacity and Q− j the combined capacity of all other firms except firm j. Thus, Q = q j + Q− j. Firm j receives uncertain profit flow πɶ j = πɶ j ( Xɶ t ; Q) = πɶ j ( Xɶ t ; q j , Q− j ). The profit flow is the same for all identical incumbent firms.10 Let I (I ≡ i × dq j) be the investment cost paid by firm j to expand capacity incrementally and i the investment cost per unit of capacity. Once again, a firm’s investment strategy consists in choosing ex ante a future target level XT for the stochastic variable Xɶ t and investing when this target value is first reached at random time Tɶ = inf t ≥ 0⏐ Xɶ t ≥ XT .11 VT [πɶ j ] denotes firm j’s forward value as a perpetuity of profit flows πɶ j ( Xɶ t ; q j , Q− j ) from time Tɶ onward.12 When firm j invests incrementally in new capacity (by small amount dq j), it receives an extra profit flow amounting to πɶ j ′ dq j , with

{

}

8. We here examine expansion options in the context of production capacity problems. Pindyck (1988) and Dixit and Pindyck (1994) take a broader approach, interpreting them in terms of capital stocks, including capacity, human capital (new labor, enhanced intellectual capital), and technological expertise. 9. We do not intend to give a mathematical treatment of this problem here but rather to stress the economic intuition. 10. Capacity is infinitely divisible so that the stage action set is continuous. The profit-flow function is differentiable with respect to capacity. 11. For simplicity, assume that an industry organization is responsible for the industry’s common interests and wields sufficient power to enforce its investment-timing decisions. 12. To avoid use of two subscripts, we drop the use of the subscript j henceforth, with the understanding that these formulas concern an individual firm, not the industry as a whole.

Oligopoly: Simultaneous Investment

319

∂πɶ j ɶ πɶ j ′ = πɶ j ′ ( Xɶ t ; q j , Q− j ) ≡ ( X t ; qj , Q− j ). ∂q j At random time Tɶ when firm j raises its capacity by dq j, it receives the forward perpetuity value vT (Q) = v ( XT ; Q) ≡ VT [πɶ j ′ dq j ]

(10.10)

or net forward value of vT (Q) − I . In equation (10.10), VT [⋅] is a forward perpetuity value operator. Let c0 ( XT ; Q) denote firm j’s time-0 value of investing at target level XT given industry capacity Q. This value for firm j is ∞ ∞ ɶ c0 ( XT ; Q) = Eˆ 0 ⎡ ∫ πɶ j e − rt dt + ∫ ɶ (πɶ j ′ dq j ) e − rt dt − I × e − rT ⎤. T ⎣ 0 ⎦ The first term in the expectation is a perpetuity value that is independent of the investment strategy choice. The equation above can simplify to

c0 ( XT ; Q) = V0 (Q) + B0 (Tɶ ) [vT (Q) − I ].

(10.11)

To determine the optimal investment strategy, we need to determine the optimal joint cutoff value X n* = X*n (Q) that maximizes the expression above. This is found at the point where v*(Q) = v ( X*; Q) satisfies v* (Q) = Π* (Q), I

(10.12)

where Π*(Q) = Π ( X n *; Q), with Π (⋅; Q) being the profitability index (function) given by Π ( XT ; Q ) =

ε B ( XT ) . ε B ( XT ) + ε v ( XT ; Q )

(10.13)

In equation (10.13), the elasticity measures ε B(⋅) and ε v(⋅; Q) are given by XT ε B( XT ) = − BX (Tɶ ) × , B0 (Tɶ )

ε v( XT ; Q) = − vX (Q) ×

XT , v ( XT ; Q )

where BX (⋅) = ∂B ∂XT and vX (Q) = ∂v ∂XT . This expression is analogous to equation (9.38) derived earlier for a monopolist having the option to invest incrementally in capacity. A significant difference exists, however. Here the incremental value change (due to capacity addition), vT , is lower than the incremental value increase in the case of a monopoly

320

Chapter 10

due to the presence of competing firms acting as a negative externality. Given that all firms invest simultaneously, when firm j increases its capacity by a certain amount (dq j), the equilibrium value is not affected solely by firm j’s capacity increase but also by the entire industry’s capacity expansion (dQ) as all rivals follow suit at the same time. Case of Multiplicative Shock Suppose now that the total uncertain profit flow, πɶ j ( Xɶ t ; Q), consists of a deterministic reduced-form component, π j (Q), and an industrywide multiplicative shock that follows an Itô process as per equation (10.1). The stochastic profit flow for firm j is now given by

πɶ j ( Xɶ t ; q j , Q− j ) = Xɶ t π j ( q j , Q− j ). Following Grenadier (2002), assume that incremental production costs are negligible or that firms maximize revenues. The deterministic profit then equals firm j’s quantity, q j, multiplied by the market-clearing price, p (Q):13

π j (Q) = π j (q j , Q− j ) = q j p (Q). Since all firms invest simultaneously and by the same increments, total industry capacity rises by n times this increment when joint investment occurs. In other words, when each firm j increases capacity by dq j, the industry increases total capacity by dQ = n dq j , so Q = nq j at all times. Since the (inverse) demand function is downward sloping, the market price decreases in proportion to dQ. The first-order derivative of firm j’s profit function is π j ′ (Q) = p (Q) + q j p′ (Q). Since all firms are symmetric and q j = Q n, this simplifies to

π j ′ (Q) ≡

dπ j Q (Q) = p(Q) + p′ (Q). dq j n

(10.14)

We can now specify the incremental forward value given in equation (10.10). Since the shock is multiplicative, equation (10.10) obtains as vT ( q j , Q− j ) = [π j ′ dq j ]V[ XT ], where V[ XT ] denotes the perpetuity (expected discounted) value of the shock from time Tɶ going forward and π j ′ is the incremental profit given in equation (10.14). The optimal investment trigger X n* (Q) for each firm 13. We assume the (inverse) market demand function is downward sloping and twice continuously differentiable.

Oligopoly: Simultaneous Investment

321

in an oligopoly in equilibrium with n symmetric firms obtains from equations (10.12) and (10.14) as V [ X n*(Q)] =

Π* i , p (Q) + (Q n) p′ (Q)

(10.15)

where i is the investment cost per unit of capacity. The result above is fairly general and tractable. The value expression in equation (10.15) depends on the industry structure through the number of firms, n. Two polar cases are of interest: (1) monopoly, with n = 1, and (2) perfect competition, where n → ∞. To illustrate, consider the case of geometric Brownian motion based on equation (10.3) where the perpetuity value, V [ X n*(Q)], the expected discount factor, B0 (Tɶ ), and the elasticities readily obtain.14 In this case the perpetuity value is obtained from equation (A.30) in the appendix as V [ X n*(Q)] = X n*(Q) δ , where δ ≡ k − g = r − gˆ . The optimal investment trigger for each of the n oligopolist firms is obtained from equation (10.15) as X n*(Q) =

Π*i , π j ′ (Q) δ

(10.16)

where π j ′ (Q) is given in equation (10.14), and Π* = β1 (β1 − 1) is the profitability index in case of geometric Brownian motion. For a monopolist, being a special case of the expression in (10.16) with n = 1, X 1*(Q) =

Π*I , π ′ (Q) δ

with π ′ (Q) = p(Q) + Qp′(Q). This confirms the investment trigger for the monopolist based on the profitability index investment rule derived in chapter 9, equation (9.40). The preceding general model for oligopoly is in line with previous literature on real options, including the case of monopoly. Example 10.2 Oligopoly with Isoelastic Demand Suppose again that the shock process follows the geometric Brownian motion of equation (10.3), but that now firms face an (inverse) demand function of the constant-elasticity form 14. Pindyck (1988) also assumes a multiplicative industry stock that follows a geometric Brownian motion, that capital has no scrap value, and unit production costs are negligible in a monopoly context. We can thus compare these results.

322

Chapter 10

p (Q) = Q−1 ε p , where −ε p is the price elasticity of demand.15 The first-order derivative of the demand function is p′ (Q) = − (1 ε p ) Q− (1 ε p )− 1 (< 0 ). When this value is substituted into equation (10.14) and π j ′ (Q) into equation (10.16), the optimal investment trigger for each of the n oligopolist firms in the case of isoelastic demand becomes ⎛ nε p ⎞ 1 ε p ⎛ nε p ⎞ 1 ε p ⎛ 1 = ⎜ r + β1σ 2 ⎞⎟ i ⎜ X n*(Q) = Π* i δ ⎜ Q Q . ⎠ ⎝ nε p − 1⎟⎠ ⎝ ⎝ nε p − 1⎟⎠ 2

(10.17)

This confirms the result obtained in Grenadier (2002), equation (21). Under certain conditions for the constant elasticity of demand (see footnote 15), ∂X n* (Q) ∂Q ≥ 0. Therefore the investment trigger increases with existing capacity so that a small firm is more reactive to small shocks and likely to invest earlier than a firm with large capacity. An industry with a high level of existing capacity is less likely to invest in added capacity. Figure 10.2 illustrates the sensitivity of the investment trigger to the number of firms n for a given installed industry capacity Q. It confirms that the investment trigger decreases with the number of firms, so that more competition hastens rather than delays investment.16 This holds under most typical demand functions and stochastic process specifications.17 For a very large number of firms (approximating perfect competition) the option to wait to invest almost vanishes.18 Appendix 10A provides an alternative derivation based on dynamic programming. 10.3

Perfect Competition and Social Optimality

The general model above for analyzing investment in capacity can also be used to obtain the polar case of perfect competition as the number 15. We assume that ε p > 1 n > 0 . 16. The investment trigger in equation (10.17) has the following first-order derivative: ∂ X n* X *(Q) (Q) = − n (< 0 ) . ∂n n ( nε p − 1) This result contrasts with the previous case involving lump sums. There we assumed that firms select their Nash equilibrium actions at each stage. Here the profit is given exogenously by a function continuous in capacity Q. 17. See Grenadier (2002) for derivation of the first-order derivative of the trigger function under other stochastic processes and demand functions. 18. It is still not correct to say that in perfect competition the NPV rule holds. The asymptote to the optimal trigger value curve is the trigger in perfect competition. This is not tantamount to the NPV rule as the profitability index in perfect competition is constant and higher than 1.

Oligopoly: Simultaneous Investment

323

Optimal trigger value 80

60

Monopoly (n = 1)

Duopoly (n = 2)

40

Oligopoly (n = 5)

20

0 1

2

3

4

5

6

Initial installed industry capacity (Q)

(a)

Expected investment time

Optimal trigger value

(years to investment)

80

60

60

Expected investment time

40

40 Optimal trigger

20 20

0

0 1

2

3

4

5

6

7

8

9

10

Number of firms (n)

(b) Figure 10.2 Sensitivity of the optimal investment strategy (trigger) in oligopoly Assume isoelastic demand p(Q) = Q−1 ε p , with constant ε p = 2 . The discount rate is k = 12 percent, growth g = 8 percent, and volatility σ = 20 percent. Investment cost i = 100 (per firm). For panel b, initial installed industry capacity Q = 2.

324

Chapter 10

of firms, n, increases. Consider again the case of geometric Brownian motion as in equation (10.3). The investment trigger of a firm operating in perfect competition can be obtained as the limit of X n*(Q) from equation (10.16) as n approaches infinity: X ∞ *(Q) = lim X n*(Q) = lim n →∞

n →∞

Π* i δ Π*i , = p (Q) + (Q n) p′ (Q) p (Q) δ

(10.18)

where the profitability index Π* is given in equation (10.6). The optimal threshold in perfect competition is thus defined explicitly in terms of the current industry capacity Q, the market-clearing price p (Q) given the existing capacity, the profitability index Π*, the capital investment cost i , and the opportunity cost of waiting δ . Equivalently, in perfect competitive equilibrium, the return on investment satisfies the modified Jorgensionian rule of investment obtained from equation (10.18) as p (Q) X ∞ * (Q) 1 = Π*δ = r + β1σ 2. 2 i This is analogous to equation (9.25) seen in the monopoly case. The preceding general oligopoly model with n firms thus makes it possible to analyze a continuum of oligopolistic structures including the classic polar cases of monopoly and perfect competition. The analysis above relates to a number of key results in the real options literature. The connection between the socially optimal investment threshold and the trigger obtained under perfect competition was first noted by Leahy (1993). Leahy showed that firms’ optimal exercise strategies exhibit some form of myopia in that a perfectly competitive firm should invest at the same time as a myopic (or monopolist) firm ignoring potential future capacity expansions by rivals, considering the market price dynamics as exogenous. A myopic firm behaves as if industrywide production capacity (going forward) remains fixed and that the future price process is solely driven by exogenous shocks, not by rival capacity adjustments. Such myopia makes firms behave as if they are the last entrant setting their optimal investment trigger in isolation. Leahy (1993) compares this investment strategy with the one formulated by a social planner imposed on decentralized firms. Myopic investment policies turn out to be socially optimal. These rather striking results have an important implication: the optimal investment behavior by firms in perfect competition is the same as the one ignoring the effect of rivals’ capacity expansion decisions.

Oligopoly: Simultaneous Investment

325

Grenadier (2000b) additionally analyzes the effect of completion delays (time to build) on firm investment behavior in perfect competition extending the work of Leahy (1993).19 Baldursson and Karatzas (1996) consider capacity expansion decisions involving non-Markovian stochastic processes, establishing the correspondence among perfect competition, myopic strategies, and social optimality for a larger family of processes using a probabilistic approach to stochastic control theory. Baldursson (1998) considers both expansion and downsizing decisions, comparing the results obtained in Nash equilibrium with the choice of a social planner. Some special cases admit closed-form solutions. Dixit (1991) further examines investment behavior in a perfectly competitive market with exogenous price ceilings.20 Conclusion We analyzed here oligopolistic market structures considering the investment decision of firms when they invest all at the same time and by the same lump sum or increment. We first considered a case where firms make lumpy investments and agree to cooperate on joint investment timing. We then extended the analysis to settings where firms can expand capacity by any small or incremental amount. We discussed also the special case of perfect competition, noting the optimality of myopic behavior that ignores rivals’ capacity expansion decisions. Selected References Pindyck (1988) paves the way for the analysis of irreversible investment in incremental units of capital. Leahy (1993) introduces strategic 19. The presence of completion delays or time to build still allows use of a Markov state space provided one uses a new, simplified Markov state that keeps track of both assets in place and capacity under construction, an aggregate index of “committed capacity.” 20. Back and Paulsen (2009) discuss the appropriateness of the Nash or open-loop equilibrium concept employed in models of oligopoly and perfect competition (e.g., Grenadier 2002; Baldursson 1998; Baldursson and Karatzas 1996). Open-loop strategies allow firms to respond to the resolution of uncertainty with respect to the exogenous shock but not to the observed actions by rivals. Optimal open-loop strategies form a Nash equilibrium as part of the open-loop equilibrium. If firms could in effect respond to their rivals’ actions (i.e., formulate closed-loop strategies), the equilibrium strategies obtained by Grenadier (2002) would fail to be subgame perfect. If firms were to pursue such Nash equilibrium open-loop strategies even though they observe their rivals’ actions and can revise their strategies accordingly, they would face the risk of preemption. Formulating the dynamic capacity-expansion problem in closed-loop strategies is rather difficult. Back and Paulsen show that in the limit, the perfect competition outcome derived in Leahy (1993) is part of a perfect closed-loop equilibrium.

326

Chapter 10

considerations to this setting and points out their irrelevance in the context of perfect competition. Grenadier (2002) extends the analysis considering oligopolistic industry structures and completion delays. Grenadier, Steven R. 2002. Option exercise games: An application to the equilibrium investment strategies of firms. Review of Financial Studies 15 (3): 691–721. Leahy, John V. 1993. Investment in competitive equilibrium: The optimality of myopic behavior. Quarterly Journal of Economics 108 (4): 1105–33. Pindyck, Robert S. 1988. Irreversible investment, capacity choice and the value of the firm. American Economic Review 78 (5): 969–85. Appendix 10A: Derivation Based on Dynamic Programming An investment opportunity is commonly treated in the real options literature as being analogous to a perpetual American call option written on a real asset. When the underlying asset is not affected by other investors’ exercise policies, a reasonable assumption in highly competitive markets, we can analyze such investment opportunities by means of contingent-claims analysis. Under uncertainty the underlying profit flow has both an exogenous (demand) and an endogenous (strategic) value component. Here we sketch a derivation based on dynamic programming to obtain the partial differential equation and boundary conditions derived by Grenadier (2002).21 Consider an oligopoly consisting of n symmetric firms competing over a single, nonstorable homogeneous product. The output produced by n firm j is denoted q j . Q = ∑ j = 1 q j is the total industry output, and Q− j denotes the output produced by all other firms except firm j. Output is assumed infinitely divisible. The uncertainty is modeled by Xɶ t , an exogenous industrywide demand shock process. Suppose that the exogenous shock is of the multiplicative form and follows the Itô process of equation (10.1). For simplicity we disregard variable unit production costs obtaining:

πɶ j (Q) = Xɶ t π j (Q) = Xɶ t q j p (Q).

(10.19)

21. The appendix at the end of the book is more precise on how to derive HJB equations and the relevant boundary conditions. We refer more analytically minded readers to this appendix.

Oligopoly: Simultaneous Investment

327

In such a symmetric oligopoly, firms increase output all at the same time in response to a favorable market development, so at all times qj =

Q n

∀j = 1, . . ., n.

Firms at any time may invest in extra capacity, each increasing their output by a small increment dq j. Investing in added capacity involves a capital expenditure by firm j of I ≡ i × dq j, where i ( > 0 ) is the price of one unit of capacity. Firms share a perpetual American call option to invest with underlying asset being the stream of incremental profit flows from increased capacity and exercise price the added capital investment cost I . The optimal exercise strategy can be found in terms of trigger policies. Firm j will exercise its option to invest in added capacity the first moment a specified investment threshold is reached by the stochastic process Xɶ t . This threshold, denoted by X n* (Q), is a function of the number of firms and the total industry capacity, Q. In Nash equilibrium, firm j determines its optimal investment strategy taking other firms’ (optimal) strategies as given. The value of the firm if it follows the Nash equilibrium strategy to invest when X* ≡ X n * (Q) is first reached is denoted as c ≡ c ( X 0 , X *; Q). Over a small time period of length dt , the return to firm j consists of (1) the profit flow from existing capacity πɶ j ( Xɶ t ; Q), analogous to a “dividend yield,” and (2) the expected value increase or “capital gain,” E ( dc ). In equilibrium, the instantaneous total expected return should equal the continuously compounded cost of capital. This leads to the following Bellman equation in continuous time (or HJB equation):

πɶ j ( Xɶ t ; Q) dt + E (dc ) = rcdt.

(10.20)

From Itô’s lemma given in equation (A1.2), the change in the value of firm j over an infinitesimal time period dt is given by 1 dc = ⎛⎜ cX gt + cXX σ t ² ⎞⎟ dt + cX σ t dzt ⎠ ⎝ 2

(10.21)

with gt ≡ g( Xɶ t ), σ t ≡ σ ( Xɶ t ), zt being a standard Brownian motion. The first- and second-order derivatives in equation (10.21) are the derivatives of c ( X 0 , X *; Q) with respect to X 0. When taking expectation, the second right-hand term in equation (10.21) drops out since E [ dzt ] = 0. After substituting the result derived from Itô’s lemma in equation (10.21) into

328

Chapter 10

the HJB equation (10.20), we obtain the following partial differential equation: 1 πɶ j ( Xɶ t ; Q) + cX gt + cXX σ t 2 = rc. 2

(10.22)

This equation describes the option value dynamics under the optimal investment policy. Firm j’s optimal investment strategy is to increase capacity when the shock variable Xɶ t reaches the threshold level X*. The investment opportunity value c must satisfy the equation above, subject to specific boundary conditions, discussed below. Value-Matching Condition If firm j invests in dq j capacity units, its value will be c( X 0 , X*; q j + dq j , Q− j ) − I . When the state variable reaches the threshold X*, firm j is indifferent between investing or keeping open its option to wait, so that c ( X*, X*; q j + dq j , Q− j ) − I = c ( X*, X*; q j , Q− j ), yielding cq j ( X*, X*; q j , Q− j ) = i ,

(10.23)

where cq j (⋅, X*; q j , Q− j ) ≡ ∂c (⋅, X*; q j , Q− j ) ∂q j . If we assume that symmetric firms invest at the same time and by the same increments, at optimal exercise, c ( X*, X*; q j , Q− j + dQ− j ) = c ( X*, X*; q j , Q− j ). This leads to cQ− j ( X*, X*; q j , Q− j ) = 0,

(10.24)

where cQ− j ( X*, X*; q j , Q− j ) ≡ ∂c ( X*, X*; q j , Q− j ) ∂Q− j . Conditions (10.23) and (10.24) are the value matching conditions. Smooth-Pasting Condition At the optimal investment trigger point, the smooth-pasting condition for cq j holds, namely cX ( X*, X*; q j + dq j , Q− j ) = cX ( X*, X*; q j , Q− j ),

Oligopoly: Simultaneous Investment

329

obtaining ∂c X ( X*, X*; qj , Q− j ) = 0. ∂q j

(10.25)

This boundary condition is the super-contact or smooth-pasting condition.22 No general analytical solution exists for partial differential equation (10.22) subject to boundary conditions (10.23), (10.24), and (10.25). Special choices for the stochastic process, however, such as the geometric Brownian motion, make the analysis amenable to closed-form solutions. 22. Rigorously speaking, this condition is the super-contact condition since it is of second order. Sometimes the term smooth-pasting condition is used in this context. Smoothpasting condition is a first-order condition that here applies to the first-order derivative of the value function with respect to the capacity level. Refer to Dumas (1993) for details.

11

Leadership and Early-Mover Advantage

In the previous chapter, models of simultaneous investment among oligopolists were discussed. These models assumed that investment timing was decided collectively (or by a social planner able to enforce its investment-timing decision). Here we consider strategic interactions among firms, so collusive simultaneous investment is ruled out as a Nash equilibrium of the investment game. Investment may occur sequentially, particularly if the leader has distinctive capabilities with sufficiently high competitive advantage that makes it possible to disregard the competitor’s investment decision.1 Section 11.1 discusses the basic deterministic game-theoretic framework of Reinganum (1981a) that shows why sequential, rather than simultaneous, investment emerges as the equilibrium when duopolist firms hold a shared investment option. The following sections extend this basic, deterministic framework allowing for a market evolving stochastically. Sections 11.2 and 11.3 focus on the option to invest in a duopoly and in an oligopoly setting, respectively. Section 11.4 deals with the option to expand production capacity. Such investment-timing games can help explain firm leadership and early-mover advantage. 11.1 A Basic Framework for Sequential Investment in a Duopoly In part I of the book we considered separately the two main underlying theories, real options analysis and game theory. In chapter 9 on the 1. We focus here on models involving open-loop strategies; that is, firms precommit and are not permitted to revise their strategies in view of rivals’ actions over the play of the game. In such cases preemption does not occur. It is simpler first to deal with option games involving no risk of preemption as in open-loop models. Reinganum’s (1981a, b) approach to open-loop equilibrium provides the basic (deterministic) setting underlying option games involving sequential investment, such as Joaquin and Butler (2001). Reinganum (1981a) shows that simultaneous investment does not arise in equilibrium in such games.

332

Chapter 11

investment option of a monopolist, we elaborated how one can obtain analytic solutions for certain real options problems (e.g., the option to invest and the option to expand) when the option holder faces no competition or when the investment opportunity is protected by high structural entry barriers. Although standard real options analysis gives interesting insights into how a monopolist firm should behave, it does not adequately address situations when real options are “shared” among several rivals. Nonetheless, the methodology we mastered along the way to solve such decision-theoretic models will serve us well as a first building block for analyzing such shared option games. The second building block is game theory and industrial organization, with a particular focus on games of timing. These settings are typically modeled in continuous time, so it is useful to develop some understanding of game theory in continuous time before dealing with uncertainty in integrated realoptions and game-theoretic models. Two key articles that have had a major impact on this field are those by Reinganum (1981a) and Fudenberg and Tirole (1985). Both deal with investment timing when there is no stochastic uncertainty regarding the payoffs firms will receive upon acting. Although these authors explicitly deal with technology adoption, their results and insights are tractable and applicable to other classes of problems, such as timing of market entry or of (lumpy) capacity expansion. We extend these results within the real options framework and apply them to provide insights into the strategic investment challenge under market uncertainty.2 In contrast to Fudenberg and Tirole (1985), Reinganum’s (1981a) model does not require the use of mixed-strategy equilibria in continuous time and is less technical as a starting point. We present it first and discuss Fudenberg and Tirole’s model in the upcoming chapter. Assume that firms’ investment strategy consists in choosing an investment date a priori and committing to it. Since for now we assume that there is no stochastic uncertainty concerning the underlying market development, assuming a fixed investment schedule is reasonable. Suppose that two identical firms (firms i and j) are active in the market (at time t = 0) and behave rationally by selecting profit-maximizing outputs. Both firms have the option to make an investment (e.g., in capacity expansion) and increase their profits accordingly (existing market model). Supposing that we deal with capacity expansion, it is clear that capacity expansion by one of the firms is made at the expense of its rival: 2. For a treatment of option games with a focus on technology adoption, see Huisman (2001).

Leadership and Early-Mover Advantage

333

as price declines, the rival firm whose capacity has remained fixed loses revenues, while the expanding firm may earn higher profits due to the combined effect of increased quantity (market share) and lower price.3 The problem for the option holder lies in selecting the right time to stop waiting and initiate the expansion project. The duopolists have an infinite planning horizon and can choose to invest at anytime. Firms face the same interest rate r.4 Assume a constant investment cost I . At the beginning of the game, each firm earns a profit denoted π 0, growing (compoundly) over time at a constant growth rate g (percent) per unit time, with g < r. There is an incentive to delay, making the investment at a time when the market is larger and more profitable.5 A differentiating feature between Reinganum’s (1981a) model and our option games approach is that Reinganum does not consider uncertainty relating to the underlying process. In Reinganum’s framework the investment time is deterministic. The notion of investment strategy is thus somewhat different: rather than selecting an investment threshold XT and investing at the (random) time Tɶ when this trigger has first been reached, the problem here simplifies to selecting a certain time T ( ≥ t0 ) at which the firm invests. This brings out a key difference between deterministic game-theoretic models of timing (e.g., Reinganum 1981a; Fudenberg and Tirole 1985) and option models of investment timing under uncertainty (e.g., McDonald and Siegel 1986; Dixit and Pindyck 1994). In the deterministic game-theoretic models the investment strategy directly relates to the investment timing, whereas in option models of investment under uncertainty the investment trigger is a strategic choice parameter that, in turn, defines the random time of investment. For option games a necessary useful step is to consider that players select an investment trigger, rather than a predetermined investment timing directly. Consider first the deterministic case. The investment strategy for firm i here consists in choosing a time Ti (Ti ≥ t0) at which to invest and incur the sunk investment cost I . By contrast to the previous chapter where we assumed competitors invest (collusively) simultaneously, here the two firms are symmetric but may choose distinct investment times (Ti and 3. This holds if there are negative externalities from capacity expansion or, equivalently, if the inverse demand function is downward sloping. 4. Assuming no arbitrage opportunities in a complete market, the appropriate discount rate is the risk-free interest rate, r. 5. Reinganum (1981a) models the incentive to delay investment in a setting involving a constant profit flow and an investment cost decreasing over time. Here these assumptions are reversed: profits are growing with time and investment cost is constant whether investment occurs today or in the future, independently of the discounting effect. The net effect on the incentive to wait is analogous.

334

Chapter 11

Tj). When the first of the two thresholds (min {Ti , Tj }) is reached, at least one of the firms invests. Three industry structures may occur: 1. No one invests in additional capacity. When no firm has yet invested in additional capacity, each firm already earns at time t a profit amounting to π 0 exp ( gt ), with π 0 ≥ 0. This industry structure occurs for t such that t0 ≤ t ≤ min {Ti , Tj }. 2. Only one firm invests. If the lowest investment time threshold is reached, but not yet the second one, there emerges an industry structure involving a “leader” and a “follower” wherein only one firm (the leader) expands capacity. The leader earns a higher profit (π L > π 0) as it benefits from capacity expansion, the follower being unfavorably affected by its rival’s investment through negative externalities, such as price decrease in Cournot quantity competition. The follower receives a lower profit π F ( < π 0 ).6 There are two types of possible leader–follower industry structures: for Ti ≤ t ≤ Tj, firm i is the leader and firm j the follower; for Tj ≤ t ≤ Ti, the roles are reversed. 3. Both firms invest (expand capacity). If both time thresholds are exceeded, both firms invest in additional production capacity. For t ≥ max {Ti , Tj } ≥ 0 both firms earn at time t a profit equal to π C e gt, with π L > π C > π F. The leader suffers somewhat from a capacity expansion by its rival.7 Table 11.1 summarizes the profits for the duopolist firms depending on the time elapsed. Note that the profit firms earn depends on their rival’s investment strategy. The value (payoff) is consequently affected by strategic interactions. The optimal investment strategies must thus be part of an industry Nash equilibrium. Suppose further that there is a higher incentive to invest as a leader than as a follower; that is, the value increment from leadership (π L − π 0 ) is larger than the value increment from followership (π C − π F ). Suppose as well that no firm has an incentive to invest right at the outset.8 Consider next the time thresholds of the leader (TL) and the follower (TF ). Which player actually becomes the leader or the follower is discussed later. Suppose a weak ordering of firm roles with firm i being 6. These profit flows substitute for the initial profit flow π 0. They do not represent the additional profit from the new investment, but the overall profit after the new investment. In case of capacity expansion, π L is the total profit of the leader stemming from the old and the new capacities, and π F from the old capacity of the follower. L stands for leader and F for follower. Note that the notions of leader and follower here are different than in a Stackelberg game setting. 7. The subscript C stands∞for competition. 8. To ensure this, we set ∫ 0 π 0 e −(r − g )t dt − I < 0 .

Leadership and Early-Mover Advantage

335

Table 11.1 Profits for capacity-expanding duopolists depending on timing Profits Time

Industry structure

Firm i

Firm j

t ≤ min {Ti , Tj }

No one invests

π 0 e gt π L e gt π F e gt π C e gt

π 0 e gt π F e gt π L e gt π C e gt

Ti ≤ t ≤ Tj Tj ≤ t ≤ Ti t ≥ max {Ti , Tj }

Only one invests Both firms invest

Note: π L > π C > π F and π L > π 0 > π F .

the leader. This occurs if Ti ≤ Tj. In the “history” of the game (with t0 = 0 < Ti = TL), the leader will go through three different time stages characterized by distinct profit flows.9 When the market is far from being sufficiently profitable for a new investment to occur (for t ≤ min {Ti , Tj }), the leader earns π 0 exp ( gt ) at time t. The present value of the deterministic profits earned before the leader invests in additional TL TL capacity at time TL is ∫ π 0 e gt e − rt dt = ∫ π 0 e −δ t dt with δ ≡ r − g ( > 0 ) being 0 0 some form of dividend yield or opportunity cost of waiting. In the following stage, when Ti ≤ t ≤ Tj, firm i gains a leader status earning π L e gt at time t. The value of being (during that period) the stand-alone TF investor in new production capacity is ∫ π L e −δ t dt. In the third stage TL (t ≥ max {Ti , Tj } or t ≥ Tj ≥ Ti), firm i competes head-on with firm j, earning duopoly profits π C e gt.10 The value for firm i in this stage where ∞ both duopolists have expanded capacity is ∫ π C e −δ t dt. The leader incurs TF at time of investment (TL) the given investment outlay I , which must be discounted back at the present time.11 The present value accruing to the leading firm i (in case t0 < TL) is the sum of the values in each of the three stages: 9. If firms invest at the same time, meaning Ti = Tj , the second region (Ti , Tj ) reduces to a null set. 10. The leader does not have a sustainable competitive advantage (first-mover advantage) through being the sole investor in the previous stage; that is, it earns exactly the same profit as its rival in the third stage. Whether a firm has gained a sustainable advantage by being the first investor is a tricky issue from a game-theoretic viewpoint. This question is central to understanding the Stackelberg model of duopoly. In the competition stage, the Cournot outcome for the reduced-form profit may be more realistic than the result obtained under the Stackelberg leader-follower model since the latter may be timeinconsistent (see section 8.2 in Tirole 1988). This inconsistency justifies the assumption of the model by Reinganum (1981a). 11. Since there is no uncertainty concerning the market development, there is no need to use expectation in the value expression. Models involving uncertainty differ on that dimension.

336

Li (TL , TF ) = ∫

Chapter 11

TL 0

π 0 e −δ t dt + ∫

TF TL

π L e −δ t dt + ∫

∞ TF

π C e −δ t dt − I e− rTL .

(11.1)

For the follower there are also three distinguished stages. Table 11.1 also identifies those stages. The value of firm j as a follower is given by Fj (TL , TF ) = ∫

TL 0

π 0 e −δ t dt + ∫

TF TL

π F e −δ t dt + ∫

∞ TF

π C e −δ t dt − I e− rTF .

(11.2)

By symmetry, firm j (firm i)’s value as leader (follower) is identical. As before, the value function of the firm is strictly concave in its own action due to the contrarian effects of growth (g > 0) and discounting (r > 0). The condition δ ≡ r − g > 0 ensures that the investment time is finite. The optimal investment time is found at the point where the first-order derivative of the value function equals zero. In other words, TL* ( > 0 ) is such that ∂Li (TL*, TF ) = − (π L − π 0 ) e−δ TL* + rI e− r TL* = 0 , ∂TL or ⎛ π L − π 0 ⎞ e gTL* = r , ⎟ ⎜⎝ I ⎠

(11.3)

with TL* = ln (rI (π L − π 0 )) g. This is the Jorgensonian rule for expanding investment under certainty. At the optimal time of investment, the project’s excess return on investment equals the cost of capital, or equivalently one should invest in additional capacity the first time the project’s excess return on investment exceeds the cost of capital, r. Alternatively, one can obtain from equation (11.3) the optimal time for the leader TL* based on the following profitability index investment rule: ⎛ VL − V0 ⎞ e gTL* = b , ⎟ ⎜⎝ I ⎠ b−1

(11.4)

where b ≡ r g . The optimal investment time for the follower can be determined and interpreted similarly: ⎛ π C − π F ⎞ e gTF * = r ⎟⎠ ⎜⎝ I

V − VF ⎞ gTF * b . or ⎛⎜ C = ⎟⎠ e ⎝ I b−1

(11.5)

In this case as well, it is optimal for the follower to invest at the time when the excess return on investment equals the interest rate, r. TL* and TF * are not explicit (reaction) functions of the rival’s investment time.

Leadership and Early-Mover Advantage

337

Consider also a third time threshold TP * at which the two firms are indifferent between being a leader investing at TP * or a follower with the rival investing at TP *:12 Li (TP *, TF *) = Fi (TP *, TF *).

(11.6)

Before this time, that is for Ti < TP *, a firm is better off delaying investment than investing right now since Li (Ti , TF *) < Fi (Ti , TF *). After this indifference point (Ti > TP *) there is an incentive to become leader as Li (Ti , TF *) > Fi (Ti , TF *). It also holds that TP * < TL* < TF *.13 To determine the Nash equilibria, we need to consider the best-reply functions for firms i and j. Suppose that firm j selects beforehand an investment time Tj strictly higher than TP *. During the period (TP *, Tj ) there exists a first-mover advantage for firm i, in that firm i is better off investing now as a leader than delaying investment further; in such a case firm i optimally chooses to invest at time TL* as per equation (11.4) above. If, however, firm j were to invest early, before time TP * (Tj < TP *), firm i will wait, optimally investing as a follower at time TF * as given in equation (11.5). If firm j chooses to invest at exactly time TP *, firm i maximizes its value by either selecting TL* as in equation (11.4) or TF * as in equation (11.5). Thus the reaction function of firm i with respect to the investment time chosen by its rival, firm j, is given by if Tj < TP *, ⎧TF * ⎪ Ri (Tj ) ≡ Ti * (Tj ) = ⎨{TL*, TF *} if Tj = TP *, ⎪T * if Tj > TP *. ⎩ L The reaction functions for firms i and j are illustrated in figure 11.1, with firm j’s best-reply function being obtained symmetrically. As seen in figure 11.1, there are two Nash equilibria in pure strategies. The best-reply functions intersect at two distinct points: (TL*, TF *) and (TF *, TL*). The first Nash equilibrium is that firm i invests as a leader and firm j as a follower, namely (TL *, TF *) *. The second one is that firm j takes the lead with firm i following suit, namely (TF *, TL*) *. These two 12. The subscript P stands for preemption. This point is discussed in detail in the following chapter. 13. Set f (T ) ≡ Li(T , TF *) − Fi(T , TF *) . f (⋅) is continuous and strictly increasing on (t0 , TF *) . By assumption, Li(t0 , TF *) < Fi( t0 , TF *) or f (t0 ) < 0 . As noted, the leader enjoys a firstmover advantage with Li(TL*, TF *) > Fi(TL*, TF *) or f (TL*) > 0. By strict monotonicity, the root of f (⋅)—which, according to equation (11.6), corresponds to the (preemption) point TP *—obtains to be unique with t0 < TP* < TL*.

338

Chapter 11

Firm i’s strategy Ti Firm i ’s best-reply function

Ri (Tj) TF *

(TF*, TL*)*

TP* Firm j ’s best-reply function

Ri (Ti)

TL*

(TL*, TF*)*

TL*

Figure 11.1 Best-reply functions for symmetric firms

TP*

i

and

TF *

Firm j’s strategy Tj

j

Nash equilibria share a common feature: they both involve sequential investment where one firm invests as leader and the second one as follower. Such a sequential ordering is coined diffusion by Reinganum (1981a) who applied this framework to the analysis of the adoption of new technologies by competing firms. Time TL* is determined by the modified Jorgensonian rule of investment in equation (11.4), whereas TF * satisfies equation (11.5). Which of the two Nash equilibria in pure strategies is the most reasonable to expect is not specified a priori. Note the difference between this model and the situation in chapter 10. Previously we assumed that symmetric firms invest simultaneously. Here, in contrast, given the pure-strategy equilibria for symmetric firms—based on the model assumptions of Reinganum (1981a)—simultaneous investment never happens due to strategic interaction. Simultaneous investment is not a pure-strategy Nash equilibrium in a duopoly with identical firms and can only be sustained by collusive behavior in the marketplace. Another interesting outcome is that the values the firms receive are decreasing in the order of entry, meaning that the leader receives a higher

Leadership and Early-Mover Advantage

339

value than the second entrant in equilibrium.14 The leader thus enjoys a first-mover advantage stemming from the monopoly “rents” it earns before the rival’s expansion. As suggested by Reinganum (1981a), this deterministic analysis can be generalized in three directions: It can be extended to an oligopoly consisting of n firms (rather than two firms in a duopoly). Such a situation is discussed by Reinganum (1981b). Although such an analysis is more involved, the end result is essentially the same: there are several Nash equilibria in pure strategies but none involves simultaneous investment. Each Nash equilibrium involves a sequential ordering of firms’ investment.15

•

•

It can be extended to asymmetric firms, as in Flaherty (1977).

It can be extended by considering stochastically evolving profit values (πɶ 0, πɶ L, πɶ F, and πɶ C).

•

In the following sections we discuss these extensions and provide refinements allowing for uncertainty in market development. We also extend the analysis by assuming asymmetric firms to allow for a more natural (focal-point) ordering of firms’ investment timing.16 Finally, we extend the previous analysis to the case of a large oligopoly facing uncertain market development. 11.2

Duopoly with Sequential Investment under Uncertainty

In this section we consider the option to invest in a duopoly under uncertainty (with stochastic profits) and examine the industry dynamics when one of the firms has a substantial competitive (e.g., cost) advantage justifying a natural leader–follower industry structure. Joaquin and Butler (2000) follow a similar analysis based on a specific stochastic process—geometric Brownian motion—assuming that (certain) 14. By definition of the Nash equilibrium, L (TL*, TF *) ≥ L (TF *, TF *) . Since firms are symmetric, L (TF *, TF *) = F (TF *, TF *). Since the follower’s value is increasing in its rival’s entry time, F (TF *, TF *) > F (TL*, TF *), and so the value decreases in the order of entry L (TL*, TF *) > F (TL*, TF *) . 15. Within a sequence of n firm investments, roles are interchangeable. Thus there are n! pure-strategy Nash equilibria. 16. If Pareto optimality is defined from the firms’ perspective, the Pareto-optimal sequence corresponds to the open-loop sequence with the advantaged firm investing first. In chapter 12 we show that for large firm asymmetry, the outcome of the Pareto-optimal investment sequence corresponds to the outcome of the perfect equilibrium in closed-loop strategies. To avoid technical discussions early on, we assume for now that the investment sequence with the advantaged firm investing first is more “natural” or “focal” as being Pareto optimal. As shown later, this assumption is reasonable when asymmetry is substantial, namely if the difference among firms is higher than a certain threshold.

340

Chapter 11

profits are Cournot outcomes in simultaneous quantity competition. Their model is therefore a special case of the one we elaborate herein. The approach herein is general and tractable to different stochastic processes and market demand functions. Consider two firms in a duopoly sharing an option to export to a developing country (where they are not yet present) by expanding the size of their domestic production plant (new market model). The main source of uncertainty here is the exchange rate at which firms repatriate profits from the foreign country into domestic currency. Suppose that due to a cost advantage one of the firms (firm i) is more likely to become a leader. When the market is just large enough for only one player, firm i due to its competitive advantage is the first (only) one to invest; the follower (firm j) will wait for the market to develop further before deciding whether to also invest. When the market is too small, no one invests and the two firms earn no profits. At the market’s early stage, only one firm (the leader) can invest and earn positive economic profits. Firm i invests, earning monopoly rents, while the follower waits, receiving nothing for the time being. When the market becomes more mature (i.e., large enough to accommodate both firms), the follower also enters and both compete in the foreign marketplace, earning Cournot duopoly profits. Table 11.2 summarizes the profits in the resulting three stages. Uncertainty enters as a multiplicative stochastic exogenous shock (the exchange rate) Xɶ t , which follows a (time-homogeneous) Itô process according to the stochastic differential equation dXɶ t = g ( Xɶ t ) dt + σ ( Xɶ t ) dzt.

(11.7)

The value of the process at the beginning (i.e., at time t0 = 0) is denoted by X 0. We assume as before that stochastic profits can be decomposed into two components, the multiplicative shock Xɶ t and a deterministic profit component, π L or π C. We proceed by considering the follower’s Table 11.2 Profits for duopolist firms in three development stages Stochastic profits

Certain profits

Time

Industry structure

Firm i

Firm j

Firm i

Firm j

t ≤ TɶL TɶL ≤ t ≤ TɶF

No one invests Both firms invest

0 πɶ Cj

0 π Li π Ci

0

t ≥ TɶF

0 πɶ Li πɶ Ci

0

Only one invests (firm i leads)

0

π Cj

Leadership and Early-Mover Advantage

341

optimal strategy first as it does not depend on the leader’s market-entry choice. We analyze the leader’s strategy formulation subsequently. Investment Decision of the Second Entrant (Follower) The follower contemplates investing in the foreign country when the trigger exchange rate it has chosen, XT , is first reached at random time Tɶ ≡ inf t ≥ t0 ⏐ Xɶ t ≥ XT . The forward net present value at time Tɶ is NPVT ≡ VT − I with VT ≡ V ( XT ) . For X 0 ≤ XT , the expected present value accruing to the follower Fj (⋅), given that it selects the strategy to invest at time Tɶ , is

{

}

Fj ( XT ) = NPVT B0 (Tɶ ) = (VT − I ) B0 (Tɶ ).

(11.8)

The follower may either invest immediately (at time t0 = 0) and get the static NPV0 ≡ V0 − I or defer the investment until time Tɶ , receiving today Fj ( XT ) as per (11.8) instead. The firm has an option to postpone investment and will optimally invest when the investment trigger X F * that maximizes the follower’s value Fj (⋅) in equation (11.8) is first reached, provided this critical target value has not previously been reached (i.e., provided that X 0 ≤ X F *). The optimal trigger X F * for the follower satisfies the first-order condition V* = Π ( X F *), I

(11.9)

where, as before, Π ( XT ) measures the profitability of the strategy that prescribes to invest at a specified trigger level XT . This measure is given by Π ( XT ) =

ε B ( XT ) . ε B ( XT ) + ε V ( XT )

(11.10)

Π ( X F *) is the profitability required when the firm formulates the optimal investment strategy. Equation (11.9) above shows that the follower can choose its trigger myopically, meaning its investment trigger is exactly the same as the one it would have chosen as a monopolistic option holder earning πɶ Fj ( Xɶ t ) in perpetuity from time TF * on. The value of the option for follower firm j given that it follows the optimal investment strategy given in equation (11.9) reads therefore Fj ( X F *) = [Π ( X F *) − 1] IB0 (TɶF *),

{

}

(11.11)

where TɶF * = inf t ≥ 0⏐ Xɶ t ≥ X F * and Π ( X F *) − 1 is the excess profitability index (higher than zero). This is the same value expression for the

342

Chapter 11

monopolist in equation (9.20). The follower thus invests as if it were a monopolist ignoring rivals. Investment Decision of the First Entrant (Leader) The value of the leader, however, is linked to the follower’s decision concerning investment timing. When the incumbency profits are sufficiently high for the leader to profitably enter (but not large enough for the follower to invest), it will earn temporary monopoly profits until the follower also enters. If X 0 < X L, the history of the market for the leader is characterized by three sequential stages. In the first stage (t0 ≤ t ≤ TɶL), the leader earns no profit (while no one invests). ɶL ≤ t ≤ TɶF *), the leader invests at time TɶL and • In the second stage (T earns temporary monopoly profits until the follower (firm j) also invests at future (random) optimal time TɶF *. ɶF *), the leader • Once the follower has also entered the market (t ≥ T •

receives reduced-form duopoly profits. There is a discontinuity in the leader’s profit function at the time the follower’s investment takes place: before TɶF * the leader receives monopoly profits but just after the rival enters the leader’s profit jumps down to competitive duopoly rents (πɶ Ci (⋅) < πɶ Li (⋅)).17 The leader’s value—given that it invests at time TɶL (and that t0 < TɶL)—is the sum of the values earned in these stages. It is given by18 TF * ∞ ɶ Li ( X L , X F *) = Eˆ 0 ⎡ ∫ ɶ πɶ Li ( Xɶ t ) e − rt dt + ∫ πɶ Ci ( Xɶ t ) e − rt dt − I e− rTL ⎤. TF * ⎣⎢ TL ⎦⎥ ɶ

(11.12)

In the first stage [ t0 , TL ), no one invests and the leader earns zero profit; in the second stage [TL, TF *), the leader receives stochastic monopoly rents πɶ Li (discounted back to present time t0 = 0); in the third stage [TF *, ∞ ), profits are eroded due to the arrival of competition with profits reduced to duopoly competition rents πɶ Ci. Alternatively, one can reformulate equation (11.12) as Li ( X L , X F *) = f1(TɶL ) − f2 (TɶF *)

(11.12′)

where the functions f1(⋅) and f2(⋅) are given by 17. There is no sustainable first-mover competitive advantage for the first entrant. 18. Eˆ 0[⋅] stands for the risk-neutral expectation conditional on the information available at time t = 0 .

Leadership and Early-Mover Advantage

343

∞ f1(TɶL ) = Eˆ 0 ⎡ ∫ ɶ πɶ Li ( Xɶ t ) e − rt dt ⎤ − I B0 (TɶL ) = [VTɶL (πɶ Li ) − I ] B0 (TɶL ) ⎣ TL ⎦ ɶ = NPVTɶL B0 (TL ) ∞ f2 (TɶF *) = Eˆ 0 ⎡ ∫ ɶ (πɶ Li − πɶ Ci ) ( Xɶ t ) e − rt dt ⎤ = [VTF * (πɶ Li − πɶ Ci )] B0 (TɶF *). ⎣ TF * ⎦ Note that function f2(⋅) drops out in the first-order derivative of Li (⋅, X F *). In other words, the profit in the second period (duopoly profit) does not impact the leader’s optimal investment strategy X L* or, equivalently, the trigger is selected myopically. The first-order condition leads to the following profitability-index formula:

V * = Π ( X L*) I , with Π (⋅) defined as in equation (11.10). The Nash equilibrium value (expanded NPV) of the leader is thus Li ( X L*, XC *) = [VTL * (πɶ Li ) − I ] B0 (TɶL*) − VTF *(πɶ Li − πɶ Ci ) B0 (TɶF *)

(11.13)

for X 0 < X L*. The option value for the first entrant (leader) consists of two terms. The first term, [VTL * (πɶ L ) − I ] B0 (TɶL*), is the standard deferral option value for a monopolist since the profit as leader in the second time period (TɶL * ≤ t ≤ TɶF *) equals the monopoly rent. The forward NPV, VTL * (πɶ L ) − I , received at random time TɶL* is discounted back with use of the expected discount factor linked to the optimal investment time TɶL* of the leader, B0 (TɶL*). This expanded NPV for the leader (as monopolist) is, however, eroded due to the competitive (follower’s) arrival.19 Two effects must be accounted for. On one hand, the first entrant loses its monopoly profit stream in perpetuity when the second entrant decides to invest (at random time TɶF *); on the other hand, from that time on, it earns competitive duopoly profits in perpetuity. This second (competitive-erosion) component, VTF * (πɶ L − πɶ C ), is discounted back with use of the expected discount factor B0 (TɶF *), which depends on when the second entrant invests—not on the investment decision of the first entrant. This is the reason why the second component “disappears” when one computes the first-order derivative of the leader’s function and why the investment trigger is selected myopically in this setup. Figure 11.2 presents the market structure regions depending on the value reached by the exogenous stochastic demand shock Xɶ t . Consider an application from the energy sector. Suppose that two European electric utilities, Enel of Italy and Eon of Germany, 19. Note that VTF *(πɶ L − πɶ C ) is positive as πɶ L > πɶ C.

344

Chapter 11

No investment

−∞

Monopoly

XL*

Duopoly

XF *

∞

Figure 11.2 Exogenous demand regions and market structure Firms are operating under the indicated market structures once the exogenous demand shock variable ( Xɶ t ) enters one of these demand regions.

Table 11.3 Profits in asymmetric Cournot duopoly Industry structure

Monopoly profits

Leader

π Li =

Follower

NA

(a − ci ) ² 4b

Duopoly profits

π Ci = π Cj

(a − 2ci + c j )2 9b

(a − 2c j + ci )2 = 9b

contemplate investing in an eastern European market, sharing an option to invest. The local currency has had an annual drift of g percent and is expected to move forward similarly, though realized future growth will likely fluctuate around this long-term average depending on the volatility σ . Suppose that the exchange rate follows geometric Brownian motion of the form dXɶ t = ( gXɶ t ) dt + (σ Xɶ t ) dzt,

(11.14)

where zt is a standard Brownian motion.20 The (deterministic) profit component in local currency is driven by the (inverse) linear demand function p (Q) = a − bQ,

(11.15)

where Q is total industry output. Each firm’s strategy once in the market consists in selecting its output given its rival’s strategy. Equilibrium (reduced-form) profits are Cournot duopoly outcomes as obtained previously in chapter 3. They are summarized in table 11.3. The case above corresponds to a continuous-time extension of the model of Smit and Trigeorgis (1997) by Joaquin and Butler (2000), which provides valuable insights into real-world investment games. Here the sequence of investment is driven by cost asymmetry in the industry. If 20. For technical reasons (finiteness of the first-hitting time), we assume that gˆ ≡ r − δ > (σ 2 2).

Leadership and Early-Mover Advantage

345

one of the two firms invests first, it will enter at the time when the leader’s optimal investment threshold is first reached. The first entrant will temporarily earn monopoly profits up to the random time when the follower’s trigger value X F * is reached. Thereafter, as both firms are operating in the market, they will earn asymmetric Cournot profits. The first entrant will earn Cournot duopoly rents, just as its rival. Such sequential investment is characteristic of an industry with asymmetric costs. In such a sequential game there exist two Nash equilibria in pure strategies: (1) the low-cost firm enters first and the high-cost firm second and (2) the high-cost firm enters first and the low-cost firm second. The reader can intuit which of the two equilibria is likely to occur. As discussed in the discrete-time analysis in the Cournot setup, use of Schelling’s (1960) focal-point argument suggests a more likely Nash equilibrium solution for this sequential investment game. The two competing firms may agree that the Pareto-optimal equilibrium is a focal point of the game. Social optimality is reached when the low-cost firm invests first since higher cumulative profits are reached. A social planner would have an incentive to promote this equilibrium and enforce it on the two firms.21 Suppose that here firm i has a substantial cost advantage ensuring it the position of cost leader. Firm j is the follower. In this case the equilibrium value for firm j as follower from (11.11) is ⎛πj ⎞⎛ X ⎞ Fj ( X F *) = ⎜ C X F * − I ⎟ ⎜ 0 ⎟ ⎝ δ ⎠ ⎝ X F *⎠

β1

β1

⎛ X ⎞ = (Π* −1) I ⎜ 0 ⎟ , ⎝ X F *⎠

(11.16)

where β1 (in the risk-neutral case) is given by

β1 = −

2 αˆ r ⎛ αˆ ⎞ + +2 2 ⎜ ⎟ 2 2⎠ ⎝ σ σ σ

( > 1)

(11.17)

with αˆ = gˆ − (σ 2 2), gˆ = r − δ , Π* = β1 ( β1 − 1) is the profitability index and the investment trigger X F * satisfies the modified Jorgensonian rule of investment, 9 bδ ⎛ β ⎞ XF * = ⎜ 1 ⎟ I , or ⎝ β 1 − 1 ⎠ ( a − 2 c H + cL ) 2

π Cj X F * 1 = r + β1σ 2 . 2 I

(11.18)

The equilibrium value for the leader (firm i) from (11.13) is (for X 0 < X L*); then 21. Later in chapter 12 it is shown that this focal-point equilibrium is obtained as a perfect equilibrium in mixed-strategies if the cost advantage of the low-cost firm is sufficiently high.

346

Chapter 11

⎛πi ⎞⎛ X ⎞ Li ( X L*, X F *) = ⎜ L X L* − I ⎟ ⎜ 0 ⎟ ⎝ δ ⎠ ⎝ X L* ⎠

β1

β1

X *⎤ ⎛ X ⎞ − ⎡⎢(π Li − π Ci ) F ⎥ ⎜ 0 ⎟ , δ ⎦ ⎝ X F *⎠ ⎣ (11.19)

with δ ≡ k − g = r − gˆ ( > 0 ) being a form of convenience or dividend yield, and X L* such that ⎛ β ⎞ 4 bδ X L* = ⎜ 1 ⎟ I , or ⎝ β1 − 1⎠ ( a − cL )2

π Lj X L* 1 = r + β1σ 2 . 2 I

(11.20)

The two equations above confirm the results obtained by Joaquin and Butler (2000). They are analogous to equation (9.25). Example 11.1 Duopoly Consider two firms competing in quantity. The linear (inverse) demand in equation (11.15) is characterized by a = 50 and b = 5. The duopolists face cost asymmetry: the unit cost for the low-cost producer is cL = 18, and cH = 20 for the high-cost firm. The investment cost (exercise price) I amounts to &500 m. The risk-free rate is r = 6 percent. The opportunity cost of delaying or dividend yield is δ = 4 percent. The risk-neutral drift for the foreign country is thus gˆ ≡ r − δ = 0.02. The volatility of the exchange rate is σ = 0.10 and αˆ = gˆ − (σ 2 2) = 0.015. Given these parameters, the elasticity of the investment option based on equation (11.17) is

β1 = −

0.015 0.015 ⎞ 2 (0.06 ) ≈ 2.27. + + ⎛⎜ ⎟ 2 2 ⎝ 0.10 ⎠ 0.10 0.10 2 2

The investment trigger of the second-entrant obtained from equation (11.18) is 2.27 ⎞ 9 × 5 × 0.04 X F * = ⎛⎜ × 500 = 2.05. ⎝ 2.27 − 1⎟⎠ ( 50 − 2 × 20 + 18 )2 From equation (11.20), the investment trigger for the low-cost firm is X L * = 0.70.22 By substituting these values into the expanded-NPV expression for the follower and the leader in equations (11.16) and (11.19), one gets the deferral option value of the two firms as a function of the initial value X0. Figure 11.3 shows simulated results for this example. The y-axis represents the value functions of the leader and the follower 22. We consider here the focal-point equilibrium where the first entrant is the low-cost firm and the high-cost firm is the second entrant. Readers may derive the second Nash equilibrium as an exercise.

Leadership and Early-Mover Advantage

347

Project value (in millions of euros)

1,250

1,000 E-NPV leader

750

500 E-NPV follower

250 NPV follower

0 0.0

0.5

1.0

XL*

1.5

2.0

2.5

XF* Initial value X0

Figure 11.3 Leader and follower values in asymmetric duopoly The deterministic linear demand and profit function parameters are a = 50, b = 5, cL = 18 , and cH = 20 . The parameters of the stochastic (risk-neutral) process (GBM) are r = 6 percent, δ = 4 percent, and σ = 10 percent. Investment cost is I = 500.

in equilibrium, with the low-cost firm being the leader and the high-cost firm the follower, for the relevant exchange-rate regions. A kink in the leader’s value at the follower’s investment trigger X F * = 2.05 is readily seen. The shape of the value function for the follower is similar to the one in the case of a monopolist having a deferral call option; this stems from the myopic stance of the follower. The follower’s NPV is tangent to this curve at XF*. When the cost differential between the two firms gets very high (approaching infinite), the result of the previous model reduces to McDonald and Siegel’s (1986) result for the investment-timing problem of a monopolist. In this case the high-cost firm never enters and the low-cost firm (as first entrant) enjoys monopoly profits in perpetuity. The preceding model of sequential investment may be realistic than the previous models involving simultaneous investment—which assumed simultaneous investment is the industry equilibrium. The asymmetric model presented here stresses the importance of asymmetric variable

348

Chapter 11

costs and cost leadership in investment timing games.23 In addition to the traditional, profit-flow benefits a firm obtains from following a generic cost-leadership strategy, the asymmetric nature of the cost advantage has a critical impact on the firms’ future behavior and investment strategies and is therefore also beneficial from a dynamic perspective. A delayed competitive entry may result due to a substantial cost advantage enjoyed by the leader, resulting in higher firm value. 11.3

Oligopoly with Sequential Investment under Uncertainty

In this section we generalize the insights from the previous duopoly model involving sequential investment under uncertainty to the case of an oligopoly consisting of n active firms. Here the firms’ investment roles are given exogenously and the investment game is characterized by a sequence of investments where roles are pre-assigned to firms. This “natural” sequential investment may result, for instance, from “substantial” cost asymmetries among firms active in the industry.24 The uncertain profit accruing to firm i (i = 1, . . ., n ) is denoted by πɶ i ( m), where m is the number of firms which have already invested. The initial value of the stochastic process is X 0. The risk-free interest rate is r and optionholding firms have the same beliefs about the underlying exogenous shock evolution (drift and diffusion terms). If firm i has already invested, its profit will drop if a new rival firm enters the market, causing discontinuity in the profit flow. Formally, πɶ i ( m) is such that πɶ i ( m + 1) ≤ πɶ i ( m) for all m = 1, . . ., n − 1.25 The investment cost (exercise price) I i is specific to firm i ( i = 1, . . ., n). Firm i’s investment strategy consists in choosing a future target value X i ( ≥ X 0 ) for the stochastic variable Xɶ t and investing when this 23. When the low-cost firm is the first entrant (as ∂X Li* ∂ci > 0 and ∂X Fj * ∂ci < 0 ), a decrease in cost creates a multiple cost advantage for the low-cost firm: (1) it lowers its investment trigger such that it is more likely to invest earlier, (2) it increases the investment trigger of the second entrant (high-cost firm) such that the first entrant enjoys temporary monopoly profits for a longer period (it enters earlier and faces competition later), (3) it increases its profits as Cournot duopolist in the second stage once the high-cost firm enters. 24. See the discussion in note 16 above. 25. In case of quantity competition in a Cournot oligopoly, the certain equilibrium profits for an oligopoly with m identical firms are given in equation (3.20) as

π C ( m) =

( )

1 a−c

b m+1

2

.

In case of an additional multiplicative shock, this deterministic profit flow can be readily transformed to a stochastic profit flow by multiplying by Xɶ t .

Leadership and Early-Mover Advantage

349

investment trigger is first reached at random time Tɶi. The value of firm i when it follows the strategy to invest upon first reaching X i is denoted by Fi ( X i , X − i ), where X − i stands for the investment triggers of all other firms except firm i. What really matters is to determine the investment trigger X i * when firm i should optimally invest. In Nash equilibrium, X i * maximizes Fi (⋅, X − i *) given that rivals also act optimally following investment triggers X − i *. In such an oligopoly game where followers’ entries affect the investment value of incumbent firms, the values of all entrants (except the last one) depend on when and how many other firms enter afterwards. Assuming X 0 < X i *, the value for firm i is given by Ti +1 Ti + 2 ɶ Fi ( X i , X − i ) = Eˆ 0 ⎡ ∫ ɶ πɶ i (i ) e − rt dt − I i e − rTi + ∫ ɶ πɶ i (i + 1) e − rt dt + … + ⎢⎣ Ti Ti +1 ∞ ∫Tɶn πɶ i ( n) e− rt dt ⎤⎦ ⎤ ⎡ n Tɶm+1 = Eˆ 0 ⎢ ∑ ∫ ɶ πɶ i ( m) B0 ( t ) dt ⎥ − I i B0 (Tɶi ) Tm ⎦ ⎣ m=i ɶ ɶ where Tn+ 1 = ∞ by convention and B0 (Ti ) is the expected discount factor appropriate to discount flows received at random time Tɶ into today’s value (time t0 = 0). The value of firm i (ith investor) obtains as26 ɶ

ɶ

(

)

Fi ( X i , X − i ) = [VTɶi ( i ) − I i ] B0 (Tɶi ) −

n

∑ [V

m= i +1

Tɶm

( m − 1) − VTɶm ( m)] B0(Tɶm ),

(11.21)

where VTm ( m) is the perpetuity value to firm i valued at future (random) time Tɶm when the mth active firm enters. The expression in equation 26. Letting Tɶ ≡ (Tɶ1 , . . ., Tɶn ) and ⎡ n g (Tɶ ) = Eˆ 0 ⎢ ∑ ⎣ m =i

(∫

Tɶm+ 1 Tɶm

)

⎤ πɶ i (m) B0 (t )dt ⎥, ⎦

one obtains n

∞ Tm Ti g (Tɶ ) = Eˆ 0 ⎡⎢ ∫ πɶ i ( n) B0 (t ) dt ⎤⎥ + ∑ Eˆ 0 ⎡∫ {πɶ i ( m − 1) − πɶ i ( m)} B0 ( t ) dt ⎤ − Eˆ 0 ⎡∫ πɶ i ( i ) B0 ( t ) dt ⎤ . ⎣ 0 ⎦ m=i+1 ⎣⎢ 0 ⎦⎥ ⎣⎢ 0 ⎦⎥ ɶ

ɶ

∞ Let VT ( n) ≡ Eˆ 0 ⎡⎢∫ πɶ i ( n ) BT (t ) dt ⎤⎥ . From equation (A.45) in the appendix we have ⎣ T ⎦ T Eˆ 0 ⎡⎢ ∫ πɶ ( Xɶ t ) B0( t ) dt ⎤⎥ = V0 − B0(Tɶ ) VT . ⎣ 0 ⎦

Hence n n ⎧ ⎫ g (Tɶ ) = ⎨V0(n) + ∑ [V0(m − 1) − V0( m)] − V0 (i )⎬ + ∑ B0(Tɶm ) [VTm (m) − VTm ( m − 1)] + B0(Tɶi ) VTi (i ) . ⎩ ⎭ m= i + 1 m =i +1

Recognizing that the term in brackets {⋅} is zero, we see that equation (11.21) above results.

350

Chapter 11

(11.21) can be interpreted as follows. At time Tɶi, firm i invests and receives the forward net present value, VTi (i ) − I i, consisting of the perpetuity profit value VTi (i ) minus the investment cost (I i) incurred by firm i at that time. Given that the investment occurs at random time Tɶi, the appropriate expected discount factor is B0 (Tɶi ). Subsequently, at each random time Tɶm ( m ≥ i + 1) when a new competitor enters, the incumbent firm i has to give up or “exchange” its perpetuity profit value under the old industry setting (with m − 1 operating firms) for a new, reduced perpetuity value under the new industry structure (with m firms). The value of this “exchange,” VTm ( m − 1) − VTm ( m) , occurring at each random entry time Tɶm is discounted to the present time (time t0 = 0) by the expected discount factor B0 (Tɶm ). This occurs for all subsequent competitive arrivals, hence the summation. In effect the second term in equation (11.21) represents the present value of competitive erosion. Note that this competitive loss does not depend on the investment time Tɶi of firm i and is consequently independent of firm i’s investment strategy. It only depends on the timing of other rival firm entries (Tɶ− i). In this sense it can be treated as exogenous by firm i. When firm i selects its optimal investment strategy, terms depending on its followers’ investment strategies drop out. Therefore firm i can behave in a myopic way and choose its optimal strategy regardless of followers’ investment time schedules. The optimal investment triggers are given by the usual first-order condition that holds for myopic firms in equilibrium, namely V* − I ε ( X *) =− V i . V* ε B( X i *)

(11.22)

The optimal investment rule is, equivalently, V * = Π ( X i *) I ,

(11.22′)

where the elasticity of the discount factor, of the forward value and the profitability index are given, respectively, by XT , B0 ( XT ) XT , ε V ( XT ) = −VX ( XT ) × VT ( XT ) ε B ( XT ) . Π ( XT ) = ε B ( X T ) + ε V ( XT )

ε B ( XT ) = − BX ( XT ) ×

Leadership and Early-Mover Advantage

351

Suppose that the industrywide shock Xɶ t is multiplicative (e.g., being an exchange rate) and follows the geometric Brownian motion of equation (11.14). The stochastic profit flow then is πɶ i ( m) = Xɶ t π i ( m) and the elasticity of the terminal value becomes ε V = −1. The elasticity of the expected discount factor ε B is β1. The optimal investment trigger for firm i, X i *, from equation (11.22′) is then given by Xi * =

Π*I , Vi (i)

(11.23)

where Vi (i) ≡ π i ( i ) δ and Π* ≡ β1 ( β1 − 1). For geometric Brownian β motion, B0 (Tɶ ) = ( X 0 XT ) 1 . In case X 0 ≤ X i *, the expanded net present value under the optimal investment strategy for firm i (ith investor) is thus β1 β1 n ⎛ X0 ⎞ ⎛ X0 ⎞ F ( X i *, X − i *) = ⎡⎣Vi (i) X i * − I i ⎤⎦ ⎜ − ⎡⎣Vi (m − 1) − Vi (m) ⎤⎦ X m* ⎜ ⎝ X i * ⎟⎠ m∑ ⎝ X m* ⎟⎠ =i+1 (11.24) The first right-hand term represents the option value to wait to invest by a monopolist. Since π i ( m) ≥ π i ( m + 1) for all m = 1, . . ., n − 1, it obtains Vi (m − 1) − Vi (m) ≥ 0 for all m = 2, . . ., n, so that the second right-hand term in equation (11.24) is negative. This term represents the present value of exogenous competitive erosion that negatively affects the option value of a stand-alone monopolist option holder. Example 11.2 Duopoly Investment Case Consider the duopoly case (n = 2) we analyzed previously. The perpetuity duopoly value of the high-cost firm is lower than that of the low-cost firm and therefore the investment trigger of the follower X F * is higher. The follower will invest later when the trigger X F * is first reached; from equation (11.23) with n = 2, this is XF * =

Π* I , VF ( 2 )

where VF (2) is the present value of Cournot profits in a duopoly (n = 2 ) received by the high-cost firm (follower) in perpetuity. From equation (11.24) with n = 2, the expanded NPV for the second entrant (follower) is F ( X F *) = ⎡⎣VF ( 2 ) X F * − I ⎤⎦ B0 (TɶF *) ,

352

Chapter 11

β where B0(TɶF *) ≡ ( X 0 X F *) 1 and VF (2 ) = π CF δ . This confirms the result for the follower obtained in a duopoly in equation (11.16) previously. The leader’s investment threshold obtains similarly

XL * =

Π* I , VL (1)

with VL (1) being the perpetuity value of monopoly profits for the leader. The expanded NPV for the leader (in case X 0 ≤ X L* ≤ X F *) is L ( X L*, X F *) = ⎡⎣VL (1) X L* − I ⎤⎦ B0 (TɶL*) − ⎡⎣VL (1) − VL (2)⎤⎦ X F * B0 (TɶF *), (11.25) β with B0 (TɶL*) ≡ ( X 0 X L*) 1 , VL (1) = π L δ , VL (2 ) = π C δ . The expression above confirms the result for the leader derived previously in the duopoly case in equation (11.19).

11.4

Option to Expand Capacity

Consider now the duopoly case where firms are already operating in the market and have the opportunity to invest in additional capacity. Box 11.1 gives some flavor to the problem of expanding capacity in lump sum or incrementally in the context of commercial airlines. For expositional simplicity, suppose stochastic profits consist of deterministic reduced-form profits (given in table 11.4) times a multiplicative shock Xɶ t following the geometric Brownian motion of equation (11.14). Denote by firm i the leader and firm j the follower. Following analogous steps as before (and assuming X 0 ≤ X Li* ≤ X Fj *), it can be seen that the leader and follower’s expanded NPVs are as follows:27 Li ( X Li*, X Fj *) = V0i X 0 + ⎡⎣(Δ 1VLi ) X Li* − I i ⎤⎦ B0 (TɶLi*) + (Δ 2 VLi ) X Fj * B0 (TɶFj *) , (11.26) Fj ( X Li*, X Fj *) = V0j X 0 + ( Δ 1VFj ) X Li* B0 (TɶLi*) + ⎡⎣( Δ 2 VFj ) X Fj * − I j ⎦⎤ B0 (TɶFj *), where the optimal investment triggers are given by 27. In the preceding formulation, firms may have distinct deterministic profit values and investment costs. To allow for a natural ordering of firms, the leader must have competitive advantage with respect to both the product-stage competition (profit values) and access to the market (investment cost). The firm may have a disadvantage with respect to one of these and still be a leader if its overall advantage overweights its specific disadvantage.

Leadership and Early-Mover Advantage

353

Box 11.1 Lump-sum versus incremental capacity expansion—or big versus small expansion in aircraft fleet

In August 2004, Virgin Atlantic Airways announced it will increase its fleet with the addition of 13 Airbus 340 aircraft, a large four-engine airplane that seats more than 300 passengers (The New York Times, August 6, 2004). The airline had an option for an additional 13 such big planes. This expansion would nearly double Virgin’s fleet in an effort to beat rival British Airways (BA). Virgin would “like to fly every route British Airways flies,” Sir Richard Branson, Virgin’s chairman stated. The first planes would fly Australian and Carribean routes, competing directly with BA. The company plans to add 6,000 jobs as it expands, he said. The airline had previously ordered six huge Airbus A380 planes, which seat more than 500 passengers. Virgin’s new deal represents “a big increase in capacity,” said Chris Avery, an airline analyst. Other analysts said that Sir Richard’s big order was a big gamble, given the uncertain future for European airlines. By contrast, Air Canada, which emerged from bankruptcy protection just half a year earlier, announced in April 2005 that it would buy new lean Boeing aircraft that are more modern and fuel efficient (The New York Times, April 26, 2005). The agreement included firm orders for 14 Boeing 787 Dreamliner jets, with options and purchase rights for 46 more 787s. Boeing said it would be the largest deal so far for its new Dreamliner aircraft if Air Canada buys all 60 planes. Robert Milton, chairman of Air Canada’s parent company, said the new fleet “would save the company hundreds of millions of dollars” by lowering its fuel costs and eliminating the need to upgrade its current aging wide-body aircraft. The company plans to dedicate the aircraft primarily to long-distance one-stop flights between Canada and destinations in Asia, including China and India. The company would also expand its international cargo service, eliminating costly stopovers in Alaska. “They are trying to reinvent themselves,” said Richard Aboulafia, an aviation analyst. “If you are striving for the best, this is how you would do it.” These airlines’ different business models and strategies, a heavy lumpsum capacity expansion commitment with large aircraft vs. a more incremental and flexible strategy with more lean aircraft to be more adaptive to an uncertain business environment, are also reflective of the different business strategies of the two main commercial aircraft manufacturers, Airbus and Boeing, discussed in box 1.2 of chapter 1. Sources: The New York Times, August 6, 2004, “Virgin Air Picks Airbus Over Boeing”; and April 26, 2005, “Air Canada, Out of Bankruptcy, to Buy Up to 96 Boeing Planes.”

354

Chapter 11

Table 11.4 Profits in duopoly with expanded capacity option (existing market model) Deterministic profits Time

Industry structure

Firm i

Firm j

t ≤ Ti Tɶi ≤ t ≤ Tɶj

No one invests

π 0i π Li π Ci

π 0j π Fj π Cj

Only one invests (leader)

t ≥ Tj

Both firms invest

Note: π > π > 0, π > π 0i > 0, π Cj > π Fj > 0, π 0j > π Fj > 0. i L

X Li * =

i C

i L

Π*I Π*I ; X Fj * = , i Δ 1VL Δ 2 VFj

(11.27)

with ⎧ Δ 1VLi ⎪Δ V i ⎪ 2 L ⎨ j ⎪ Δ 1VF ⎪⎩Δ 2 VFj

≡ VLi − V0i ≡ VCi − VLi , ≡ VFj − V0j ≡ VCj − VFj

⎧ V0i ≡ π 0i δ ; V0j ≡ π 0j δ ⎪ i j j i ⎨VL ≡ π L δ ; VF ≡ π F δ , j j ⎪VCi ≡ π Ci δ ; V ≡ π δ C C ⎩

and Π* = β1 ( β1 − 1) is the profitability index reached at the level of optimal investment. The value expressions for the leader and for the follower in equation (11.26) above can be interpreted as follows. At time t0 = 0, the leader (firm i) receives the deterministic perpetuity profit value from being already active in the market (V0i ) multiplied by the initial value of the shock X0. When it invests in additional capacity at time TɶLi*, it effectively exchanges its old perpetuity profit value for a higher perpetuity value VLi, which mirrors firm i’s temporary “monopoly” profit stream during the period TɶLi * ≤ t ≤ TɶFj *. To make this “exchange,” firm i incurs an investment outlay I i. The leader’s investment occurs at random time TɶLi*, so the extra value is discounted at the appropriate expected discount factor B0 (TɶLi *). Subsequently, at random time TɶFj * when the follower also invests in added capacity, the leader gives up its previous perpetuity profit value VLi for the lower perpetuity value of profits as a Cournot duopolist when both firms have expanded capacity, VCi . At the random times of added capacity investment, TɶLi* and TɶFj *, the deterministic profits are multiplied by the value of the stochastic shock (exchange rate), namely by X Li* and X Fj *.

Leadership and Early-Mover Advantage

355

The interpretation for the follower value in equation (11.26) is analogous. At time t0 = 0, the follower receives the certain perpetuity value corresponding to no additional capacity investment; that is, it earns V0j multiplied by the initial value of the shock X 0 . Once the leader has invested, the follower is affected adversely due to the negative externalities of the leader’s investment; the follower effectively gives up its old perpetuity value in exchange for a lower one (VFj ≤ V0j). When the follower subsequently invests at random time TɶFj *, it pays the investment cost I j and “substitutes” its old follower profit value VFj for the new Cournot–Nash equilibrium profit value VCj in a “simultaneous” game.28 Oligopoly Case Consider now the more general oligopoly case involving n firms active in the market (existing market model), of which m ( ≤ n) firms have already invested in additional capacity. At the outset (time t0 = 0) no firm has invested in additional capacity. Let πɶ i ( m) represent the profit of firm i provided that m out of the n active firms have invested in additional capacities. Firm i may receive profits from the existing market even if it has not yet invested in additional capacity, meaning πɶ i ( m) ≥ 0 for m = 0, . . ., i − 1. Once again, there are negative externalities (reduced equilibrium profit values) resulting from competitive arrivals, so that πɶ i ( m + 1) ≤ πɶ i ( m) for all m = 1, . . ., n − 1. Investment costs are firmspecific, with I i denoting firm i’s investment cost. An investment strategy for firm i consists in selecting a target value X i and investing when this trigger is first hit. We again assume that uncertain profit is made of two components: a certain profit flow and a stochastic multiplicative shock Xɶ t (exchange rate) following the geometric Brownian motion given in equation (11.14). The value of firm i (ith firm) with an option to expand capacity is given by Fi ( X i , X − i ) =

n

∑π

m= 0

( m) Eˆ 0 ⎡⎢ ∫Tɶ

Tɶm+1

i

⎣

m

Xɶ t B0 ( t ) dt ⎤ − I i B0 (Tɶi ), ⎦⎥

(11.28)

resulting in n

Fi ( X i , X − i ) = Vi ( 0 ) X 0 − ∑ ⎡⎣Vi ( m − 1) − Vi ( m)⎤⎦ X m B0 (Tɶm ) − I i B0 (Tɶi ), m=1

(11.29) 28. Given the time inconsistency of the Stackelberg quantity model that assumes that the stage output by the Stackelberg leader is not on its reaction curve, the Cournot quantity model is best as reduced-form profit in such a setting.

356

Chapter 11

where Tɶ0 = t0 = 0, Tɶn+ 1 = ∞ (by convention), and Vi ( m) = π i ( m) δ for m = 0, . . ., n.29 Firm i optimizes its value by selecting the investment trigger Xi at the point where the first-order derivative equals zero. The first-order derivative is ∂Fi ( X i , X − i ) = ΔVi ⎡⎣ B0 (Tɶi ) + X i BX ( X i ) ⎤⎦ − I i BX ( X i ) ∂X i with ΔVi ≡ Vi (i ) − Vi ( i − 1) and BX = ∂B ∂X i . Given the presumed “natural” sequencing of investment, the investment timing decision of the ith firm is independent of its competitors’ investment strategies. Effectively, each firm can behave myopically and invest as if it were a monopolist; the only difference lies in the perpetuity profit value functions, which differ depending on the industry structure. The optimal investment rule for firm i prescribes to invest when the investment value X i ΔVi exceeds the investment cost I i by a factor Π* = β1 ( β1 − 1) > 1. This optimal investment strategy can be restated based on the modified Jorgensonian rule of investment: “invest when the project’s additional return on investment Xɶ t Δπ i I (with Δπ i ≡ π i ( i ) − π i ( i − 1)) equals or exceeds the firm’s interest rate (r) plus an additional term capturing the impact of irreversibility in an uncertain world (r + (β1σ 2 2)).” Example 11.3 Duopoly Expansion Case Consider again the special case of only two incumbent firms active in the market (n = 2), holding a shared option to invest in additional capacity. From above equation (11.27), the investment trigger for the follower (firm j) is given by X Fj * =

Π*I , Δ 2 VFj

29. From equation (11.28), it obtains n

Tm+ 1 Tm Fi ( X i , X − i ) = ∑ π i (m) Eˆ 0 ⎡ ∫ Xɶ t B0 (t ) dt − ∫ Xɶ t B0 (t ) dt ⎤ − I i B0 (Tɶi ) ⎢ 0 0 ⎣ ⎦⎥ m= 0 ɶ

ɶ

π i (m) ⎡ X 0 − X m+1B0 (Tɶm+1 ) − X 0 + X m B0 (Tɶm )⎤⎦ − I i B0 (Tɶi ) δ ⎣ m= 0 n

=∑ n

= ∑ Vi ( m) ⎡⎣ B0 (Tɶm ) X m − B0 (Tɶm+1 ) X m+1 ⎤⎦ − I i B0 (Tɶi ) . m= 0

Since lim XT →∞ B ( X 0 ; XT ) V ( XT ) = 0 (as δ > 0), the value of firm i for a given target X i is n

n+ 1

m= 0

m=1

Fi ( X i , X − i ) = ∑ Vi (m) ⎡⎣ B0 (Tɶm ) X m ⎤⎦ − ∑ ⎡⎣Vi (m − 1) X m B0 (Tɶm )⎤⎦ − I i B0 (Tɶi ) . Equation (11.29) obtains by summation.

Leadership and Early-Mover Advantage

357

where Δ 2VFj ≡ VCj − VFj , VCj ≡ π Cj δ and VFj ≡ π Fj δ , with δ ≡ k − g = r − gˆ . The investment trigger of the leader (firm i) is X Li* =

Π*I , Δ 1VLi

where Δ 1VLi ≡ VLi − V0i , VLi ≡ π Li δ , and V0i ≡ π 0i δ . The value for the follower (firm j), assuming X 0 ≤ X Li* ≤ X Fj *, from equation (11.29) with n = 2 is Fj ( X Fj *) = V0j X 0 + (Δ 1VFj ) X Li* B0 (TɶLi*) + ⎡⎣(Δ 2VFj ) X Fj * − I j ⎤⎦ B0 (TɶFj *) and for the leader (firm i) Li ( X Li*) = V0i X 0 + ⎡⎣( Δ 1VLi ) X Li* − I i ⎤⎦ B0 (TɶLi*) + ( Δ 2VLi ) X Fj * B0 (TɶFj *), with Δ 1VFj ≡ VFj − V0j and Δ 2VLi ≡ VCi − VLi. The result above confirms the results obtained in equation (11.26) for the duopoly case. Conclusion In this chapter we have shown how the pure-strategy Nash equilibria in investment games can exhibit a sequencing of investment timing. Moreover in the present context, even for a priori symmetric firms, early investors enjoy an early-mover advantage as their value function as leader exceeds their later entrants’ value. When a firm has a “substantial” competitive (e.g., cost) advantage over its rivals, the sequence of investment may involve a natural ordering, with the lowest cost firm entering first. We analyzed the option to invest in a new market as well as the option to expand (by a lumpy amount) an existing market, deriving the optimal value and trigger strategies for competing firms in oligopoly. In the next chapter we revisit the main result according to which early movers can enjoy first-mover advantages by allowing firms to use mixed strategies. Selected References The formulation of games of sequential investment timing has been introduced by Reinganum (1981a,b) in a deterministic setting. Joaquin and Butler (2000) discuss an asymmetric duopoly quantity competition model where firms are ordered in their entries. Their model is a continuous-time version of Smit and Trigeorgis’s (1997) discrete-time asymmetric duopoly model.

358

Chapter 11

Joaquin, Domingo C., and Kirt C. Butler. 2000. Competitive investment decisions: A synthesis. In Michael J. Brennan and Lenos Trigeorgis, eds., Project Flexibility, Agency, and Competition: New Developments in the Theory and Application of Real Options. New York: Oxford University Press, pp. 324–39. Reinganum, Jennifer F. 1981a. On the diffusion of new technology: A game-theoretic approach. Review of Economic Studies 48 (3): 395–405. Reinganum, Jennifer F. 1981b. Market structure and the diffusion of new technology. Bell Journal of Economics 12 (2): 618–24. Smit, Han T. J., and Lenos Trigeorgis. 1997. Flexibility and competitive R&D strategies. Working paper. Columbia University.

12

Preemption versus Collaboration in a Duopoly

In the previous chapter we analyzed oligopoly models involving sufficient competitive advantage or asymmetry such that firm roles (who is the leader and who the follower) were arguably rather clear and determinable a priori. When firms are nearly identical, however, there are multiple Nash equilibria in pure strategies and the more likely outcome of the game cannot be readily determined.1 When no firm has a clear competitive (e.g., cost) advantage, there appears to be a “coordination problem” in determining who acts first and becomes the leader. Mixed strategies may give further insights and help determine what might happen when firms are not sufficiently distinct. Fudenberg and Tirole (1985) provide a solution for this coordination problem using symmetric mixed-strategy equilibria in a deterministic, continuous-time setting. We discuss this approach next and subsequently extend the analysis to option game situations under uncertainty, deriving implications for investment strategies and optimal investment timing in competitive settings. For simplicity, we focus the discussion on duopolistic markets. The chapter is organized as follows. In section 12.1 we present the original deterministic model by Fudenberg and Tirole (1985) to help analyze preemption versus cooperation. We then extend this in more complex settings to account for stochastic market uncertainty. We discuss the option to invest in a new market in section 12.2, and the option to expand an existing market in section 12.3. Throughout the chapter, we look at the impact of firm asymmetry on the equilibrium investment behavior of firms. 1. Investments are still made in sequence but in an industry with n firms there are n! different pure-strategy Nash equilibria.

360

12.1

Chapter 12

Preemption versus Cooperation

Consider first the simpler deterministic case when there is no uncertainty concerning the underlying market. In the previous chapter we discussed how the model by Reinganum (1981a) might fit certain option game situations, especially those involving a natural sequencing of investment. In the present chapter where coordination problems are explicitly addressed, we apply a refinement of Reinganum’s model of investment timing proposed by Fudenberg and Tirole (1985) who adopt a continuous-time mixed-strategy approach. The model by Reinganum was based on certain simplifying assumptions. Reinganum (1981a), as well as Scherer (1967), consider investment-timing decisions based on the notion that firms are precommitted to invest at a certain (deterministic) future time and whatever happens they are to stick to their plans. This assumption may be justified in some cases when implementing an investment decision is fairly timeconsuming or when altering the investment plan is prohibitively costly. Reinganum finds that there is a sequencing of investments even if firms are identical, namely one of the firms (the leader) invests first and the other (follower) invests later. In a duopoly setup where firms decide on their investment timing beforehand, there exist two pure-strategy Nash equilibria (of the sequential-investment type) that involve a first-mover advantage.2 This setup changes if there is a sufficiently large competitive advantage by one of the duopolists; a natural leader-follower equilibrium results where the firm with a large competitive advantage invests first as part of a focal-point equilibrium. Preemption is ruled out in Reinganum’s (1981a) model given the assumed precommitment to a stringent investment timing schedule by both firms. Preemption is, however, possible in Fudenberg and Tirole’s (1985) approach. This form of rivalry results from the strategic timing interplay between competitors eager to capture the lion’s share when there is a first-mover advantage. If there is a large first-mover advantage tapped by the leader, both firms will want to invest as the leader to grasp this advantage. In this strategic setting, precommitment to a fixed investment-timing schedule seems rather unrealistic. Precommitment (open-loop) equilibria fail to capture the fact that firms may be tempted 2. The first-mover advantage refers to the investment stage only, not the market or commercialization stage. In a given region the leader’s value exceeds the follower’s since the former earns first-stage monopoly rents (new market model) or higher incumbency rents (existing market model). In the duopoly competition stage both profit flows are equal.

Preemption versus Collaboration in a Duopoly

361

to undermine their rivals and preempt them to obtain a higher value. The precommitment outcome does not make as much sense in settings where firms can instantaneously respond to their rivals’ actions and do not pay prohibitively high costs if they revise their planned investment schedule. Even if firms had to precommit to a strict investment-timing schedule, the existence of an early-mover advantage may still lead to timing rivalry as each firm tries to precommit first, announcing its commitment decision promptly to alter the rival’s investment schedule.3 This results once again in a coordination problem between option holders: if precommitment to an investment-timing schedule is not enforceable, both firms will attempt to be the first entrant to seize the first-mover advantage. In order to incorporate this feature of endogenously determined firm roles in equilibrium, we need to resort to the closed-loop equilibrium concept of investment timing games.4 To tackle this coordination issue, Fudenberg and Tirole (1985) refine Reinganum’s model by relaxing the assumption of precommitment to a pre-specified future time.5 The revised setup by Fudenberg and Tirole (1985) makes it possible to determine firm roles endogenously, in contrast with the focal-point argument used in chapter 11 where firm roles are determined exogenously in case of a large competitive advantage. Their model implies that a firm can only become a leader by actually investing first, not simply by having some prior cost advantage. Leadership is not an inherited “quality” but the result of a move as the actual first investor. Endogenous firm roles imply that in making its investment timing decision, a firm should compare now (i.e., at each time and state) the relative values of being a leader, a follower, or a simultaneous investor—not simply consider the 3. This situation does not emerge in the model by Reinganum (1981a) since firms are assumed to precommit to an investment plan simultaneously. 4. Fudenberg and Tirole (1985) introduce two notions along which they differentiate their model from Reinganum’s (1981a, b). Reinganum formulates open-loop strategies where the firms’ investment decisions do not depend on the previous play by rivals but only on calendar time; these are essentially myopic strategies selected regardless of rival’s reactions. By this assumption the industry equilibrium is a Nash equilibrium in open-loop strategies, namely an open-loop equilibrium. Fudenberg and Tirole (1985) instead consider closedloop strategies that permit the firms to condition their play on previous actions given the history of the industry. They also require that at every state/ subgame the industry could possibly reach (even off the equilibrium path) the continuation strategies form a Nash equilibrium going forward. That is, they require that the optimal closed-loop strategies form a perfect equilibrium. The underlying solution concept is known as perfect closedloop equilibrium. 5. The equilibrium concept used in case of preemption was introduced by Fudenberg and Tirole (1985) for symmetric players and by Simon (1987a, b) for asymmetric players. We here deal only with the case involving identical players and symmetric investment strategies.

362

Chapter 12

value of leadership at the beginning of the game, as was done in the previous chapter (where roles were pre-assigned).6 We next discuss the coordination problem in the deterministic case as originally analyzed by Fudenberg and Tirole (1985). To address the above coordination problem, the authors use continuous-time mixed strategies and derive the perfect closed-loop equilibrium.7 They adopt a new formalization (presented in appendix 12A) enabling studying such games of timing.8 Strategy is redefined in terms of two functions in order to describe appropriate continuous-time mixed strategies. A first term Gi ( t ) tracks whether firm i has invested before (or at) time t given that the other firm has not yet invested, while a second term qi ( t ) measures the instantaneous probability or “intensity” of investing at time t.9 Consider first two identical firms following symmetric strategies. L (⋅), F (⋅), and C (⋅) are, respectively, the values of being the leader, the follower, or a simultaneous Cournot duopolist (discounted back to the outset, time t0).10 For firm i or j, the payoffs and action choices at time t (for Gi (t ) = Gj (t ) = 0) are depicted in strategic form as shown in figure 12.1. The strategic-form game is repeated in “rounds,” with no time elapsing between “rounds.” At each “round,” firms play randomly.11 The probability of occurrence of various industry structures can be determined from the strategic-form representation as illustrated in box 12.1. Preemption will occur if at some time t ( ≥ t0 ) there is a first-mover advantage, that is, if there exists at least a small time interval (including t) such that the value received by the leader is larger than the 6. An option to wait by the leader contradicts the model assumptions. 7. Deriving mixed-strategy equilibria in continuous time as the limit of discrete-time mixed-strategy equilibria leads to some information loss as discussed in Fudenberg and Tirole (1985, pp. 389–92). Their strategy formulation makes it possible to circumvent this problem by introducing a second function in the strategy definition. 8. Fudenberg and Tirole (1991, sec. 4.5) give a brief overview on timing games including wars of attrition. 9. qi(⋅) is called atoms function in the optimal control literature. Essentially the control and action taken at time t consumes no time to take and implement. qi(⋅) allows capturing information that is lost when considering Gi (⋅) only. See appendix 12A for a precise definition of the strategy space. 10. More precisely, L ( t ) is the payoff for the firm that succeeds in preempting its rival at time t ≥ t0 (the “leader”), F (t ) is the payoff for the preempted firm that invests at a later date (the “follower”), and C (t ) is the individual payoff for each of the two firms if they invest simultaneously at time t. For the sake of generality, we do not characterize the payoffs as functions of model primitives; they may describe new market, existing market (discussed next), or technology adoption models (as in Fudenberg and Tirole, 1985). 11. The two-by-two matrix shown in figure 12.1 represents a repeated game that takes no time to repeat or a game in which rounds are played instantaneously with no discounting. This representation is allowed by the definition of qi(⋅) as an atoms function.

Preemption versus Collaboration in a Duopoly

363

Firm j Invest qj (t)

Wait 1 – qj (t)

Simultaneous investment

Sequential investment

C (t) C (t)

L (t) F (t)

Sequential investment

No investment (waiting)

F (t) L (t)

Repeat game

Invest qi (t)

Firm i

Wait 1 – qi (t)

Figure 12.1 Strategic form of the investment timing game at time t qi( t ) measures the instantaneous investing “intensity” by firm i at time t.

value received as a follower, meaning L ( t ) > F ( t ). In such a situation there is an incentive for a firm to preempt its rival and become the leader. A different type of coordination problem involving a war of attrition may occur if instead there is a second-mover advantage, that is, if there exists an interval of time t ( ≥ t0 ) such that F (t ) > L ( t ). A continuous-time game-theoretic model involving a war of attrition was developed by Hendricks, Weiss, and Wilson (1988). Here we focus on preemption. A number of cases are interesting to consider further: (1) L(TL* ) > C(TC* ) with TL* ≡ arg max t ≥ t0 L ( t ) and TC * ≡ arg max t ≥ t0 C ( t ), and (2) L(TL* ) ≤ C(TC* ). The first case involves preemptive timing equilibria, while the second allows for tacit collusion outcomes that are beneficial to both firms. 12.1.1

Preemption

Preemption is characteristic of markets with a large first-mover advantage and high excess profits to be earned by the firm that enjoys temporary monopoly rents. Even when firms may cooperate by waiting to invest at a later date, TC *, receiving C (TC *), they have no incentive to do so, instead investing earlier at TL* to become the leader, receiving a higher value L (TL*) > C (TC *). In case of investment in a new market the above inequality clearly holds. The follower value is not negatively

364

Chapter 12

Box 12.1 Probability of occurrence of various industry structures

Given investment intensity qi (t ) ≡ qi , for firm i one can determine the probability of occurrence of certain events at time t . For example, the probability that firm i is the market leader ( pLi) equals the probability of firm i being the leader now, qi (1 − q j ), plus the probability of waiting and becoming the leader in the next “round,” (1 − qi ) (1 − q j ) pLi.a This leads to the following recursive expression: pLi = qi (1 − q j ) + (1 − qi ) (1 − q j ) pLi , yielding pLi =

qi (1 − q j ) qi + q j − qi q j

(12.1.1)

with qi , q j ≠ 0. Identical firms are assumed to follow symmetric (mixed) strategies (q = qi = q j ). In this case equation (12.1.1) reduces to pL ≡ pLi = pLj =

1− q . 2−q

(12.1.2)

The probability that firm i is the follower (with firm j being the leader) is 1 − pLj. From figure 12.1 the probability of simultaneous investment by both firms as Cournot duopolists, pC, is given by the probability of immediate simultaneous investment by both firms, qi q j , and the probability of both firms waiting until the next “round” and simultaneously investing then.b This probability satisfies pC = qi q j + (1 − qi ) (1 − q j ) pC , or pC =

qi q j qi + q j − qi q j

(12.1.3)

for qi , q j ≠ 0. In case of symmetric strategies, the expression above simplifies toc pC =

q 2−q

(≥ 0).

(12.1.4)

a. Over an infinitesimal time interval, we can reasonably assume that no change in the market environment will occur so that pLi is stationary over such a short time period. b. The subscript C is being used in the text to stand for Cournot competition or cooperation, depending on the context. The connection between these notions will be made clearer in later sections. c. For q ≠ 0 , the probability that both firms invest simultaneously is strictly positive. For q = 0 , the probability of simultaneous investment is zero, so equation (12.4) holds for all values of q ∈[0, 1].

Preemption versus Collaboration in a Duopoly

365

Present (time-0) value A

B

L(t)

F

F (t)

L(t), F (t), C(t)

C D

E

G

TP*

C(t)

TL*

TF *

TC*

Time (t)

Figure 12.2 Preemption case: L(TL *) > C(TC *) L (t ) is the expected value of being the leader; F (t ) of being the follower; C ( t ) of simultaneous investment as a Cournot duopolist. The graph is based on the technology-adoption problem described in Fudenberg and Tirole (1985) with specific assumptions and does not necessarily accurately represent the new market model discussed here.

affected by the leader’s investment decision (π F = π 0 = 0) and the value of simultaneous Cournot investment equals the value of being a follower. Figure 12.2 illustrates this preemption case, described in Fudenberg and Tirole (1985), showing the present (time-0) expected value of being a leader (L), a follower (F ), or investing simultaneously as a Cournot duopolist (C).12 In the graph above, TF * is the investment time for the follower. Once the follower enters and both (symmetric) firms compete in the market as Cournot duopolists, they earn the same profits hence values are equalized. TP * is the earliest (or first) time that the values of being a leader or a follower are equal, that is, TP * ≡ inf 0 ≤ t ≤ TL *L (t ) = F (t ) . In the region TP * < t < TF * there is a first-mover advantage that the leader can exploit; that is, the leader’s value curve is above the follower’s or L ( t ) > F ( t ). TC * is the later investment time that maximizes joint investment value under simultaneous investment (i.e., it maximizes C (⋅)).13

{

}

12. For the sake of generality, the payoffs accruing to the players are, for the time being, expressed in a generic form. 13. The optimal investment times (except the preemption point) can be determined by use of the Jorgensonian rule of investment (see chapter 11).

366

Chapter 12

Provided that L (TL*) > C (TC *), or point A is above B, neither firm can do better than receiving the leader value L (TL*) at point A. Ignoring strategic interactions, each firm would like to invest exactly at optimal time TL* that maximizes the leader’s value. Given the coordination problem resulting from the lack of a natural leader-follower entry sequencing, a simultaneous investment by both firms as Cournot duopolists at time TL*, or point C, would be detrimental to both firms since C (TL*) < F (TL*) < L (TL*). As a result of the first-mover advantage, each firm will try to preempt its rival and invest just before the rival does.14 This preemption process would continue and will stop at time TP * when the expected values of being a leader or a follower are exactly equal, namely when L (TP *) = F (TP *). We refer to TP * as the preemption time. Prior to this (t < TP *), L ( t ) < F ( t ) so no one has an incentive to invest earlier than the preemption time TP * as preemption at this stage is not a value-enhancing strategy. Fudenberg and Tirole (1985) refer to this phenomenon above arising in the case of timing rivalry or preemption as “rent equalization.” At the preemption time TP *, there is neither a first- nor second-mover advantage for either firm. Competing firms are just indifferent between being the first or the second investor, meaning “rents” are equalized. Box 12.2 discusses an analogous problem in the context of the first recorded auction for the highest “prize.” At the preemption time TP * one of the firms will invest first. Since the rival firm is worse off by investing now than by waiting, that is, C (TP *) < F (TP *) or point G is below D, it is optimal for the rival to wait until optimal time TF * when being a follower results in a higher value. In the preemption case, the perfect (closed-loop) equilibrium results in an ordering of firm roles with deterministic adoption times, TP * (for the leader) and TF * (for the follower). However, compared with the (openloop) model we discussed in chapter 11 based on Reinganum (1981a), the investment timing trigger for the first-mover is no longer the myopic investment time, TL*, but the preemption time, TP *. This investment trigger does not directly maximize the leader’s value but rather is the outcome of strategic interactions that in equilibrium result in indifference between the leader and the follower roles. In this context one of the firms (the leader) invests at preemption time TP * and the other (follower) invests at a later date (at TF *). 14. If firm j would invest at TL*, firm i will want to invest at time TL* − ε (where ε is an infinitesimal amount). Firm j will then invest just before that, at time TL* − 2ε , and so on and on.

Preemption versus Collaboration in a Duopoly

367

Box 12.2 The first auction and first-mover advantage

The “Father of History,” Herodotus, made the first written reference to an auction in the first book of The Histories as follows: Once a year in every village all the maidens as they attained marriageable age were collected and brought together into one place, with a crowd of men standing around. Then a crier would display and offer them for sale one by one, first the fairest of all; and then, when she had fetched a great price, he put up for sale the next most attractive, selling all the maidens as lawful wives. Rich men of Assyria who desired to marry would outbid each other for the fairest; the ordinary people, who desired to marry and had no use for beauty, could take the ugly ones and money besides; for when the crier had sold all the most attractive, he would put up the one that was least beautiful and offer her to whoever would take her as wife for the least amount, until she fell to one who promised to accept the least; the money came from the sale of the attractive ones, who thus paid the dowry of the ugly. (Source: Herodotus, The Histories, I:196 [describing a custom of Eneti in Illyria], fifth century BC.)

Besides documenting the first known auction, this reference introduces the notion of first-mover advantage in this context. “Maidens” are ranked in decreasing value in terms of beauty, with the first “bidder” receiving the “fairest maiden,” gaining a first-mover advantage. This suggests (as in chapter 11) an ordering of advantages, with the last “entrant” getting the “maiden of least fairness.” Rich Assyrians outbid each other in an attempt to obtain the highest “prize.” Albeit rich, Assyrians might bid no more than the “fairest maiden” was worth. The “rent equalization” principle might apply here. The firstmover advantage might go away as rich Assyrians would outbid each other up to the point where they are indifferent between receiving the “fairest maiden” for a high price or receiving the next “fairest maiden” for a lower price. This suggests an extension of the rent-equalization principle to more than two “prizes” or roles, namely leader and follower. At the end of the process, the Assyrian left with the “maiden of least fairness” is compensated by a monetary payoff so that roles are equally attractive.

As shown in figure 12.2, the optimal investment triggers arising out of this strategic equilibrium are ranked as follows15 TP * ≤ TL* ≤ TF * ≤ TC *. The payoff for firm i in each region depends both on the intensity (probability) of investment by firm i, qi ( t ), and that of rival firm j, q j ( t ). Let us next determine the equilibrium mixed-strategy investment qi * (t ) of 15. This can be shown by way of contradiction. Suppose TP* > TL* and TP * < TF *. Since F (⋅) is increasing in [t0 , TF *] and by definition of TP*, it follows that L (TP *) = F (TP *) ≥ F (TL*) > L (TL*). This is in contradiction with the definition of TL*. For the inequality TC * ≥ TF *, see the proof of lemma 4.2 in Huisman (2001, p. 91).

368

Chapter 12

firm i in the subgame starting at time t ∈[TP *, TF *]. The strategic-form representation of figure 12.1 helps illustrate this. A formal description of the subgame perfect equilibrium is given in appendix 12B.1. In a Nash equilibrium mixed-strategy profile, each player is willing to randomize so that it is indifferent, given the mixed strategy played by its rival, between its own pure-strategy choices (here “invest” or “wait”). We suppose that in equilibrium, firm i invests with probability qi * = qi * ( t ) and adopt firm j’s point of view. If firm j decides to invest, it will operate as a Cournot duopolist with probability qi * and as a leader with probability 1 − qi *. If firm j delays investment, it will receive F j (t ). In other words, firm j is indifferent among the two pure-strategy actions if16 qi * C j ( t ) + [1 − qi *] Lj ( t ) = F j (t ) . (invest) (wait) The resulting equilibrium investment intensity for firm i in the subgame starting at time t ∈[TL*, TF *] is qi ( t ) = φ j ( t ) =

Lj (t ) − F j ( t ) . Lj ( t ) − C j (t )

(12.1)

Interestingly a firm’s equilibrium investment intensity, qi ( t ), generally depends on the value functions (as a leader, follower or Cournot duopolist) of its rival, not its own. In the symmetric case, qi ( t ) = q j (t ) = q(t ), equation (12.1) simplifies to q (t ) = φ (t ) ≡

L (t ) − F (t ) . L (t ) − C (t )

(12.2)

At the preemption time TP *, each symmetric firm is just indifferent between being the leader or a follower, meaning L (TP *) = F (TP *), so the intensity or probability of investing at TP *, obtained from box equation (12.1.1), is q (TP *) = 0. From equation (12.1.2) the probability that a firm becomes the leader at the preemption time TP * (for t0 ≤ TP *) is thus pL = 1 2. At this indifference point it obtains from equation (12.1.4) with q = 0 that the probability of simultaneous Cournot investment, pC, is zero. The resulting equilibrium once again leads to a sequencing of firm investments, with one of the firms investing as a leader (at preemption time TP *) and the other as a follower (at later date TF *) but here the first investment takes place earlier than in the open-loop case (TP * < TL*). 16. Here discounting plays no role since we define qi (·) as an atoms function whereby the action takes no time to implement.

Preemption versus Collaboration in a Duopoly

369

Probability 100%

Monopoly (firm i)

75%

Cournot duopoly

No investment

50%

Monopoly (firm j)

25%

0%

t0

TP*

TF *

∞ Time (t)

Figure 12.3 Probability of industry structure along market development (for t0 £ TP *)

For t0 ≤ TP *, each symmetric firm has a 50 percent chance to become the leader over the play of the game as is illustrated in figure 12.3. For TP * < t0 < TF *, there is a first-mover advantage that each firm wants to grasp immediately (as L ( t0 ) > F ( t0 )). A coordination-timing problem arises in determining who leads and who follows suit. Since in this region we have L (t0 ) > F (t0 ), it results from equation (12.2) that the investment intensity for symmetric firms is q ( t0 ) = φ ( t0 ) ≡

L ( t0 ) − F ( t0 ) > 0. L ( t0 ) − C ( t0 )

Therefore someone will invest in this market with positive probability q ( t0 ). The probability of simultaneous investment at time t0, obtained from equation (12.1.4), is also strictly positive ( pC > 0). Thus, if t0 ∈ (TP *, TF *) coordination may fail to result in an industry equilibrium where one firm enters as a leader and the other firm follows. This “coordination failure” is detrimental to both firms as C ( t0 ) < F ( t0 ) < L ( t0 ); nonetheless if t0 ∈ (TP *, TF *), simultaneous investment may obtain as an industry equilibrium. For t0 ≥ TF *, both firms invest simultaneously resulting in a Cournot duopoly. The resulting market structure for differing values of t0 is illustrated in figure 12.4.

370

Chapter 12

Probability 100%

Monopoly (firm i )

75%

50%

No investment

Cournot duopoly

“Coordination failure”

25%

Monopoly (firm j )

0%

TP*

TF *

∞ Starting time (t0)

Figure 12.4 Probability of market structure occurring at the outset (t 0 ) The graph is for illustrative purposes. The value functions actually used for L (⋅), F (⋅), and C (⋅) may lead to nonlinear curves separating the monopoly regions from the “coordination failure” region.

There is some persistence of the industry structure over time. If t0 < TP *, as time passes one firm invests as a leader at time TP * and the other invests later at time TF *. Just after the preemption time TP *, the second firm has no incentive to invest immediately as its value as a Cournot duopolist is lower than that of waiting to invest at time TF * as a follower. The probability of ending up in a Cournot duopoly in the region TP * < t < TF * (“coordination failure”) is zero (for t0 ≤ TP * < TF *). 12.1.2

Cooperation in an Existing Market

In the case of an existing market, if the first-mover profit advantage for the leader, π L − π 0, is low (close to π C − π F ), the gain from preempting its rival is small. In this case collaboration in terms of a jointly selected investment timing may be preferable for each firm than the sequential preemptive sequence discussed above. Such a pattern may emerge when the follower’s value differs from the simultaneous Cournot investment value significantly. This happens when the follower is unfavorably affected in the intermediate region by its rival’s investment. Existing

Preemption versus Collaboration in a Duopoly

371

Present (time-0) value

A

B

C

L (t)

L (t), F (t), C (t)

F

F (t) D

C (t)

TP*

TL*

TF *

TC

TC*

Time t

Figure 12.5 Joint investment/collaboration: C(TC*) ≥ L(TL*) L (t ) is the expected value of being the leader; F (t ) of being the follower; C ( t ) of simultaneous investment as a Cournot duopolist. The graph is based on the technology-adoption problem described in Fudenberg and Tirole (1985) with specific assumptions and does not necessarily accurately represent the existing market model.

market models may exhibit such tacit collusion or cooperation among option holders, while new market models do not since the follower earns no profit flow in the intermediary region. Figure 12.5 illustrates such a case of joint investment (collaboration) potentially resulting in higher benefits than preemption when C (TC *) ≥ L (TL*), where TC * is the optimal joint investment trigger from the point of view of the duopolist option holders. Let TC be the earliest time within [TF *, TC *] when the value of simultaneous (joint) investment just equals the optimal leader value (point C in figure 12.5), that is, when TC ≡ inf TF * ≤ t ≤ TC *C(t ) = L(TL*) . There exists a region t ≥ t0 C (t ) > L (TL*) where the value of simultaneous or joint investment exceeds the value of being a leader. In this case neither firm has an incentive to invest earlier than its competitor, which rules out preemption. In this region both firms invest immediately at time t (rather than invest earlier at TL* for a lower value), resulting in numerous simultaneous investment equilibria ( t ≥ t0 C (t ) > L (TL*) ). Although there is a continuum of joint-investment equilibria starting at time TC , one of them appears most reasonable. The equilibrium involving joint investment at time TC * Pareto-dominates all other joint-investment equilibria. As

{

{ }

}

{

}

372

Chapter 12

suggested by Fudenberg and Tirole (1985), TC * is the most attractive among all possible equilibria. A formal description of the tacit collusion perfect equilibria (including the Pareto-superior one) is provided in appendix 12B.2. For TF * < t < TC , the value of being a leader is higher and there is a first-mover advantage leading to preemption. This creates a motive to invest before one’s rival, resulting in a sequential preemptive ordering equilibrium where the leader invests at TP * and the follower at TF *. Consequently, if L (TL*) ≤ C (TC *), there exist two classes of equilibria: the first class is the sequential ordering equilibrium discussed earlier with investment times TP * and TF *;17 the second is a continuum of joint-investment equilibria for t ≥ TC, the most reasonable of which is joint investment at common time TC * that maximizes joint profits.18 12.2

Option to Invest in a New Market under Uncertainty

The previous analysis based on Fudenberg and Tirole (1985) was developed in a deterministic environment. It provides useful insights into how firms interact in games of timing and how preemption or tacit collusion may result, but it does not address the coordination or entry sequencing problem under (exogenous) market uncertainty. The preceding deterministic setting needs to be extended, leveraging the insights from real options analysis to add more realistic guidance to strategic investment under uncertainty. We address this challenge in sections 12.2 and 12.3. Section 12.2 discusses entry into a new market, and section 12.3 extends the analysis to cases where firms earn an initial profit flow before expanding investment (existing market models). We discuss here coordination issues arising in new market models, first when firms are identical and follow symmetric investment strategies, and then consider the impact of firm asymmetry on the optimal investment strategies. In case of entry into a new market, the follower—which is not operating at the outset—does not suffer a profit value drop upon entry by the rival. The values of the 17. If TP * < t0 < TF *, two types of industry equilibrium may emerge. The first is a sequential preemptive equilibrium with one firm investing at time t0 and the other at TF *; the leader and follower roles can be reversed. This type of structure occurs with positive probability 2 pL , with pL obtained from equations (12.1.2) and (12.2). The second is simultaneous investment at the outset ( t0 ), occurring with positive probability pC obtained from equations (12.1.4) and (12.2). 18. From the Jorgensonian rule of investment, TC * = 1 g × ln (rI (π C − π 0 )), provided that the market grows at a constant growth rate g (per unit time). Here r is the discount rate, π 0, π C are, respectively, the profit flows before and after both firms have invested, and I is the required investment outlay.

Preemption versus Collaboration in a Duopoly

373

follower and of the simultaneous Cournot investment are equal. Thus, cooperation or tacit collusion (described in previous section 12.1.2) does not occur. 12.2.1

Symmetric Case

The first model of option games extending Fudenberg and Tirole’s (1985) framework under uncertainty was developed by Smets (1991) in the context of multinational firms. Smets discusses the case of a duopoly where two firms have a shared option to make an irreversible investment to enhance their incumbency profits. We discuss Smets’s (1991) actual model in the next section as it deals with the option to expand an existing market. We introduce here a simplification of that model involving the option to invest in a new market. This simplified model was discussed in Dixit and Pindyck (1994) and Nielsen (2002).19 Consider two identical firms that follow symmetric (mixed) strategies. Assume that the “risky” profit flow πɶ consists of a certain or deterministic profit flow component, π ( = π L , π C ), and a multiplicative stochastic process component, Xɶ t , accounting for industrywide shocks. The uncertain profit flow is thus πɶ = Xɶ t π , where Xɶ t follows the geometric Brownian motion dXɶ t = ( gXɶ t ) dt + (σ Xɶ t ) dzt ,

(12.3)

with constant drift g and instantaneous volatility σ . π L ( ≥ 0 ) stands for the deterministic profit flow of the first investor (leader) when it is the only firm in the market, and π C ( ≥ 0 ) the deterministic profit flow earned by each firm when they are both operating as Cournot duopolists. The profit flow of the monopolist (π L) is higher than that of a Cournot duopolist (π C). As before, there is no lasting first-mover advantage with respect to the profit flow; at the time the follower invests, the profit of the leader drops and subsequently equals that of the follower. Consider first the problem faced by the follower. Suppose that the leader has already invested and currently the second investor contemplates entry. The optimal investment trigger for the follower (F ) is given by the markup rule (see chapter 11, equation 11.9) VC X F * = Π* I

(> 1)

(12.4)

19. Fudenberg and Tirole’s (1985) model presented earlier is an existing market model in a deterministic setup. The models by Smets (1991), Grenadier (1996), and Huisman and Kort (1999) involve an option to expand (existing market models).

374

Chapter 12

with VC ≡ π C δ , Π* ≡ β1 (β1 − 1), and β1 given by

β1 = −

2 αˆ r  αˆ  + +2 2   2 2  σ σ σ

( > 1)

(12.5)

with αˆ = gˆ − (σ 2 2), gˆ = r − δ , r being the risk-free interest rate, and I being the investment cost. Equivalently, X F * is the first threshold at which the return on investment equals the modified Jorgensonian discount rate,

πC XF * 1 = r + β1σ 2 . 2 I At this threshold, X F *, the follower is indifferent between investing or keeping its deferral option alive (value-matching condition). The optimal trigger X F * is reached at random time TɶF *, with TɶF * ≡ inf t ≥ 0 Xɶ t ≥ X F * . The value for the follower pursuing the optimal investment strategy is given by

{

}

(VC X F * − I ) B0 (TɶF *) if F ( X 0 , X F *) =  if VC X 0 − I

X 0 < X F *, X 0 ≥ X F *,

(12.6)

with the expected discount factor given by β1

 X  B0 (TɶF *) ≡  0  .  X F *

(12.7)

For X 0 < X F *, the follower receives no profit prior to investing. The entire value consists solely of its option to defer the investment. For X 0 ≥ X F *, the follower has a dominant strategy to invest immediately, receiving the committed project value or static NPV, VC X 0 − I . When considering endogenous firm roles, it is contradictory to assume an option to defer for the leader since firms take the lead by actually investing first, thereby killing their option to defer. As a result, since the leader will invest early (at t0), there is no need to determine an optimal investment timing trigger X L* for the leader. Thus the expected discount factor for the term representing the defer option value does not appear in the leader’s value expression—contrary to the case in equation (11.19). This is a major difference between the model we discussed in chapter 11 and the model presented here. When firm roles are endogenous, the value of investing I and becoming leader at the outset (in state X 0) is (VL X 0 − I ) − (VL − VC ) X F * B0 (TɶF *) if L ( X0 ) =  if VC X 0 − I

X 0 < X F *, X 0 ≥ X F *,

(12.8)

Preemption versus Collaboration in a Duopoly

375

with VL ≡ π L δ and B0(TɶF *) as given in equation (12.7). For X 0 < X F *, the value of investing now (at t0) as the leader equals the net present value as a monopolist, VL X 0 − I , minus the discounted value difference (competitive value erosion) resulting from the rival’s entry at random time TɶF *. Once the leader has invested at current time t0 , it has no further growth investment opportunities and merely suffers from any adverse developments in the market, such as entry by the follower. For X 0 ≥ X F *, both firms invest immediately (simultaneously) as Cournot duopolists: for symmetric firms the value of the leader and the follower are equal(ized), both earning VC or a net present value VC X 0 − I . A special case is discussed next. Example 12.1 Symmetric Preemption Suppose that two firms share the option to invest in a new market for a fixed investment cost I = 100. If both firms enter, they compete in quantity (Cournot competition) and face deterministic linear demand p(Q) = 50 − 5Q. Firms have symmetric variable production cost, c = 10. According to the analysis in chapter 3 (see table 3.2, panel A), a monopolist firm would make excess profit equal to

πL =

(a − c )2 4b

=

( 50 − 10 )2 4×5

= 80.

In Cournot competition, each firm would earn

πC =

( a − c )2 9b

=

(50 − 10 )2 9×5

≈ 35.56 .

Deterministic profits are subject to an exogenous multiplicative shock that follows a (risk-adjusted) geometric Brownian motion with gˆ = 5 percent and σ = 10 percent. The risk-free rate is r = 7 percent (and the dividend yield is δ = 2 percent). The value of being a monopolist forever in perpetuity is VL =

πM 80 = = 4, 000 , 0.02 δ

while Cournot duopoly value is VC =

πC ≈ 1, 778 . δ

Note that αˆ = gˆ − (σ 2 2) = 0.05 − (0.01 2) = 0.045, and

376

Chapter 12

β1 = −

2 αˆ r  αˆ  +  2  + 2 2 2 σ σ σ 2

=−

0.045 0.045  0.07 ≈ 1.35. +   + 2 ×  0.01 0.01 0.01

The required profitability index is Π* =

1.35 β1 ≈ ≈ 3.84 . β1 − 1 1.35 − 1

We can now determine the follower’s entry threshold from equation (12.4): X F * = Π*

I 100 ≈ 3.84 × ≈ 0.22. C 1778 V

The entry threshold for the leader in case of preemption cannot be readily obtained. Given the parameter values obtained above, we can, however, specify the follower’s and leader’s values from equations (12.6) and (12.8), respectively, as a function of the starting value X 0 . By rent equalization, the leader’s threshold is obtained at the process value X P * that equalizes L( X P *) = F ( X P *, X F *), that is, X P * ≈ 0.057. Figure 12.6 illustrates these value functions for the follower and the leader as given in equations (12.6) and (12.8). For X 0 ≥ X F *, both firms enter the market immediately earning the same economic profits (π C), for a net present value VC X 0 − I . For X P * < X 0 < X F *, the leader’s value, L ( X 0 ), exceeds the follower’s, F ( X 0 , X F *), as the leader enjoys a higher profit while being the sole firm active in the market. For X 0 < X P *, the option value as a follower exceeds the net present value of investing early as a leader. The leader incurs an investment cost I prohibitively large compared to the profit flow accruing to it and the value of investment commitment is largely negative. At the preemption time TɶP * ( X P * ≈ 0.057), the value as a leader exactly equals the value as follower (rent equalization). In this case (with X 0 ≤ X P *) the two firms are indifferent at time TɶP * between being the leader or the follower and would accept to be the leader (respectively, the follower) randomly, for example, on the flip of a fair coin. The actual first investor will get the leader’s value and the follower will wait until the follower’s trigger X F * (≈ 0.22 ) is first reached. There are two equilibria where the firm roles are permuted.

Preemption versus Collaboration in a Duopoly

377

L (⋅), F (⋅, XF*) 400

300

Leader’s value

200

L(⋅) Follower’s value

100

F (⋅, XF*)

0 0.05

XP* (100)

0.10

0.15

0.20

0.25

0.30

XF * ≈ 0.22 Initial value (X0)

Figure 12.6 Values and investment thresholds for the leader and follower in a new market Assume Cournot quantity competition with linear (inverse) demand p (Q) = 50 − 5Q. Firms have symmetric variable production cost, c = 10 . Investment cost is I = 100 for both firms, gˆ = 5 percent, r = 7 percent, δ = 2 percent, and σ = 10 percent. Threshold values are derived in example 12.1.

A formal description of the perfect equilibrium strategy profile is given in appendix 12B.1. For very low values of the stochastic process, there is no advantage gained from investing first. Considering mixed strategies, at the preemption time TɶP * the investment probability (intensity) is q(TɶP *) = φ ( X P *) = 0. Since equation (12.1.2) still holds in the stochastic case, the probability that each firm takes the lead at the preemption point X P * is pL = 1 2. Moreover at random time TɶP * the probability of a simultaneous investment (determined as the stochastic variant of equation (12.1.4)) is pC = 0. The expected value for the leader at the preemption point equals F ( X P *, X F *). In expectation no firm is better off in this region ( X P * < Xɶ t < X F *) since the expected value for each firm (even as the actual leader) equals the value of the follower.20 If the market starts at low levels ( X 0 < X P *), the industry 20. Let V( qi , q j ) be the expected value as a function of the equilibrium strategies of the option-holding firms. It obtains V( qi , q j ) = pL ( L + F ) + pCC . From equations (12.1.2) and (12.1.4), it results that V(qi , q j ) = [ L + (1 − q ) F − q ( L − C )] ( 2 − q ). From equation (12.2), V(qi , q j ) = F obtains.

378

Chapter 12

Probability of industry structure 100%

Monopoly (firm i)

75%

50% No investment

Duopoly

Monopoly (firm j)

25%

0% X0

0.05

XP* ≈ 0.06

0.10

0.15

0.20

0.25

X F * ≈ 0.22

0.30 ~

Market development (Xt)

Figure 12.7 Market structure evolution ( X 0 < X P *) Assume Cournot quantity competition with linear (inverse) demand p (Q) = 50 − 5Q and symmetric marginal cost, c = 10 . Investment cost is I = 100 , gˆ = 5 percent, r = 7 percent, δ = 2 percent, and σ = 10 percent. Threshold values are derived in example 12.1.

structure evolution along the market development might be as seen in figure 12.7. If the market starts at a value X 0 higher than the preemption point but lower than the follower’s investment trigger (i.e., X P * < X 0 < X F *), however, the value as a leader strictly exceeds the value as a follower. In this case the probability to invest (from equation 12.2 or 12.23 in appendix 12C.1) is positive (q ( X 0 ) > 0), and both firms invest simultaneously with positive probability ( pC = q ( 2 − q) > 0). If simultaneous Cournot investment occurs due to a “coordination failure,” each firm would receive lower value C ( X 0 ).21 The industry structure emerging at starting time t0 for different starting values X 0 is illustrated in figure 12.8. 21. Dixit and Pindyck (1994) disregard this risk, assuming that each firm becomes leader with probability one-half. Their result that each firm becomes leader over the flip of a fair coin, however, only holds if the market starts at a low value ( X 0 ≤ X P* ).

Preemption versus Collaboration in a Duopoly

379

Probability of industry structure 100% Monopoly (firm j)

75%

50% No investment

“Coordination failure”

Duopoly

25% Monopoly (firm i)

0%

X0

0.05 Low

0.10

XP* ≈ 0.06

0.15 Intermediate

0.20

X F * ≈ 0.22

0.25 High

0.30

Initial value (X0)

Figure 12.8 Market structure emerging at the beginning of the game X 0 Assume Cournot quantity competition with linear (inverse) demand p (Q) = 50 − 5Q and symmetric marginal cost c = 10 . Investment cost is I = 100 , gˆ = 5 percent, r = 7 percent, δ = 2 percent, and σ = 10 percent. Threshold values are derived in example 12.1.

The leader will only invest (at time TɶP *) if the current profit flow provides a sufficiently high return on the invested capital.22 Therefore the standard NPV rule does not hold in case of preemption.23 12.2.2 Asymmetric Case In the intermediate demand region, many option games typically involve coordination problems in determining who is the leader or follower. In pure strategies there is only one type of equilibrium characterized by a leader-follower ordering whereby firm roles are permuted. In chapter 11 22. At the preemption trigger X P*(< X F *), L ( X P *) = F ( X P *, X F *) ( > 0 ). From equations (12.6) and (12.8), L ( X P *) − F ( X P *, X F *) = (VL X P * − I ) − [VL X F * − I ] ( X P * X F *)β1 ; therefore β VL X P * = I + [VL X F * − I ] ( X P * X F *) 1 > I . This confirms the result shown in Dixit and Pindyck (1994, p. 313). 23. Assuming the follower does not make any profit (i.e., there is full preemption as in the case of a technology firm acquiring a perpetual patent), Lambrecht and Perraudin (2003) argue that the NPV threshold corresponds to the preemption point.

380

Chapter 12

we solved this coordination problem by use of a focal-point argument for asymmetric settings when one firm has a substantial competitive advantage. Using subgame perfect mixed strategies makes it possible to solve this coordination problem more generally. Here we illustrate these insights in the case of an asymmetric duopoly without relying on a focalpoint argument. We previously discussed perfect equilibrium in case of symmetric duopolists. In such a case identical firms can be reasonably assumed to follow symmetric mixed strategies. Here, following Días and Teixeira (2010), we extend the previous (new market) model to account for production cost differentials.24 Assuming asymmetry among firms is more realistic and descriptive of many duopolies. When firms are asymmetric (e.g., have asymmetric variable or fixed costs), they may follow asymmetric mixed strategies. Assume again that the “risky” profit flow πɶ consists of a deterministic component π and a multiplicative stochastic shock, Xɶ t , following the geometric Brownian motion of equation (12.3). Firms are characterized by distinct profit flows depending on the industry structure. These are summarized in table 12.1. Consider first the investment decision by the second investor (firm j ). The follower does not fear subsequent investment and therefore selects its investment strategy myopically. The optimal investment trigger for firm j as a follower, X Fj *, as in (12.4), is given by the first-order condition VCj X Fj * = Π*, I

(12.4′)

Table 12.1 Deterministic profit flows for asymmetric duopolists Certain profits Time

Industry structure

Firm i

Firm j

0 ≤ t ≤ min {Tɶi , Tɶj } Tɶi ≤ t ≤ Tɶj

No one invests

0

0 π Lj 0 π Cj

Tɶj ≤ t ≤ Tɶi

t ≥ max {Tɶi , Tɶj }

Only one invests (leader) Both firms invest

0

π Li π Ci

24. Días and Teixeira (2009, 2010) provide a review of option games in continuous time. Días and Teixeira (2009) discuss a chicken game applied to oil exploration. Pawlina and Kort (2006) study asymmetry in a setting where firms differ in the magnitude of the required investment outlay.

Preemption versus Collaboration in a Duopoly

381

where VCj ≡ π Cj δ and Π* ≡ β1 ( β1 − 1). Equivalently, in equilibrium the modified Jorgensonian rule of investment prescribes to invest at the first (random) time the process Xɶ t exceeds X Fj *, where X Fj * satisfies the relationship X Fj * π Cj I = r + (β1σ 2 2). The optimal investment trigger for firm i as a follower is determined symmetrically. The value for firm j as follower (given its optimal investment strategy), similar to (12.6), is (VCj X Fj * − I ) B0 (TɶFj *) if F j ( X 0 , X Fj *) =  j if VC X 0 − I

X 0 < X Fj *, X 0 ≥ X Fj *,

(12.9)

β1 with B0 (TɶFj *) = ( X 0 X Fj *) and β1 as given in equation (12.5). For X 0 higher than the threshold max {X Fi *, X Fj *}, each firm has a dominant strategy to invest immediately, resulting in a Cournot duopoly. The leader (firm i) has no option to wait in case of endogenous firm roles. If it invests now (at time t0), it receives the value of an irreversible investment commitment given by j j i i i (VL X 0 − I ) − (VL − VC ) X F * B0 (TɶF *) if Li ( X 0 ) =  i if VC X 0 − I

X 0 < X Fj *, X 0 ≥ X Fj *,

(12.10)

with VLi ≡ π Li δ and VCi ≡ π Ci δ . Consider again the quantity competition game discussed in earlier chapters involving a high-cost firm (denoted henceforth as firm H) and a low-cost firm (firm L). Two cases can be distinguished depending on whether the cost advantage is large or small. Example 12.2 illustrates these two cases. Example 12.2 Asymmetric Preemption We extend example 12.1 to allow for asymmetry between firms. We here consider asymmetry in terms of variable production costs, rather than in terms of fixed investment costs. Again, demand is linear with p(Q) = 50 − 5Q but subject to a multiplicative demand shock (GBM) with gˆ = 5 percent and σ = 10 percent. The risk-free rate is r = 7 percent, so δ = 2 percent. Firm L has a cost advantage, with cL = 10 < cH = 15. In asymmetric Cournot quantity competition (see table 3.2., panel B, in chapter 3), the high-cost firm would earn

π CH =

(a − 2cH + cL )2 9b

=

( 50 − 2 × 15 + 10)2 9×5

= 20.

The perpetuity value for the follower is VCH = π CH / δ = 20 0.02 = 1, 000. From this value and the required profitability index Π* derived in

382

Chapter 12

example 12.1, we can determine the high-cost firm’s investment threshold as a follower: X FH * = Π* ×

I 100 ≈ 3.84 × ≈ 0.38 . H 1, 000 VC

As a monopolist, the low-cost firm (L) would earn

(a − cL )2

π LM =

4b

=

( 50 − 10)2 4×5

= 80 ,

which yields value VLM = 80 0.02 = 4, 000 in perpetuity. The low-cost firm’s threshold as a myopic leader is X LL* = Π* ×

I 100 ≈ 3.84 × ≈ 0.10 . 4, 000 VML

In Cournot competition, the low-cost firm makes an excess profit of

π CL =

(a − 2cL + cH )2 9b

=

(50 − 2 × 10 + 15)2 9×5

= 45 ,

resulting in a perpetuity value of VCL = 45 0.02 = 2, 250. The low-cost firm’s threshold as a follower is X FL* = Π* ×

I 100 ≈ 3.84 × ≈ 0.17 . L 2, 250 VC

If the high-cost firm’s disadvantage reduces to cH = 12 , we have

π CH ′ =

(a − 2cH + cL )2 9b

=

(50 − 2 × 12 + 10 )2 9×5

= 28.8 ,

or perpetuity value VCH ′ = 28.8 0.02 = 1, 440. In this case the follower will invest earlier, at X FH * ≈ 3.84 × (100 1, 440) ≈ 0.27 (< 0.38). The leader’s threshold cannot be readily obtained but results from rent equalization. 12.2.3

Size of Competitive (Cost) Advantage

Below we consider distinct magnitudes of cost advantages and their impact on preemption and coordination in the form of joint or collaborative investment. Large Cost Advantage Consider as in example 12.2 the case of a large cost disadvantage, cH − cL = 5. In this case (see figure 12.9), assuming that firms pursue their

Preemption versus Collaboration in a Duopoly

383

Firm H value

(

)

LH(⋅), F H ⋅, XFH* 400

300

200 Firm H as a follower

100 Firm H as a leader

0

0.05

0.10

0.15

0.20

XFL * ≈ 0.17 (100)

0.25

0.30

0.35

0.40

0.45

XFH* ≈ 0.38

Initial value (X0)

Figure 12.9 Project value as a leader or follower for the high-cost firm ( H ) The demand function is p (Q) = 50 − 5Q . Costs are asymmetric, with cL = 10 and cH = 15. I = 100 , gˆ = 5 percent, r = 7 percent, δ = 2 percent, and σ = 10 percent. Threshold values are derived in example 12.2.

value-maximizing strategies, the high-cost firm (H) never has any firstmover advantage since its leader value curve is always located below its follower value curve. That is, for a cost differential higher than a given threshold, the low-cost firm (L) does not have to ever fear preemption by its competitor because the high-cost firm can never be better off preempting it; the high-cost firm thus will wait as a patient follower until the optimal threshold X FH * is first reached.25 The low-cost firm can therefore select its optimal investment trigger myopically, ignoring the investment policy of its rival. This is analogous to the open-loop equilibrium discussed in chapter 11 obtained using the focal-point argument in selecting among two pure-strategy Nash equilibria. This result confirms that the focal-point equilibrium considered earlier is appropriate and characteristic of certain industry situations where heterogeneity or asymmetry among firms is sufficiently high. Figure 12.9 depicts the project value as a leader or follower (as a function of the initial value X 0) for 25. High-cost firm H ’s value as leader is low because in equilibrium the low-cost firm enters early, making it difficult for firm H to earn sufficient temporary monopoly profits. A lower cost disadvantage would make firm L invest later. Firm H ’s value curve as leader would then come “closer” to its value curve as follower and may eventually cross it.

384

Chapter 12

the high-cost firm H given its optimal investment threshold X FH *. The value of high-cost firm H is nonsmooth at X FL* when the low-cost firm enters. The optimal investment threshold for the follower (high-cost firm H) X FH * satisfies the profitability criterion VCH X FH * = Π*, I

(12.4′′)

resulting in follower value from (12.9) of (V H X H * − I ) ( X 0 X FH *)β1 F H ( X 0 , X FH *) =  HC F VC X 0 − I

if if

X 0 < X FH *, X 0 ≥ X FH *.

(12.11)

In case of a large cost differential, the optimal investment trigger for the leader (L) or low-cost firm, X LL*, is determined by the same decisiontheoretic techniques as before. It satisfies VLL X LL* = Π*. I

(12.4′′′)

As there is no risk of preemption by its rival in this case, the expected discount factor representing the deferral option in the leader’s value expression re-appears. The leader’s value (starting in state X 0) is (VLL X LL* − I ) ( X 0 X LL*)β1  β1  − (VLL − VCL ) X FH * ( X 0 X FH *)  LL( X 0 , X LL*) = (VLL X 0 − I ) − (VLL − VCL ) X FH *  H β1  × ( X 0 X F *) V L X − I  C 0

if

X 0 < X LL*,

if

X LL * ≤ X 0 < X FH *,

if

X 0 ≥ X FH *. (12.12)

If X 0 ≥ X FH * = 0.38, both firms invest simultaneously as Cournot duopolists. The value curves for both firms are depicted in figure 12.10. The expanded-NPV for the high-cost firm (follower) exceeds its immediatecommitment NPV for X 0 < X FH *. The expanded-NPV for the low-cost firm (leader) lies above that of the follower. Small Cost Advantage Consider now the case where the cost differential is not large. Assume that either firm can acquire a first-mover advantage in the investment stage. That is, there exists a region for Xɶ t such that LH ( Xɶ t ) > F H ( Xɶ t ). In this case, from the stochastic equivalent of (12.1), the high-cost firm will

Preemption versus Collaboration in a Duopoly

(

LL(⋅), F H ⋅, XFH *

385

)

1,000

800

600 E-NPV for the low-cost firm (leader)

400 E-NPV for the high-cost firm (follower)

200

0

0.05

0.10

0.15

0.20

XLL * ≈ 0.10

0.25

0.30

0.35

0.40

0.45

XFH* ≈ 0.38

Initial value (X0)

Figure 12.10 Value curves of the low- and high-cost firms for large cost advantage The demand function is p (Q) = 50 − 5Q, cL = 10 , cH = 15 , I = 100 , gˆ = 5 percent, r = 7 percent, δ = 2 percent, and σ = 10 percent. Threshold values are derived in example 12.2.

invest with a (strictly) positive probability qH (t ). As the investment probability (density) of the low-cost firm is also positive, a coordination problem—potentially inducing a “coordination failure” in the form of joint investment—may arise. In contrast to the previous case involving a low-cost firm with a large cost advantage enabling it to ignore the risk of preemption by the weaker rival, here a preemption threat emerges that can induce an earlier investment by the low-cost firm. The threat of preemption arises because firms are not precommited, and strategic interactions are accounted for via mixed (closed-loop) strategies. Owing to this threat of competitive preemption, firm i cannot simply wait until TɶLi * ≡ inf t ≥ 0 Xɶ t ≥ X Li* to invest. To see this, suppose that firm i decides to invest at the myopic investment trigger of the leader X Li* ( < X Fj *). It could then be preempted by firm j investing at an earlier threshold, which would force firm i to invest even earlier, and so on. The ensuing preemption war would end when the rival’s rents are equalized. The preemption point for firm i is such that

{

}

386

Chapter 12

{

}

X Pi * = inf Xɶ t < X Fj *Lj ( Xɶ t ) = F j ( Xɶ t ) . Firm j would not be willing to select a lower investment trigger than X Pi * at which a second-mover advantage rather than a first-mover advantage would result for it; that is, Lj ( X j ) < F j ( X j ) if X j < X Pi *. In the intermediate demand region where there is a risk of preemption, the coordination problem can be solved by use of mixed strategies. A formal description of the solution is given in appendix 12C.1. Three possible industry structures may emerge. The probability of occurrence of each can be determined from equations (12.1.1) and (12.1.3′) and from the investment density in equation (12.21) given in appendix 12C.1. Preemptive investment timing sequences where one firm temporarily earns monopoly profits arise with positive probability (two distinct orderings). Firm i becomes leader with probability pLi, with firm j being the follower. Conversely, firm j takes the lead with probability pLj. The probabilities for the leader-follower orderings from (12.1.1) are pLi =

qi (1 − q j ) qi + q j − qi q j

and

pLj =

q j (1 − qi ) (≠ pLi ). qi + q j − qi q j

(12.13a)

A third possible industry structure occurs when firms invest simultaneously as Cournot duopolists; from (12.1.3) this occurs with probability pC =

qi q j . qi + q j − qi q j

(12.13b)

If one of the firms, firm L, has a cost advantage over its rival, cL < cH, its investment intensity will be higher than its rival’s, that is, qL ( X 0 ) ≥ qH ( X 0 ). This results in pLi ≥ pLj; hence the low-cost firm is more likely to be the leader. Nonetheless, it might still happen that the high-cost rival enters first and the low-cost firm invests second (with probability pLH ≥ 0). In the intermediate region (with X PL* < X 0 < X FL* and X PH * < X 0 < X FH *), both firms’ investment densities are strictly positive. Therefore there is a positive probability of simultaneous joint investment, resulting in lower value for both firms since F L ( X 0 , X FL*) > C L ( X 0 , X CL*) and F H ( X 0 , X FH *) > C H ( X 0 , XCH *). Depending on the parameters of the investment triggers and on the starting demand region, the resulting industry structures may differ. We distinguish two subcases. A. Low-cost firm’s preemption point above leader’s myopic trigger If the starting value is lower than the low-cost firm’s myopic investment trigger, meaning X 0 ≤ X LL*, the low-cost firm invests as a leader when

Preemption versus Collaboration in a Duopoly

387

X LL* (< X LH *) is first reached. The high-cost firm invests later as a follower at random time TɶFH * ≡ inf t ≥ t0  Xɶ t ≥ X FH * . If X LL* ≤ X 0 ≤ X PL*, the low-cost firm’s myopic threshold is exceeded and there is no preemption threat from the high-cost firm. The low-cost firm invests immediately (at t0) and the high-cost firm waits to invest until random time TɶFH *. If X PL* < X 0 < X FH *, there exists a preemption threat and both firms have a first-mover advantage. The low-cost firm invests as the (sole) leader with probability

{

pLL =

}

qL (1 − qH ) > 0, qL + qH − qL qH

(12.14a)

as a follower with probability pFL = pLH =

qH (1 − qL ) > 0, qL + qH − qL qH

(12.14b)

or as a simultaneous Cournot investor with probability pC =

qL qH > 0, qL + qH − qL qH

(12.14c)

where the equilibrium investment densities are qL = φ H ( X 0 ) and qH = φ L( X 0 ), with φ j (⋅) defined in appendix 12C.1, equation (12.2.1). If X 0 ≥ X FH *, both firms invest immediately resulting in a Cournot duopoly with asymmetric payoffs. B. Low-cost firm’s preemption point below leader’s myopic trigger If X 0 < X PL* ≤ X PH *, both firms will wait. When X PL* is first reached at time TɶPL*, the low-cost firm invests as a leader; before this point there is no investment. The high-cost firm invests at time TɶFH *. If X PL* < X 0 < X PH *, there is a coordination problem. Three industry structures may emerge. The low-cost firm becomes the leader and the high-cost firm enters as follower at time TɶFH *, with probability pLL as given in equation (12.14a). The high-cost firm takes the lead and the low-cost firm becomes follower and invests at time TɶFL*, with probability pFL = pLH as per equation (12.14b). Simultaneous Cournot investment occurs with probability pC, as per equation (12.14c). If X PH * < X 0 < X FH *, the low-cost firm invests immediately and the high-cost firm waits until time TɶFH *. If X 0 ≥ X FH *, both firms invest immediately resulting in a Cournot duopoly. This is illustrated in figure 12.11. In the asymmetric case described above, the preemption equilibrium where the leader invests earlier at time TɶPL* (instead of at the myopic

388

Chapter 12

(

)

(

)

LL(⋅), F H ⋅, XFH * , LH(⋅), F L ⋅, XFL * 600 500 400

Firm L as follower

300 Firm L as leader

200 100

Firm H as follower Firm H as leader

0 (100)

0.05

X PH* X PL*

0.10

0.15

0.20

X FL* ≈ 0.20

0.25

0.30

0.35

X FH * ≈ 0.27 Initial value (X0)

Figure 12.11 Value curves of the low- and high-cost firms for small cost advantage Here ( X PL * < X LL *) . The demand function is given by p (Q) = 50 − 5Q, cL = 10 , cH = 12 , I = 100 , gˆ = 5 percent, r = 7 percent, δ = 2 percent, andσ = 10 percent. Threshold values are derived in example 12.2.

trigger TɶLL*) occurs if the cost advantage of the low-cost firm is relatively small. When this is the case, firms must take account of the strategic interaction (preemption threat), leading to earlier investment. As the cost difference (asymmetry) approaches zero, the result converges to the symmetric case where firm values are equalized (rent dissipation). By contrast, the open-loop sequential ordering equilibrium (where the leader invests when the myopic trigger X LL* is first reached) results if there is a sufficiently large cost advantage for the low-cost firm. In this case the low-cost firm acts myopically as if it has a proprietary investment option (as a monopolist) with no threat of preemption by its rival and invests at random time TɶLL*. Nonetheless, its option value is eroded by the subsequent competitor’s entry, occurring at random time TɶFH *. The analysis above helps stress the importance of attaining sufficient cost advantage in a dynamic setting. In a static setting, such an advantage enables the firm to extract more value from the market than its competitor. From a dynamic viewpoint, it also allows it to extract monopoly rents for a longer period of time. If the cost advantage is sufficiently large, it renders the threat of preemption by the weaker rival irrelevant.

Preemption versus Collaboration in a Duopoly

389

The highly cost-advantaged firm can confidently invest at the myopic random time TɶLL*, ignoring its rival’s entry timing altogether, though it is still subject to damage from competitive erosion. If its cost advantage is small, however, it needs to enter at an earlier preemption time TɶPL* due to strategic interaction and timing rivalry (closed-loop equilibrium). 12.3

Option to Expand an Existing Market

We next consider expansion decisions in an existing market involving two firms already active in the marketplace. Both firms have a “shared” option to make an irreversible investment to increase their current profit flow by expanding their existing market. For example, the firms may have the possibility to adopt a new, more efficient technology that improves the quality or reduces the production cost of their existing products through enhanced processes. Without loss of generality, we focus on the situation where firms have the opportunity to invest in additional production capacities. In section 12.3.1 we consider the symmetric duopoly case, and outline the results for the asymmetric case in section 12.3.2. 12.3.1

Symmetric Case

Suppose that the stochastic profit flow is again made of two components: Xɶ t is a multiplicative exogenous industry shock modeled as a geometric Brownian motion as per equation (12.3), while π (= π 0 , π L , π F , π C ) indicates the deterministic (reduced-form) profit flow under a given industry structure. Upon expanding its production capacity by making additional investment I , a firm can potentially make a higher profit π L ( > π 0 ). Once the leader has invested in additional capacity, the follower—previously on an equal footing with the leader (earning π 0)—will experience a lower profit π F (< π 0 ) if it has not also invested in new capacity. If both firms have invested, they will face once again symmetric Cournot competition—though now duopoly profits are higher than under the old industry structure (π C > π 0) due to enhanced production.26 The leader suffers from capacity additions by its rival as π C < π L. Suppose that there is a higher incentive to invest as a leader rather than as a follower in that the profit value increment from leadership (π L − π 0) is larger than the profit increment received as follower upon expanding capacity (π C − π F ). The 26. Boyer, Lasserre, and Moreaux (2010) model capacity expansion more explicitly in a model where capacities can be expanded repeatedly in lump sums. They obtain, under certain circumstances, that the deterministic profit under symmetric capacity expansion is lower than under the initial industry capacity. In this case tacit collusion takes the form of simultaneous investment being delayed forever.

390

Chapter 12

above described differences with respect to profit values drive somewhat different results for “existing market models” involving the option to expand capacity compared to “new market models” involving merely the option to invest. In particular, the drop in the follower’s profit between the old industry structure (π 0) and the industry state following the leader’s added investment (π F) may induce collaborative (tacit-collusion) equilibrium. This feature also held in Fudenberg and Tirole’s (1985) deterministic setting discussed above. This effect is not present when firms are not yet invested (wait to invest). The optimal joint-investment trigger X C * at which collaboration (or tacit collusion) results in joint-value maximization satisfies

(VC − V0 ) X I

C

* = Π *,

(12.15)

where VC ≡ π C δ and V0 ≡ π 0 δ are the perpetuity values of the deterministic profit flows π C and π 0, δ ≡ k − g = r − gˆ , and Π* ≡ β1 (β1 − 1) is the profitability index, with β1 given in equation (12.5). When the firms choose to invest in additional capacities simultaneously, they both effectively “exchange” their current profit flow π 0 for the higher profit π C. The resulting positive value increment in case of joint investment is VC − V0 . The profit flow for the follower decreases once the leader has invested. The investment trigger for the follower is driven by the value differential between the region where it has not yet invested and the region where it is once again on an equal footing with the leader (after expansion) invests. This value differential is VC − VF. The (myopic) investment trigger for the follower, X F *, satisfies

(VC − VF ) X I

F

* = Π*,

(12.16)

where VF ≡ π F δ is the perpetuity profit value as a follower. Under the new market model involving the option to invest discussed in the previous section, the follower earns no profits while waiting to invest, that is, π F = π 0 = 0. Hence it suffers no damage when the leader invests. For this reason the trigger values, X F * and XC *, were the same. Moreover the value curve of the follower coincided with the jointinvestment value curve. Thus, in the case of new market models, joint investment and collaboration (tacit collusion) do not emerge as possible equilibria. By contrast, in the existing market model involving the option to expand capacity, X F * < XC *.

Preemption versus Collaboration in a Duopoly

391

In a Pareto-dominant equilibrium, firms would coordinate (or tacitly collude) and refrain from investing until the stochastic state variable X t has reached the larger threshold XC * ( > X F *) given in equation (12.15). We refer to this Pareto-superior joint-investment equilibrium as “collaborative” or “tacit collusion” equilibrium (with subscript C). Among all possible joint-investment equilibria, it is the more plausible. The value functions as a leader, follower, or collaborator are given by (VL X 0 − I ) − (VL − VC ) X F * B0 (TɶF *) if L ( X0 ) =  if VC X 0 − I

X 0 < X F *, X 0 ≥ X F *,

VF X 0 + (VC − VF ) X F * − I  B0 (TɶF *) if F ( X 0 , X F *) =  if VC X 0 − I V0 X 0 + (VC − V0 ) XC * − I  B0 (TɶC *) if C ( X 0 , XC *) =  if VC X 0 − I

(12.17a)

X 0 < X F *, X 0 ≥ X F *, (12.17b) X 0 < XC *, X 0 ≥ X C *, (12.17c)

β β where for GBM B0(TɶF *) = ( X 0 X F *) 1 and B0 (TɶC *) = ( X 0 XC *) 1 , with β1 as in equation (12.5). It is interesting to examine two benchmark cases. In the first case, there is a first-mover advantage; namely the leader’s value from immediate investment, L ( X 0 ), exceeds the collaboration value from jointly waiting until the common trigger XC *, C ( X 0 , XC *), which corresponds to the first case considered in the deterministic setting. In the second case, such a first-mover advantage does not exist and L ( X 0 ) ≤ C ( X 0 , XC *).

A. First-mover advantage and preemptive investment In this case, where L ( X 0 ) > C ( X 0 , XC *), the probability of having invested depends on the overall industry history. For low past values of the process, no firm has already invested, but if the process value gets higher than the preemption point X P *, at least one firm will have expanded capacity. It is not clear, however, which firm will actually be the first investor. For very low state value Xɶ t , there is no advantage to investing first. At time TɶP * the values as leader and follower are equal. At this preemption time qi (TɶP *) = 0 (from equation 12.21 in appendix 12C.1) and the probability of firm i being leader and firm j being follower is 1/2, as seen from equation (12.1.2). The probability of simultaneous investment from equation (12.1.4) is zero. The expected value for the leader is L ( X P *). The perfect equilibrium strategy profile is given in appendix 12C.1.

392

Chapter 12

For state values in the intermediate region, X P * < X 0 < X F *, the value as leader exceeds the value as follower. From equation (12.23) in appendix 12C.1 the probability of investing is strictly positive with a positive probability of simultaneous investment. If such a “coordination failure” happens, each firm receives the lower joint value C ( X 0 , X C *). No firm is better off since the expected value for each firm (even for the actual leader) equals the value of the follower. For X 0 ≥ X F *, the equilibrium outcome is joint investment at time t0. The value to each firm corresponds to the net present value from immediate investment commitment, C ( X 0 , XC *) = VC X 0 − I. B. Collaborative investment In the case when the collaborative value exceeds the leadership value, C ( X 0 , X C *) ≥ L ( X 0 ), there are many symmetric equilibrium strategies. These can be divided into two classes. The first class consists of the sequential role orderings described in case A. The second class consists of strategies where firms invest simultaneously in a collaborative fashion. This second class of equilibrium strategies forms a continuum; these equilibrium profiles are described in appendix 12C.2. The interpretation is basically the same as in the deterministic case. Simultaneous investment as part of a collaborative equilibrium will take place the first time the common threshold that maximizes joint value is reached (the Paretooptimal equilibrium is the most likely among the simultaneous investment equilibria). In comparing cases A and B above, it may not be readily clear under which circumstances investment takes place in a preemptive sequence or when simultaneous joint investment at a later date occurs as part of a cooperative relationship.27 As the reader might intuit, the fixed investment cost I does not alter the relative attractiveness of the preemptive timing equilibrium outcome compared to the tacit collusion one—it only affects the level of the investment thresholds but in the same proportion. In cases of large uncertainty (high volatility σ), a high growth rate g , or low discount rate (high discount factor), the outcome from strategic interaction is typically that of the collaborative type. Conversely, for a low degree of uncertainty, low growth, or a high discount rate (low discount factor), firms may find it appealing to do their utmost to grasp a first-mover advantage, likely resulting in a preemptive industry equilibrium. The speed of investment increases if market conditions favor the preemptive type of equilibrium. 27. For that it is useful to perform comparative statics analysis as done by Huisman and Kort (1999) and Huisman (2001, app. B).

Preemption versus Collaboration in a Duopoly

393

12.3.2 Asymmetric Case As noted, heterogeneity among option holders may change equilibrium outcomes altogether. First, a myopic (open-loop) sequential investment equilibrium where the leader invests at time TɶL* ignoring its rivals may re-emerge as the industry equilibrium if the cost-advantage of the lowcost firm is sufficiently high. If the cost advantage is relatively small, the preemption threat precipitates the investment decision of the low-cost firm leading it to invest at an earlier time that does not necessarily maximize its stand-alone value. Existing market models may additionally exhibit the third type of equilibrium, that of a joint or collaborative simultaneous investment. This case is analyzed by Pawlina and Kort (2006).28 Asymmetry among firms results in distinct beliefs about the optimal joint investment thresholds, with the low-cost firm having a “collaborative” investment trigger lower than that of the high-cost firm. Actual investment thus occurs at the time the lower (low-cost firm’s) trigger is first reached. What matters from a strategic perspective is to understand the key drivers that lead to these different types of equilibria. A primary driver is the presence of a first-mover advantage, typically a market characteristic. A second key driver relates to the size of the competitive advantage by the advantaged firm, a firm characteristic. When the competitive (e.g., cost) advantage is relatively small and there is only a slight first-mover advantage, the outcome tends to be that of a collaborative equilibrium involving joint investment at a later (random) investment time TɶCL*. When no firm has a significant first-mover advantage, it is preferable to invest simultaneously at a later optimal time, jointly appropriating the option value of waiting against exogenous demand uncertainty, than to engage in detrimental timing rivalry resulting in cutthroat preemption. If the cost advantage is small but there is a substantial first-mover advantage, the low-cost firm may opt to reject the collaborative joint investment alternative and, driven by fear of preemption, invest early at time TɶPL*; the high-cost firm will invest when the follower’s investment trigger X FH * is first reached. These results are in line with the ones obtained under firm symmetry in the previous section. If there is both a sizable first-mover advantage and a substantial cost advantage, the 28. Pawlina and Kort (2006) consider an asymmetric model where firms have different fixed investment costs. Their results may be extended to cases where firms have asymmetric variable production costs, as in Días and Teixeira (2010), for the new market model case. Different variable production costs allow for distinct reduced-form profits.

394

Chapter 12

Large

Sequential investment ~ (open loop,TL∗) Competitive (cost) advantage

Collaborative (joint) investment ~ (TC∗)

Preemptive timing investment ~ (closed loop,TP∗≤TL∗)

Low Low

First-mover advantage

Large

Figure 12.12 Investment strategies and different types of equilibria depending on first-mover advantage and the size of competitive (cost) advantage “Collaborative (joint) investment” involves simultaneous investment at (random) time TɶC * ; In case of “preemptive timing investment,” the leader invests at an early preemption time TɶP * ( ≤ TɶL*); “sequential investment” refers to the (“soft”) case where the leader need not consider the preemption risk. The separating curves are not necessarily linear.

open-loop sequential investment where the low-cost firm invests as a monopolist ignoring its rivals will likely be the industry equilibrium.29 In specific market settings, the open-loop sequential investment is more likely to occur than either preemption or delayed collaborative simultaneous investment. Figure 12.12 summarizes the different types of equilibrium outcomes that may result depending on certain market or firm characteristics, such as the magnitude of the first-mover advantage and the size of competitive (cost) advantage. Collaborative or joint investment refers to the decision made by duopolists to invest simultaneously at a later random optimal time TɶC *; preemptive timing investment is characterized by a sequential investment where the leader invests at an early time TɶP * ( ≤ TɶL*) due to strategic interactions arising from the threat of preemption (closed-loop equilibrium); sequential investment refers to the 29. The outcome that sufficiently large cost advantage may induce firms to invest in a “soft” (accommodating) investment sequence that does not exhibit preemption was discussed in sections 12.2.2 and 12.2.3. Here we put this result in relative perspective and compare it to the collaborative outcome.

Preemption versus Collaboration in a Duopoly

395

case (discussed in chapter 11) where a firm having a substantial competitive advantage invests myopically as a monopolist at time TɶL*, ignoring rival effects (open-loop equilibrium). Trigeorgis and Baldi (2010) discuss a slightly different option game setting and derive similar insights. The authors assess the value of optimal patent leveraging strategies under demand and strategic uncertainty, illustrating how the magnitude of competitive advantage arising from the patented innovation and the level and volatility of demand affect the optimal patent strategy. They show the following: 1. When there is no competitive advantage and rivals are symmetric, collaboration (e.g., via cross-licensing) is a natural equilibrium across demand states. 2. When one of the rivals has a large comparative advantage via a superior patent, a fight mode is likely as the equilibrium strategy. However, the precise patent leveraging strategy may differ across demand states. It may range from offensive fighting (via bracketing each other’s patents resulting in a patent war) if demand is high, to raising a defensive patent wall by the advantaged firm (to strengthen its relative position and drive out the rival) if demand is medium (with room for just one firm in the market), to a defer (patent sleep) strategy maintaining an option on a future monopoly position should the market recover if current demand is low, with the disadvantaged rival abandoning the market if demand is insufficient. 3. In case of a small competitive advantage at intermediate levels of demand, the firm is better off to pursue a flexible hybrid strategy, switching from a fight mode (e.g., bracketing) at high demand to collaboration (licensing out) as demand declines. Conclusion In this chapter we provided an overview of several important investment timing and equilibrium issues arising in option games. We first discussed the simpler deterministic setting and explained how preemption threat can impact the optimal investment timing decisions of firms. We subsequently discussed how to extend the analysis to take account of uncertainty and strategic aspects arising in real options analysis. We analyzed in turn the investment timing option (new market model) and the option

396

Chapter 12

to expand (existing market model) in a competitive duopoly setting under uncertainty.30 We have seen that the presence of a first-mover advantage and the size of a comparative cost advantage are the main drivers in such strategic investment decisions. We have shown that an aggressive stance toward one’s rival (preemption) is not always the preferable modus vivendi among firms in a duopolist industry. Firms may sometimes find it preferable to coordinate their entry or investment timing decisions depending on the state and evolution of the market. Investing in a more efficient production technology (achieving lower marginal operating cost) may result in a “soft” ordering (open-loop investment sequence) where the leader does not suffer from preemption threat. But the preferred strategy is not obvious as it depends on both market and firmspecific characteristics that merit closer assessment. This chapter provided useful insights on when to lead, when to follow, and when to collaborate with rivals under uncertainty. Selected References Fudenberg and Tirole (1985) refine Reinganum’s (1981a) model discussing continuous-time mixed-strategy equilibria. They set the foundations for deterministic games of timing, including the analysis of preemption. Smets (1991) extends Fudenberg and Tirole’s framework to a stochastic setting analyzing an existing market model (expansion option). Dixit and Pindyck (1994) present a simplified version based on a new market model (investment timing option). Huisman and Kort (1999) provide a comprehensive view on this problem. Días and Teixeira (2010) analyze production cost asymmetries in such a context, while Pawlina and Kort (2006) consider asymmetric investment costs. Thijssen, Huisman, and Kort (2002) discuss an extended definition of strategy spaces that enables handling timing issues in stochastic environments. Huisman et al. (2004) review option games contributions dealing with new versus existing market models in a lumpy-investment context. Días, Marco A. G., and José P. Teixeira. 2010. Continuous-time option games: Review of models and extensions. Multinational Finance Journal 14 (3/4): 219–54. 30. Bouis, Huisman, and Kort (2009) recently investigate such problems in an oligopoly context with more than two firms.

Preemption versus Collaboration in a Duopoly

397

Dixit, Avinash K., and Robert S. Pindyck. 1994. Investment under Uncertainty. Princeton: Princeton University Press. Fudenberg, Drew, and Jean Tirole. 1985. Preemption and rent equalization in the adoption of new technology. Review of Economic Studies 52 (3): 383–401. Huisman, Kuno J. M., and Peter M. Kort. 1999. Effects of strategic interactions on the option value of waiting. CentER discussion paper 9992, Tilburg University, The Netherlands. Huisman, Kuno J. M., Peter M. Kort, Grzegorz Pawlina, and Jacco J. J. Thijssen. 2004. Strategic investment under uncertainty: Merging real options with game theory. Zeitschrift für Betriebswirtschaft 67 (3): 97–123. Pawlina, Grzegorz, and Peter M. Kort. 2006. Real options in an asymmetric duopoly: who benefits from your competitive disadvantage? Journal of Economics and Management Strategy 15 (1): 1–35. Reinganum, Jennifer F. 1981a. On the diffusion of new technology: A game-theoretic approach. Review of Economic Studies 48 (3): 395–405. Smets, Frank. 1991. Exporting versus FDI: The effect of uncertainty, irreversibilities and strategic interactions. Working paper. Yale University. Thijssen, Jacco J. J., Kuno J. M. Huisman, and Peter M. Kort. 2002. Symmetric equilibrium strategies in game-theoretic real option models. Discussion paper 2002–81, CentER, Tilburg University, Tilburg, The Netherlands. Appendix 12A: Strategy Space and Solution Concept Players’ strategies are defined in the following fashion. A continuation strategy si (t ) for firm i in a subgame starting at time t consists of a pair of strategy functions si ( t ) ≡ (Gi ( t ) , qi ( t )) such that:31 1. Gi (⋅) is a cumulative distribution function reflecting the history of the industry. It represents the probability that firm i has invested before or at time t ( ≥ t0 ) given that the other firm has not yet invested. 31. For a more detailed discussion on the definition of strategy space and closed-loop strategy, refer to Fudenberg and Tirole (1985) or Thijssen, Huisman, and Kort (2002). We use here the term “continuation strategy” instead of “simple strategy” as in Fudenberg and Tirole (1985) or Thijssen, Huisman, and Kort (2002) since, in our opinion, the term “continuation” better captures the fact that the strategy applies to the subgame starting at t ≥ t0 that is concatenated in the whole game starting at t0 .

398

Chapter 12

2. qi (⋅) represents the instantaneous (mixed) action taken by firm i at time t. It is a notion of the probability (“intensity” or “density”) with which firm i invests at time t. qi (⋅) is an atoms function in optimal control theory. It represents discrete-time mixed-strategy measures which are lost when one considers only the distribution function Gi (⋅).32 The strategic-form game depicted in figure 12.1 is played as soon as one of the firms invests with positive probability (qi ( t ) > 0 or q j (t ) > 0), namely at time T ≡ min {Ti ,Tj }, where Ti ≡ inf s ≥ tqi ( s ) > 0 or Ti ≡ ∞ if qi ( t ) = q j ( t ) = 0 for all t ≥ t0 . The game is played repeatedly in “rounds” at time t, with no time elapsing between “rounds.” A pair of continuation strategies ( si * (t ) , s j * (t )) is a Nash equilibrium of the game starting at time t if each player’s strategy maximizes her payoff given the optimal strategy of its rival. A closed-loop strategy si = {si ( t )}t ≥ t0 is a collection of continuation strategies specifying for each subgame at time t, which continuation strategy to pursue.33 A pair of closed-loop strategies ( si *, s j *) forms a subgame perfect Nash equilibrium if for every time t ( ≥ t0 ) the pair of continuation strategies ( si * (t ) , s j * (t )) is a Nash equilibrium. The definition above of closedloop equilibrium is a continuous-time translation of the subgame perfection solution concept prescribing that players act optimally (as part of Nash equilibrium) at every subgame (here, the subgame at future time t ≥ t0).

{

}

Appendix 12B: Perfect Equilibrium in Deterministic Setting Here we provide a formal representation for the equilibrium investment density and specify the perfect equilibrium strategy profile. We define TP* ≡ inf t ≥ t0 L(t ) = F (t ) , TF* ≡ arg max t ≥ t0 F (t ), and TC* ≡ arg max C( t ).

{

}

32. The way used, for example by Pitchik (1982), to describe a player’s strategy involves the function Gi (·) as a nondecreasing, right-continuous cumulative probability that player i has invested conditional on the other player not having invested before. This approach allows for mixed strategies but is refined by Fudenberg and Tirole (1985) for two reasons. First, it does not specify what happens in all possible subgames, hindering the use of the subgame perfect equilibrium concept that prescribes continuation strategies that form Nash equilibrium for all subgames, even those off the equilibrium path. Second, the function Gi (·) fails to be a continuous-time analogue to the equivalent discrete-time game of timing because of loss of information when taking the limit. 33. Fudenberg and Tirole (1985, p. 393) and Thijssen, Huisman, and Kort (2002, p. 11) impose some intertemporal consistency conditions on the closed-loop strategies.

Preemption versus Collaboration in a Duopoly

12B.1

399

Case L(TL* ) > C(TC* )

Since playing the strategic-form game takes no time, the value for firm i, V i (qi , q j ), as a function of the instantaneous investing probabilities qi = qi (t ) and q j = q j ( t ), can be expressed in a recursive manner as V i ( qi , q j ) = qi q j C i (t ) + qi (1 − q j )Li (t ) + (1 − qi )q j F i (t ) + (1 − qi )(1 − q j )V i ( qi , q j ) obtaining, for (qi , q j ) ≠ (0, 0), V i ( qi , q j ) =

qi q j C i ( t ) + qi (1 − q j ) Li (t ) + (1 − qi ) q j F i (t ) . q j + qi − qi q j

The first-order optimization condition gives ∂V i (qi *, q j *) = 0, or qj * [(1 − qj *) Li (t ) − F i( t ) + qj * C i (t )] = 0. ∂qi The concavity of V i (⋅, q j ) is confirmed from the second-order derivative. Applying these conditions to the above (asymmetric) expression leads to equation (12.1). The cumulative distribution function for the symmetric equilibrium when L(TP* ) > C(TC* ) is given by 0 if t < TP *, G (t ) ( = Gi ( t ) = Gj ( t )) =   1 if t ≥ TP *. Together with the (symmetric) instantaneous investing intensity (probability) if t < TP *, 0  q (t ) = φ ( t ) if TP * ≤ t < TF *, 1 if t ≥ TF *,  with φ (⋅) given in (12.2) this constitutes the perfect closed-loop equilibrium in the preemption case involving symmetric firms. This equilibrium strategy is interpreted as follows. In the period prior to the preemption time TP *, there is no incentive for either firm to invest, so no one invests (G ( t ) = 0 and q ( t ) = 0 if t < TP *). In the period after the preemption time TP *, one firm in the industry will invest, such that G( t ) = 1 for t ≥ TP*. For t > TF *, the follower firm invests. qi (·) indicates the equilibrium intensity of investment: for a burgeoning market, firms stay out; for t ∈[TP *, TF *] they mix their investment decision; for a mature market, they invest with probability 1.

400

Chapter 12

Case L(TL* ) £ C(TC* )

12B.2

In a duopoly with identical firms where L (TL*) = C (TC ) ≤ C (TC *), the following symmetric strategies result in a perfect equilibrium: G ( t ) ( = Gi (t ) = Gj ( t )) =

{

0 if 1 if

t < T, t ≥ T,

and q (t ) =

{

0 if 1 if

t < T, t ≥ T,

(12.18)

for any T ∈[TC , TC *]. Among them, the Pareto-superior equilibrium is characterized by the following symmetric strategy profiles: G ( t ) ( = Gi ( t ) = Gj (t )) =

{

0 if 1 if

t < TC *, t ≥ TC *,

and q (t ) =

{

0 if 1 if

t < TC *, t ≥ TC *.

(12.19)

Appendix 12C: Perfect Equilibrium in Stochastic Setting 12C.1

Case L( X 0 ) > C( X 0 , XC* )

The closed-loop equilibrium investment strategy for firm i again consists of two functions: a cumulative distribution function Gi (⋅) and an investment intensity function qi (⋅). In the asymmetric case the perfect equilibrium strategy for firm i (firm j’s optimal strategy being obtained symmetrically) is given by 0 if t < TɶPi * (or max { X s ; 0 ≤ s ≤ t} < X Pi *), Gi ( t ) =  i i  1 if t ≥ TɶP * (or max { X s ; 0 ≤ s ≤ t} ≥ X P *), and 0 if t < TɶPi * (or max { X s ; 0 ≤ s ≤ t} < X Pi *),  qi ( t ) = φ j ( t ) if TɶPi * ≤ t < TɶFi * (or X Pi * ≤ max {X s ; 0 ≤ s ≤ t} < X Fi *),  if t ≥ TɶFi * (or max { X s ; 0 ≤ s ≤ t} ≥ X Fi *), 1 (12.20)

Preemption versus Collaboration in a Duopoly

401

where in the intermediate region

φ j (⋅) ≡

Lj (⋅) − F j (⋅) . Lj (⋅) − C j (⋅)

(12.21)

For the symmetric case the cumulative distribution term simplifies to 0 if t < TɶP * (or max {X s ; 0 ≤ s ≤ t} < X P *), G(t ) ( = Gi (t ) = Gj (t )) =   1 if t ≥ TɶP * (or max { X s ; 0 ≤ s ≤ t} ≥ X P *). The probability of entering during the next instant t becomes 0 if t < TɶP * (or max {X s ; 0 ≤ s ≤ t} < X P *),  q ( t ) = φ ( t ) if TɶP * ≤ t < TɶF * (or X P * ≤ max {X s ; 0 ≤ s ≤ t} < X F *),  if t ≥ TɶF * (or max {X s ; 0 ≤ s ≤ t} ≥ X F *), 1 (12.22) where

φ (t ) ≡

L (t ) − F (t ) . L (t ) − C (t )

(12.23)

The probability of having invested is related to the overall industry history. For low current values of the process Xɶ t , namely for Xɶ t such that max {X s ; 0 ≤ s ≤ t}, no firm has already invested. If the process reaches values higher than the preemption point X P *, at least one firm has invested. 12C.2

Case C( X 0 , XC* ) ≥ L( X 0 )

Symmetric collaborative (or tacit-collusion) equilibria are pairs of strategies of the form 0 if t < Tɶ Gi (t ) =  1 if t ≥ Tɶ

(or max {X s ; 0 ≤ s ≤ t } < X ), (or max {X s ; 0 ≤ s ≤ t } ≥ X ),

and 0 if t < Tɶ qi (t ) =  1 if t ≥ Tɶ

(or max {X s ; 0 ≤ s ≤ t } < X ), (or max {X s ; 0 ≤ s ≤ t } ≥ X ),

where X ∈[ XC , X C *] and XC is such that L ( XC ) = C ( XC , XC ) and Tɶ = inf {t ≥ 0 | Xɶ t ≥ X }. The choice XC * ∈[ XC , XC *] that maximizes joint value is Pareto optimal and may be considered more likely to be implemented in the industry. The Pareto-optimal collaborative

402

Chapter 12

equilibrium consists of the following strategy profiles (for i and j symmetrically): 0 if t < TɶC * (or max {X s ; 0 ≤ s ≤ t} < X C *), Gi ( t ) =  1 if t ≥ TɶC * (or max {X s ; 0 ≤ s ≤ t} ≥ XC *),

(12.24)

and 0 if t < TɶC * (or max {X s ; 0 ≤ s ≤ t} < XC *), qi ( t ) =  1 if t ≥ TɶC * (or max {X s ; 0 ≤ s ≤ t} ≥ XC *).

(12.25)

13

Extensions and Other Applications

In the previous chapters we discussed how to analyze option games and discussed how they can provide powerful insights into how firms (should) behave when they face an option to defer investment as well as strategic interactions, potentially leading to early preemptive investment. We restricted the discussion primarily to models of complete information and ignored potential time lags between the investment decision and effective entry in the market. In this chapter we discuss extensions of the option games framework that allow for a time lag or time to build, for technological uncertainty and for information asymmetry. This chapter also serves to provide an overview of other important contributions to the analysis of real options and strategic competition in a dynamic setting.1 The analysis of option games in such contexts brings about additional insights and helps explain various real-world industry phenomena, such as waves in real-estate markets. Below we review briefly specific research contributions to option games analysis. Section 13.1 deals with early approaches where competitive entry decisions are treated exogenously. Applications to the realestate sector are presented in section 13.2. Section 13.3 elaborates on multistage R&D or patent strategies. Section 13.4 addresses information asymmetry among option holders and how it may affect investment strategies. Models dealing with exit are also addressed in section 13.5. When a market or industry declines, a firm may wait for its rival to exit first hoping to enjoy monopoly rents. Situations where firms can reduce their capacity utilization to cushion against 1. The edited book by Grenadier (2000a) provides a good collection of articles on gametheoretic option models. Grenadier (2000b) illustrates how the intersection of real options and game theory provides powerful insights into the behavior of economic agents under uncertainty with applications in real estate and oil exploration.

404

Chapter 13

market downturns are addressed in section 13.6. Section 13.7 considers situations where firms make sequential lumpy investment decisions. The last section provides a broad overview of recent developments in the field and other extensions or applications. 13.1

Exogenous Competition and Random Entry

Early research focused on aspects of competition that can be modeled exogenously. This approach helped identify certain major drivers underlying option games. Trigeorgis (1991) studies the impact of competition on optimal investment timing using standard contingent-claims analysis based on the geometric Brownian motion. Consider an option-holding firm facing the introduction of close substitute products. Such “competitive arrivals” may reduce the value of the firm’s own investment opportunity by taking away market share. Competition in this context can be modeled in one of two ways, depending on whether competitive entry is anticipated or random: Anticipated competitive erosion This can be treated analogous to an opportunity cost or dividend yield reducing the firm’s incentive to defer investment.2 If an option-holding firm decides to wait longer, it risks suffering competitive damage analogous to forgone “dividends,” just as the holder of a financial call option on a dividend-paying stock forgoes cash dividend payments if it holds the option and does not exercise early. If instead the firm invests early and thereby preempts its rivals, it “keeps” the “dividends” that would otherwise be lost. •

• Random rival arrivals Random arrivals of substitute products by competitors can be modeled via the Poisson term in a mixed-jumpdiffusion process with mean arrival rate λ . At the time of a random rival entry, the incumbent’s profit value suddenly drops from monopoly rents to duopoly profits, creating a value discontinuity captured by the jump term. The mean arrival rate λ represents the random competitive arrival rate or the instantaneous probability of a random competitive entry causing a downward jump. Trigeorgis (1991) shows that the value of a shared investment opportunity characterized by such exogenous random competitive entry is a weighted sum (or expected value over a Poisson distribution) of Black–Scholes option values with a dividend yield

2. In market equilibrium the total return of a project ( k ) equals the expected capital gain ( g ) plus the dividend payout (δ ). A higher dividend yield implies a lower proportion of capital gains, g / k .

Extensions and Other Applications

405

increased by an additional “dividend payout” term whose magnitude depends on λ (competitive intensity).3 This simplified framework modeling competitive entry exogenously can help an option-holding firm determine whether to wait despite anticipated or random competitive value erosion and assess the value of the deferral option in the midst of competition erosion. Reiss (1998) develops a framework to determine whether, and when, a firm should patent or adopt an innovation in a setting where rivals arrive randomly following a Poisson process. She identifies several option exercise strategies applied in a context where competition is exogenously determined. The exogenous approach, however, is limited in that it does not explain what drives competitors’ entry decisions. Competitive arrivals are simply assumed exogenous driven by some external process in competitive markets. An endogenous modeling approach focused on explaining the drivers behind competitors’ entry decisions and strategic interaction from a game-theoretic perspective is more appropriate for oligopolistic industries. 13.2

Real-Estate Development

Grenadier (1996) develops a duopoly model involving time lags or “time to build” that provides insights into the forces that shape certain market behaviors in the real-estate market. The approach helps explain several puzzling real-estate phenomena, such as booms and bursts. Real-estate markets are characterized by sudden large development efforts in some periods while smoother development patterns are observed in other periods. The analysis helps identify the factors that make some markets prone to bursts of concentrated development, explaining why such markets sometimes experience building booms in the face of declining demand and property values. Consider two symmetric real-estate developers having the possibility to refurbish their buildings for an investment outlay I , increasing their profit stream accordingly. The two buildings currently yield a constant profit stream of π 0 per unit of time. If only one building is refurbished, this building will earn a stochastic profit stream of Xɶ t π L , with Xɶ t following a geometric Brownian motion (where π L > π 0). This action nevertheless will affect the rival as the deterministic profit flow for its (nonrefurbished) building changes 3. Trigeorgis (1991) assumes a finite planning horizon, making it possible to use the Black– Scholes option pricing formula.

406

Chapter 13

from π 0 to π F ( Xɶ t ) ≡ π F , where π F < π 0 . If the second developer also decides to renovate its building, both firms will earn a stochastic profit stream of X t π C per unit of time (where π C < π L). Grenadier takes into account that real estate development involves time-to-build delays, thereby affecting the standard real options investment rule under uncertainty. In many situations, undertaking an investment takes time. There may be several years between the decision to invest and the time at which the project gets completed and revenues get generated. Suppose that refurbishing the old building takes D years until completion. During this delay period the owner cannot receive rents from the building. The follower’s optimal threshold previously given by equation (12.4) is now adjusted to VC X F * = Π* e −δ D , I′

(13.1)

where VC ≡ π C δ and I′ is the total cost involved in exercising the option, namely the investment cost I plus the value of the foregone perpetual rent stream from the old building, π F δ . The profitability index is again given by Π* =

β1 . β1 − 1

(13.2)

Depending on initial demand, the equilibrium strategies pursued by the two developers lead to four distinct scenarios. For Xˆ t < X P * with Xˆ t ≡ max {X s ; 0 ≤ s ≤ t}, the follower’s value strictly exceeds the leader’s and no refurbishment investments will take place in equilibrium. At the preemption threshold, X P *, the two real-estate developers are indifferent between the leader role or the follower role (rent equalization). In Grenadier’s model the actual leader is chosen randomly with probability one half.4 For X P * ≤ Xˆ t < X F *, only one firm invests. For Xˆ t ≥ X F *, both developers decide to renovate their premises and their values are equal. A tacit collusion scenario may arise for certain values of the underlying process.5 Grenadier (1996) establishes the existence of two classes of equilibria. Depending on initial conditions, some equilibria are characterized by sequential development while others by simultaneous 4. Grenadier (1996) assumes a strictly sequential equilibrium entry ordering where one firm is selected as leader over the flip of a fair coin. He suggests that this mechanism resembles real-estate developers applying for permits with only one receiving an initial approval. Huisman and Kort (1999) point out that this result holds only for X 0 < X P . 5. In reality there exists a continuum of tacit-collusion equilibria. Game theorists (e.g., Fudenberg and Tirole, 1985) often assume that the Pareto-efficient equilibrium is selected.

Extensions and Other Applications

407

investment. For a burgeoning market (starting at X 0 < X L*), a sequential investment equilibrium occurs with one firm investing at time TɶP * and its rival waiting to invest until time TɶF *. For a mature market ( X 0 ≥ X F *), the sequential investment equilibrium is ruled out. Both firms will wait for the Pareto-optimal collaborative trigger XC * to be reached and invest simultaneously. Based on this framework, Grenadier (1996) derives a rationale for certain investment behaviors observed in real-estate markets, such as investment cascades and recession-induced construction booms (RCB). During an investment cascade, a number of real-estate projects are being developed concurrently over a sustained time period. This is measured by the time interval between refurbishments, namely the median time span between the leader’s and the follower’s investments. Recessioninduced construction booms characterize periods where, despite an economic downturn, brand new buildings are being completed and commercialized. With ( X F *, X C *) being the interval for initial demand, TɶF * (TɶC *) is the first time when X F * ( X C *) is hit. If TɶF * < TɶC *, a recessioninduced construction boom (RCB) results as both developers will invest in fear of being preempted, even though the market demand is low. Probability P (TɶF * < TɶC *) indicates the likelihood of occurrence of a RCB. Comparative statics for the median time span and for P (TɶF * < TɶC *) enable identifying key drivers for such real-estate market phenomena. Time to build and depreciation of old buildings have no effect on the mean time span, but volatility does. Volatile markets are more likely to exhibit construction cascades. The time-to-build delay is a major driver, increasing the likelihood of occurrence of recession-induced construction booms. The inferences above are confirmed by empirical real-estate studies, helping us understand these puzzling real-estate market phenomena. Based on data relating to more than 1,200 real-estate projects in Vancouver, Canada, between 1979 and 1998, Bulan, Mayer, and Somerville (2009) provide empirical support to the argument that competition erodes option value. 13.3

R&D and Patenting Applications

Despite much technological progress made in the last decades, R&D investment decisions remain challenging. Many technology firms put technology adoption as a primary goal. At the same time firms face more competition than ever. Option games can help provide powerful insights to understand competitive technology-driven industries, giving valuable

408

Chapter 13

investment timing guidance to management. Several authors use option games to derive insights into how firms should conduct R&D or use their patents as a strategic weapon. These models help explain how competition affects firms’ research programs and the portfolio of intellectual property rights. Huisman (2001) discusses various types of models of technology adoption. He identifies decision-theoretic models of investment under uncertainty, involving technology adoption policies by a monopolist firm. He then discusses game-theoretic models in a deterministic setting. Finally, he combines the two approaches to deal concurrently with both market as well as competitive uncertainty. Although Huisman (2001) focuses on technology adoption, some of his insights are tractable and applicable in other contexts, such as when to enter a foreign market. Huisman discusses symmetric and asymmetric models and examines the effects of negative versus positive externalities on the equilibrium outcome. In technology-driven markets the availability of new technologies is uncertain, so the expected speed of the arrival of new technologies affects the optimal investment sequence. Huisman and Kort (2004) examine technology adoption in a duopoly given the possible future arrival of an improved technology, allowing for uncertain arrival time following a Poisson process. Weeds (2002) analyzes a patent race among duopolist firms and the effect of competitive pressure on firms’ research strategies under a winner-takes-all patent system. Uncertainty takes two forms: the technological success of research activity is probabilistic, while the economic value of the patent evolves stochastically. The author compares equilibrium strategies obtained in this setting with a socially optimal benchmark, deriving implications for research policy. A central planner can steer research in one of two ways: allocating R&D investment to decentralized research units or setting up a common aggregate research center with a single investment policy. Suppose that two identical firms have the opportunity to launch an R&D project by investing I . The first firm to succeed will gain an exclusive patent yielding a stochastic profit flow Xɶ t , while the other firm will be left with nothing. A firm’s R&D strategy sets a profit trigger at which research activity is initiated. Weeds derives first the optimal behavior imposed by a social planner on two decentralized research units. The social planner would optimally choose to phase the research. In the socially optimal scenario, one firm (the leader) starts conducting research in hope of acquiring the patent when profit reaches a specified threshold level X L*; the follower initiates research later on,

Extensions and Other Applications

409

when the socially optimal threshold X F * is first attained. This threshold is such that XF * δ + 2λ ⎞ = Π* ⎛⎜ ⎝ λ ⎟⎠ I

(13.3)

with Π * as per equation (13.2). If firms behave as a single centralized research center, as opposed to independent research units, the optimal trigger is X C ′, such that X C ′ I = Π* [(δ + 2λ ) λ ]. This threshold XC ′ is between X L* and X F *. The centralized collaborative research setup leads to later investment compared to the case of independent research units (as X L* < XC ′). Weeds (2002) shows that the choice made by a centralized research unit may be socially suboptimal. This contrasts with standard antitrust thinking regarding joint research ventures. These findings challenge conventional policies that aim to foster research cooperation. Mason and Weeds (2002) consider further strategic interactions between option-holding firms. A firm considering being the leader might be hurt by the follower’s entry (negative externality) or benefit from it (positive externality, e.g., due to network effects). They consider the irreversible adoption of a technology whose returns are uncertain when many firms are active in the market, focusing on the effect of uncertainty and externalities on the type of investment schedule (sequential vs. simultaneous investment). The combination of preemption and negative externalities can hasten investment compared to the myopic benchmark. Option value is eroded when firms are faced with the fear of competitive preemption. Two patterns of adoption emerge: sequential vs. simultaneous investment. With no first-mover advantage and no preemption, the leader adopts the new technology at the simultaneous cooperative trigger point; otherwise, a preemptive sequential investment occurs where the follower adopts earlier than the cooperative solution. Miltersen and Schwartz (2004) analyze patent-protected R&D investments with imperfect competition in the development and commercialization of a product. Strategic interactions in this case substantially alter the investment decisions of a stand-alone firm. Option holders have to take into account not only the standard factors that directly affect their own decisions, but also the impact of their competitors’ decisions on their own investment policy. Competition in R&D can increase aggregate production and reduce prices. It can also shorten the time necessary to develop the product and raise the probability of successful development.

410

Chapter 13

These benefits to society are offset by increased R&D investment costs and lower value from the R&D investment for each firm. Time-to-build delays can also affect the innovative investment strategies of firms. In R&D, such delays before completion of a project are rarely known in advance as there is additional uncertainty over the innovation success. Technological uncertainty is characteristic of many industries, such as bio-tech, IT, and oil exploration. Weeds (2000) examines the uncertainty over innovation completion and its impact on the duopolists’ technology adoption decisions. She considers two-stage R&D investment processes with completion uncertainty and their implications for sleeping patents. The firm invests in R&D with the aim to acquire a patent giving it exclusive access to a new market. If the first-stage innovation is successful, the firm can make an irreversible investment to adopt the new technology and enter the market. This two-stage investment opportunity can be viewed as a compound option where the value of the first-stage research option partly derives from the second-stage commercial investment option. This framework provides a rational explanation for the existence of sleeping patents, that is, patents granted but kept in a stand-by or “sleep” mode. Policy makers typically regard sleeping patents as anticompetitive devices employed by dominant firms to erect entry barriers (blockaded entry). However, in this context sleeping patents may arise purely from optimizing behavior when option values coexist with completion uncertainty. Sleeping patents thus do not necessarily indicate anticompetitive behavior. By restricting a firm’s ability to time entry in the product market (with the possibility to let a patent sleep if optimal under uncertain conditions), compulsory licensing imposed by antitrust authorities can actually reduce option values and weaken the firm’s incentive to conduct research in the first place. Lambrecht (2000) derives optimal investment strategies for two symmetric firms sharing the option to make a two-stage sequential investment with incomplete information about the rival’s profit. In the first stage each firm is competing to acquire a patent enabling it to proceed to the second stage involving commercialization of the invention. The optimal investment policy for the first stage reveals a trade-off between the benefit of waiting to invest under uncertainty and the cost of being preempted. Lambrecht derives a condition under which inventions are likely to be patented without being put to immediate commercial use. Sleeping patents are more likely to exist in an R&D portfolio when interest rates are low, volatility is high, and the second-stage cost is high relative to the first-stage cost.

Extensions and Other Applications

411

Smit and Trigeorgis (1997) analyze an R&D investment problem where the underlying R&D value is affected by strategic interactions. Firms choose output endogenously and may have different production costs as a result of their R&D effort success, their roles (leader or follower), and their learning experience. Firm incentives to conduct R&D depend on market and technological uncertainty, the proprietary or shared nature of R&D benefits, firms’ information asymmetry or learning experience, and competition versus cooperation (e.g., joint venture) in the R&D stage. Garlappi (2004) analyzes the impact of competition on the risk premia of R&D ventures. Firms are engaged in a multiple-stage patent race under technical and market uncertainty. The case of a two-stage race admits a closed-form solution whereby the firm’s risk premium decreases due to technical progress but increases when a rival pulls ahead. R&D competition erodes the option value to delay a project, reduces the completion time, and lowers the failure rate of R&D. It generally causes higher risk premia. Azevedo and Paxson (2009) discuss investment in new technologies whose functions are complementary, determining the leader’s and the follower’s values and their investment thresholds. At the outset firms have two technologies with the possibility to adopt both. Their results challenge the common rule of thumb that a firm should upgrade or replace its production processes simultaneously if they involve complementary inputs. Faced with uncertain revenues and technology costs, the leader and the follower may be better off adopting the technologies at different times, adopting first the technology with slowly decreasing cost and later the technology with a more rapid cost decline. 13.4

Investment with Information Asymmetry

Real life is often characterized by information asymmetry among rival firms. For instance, a firm may have better knowledge of its own cost structure or the probability of success of its own R&D efforts. Through their option exercise decisions, investing firms may convey signals to other firms that enable learning or revision of their prior beliefs. The development of an office building, the drilling of an exploratory oil well, or an application by a pharmaceutical company for regulatory approval of a new drug convey (bad or good) news to competing firms, which may alter their initially planned behavior accordingly. We discuss

412

Chapter 13

discrete-time option models first, and then present continuous-time analyses. Zhu and Weyant (2003a, b) consider two firms facing stochastic linear (inverse) demand of the form given in equation (7.1) with a = 1, namely P( Xɶ t , Q) = Xɶ t − bQ, where Q = qi + q j is the total industry output and Xɶ t is an additive stochastic shock (demand intercept) as in Smit and Trigeorgis (2004). Firms face linear marginal costs, Ci (qi ) = ci qi . Firm j has complete information, whereas firm i knows its own cost ci but not its rival’s, believing it to be cH with probability P or cL with probability 1 − P. Firm i’s expectation about its rival’s cost is c j = PcH + (1 − P ) cL. If firms invest simultaneously, firm j selects equilibrium quantity q*j =

1 ɶ ( X t − 2c j + ci ) 3b

(13.4)

in knowledge of both costs, its own cost being cH or cL. Firm i instead forms expectations about its rival’s quantity and optimally selects the output based on the rival’s expected cost c j: qi* =

1 ɶ ( X t − 2ci + cj ). 3b

(13.5)

In case of sequential investment decisions, the outcome may be different. That is, the effect of incomplete information crucially depends on the order of the investment decisions. If the less-informed party (firm i) moves first, the sequential incomplete-information equilibrium is derived analogously to the simultaneous one. However, if firm j invests first, it will reveal its private cost information through its quantity choice (Cournot–Nash quantity). Information asymmetry effectively collapses, so that quantities are chosen as in the complete-information duopoly game, that is, as if having knowledge of the leader’s cost, cL or cH . Following continuous-time analysis, Lambrecht and Perraudin (2003) consider the effect of incomplete information on optimal timing in a duopoly game where stochastic profit flow Xɶ t follows a geometric Brownian motion. No firm knows the exact realization of their rival’s investment cost but each has some prior beliefs about it in the form of a known distribution G( I ). There exists a Bayesian equilibrium that maps firm i’s investment cost I i into firm i’s investment threshold. Since both firms are (ex ante) symmetric, firm i’s exercise strategy involves a

Extensions and Other Applications

413

mapping from the distribution G( I j ) to a belief Fj ( X j ) for the rival’s investment trigger. The authors assume that the market is incontestable once a leader has entered, as is the case when the market is fully protected by a patent. The patent allows disregarding the entry of the follower and concentrating solely on the leader’s optimal timing decision and the interplay between preemption and information asymmetry. Assuming a distribution function Fj (⋅) for the rival’s investment trigger X j , firm i updates its beliefs about its rival’s investment strategy by observing whether the latter decides to invest when a new (running) maximum max {X s ; 0 ≤ s ≤ t} is reached.6 Accounting for the risk of preemption, firm i’s optimal investment threshold is X Pi such that [VL ( X Pi ) I ] = Π*′, where VL ( X Pi ) = X Pi δ and Π*′ ≡

β1 + hj ( X Pi ) , β1 − 1 + hj ( X Pi )

(13.6)

where β1 is given in equation (9.12) and hj (·) is the hazard rate hj ( x) = xFj ′ ( x ) [1 − Fj ( x)]. The authors derive firm i’s myopic threshold X Li, when firm i ignores its rival’s action, yielding value VL ( X Li ) I = Π*, with Π * as per equation (13.2). They show that the threshold in case of information asymmetry is located between the zero-NPV (preemption) threshold and the monopolist’s myopic threshold.7 Given the optimal timing decision, firm i’s value is ⎛X ⎞ Li0 X t , Xˆ t = [VL ( X Pi ) − I i ] ⎜ i0 ⎟ ⎝ XP ⎠

(

)

β1

⎡ 1 − Fj ( X Pi ) ⎤ ⎥. ⎢ ⎢⎣ 1 − Fj Xˆ t ⎥⎦

( )

(13.7)

If firm i does not exactly know its rival’s investment cost, I , but knows its distribution, there exists a mapping from the firm’s investment cost to its optimal investment trigger X Pi . Murto and Keppo (2002) consider an investment game where the follower loses any possibility to enter. They characterize the resulting Nash equilibrium under different assumptions concerning the information that the firms have about their rivals’ valuation. Thijssen, Huisman, and 6. Rival’s inaction at a new high allows a firm to update its belief about the rival’s investment trigger and thus about its rival’s investment cost (inaction by the rival is “good” news). While the updating process raises the value of each firm, it does not alter the firms’ investment strategies. When finally one of the firms invests, the rival’s value drops to zero. 7. As in the case of preemption with complete information, the rent-equalization principle suggests that at the preemption point the leader’s value equals the follower’s (here equal to zero) so that the complete-information preemption trigger corresponds to the zero-NPV threshold.

414

Chapter 13

Kort (2006) consider a market where two firms compete for investing in a risky project. They incorporate both a first-mover advantage and a second-mover advantage in terms of information spillovers resulting from option exercise. Depending on specific parameter choices, either the first or the second-mover advantage may dominate, leading to preemption or a war of attrition game. Interestingly, more competition does not necessarily lead to higher social welfare. This ultimately relates to the intensity and informativeness of signals. Duopoly leads to higher welfare than monopoly if there are few and relatively noninformative signals, whereas the opposite holds if there are many and relatively informative signals. Grenadier (1999) analyzes a general setting where agents formulate option exercise strategies under imperfect information. Suppose that the payoff received upon entry is not perfectly known to the firms, each having received an independent private signal concerning the true underlying value. A firm has a prior belief about the distribution of its rivals’ signals. The firm may nevertheless infer its rivals’ private signals by observing their entry decisions, updating its beliefs when new entry triggers are attained. This setting provides a general foundation for solving many problems arising under imperfect information. In such settings investment strategies can generate equilibrium outcomes that differ significantly from the standard full-information equilibrium outcome. However, while in many cases observed exercise decisions may convey valuable private information, there are other instances when no useful information can be inferred. In such cases an information cascade may emerge where firms disregard their private information and invest in a rush following others as in a herd behavior. This may occur when optionholding firms find their private information overwhelmed by recent information conveyed by others and adapt their behavior accordingly. Markets with large information asymmetry may experience smooth exercise patterns over time, while markets with milder information asymmetries may sometimes experience a rapid series of investments or information cascades. Martzoukos and Zacharias (2001) study project value enhancement in the presence of incomplete information and R&D spillover effects. Duopolists may raise project value by conducting R&D and/or gathering more information about the project. Due to information spillovers, they act strategically by optimizing their behavior conditional on the actions of their rival. Maeland (2010) combines real options theory with auction theory to develop a winner-takes-all investment model for

Extensions and Other Applications

415

markets with two or more players with asymmetric information about the cost of investment. Each investor knows its own costs but ignores its rival’s cost. Décamps and Mariotti (2004) consider a duopoly in which firms learn about the attractiveness of a project by observing some public signal and their rival’s actions. Firms have symmetric information about the signal realization but asymmetric information about their rival’s investment cost. The authors examine the learning externality. By delaying investment, each firm tries to convince its rival that its own cost is high so that the rival should invest first. The resulting “war of attrition” results in a unique symmetric equilibrium. 13.5

Exit Strategies

Exit decisions in declining markets may also involve strategic interactions among incumbent firms. Once again, the decisions made by duopolists may differ from what a monopolist would do. Firms may wait for their rival to exit first, hoping to enjoy rents in a monopolistic industry structure. This may lead to a war of attrition. Fine and Li (1989) complement deterministic duopoly models of exit, such as Ghemawat and Nalebuff (1985, 1990) or Fudenberg and Tirole (1986), by allowing for the stochastic decline of a market. The authors show that the sequence of exit is not unique due to “jumps” in the demand process. Sparla (2004) analyzes closure or exit options for a duopoly where firms face a stochastically declining market in continuous time. The optimal timing game is viewed as a war of attrition or chicken game. In this setting an aggregate shock (modeled as a geometric Brownian motion) affects the profit value received by the two firms.8 It is seen that the equilibrium exit policies in a symmetric duopoly differ significantly from the disinvestment trigger of a monopolist or a firm holding a proprietary option. Duopolists disinvest or exit later than a monopolist.9 The follower is faced with a decision-theoretic optimal-stopping problem wherein the timing decision is not affected by the leader’s behavior because the follower’s value does not depend on the leader’s exit policy. 8. Similar to previous models where investment is considered irreversible, re-entering the market after having exited is ruled out. 9. A firm’s exit strategy consists of a cumulative distribution G ( Xɶ t ) indicating whether the firm has exited at state Xɶ t or not. In cases involving perpetual American call options (e.g., option to invest) the threshold is attained from below. Inversely, in the present case where we deal with a put option the threshold is reached from above.

416

Chapter 13

The follower (firm j) will exit the first time X Fj is attained, where the threshold X Fj is given by10 X Fj VFj = Π* S

(13.8)

with VFj = π Fj / δ where the “convenience yield” is δ = k − g = r − gˆ . Π* is a lower cutoff profitability level to be reached at the time of exit. It is given by Π* ≡

β2 , β2 − 1

(13.9)

where β 2 is the negative root of the fundamental quadratic given in equation (A2.3) in the appendix at the end of the book:

β2 ≡

−αˆ − αˆ 2 + 2rσ 2 σ2

(< 0 )

with αˆ ≡ gˆ − (σ 2 2). In this war of attrition both firms have an incentive to wait until the rival exits first or until market conditions deteriorate so badly that both firms are better off leaving the market irrespective of their rival’s action. For identical firms the unique symmetric strategy profile in equilibrium is for both firms to exit the first time X F , given by equation (13.8), is hit. The symmetric equilibrium profile where both firms exit at TɶF is the best achievable outcome from the viewpoint of the industry. In case of asymmetric firms with small cost differential, the low-cost firm may possibly exit earlier. If the cost differential is large, however, the high-cost firm exits first and the low-cost firm stays longer. Murto (2004) considers a similar problem in which duopoly firms differ in terms of production scale (with firm i smaller than firm j). Here firms face a multiplicative aggregate demand shock Xɶ t following a geometric Brownian motion. The deterministic component of the market’s (inverse) demand function is of the constant-elasticity type:

π ( Xɶ t , Qt ) = Xɶ t Qt−1 η,

(13.10)

10. For the sake of comparability between models, Sparla’s (2004) model was simplified by assuming full closure, that is, exit rather than partial closure (as in the original paper). We also simplify the expression for the payoff received upon exiting. Sparla (2004) and Murto (2004) (discussed later) are more explicit, decomposing this value into operating cost savings made upon exiting the market (positive value) and costs incurred to make exit effective (e.g., layoff costs). We simplify using a single aggregated salvage value S.

Extensions and Other Applications

417

where η ( > 1) is the elasticity of demand. The resulting exit thresholds, X Li for the first exiting firm (leader) and X Fj for the last one (follower), are such that X Fj VFj = Π* S

(13.11)

and X Li VLi = Π*, S

(13.12)

where the value functions are VFj = π Mj / δ and VLi = π Ci / δ , with δ = k − g = r − gˆ and Π* as given in equation (13.9). The leader’s willingness to stay in the market increases with volatility because its put option value is increased. For low levels of volatility, there is a unique exit sequence where the smaller firm (i) exits first (leads) when threshold X Li obtained from equation (13.12) is first reached and the largest firm ( j) follows suit when X Fj from equation (13.11) is reached for the first time. For high volatility levels, however, this equilibrium is no longer unique and the reverse ordering with the largest firm exiting first may also obtain as industry equilibrium. 13.6

Optimal Capacity Utilization

Models dealing with investment in incremental capacity generally assume that output production is costless and capacity is fully utilized (constant returns to scale). Aguerrevere (2003) relaxes this assumption and obtains a mean-reverting price process exhibiting volatility spikes, even when the (additive) stochastic demand shock or demand intercept Xɶ t follows a geometric Brownian motion.11 He assumes an oligopolistic market with n identical firms facing a linear (inverse) demand for a perishable good of the form P( Xɶ t , Q) = Xɶ t − bQ.12 At each period firms choose their optimal level of capacity utilization. Firms can incur a cost of I per unit, expanding capacity incrementally at any point in time. Capacity expansion takes D years to implement (complete). To account for time to build, a new (Markov) committed capacity state, K t = Qt + Qt ′ = Qt + D, tracks 11. Such price evolutions have traditionally been explained through the dynamics of storage. Aguerrevere’s (2003) model complements this explanation. In his model the price behavior arises from the interplay among installed capacity, capacity utilization, and time to build. 12. Firms operate identical production facilities. The variable cost incurred by a firm is a function of capacity utilization υ ∈[0; 1]; the cost function is C (υ ) = c1υ + (c2υ 2 2).

418

Chapter 13

the sum of capacity units currently operational Qt and units currently under construction Qt ′ (“in the pipeline”). If an extra unit of capacity always at least breaks even (as it can be shut down at no cost), capacity units under construction can effectively be considered as a set of European call options (with maturities t + D) on the net profit from an extra unit of capacity. The option to launch the construction of a unit of capacity is thus analogous to an American call on this set of European calls. The value of each European call option depends only on Xɶ t and K t, as all committed capacity will be operational prior to the earliest maturity t + D.13 The trigger demand level X M (Kt ) for exercising an expansion option given the currently committed capacity is obtained. Whenever the current demand level Xɶ t exceeds X M (K ), a monopolist firm will commit additional capacity. For an n-firm oligopoly, a symmetric equilibrium results such that the expansion threshold is identical to the monopoly case. That is, oligopolistic firms will add capacity at the exact same time as a monopolist would. However, both the committed and operational oligopoly capacity are strictly greater.14 Surprisingly, utilization of operational capacity is independent of the number of firms. Without time-tobuild delays, aggregate committed capacity Kt is decreasing in the underlying volatility. However, with time to build, firms may provide more capacity if faced with greater uncertainty. This arises from the trade-off between the increased risk of capacity underutilization and the higher uncertainty increasing the value of capacity under construction.15 The output price paths exhibit mean reversion and significant spikes in times of full capacity utilization. Due to time-to-build delays, completion of capacity expansion is preceded by a phase of high utilization and high prices. Industry capacity increases with the number of firms, but the timing of expansion and the utilization rate are independent. Aguerrevere (2009) relaxes the assumption of time to build and uses linear operating costs. He examines the firm’s systematic risk (beta) in a competitive setting. The firm’s beta is determined as the weighted average of the beta of assets in place and the beta of the firm’s growth options.16 In line with Grenadier (2002), the value of growth options decreases with the number of firms and approaches zero when n tends to infinity (as in 13. Given this relationship, the partial differential equation describing the value evolution of the monopolist’s option to invest in an extra unit of capacity can be derived. 14. The oligopoly quantities can be expressed as the corresponding monopoly capacity times a constant multiplier. 15. Capacity under construction is analogous to a set of European call options whose value is strictly increasing in volatility. 16. Weights are determined based on the present values of assets in place and of growth options.

Extensions and Other Applications

419

perfect competition). This affects the firm’s beta since the beta of the assets in place and their weight increases with the number of firms active in the industry, n. Under intensified competition, the capacity held in the industry is utilized more in response to demand increases. Regardless of the number of firms, assets in place are generally less risky when demand is high as capacity utilization is increased. In case of geometric Brownian motion, the growth option’s beta is constant, meaning it is independent of industry capacity, demand level or the number of firms. For high demand, the firm’s beta decreases with the number of rivals, but for low demand it increases. 13.7

Lumpy Capacity Expansion (Repeated)

Previous models typically assumed that each firm has only one investment option. Murto, Näsäkkälä, and Keppo (2004) consider multiple investment opportunities available to firms. Consider a data bandwidth market with two incumbent firms. Suppose that the demand function is of the constant-elasticity type as in equation (13.10).17 Once the infrastructure (transmission capacity) is installed, providing bandwidth is effectively costless. Starting with zero initial capacity, firm i (symmetrically firm j) can decide at any time to invest I i (I j) to increase capacity by a lump sum ΔQi (ΔQj). In each period the firms set their output and earn the market-clearing price for each unit sold. Investment decisions are sequential, with the first mover being randomly chosen. In any given state firms maximize current profits by selling the Cournot–Nash quantity under capacity constraints. While this quantity choice constitutes a tactical decision, firms also decide on whether to expand capacity. Any single investment subgame is fully described by current firm capacities, the level of market demand and the current time, enabling derivation of the optimal expansion (Markov) strategies via dynamic programming. Investment-inducing demand thresholds characterize these expansion strategies. As these thresholds are increasing in a firm’s installed capacity, the smallest firm is more likely to respond to small demand shocks by expanding capacity. The authors examine the asymmetric case where firm j must invest in large lumps compared to its rival, but this disadvantage is compensated by a lower outlay per unit of capacity added. The outcome in this asymmetric case stands in sharp contrast to the case 17. The GBM is discretized in a CRR binomial lattice. The firms’ continuation values in the final period are determined as perpetuities assuming steady state for future capacity.

420

Chapter 13

involving symmetric firms.18 Firm i’s payoff distribution has distinctly more probability mass at higher payoff levels than firm j’s. Firm i benefits from the asymmetry although it has higher costs per unit of capacity because it can react quicker to changes in demand, effectively investing more often than its rival. This response flexibility advantage exceeds the cost disadvantage regarding the necessary investment outlay. Boyer, Lasserre, and Moreaux (2010) refine previous contributions on strategic investment developed in a deterministic setting, notably Gilbert and Harris (1984), Fudenberg and Tirole (1985), and Mills (1988). The authors characterize the development of a stochastically growing duopoly market and analyze industry dynamics in a setting where firms build capacities through multiple irreversible lumpy investments. The firm roles are endogenously determined and commitment to a rigid long-term development program is not credible.19 Initially, two firms have low capacities. When firms do not hold any existing capacity, tacit-collusion equilibria are ruled out because firms are not threatened by the loss of existing rents. The only possible equilibrium in such burgeoning industries is the preemptive sequential (closed-loop) equilibrium. This timing rivalry causes the first industrywide investment to occur earlier than what would be socially optimal from the viewpoint of industry participants. This distortion implies riskier entry and lower expected returns. Rent equalization occurs irrespective of the volatility or the speed of market development, but higher volatility may cause the first industry investment to occur earlier. Later if firms hold capacity, two types of equilibria may arise. The threat of preemption is real and may lead to the complete dissipation of any first-mover advantage as in a preemptive sequential equilibrium. Simultaneous investment is also possible and sustainable as industry equilibrium: such tacit-collusion episodes or investment waves take the form of postponed simultaneous investments by both firms. Such equilibria are more likely to exist in highly volatile or fast-growing markets. When firms are of equal size, this is compatible with joint-profit maximization. However, when firm asymmetry is large the joint investment threshold is beyond the level that maximizes value for the disadvantaged firm. The possibility of collusion or cooperation is more attractive to symmetric firms than to (sufficiently) asymmetric 18. In the symmetric case the firms’ payoffs are distributed identically. Compared to the payoffs of a monopolist, the aggregate duopoly payoff in the asymmetric case is lower and more skewed to the left; both the lower market power in duopoly and the threat of preemption (hastened increased capacity buildup) drive this result. 19. Boyer et al. (2004) assume that reduced-form stage profits are the outcome of Bertrand price competition, whereas Boyer, Lasserre, and Moreaux (2010) consider Cournot quantity competition.

Extensions and Other Applications

421

ones. At later stages of development, when firms hold substantial capacity, competition is weaker and tacit-collusion equilibria may persist. In such a context, the conventional real options result that higher volatility leads to later investment is reinforced by the switch from the preemptive to the tacit-collusion equilibrium at higher volatility levels. 13.8

Other Extensions and Applications

In this section we provide more breadth of application covering other related literature. Cottrell and Sick (2002) examine competitive value erosion arising when decision makers anticipate preemptive entry by rivals. Although market pioneers may gain first-mover advantage, the authors show that followers may have important advantages as well. Boyer and Clamens (2001) examine why in the US reengineering projects often fail due to internal resistance to corporate changes or lack of commitment from management. Extending a model proposed by Stenbacka and Tombak (1994), the authors show how efficient implementation programs affect the adoption-timing decisions in a duopoly. They also examine the effect of first vs. second-mover advantages and adoption costs on adoption timing. Kong and Kwok (2007) examine a duopoly involving asymmetric firms in terms of investment costs and uncertain revenue flows. Paxson and Pinto (2003a) consider a duopoly where the leader’s market share follows a birth and death process, determining the follower’s and leader’s values. They examine the sensitivity of the firms’ value to changes in market shares, process parameters and volatility. Paxson and Pinto (2003b) discuss two distinct duopoly models. In the first, both the unit price and demand (number of units sold) are random. In the second, returns and investment costs are subject to economic shocks. Kulatilaka and Perotti (1998) discuss growth options under uncertainty and imperfect competition. A first-mover advantage may lead to capture a greater market share, either by deterring entry or by inducing rivals to leave more room for a stronger competitor. When the strategic advantage is strong, increased volatility fosters investment in growth options, while lower volatility might reduce the incentive to do so. This challenges the common result that higher volatility delays investment. Nielsen (2002) considers positive externalities and scenarios where a firm has multiple investment opportunities. With decreasing profit flow, a monopolist invests later than a leader in a preemption scenario. The

422

Chapter 13

number of available investment projects does not affect the timing of the first investment. A monopolist will make its second investment earlier than the follower. Wu (2006) analyzes capacity expansion in an optiongame setting where two symmetric firms choose both investment timing and capacity levels. Demand grows until an unknown date and declines thereafter. Firms may enter or exit depending on demand realization. Williams (1993) considers the option to develop an asset under stochastic demand uncertainty, deriving the optimal investment strategies and resulting firm values under perfect equilibrium. The author assesses the impact of development capacity, of the supply of underdeveloped assets, and the concentration of developers. In Nash equilibrium, symmetric developers build at the maximum feasible rate whenever income rises above a critical value. The optimal building rate depends on a stochastic exogenous factor and affects aggregate demand. Weyant and Yao (2005) study the sustainability of tacit collusion equilibrium in case of information time lags. For a longer time lag, preemptive equilibrium is more likely to arise. Inversely, for a sufficiently short time lag, tacit collusion may sustain whereby firms delay investment more than a monopolist. Odening et al. (2007) show that myopic planning may lead to suboptimal investment strategies. They identify the degree of suboptimality and propose measures to reduce the discrepancy. Botteron, Chesney, and Gibson-Asner (2003) model production and sales delocalization flexibility for multinational firms under exchangerate risk. Depending on industry structure, firms may act strategically and exercise their delocalization options preemptively at an endogenously set exchange rate. Shackleton, Tsekrekos, and Wojakowski (2004) analyze the entry decisions of competing firms in a duopoly when rival firms earn distinct but correlated economic profits. In the presence of sunk entry costs, firms face hysteresis or delay effects, the range of which is decreasing in the correlation parameter. They determine the expected entry time and the probability that both firms enter within a given time interval. They illustrate their model in the duopoly situation faced by Boeing and Airbus, focusing on Airbus’s A380 lunch and Boeing’s best strategic response. Pineau and Murto (2004) apply similar thinking to the deregulated Finnish electricity market subject to stochastic demand growth, determining firm strategies in terms of investment and production levels for base and peak-load periods. Baba (2001) considers a duopoly option game to examine a bank’s entry decisions into the Japanese loan market, shedding light on the prolonged slump in this market over the 1990s.

Extensions and Other Applications

423

Conclusion In this chapter we synthesized and discussed new developments and applications concerning option games. We have shown how integrating game theory and real options in a unified framework provides new insights into competitive strategies and how firms behave in uncertain markets. These tools are useful for the understanding of real-world industry phenomena and for predicting how firms (should) behave when faced with both market and competitive or strategic uncertainty. The option games approach developed herein considers optimal investment strategies as part of an industry equilibrium. We considered both symmetric and asymmetric firm conditions. Asymmetric models are more involved but provide a more natural ordering of first and second-investor timing in sequential decisions. Sequential investment in case of cost asymmetry involves substantial option value erosion due to competitive entry. We also glimpsed at important extensions and applications with thought-provoking implications for firm competitive strategy and public policy. These new insights help us understand puzzling phenomena, such as why there may be strategic delays in patent races while in other situations (e.g., involving positive externalities) investment might be expedited. They also help us understand industry phenomena like delayed exit, delayed implementation (sleeping patents) or real-estate market quirks. Option games are at the forefront of developments in corporate finance, economics and strategy in both the academic and managerial realms. Managers willing to consider managerial flexibility and strategic interactions as a cornerstone of business decisions under uncertainty will find the option games approach most valuable. Selected References Boyer, Gravel, and Lasserre (2004) discuss a set of option games contributions. Huisman et al. (2004) give an overview of continuous-time models dealing with lumpy investments in a competitive setting. Chevalier-Roignant et al. (2011) discuss a number of research contributions and the managerial insights derived. Grenadier (1996) develops a general approach for dealing with time lags with application to the real estate market, and Grenadier (1999) discusses informational asymmetry.

424

Chapter 13

Boyer, Marcel, Eric Gravel, and Pierre Lasserre. 2004. Real options and strategic competition: A survey. Mimeo. CIRANO, Montreal. Chevalier-Roignant, Benoît, Christoph M. Flath, Arnd Huchzermeier, and Lenos Trigeorgis. 2011. Strategic investment under uncertainty: A synthesis. European Journal of Operational Research 215 (3): 639–50. Grenadier, Steven R. 1996. The strategic exercise of options: Development cascades and overbuilding in real estate markets. Journal of Finance 51 (5): 1653–79. Grenadier, Steven R. 1999. Information revelation through option exercise. Review of Financial Studies 12 (1): 95–129. Huisman, Kuno J. M., Peter M. Kort, Gregorcz Pawlina., Jacco J. J. Thijssen. 2004. Strategic investment under uncertainty: Merging real options with game theory. Zeitschrift für Betriebswirtschaft 67 (3): 97–123.

Appendix: Basics of Stochastic Processes

The strategy of a firm must generally be adapted to actual, not expected, market developments.1 As noted, real options analysis allows for the determination of such optimal strategies. This is enabled by using stochastic calculus and control theory, topics in advanced mathematics that have many applications in applied sciences, such as physics, finance, and economics. The underlying theories are often mathematically involved, so we refer to the dedicated literature.2 Nevertheless, we take care to interpret the assumptions and provide a compendium of key properties. The raison d’être of this appendix is to fill prerequisite gaps and help smooth the exposition in previous chapters. At places we refer to the following results from this appendix to avoid having lengthy demonstrations in the chapters. In this appendix we briefly describe in section A.1 the main stochastic processes that admit “nice” mathematical properties useful for applications in economics and finance. These stochastic processes are made up 1. There may be a substantial difference between the outcome deduced from a model taking account of the actual values (with possible deviation from the expected development) and an equivalent model based on expected values. This difference can be seen from Jensen’s inequality applied to convex or concave functions. Since the payoff function f (⋅) of a call option is convex in the underlying factor X , Jensen’s inequality states that E [ f ( X )] ≥ f ( E [ X ]). For a put option whose payoff function is concave in the underlying factor, the opposite inequality holds. Refer to Savage (2009) for an intuitive treatment of Jensen’s inequality and how to circumvent the “flaw” arising when using averages for decision-making under uncertainty. 2. We here concentrate primarily on Itô’s theory of integration. An alternative approach— sketched in Øksendal (2007)—is based on the Stratonovitch integral rather than the Itô integral. When Merton identified stochastic calculus as a useful tool for the continuous-time analysis of capital markets, he considered Itô calculus a more appropriate tool for economics since it precludes foresight. Merton (1998, pp. 327–28n. 6) argues: “A Stratonovich-type formulation of the underlying price process implies that traders have a partial knowledge about future asset prices that the nonanticipating character of the Itô process does not.”

426

Appendix: Basics of Stochastic Processes

of an expected-value term (growth trend or drift) and a “diffusion” term capturing the stochastic movement of the process around its long-term expected growth trend. One interpretation is to think of the drift term as representing a forecast value (mean) for the underlying random factor and the volatility term as a sort of disturbance, noise or error term that reflects deviation from the expectation, proxying for uncertainty. In section A.2 that follows we discuss the notion of “forward net present value” and provide analytical solutions in a number of interesting situations. In section A.3 we discuss the concept of first-hitting time, providing explicit solutions for expected discount factors for certain Itô processes. Finally, we sketch the mathematical theory of optimal stopping/timing under uncertainty in section A.4. A.1

Continuous-Time Stochastic Processes

Stochastic processes represent a mathematical cornerstone of optionpricing theory and, by extension, of real options analysis. They are also essential as a building block for the evaluation of real options in a competitive setting. In such models the underlying asset or factor (e.g., stock price or project value) is assumed to follow a specific stochastic process; that is, the changes in the value of the asset evolve over time in an uncertain or unpredictable manner. A stochastic process is a collection of random variables such that the value of the process at each time t is random but determined via a known probability distribution.3 Alternatively, one could think of the changes between the value of the process at instant t and the value at the next instant t + h as following a certain probability law. Stochastic processes are often classified based on the assumed probabilistic law of motion governing their increments.4 Stochastic processes can be classified in two categories. They can be formulated in discrete time if they change only over certain discrete time intervals (countable time set), or in continuous time if they are subject to change at any time (continuous time set). Here we primarily focus on Itô processes, which are continuous-time stochastic processes with 3. In the following discussion we focus on one-dimensional stochastic processes of the Itô family. The study of multidimensional processes allows the analysis of economic problems where value functions may depend on several, possibly correlated, stochastic processes. 4. The Brownian motion discussed below, for instance, is characterized by stationary, individually independent, normally distributed increments. Any continuous-path process whose increments share these three properties is a (drifted) Brownian motion (see proof in Breiman 1968, prop 12.4).

Appendix: Basics of Stochastic Processes

427

continuous sample paths.5 Itô processes are memoryless or Markov, meaning that all relevant information governing future increments are summarized in the latest state of the process. When applied to traded assets, this property is consistent with the weak form of the efficientmarket hypothesis (EMH).6 Technically we assume the following (filtered) probability space (Ω, F, P ), where F ≡ {Ft }t ≥ 0 is a filtration, namely a family of tribes (or σ algebra) such that Fs ⊆ Ft for all s ≤ t . The tribe Ft corresponds to the information set on which the decision maker bases her decision at time t. Our discussion is based on the premise that stochastic processes and their functions are adapted to the filtration, namely that decision makers cannot make decisions based on information that is not yet revealed.7 A.1.1

Brownian Motion

A cornerstone stochastic process is the standard Brownian motion or Wiener process.8 A multitude of continuous-path Markov processes can be represented in terms of a standard Brownian motion, the most common being arithmetic and geometric Brownian motions. Standard Brownian Motion The term dzt , often entering the description of Itô processes, corresponds to the increment of a standard Brownian motion. Definition (adapted from Karatzas and Shreve 1988, p. 47) A standard one-dimensional Brownian motion is a continuous, adapted process z = {zt } defined on a given probability space (Ω, F, P ) such that:9 5. Poisson and jump-diffusion processes are not a primary subject of examination here since they have discontinuous sample paths. 6. The efficient market hypothesis (EMH) asserts that competition among hundreds of competent, rational investors ensures that the latest information is immediately incorporated into the current price. If the current price already reflects all the information contained in past prices, prices will change only in response to new, unpredictable information—which is unrelated to past unexpected information that drove past price movements. Asset price changes are thus independent over time. The EMH challenges the foundation of technical analysis. 7. To prevent foresight, expectations formed at time t are conditional on the information revealed up to now, Ft ; conditional expectations are written E ⎡⎣⋅⏐Ft ⎤⎦ or Et[⋅]. 8. This process has been formulated in physics to study the motion of small particles in liquids or gas. The origin of the concept explains why still today the graph of the values taken by the process is referred to as a “path.” The physical phenomenon has been discovered by the English botanist Robert Brown in 1827. It has been mathematically formulated by Louis Bachelier (1901) and Albert Einstein (1905). Bachelier’s (1901) treatise provides the first known application of the Brownian motion to describe financial or economic phenomena. 9. The standard Brownian motion and Wiener processes differ in their definitions, but the mathematical object is essentially the same, owing to Lévy characterization theorem. See Neftci (2000, pp. 177–78).

428

•

Appendix: Basics of Stochastic Processes

the starting value z0 = 0 almost surely;

the value increment zt + h − zt is normally distributed with mean zero and variance h;

•

the value increment zt + h − zt is independent of information revealed up to time t, that is, independent of the information set Ft (and zt a fortiori). •

Selected Properties The standard Brownian motion is characterized by the following properties: •

It has continuous sample paths.

It is a martingale, meaning that the best estimate (expectation) of its future value is its present value. Formally E [ zt + h | Ft ] = zt for all h > 0. In the present case the expected value is zero since the starting value of the process is zero (by definition).

•

•

It is a Markov process, so that10

E ⎡⎣zt + h ⏐Ft ⎤⎦ = E ⎡⎣zt + h ⏐zt ⎤⎦

∀h > 0 .

A measure of dispersion or variation over a time interval h is var(zt + h − zt )2 = h.11 One can also (informally) state that dzt 2 = dt over any infinitesimal increment (h → dt ) and var (zt ) = t. •

Under certain conditions one can define an integral with respect to a standard Brownian motion (Itô integral) rather than with respect to time. The Itô integral captures the noise of the phenomenon around its expected trend. Such integrals have zero expected value. •

A good understanding of the standard Brownian motion is important because it serves as a building block for a multitude of continuous-path Markov processes (and all continuous-path martingales).12 Unfortunately, the standard Brownian motion, and all processes based on it, do not admit a time derivative. This feature is of cardinal significance for stochastic calculus, as standard differentiation techniques cannot be readily employed. 10. The equation here relates to the Markov property, not the strong Markov property that involves stopping times. 11. The mathematical subject underlying this property is more complex than implied here. It relates to the quadratic variation of the Brownian motion. The quadratic variation drives much of the differences between deterministic and stochastic processes. We do not intend to discuss this measure of dispersion in detail, sufficing to use a loose definition of it. 12. Martingales are not discussed at length here but they are regarded essential for a mathematical understanding of continuous-time finance (e.g., risk-neutral valuation).

Appendix: Basics of Stochastic Processes

429

Arithmetic Brownian Motion The drifted or arithmetic Brownian motion x = {xt }t ≥0 , or xt as shorthand, can be constructed based on the standard Brownian motion described above. It admits a nonzero starting value and an expected trend. Formally this process is of the form xɶ t = x0 + α t + σ zt ,

(A.1)

where zt is a standard Brownian motion as defined previously. The parameter α, called the “drift parameter,” measures the growth (expected) trend of the process, and σ is its volatility or standard deviation. The arithmetic Brownian motion (ABM) is a continuous-time Markov process such that, given its initial value x0, the random variable xɶ t for any future time t ( > 0 ) is normally distributed with mean x0 + α t and variance σ ²t (or equivalently with standard deviation σ t ).13 One may alternatively think of drifted Brownian motion as the accumulation of independent, stationary, and identical normally distributed increments dxɶ t over many nonoverlapping small time intervals of length dt , with each increment having mean α dt and “variance” σ ²dt. This representation of the arithmetic Brownian motion permits a description as a stochastic differential equation (SDE) of the form dxɶ t = α dt + σ dzt .

(A.2)

Figure A.1 depicts a sample realization or sample path of an arithmetic Brownian motion. A linear function with slope α represents the expected growth trend. The actual sample path moves around the expected (long-term) trend due to volatility σ . Most economic variables can be reasonably described by their drift and volatility. For a large time horizon, the growth trend is the dominant determinant, whereas in the short term volatility is what really matters.14 When applied to financial assets, this property is in line with the common saying that in the long term, stock prices are driven by expected real growth trends, but in the short run they may fluctuate due to random factors or volatile capital markets. 13. Since the standard Brownian motion is a martingale with starting value zero, the third right-hand term disappears in the expectation. The “variance” of the arithmetic Brownian motion comes from the “variance” of the standardized Brownian motion, namely var (zt ) = t. 14. Since the mean of the drifted Brownian motion grows with t and the standard deviation with t , the standard deviation dictates the overall nature of the process in the short run (as t becomes negligible compared to t ) but the drift dominates in the long run (as t becomes negligible compared to t).

430

Appendix: Basics of Stochastic Processes

5

4

Expected growth trend (linear)

3

2

Sample path 1

0

t Figure A.1 Sample path of an arithmetic Brownian motion ABM is discretized by xɶ t + h = xɶ t + α h + σ h × εɶ , where εɶ is a standard normal random variable generated by a computer program. We assume h = 0.19685, α = 4 percent, σ = 30 percent, x0 = 1 ( t0 = 0 ).

Example A.1 Probability of Option Exercise at Maturity (or of “Being in the Money”) Suppose that stock price vt follows an arithmetic Brownian motion as in equation (A.2) above. A European call option on this stock may be exercised at maturity (time T ) by paying an exercise price I . What is the probability that the option will end up “in the money” at maturity T , θ t ≡ Pt [ vT ≥ I ]? Here Pt [·] stands for the probability conditional on time-t information. From equation (A.1) and the definition of the standard Brownian motion, we have

θ t = Pt ⎡⎣vt + ατ + σ τ εɶ ≥ I ⎤⎦ , where τ ≡ T − t and εɶ is a random variable with standard normal distribution. It follows that

θ t = Pt [ εɶ ≥ −d2 ]

(A.3)

with d2 ≡

vt − I + ατ

σ τ

.

(A.4)

Appendix: Basics of Stochastic Processes

431

Since the cumulative standard normal distribution N (⋅) is symmetric (at zero),

θ t = Pt [ εɶ ≥ −d2 ] = Pt [ εɶ ≤ d2 ] = N(d2 ).

(A.5)

The modeling of stock prices as an arithmetic Brownian motion (as in example A.1) is ill-advised. As pointed out by Samuelson (1965), this process has a severe flaw for modeling financial assets since prices may possibly become negative (owing to the normally distributed diffusion term), a feature hardly descriptive of real price dynamics.15 Stock prices Xɶ t are instead typically assumed to be log-normally distributed to avoid this flaw exhibited by the arithmetic Brownian motion. That is, the (natural) logarithm of price is assumed to follow a (drifted or arithmetic) Brownian motion. The increment of the log of the price can be thought of as the log-return: ⎛ Xɶ ⎞ ln ( Xɶ t + h ) − ln ( Xɶ t ) = ln ⎜ t + h ⎟ . ⎝ Xɶ t ⎠ Log-returns are normally distributed. This different angle calls for the modeling of prices as a geometric Brownian motion. Geometric Brownian Motion and Other Exponentials The geometric Brownian motion (GBM), commonly used for modeling asset prices, is an exponential form of Brownian motions. Such exponentials are characterized by Xɶ t = f ( xɶ t ) = exp (β xɶ t ) = e β xɶt ,

(A.6)

where xɶ t follows the arithmetic Brownian motion of equation (A.1). β is a constant parameter specifying the shape of the log-return distribution.16 The starting value of the process is X 0 = e β x0 . One can 15. If prices were negative, investors would purchase the asset and receive funds from the purchase! In the end the law of demand and supply would force the price to rise to zero or above. 16. When log-return errors are normally distributed, β = 1. If returns exhibit higher moments (skewness and leptokurtosis), β may differ from 1. For positively skewed growth or distress stocks, β may be less than 1 (leading to lower average returns); for negatively skewed indexes, β may exceed 1. The daily log-returns of most financial assets or indexes (e.g., stock market indexes, foreign exchange rates, metal and gold prices) exhibit significant skewness and kurtosis. This evidence is consistent with the premise that log-returns of financial assets follow a more general distribution than the normal distribution implied by the geometric Brownian motion. Subbotin (1923) proposed a more general error distribution than the normal, the power exponential or generalized error distribution (GED). The symmetric case of the GED includes the double exponential, the normal and the uniform distributions as special cases. The double exponential underlies Laplace’s (as well as Poisson’s) first law of random measurement error with the most-likely estimate being

432

Appendix: Basics of Stochastic Processes

readily associate the exponential Xɶ t with its corresponding Brownian motion by inverting (A.6), obtaining xɶ t =

1 ln ( Xɶ t ). β

(A.7)

Applying Itô’s lemma (see equation A1.2 in box A.1) to the exponential function and Brownian motion xɶ t yields17 dXɶ t = ( gXɶ t ) dt + (βσ Xɶ t ) dzt ,

(A.8)

where g is given by 1

g ≡ g (β ) = αβ + β 2σ 2. 2

(A.9)

The term gXɶ t is the drift of the process X t and βσ Xɶ t is its diffusion term. The drift parameter g of X t is the drift of the corresponding Brownian motion incremented by β 2σ 2 2, while its volatility term is simply a multiple (β ) of the volatility of the arithmetic Brownian motion (σ). The drift parameter g is increasing in the volatility (σ ). The expected value of the process at future time t follows an exponential growth path given by18 E0 ⎡⎣ Xɶ t ⎤⎦ = X 0 e gt .

(A.10)

A special case of exponentials of Brownian motion is the geometric Brownian motion, which assumes that log-return errors are normally distributed. It is the special case obtained for β = 1. Equation (A.9) with β = 1 establishes the relationship between the drift of the geometric Brownian motion and that of ABM or its logarithm, xɶ t = ln ( Xɶ t ): the median of observations. When the arithmetic mean is used instead (as the most probable value for a number of separate and similar observations in a symmetric distribution), the normal distribution is obtained instead (Gauss’s law and Laplace’s second law). Subbotin (1923) showed that Gauss’s law fails to be universal if one relaxes the assumption that the distribution may be expanded in the power series. Theodossiou and Trigeorgis (2010) extended this to the “skewed GED.” Two additional parameters control the degree of skewness and leptokurtosis. Here β > 1 corresponds to skewness and kurtosis being present in financial asset return errors, which is more accurate for real-world asset prices. The special case β = 1 corresponds to the theoretical assumption that log-return distribution errors are normally distributed as per the efficient market hypothesis. 17. f : x  exp ( x ) is smooth. 18. The mean and variance (as well as higher moments) can be derived by using the fact that the diffusion term drops out when taking expectations (the stochastic integral with respect to the standard Brownian motion is a martingale with starting value zero). Let h ( s ) ≡ E0 ⎡⎣ Xɶ s ⎤⎦ . From (A.8) expressed in the stochastic integral form and Fubini’s theorem that t allow for permutation of time and state integrals, it obtains h (t ) = h (0 ) + g ∫ 0 h ( s ) ds . In other words, h (⋅) solves the differential equation h′ (t ) = gh (t ) with initial boundary condition h (0 ) = X 0. Hence, h (t ) = X 0 e gt , as in (A.10).

Appendix: Basics of Stochastic Processes

433

Box A.1 Itô’s lemma

To describe the dynamics of functions of stochastic processes, one needs a sort of “differentiation” rule. Since standard differentiation approaches do not apply for stochastic processes, a common approach is to use Itô’s lemma. This is analogous to the “chain-rule” of stochastic calculus. We here state Itô’s lemma in the context of options though it is applicable to a broader range of problems.a Suppose an underlying asset (e.g., stock price or project value) modeled as an Itô process that follows the stochastic differential equation (A1.1) dXɶ t = gt dt + σ t dzt , with drift gt = g( X t , t ) and diffusion σ t = σ ( X t , t ). Consider an option on this asset whose payoff function f (⋅, ⋅) is twice continuously differentiable in the asset price and continuously differentiable in time. Itô’s lemma asserts that the option’s price dynamics { f ( Xɶ t , t )}t ≥0 also follows an Itô process and that its value increment is described by the stochastic differential equation 1 (A1.2) df ( Xɶ t , t ) = ⎡⎢ ft + gt fX + σ t2 fXX ⎤⎥ dt + σ t f X dzt , 2 ⎣ ⎦ where ft = ft ( Xɶ t , t ); fX = fX ( Xɶ t , t ) and f XX = f XX ( Xɶ t , t ) stand for the firstorder and second-order derivatives of f (⋅, ⋅) with respect to their subscript. The expected capital gain of the option over an infinitesimal time interval is given by

Γf ( X t , t ) ≡ lim h→0

Et [ f ( X t + h , t + h)] − f ( X t , t ) h 1

(A1.3)

= ft + gt fX + σ t2 fXX . 2

The operator Γ , called the infinitesimal generator, is given by Γ≡

1 ∂ ∂ ∂2 + gt + σ t2 . ∂t ∂X 2 ∂X 2

(A1.4)

This operator is useful henceforth for the understanding of Bellman or HJB equations. a. Doeblin (1940) and Itô (1951) independently discovered this formula. Doeblin’s variant was lost for almost sixty years until the early 2000s. This formula should properly be referred to as Itô–Doeblin formula, although the term “Itô’s lemma” has become standard in financial economics. Karatzas and Shreve (1988, ch. 3) discuss generalization of Itô–Doeblin formula to multidimensional processes (integrand) and to functions having less restrictive “smoothness” conditions (with continuous martingales as integrator). Protter (2004, ch. 2) defines stochastic integrals with respect to (nonnecessarily continuous) semimartingales.

434

Appendix: Basics of Stochastic Processes

2.5

Expected growth trend (exponential)

2.0

1.5

1.0 Sample path

0.5

0.0

t

Figure A.2 Sample path of a geometric Brownian motion In the graph the geometric Brownian motion is approximated in discrete time by Xɶ t + h = Xɶ t exp α h + σεɶ t h with α ≡ g − (σ 2 2). h = 0.039526, g = 9 percent, σ = 20 percent, and X 0 = 1 ( t0 = 0 ). εɶ t is (standard) normally distributed.

(

)

1

g ≡ g (1) = α + σ 2 . 2

(A.11)

The volatilities are the same, σ . The stochastic differential equation for the geometric Brownian motion can be written (from equation A.8 with β = 1) as dXɶ t = ( g Xɶ t ) dt + (σ Xɶ t ) dzt,

(A.12)

where zt is a standard Brownian motion. The geometric Brownian motion is also referred to as a proportional Brownian motion since the expected change per time interval h, Et ⎡⎣ Xɶ t + h − Xɶ t ⎤⎦ Xɶ t , equal to gh , is proportional to the time interval considered.19 Figure A.2 depicts a realized sample path for the geometric Brownian motion. In contrast to the arithmetic Brownian motion where the growth trend is linear, we here have an exponential growth trend due to the continuously compounded nature of growth. The actual sample path moves 19. Dixit (1993) employs the term “proportional Brownian motion.”

Appendix: Basics of Stochastic Processes

435

around this exponential expected growth trend. The expected future value of Xɶ t , obtained by substituting equation (A.11) in equation (A.10), is 1 ⎞⎤ ⎡ E0 ⎡⎣ Xɶ t ⎤⎦ = X 0 exp ⎢⎛⎜ α + σ 2 ⎟ ⎥ t . 2 ⎠⎦ ⎣⎝

(A.13)

X 0 ≡ e x0 is the initial (time-0) value of the geometric Brownian motion. The expected future value depends on the initial value X 0 increasing exponentially (geometrically) at a constant growth rate g = α + (σ 2 2). Growth is compounded since future value is driven by both growth on the initial value X 0 and growth on recent growth. The expected value for powers of the geometric Brownian motion is derived in a similar manner: E0 ⎡⎣ Xɶ t n ⎤⎦ = X 0 n eγ t,

(A.14)

with γ ≡ ng + [ n (n − 1) σ 2 2 ] and n a positive integer.20 The variance of the geometric Brownian motion is given by21

(

)

2 var0 ⎡⎣ Xɶ t ⎤⎦ = X 0 2 e 2 gt eσ t − 1 .

(A.15)

Uncertainty affects the dispersion around the growth trend, with the variance of Xɶ t increasing with instantaneous volatility σ . The actual realized value of the process at time t, Xɶ t , might differ from the expected value E0 ⎣⎡ Xɶ t ⎦⎤ since the actual value depends on noise or randomness as well. In the deterministic case (σ → 0), i.e., when volatility is zero, only the expected value matters (as var0 ⎡⎣ Xɶ t ⎤⎦ → 0). Example A.2 Black–Scholes European Call Option Pricing Suppose that stock price Vt follows a geometric Brownian motion as in equation (A.12). Consider a European call option with maturity T (> t ) and exercise price I . Example A.1 derived the probability of being in the money for a European call option on a stock that follows an arithmetic Brownian motion. Replace vt by ln (Vt ) and the exercise price I by ln ( I ) in equation (A.4), so the probability of the option being exercised at 20. The function f : x  x n is twice continuously differentiable with f x ( x ) = nx n−1 and f xx ( x ) = n (n − 1) x n−2. Apply Itô’s lemma from box A.1 equation (A1.2) to f (⋅) and employ the infinitesimal generator notation in (A1.4); then the instantaneous expected value change is Γf ( Xɶ t ) = γ f ( Xɶ t ) . Set h ( s ) = E0 ⎡⎣ Xɶ s n ⎤⎦. From Dynkin’s formula and t Fubini’s theorem, h (t ) = X 0 n + γ ∫ 0 h ( s) ds. h (⋅) thus solves the ordinary differential equation h ′ (t ) = γ h (t ) with initial condition h (t ) = X 0 n . Equation (A.14) follows. 2 21. As var0 ⎣⎡ Xɶ t ⎦⎤ = E0 ⎣⎡ Xɶ t 2 ⎦⎤ − E0 ⎣⎡ Xɶ t ⎦⎤ , the result follows from equations (A.14) (with n = 2 ) and (A.13) [first moment] by factorization.

436

Appendix: Basics of Stochastic Processes

maturity (ending up in the money) in the case of the geometric Brownian motion obtains as

θ t = Pt [VT ≥ I ] = N(d2 )

(A.16)

with ln (Vt I ) + ατ

d2 ≡

σ τ

,

(A.17)

where α = g − (σ 2 2) and τ = T − t. The expected value of the underlying asset conditional on being “in the money” at maturity is

ξt ≡ Et ⎡⎣VT ⏐VT ≥ I ⎤⎦ ∞ 1 = ∫ Vt exp ⎛⎜ g − σ 2 ⎞⎟ τ + σ τ εɶ χ {VT ≥ I } dN(εɶ ) , −∞ ⎝ 2 ⎠

{

}

where χ {}⋅ is the indicator function— χ {VT ≥ I } is 1 if VT ≥ I (i.e., if εɶ ≥ −d2) and zero otherwise—and the random variable εɶ is standard normally distributed. dN(⋅) is the probability density of the standard normal distribution ⎧ εɶ 2 ⎫ exp ⎨− ⎬ dεɶ . 2π ⎩ 2⎭ 1

dN (εɶ ) ≡

It follows that

ξt =

Vt e gτ 2π

∫

∞

− d2

{

exp −

1 2

(εɶ

2

}

)

− 2σ τ εɶ + σ 2 τ dεɶ .

Let εɶ ′ ≡ εɶ − σ τ , with εɶ ′ 2 = εɶ 2 − 2σ τ εɶ + σ 2 τ . Since εɶ ≥ −d2 implies εɶ ′ ≥ −d1 with d1 ≡ d2 + σ τ ,

(A.18)

then

ξt =

Vt e gτ 2π

∫

∞

− d1

⎧ εɶ ′ 2 ⎫ ɶ gτ exp ⎨− ⎬dε ′ = Vt e N (d1 ) 2 ⎩ ⎭

(A.19)

by symmetry (at zero) of the standard normal distribution (εɶ ′ ). In complete capital markets that preclude arbitrage opportunities, we can use the insights from risk-neutral pricing (e.g., see Cox and Ross 1976 or Harrison and Kreps 1979), whereby we replace the growth rate g in equation (A.19) by (a certainty equivalent growth gˆ analogous to) the risk-free rate r. At maturity T the payoff of the European call option on

Appendix: Basics of Stochastic Processes

437

a (non–dividend-paying) asset is CT = max {VT − I ; 0}. Therefore the (time-t) value of a European call option, obtained as the discounted expected future value under risk-neutral expectation, is Ct = e − rτ Eˆ t [CT ] = e − rτ Eˆ t [VT − I | VT ≥ I ] = e − rτ Eˆ t [VT | VT ≥ I ] − e − rτ IPt [VT − I ] = e − rτ ξt − e − rτ Iθ t .

From equations (A.16) and (A.19) with discount rate r, this readily results in the Black–Scholes formula given in equation (5.7): Ct = Vt N (d1 ) − Ie − rτ N (d2 ) with d1 and d2 given in equations (A.18) and (A.17), with α = r − (σ 2 2). The geometric Brownian motion is probably the most widely used price process in finance and economics because it provides a good starting proxy for the dynamics of stock prices, exchange rates, prices of natural resources, and other financial time series, while it remains tractable mathematically. Consider a traded asset whose price dynamics can be modeled via geometric Brownian motion. The future value Xɶ t + h of the asset price (considered from time-t perspective) is a log-normally distributed random variable with mean X t e gh and volatility σ h increasing in the time horizon (h). Given the relationship between the geometric Brownian motion and the arithmetic Brownian motion, the logarithm of the asset price, ln ( Xɶ t + h ) ≡ xɶ t + h, is a normally distributed random variable with mean xt + α h (with xt ≡ ln ( X t ) and α = g − (σ 2 2)) and variance σ 2 h. Alternatively, log-returns ( xɶ t + h − xt = ln ( Xɶ t + h X t )) are normally distributed with mean α h and variance σ 2 h. Although market participants may observe the past values of Xɶ t as it evolves over time (investors may possess a long time series of past prices), the actual future value of the process always remains uncertain to them. Market participants naturally form expectations about future developments.22 The drift of the geometric Brownian motion (used to model the expected asset price) can be seen in perspective vis-à-vis the total return of the asset and the dividend payout. In capital markets the appropriate risk-adjusted discount rate k —which equals the total return on the asset in equilibrium—is typically higher than the expected price growth rate g (k ≥ g). The total return k consists of the asset price increase or capital gain, g, plus any benefit or “dividend yield,” δ , the 22. Henceforth to avoid any confusion, the value of the process at time t (once realized and observed) is denoted by X t , whereas Xɶ t refers to the unknown random variable.

438

Appendix: Basics of Stochastic Processes

asset holder receives over time. To ensure that there is no arbitrage opportunity, k = g + δ must hold.23 Consider a time series of the asset price, Xɶ t = exp ( xɶ t ), modeled as geometric Brownian motion. Even though at first sight the drift and volatility parameters of the geometric Brownian motion may appear difficult to estimate given the exponential nature of the process, they can be readily determined when one considers the logarithm of the price ( xɶ t = ln( Xɶ t )) or the evolution of the log-return (ln ( Xɶ t + h Xɶ t ) for h small). It is generally easier to estimate the drift parameter α by taking the average value of the log-returns Δxɶ t ≡ xɶ t + h − xɶ t = ln ( Xɶ t + h Xɶ t ) over small, equally sized time intervals of length h. The volatility of asset price changes equals the volatility of Δxɶ t or xɶ t . The latter can be readily assessed as the standard deviation of the asset’s log-returns. The growth trend g of Xɶ t is readily established in equation (A.11) from past data concerning the asset’s log-return.24 A.1.2

Mean-Reversion Process

Arithmetic or geometric Brownian motions characterize asset prices whose future value may potentially attain values far away from the starting one and may increase or decline dramatically for long time horizons. Although such processes may serve as useful approximations in many circumstances, they may not represent adequately equilibrium dynamics in many situations where capacity adjusts to meet demand. Certain asset prices, such as commodity prices or interest rates, tend to move toward a natural long-run equilibrium with actual values evolving randomly up and down around the long-term mean. An example of a commodity whose price follows a mean-reverting process is copper. Although the actual price is stochastic, in the long term it tends to revert to a historical mean observable in long-term time series. For such processes a meanreverting process may be more suitable as a modeling device. Arithmetic Ornstein–Uhlenbeck Process The simplest and best known mean-reverting process is the simple or arithmetic Ornstein–Uhlenbeck process represented by the following stochastic differential equation: 23. For American call options, Merton (1973) shows that the option will not be exercised prior to maturity unless the dividend yield is positive (δ > 0). 24. A better estimate of the drift parameter is obtained by considering a long-time average since the trend is the determinant factor in the long run. For the volatility parameter, it is possible to use a series over a number of days if the number of estimates is high enough. The volatility of the geometric Brownian motion is constant whatever the time horizon considered.

Appendix: Basics of Stochastic Processes

dXɶ t = η ( X − Xɶ t ) dt + σ dzt.

439

(A.20)

Here η describes the speed or strength of reversion toward the “natural level” X of the process (the long-term mean). In case of commodities, X might represent the long-run marginal production cost. When there is no force toward the long-term mean (η → 0), the process Xɶ t becomes a Brownian motion without drift (α = 0), with the future value having variance σ 2 t . Given the initial value of the process at time 0, X 0 , the expected value for Xɶ t at future time t is given by E0 ⎡⎣ Xɶ t ⎤⎦ = wη (t ) X 0 + (1 − wη (t )) X ,

(A.21)

with wη (t ) ≡ exp (−ηt ). Since wη (t ) ∈ (0, 1), the expectation in equation (A.21) can be interpreted as a weighted average of the initial value X 0 and the long-term equilibrium mean value X . As t goes to infinity, the expected value in equation (A.21) converges to the long-term average X (since wη (t ) → 0). The variance of the mean-reversion process above is given by25 var ( Xɶ t − X ) =

σ2 (1 − wη(t )2 ). 2η

(A.22)

If the speed of mean-reversion gets large (η → ∞), the process tends rapidly to its natural mean level X (since wη (t ) → 0) and the variance around the mean becomes negligible. Geometric Ornstein–Uhlenbeck Process An extension of the preceding mean-reverting process is the geometric Ornstein–Uhlenbeck process solving the stochastic differential equation dXɶ t = η ( X − Xɶ t ) Xɶ t dt + σ Xɶ t dzt .

(A.23)

The key differentiating feature between the geometric Ornstein–Uhlenbeck process and the geometric Brownian motion discussed previously lies in the drift parameter. In mean-reversion the difference Xɶ t − X between the current level at time t ( Xɶ t ) and the “natural level” X influences the drift η ( X − Xɶ t ) Xɶ t , being positive if the current value is below its long-term average and negative otherwise. There is no expected change in the process if the present value exactly equals the long-term mean X . For geometric Brownian motion, however, the drift gXɶ t is proportional to the latest value of the process at time t . Its sign is always either positive or negative. The diffusion term (σ > 0) here suggests that even 25. See Dixit and Pindyck (1994, ch. 3, app. A) for a derivation of equations (A.21) and (A.22) above.

440

Appendix: Basics of Stochastic Processes

if the process has reached its long-term average, it may still deviate from it. The higher the volatility, the higher is the probability of a deviation from the average. A.1.3

General Itô Processes

Broader families of processes may sometimes be more appropriate to describe certain price dynamics. Such processes are also used to derive useful properties and insights applicable to the whole family. So far we have discussed specific cases of memoryless or Markov processes, namely arithmetic and geometric Brownian motion and the mean-reverting processes. All these belong to a broader class of processes, called Itô processes. These processes are such that decision makers cannot rely on information not yet revealed (are “adapted to the filtration”) and can be expressed as the sum of a drift (an integral with respect to time) and an Itô integral involving the standard Brownian motion. For such processes, the drift and the diffusion term, gt ≡ g ( Xɶ t , t ) and σ t ≡ σ ( Xɶ t , t ), are allowed to depend on the latest value of the asset price Xɶ t and the time period (t). An Itô process is a stochastic process of the (integral) form t t Xɶ t = X 0 + ∫ g s ds + ∫ σ s dzs . 0

0

(A.24)

It is common to write the Itô process (A.24) in its differential form dXɶ t = gt dt + σ t dzt ,

(A.25)

where zt is a standard Brownian motion.26 An Itô process whose drift and volatility do not depend on time but only on the latest asset price is called a time-homogeneous Itô process or a diffusion process. It can be expressed (in its differential form) as dXɶ t = g ( Xɶ t ) dt + σ ( Xɶ t ) dzt .

(A.26)

The general Itô process covers a fairly broad family of stochastic processes used in economic analysis and enables the modeling of Markov processes with continuous sample paths.27 All processes considered 26. For technical reasons (existence) it is further assumed that the drift and the volatility terms are adapted to the filtration, have finite variations and that they comply with the linear growth and Lipschitz conditions (see Karatzas and Shreve 1988). 27. Itô processes belong to a larger family of processes. They are subsumed into Lévy processes and càdlàg processes. Càdlàg processes share the property of being rightcontinuous and admitting left-limits along all sample paths. Lévy processes are càdlàg processes with the additional property of having independent, identically distributed increments. The Itô, Poisson, and mixed jump-diffusion processes are all Lévy processes. Lévy and càdlàg processes are beyond our scope of analysis here.

Appendix: Basics of Stochastic Processes

441

previously belong to this general class. The arithmetic Brownian motion described in equation (A.2) is obtained for gt = α and σ t = σ (α and σ being constant). For the geometric Brownian motion of equation (A.12), the drift is gt = gXɶ t = [α + (σ 2 2)] Xɶ t and the diffusion term σ t = σ Xɶ t (α and σ being constant). Mean-reverting processes are also Itô processes. For the simple Ornstein–Uhlenbeck process given in equation (A.20), gt = η ( X − Xɶ t ) and σ t = σ , where X is the long-term average, η the speed of mean reversion and σ a constant volatility. The geometric Ornstein–Uhlenbeck process in equation (A.23) is characterized by gt = η ( Xɶ t − X ) Xɶ t and σ t = σ Xɶ t. Itô’s lemma, given in box A.1 equation (A1.2), applies to this general class of processes. It is often desirable to develop models that admit closed-form solutions to help deduce clear-cut investment rules to be followed by analysts or decision makers. Unfortunately, it is not always easy or possible to obtain analytical solutions for an Itô process. This is the case, for example, when the underlying factor is mean reverting. In such settings one must resort to numerical methods instead. Trigeorgis (1996, ch. 10) summarizes some of them. These methods take advantage of the Markov property of the Itô process. Simple models involving the arithmetic Brownian motion generally admit analytical solutions. Nonetheless, as noted, the arithmetic Brownian motion is ill-suited to describe such phenomena as equilibrium asset price dynamics. The geometric Brownian motion corrects some of these flaws while maintaining some of the “nice” mathematical properties of the simple Brownian motion. This is the reason why this process is generally preferred in much of real options analysis. A.2

Forward Net Present Value

In valuing investment timing options, one can follow a backward valuation process, first determining the forward expected net value of the project (“reward function”) at maturity and then discounting this using an appropriate expected discount factor. In assessing the current value of the option, we first need to determine the forward value of the underlying project received upon exercise in the future.28 28. Deriving this value rests on a number of equilibrium conditions applied to dynamic problems. The underlying notion is Bellman’s (1957) principle of optimality. It asserts that: “An optimal policy has the property that, whatever the initial action, the remaining choices constitute an optimal policy with respect to the subproblem starting at the state that results from the initial actions.” This principle is at the core of dynamic programming and is used to derive optimal behavior when faced with dynamic problems.

442

Appendix: Basics of Stochastic Processes

Consider the deferral or investment option in discrete time, where VTs denotes the value of the underlying asset (project) in state s at future time T . The exercise price of the investment option is the investment cost I . The decision whether to invest at time T depends on the value of VTs − I in state s. This value represents the forward net present value obtained upon exercise at time T . In the simplest setting, the payoff at time of exercise T is determined given the assumed irreversible nature of investment. This value is the stream of all expected profits to be received onward, net of costs, with no possibility to take corrective action. This process for assessing the forward net present value is analogous to discounted cash flow. Suppose that the underlying stochastic factor Xɶ t follows the diffusion or time-homogeneous Itô process of equation (A.26). Owing to the Markov property, the time-T perpetuity value of a cash-flow stream starting in a given state equals the time-0 perpetuity value of an identical flow stream starting in the same state. For simplicity, consider the time-0 perpetuity value. Suppose that upon exercising the investment option at time 0, the decision maker receives onward an instantaneous profit flow π ( Xɶ t ) in state Xɶ t in perpetuity.29 Suppose the appropriate risk-adjusted discount rate is k . What is the perpetuity value V [π ( X 0 )] received upon exercising the investment option at time 0 when the firm will receive profit flow π (⋅) in perpetuity? This time-0 state-contingent project value is given by ∞ (A.27) V0 [π ( X 0 )] ≡ E0 ⎡ ∫ π ( Xɶ t ) e − kt dt ⎤ . ⎣ 0 ⎦ Over an infinitesimal time interval dt, an operating firm can expect to receive the instantaneous profit or dividend flow π = π ( Xɶ t ) plus additional capital gain. The expected (instantaneous) capital gain is E (dV ) dt . This expected capital gain corresponds to the drift term in the stochastic differential equation descriptive of the value increment. It is given by the infinitesimal operator Γ in box A.1 equation (A1.4):

1

ΓV = gt VX + σ t2 VXX , 2

where VX , VXX refer to the first and second-order derivatives of the value function with respect to the shock X t .30 In the case of a time-homogeneous Itô process, gt ≡ g ( Xɶ t ) and σ t ≡ σ ( Xɶ t ). If there are no arbitrage opportunities, the firm should receive during this time length the same total return it would have obtained from holding an asset in the capital 29. The profit function π (⋅) is twice continuously differentiable in the shock. 30. The value function does not depend on time, so that Vt ≡ ∂V ∂t = 0 .

Appendix: Basics of Stochastic Processes

443

markets with the same risk profile; that is, it should receive the (instantaneous) total return k . Thus in equilibrium it must hold that kV = π + ΓV ,

(A.28)

where for notational simplicity we drop the dependence on the underlying process X t . This equation is known in stochastic (Itô) calculus as the Hamilton–Jacobi–Bellman (HJB) equation.31 In general, there is no analytical closed-form solution to this partial differential equation (PDE) since the solution depends on the functional form of gt and σ t . In specific cases, however, analytical solutions can be obtained.32 We discuss such cases next. Example A.3 Forward NPV for Arithmetic Brownian Motion From equation (A.1), E0 [ xɶ t ] = x0 + α t . Applying Fubini’s theorem to (A.27) in the case of the arithmetic Brownian motion of equation (A.1) yields33 V0 ≡ V [ x0 ] = ∫

∞

0

= x0 ∫

∞

0

Since ∫

∞

0

V0 =

e − kt E0[ xɶ t ] dt

e − kt dt + α ∫

∞

0

te − kt dt.

e − kt dt = 1 k and ∫

∞

0

te − kt dt = 1 k 2 , it follows that34

x0 α + . k k2

(A.29)

31. We derive here the HJB equation more formally. For notational simplicity, we denote Vt = V [π ( X t )]. Equation (A.27) above can be approximated for a small time interval h by ∞ 1 V0 ≈ π ( X 0 ) h + E0 ⎡⎢ e − kh ∫ π ( X t ) e − k(t −h)dt ⎤⎥ ≈ π ( X 0 ) h + E0 [Vh ] . h ⎣ ⎦ 1 + kh

By multiplying the expression above by 1 + kh and substracting V0 from both sides, we obtain khV0 ≈ (1 + kh) π ( X 0 ) h + E0 [Vh ] − V0, or kV0 ≈ (1 + kh) π ( X 0 ) +

1 {E0 [Vh ] − V0 }. h

Using the infinitesimal generator notation in (A1.4) of box A.1, equation (A.28) obtains by taking the limit h → 0 . The HJB equation above does not allow corrective action/control over time interval h. 32. The examples below are solved from an alternative probabilistic perspective, not from a functional-analysis approach involving the ordinary differential equation (A.28). For other settings the functional approach might be more useful. 33. In the context of stochastic integrals, Fubini’s theorem states that it is permissible to interchange the time (Lebesgue) and the∞ Itô integrals. In this case we apply this to the ∞ expectation operator: E0 ⎡⎣ ∫ 0 xɶ t e − kt dt ⎤⎦ = ∫ 0 e − kt E0[ xɶ t ] dt . See Karatzas and Shreve (1988, p. 209). 34. Based on a different approach, Dixit (1993, pp. 11–12) examines power functions of the arithmetic Brownian motion.

444

Appendix: Basics of Stochastic Processes

Example A.4 Forward NPV for Geometric Brownian Motion and Other Exponentials Consider the exponentials Xɶ t = e β xɶt of the arithmetic Brownian motion in (A.1). From Fubini’s theorem ∞ ∞ V0 ≡ V [ X 0 ] = E0 ⎡ ∫ e − kt Xɶ t dt ⎤ = ∫ e − kt E0 ⎡⎣ Xɶ t ⎤⎦ dt . 0 0 ⎣ ⎦ It follows from equation (A.10) that ∞

V0 = X 0 ∫ e − {k − g (β )}t dt . 0

For δ (β ) ≡ k − g (β ) > 0, this converges to V0 =

X0 , δ (β )

(A.30)

where δ (·) is the fundamental quadratic function defined in equation (A2.1) of box A.2. The terminal value V [ X 0 ] exists if and only if the dividend yield δ (β ) is strictly positive, namely if β ∈ (β1 , β 2 ), where β1 and β 2 are the positive and negative roots of the fundamental quadratic function. Note that 1 is always contained inside the two roots of the fundamental quadratic (since β 2 < 0 < 1 < β1). The choice of the appropriate type of fundamental quadratic function is discussed in box A.2. In the special case where β = 1, δ (1) = δ = k − g , and Xɶ t = exp ( xɶ t ) so that Xɶ t follows a geometric Brownian motion with drift parameter g = α + (σ 2 2) and volatility σ; the terminal value expression in equation (A.30) then becomes analogous to the Gordon perpetuity formula, giving the forward present value of a cash flow stream starting at time T + 1 (with value at time T being XT ), growing exponentially at expected constant rate g in perpetuity. This formula—usually thought of in discrete time—reads t

∞ XT + 1 XT + 1 1+ g⎞ VT ≡ V [ XT ] = ∑ ⎛⎜ , = ⎟⎠ XT = ⎝ δ k−g t =1 1 + k

where XT + 1 = (1 + g ) XT . The (improper) integral underlying the equation above converges if and only if there is a positive dividend yield δ , namely for δ ≡ k − g > 0 (or k > g). Example A.5 Forward NPV for Powers of Geometric Brownian Motion The expected present value for powers of geometric Brownian motion can be derived in a similar manner. From equation (A.14) and Fubini’s theorem,

Appendix: Basics of Stochastic Processes

445

Box A.2 The fundamental quadratic

We consider two versions of the “fundamental quadratic” for Brownian motion, the choice of which depends on model assumptions. The first “general” expression applies when one uses the actual drift g, actual probabilities and the risk-adjusted discount rate k. The second version, using risk-neutral drift gˆ , martingale or risk-neutral probability measures and the risk-free rate r , is anchored in option-pricing theory and risk-neutral valuation.a The first, more general expression for the fundamental quadratic of Brownian motion is 1

δ (β ) ≡ k − g (β ) = k − αβ − β 2σ 2 , 2

(A2.1)

where α is the drift parameter of the arithmetic Brownian motion xɶ t corresponding to exponentials of Brownian motion Xɶ t ( xɶ t = ln ( Xɶ t ) β ) with drift g = αβ + (β 2σ 2 2). The fundamental quadratic expression can be thought of as describing some sort of dividend yield, namely the difference between the total equilibrium return k and capital gains g ( = αβ + (β 2σ 2 2)). The roots of this quadratic function (setting δ (⋅) = 0) in the general case are

β1 = −

α α 2 k + ⎛⎜ 2 ⎞⎟ + 2 2 2 ⎝σ ⎠ σ σ

(> 1) ,

(A2.2) α α ⎞2 k ⎛ β2 = − 2 − ⎜ 2 ⎟ + 2 2 (< 0 ) . ⎝σ ⎠ σ σ β1 and β 2 above are the positive and negative roots of the fundamental quadratic. The fundamental quadratic is strictly concave in β and strictly positive in the interval (β 2 , β1 ). Since 1 ∈ (β 2 , β 1 ), the expected present value exists for the geometric Brownian motion. An example of the fundamental quadratic function is shown in figure A.3. When risk-neutral valuation holds (in complete markets with no arbitrage opportunities), the total return k is replaced by the risk-free rate r and the risk-neutral drift αˆ is used (instead of α ).b The risk-neutral version of the fundamental quadratic above becomes

()

1 ˆ ˆ − βˆ 2σ 2 , δ βˆ = r − αβ 2

(A2.3)

a. The dynamic programming approach is “general” as long as one can identify the correct discount rate—which may not be an exogenous constant as Dixit and Pindyck (2004) assume. It applies to both incomplete markets (where there is a range of solutions) and complete markets (where risk-free arbitrage ensures a single unique solution, the risk-neutral version). b. Trigeorgis (1996, pp. 101–106) discusses how to obtain the risk-neutral drift.

446

Appendix: Basics of Stochastic Processes

Box A.2 (continued)

Fundamental quadratic

d (b) = k − ab − 1 b 2s 2 2

b2

d (0) = k

0

Slope

∂d (0) = −a ∂b

b1

b

Figure A.3 Fundamental quadratic of Brownian motion

ˆ ˆ + (βˆ 2σ 2 2)). The roots of the where αˆ is the risk-neutral drift (with gˆ = αβ risk-neutral quadratic are 2 r αˆ ⎛ αˆ ⎞ βˆ 1 = − 2 + ⎜ 2 ⎟ + 2 2 ⎝σ ⎠ σ σ

(> 1),

2 r αˆ ⎛ αˆ ⎞ βˆ 2 = − 2 − ⎜ 2 ⎟ + 2 2 ⎠ ⎝ σ σ σ

(< 0 ) .

(A2.4)

Both variants of the fundamental quadratic have been used in the literature. McKean (1965), Karlin and Taylor (1975), and McDonald and Siegel (1986) discuss the problem of early exercise of the perpetual American call option. The former authors use the general expression of the fundamental quadratic, whereas McDonald and Siegel use the risk-neutral version. We generally prefer the risk-neutral version since it is in line with modern financial theory (option valuation). Using the first expression is justified, in general, if the assumptions underlying risk-neutral valuation do not hold.

Appendix: Basics of Stochastic Processes

∞

∞

0

0

447

V [ X 0n ] = ∫ e − kt E0 ⎡⎣ Xɶ t n ⎤⎦ dt = X 0 n ∫ e − (k −γ )t dt ,

with γ ≡ ng + [ n ( n − 1) σ 2 ] 2. For k > γ , the expression above converges to V [ X 0n ] = A.3

X0n . k −γ

(A.31)

First-Hitting Time and Expected Discount Factor

In part III we viewed deferral option valuation as an optimal investment-timing problem involving the forward perpetuity value received upon exercise and an expected discount factor associated with the timing decision. In the previous section we discussed how to obtain the forward expected present value of a perpetuity flow. We next consider the second component of this call option value. Section A3.1 deals with the notion of first-hitting time, and section A3.2 is concerned with the discounting of a flow received at such a random firsthitting time. A.3.1

Exercise Timing and First-Hitting Time

First-hitting time refers to the first time a stochastic process reaches a given or specified (here an absorbing) barrier.35 Once the process reaches this critical threshold (from above or below), the decision maker takes action. An important issue for practical application thus concerns when the process reaches the critical threshold. The random time the stochastic process Xɶ t first reaches the critical threshold XT (from below) is Tɶ ≡ inf t ≥ t0 ⏐ Xɶ t ≥ XT . The first-hitting time is an important factor in deciding when one should actually invest. Unfortunately, the first-hitting time is a random variable per se, so a clear-cut prescription cannot be given. An expected value for the first time when the underlying asset price reaches the critical threshold, however, can be determined and may admit a closed-form solution in certain settings. Uncertainty or randomness may of course lead to an actual first-passage time deviating from the expected one. If the expected first-hitting time is, say, ten years, it does not mean that management should launch the project exactly in ten years. This just gives an early warning proxy.

{

}

35. The first-hitting time is a stopping time adapted to the filtration F , meaning that at each time the decision maker knows whether the “passage” though the barrier has occurred.

448

Appendix: Basics of Stochastic Processes

Example A.6 Expected First-Hitting Time for Arithmetic Brownian Motion For the arithmetic Brownian motion given in equation (A.1) with drift α (> 0), the expected time for the Brownian motion first reaching (absorbing) barrier xT ( xT > x0) is given by x − x0 E0 ⎡⎣Tɶ ⎤⎦ = T . α

(A.32)

The expected time equals the “gap” between the time-0 value of the process ( x0) and the terminal value to be reached ( xT ), divided by the expected (arithmetic) growth α per unit of time. The variance of the firstpassage time is given by

σ 2 ( xT − x0 ) var0 ⎡⎣Tɶ ⎤⎦ = . 2α 3

(A.33)

The likelihood of actual investment occurring before or after the expected time is increasing in the underlying volatility.36 Example A.7 Expected First-Hitting Time for Geometric Brownian Motion The expectation of the first-hitting time for the geometric Brownian motion can be readily obtained from equation (A.32) above by transforming the geometric into an arithmetic Brownian motion. Suppose that Xɶ t follows geometric Brownian motion with drift parameter g and volatility σ . The first time that the threshold XT is reached by the geometric process Xɶ t (starting at X 0) corresponds to the first time that the corresponding arithmetic Brownian motion xɶ t = ln ( Xɶ t ), starting at x0 = ln( X 0 ), reaches the threshold xT = ln( XT ). The arithmetic Brownian motion has drift α ≡ g − (σ 2 2) and volatility σ . The expected first-hitting time (upper barrier) is finite if g > σ 2 2. Substituting these parameters in equation (A.32) gives the expected first-passage time to hit an upper barrier ( XT > X 0) for geometric Brownian motion as E0 ⎡⎣Tɶ ⎤⎦ = A.3.2

1 ⎛ XT ⎞ ln ⎜ ⎟. α ⎝ X0 ⎠

(A.34)

Expected Discount Factor

The second step in the valuation process involves determining the present (time-0) value of receiving at a future random time (Tɶ ) a given forward 36. Cox and Miller (1965, pp. 221–22) derive formulas for the expected value and the variance of the first hitting time.

Appendix: Basics of Stochastic Processes

449

net present (time-Tɶ ) value VT − I. This requires a formula for the expected discount factor that allows transforming tomorrow’s uncertain (timewise) payoff into present (time-0) value. Since the investment time Tɶ is a random variable, classical discounting tools relying on deterministic timing cannot be utilized. We next describe the (stochastic) expected discount factor, a notion used extensively in the latter chapters of the book. We express the discount factor using risk-neutral or risk-adjusted expectations Eˆ t [⋅] while discounting at the risk-free rate (r). We define the expected discount factor as follows: Bt (Tɶ ) ≡ Eˆ t ⎡⎣e− r (T − t ) ⎤⎦ ɶ

∀t ≥ 0 .

(A.35)

This expected discount factor can be used to convert tomorrow’s value (received at unknown time Tɶ ) into today’s terms (at known time t). Equivalently, consider a bond with a nomination of /1 paid at a future random time Tɶ . The expected discount factor Bt (Tɶ ) refers to the expected present value (as of current time t) of /1 to be paid or received at an uncertain time Tɶ in the future. Bt (Tɶ ) is jointly a function ɶ XT , and the value of the of the future value to be reached at time T, process at current time t, X t . It may be understood as a discount factor over states X t and XT . Therefore, for some applications, we express it as B ( X t ; XT ) . In the case of a time-homogeneous Itô process or diffusion as per equation (A.26) with drift gt ≡ g ( Xɶ t ) and volatility σ t ≡ σ ( Xɶ t ), the expected discount factor has the following properties:37 B ( X 0 ; XT ) = B ( X 0 ; X t ) × B ( X t ; XT ) or B ( X 0 ; XT ) =

1 B ( XT ; X 0 )

or

B0 (Tɶ ) =

B0 (Tɶ ) = B0 (t ) × Bt (Tɶ ), (A.36)

1 . BT (0 )

(A.37)

We now sketch the necessary steps that enable the derivation of the expected discount factor. As long as Xɶ t < XT, for a very small time interval h the event ( X t + h = XT ) is highly unlikely, resulting in the following recursion expression B ( X t ; XT ) ≈

1 × Et ⎡⎣ B ( Xɶ t + h ; XT )⎤⎦ . 1 + rh

37. Given the strong Markov property of Itô processes, the increments Xɶ t − X 0 and Xɶ T − Xɶ t are independent for T ∉[0; t ]. Hence B ( X 0 ; XT ) = B ( X 0 ; X t ) × B ( X t ; XT ), confirming equation (A.36). For XT = X 0 , it follows from (A.36) that B ( X 0 ; X t ) × B ( X t ; X 0 ) = B ( X 0 ; X 0 ) = 1, obtaining (A.37).

450

Appendix: Basics of Stochastic Processes

Multiplying by 1 + rh and substracting B ( X t ; XT ) from both sides, this yields rhB ( X t ; XT ) ≈ Et ⎡⎣ B ( Xɶ t + h ; XT )⎤⎦ − B ( X t ; XT ). From box A.1 equation (A1.3), it obtains that 1

rB − gt BX − σ t2 BXX = 0 , 2

(A.38)

where BX and BXX denote the first- and second-order derivatives of the expected discounted factor with respect to X t (Bt = 0). This equation can be solved subject to certain boundary conditions. The first boundary condition is that when the threshold is reached, the stochastic bond pays /1 so the discount factor equals 1: B ( XT , XT ) = 1.

(A.39)

The second condition is that the higher the distance from the preselected investment target XT , the lower the likelihood the cutoff value XT will be reached. That is, when the value process approaches zero, the discount factor tends to zero as well: lim B ( Xɶ t ; XT ) = 0 .

Xɶ t → 0

(A.40)

The partial differential equation in (A.38) and the boundary conditions in (A.39) and (A.40) help obtain the expected discount factor in a general setting. In the following, we consider some particular cases of interest. Example A.8 Expected Discount Factor for Arithmetic Brownian Motion Following Dixit (1993, pp. 16–17), the general solution of equation (A.38) for the arithmetic Brownian motion is B ( xt ; xT ) = Aeβ1 xt + Beβ2 xt , where β1 and β 2 are the positive and negative roots of the fundamental quadratic given in box A.2 equation (A.2.2). Boundary conditions (A.39) and (A.40) help identify the constants A and B. Since β 2 < 0, we have B = 0 from condition (A.40). From condition (A.39), A = e − β1 xT. The expected discount factor, giving the value at time t of receiving 1 euro at random time Tɶ (for an upper threshold xT > xt) is thus Bt (Tɶ ) = e β1 ( xt − xT ).

(A.41)

Appendix: Basics of Stochastic Processes

451

Example A.9 Expected Discount Factor for Geometric Brownian Motion and Other Exponentials Suppose that the underlying asset now follows an exponential of the Brownian motion Xɶ t = exp (β xɶ t ), where the process xɶ t satisfies the stochastic differential equation (A.2). Given that xɶ t = ln ( Xɶ t ) β , equation (A.41) results in the following expected discount factor (used to discount to present time t a flow received at random future time Tɶ ):38 b

⎛X ⎞ Bt (Tɶ ) = ⎜ t ⎟ , ⎝ XT ⎠

(A.42)

where b ≡ β1 β . For the special case of a geometric Brownian motion (β = 1), the expected discount factor becomes39 β

1 ⎛X ⎞ Bt (Tɶ ) = ⎜ t ⎟ . ⎝ XT ⎠

(A.43)

Example A.10 Expected Discount Factor for Mean-Reverting Process In case the stochastic process follows mean-reversion (geometric Ornstein–Uhlenbeck) as in equation (A.23), Dixit, Pindyck, and Sødal (1999) show that the discount factor is given by40 ⎛X ⎞ Bt (Tɶ ) = ⎜ t ⎟ ⎝ XT ⎠

θ1

H (ζ X t ) , H (ζ XT )

(A.44a)

where ζ = 2η σ 2 , δ = k − g = r − gˆ , and

θ1 =

2

r δ + ηX ⎛ 1 δ + ηX ⎞ +2 2 . + ⎜ − ⎝2 2 σ2 σ 2 ⎟⎠ σ 1

−

(A.44b)

The function H (⋅) is defined by H ( y) = 1 +

θ1 θ (θ + 1) y2 θ 1 (θ 1 + 1) (θ 1 + 2 ) y3 . . . y+ 1 1 + + , b1 b1 (b1 + 1) 2 ! b1 (b1 + 1) (b1 + 2 ) 3!

(A.44c)

where

δ + ηX ⎞ ⎛ b1 = 2 ⎜ θ 1 + . ⎝ σ 2 ⎟⎠

(A.44d)

38. This is subject to β > 0 , otherwise growth g (β ) is negative and the expected discount factor is not well defined. 39. The results here are based on a functional approach. Harrison (1985) employs a martingale approach to derive the expected discount factor, whereas Karlin and Taylor (1975, p. 364) derive it based on the density function of the first-hitting time of the arithmetic Brownian motion and its Laplace transform. 40. See page 10 of the 1997 working paper version of Dixit, Pindyck, and Sødal (1999).

452

Appendix: Basics of Stochastic Processes

A.3.3

Profit-Flow Stream with Stochastic Termination

The discussion above on expected discount factors enables us to derive useful properties concerning the (expected) forward present value. Let ⎤ ⎤ ⎡∞ ⎡∞ V [ XT ] ≡ ET ⎢ ∫ X t exp ( −k (t − T )) dt ⎥ = ET ⎢ ∫ X t BT (t ) dt ⎥ ⎦ ⎦ ⎣T ⎣T be the terminal (forward) value once the process first reaches trigger level XT . For simplicity, we denote VT = V [ XT ]. Whether closedform solutions for terminal values exist or not depends on the stochastic process followed by the underlying factor. Exponentials (and polynomial functions) of the Brownian motion do admit analytical solutions. Consider the present value of a continuous profit flow, πɶ (⋅), starting now (t = 0) and continuing until termination at time Tɶ . This value is important when we deal with the option to expand capacity (by lump sums) and for option games in which profit flows are earned in the continuation region (or terminated when a competitor enters at random time). The present value of profit flows earned in the continuation region is given by ∞ ∞ T E0 ⎡ ∫ πɶ ( Xɶ t ) e− kt dt ⎤ = E0 ⎡ ∫ πɶ ( Xɶ t ) e− kt dt ⎤ − E0 ⎡ ∫ ɶ πɶ ( Xɶ t ) e− kt dt ⎤ . ⎣T ⎦ ⎣ 0 ⎦ ⎣⎢ 0 ⎦⎥ ɶ

Given the (strong) Markov property for the Itô process, it follows that41 T ∞ ∞ k t Tɶ E0 ⎡ ∫ πɶ ( Xɶ t ) e − kt dt ⎤ = E0 ⎡ ∫ πɶ ( Xɶ t ) e − kt dt ⎤ − B0 (Tɶ ) ET ⎡ ∫ ɶ πɶ ( Xɶ t ) e − ( − ) dt ⎤ 0 T ⎢⎣ 0 ⎥⎦ ⎣ ⎦ ⎣ ⎦ = V0 − B0 (Tɶ ) VT . ɶ

Alternatively, T V0 = E0 ⎡ ∫ π( Xɶ t ) e − kt dt ⎤ + B0 (Tɶ ) VT . ⎣⎢ 0 ⎦⎥ ɶ

(A.45)

Example A.11 Present Value with Stochastic Expiration for Geometric Brownian Motion Consider the special case of geometric Brownian motion. For this diffusion process, from (A.30) V0 = X 0 δ , VT = XT δ , and from (A.43) β B0 (Tɶ ) = ( X 0 XT ) 1, where δ ≡ k − g = r − gˆ with gˆ being the constant (riskneutral) drift parameter of the process and β1 as given in equation (A2.2) of box A.2. From equation (A.45), in case of the geometric Brownian 41. For a formal derivation of this property, the reader may refer to Harrison (1985, pp. 44–46).

Appendix: Basics of Stochastic Processes

453

motion the present value of a profit flow stream with stochastic expiration Tɶ becomes42 β −1 Tɶ X ⎛ ⎛X ⎞ 1 ⎞ E0 ⎡ ∫ X t e− kt dt ⎤ = 0 ⎜ 1 − ⎜ 0 ⎟ ⎟⎠ . ⎣⎢ 0 ⎦⎥ δ ⎝ ⎝ XT ⎠

(A.46)

The expression above is illustrated for a sample path example in figure A.4.43 A.4

Optimal Stopping

Obtaining formulas for forward values is generally useful in valuing real options problems. In the case of the investment timing option, the option holder has the opportunity to delay investment (the investment cost I , once incurred, is sunk). Upon investing at stochastic time Tɶ , the firm will receive the forward value of the project (VT ) by paying investment cost I . Assuming the project’s profit flow stream is stochastic (e.g., the profit is a derivative on a stochastically evolving underlying factor), forward value formulas enable us to determine the value of the committed investment (the forward NPV) at the future random time Tɶ the investment decision will take place. The problem for the investor consists in determining the optimal investment timing, given that the project value evolves stochastically. This type of problem is called optimal stopping as it involves deciding when to optimally stop waiting and start investing. Before making the stop/invest decision, the process “continues” until it reaches a critical barrier or trigger value. The range of state values for which the stopping decision is not (yet) made is the continuation or inaction region. Suppose that before the investment is undertaken (e.g., a lumpy expansion investment is made), the firm receives initial stochastic profit flow πɶ 0 ( Xɶ t ) . This profit flow continues until the firm decides to make a new investment (expand). The first region before the new investment occurs during which the firm receives a base profit flow, πɶ 0 ( Xɶ t ), is the continuation or inaction region; in this region the firm still waits. The second region in which the firm invests (in added capacity) is the stopping or action region. Once the new investment threshold has been 42. This is identical to the result in Dixit and Pindyck (1994, p. 316). Symbols are changed for notational consistency. 43. Rigorously speaking, (expected) present values V0 and VT are not depicted in the graph; here only a sample path is considered.

–6

~

T

VT

2.0

2.5

( ~)

V0 − B0 T VT ~

T

( ~)

B0 T VT

Time t

VT

Time t

p t × exp(−k(t−T ))

~

Time t

p t × exp(−kt)

Figure A.4 Value in the continuation region for a specific sample path of geometric Brownian motion We consider the geometric Brownian motion of equation (A.12) with g = 9 percent and σ = 20 percent. The (risk-adjusted) discount rate is k = 15 percent. In the discretized version of the GBM, time increments are h = 0.04 . For illustrative simplicity, we consider a ten-year time period.

0.0

Time t

0.0

–6

0.5

0.5

1.0

~

p t × exp(−k(t−T ))

V0

p t × exp(−kt)

Profit value in the continuation region (in millions of euros)

0.0

0.5

1.0 p (X *)

pt

Time t

exp(− kt)

1.0

1.5

2.0

2.5

1.5

pt

p 0 exp (gt)

Present value V0 (in millions of euros)

1.5

2.0

Stochastic profit flow (in millions of euros) 2.5

0.0

0.5

1.0

1.5

2.0

2.5

Stochastic profit flow (in millions of euros)

454 Appendix: Basics of Stochastic Processes

Appendix: Basics of Stochastic Processes

455

reached, the firm stops waiting and invests immediately. Once the firm undertakes the added investment, it receives a larger profit flow πɶ 1 ( Xɶ t ). Let VT denote the value of the project at time T in state XT . Let F ( X t ) be the time-t value of a perpetual American call option on this project. Depending on the region in which the process is found, the project value equals either the present value of receiving the new stochastic profit flow π 1 ( Xɶ t ) in perpetuity from time T onward, resulting in gross project value VT minus the necessary investment outlay I , or the value of waiting and deferring the investment (expansion) decision for a time period of length h. There is actually an investment threshold X* that divides the state space into two regions, the inaction and action regions.44 This critical cutoff level X* provides guidance in deciding whether and when to invest in the project. If the current state is in the region ( −∞, X*), it is optimal not to act (i.e., wait to invest). In the region ( X*, ∞ ), it is strictly dominant to act (invest). At the cutoff point, X*, the firm is just indifferent between action (investing) and inaction (waiting). This threshold ɶ are is unique.45 The two regions and the optimal investment timing T* illustrated in figure A.5 for a sample realization of a stochastic process with positive drift. ɶ ), the continuIn the period prior to investment (at stochastic time T* ation value is characterized as follows. Over a small time interval, h, the profit flow π 0 ( Xɶ t ) h received by the firm plus the instantaneous capital gain Eˆ t ⎡⎣ F ( Xɶ t + h )⎤⎦ − F ( Xɶ t ) h must equal the total equilibrium return, rF ( X t ) h. For h → 0, this leads to the HJB equation

{

rF = πɶ 0 + ΓF .

}

(A.47)

There are various approaches used to derive the unique optimal critical level X* that divides the two regions. A common approach (see Dixit and Pindyck 1994) involves deriving two related “boundary conditions”: 1. Value-matching At the optimal threshold, X*, the option holder is just indifferent between still waiting and holding the option, F ( X*), 44. This threshold level is an “absorbing barrier.” 45. For the uniqueness of the threshold, see Dixit and Pindyck (1994, pp. 128–30). Dixit and Pindyck’s (1994) model is more general as it addresses concurrently the stopping problem for investment and exit. Huisman (2001, pp. 34–35) is more in line with our present setup as only the option to defer is considered.

456

Appendix: Basics of Stochastic Processes

Action region (invest)

X*

Inaction region (wait)

Sample path of the stochastic process

~ T*

t

Figure A.5 Optimal stopping and first-hitting time

or investing immediately and receiving the forward value VT ( X*) − I. That is, F ( X*) = VT ( X*) − I .

(A.48)

2. Smooth-pasting (optimality)46 This condition ensures that the option to wait approaches smoothly the committed project value when the process Xɶ t approaches the optimal cutoff value X*. At the point of optimal investment, X*, the project value function (V − I or project NPV) is tangent to the option value function, that is, FX ( X*) = VX ( X*).

(A.49)

The investment-timing problem is crucial to analyzing a number of real options in continuous time. The investor needs to deduce (1) the value of the investment opportunity given a specified investment rule and (2) the optimal investment threshold level, X*, at which the investor should take action. By following this optimal investment rule, the firm receives 46. This condition, also called high-contact or smooth-fit condition, was first applied by McKean (1965) and Merton (1973) in option-pricing models. Dixit and Pindyck (1994, pp. 130–32) discuss it informally, whereas Øksendal (2007, ch. 10) provides a more advanced treatment. Peskir and Shiryaev (2006) provide an advanced mathematical treatise on optimal stopping. Dumas (1991) discusses the appropriate optimality conditions in stochastic control models: the smooth-pasting condition applies to stopping and impulse control models (e.g., lumpy investment decisions), while the supercontact condition (a higher order condition) to instantaneous control models (e.g., incremental capacity investment problems).

dxɶ t = α dt + σ dzt dXɶ t = gXɶ t dt + σ Xɶ t dzt Xɶ t = exp (β xɶ t ) with xɶ t ABM dXɶ t = η ( X − Xɶ t ) dt + σ Xɶ t dzt

Arithmetic Brownian motion (ABM)

Geometric Brownian motion (GBM)

Exponentials of Brownian motion (EBM)

Geometric Ornstein–Uhlenbeck (GOU)

b

⎛ X0 ⎞ ⎝⎜ XT ⎟⎠ θ1 ⎛ X 0 ⎞ H(ζ X 0 ) ⎟ ⎜⎝ XT ⎠ H(ζ XT )

β1

NA

xT − x0 α 1 ⎛X ⎞ ln ⎜ T ⎟ g − (σ 2 2) ⎝ X 0 ⎠ 1 ⎛X ⎞ ln ⎜ T ⎟ αβ ⎝ X 0 ⎠

e β1( X0 − XT )

α xT + k2 k XT k−g XT δ (β ) NA ⎛ X0 ⎞ ⎟ ⎜⎝ XT ⎠

Expected firsthitting time E0(Tɶ )

Expected discount factor B0 (Tɶ )

Terminal value VT

Note: β1 , θ 1, and b1 are given in (A.2.2), (A.44b), and (A.44d) respectively; ζ = 2η σ 2 ; b = β1 β ; H (⋅) is defined in (A.44c); NA: Not available. We use the risk-neutral version throughout chapters 9 to 12.

Description

Stochastic process

Table A.1 Characteristics of basic stochastic processes

Appendix: Basics of Stochastic Processes 457

458

Appendix: Basics of Stochastic Processes

the highest option value (optimal investment policy). Finally, the investor can ascertain when the optimal decision is expected to take place. Once the investment trigger is found, actual investment may possibly occur far in the future if the current state does not evolve to offer sufficient profitability for the project. Conclusion In this appendix we described several important stochastic processes and reviewed the basics of stochastic calculus. We discussed useful properties for the most widely used diffusion processes, namely arithmetic Brownian motion, geometric Brownian motion, and mean-reversion. We also reviewed important notions concerning forward net present value, firsthitting time, and expected discount factor. The developed concepts and derived formulas serve as useful building blocks for various chapters in part III of the book. In chapter 9, for example, we recast the investment option in view of the trade-off between obtaining a larger forward net present value (discussed in section A.2) versus a lower expected discount factor (discussed in section A.3.2). Table A.1 offers a compendium of useful formulas for easy reference. Selected References Harrison (1985), Dixit (1993), and Stokey (2008) summarize useful properties on several continuous-time processes and discuss how to apply them in economic contexts involving stochastic control models. Seasoned readers may also find useful the more advanced mathematical treatment of Brownian motion and stochastic Itô calculus by Øksendal (2007) and Karatzas and Shreve (1988). Merton (1990), Duffie (1992), and Shreve (2004) apply these notions to mathematical finance, deriving, for example, the Black–Scholes formula. Dixit, Avinash K. 1993. The Art of Smooth Pasting. New York: Routledge. Duffie, Darrell. 1992. Dynamic Asset Pricing Theory. Princeton: Princeton University Press. Harrison, J. Michael. 1985. Brownian Motion and Stochastic Flow Systems. New York: Wiley.

Appendix: Basics of Stochastic Processes

459

Karatzas, Ioannis, and Steven E. Shreve. 1988. Brownian Motion and Stochastic Calculus. Berlin: Springer. Merton, Robert C. 1990. Continuous-Time Finance. Malden, MA: Wiley-Blackwell. Øksendal, Bernt. 2007. Stochastic Differential Equations: An Introduction with Applications, 6th ed. Berlin: Springer. Shreve, Steven E. 2004. Stochastic Calculus for Finance II: ContinuousTime Models. Springer Finance Series. New York: Springer. Stokey, Nancy L. 2008. The Economics of Inaction: Stochastic Control Models with Fixed Costs. Princeton: Princeton University Press.

References

Aguerrevere, Felipe L. 2003. Equilibrium investment strategies and output price behavior: A real-options approach. Review of Financial Studies 16 (4): 1239–72. Aguerrevere, Felipe L. 2009. Real options, product market competition, and asset returns. Journal of Finance 64 (2): 957–83. Aitchison, John, and James A. C. Brown. 1957. The Lognormal Distribution. New York: Cambridge University Press. Arrow, Kenneth J. 1962. Economic welfare and the allocation of resources for invention. In Richard R. Nelson, ed., The Rate and Direction of Inventive Activity: Economic and Social Factors. Princeton: Princeton University Press, pp. 609–26. Aumann, Robert J., and Sergiu Hart. 1992. eds. Handbook of Game Theory with Economic Applications, vol. 1. Amsterdam: Elsevier North Holland. Azevedo, Alcino, and Dean A. Paxson. 2009. Uncertainty and competition in the adoption of complementary technologies. Working paper. Manchester Business School, UK. Baba, Naohiko. 2001. Uncertainty, monitoring costs, and private banks’ lending decisions in a duopolistic loan market: A game-theoretic real options approach. Monetary and Economic Studies 19 (May): 21–46. Bachelier, Louis. 1901. Théorie de la spéculation. Annales Scientifiques de l’École Normale Supérieure 3 (17): 21–86. Back, Kerry, and Dirk Paulsen. 2009. Open-loop equilibria and perfect competition in option exercise games. Review of Financial Studies 22 (11): 4531–52. Bain, Joe S. 1956. Barriers to New Competition. Cambridge: Harvard University Press. Baldursson, Fridrik M. 1998. Irreversible investment under uncertainty in oligopoly. Journal of Economic Dynamics and Control 22 (4): 627–44. Baldursson, Fridrik M., and Ioannis Karatzas. 1996. Irreversible investment and industry equilibrium. Finance and Stochastics 1 (1): 69–89. Beaudry, Paul, and Michel Poitevin. 1994. The commitment value of contracts under dynamic renegotiation. Rand Journal of Economics 25 (4): 501–17. Bellman, Richard E. 1957. Dynamic Programming. Princeton: Princeton University Press. Bernheim, B. Douglas. 1984. Rationalizable strategic behavior. Econometrica 52 (4): 1007–28.

462

References

Bertola, Giuseppe. 1988. Adjustment costs and dynamic factor demands: Investment and employment under uncertainty. PhD dissertation. MIT. Bertola, Giuseppe. 1989. Irreversible investment. Working paper. Princeton University. Bertrand, Joseph L.F. 1883. Théorie mathématique de la richesse sociale. Journal des Savants 67: 499–508. Besanko, David, David Dranove, Mark Shanley, and Scott Schaefer. 2004. Economics of Strategy, 3rd ed. New York: Wiley. Black, Fischer, and Myron S. Scholes. 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81 (3): 637–54. Botteron, Pascal, Marc Chesney, and Rajna Gibson-Asner. 2003. Analyzing firms’ strategic investment decisions in a real options framework. Journal of International Financial Markets, Institutions and Money 13 (5): 451–79. Bouis, Romain, Kuno J. M. Huisman, and Peter M. Kort. 2009. Investment in oligopoly under uncertainty: The accordion effect. International Journal of Industrial Organization 27 (2): 320–31. Boyer, Marcel, and Séverine Clamens. 2001. Strategic adoption of a new technology under uncertain implementation. Mimeo. CIRANO, Montreal. Boyer, Marcel, Eric Gravel, and Pierre Lasserre. 2004. Real options and strategic competition: A survey. Mimeo. CIRANO, Montreal. Boyer, Marcel, Pierre Lasserre, Thomas Mariotti, and Michel Moreaux. 2004. Preemption and rent dissipation under Bertrand competition. International Journal of Industrial Organization 22 (3): 309–28. Boyer, Marcel, Pierre Lasserre, and Michel Moreaux. 2010. A dynamic duopoly investment game under uncertain market growth. Working paper 617. IDEI, Toulouse. Brandenburger, Adam M., and Barry Nalebuff. 1996. Co-opetition. New York: Broadway Business. Brandenburger, Adam M., and Harborne Stuart. 2007. Biform games. Management Science 53 (4): 537–49. Breiman, Leo. 1968. Probability. Reading, MA: Addison-Wesley. Brennan, Michael J., and Lenos Trigeorgis, eds. 2000. Project Flexibility, Agency and Competition. New York: Oxford University Press. Bulan, Laarni, Christopher Mayer, and C. Tsuriel Somerville. 2009. Irreversible investment, real options, and competition: Evidence from real estate development. Journal of Urban Economics 65 (3): 237–51. Bulow, Jeremy I., John D. Geanakoplos, and Paul D. Klemperer. 1985. Multimarket oligopoly: Strategic substitutes and complements. Journal of Political Economy 93 (3): 488–511. Caballero, Ricardo J., and Robert S. Pindyck. 1996. Uncertainty, investment and industry evolution. International Economic Review 37 (3): 641–62. Chamberlin, Edward H. 1933. The Theory of Monopolistic Competition. Cambridge: Harvard University Press. Chandler, Alfred D., Jr. 1962. Strategy and Structure. Cambridge: MIT Press.

References

463

Chevalier-Roignant, Benoît, Christoph M. Flath, Arnd Huchzermeier, and Lenos Trigeorgis. 2011. Strategic investment under uncertainty: A synthesis. European Journal of Operational Research 215 (3): 639–50. Collins, David J., and Cynthia A. Montgomery. 1995. Competing on resources: Strategy in the 1990s. Harvard Business Review 73 (July–August): 119–28. Copeland, Thomas E., and Vladimir Antikarov. 2001. Real Options: A Practitioner’s Guide. New York: Texere. Copeland, Thomas E., J. Fred Weston, and Kuldeep Shastri. 2005. Financial Theory and Corporate Policy, 4th ed. Boston: Pearson Addison Wesley. Cottrel, Tom J., and Gordon A. Sick. 2001. First-mover (dis)advantage and real options. Journal of Applied Corporate Finance 14 (2): 41–51. Cournot, Antoine A. 1838. Recherches sur les principes mathématiques de la théorie des richesses. Paris: M. Riviere. Cox, John C., and Stephen A. Ross. 1979. The valuation of options for alternative stochastic processes. Journal of Financial Economics 3 (1–2): 145–66. Cox, John C., Stephen A. Ross, and Mark E. Rubinstein. 1979. Option pricing: A simplified approach. Journal of Financial Economics 7 (3): 229–63. d’Aspremont, Claude, and Alexis Jacquemin. 1988. Cooperative and non-cooperative R&D in duopoly with spillovers. American Economic Review 78 (5): 1133–37. Décamps, Jean-Paul, and Thomas Mariotti. 2004. Investment timing and learning externalities. Journal of Economic Theory 118 (1): 80–102. Décamps, Jean-Paul, Thomas Mariotti, and Stéphane Villeneuve. 2006. Irreversible investment in alternative projects. Economic Theory 28 (2): 425–48. Días, Marco A.G. 2004. Valuation of exploration and production assets: An overview of real options models. Journal of Petroleum Science Engineering 44 (1–2): 93–114. Días, Marco A. G., and José P. Teixeira. 2010. Continuous-time option games: Review of models and extensions. Multinational Finance Journal 14 (3–4): 219–54. Días, Marco A. G., and José P. Teixeira. 2009. Continuous-time option games: War of attrition and bargaining under uncertainty in oil exploration. In Edward R. Pitt and Christopher N. Leung, eds., OPEC, Oil Prices and LNG (Energy Policies, Politics and Prices). Hauppauge, NY: Nova Science Publishers, pp. 73–105. Dixit, Avinash K. 1980. The role of investment in entry deterrence. Economic Journal 90 (357): 95–106. Dixit, Avinash K. 1989a. Hysteresis, import penetration and exchange rate pass-through. Quarterly Journal of Economics 104 (2): 205–28. Dixit, Avinash K. 1989b. Entry and exit decisions under uncertainty. Journal of Political Economy 97 (3): 620–38. Dixit, Avinash K. 1991. Irreversible investment with price ceilings. Journal of Political Economy 99 (3): 541–57. Dixit, Avinash K. 1993. The Art of Smooth Pasting. New York: Routledge. Dixit, Avinash K., and Barry Nalebuff. 1991. Thinking Strategically: The Competitive Edge in Business, Politics and Everyday Life. New York: Norton.

464

References

Dixit, Avinash K., and Robert S. Pindyck. 1994. Investment under Uncertainty. Princeton: Princeton University Press. Dixit, Avinash K., Robert S. Pindyck, and Sigbjørn Sødal. 1999. A mark-up interpretation of optimal investment rules. Economic Journal 109 (455): 179–89. Dockner, Engelbert J., Steffen Jorgensen, Ngo Van Long, and Gerhard Sorger. 2000. Differential Games in Economics and Management Science. Cambridge: Cambridge University Press. Doeblin, Vincent. [1940] 2000. Sur l’équation de Kolmogoroff. Académie des Sciences de Paris Series 1 (331): 1031–1187. (Submitted in February 1940 but lost until the late 1990s.) Dresher, Melvin, Lloyd S. Shapley, and Albert W. Tucker, eds. 1964. Advances in Game Theory. Annals of Mathematics Studies AM-52. Princeton: Princeton University Press. Duffie, Darrell. 1992. Dynamic Asset Pricing Theory. Princeton: Princeton University Press. Dumas, Bernard. 1991. Super contact and related optimality conditions. Journal of Economic Dynamics and Control 15 (4): 675–85. Einstein, Albert. 1905. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Annalen der Physik 17 (549): 549–60. Encaoua, David, and Alexis Jacquemin. 1980. Degree of monopoly, indices of concentration and threat of entry. International Economic Review 21 (1): 87–105. Ferreira, Nelson, Jayanti Kar, and Lenos Trigeorgis. 2009. Option games: The key to competing in capital-intensive industries. Harvard Business Review 87 (3): 101–107. Fershtman, Chaim, and Kenneth L. Judd. 1987. Equilibrium incentives in oligopoly. American Economic Review 77 (5): 927–40. Fine, Charles H., and Lode Li. 1989. Equilibrium exit in stochastically declining industries. Games and Economic Behavior 1 (1): 40–59. Flaherty, M. Thérèse. 1977. Timing patterns of new technology adoption in duopolistic industries. Studies in Industry Economics 81. Stanford University. Fabozzi, Frank J., ed. 1990. Advances in Futures and Options Research, vol. 4. Greenwich, CT: JAI Press. Friedman, James W. 1971. A non-cooperative equilibrium for supergames. Review of Economic Studies 38 (1): 1–12. Friedman, James W. 1977. Oligopoly and the Theory of Games. Amsterdam: North-Holland. Fudenberg, Drew, and Eric S. Maskin. 1986. The folk theorem in repeated games with discounting or with incomplete information. Econometrica 54 (3): 533–54. Fudenberg, Drew, and Jean Tirole. 1983. Capital as a commitment: Strategic investment to deter mobility. Journal of Economic Theory 31 (2): 227–50. Fudenberg, Drew, and Jean Tirole. 1984. The fat cat effect, the puppy dog ploy and the lean and hungry look. American Economic Review 74 (2): 361–66. Fudenberg, Drew, and Jean Tirole. 1985. Preemption and rent equalization in the adoption of new technology. Review of Economic Studies 52 (3): 383–401. Fudenberg, Drew, and Jean Tirole. 1986. A theory of exit in duopoly. Econometrica 54 (4): 943–60.

References

465

Fudenberg, Drew, and Jean Tirole. 1991. Game Theory. Cambridge: MIT Press. Gal-Or, Esther. 1985. First-mover and second-mover advantages. International Economic Review 26 (3): 649–53. Garlappi, Lorenzo. 2004. Risk premia and preemption in R&D ventures. Journal of Financial and Quantitative Analysis 39 (4): 843–72. Ghemawat, Pankaj, and Barry Nalebuff. 1985. Exit. Rand Journal of Economics 16 (2): 184–94. Ghemawat, Pankaj, and Barry Nalebuff. 1990. The devolution of declining industries. Quarterly Journal of Economics 105 (1): 167–86. Ghemawat, Pankaj. 1991. Commitment: The Dynamic of Strategy. New York: Free Press. Gibbons, Robert. 1992. Game Theory for Applied Economists. Princeton: Princeton University Press. Gilbert, Richard J., and Richard G. Harris. 1984. Competition with lumpy investment. Rand Journal of Economics 15 (2): 197–212. Grant, Robert M. 2005. Contemporary Strategy Analysis, 5th ed. Malden, MA: Blackwell. Grenadier, Steven R. 1996. The strategic exercise of options: Development cascades and overbuilding in real estate markets. Journal of Finance 51 (5): 1653–79. Grenadier, Steven R. 1999. Information revelation through option exercise. Review of Financial Studies 12 (1): 95–129. Grenadier, Steven R., ed. 2000a. Game Choices: The Interaction of Real Options and Game Theory. London: Risk Books. Grenadier, Steven R. 2000b. Equilibrium with time-to-build: A real options approach. In Michael J. Brennan and Lenos Trigeorgis, eds., Project flexibility, Agency and Competition. Oxford: Oxford University Press, pp. 275–96. Grenadier, Steven R. 2002. Option exercise games: An application to the equilibrium investment strategies of firms. Review of Financial Studies 15 (3): 691–721. Harrison, J. Michael, and David M. Kreps. 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20 (3): 381–408. Harrison, J. Michael. 1985. Brownian Motion and Stochastic Flow Systems. New York: Wiley. Harsanyi, John C. 1964. A general solution for finite noncooperative games based on risk dominance. In Melvin Dresher, Lloyd S. Shapley, and Albert W. Tucker, eds., Advances in Game Theory, Annals of Mathematics AM-52. Princeton: Princeton University Press, pp. 627–50. Harsanyi, John C. 1967. Games with incomplete information played by “Bayesian” players, I–III. Management Science 14 (3):159–182, 320–334, 486–502. Harsanyi, John C., and Reinhard Selten. 1988. A General Theory of Equilibrium Selection in Games. Cambridge: MIT Press. Hendricks, Ken, Andrew Weiss, and Charles Wilson. 1988. The war of attrition in continuous time with complete information. International Economic Review 29 (4): 663–80. Herfindhal, Orris C. 1950. Concentration in the steel industry. PhD dissertation. Columbia University.

466

References

Hicks, John R. 1935. Annual survey of economic theory: The theory of monopoly. Econometrica 3 (1): 1–20. Hirschman, Albert O. 1945. National Power and the Structure of Foreign Trade. Los Angeles: University of California Press. Hirschman, Albert O. 1964. The paternity of an index. American Economic Review 54 (5): 761–62. Huisman, Kuno J. M. 2001. Technology Investment: A Game-Theoretic Real Options Approach. Dordrecht: Kluwer Academic. Huisman, Kuno J. M., and Peter M. Kort. 1999. Effects of strategic interactions on the option value of waiting. CentER Discussion paper 9992. Tilburg University, The Netherlands. Huisman, Kuno J. M., and Peter M. Kort. 2004. Strategic technology adoption taking into account future technological improvements: A real options approach. European Journal of Operational Research 159 (3): 705–28. Huisman, Kuno J. M., Peter M. Kort, Grzegorz Pawlina, and Jacco J. J. Thijssen. 2004. Strategic investment under uncertainty: Merging real options with game theory. Zeitschrift für Betriebswirtschaft 67 (3): 97–123. International Energy Agency. 2007. Tackling Investment Challenges in Power Generation. Paris: IEA Publications. Itô, Kiyoshi. 1951. On stochastic differential equations. Memoirs of American Mathematical Society 4: 1–51. Jarrow, Robert A., and Andrew Rudd. 1983. Option Pricing. Homewood, Ill.: Dow Jones-Irwin. Joaquin, Domingo C., and Kirt C. Butler. 2000. Competitive investment decisions: A synthesis. In Michael J. Brennan and Lenos Trigeorgis, eds., Project Flexibility, Agency, and Competition: New Developments in the Theory and Application of Real Options. New York: Oxford University Press, pp. 324–39. Kamien, Morton I. 1992. Patent licensing. In Robert J. Aumann and Sergiu Hart, eds., Handbook of Game Theory with Economic Applications 1. Amsterdam: Elsevier NorthHolland, pp. 331–54. Kamien, Morton I., and Yair Tauman. 1986. Fees versus royalties and the private value of a patent. Quarterly Journal of Economics 101 (3): 471–92. Karatzas, Ioannis, and Steven E. Shreve. 1988. Brownian Motion and Stochastic Calculus. Berlin: Springer. Karlin, Samuel, and Howard M. Taylor. 1975. A First Course in Stochastic Processes. London: Academic Press. Kester, W. Carl. 1984. Today’s options for tomorrow’s growth. Harvard Business Review 62 (2): 153–60. Kong, Jean J., and Yue Kuen Kwok. 2007. Real options in strategic investment games between two asymmetric firms. European Journal of Operational Research 181 (2): 967–85. Kreps, David M., and Jose A. Scheinkman. 1983. Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell Journal of Economics 14 (2): 326–37. Kreps, David M., and Robert Wilson. 1982. Sequential equilibria. Econometrica 50 (4): 863–94.

References

467

Kulatilaka, Nalin, and Enrico C. Perotti. 1998. Strategic growth options. Management Science 44 (8): 1021–31. Lamb, Robert B., ed. 1984. Competitive Strategic Management. Englewood Cliffs, NJ: Prentice-Hall. Lambrecht, Bart M. 2000. Strategic sequential investments and sleeping patents. In Michael J. Brennan and Lenos Trigeorgis, eds., Project Flexibility, Agency, and Competition: New Developments in the Theory and Application of Real Options. New York: Oxford University Press, pp. 297–323. Lambrecht, Bart M. 2001. The Impact of debt financing on entry and exit in a duopoly. Review of Financial Studies 14 (3): 765–804. Lambrecht, Bart M., and William R. Perraudin. 2003. Real options and preemption under incomplete information. Journal of Economic Dynamics and Control 27 (4): 619–43. Leahy, John V. 1993. Investment in competitive equilibrium: The optimality of myopic behavior. Quarterly Journal of Economics 108 (4): 1105–33. Lerner, Abba P. 1934. The concept of monopoly and the measurement of monopoly power. Review of Economic Studies 1 (3): 157–75. Mæland, Jøril. 2010. Asymmetric information and irreversible investments: an auction model. Multinational Finance Journal 14 (3–4): 255–89. Mailath, George J., and Larry Samuelson. 2006. Repeated Games and Reputations: LongRun Relationships. Oxford: Oxford University Press. Margrabe, William. 1978. The value of an option to exchange one asset for another. Journal of Finance 33 (1): 177–86. Martzoukos, Spiros H., and Eleftherios Zacharias. 2001. Real options with incomplete information and spillovers. Working paper. University of Cyprus. Mason, Robin, and Helen Weeds. 2002. Reverse hysteresis: R&D investment and stochastic innovation. Discussion paper. CEPR, London. McDonald, Robert L. 2000. Real options and rules of thumb in capital budgeting. In Michael J. Brennan and Lenos Trigeorgis, eds., Project Flexibility, Agency, and Competition: New Developments in the Theory and Application of Real Options. New York: Oxford University Press, pp. 13–33. McDonald, Robert L. 2006. Derivatives Markets, 2nd ed. Boston: Pearson Addison Wesley. McDonald, Robert L., and Daniel Siegel. 1985. Investment and the valuation of firms when there is an option to shut down. International Economic Review 26 (2): 331–49. McDonald, Robert L., and Daniel Siegel. 1986. The value of waiting to invest. Quarterly Journal of Economics 101 (4): 707–28. McKean, H. C., Jr. 1965. Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Management Review 6 (2): 32–39. Merton, Robert C. 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4 (1): 141–83. Merton, Robert C. 1990. Continuous-Time Finance. Malden, MA: Wiley-Blackwell. Merton, Robert C. 1998. Applications of option-pricing theory: Twenty-five years later. American Economic Review 88 (3): 323–49.

468

References

Miltersen, Kristian R., and Eduardo S. Schwartz. 2004. R&D investments with competitive interactions. Review of Finance 8 (3): 355–401. Mintzberg, Henry, Joseph B. Lampel, James B. Quinn, and Sumantra Ghoshal. 2002. The Strategy Process: Concepts, Context, Cases, 4th ed. Harlow, UK: Prentice Hall. Murto, Pauli. 2004. Exit in duopoly under uncertainty. Rand Journal of Economics 35 (1): 111–27. Murto, Pauli, and Jussi Keppo. 2002. A game model of irreversible investment under uncertainty. International Game Theory Review 4 (2): 127–40. Murto, Pauli, Erkka Näsäkkälä, and Jussi Keppo. 2004. Timing of investments in oligopoly under uncertainty: A framework for numerical analysis. European Journal of Operational Research 157 (2): 486–500. Myers, Stewart C. 1977. Determinants of corporate borrowing. Journal of Financial Economics 5 (2): 147–75. Myers, Stewart C. 1984. Finance theory and financial strategy. Interfaces 14 (1): 126–37. Myers, Stewart C., and Saman Majd. 1990. Abandonment value and project life. In Frank J. Fabozzi, ed., Advances in Futures and Options Research, vol. 4. Greenwich, CT: JAI Press, pp. 1–21. Myerson, Roger B. 1992. Game Theory: Analysis of Conflict. Cambridge: Harvard University Press. Myerson, Roger B. 1999. Nash equilibrium and the history of economic theory. Journal of Economic Literature 37 (3): 1067–82. Nash, John F. 1950a. The bargaining problem. Econometrica 18 (2): 155–62. Nash, John F. 1950b. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America 36 (1): 48–49. Nash, John F. 1951. Non-cooperative games. Annals of Mathematics 54 (2): 286–95. Neftci, Salih N. 2000. An Introduction to the Mathematics of Financial Derivatives, 2nd ed. San Diego: Academic Press. Nelson, Richard R., ed. 1962. The Rate and Direction of Inventive Activity: Economic and Social Factors. Princeton: Princeton University Press. von Neumann, John, and Oskar Morgenstern. 1944. The Theory of Games and Economic Behavior. New York: Wiley. Nielsen, Martin J. 2002. Competition and irreversible investments. International Journal of Industrial Organization 20 (5): 731–43. Novy-Marx, Robert. 2007. An equilibrium model of investment under uncertainty. Review of Financial Studies 20 (5): 1461–1502. Odening, Martin, Oliver Mußhoff, Norbert Hirschauer, and Alfons Balmann. 2007. Investment under uncertainty: Does competition matter? Journal of Economic Dynamics and Control 31 (3): 994–1014. Øksendal, Bernt. 2007. Stochastic Differential Equations: An Introduction with Applications, 6th ed. Berlin: Springer. Osborne, Martin J. 2004. An Introduction to Game Theory. New York: Oxford University Press.

References

469

Pawlina, Grzegorz, and Peter M. Kort. 2006. Real options in an asymmetric duopoly: Who benefits from your competitive disadvantage? Journal of Economics & Management Strategy 15 (1): 1–35. Paxson, Dean A., and Helen Pinto. 2003a. Leader-follower value functions if the market share follows a birth and death process. In Dean A. Paxson, ed., Real R&D Options. Oxford: Butterworth-Heinemann, pp. 208–27. Paxson, Dean A., and Helen Pinto. 2003b. Competitive games in duopoly settings with two stochastic factors. Working paper. University of Manchester, UK. Paxson, Dean A., ed. 2003. Real R&D Options. Oxford: Butterworth-Heinemann. Pearce, David G. 1984. Rationalizable strategic behavior and the problem of perfection. Econometrica 52 (4): 1029–50. Penrose, Edith T. 1959. The Theory of the Growth of the Firm. New York: Wiley. Peškir, Goran, and Albert Shiryaev. Optimal Stopping and Free-Boundary Problems. Basel: Birkhauser. Pindyck, Robert S. 1988. Irreversible investment, capacity choice and the value of the firm. American Economic Review 78 (5): 969–85. Pineau, Pierre-Olivier, and Pauli Murto. 2004. An oligopolistic investment model of the Finnish electricity market. Annals of Operations Research 121(1–4): 123–48. Pitchik, Carolyn. 1982. Equilibria of a two-person non-zero sum noisy game of timing. International Journal of Game Theory 10: 207–21. Pitt, Edward R., and Christopher N. Leung, eds. 2009. OPEC, Oil Prices and LNG (Energy Policies, Politics and Prices). Hauppauge, NY: Nova Science Publishers. Porter, Michael E. 1980. Competitive Strategy. London: Macmillan. Prahalad, Coimbatore K., and Gary Hamel. 1990. The core competence of the corporation. Harvard Business Review 68 (May–June): 79–91. Protter, Philip E. 2004. Stochastic Integration and Differential Equations, 2nd ed. Heidelberg: Springer. Rasmussen, Eric. 2005. Games and Information: An Introduction to Game Theory. Oxford: Blackwell. Reinganum, Jennifer F. 1981a. On the diffusion of new technology: A game-theoretic approach. Review of Economic Studies 48 (3): 395–405. Reinganum, Jennifer F. 1981b. Market structure and the diffusion of new technology. Bell Journal of Economics 12 (2): 618–24. Reiss, Ariane. 1998. Investment in innovations and competition: An option-pricing approach. Quarterly Review of Economics and Finance 38 (3): 635–50. Rubinstein, Ariel. 1982. Perfect equilibrium in a bargaining model. Econometrica 50 (1): 97–109. Rumelt, Richard P. 1984. Towards a stategic theory of the firm. In Robert B. Lamb, ed., Competitive Strategic Management. Englewood Cliffs, NJ: Prentice-Hall, pp. 556–70. Saloner, Garth. 1991. Modeling, game theory, and strategic management. Strategic Management Journal 12 (Winter): 119–36.

470

References

Samuelson, Paul A. 1965a. A rational theory of warrant pricing. Industrial Management Review 6 (2): 32–39. Samuelson, Paul A. 1965b. Proof that properly anticipated prices fluctuate randomly. Industrial Management Review 6 (2): 41–49. Samuelson, Paul A. 1973. Mathematics of speculative price. SIAM Review 15 (1): 1–42. Savage, Sam L. 2009. The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty. Hoboken, NJ: Wiley. Schelling, Thomas C. 1960. The Strategy of Conflict. Cambridge: Harvard University Press. Scherer, Frederic M. 1967. Research and development resource allocation under rivalry. Quarterly Journal of Economics 81 (3): 359–94. Schwartz, Eduardo S., and Lenos Trigeorgis, eds. 2001. Real Options and Investment under Uncertainty: Classical Readings and Recent Contributions. Cambridge: MIT Press. Selten, Reinhard. 1965. Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit. Zeitschrift fur die Gesamte Staatswissenschaft 121: 301–24, 667–89. Selten, Reinhard. 1975. Reexamination of the perfectness concept for equilibrium points in extensive games. International Journal of Game Theory 4 (1): 25–55. Shackleton, Mark, Andrianos E. Tsekrekos, and Rafal Wojakowski. 2004. Strategic entry and market leadership in a two-player real option game. Journal of Banking and Finance 28 (1): 179–201. Shaked, Avner, and John Sutton. 1984. Involuntary unemployment as a perfect equilibrium in a bargaining model. Econometrica 52 (6): 1351–64. Shapiro, Carl. 1989. The theory of business strategy. Rand Journal of Economics 20 (1): 125–37. Shreve, Steven E. 2004. Stochastic Calculus for Finance II: Continuous-Time Models. Springer Finance series. New York: Springer. Simon, Leo K. 1987. Basic timing games. Working paper 8745. University of California at Berkeley. Smets, Frank. 1991. Exporting versus FDI: The effect of uncertainty, irreversibilities and strategic interactions. Working paper. Yale University. Smit, Han T. J., and L. A. Ankum. 1993. A real options and game-theoretic approach to corporate investment strategy under competition. Financial Management 22 (3): 241–50. Smit, Han T. J., and Lenos Trigeorgis. 1997. Flexibility and competitive R&D strategies. Working paper. Columbia University. Smit, Han T. J., and Lenos Trigeorgis. 2001. Flexibility and commitment in strategic investment. In Eduardo S. Schwarz and Lenos Trigeorgis, eds., Real Options and Investment under Uncertainty: Classical Readings and Recent Contributions. Cambridge: MIT Press, pp. 451–98. Smit, Han T. J., and Lenos Trigeorgis. 2004. Strategic Investment: Real Options and Games. Princeton: Princeton University Press. Smit, Han T.J., and Lenos Trigeorgis. 2009. Valuing infrastructure investment: An option games approach. California Management Review 51 (2): 79–100. Sødal, Sigbjørn. 2006. Entry and exit decisions based on a discount factor approach. Journal of Economic Dynamics and Control 30 (11): 1963–86.

References

471

Sødal, Sigbjørn. 2011. Real options in continuous time: An overview. In Lenos Trigeorgis, ed., Operating Flexibility in an International Context: New Developments and Applications in Real Options. Oxford: Oxford University Press, forthcoming. Sparla, Thomas. 2004. Closure options in a duopoly with strong strategic externalities. Zeitschrift für Betriebswirtschaft 67 (3): 125–55. Spence, A. Michael. 1977. Entry, capacity, investment and oligopolistic pricing. Bell Journal of Economics 8 (2): 534–44. Spence, A. Michael. 1979. Investment strategy and growth in a new market. Bell Journal of Economics 10 (1): 1–19. von Stackelberg, Heinrich. 1934. Marktform und Gleichgewicht. Vienna: Julius Springer. Ståhl, Ingolf. 1972. Bargaining theory. Economic Research Institute, Stockholm. Stenbacka, Rune, and Mihkel M. Tombak. 1994. Strategic timing of adoption of new technologies under uncertainty. International Journal of Industrial Organization 12 (3): 387–411. Stokey, Nancy L. 2008. The Economics of Inaction: Stochastic Control Models with Fixed Costs. Princeton: Princeton University Press. Subbotin, Mikhail F. 1923. On the law of frequency of error. Matematicheskii Sbornik 31 (2): 296–301. Teece, David J., Gary Pisano, and Amy Shuen. 1997. Dynamic capabilities and strategic management. Strategic Management Journal 18 (7): 509–33. Theodossiou, Panayiotis, and Lenos Trigeorgis. 2009. Option pricing when log-returns are skewed and leptokurtic. Working paper. University of Cyprus. Thijssen, Jacco J. J., Kuno J. M. Huisman, and Peter M. Kort. 2002. Symmetric equilibrium strategies in game-theoretic real option models. Discussion paper 2002–81. CentER, Tilburg University, Netherlands. Thijssen, Jacco J. J., Kuno J. M. Huisman, and Peter M. Kort. 2006. The effects of information on strategic investment and welfare. Economic Theory 28 (2): 399–424. Tirole, Jean. 1988. The Theory of Industrial Organization. Cambridge: MIT Press. Trigeorgis, Lenos. 1991. Anticipated competitive entry and early preemptive investment in deferrable projects. Journal of Economics and Business 43 (2): 143–56. Trigeorgis, Lenos. 1996. Real Options: Managerial Flexibility and Strategy in Resource Allocation. Cambridge: MIT Press. Trigeorgis, Lenos, and Francesco Baldi. 2010. Patent leveraging strategies: Fight or cooperate? Working paper. University of Cyprus. Trigeorgis, Lenos, and Scott P. Mason. 1987. Valuing managerial flexibility. Midland Corporate Finance Journal 5 (1): 14–21. Viner, Jacob. 1932. Cost curves and supply curves. Journal of Economics 3 (1): 23–46. Wang, X. Henry. 1998. Fee versus royalty licensing in a Cournot duopoly model. Economics Letters 60 (1): 55–62. Weeds, Helen. 2000. Sleeping patents and compulsory licensing: An options analysis. Working paper. University of Warwick, UK.

472

References

Weeds, Helen. 2002. Strategic delay in a real options model of R&D competition. Review of Economic Studies 69 (3): 729–47. Wernerfelt, Birger. 1984. A resource-based view of the firm. Strategic Management Journal 5 (2): 171–80. Weyant, John P., and Tao Yao. 2005. Strategic R&D investment under uncertainty in information technology: Tacit collusion and information time lag. Working paper. Stanford University. Williams, Joseph T. 1993. Equilibrium and options on real assets. Review of Financial Studies 6 (4): 825–50. Wu, Jianjun. 2006. Credible capacity preemption in a duopoly market under uncertainty. Working paper. University of Arizona. Zhu, Kevin, and John P. Weyant. 2003a. Strategic decisions of new technology adoption under asymmetric information: A game-theoretic model. Decision Sciences 34 (4): 643–75. Zhu, Kevin, and John P. Weyant. 2003b. Strategic exercise of real options: Investment decisions in technological systems. Journal of Systems Science and Systems Engineering 12 (3): 257–78.

Index

Abandon or contract option, 153, 163, 165, 167 Accommodated entry, xi, 65, 119, 120, 122–23, 132 Action region, 312–313, 453, 455 Advertising strategies, 264–70 soft investment in, 129, 266–67, 269 Aguerrevere, Felipe, 418 Airbus, 6, 7, 8, 12, 63, 99, 353, 422 Akerlof, George, 31, 40, 41n Altruism, reciprocal, 143, 144 American call option, xiv-xv, 294n, 307, 308, 326 perpetual (early exercise of), 446 American investment option, 285 Antitrust regulation and German chocolate, 140–41 and joint research ventures, 409 and sleeping patents, 410 Arithmetic Brownian motion, 297–98, 429–31, 457 expected discount factor for, 450 expected first-hitting time for, 448 forward NPV for, 443 Arithmetic Ornstein–Uhlenbeck process, 438–39 Asymmetric Cournot duopoly/oligopoly, 106, 235–38, 381, 382 Asymmetric or incomplete information, 30–31, 424 games of, xiii, 83n investment with, 411–15 market structure under, 102–105, 107 and R&D spillover, 414 Atoms function, 362n, 398 Auction, 367, 414–15 Aumann, Robert, 24, 40, 40–41 interview with, 145, 146–47 Backward induction, 109, 112, 131, 135 Baldursson, Fridrik, 325

Bargaining, 135–38 Bargaining power, of suppliers and customers, 65–66 Barriers to entry, xi, 51, 63, 72 strategic (endogenous), 64, 65 structural (exogenous), 64–65 Basic option valuation, 169, 174 Bayesian Nash equilibrium, 116 BCG matrix, 71, 71n Bellman equation, 327, 433 Bellman’s principle of optimality, 441 Benefit parity/proximity, 71 Bertola, Giuseppe, 305n Bertrand, Joseph, 84 Bertrand paradox, 84, 87, 90, 266, 267, 270 Bertrand price competition, 33n, 84, 86–92, 97, 126–28 with cost symmetry, 106 differentiated, 87–88, 90–92 equilibrium profits in, 270 and investment options, 238–40, 241, 242 tough vs. soft commitment in, 127 “Best-practices” approach, 10 Binomial tree/lattice, 177, 183, 220n on evolution of demand, 210, 211 on evolution of market prices, 210, 212 on evolution of market uncertainty, 201 for patent fighting strategies, 206 Black, Fischer, 40, 175. See also Black– Scholes model Black–Scholes (BS) model, 174, 176, 185–187, 189, 404–405, 435–38, 458–59 Blockaded entry, xi, 65, 119, 410 Boeing in duopoly, 63, 99, 422 interview with Scott Matthews, 11–12 787 Dreamliner, 6–9, 353 Boston Consulting Group (BCG), 198–99 BCG matrix, 71, 71n interview with Rainer Brosch and Peter Damisch, 198–99

474

Boundary conditions smooth-pasting, 293n, 328–29, 329n, 456, 458 value-matching, 293n, 328, 374, 455–56 Bounded rationality, 86 Boyer, Marcel, 389n, 420, 421, 423, 424 Brand-name reputation and early-mover advantage, xii as soft commitment, 269 Brazil, real options applications, 172–73 Brennan, Michael, 358 Brosch, Rainer, 13, 198–99 interview with, 198–99 Brownian motion, 427, 458–59 arithmetic (ABM), 297–98, 429–31, 443, 448, 450, 457 fundamental quadratic for, 416, 445–46 geometric (GBM), 298, 431–38, 457 (see also Geometric Brownian motion) standard, 287, 312, 327, 344, 427–28 Calculus, stochastic, 42, 227, 294n, 425, 426, 428, 433, 458–59 Call option, xi American, xiv-xv, 294n, 307, 308, 326 perpetual, 446 European, 193, 227, 189, 190 Black–Scholes pricing of, 185, 187, 435–38 capacity units under construction as, 418 and oil reserve development, 280 Capabilities, 3, 10, 11, 47, 50, 73, 331. See also Dynamic capabilities Capacity expansion (investment), 422 in duopoly, 332–33, 352, 354–55 in existing market, 389–95 lumpy (repeated), 419–21 lumpy vs. incremental, 299, 299n, 353 (see also Incremental capacity investment; Lumpy capacity investment) as monopolist’s option, 278, 298–306 for oligopoly and incremental investment, 317–22, 323 and lumpy investment, 312–317 options approach to and oil reserves, 279–80 and utilities, 280–81 in perfect competition, 322, 324–25 and social optimality, 324–25 Capacity utilization, optimal, 417–19 Capital, cost of, xii Cartel, 59n and Cournot duopoly, 95 Case applications, 209–16, 247–70 Cash flow, expected, 174. See also Discounted cash-flow (DCF) method

Index

Certainty equivalent, 19n, 174, 178, 436. See also Risk-neutral valuation Cheap talk, 112, 134 Chicken game. See War of attrition Closed-form solution, xi Closed-loop equilibrium, xi-xii, 361n, 366, 389, 394, 398, 399, 400 Closed-loop strategies, xi, xvi, 83n, 121n, 325n, 398, 399 Coca-Cola, 53, 266 Co-opetition, 147–50 Collaboration. See also Cooperation in investment, 392, 394 on patent, 395 Collaborative (tacit-collusion) equilibrium, 393 symmetric, 401–402 Collusion and oligopolists’ expansion, 312–17 tacit, 145, 147, 107–108, 151–52 by Cournot duopolists, 95–96 in existing market, 371–72, 389n in real-estate development, 406 perfect equilibria of, 372, 400 sustainability of, 422 symmetric, 401 Commitment (strategic), xvi, 12, 30, 107–108, 110–12, 117–18, 151–52 credibility of, 117, 118 and reaction functions, 97 and sequential Stackelberg game, 131, 133–34 strategic effect of, 271n taxonomy of strategies for, 118–119, 123–25, 126 and Bertrand competition, 126–28 (see also Bertrand price competition) and Cournot quantity competition, 128, 130–31 (see also Cournot quantity competition) entry strategies, 119–23 tough vs. soft, 121, 131, 194 (see also Soft commitment; Tough commitment) in differentiated Bertrand competition, 127 Commitment or flexibility. See Flexibility or commitment trade-off Commitment value, xii Competition and co-opetition, 147–50 exogenous, 404–405 imperfect, 24n, 421 perfect, 23, 24n, 63, 322, 324–25 (see also Perfect competition) in R&D, 409–10 and real options analysis, 14

Index

reciprocating vs. contrarian, 271, 272 (see also Soft commitment; Tough commitment) Competitive advantage, 17, 74, 471 creating and sustaining of, 66, 72–73 and generic competitive strategies, 70–72 through innovation, 54 and isolating mechanisms, xiii, 73 through value creation, 66–70 external vs. internal perspectives on, 48–50 and patent strategy, 395 static vs. dynamic, 109, 112, 151, 195 Competitive analysis, 50, 56 “five-forces” analysis, 59–66 macroeconomic, 57–58 microeconomic (industry level), 58–59 Competitive erosion, 35, 343, 350, 351, 375, 389, 404, 405, 421, 423 Competitive landscape, change in, 52 Competitive strategy. See Strategy Competitive value erosion, 35, 421 Complements. See Strategic complements Complementors, 148 Complete information, 115, 403 Compound option(s), 163, 167–68 in R&D setting, 163, 408, 410–11, 412 Concentration (industry) in “five-forces” analysis, 61 Herfindhal–Hirschman index of, 62–64 Consolidation, in corporate environment, 4 Constant-elasticity demand function, 78–79 Consumer surplus, 66–67 Contestable market, 119 Contingent-claims analysis, 307n, 308–309, 326 Continuation or inaction region, 312–13, 316, 453, 455 Continuation strategy, 397–98 Continuous-time option analysis, 275, 277, 278, 184–89, 380n Continuous-time stochastic processes, 426–427, 458 Brownian motion, 427–438 (see also Brownian motion) general Itô or diffusion process, 440–441 mean-reversion process, 438–440 Contract or abandon option, 163, 165, 167 Contrarian actions. See Strategic substitutes Cooperation, 135, 143 Aumann on, 146–147 and co-opetition, 147–50

475

between Cournot duopolists in repeated games, 138–39, 141–42, 145, 147 in existing market, 370–72 and prisoner’s dilemma, 29 and symmetric vs. asymmetric firms in capacity expansion, 420–21 Co-opetition, 147–49 Coordination failure, 235, 369, 370, 378, 384, 392. See also Coordination problem Coordination problem, 226n, 359, 384, 386, 387 in deterministic case, 362 and focal-point argument, 234, 235, 379–380 and mixed-strategy approach, 235 and precommitment, 361 and preemption, 366 and war of attrition, 363 Corporate environment, 3–10 Corporate finance. See Finance, corporate Corporate strategy. See Strategy Cost(s) of capital, xii fixed, xiii opportunity, xii, 277, 284 sunk, xvii, 117, 117n, 271n, 275 variable, xvii, 162 Cost advantage, xii large, 382–84, 385, 388–89 small, 384–89 Cost asymmetry in Cournot duopoly, 96–97, 106, 229–35 in Cournot oligopoly, 99, 100–101, 106, 235–38 Cost leadership, 70–71 and Cournot duopoly models, 92, 96 Costly reversibility, 117 Cost parity/proximity, 72 Cost symmetry in Cournot duopoly, 92–96, 106, 227–29 in Cournot oligopoly, 101, 106 Counterthreats, and game theory, 2 Cournot, Antoine Augustin, 113 Cournot quantity competition, 84, 91–92, 128, 130–31 duopoly, 33n, 92–99, 102–105, 107 cooperation in repeated games and folk theorem, 138–39, 141–42, 145, 147 with cost asymmetry, 106, 381, 382 with cost symmetry, 106 with information asymmetry, 102–105, 107 and investment options, 224–35 in new markets, 375–76 probability of simultaneous investment in, 364

476

Cournot quantity competition (cont.) and R&D investment, 247 in Stackelberg game, 134 oligopoly, 99, 100–102 with cost asymmetry, 106, 235–38 with cost symmetry, 106 and incremental investment, 318 and investment timing, 315–17 and option games, 240 and Stackelberg game, 131 Cox, John, 190, 243, 436, 448n. See also Cox–Ross–Rubinstein (CRR) binomial model Cox–Ross–Rubinstein (CRR) binomial model, 174, 184, 185, 190–91 Credibility of strategic moves, 117, 118 CRR. See Cox–Ross–Rubinstein (CRR) binomial model Customer market power, 65–66 segment, 70 switching costs, and early-mover advantage, xii value, factors in, 68 Damisch, Peter, 13, 198–99 interview with, 198–99 DFC. See Discounted cash-flow (DCF) method Decision theory, static and dynamic, 39 Decision time, 224–25 Decision-tree analysis (DTA), 19 and real options analysis, 19–20 (see also Real options analysis) Defensive patent wall, 207 Deferral or waiting (timing) option, xiv-xv, 163, 163–64 examples of, 179–180, 182–84 of follower in duopoly, 341 of monopolist, 219–24, 278, 347, 351 in deterministic case, 281–84 in stochastic case, 284–98 and NPV rule, 275 for patent, 395 Demand, isoelastic, 321–22, 323 Demand functions (inverse), 78 Demographic trends, 55 Deterministic profit, 301 Deterred entry, xii, 65, 119–20, 120–21 Deterrence, 110 Deutsche Bank, 196 Días, Marco A. G., 13, 172 interview with, 172–73 Differentiated Bertrand model, 63, 87–88, 90–92 equilibrium profits in, 270 and investment options, 238–40, 241, 242 tough vs. soft commitment in, 127

Index

Differentiation strategy, 71, 129, 265–66 horizontal, 71–72 vertical, 72 Diffusion (sequential ordering), 338 Diffusion (stochastic process), 425–26, 440, 449 Discounted cash-flow (DCF) method, 16, 22, 169. See also Net present value (NPV) method Discount factor, 282. See also Expected discount factor and Brownian motion arithmetic, 298 geometric, 287, 313 elasticity of, 292, 293, 298, 300, 302, 350 and monopolist’s option to defer, 282, 283, 286 and Wicksell model, 283 Discount rate, 174, 177, 178n, 277n, 279, 282, 286n, 287n, 294n, 298, 307n, 437, 442, 445 Discrete-time option games. See Option games (discrete-time) Discrete-time option valuation, 174, 176–84, 185 Diversification, 17, 168, 279 Dividend yield or opportunity cost, 16, 282, 289, 297, 308, 313, 335, 346, 375, 404, 435n, 437, 438n, 444, 445 Dixit, Avinash, 12, 13, 32, 33, 49, 109, 111, 117, 118, 188, 189, 242, 278n, 279–81, 285n, 291, 292n, 293, 294, 298n, 301n, 305n, 307, 312n, 318n, 325, 333, 373, 378n, 379n, 396, 397, 439n, 451, 445, 453, 455, 456n interview with, 32–33 on game theory basics, 28–32 Dixit–Pindyck model (investment timing), 333, 373 Dominant firm, 63 Dominant strategy, xii, 28–29, 111, 207–208 and cost asymmetry, 230 for investment options, 239 “Doomsday Machine,” 110, 117, 118, 121n Drift (diffusion), 425–426 Dr. Strangelove . . . (movie), 110, 117, 121n Duopoly, 81–82, 84 adoption-timing decisions in, 421 and Bertrand price competition, 33n, 84, 86–88, 90–92, 97, 106 Boeing and Airbus as, 63, 99, 422 (see also Boeing) and cooperation in existing market, 370–72 and Cournot quantity competition, 33n, 84, 91–99, 102–105, 106, 107, 134 (see also under Cournot quantity competition)

Index

exit policies in, 415 and folk theorem, 145 and expansion of existing market, 389–95 and investment in new market, 372–89 and investment in new technologies, 411 and lumpy capacity expansion, 419–21 and preemption, 360, 363, 365–70 sequential investment in, 331–39 and capacity expansion, 352, 354–55 under uncertainty, 339–48, 351–52, 356–57 and signals, 414, 415 Dynamic capabilities, 50 Dynamic decision theory, 39 Dynamic (sequential) game, 83 Dynamic game theory, 39 Dynamic models, 151 Dynamic programming, 39n, 298n, 308, 326–29 Dynamic strategic interactions, 39 Early-mover advantage. See First-mover advantage Economic profit, xii, 16, 17, 67 Economics. See also Economic sciences experimental, 85 heuristics in, 85 macroeconomic analysis, 57 Economic sciences areas of research in, 32–33 Nobel Prizes awarded in, 40 rationality assumed in, 146 Economies of scale, xii, 70, 76, 280–81 Economies of scope, xii, 70–71, 76 Efficient-market hypothesis (EMH), 427 Elasticity, xii-xiii of discount factor, 292, 293, 298, 300, 302, 350 price elasticity of demand, 78, 321–22, 323 and Lerner index (markup rule), 80–81 Elasticity markup rule, 80–81 Electricity market, 132 European, 154, 155 (see also European electricity industry) Finnish, 422 Italian (Enel), 131, 132 and real options, 162–69 reserve margins, 154–56 scale vs. flexibility in, 280, 281 uncertainties for, 156–57 firm-specific risks, 156 idiosyncratic business risks, 156, 157–58 generation technologies and business risk exposure, 159–62 technology-related business risks, 158 Energy sector, European, 154. See also European electricity industry

477

Entry accommodated, xi, 65, 119, 120, 122–23, 132 barriers to, xi, 51, 63, 64–65, 72 (see also Barriers to entry) blockaded, xi, 65, 119, 410 deterred, xii, 65, 119–20, 120–21 threat of, 64–65 Entry strategies, 119–23 Equilibrium concepts Bayesian Nash, 116 closed-loop, xi-xii path dependence, xv, 73, 285n Markov perfect, xv, 325n, 427, 452 Nash, xiv, 25–26, 28, 89, 103, 109, 113–14, 116, 239, 398 open-loop, xiv, xvi, 360–61, 393–94, 396 rationalizability, 89n solution concept, 27n, 38n, 39n, 85, 89, 92n, 112, 113, 116, 119, 135 subgame perfect Nash, xvii, 109, 114–15, 116 “trembling hand,” 115 Erosion. See Competitive erosion EU Emission Trading Scheme (EU ETS), 159 European call options, 189, 190, 193, 227 Black–Scholes pricing of, 187, 435–38 capacity units under construction as, 418 pricing of, 185 European electricity industry, 155 additional generation capacity required in, 156 business risk exposure of generation technologies in, 160–62 Emission Trading Scheme (ETS) for, 159 idiosyncratic business risks for, 158 Italian electricity authority (Enel), 4–5, 131, 132, 166 in examples, 181, 182–83, 184, 188, 202–206, 222, 343–44 liberalization in, 132 and real options, 163–64, 166–67, 168–69, 170 European energy sector, 154 European liberalization, 129, 132 European option, xiii, 220n. See also European call options Evolutionary games, 114 Exercise price or investment cost, 164, 165, 167, 188, 190, 198, 267, 280, 289n, 299, 308, 327, 346, 348, 430, 435, 442 Exercise timing, and first-hitting time, 447 Existing market model cooperation in, 370–72 and oligopolists’ lumpy expansion, 312–17

478

Existing market model (cont.) and option to expand, 389 for asymmetric case, 393–95 for symmetric case, 389–92 Exit strategies, 415–17 Exogenous competition, 404–405 Expanded or extended net present value (E-NPV), xiii, xv, 208, 288, 343, 351–52, 352, 384, 385 Expand or extend option, 163, 164–65 business applications of, 279–81 and capacity utilization, 418 in duopoly, 352, 354–55, 389–95 example of, 180–81 in existing market, 389 for asymmetric case, 393–95 for symmetric case, 389–92 and investment timing, 311 Expansion of capacity. See Capacity expansion Expected discount factor, 448–51. See also Discount factor for arithmetic Brownian motion, 297, 450 elasticity of, 351 for geometric Brownian motion, 287, 293, 451 and other exponentials, 451 for mean-reverting process, 451 and Wicksell model, 283 Expected profit function, 102 Expected-value term, 425–26 Experience effects. See Learning curve effects Experimental economics, 85 Exponential demand function, 78, 79 Exponentials of Brownian motion, 431–38, 444, 445, 451, 452, 457 Extended (expanded) net present value, xiii, xv, 208, 288, 343, 351–52, 352 “Extended-rivalry” interactions, 59 in “five-forces” analysis, 60 Extensive form, of bargaining game under complete information, 136 Externalities negative, 35, 64, 320, 333n, 334, 355, 408, 409 positive, 35, 408, 409, 421, 423 External view of firm, 48, 50 Fairness, as source of cooperation, 29 Fat cat strategy, 124–25, 267, 269 Fight mode, in patenting, 395 Finance, corporate and industrial organization, 37 and strategy, 15–20 Financial economics and strategic management, 49

Index

Financial options, 5n, 153, 174 Firm (company) external view of, 48, 50 internal view of, 50 knowledge-based view of, 47 as portfolio of businesses, 198–99 resource-based view of, 73 Firm profitability, drivers of, 56 Firm roles, 359, 361 (see also Coordination problem; Industry structure; Leadership) endogenous, 374–75 probabilities of, 364 Firm-specific risks, 156 First-hitting time, 447–48 and investment trigger, 285 First-(early-)mover advantage, xii, 34, 335n, 338–39, 342, 343, 345, 357. See also Leadership in auction, 367 and asymmetric case, 393 and capacity expansion, 354 and Nash equilibria, 360 and precommitment, 361 and preemption, xv, 362–63, 391–92 in Reinganum model, 360 Fisher separation theorem, 15 “Five-forces” analysis, 51, 59–66, 147 Fixed costs, xiii Flex-fuel technology, 172, 173 Flexibility in managerial decision-making, 19, 284 for multinational firms (exchange rate risk), 422 and optionality, 198 and real options analysis, 13–14, 21–22 and uncertainty, 271, 271n in utility planning, 280–81 Flexibility or commitment trade-off, xxv-xxvi, 5, 12, 47, 118, 195, 195n, 197, 272–73 Dixit on, 32 and integrative approach, 15 in option games, 217 quantification of, 243 Focal-point argument, xiii, 231n, 234, 235, 339, 345, 361, 380 Focal-point equilibrium, 360, 383 Focus strategy, 70, 72 Folk theorem, 145, 145n Follower or second mover, 34, 99, 106, 131, 133–137, 272, 334, 342, 344, 345–47, 350, 351–352, 357, 359, 360, 361–93, 406, 407, 408, 409, 411, 413, 415–17, 421, 422 Forward net present value, 441–47, 453 for arithmetic Brownian motion, 443 for geometric Brownian motion and other exponentials, 444

Index

for powers of geometric Brownian motion, 444, 447 France EDF electric utility in, 202–206 GDF gas utility in, 166–67 Friedman, James, 139, 145, 151 Fudenberg–Tirole model of investment timing, 332, 359, 360, 361, 362, 372, 396, 397 Fudenberg, Drew, 39n, 81n, 107, 108, 118–21, 124, 129, 132, 139n, 145, 152, 194, 243, 246n, 249, 286n, 324n, 332, 333, 359, 360, 361, 362, 365, 366, 371, 372, 373, 390, 396, 397, 406n, 415, 420 Fundamental quadratic, 416, 445–46, 450 Games of complete information, 115, 403 of imperfect information, 83n, 115, 414 of incomplete information, xiii, 83n, 102–105, 107, 115, 414 multistage, 139n of perfect information, xiii, 83n repeated (supergames), 135n, 139, 147 cooperation between Cournot duopolists in, 138–39, 141–42, 145, 147 infinitely, 139n and prisoner’s dilemma, 144 and tacit collusion, 145, 147 and “rules of the game,” 27, 82–83 sequential or dynamic, 83 simultaneous, 83, 139n, 224, 229 of timing, 332, 396, 397 for duopoly, 332–39 in Fudenberg–Tirole model, 362, 396 two-stage, 271–72 in goodwill and advertising, 264–70 innovative (R&D), 243, 246–50 in patent licensing, 262 Game theory, xiii, 1–3, 6, 12, 14, 20–34, 49, 50, 81, 84, 107–108, 143, 151–52, 153, 195, 209, 254 advantages and drawbacks in (comparison), 38 applications of, 2–3, 36–37 in business decision-making, 24–26 basics of (Dixit), 28–32 in Boeing’s strategic thinking, 12 in continuous time, 332 (see also Investment timing) development of, 113–15 fields of application of, 27 integrated with real options analysis (option games), 6, 32, 173, 195, 199, 423 and irreversible commitments, xxiii metaphorical vs. literal interpretation of, 27, 33 mixing moves in, 29–30

479

options analysis with, 39–40 origins of, 24 Selten on, 85–86 static and dynamic, 39 Gardening metaphor, for managing options portfolio, 170 General Itô or diffusion processes, 440 Generation technologies, and business risk exposure, 159–62 Generic competitive strategies, 70–72 Geometric Brownian motion (GBM), 298, 431–38, 457 and closed-form solutions, 329 and continuous-time option analysis, 184, 186 in example of Black–Scholes formula, 187 expected discount factor for, 313, 451 expected first-hitting time for, 448 forward NPV for, 444, 447 powers of, 435 present value with stochastic expiration for, 452–53, 454 Geometric Ornstein–Uhlenbeck (GOU) process, 439–40, 457 Germany antitrust actions in, 140–41 Deutsche Bank in, 196, 196n telecom market of, 129 Globalization, 53 Goodwill strategies, 264–70 Gordon perpetuity formula, 444 Grenadier, Steven, 49, 317, 320, 322, 325, 326, 373n, 403n, 405, 406, 407, 414, 418, 423, 424 Growth options, 47, 163, 167, 168, 284, 311, 318n, 418, 419, 421 Growth trend, 425–26 Hamilton–Jacobi–Bellman (HJB) equation, 433, 443 Harrison, Michael, 39n, 436, 451n, 452n, 458 Harsanyi, John C., 40, 40–41, 115 Hedging, 156, 175 Herfindhal–Hirschman index (HHI), 62–64 Heuristics,in economic analysis, 85 Horizontal differentiation, 71–72 Huisman, Kuno J. M., 49, 317n, 332n, 362n, 367n, 373n, 392n, 396, 397, 406n, 408, 414, 423, 424, 455n Hysteresis, 307n, 422 Idiosyncratic business risks, for electricity industry, 156, 157–58 Imperfect competition, games of, 24n, 421

480

Imperfect information, 83n, 115, 414. See also Information Inaction region, 312–13, 316, 453, 455 Incomplete or asymmetric information, 30–31, 424 games of, xiii, 83n, 115 investment with, 411–15 market structure under, 102–105, 107 and R&D spillover, 414 Incremental capacity investment, 303–306 Induction, backward, 109, 112, 131, 135 Industrial organization, 34, 37, 75, 107–108, 151–52 dynamic, 39 and finance, 37 and game theory, 40 static, 39 Industry analysis “five-forces” analysis, 51–52, 59–66, 147 structure–conduct–performance (SCP), 58–59 Industry structure(s) assumptions, 78 evolution of, 378 probability of occurrence of, 364, 369, 386 Infinitesimal generator, 133 Information complete, 115, 403 imperfect, 83n, 115, 414 incomplete or asymmetric, 30–31, 424 (see also Uncertainty) games of, xiii, 83n, 115 investment with, 411–15 market structure under, 102–105, 107 and R&D spillover, 414 perfect, xiii, 83n in Stackelberg model, 134 Information cascade, 414 Information economics, 31 Information set, xiii, 83 Innovation. See also R&D competitive advantage through, 54 and patents, 59 Innovative investment strategies, and time-to-build delays, 410 Integrative approach to strategy, 35–41 Intellectual property (IP) rights. See also Patent licensing of, 254 and option games, 408 Interest rate, 176, 180, 183, 186, 187, 201, 208, 211, 271n, 295, 305, 313, 333, 336, 348, 356, 374, 410, 438. See also Risk-free rate Internal rivalry, in “five forces” analysis, 61

Index

Internal view of the firm, 50 International Energy Agency, suggestions of risk from, 189 Investment. See also Commitment; Strategic investment collaborative, 392 and commitment, 360 decision-theoretic models of, 408 game-theoretic models of, 408 with information asymmetry, 411–15 joint, 371–72, 384, 394 Pareto-superior equilibrium of, 391 Jorgensonian rule of, 284, 336 modified, 295–97, 324, 356, 374, 381 modified (risk-neutral), 345 in new technologies, 411 in oil reserves, 279–80 and preemption, 366 Tobin’s q theory of, 284 sequential, 331, 357 (see also Sequential investment) simultaneous, 311, 331, 364 (see also Simultaneous investment) tough vs. soft, 266 (see also Soft commitment; Tough commitment) two-stage (R&D), 410 under uncertainty, 5 in utilities, 280–81 Investment, R&D, 243–53 Investment option(s), xiv-xv American, 285 call option, xiv-xv, 294n, 307, 308, 326, 446 for Cournot duopoly, 224–27 under cost asymmetry, 229–35 under cost symmetry, 227–29 for Cournot oligopoly (asymmetric), 235–38 for differentiated Bertrand price competition, 238–40, 242 in duopoly under uncertainty, 339 example, 186, 332 monopolist’s deferral option, 219–24 multiple, 419, 421–22 in new market, 357, 372–73 in asymmetric case, 379–89 in symmetric case, 373–79 shared, xvi, 35, 35n, 344, 356 valuation of, 284–85 Investment option value, for oligopoly, 316 Investment staging, 164 Investment strategy, 285–88 optimal, 285n, 288–98, 334, 341, 372 and time-to-build delays, 410 Investment threshold. See Investment triggers

Index

Investment timing, 195, 195n, 275, 277, 307–308, 325–26, 332, 333, 357, 360, 361, 382, 456. for capacity expansion, 278, 298–306 for duopolist, 332–39 for monopolist, 219–24, 278, 308–309, 347 in deterministic case, 281–84 in stochastic case, 284–98 for oligopoly, 311 for incremental capacity investment, 317 with lump-sum capacity expansion, 312–17 for perfect competition, 322, 324–25 under uncertainty, 196–97, 307 Investment timing game, 362–63 Investment triggers, 227, 240, 455 and capacity expansion(optimal), 327, 352, 354 for Cournot duopolist, 226, 237–38 under cost asymmetry, 229, 231, 232, 233–35, 237–38, 262, 420 under cost symmetry, 227–28 under differentiated Bertrand price competition, 241 and first-hitting time, 285 for follower in new-market investment (optimal), 341, 373–74, 380, 381 in goodwill/advertising case, 267–70 joint, 371 for leader in asymmetric new-market investment (optimal), 384 for monopolist, 221–23, 237, 288–91, 305–306, 321 incremental capacity investment, 306 for oligopoly, 311, 313, 316, 349, 350 incremental expansion, 320–21, 322 lump-sum expansion, 315, 316 perfect competition, and social optimality, 324 and R&D investment, 246–47, 248–49, 250 in duopoly with high spillover, 251–53 in duopoly with low spillover, 251, 252 Irrationality, 89n Aumann on, 146 Selton on, 85–87 Irreversible investment, 32, 280, 285n, 297, 325, 356, 373, 381, 389, 410, 415n, 420, 442 Isoelastic demand, 321–22, 323 Isolating mechanisms, xiii, 73 Italy Enel electricity authority in, 4–5, 131, 132, 166 in examples, 181, 182–83, 184, 188, 202–206, 222, 343–44

481

Itô process, 175, 288, 292, 293n, 326, 427, 440, 458–59 general Itô process, 440 Markov property for, 452 and standard Brownian motion, 427 time-homogeneous, 440, 449 Itô’s lemma, 432, 433 Joaquin, Domingo C., 331n, 339, 344, 346, 356, 358 Joint investment, 371–72, 384, 394 Pareto-superior equilibrium of, 391 Joint venture, 315 Jorgensonian rule of investment, 284, 336 modified, 295–97, 324, 345, 356, 374, 381 Kamien, Morton I., 254n, 255n, 261n Karatzas, Ioannis, 325, 458, 459 Kester, W. Carl, 35 Knowledge-based view of firm, 47 Kort, Peter M., 317n, 373n, 380n, 392n, 393, 396, 397, 406n, 408, 414, 424 Kulatilaka, Nalin, 21 Lamarre, Eric, xxvii interview with, 199–200 Lambrecht, Bart M., 379n, 410, 412 Large cost advantage, in preemption games, 382–84, 385, 388–89 Late-mover advantage, 363, 366, 386, 414, 421. See also Second-mover advantage Leader or leadership, 361 in duopoly, 334–39 capacity expansion, 354–55 new market, 374–75, 376, 377, 378 under uncertainty, 339–43, 345–48 in Stackelberg duopoly, 99, 106, 131, 133–34 Leahy, John V., 324–26 Lean and hungry look strategy, 124 Learning-curve effect, xiii, 71 and early-mover advantage, xii Lerner index (markup rule), 80–81 Licensing, Patent. See Patent licensing Linear demand function, 78, 79 Luehrman, Timothy, 170–71 Lumpy capacity investment, 299–303 repeated capacity expansion, 419–21, 423 Macroeconomic analysis, 57–58 Managerial flexibility, 5, 12–16, 19, 38, 51, 55, 169, 198, 247, 271n, 288, 423 Market, and threat of substitute products, 64 Market equilibrium, in monopoly, 81 Market share, bargaining over, 135–37

482

Market structure at beginning of investment game, 379 probabilities of, 370 Market structure games, xiii, 58–59, 217–18 dynamic approach to, 109 bargaining and cooperation, 135–145, 147–50 commitment in, 110–112, 117–28, 130–34 (see also Commitment) static approach to, 107 duopoly, 81–82, 84, 86–99, 102–105, 107 monopoly, 76–81, 106 oligopoly, 99–102 Market uncertainty, xxv, 5, 372, 373 (see also Uncertainty) Markov process, xiv, 325n, 427 arithmetic Brownian motion, 429 geometric Brownian motion, 431–38 and Itô process, 440, 452 mean-reversion process, 438–40, 451 standard Brownian motion, 427, 428 Markup rule (Lerner index), 80–81 Matthews, Scott (Boeing), xxvii interview with, 11–12 Maturity, xiv, 23, 164, 165, 183, 185, 187, 188, 190, 191, 193, 201, 202–203, 206, 208, 251n, 256, 262–63, 265–68, 271n, 289n, 418, 430, 435–36, 441 McDonald Robert L., 165n, 188n, 29, 275, 278n, 288, 294, 296n, 301n, 307, 333, 347, 446. See also McDonald–Siegel timing option value McDonald–Siegel timing option value, 275, 288, 294n, 301n, 333, 347 McKinsey & Company interview with Eric Lemarre, 199–200 Mean-reversion process, 438–40 expected discount factor for, 451 Merton, Robert C., xxvii, 40, 174, 174n, 189 interview with, 175–76 Microeconomic theory, 58–59 and strategic management, 49 Microsoft, 24, 25 Minimax solution, 113 Mining/chemicals industry, option-games application in, 209–17 Mixed strategy(ies), 113, 367–70 and coordination problem, 235, 359, 360, 362 Monopolistic differentiated competition, 63 Monopoly, 23, 24n, 58, 62–63, 76–81, 106, 231, 235, 321 vs. Cournot duopoly, 226 and differentiated Bertrand price competition, 240 and innovation, 59

Index

and investment timing, 275, 277, 278 option to expand capacity, 298–306 option to invest, 219–24, 278, 281–98, 308–309, 347 natural, xxv, 76, 129 Morgenstern, Oskar, 2, 24, 27, 40, 113 Multinational corporations, 57, 301, 302, 320–22 Multiplicative stochastic demand shock, 381, 416 Multistage games, 139n Myers, Stewart C., 43, 167n Myopic firm and strategy, xiv, 324 Nalebuff, Barry J., 33n, 49, 109, 117, 118, 148, 149 Nash, John, 26, 27, 40, 113–14. See also Nash equilibrium Nash equilibrium, xiv, 25–26, 28, 103, 109, 113–14, 116, 239, 398 and Bertrand price competition, 88, 89, 90 and Cournot quantity competition, 84, 93, 95, 100 and duopoly investment, 337, 337–38 and focal point argument, 231n mixed-strategy profile, 368 in multistage setting, 112 refinements of, 114, 115, 116 subgame perfect, xvii, 398 Natural monopolies, xxv, 76, 129 Nature (as player), xiv, 82 Net present value (NPV) method, xiv, 16, 174, 177. See also Discounted cash-flow (DCF) method advantages of, 38 drawbacks of, 18–19, 21, 22, 22–23, 38 expanded (extended) E-NPV, xiii, xv, 208, 288, 343, 351–352, 352, 384, 385 and finance-strategy gap, 17–18 forward, 441–47, 453 for arithmetic Brownian motion, 443 for geometric Brownian motion and other exponentials, 444 for powers of geometric Brownian motion, 444, 447 NPV rule, 16, 221, 227, 281, 284, 292 revision of, 275 under timing flexibility, 279 proper use of, 18 vs. real options analysis, 196 Network effects, 55–56 Neumann, John von, 2, 24, 27, 40, 113 New market, 314–17, 340, 357 option to invest in, 372–73 for asymmetric case, 379–89 for symmetric case, 373–379

Index

Niche, in focus strategy, 70, 72 NPV. See Net present value method NPV rule, 16, 221, 277, 281, 284, 292. See also Discounted cash-flow (DCF) method revision under timing flexibility, 275 Observability, of strategic move, 117 Oil reserves, investment in (options approach), 279–80 Oligopoly, 63, 81, 99–102 Cournot, 99, 100–102 under cost asymmetry, 106, 235–38 under cost symmetry, 106 and incremental investment, 318 and investment timing, 315–17 dynamic models of, 39n expansion threshold for, 418 and investment timing, 311 for incremental capacity investment, 317–22, 323 with lump-sum capacity expansion, 312–17 Open-loop equilibrium, xiv, 360–61, 393–94 Open-loop (precommitment) strategies, xiv, xvi, 83n, 121n, 325n Opportunity cost, xii and investment timing, 284 (see also Investment timing) and NPV rule, 277 Optimal investment timing, 195, 195n Optimal stopping, 453, 455–56, 458 under uncertainty, 196–97 Option(s) xiv, 153. See also Call option; Real options American, xiv-xv, 294n, 307, 308, 326 perpetual (early exercise of), 446 compound, 410 European, xiii, xiv, 193, 220n (see also European call options) proprietary, xv, 35, 35n R&D, 245–46 real, xv, 153–54, 162–63 (see also Real options) shared, xvi, 35, 35n, 344, 356 Option analysis or approach to capital investment, 279 in oil reserves, 279–80 in utilities, 280–81 continuous-time, 184–89 with game theory, 39–40 Option games, xxvi, 1, 6, 41, 195, 217, 242, 403, 423 in Boeing’s strategic thinking, 12 Brosch and Damisch (BCG) on, 199 Días (Petrobras) on, 173

483

issues arising in, 395 Lamarre (McKinsey) on, 200 Option games (continuous-time), 275, 277, 278, 184–89, 380n Option games (discrete-time), 1, 6, 35, 41, 184, 185, 193, 206–208, 210, 217–18, 253, 272, 273, 275, 333, 423 applied to mining/chemicals industry, 209–17 illustration of, 197, 201–206 Option markup formula, 290–91, 293 Option-pricing formula, 179 for multistep CRR, 190–91 Option-pricing theory, 153, 175, 178 Option valuation, xv, 169, 174, 189 discrete-time, 176–84, 185 Option value erosion, 423 Ornstein–Uhlenbeck process arithmetic, 438–39 geometric (GOU), 439–40, 457 Outsourcing of Boeing 787 Dreamliner, 7, 8–9 and value chain redesign, 148 Overinvestment, 250 Pareto optimality, and sequential investment, 339n, 345 Pareto-superior joint-investment equilibrium, 390–91 Passive management, 18 Patent(s), 253–54 and innovation, 59 in investment-timing example, 278 sleeping, 207, 410 Patent bracketing, 150 Patent leveraging strategies, 395 Patent licensing, 253–56, 263–64, 272–73, 410 and drastic innovation, 254, 256–57, 260, 262, 263–64 fixed-fee, 254, 257–58, 259, 261 and nondrastic innovation, 260, 262 royalty rate, 254, 258–59, 261 under small vs. larger cost savings, 261 and spillover effects, 249, 253 under uncertainty, 261–64 Patent race, 408–11 Patent strategy, 149–50, 206–208 Patent wall, 150 Path dependence, xv, 73, 285n Patience in bargaining, 135, 136, 138 in cooperative behavior, 142 Payoff, xv, 27, 38, 83, 244 Paxson, Dean A., 411, 421 Payoff matrix, 209

484

Perfect Bayesian equilibrium, 116 Perfect competition, 23, 24n, 63, 325–26 and investment timing, 322, 324–25 quantity produced, 99–105, 107 Perfect information, xiii, 83n Perfect Nash equilibrium. See Subgame perfect Nash equilibrium Pindyck, Robert S., 12, 49, 111, 189, 190, 242, 278n, 279–81, 285n, 289n, 291, 292n, 293, 294, 298n, 301n, 305n, 307, 312n, 318n, 321n, 325, 326, 333, 373, 378n, 379n, 396, 397, 439n, 445, 451, 453n, 455, 456n. See also Dixit–Pindyck model (investment timing) Porter, Michael, 59, 66, 70, 147 Portfolio Matrix (BCG), 199 Portfolio(s) of real options, 14, 168, 169, 170–71, 175, 176, 177–78, 185, 198–99, 285n Potential entrants, 64–65 Prahalad, Coimbatore K., 49, 51 Precommitment strategies, 360–61. See also Open-loop strategies Preemption, xv, 99, 317n, 325n, 360, 363, 365–70, 375–79, 391–92, 396, 397, 409 asymmetric, 381–82 with large cost advantage, 382–84, 385, 388–89 with small cost advantage, 384–89 fear of, 393 and first-mover advantage, 362–63 Preemption timing, 366 Present value. See also Net present value (NPV) method with stochastic expiration (for geometric Brownian motion), 452–53, 454 Price competition. See Bertrand price competition Pricing problem in Cournot duopoly, 93, 94–95, 105 in Cournot oligopoly, 101 factors determining, 65–66 in monopoly, 76, 77, 79, 291–92 Pricing, risk-neutral, 436 Prisoner’s dilemma, 24–25, 28–29, 139, 143–44 repeated, 144, 147 Producer surplus, 67 Production costs, drivers of, 69 Product redesign, 71, 72 Profit deterministic, 102, 301 economic, xii, 16, 17, 50, 63n, 64n, 67, 76n, 86–87, 99, 102, 119, 148, 162 stochastic, 301 Profitability index, 284, 294, 298, 319, 324, 336, 343, 376, 406 gross, 292–93

Index

Profit-flow stream, with stochastic termination, 452–58 Profit values, stochastically evolving, 339 Proprietary option, xv, 35, 35n Puppy dog ploy (strategy), 124, 126, 267 Pure strategy, 89, 113, 307, 338, 339, 345, 347n, 357, 359, 360, 368, 379, 383 Put option, 18, 153, 165, 167, 172, 181, 415n, 417, 425n q investment index (Tobin’s q), 284, 292–93 Quantity competition. See Cournot quantity competition Random rival entry, 404–405 Rationality, 86, 146 and game theory, 26, 84, 85–86, 89, 89n, 113–14, 145 R&D (innovation) investment, 272–273, 407–11, 423 and drastic vs. nondrastic innovation, 262 options created by, 244–46 and patent licensing, 253–64 proprietary vs. shared investment in, 248 and spillover effects, 243, 246–53, 414 Reaction function, 88, 90, 91, 93, 94, 96, 97, 97–98, 99, 102–103, 128, 337 Real estate development, 405–407, 424 waves in, 403 Real options, xv, 153–54, 162–63, 189–90, 307, 332 investmentexample of, 186 and opportunity cost, 277 portfolios of, 168–69, 170 and R&D, 244 strategy as portfolio of, 170–71 types of abandonment for salvage value, 153, 167 compound growth option, 163, 167–68 contract (scale down) or abandon, 163, 165 expand or extend, 163, 164–65, 279–81 (see also Expand or extend option) growth option, 421 investment timing, xiv-xv, 163, 163–64, 224–40, 242, 372–89, 453 (see also Deferral or waiting option; Investment options) shut down (and re-start), 165, 188 staging or time-to-build, 163, 164 switch use, 163, 165–67 Real options analysis (ROA), xv, xxiii, 5–6, 19–20, 23, 40, 153–54, 189, 195, 425 advantages and drawbacks in, 38 and basic option valuation, 169, 174

Index

at Boeing, 11 in Brazil, 172 and continuous-time option analysis, 184–89 discrete-time vs. continuous-time approach in, 197n (see also Option games (continuous time); Option games (discrete time)) and discrete-time option valuation, 176–84, 185 and dynamic programming, 39n and game theory, 32, 173 and mining-industry application, 216 and optimal investment timing, 196 and proprietary vs. shared options, 35 and R&D investment, 243 and stochastic investment timing, 285n Reciprocal altruism, 143, 144 Reciprocating actions. See Strategic complement Reinganum, Jennifer F., 286n, 331, 332, 333, 335n, 338, 339, 357, 358, 360, 361, 366, 396, 397 Reinganum model of investment timing, 331n, 332, 338, 339, 360 Rent dissipation, 388 Rent equalization, 366, 367, 376, 382, 406, 413n, 420 Repeated games, 135n, 139, 147 cooperation between Cournot duopolists in, 138–39, 141–42, 145, 147 infinitely, 139n of prisoner’s dilemma, 144, 147 and tacit collusion, 145 Reputation, 34, 65, 73, 89, 118, 147 Research and development. See R&D Resource-based view of the firm, 73 Return on investment, and investment timing, 283–84 Risk-free interest rate, 180, 183, 185, 186, 201, 208, 211, 313, 348, 374 Risk management beta as risk measure, 418 and real options analysis (Lamarre), 200 Risk neutral expected discount factor, 449 probability, 178–79 valuation, xv-xvi, 436 “Rules of the game,” 27, 82–83, 111, 112 Salvage value, 117n, 153, 167, 295n, 416n. See also Abandon or contract system Samuelson, Paul, 40, 175–76 Scale, economies of, xii, 70, 76, 280–81 Schelling, Thomas, 24, 30, 40, 40–41 Scholes, Myron, 40, 175. See also Black– Scholes model

485

Schwartz, Eduardo S., 273, 409 Scope, economies of, xii, 70–71, 76 Second-mover advantage, 34, 99, 363, 366, 386, 414, 421 Selten, Reinhard, xxvii, 30, 40–41, 114–15 interview with, 85–86 Sensitivity analysis, 19 Sequence of the play, 27 Sequential equilibrium, 115 Sequential game, 83 Sequential investment, 357. See also Investment timing and asymmetric information, 412 and asymmetric positions, 393, 393–94 and cost asymmetry, 423 in duopoly, 331–39 and capacity expansion, 352, 354–55 under uncertainty, 339–48, 351–52, 356–57 in investment timing games, 363 in oligopoly and capacity expansion, 355–56 under uncertainty, 348–52 preemptive, 409 Sequential investment equilibrium, for real estate, 406–407 Sequential Stackelberg game, 131, 133–34 787 Dreamliner (Boeing), 6–9, 353 Shared option, xvi, 35, 344, 356 Shareholder value, maximizing of, 83 Shock(s), 38n, 157 stochastic, 284, 301, 302 additive, 302 multiplicative, 301, 302, 320–22 Shutdown (and re-start) option, 165, 188 Signaling, 34 Simultaneous game, 83, 139n, 224, 229 Simultaneous investment, 311, 331 and asymmetric information, 412 Cournot, 378 in dynamic setting, 317n and identical firms, 338 in investment timing game, 363 joint or collaborative, 393 probability of, 364 in R&D, 409 vs. sequential investment, 347 Sleeping patent, 395, 410, 423 Small-cost advantage, in asymmetric preemption, 384–89 Smit, Han, 13 Smit–Trigeorgis model, 193, 197n, 217, 242, 344, 357–58, 411–12 Smooth-pasting condition, 293n, 328–29, 329n, 456, 458 Social-cultural factors, in macroeconomic analysis, 57

486

Social optimality and capacity expansion in perfect competition, 324–25 of research, 408–409 and sequential investment, 345 Sødal, Sigbjørn, 278n, 289n, 291, 292n, 307, 451 Soft commitment (accommodating stance), xvi, 121, 123 in advertising, 129, 266–67, 269 in Cournot quantity competition, 130 in differentiated Bertrand competition, 127 Solution concepts, 89, 116 Bayesian equilibrium, 116 Nash equilibrium, 25–26, 28, 109, 113–14, 116 perfect Bayesian equilibrium, 116 rationalizability, 89n refinements, 113–15 subgame perfect equilibrium, 398 “trembling-hand” equilibrium, 115 Spence, Michael, 31, 40, 41n Spillover effects, and R&D investment, 243, 246–53, 272 Stackelberg, Heinrich von, 131 Stackelberg game, sequential, 131, 133–34 Stackelberg model of duopoly, 131, 335n follower, 99, 106, 131, 133–34 leader, 99, 106, 131, 133–34 (see also Leadership) Staging or time-to-build option, 163, 164 Static decision theory or game theory, 39 Stiglitz, Joseph, 31, 40, 41n Stochastic Itô calculus, references on, 458–59 Stochastic processes, xvi, 425–26, 458–59 continuous-time, 426–27 Brownian motion, 437–38 geometric Brownian motion, 275, 298 (see also Geometric Brownian motion), 298 general Itô process, 440–41 mean-reversion process, 438–40 discrete-time, 426 expected discount factor, 448–51 first-hitting time, 447–48 forward net present value, 441–47 optimal stopping, 453, 455–56, 458 properties of (basic processes), 457 Stochastic profit, 301–303, 304, Stochastic shock, 284, 301, 302 additive, 302 multiplicative, 301, 302, 320–22 Stopping, optimal, 453, 455–56, 458 Stopping or action region, 312–13, 453, 455 Strategic commitment. See Commitment

Index

Strategic complements (reciprocating actions), xvi vs. substitutes, 97–99, 194, 271, 272 Strategic conflict. See Game theory Strategic effects, of commitment, 118, 120–21, 122 Strategic entry barriers, 65 Strategic form, of investment timing games, 363 Strategic investment, 246–53, 271, 285–88, 423 direct effect of, 120, 122 strategic effect of, 120, 122, 123–25, 126, 128, 131, 132 under uncertainty, 243, 396 Strategic management, 47, 73, 74 integrative approach to, 35–41, 195, 217 (see also Option games) paradigms of, 48–50 Strategic move, 11, 117 Strategic substitutes (contrarian actions), xvi vs. complements, 97–99, 194, 271, 272 Strategic uncertainty, xxv, 5 Strategy (competitive), xvi-xvii, 10, 12–15, 43, 82–83 in changing competitive environment, 51–56 closed-loop, xvi, 83n, 121n, 325n, 398, 399 dominant, xii, 28–29, 207–208 and cost asymmetry, 230 example, 203–204 in investment options, 239 weak, 230 exit, 415–17 generic, 70–72 open-loop, xvi, 83n, 121n, 325n perspectives on, 15 and corporate finance, 15–20 game theory, 20–34 as portfolio of real options, 170–71 and real options thinking, 198 success factors in, 10 Strategy profile, 83 Strategy space, 397–98 Strengths, weaknesses, opportunities, and threats (SWOT) analysis, 51 Structural entry barriers, 64–65 Structure–conduct–performance (SCP) paradigm, 58–59 Subgame perfect Nash equilibrium, xvii, 109, 114–15, 116 Submissive underdog strategy, 125 and underinvestment, 250n Substitutes, threat of, 64 Substitutors, 148

Index

Suicidal Siberian strategy, and overinvestment, 125, 250n Sunk costs, xvii, 117, 117n, 271n, 275 Supergames, 139. See also Repeated games Suppliers, market power of, 65–66 Sustainable competitive advantage, 17, 41, 335n, 342n Switching costs, 64, 66, 129, 165, 166n, 167n Switching option(s), 163, 165–67 Symbols used in this book, xix-xxi Technological changes, in macroeconomic analysis, 57–58 Technologies, for electric industry (and business risk exposure), 159–62 Technology portfolios, 168–169, 170 Technology-related business risks, for electricity industry, 156, 158–59 Temporary shutdown option, 188 Threats empty, 112 of entry, 64–65 in “five-forces” analysis, 60–61 and game theory, 2 counterthreats, 2 of retaliation, 145 of substitutes, 64 Time, decision vs. real, 224–25 Timing of investment. See Investment timing Tirole, Jean, xxvi, xxvii, 39n, 81n, 107, 108, 118–21, 124, 129, 132, 139n, 152, 194, 243, 246n, 249, 286n, 324n, 332, 333, 359, 361, 365, 366, 371, 372, 373, 390, 396, 397, 406n, 415, 420 interview with, 36–37 Tit-for-tat strategy, 29, 142, 143–44 Tobin, James, 40, 284, 292 Tobin’s q, 284, 292 Top dog strategy, 123–124, 128, 134 Tough commitment or investment (aggressive stance), xvii, 121, 123, 266 in advertising, 267, 269 in Cournot quantity competition, 130 “Trembling-hand” equilibrium, 115 Trigeorgis, Lenos, 13, 19, 35n, 41, 49, 119n, 149, 153n, 189, 190, 193, 197n, 206, 209, 217, 218, 219, 242, 243, 249, 272, 273, 301n, 344, 357, 358, 360, 395, 411, 412, 423n, 424, 432n, 441, 445 Trigger strategy, 142, 240, 277. See also Investment triggers Two-stage games, 271–72 in goodwill and advertising, 264–70 innovative (R&D), 243, 246–50 in patent licensing, 262

487

Uncertainty at Boeing, 11–12 and capacity expansion, 418 and competition vs. cooperation, 150 Cournot quantity competition under, 225 duopoly with sequential investment under, 339–48, 351–52, 356–57 and electric utilities, 156–57 firm-specific risks, 156 generation technologies and business risk exposure, 159–62 idiosyncratic business risks, 156, 157–58 technology-related business risks, 158–59 and flexibility–commitment trade-off, 197 and game theory, 38, 38n growth options under, 421 and investment in existing market, 392 investment opportunities under (dynamic programming), 326–29 and investment timing, 196–97, 307 and Jorgensonian rule, 295–96 market, xxv, 5 and new market investment, 372, 373 and oil reserve as option, 280 in oligopoly with sequential investment, 348–52 option games under, 6, 195, 217, 240 option models of investment under, 333 and patent leveraging strategies, 395 patent licensing under, 261–64 and patent races, 408 quantity competition under, 224–38 and R&D, 243 and real options, 20–23 strategic, xxv, 5 and strategic investment, 243, 396 technological, and sleeping patents, 410 and technology adoption, 408 in utility planning, 280–81 Underinvestment, 16n, 122, 250, 250n Utility planning. See also Electric utilities scale vs. flexibility in, 280–81 Utilization of capacity, optimal, 417–19 Valuation, risk-neutral, xv-xvi Value chain, 51, 60, 69–70 relationships along, 147, 148 Value creation, 66–70 external vs. internal perspectives on, 48–50 Value erosion, competitive, 35, 421 Value-matching condition, 293n, 328, 374, 455–56 Value net, 148–49

488

Value redistribution, 148–49 Variable costs, xvii for fuels in electric utility, 162 Vertical differentiation, 72 Volatility, 426 Waiting option. See Deferral or waiting option Waiting region, 314 War of attrition, xvii, 99, 196, 363, 414, 415 and exit strategy, 416 Weeds, Helen, 408, 409, 410 Wernerfelt, Birger, 49, 50 Wicksell model, 283 Wiener process, 427. See also Brownian motion Yahoo, Microsoft bid for, 24, 25 Zero-sum games, 27n

Index