Organization Theory and its Applications 2012019161, 9780415690393, 9780203081358

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Organization Theory and its Applications

Organization theory is a fast-developing field of microeconomics. Organizational approaches are now used in a wide range of topics in business studies. They are based on information economics, contract theory, and mechanism design. This book introduces such organizational approaches and how to adopt them as business applications. The book presents the theory in the first two chapters and proceeds to cover the applications of the theory in the three chapters that follow. The theory lays the foundation and the applications illustrate how the theory can be used in a wide range of business problems. The book concisely covers many concepts and ideas in organization theory, including complete contracts, incomplete contracts, allocation of control rights, option contracts, convertibles, and joint ventures. It will be of use to third-year undergraduates and above, as well as master’s and PhD students in business schools. Susheng Wang received his bachelor’s degree in mathematics in 1982 and his master’s degree in mathematics in 1985. He became a lecturer at Nankai University, China, in 1985. He received his PhD in economics at the University of Toronto in Canada in 1991. He was an assistant professor at Concordia University in Canada between 1991 and 1993. Since 1993, he has been with Hong Kong University of Science and Technology. His fields of research are organization theory and information economics. He has received three Franklin teaching awards for teaching excellence since 2005. He has been a Yangtze Chair Professor (an honored position) at Nanjing University in China since 2007.

Organization Theory and its Applications

Susheng Wang

First published 2013 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Simultaneously published in the USA and Canada by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2013 Susheng Wang The right of Susheng Wang to be identified as the Author of this work has been asserted by him in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Wang, Susheng. Organization theory and its application/by Susheng Wang. p. cm. Includes bibliographical references and index. 1. Organization. 2. Corporate governance. 3. Management. 4. Contracts. I. Title. HD31.W314 2012 302.3’5—dc23 2012019161 ISBN: 978-0-415-69039-3 (hbk) ISBN: 978-0-203-08135-8 (ebk) Typeset in Times New Roman by Integra Software Services Pvt. Ltd, Pondicherry, India

Contents

List of figures List of tables Preface

vi viii ix

1 Complete contracts

1

2 Incomplete contracts

61

3 Corporate finance

109

4 Corporate governance

151

5 Public governance

178

Notes References Index

200 203 208

List of figures

1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 4.1 4.2 4.3 4.4 4.5

A shift in density function First- vs. second-best contracts Optimal contract under double moral hazard, l0  l  l Optimal contract under double moral hazard, l  l1 The optimal one-step contract The incomplete contract approach Chemical dumping vs. fishing The timing of events Lemma 2.1 Optimal control zones Optimal control zones The timing of events Dominant contracts in three areas The timing of events Optimal investment A callable convertible The timeline of events The timeline of events The game tree for debt financing Decision intervals for debt Decision intervals for a straight convertible Decision intervals before a call Decision intervals after a call The game tree for callable convertible financing Decision intervals for a callable convertible with protection The timeline of events Investment with endogenous network Investment with exogenous network The timeline of events The timeline of events The timeline of events Efficiency Welfare comparison: SF vs. UF

12 12 17 17 33 66 72 93 95 97 98 100 105 111 124 125 130 135 137 138 138 139 139 140 141 143 149 150 152 159 164 169 170

List of figures 4.6 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Effort comparison The timeline of events The optimal ownership mechanism Quality curves Price curves in the first period Price curves in the second period Welfare curves The control regions depending on income levels The control regions depending on marginal costs The control regions depending on marginal utilities The sequence of events The first- and second-best solutions

vii 170 179 180 184 185 186 187 191 192 193 195 197

List of tables

2.1 4.1 4.2 4.3 4.4

Alternative assumptions on q1 and q2 Contractual joint ventures Correlations between the signal and the realized return The equilibrium solutions Summary of results

77 152 159 168 177

Preface

The aim. Organization theory is a rapidly developing field of microeconomics. Organizational approaches are now used in a wide range of topics in business studies. These approaches are based on information economics, contract theory, and mechanism design. This book applies organizational approaches to business problems. We present the theory in chapters 1 and 2, and many applications of the theory in chapters 3–5. The coverage. This book presents organization theory at the senior undergraduate, master’s and PhD levels for students in business schools. It concisely covers many concepts and ideas in organization theory. In particular, complete contracts, incomplete contracts, allocation of control rights, option contracts, convertibles, and joint ventures are all discussed. Chapter 1 focuses on traditional contracts, specifically the classical agency theory, contracts with double moral hazards, contracts with two states, optimal linear contracts, optimal one-step contracts, limit contracts, and recursive dynamic contracts. Chapter 2 covers incomplete contracts, specifically contracts that contain the allocations of income and control rights, ex-post options, renegotiation, and conditional ownership. Chapter 3 covers applications in corporate finance, specifically business financing using bonds, equity, option contracts, and convertibles. Chapter 4 covers applications in corporate governance, specifically joint ventures, staged financing, reorganization of the corporate control structure, endogenous structure of corporate governance, endogenous transparency of governance information, and contracting versus integration. Finally, Chapter 5 covers applications in public governance, specifically the allocation of control rights within the government hierarchy, private–public joint ventures, state versus private ownership, privatization, and law and economics. Special features of this book. This book differs notably from all others currently in existence. There are some on business organizations, but none of them provides as rigorous a theoretical foundation as this one does. In terms of its contents, this book covers not only the theory but also a wide range of applications. The theory lays the foundation, while the applications show how the theory can be used to solve a variety of business problems. This book is intended to be concise. Students are encouraged to attend class to learn the details and explanations. It is also good for

x Preface those instructors who prefer to fill in the details in their own words. This book can also serve as a reference for experts in the field. Supporting materials. Supporting materials, such as exercises and solutions for Chapter 1, are available at www.bm.ust.hk/~sswang/info-book/. For many of the technical terms and definitions used here, readers may refer to Wang (2008a) and Wang (2012). Errors are inevitable and corrections will be posted on the website. Other materials, such as new sections and chapters, may also be posted there. Acknowledgment. I would like to thank Alice K.Y. Cheung for English editing. Susheng WANG Hong Kong, CHINA April 2012

1

Complete contracts

1. Introduction In this chapter, we discuss income-sharing agreements only. That is, contracts in this chapter contain income rights only. Besides asymmetric information, there is another type of information problem, called the incentive problem. For example, in an employer–employee relationship, the employee’s applied effort may be observable by his employer but not verifiable to a court. If so, the effort cannot be bounded by a contract. In this type of problem, information is symmetric: both the employer and the employee have the same set of information. But the court cannot observe the information. So the issue here is: how does the employer provide sufficient incentives in a contract to motivate the employee? For convenience, in this book, a solution to an asymmetric information problem is called a policy, a solution to an incentive problem is called a contract, and both of them are referred to as mechanisms. Mechanisms include all kinds of strategic and equilibrium solutions, such as the price mechanism in general equilibrium and staged financing in venture capital investment. A policy in an asymmetric information problem has traditionally been called a mechanism, and the exercise of searching for the optimal policy is referred to as mechanism design. A policy ensures that the agent picks the offer intended for him, while a contract ensures that the agent takes the action intended for him. This is the so-called revelation principle. Efficiency is more difficult to achieve in an agency model than in an asymmetric information model. The reason is that the incentive compatible (IC) condition in an asymmetric information model is to shoot a given target (the truth), while the IC condition in an agency model is to shoot a moving target (the optimal effort, which is to be determined in equilibrium). We call contracts in this chapter complete contracts, since each of them is assumed to be the only mechanism in the model. We emphasize the completeness, since if they are treated as incomplete contracts they may not even be optimal. We will explain why they are called complete contracts in more detail in Chapter 2. This chapter will cover motivation for contract theory, the standard contract theory, and special but popular forms of contracts.

2 Complete contracts

2. Contracts in reality 2.1. Employment contracts Employment contracts are conditions and promises that determine an employer– employee relationship. The best-known employment contract is a labor contract, by which an institution hires a worker. A labor contract may specify a wage scheme, employment length, working hours, and certain requirements. It may also specify conditions under which an employee may quit his job and the employer may fire the employee. Besides labor contracts, there are many types of employment contracts. For example, a plaintiff may hire a lawyer to represent him in the court. A landlord may hire a tenant to plant and harvest crops on her land. A family may hire a babysitter to take care of a young child.

2.2. Financial assets There are many forms of financial assets. Many financial assets are financial contracts that promise a payment or a transfer of assets at certain points in time under certain conditions. Many financial contracts are created by issuers to deal with information and incentive problems. Other people, including fund managers, may hold them for their unique features. For example, since financial contracts promise to pay a certain amount at a certain time or in a certain event, financial contracts can be used to diversify consumption across states of uncertainty and smooth consumption over time. Examples of financial contracts/assets include common stock, preferred stock, bond or debt, warrant, convertible security or convertible debt, and callable convertible debt.










Common Stock. A common stock share gives the holder a piece of ownership of the firm. The holder receives a share of dividend whenever the firm distributes dividend, but there is no guarantee of when that will occur or how much will be distributed. The holder also has the right to vote in a company decision meeting. Bond or debt. A debt contract or a bond promises to the holder specified monetary payments at specified dates. Warrant. A warrant is an option to purchase a specified number of common shares at a specified price on or before a specified date. Convertible Security. A convertible security pays a guaranteed rate of interest (before conversion), and provides an option for the holder to convert his investment into the issuing company’s stock shares at anytime at a specified conversion price. A convertible security is a combination of a debt contract and a warrant. Callable Convertible Security. It is a convertible security with an option for the issuer to force conversion. When the issuer calls, the holder can choose either to convert or to take a specified amount of monetary payment.

Complete contracts

3

2.3. Relational contracts Firms may cooperate, form alliances, or engage in joint ventures. Their cooperative relationships are usually governed by contracts between them, called relational contracts. Relationships can be either horizontal or vertical. Two firms that produce or intend to produce the same product may cooperate or engage in a joint venture to produce the product – this is a horizontal relationship. There are many examples of horizontal relationships, including foreign direct investment and various forms of joint ventures. Two firms of which one is a supplier of the other may form a close relationship – a vertical relationship. There are many examples of vertical relationships in the retail industry, such as the guaranteed supply of a product and exclusive rights to sell a product. There are also cases in which a relationship bears the features of both a horizontal and a vertical relationship, such as franchising, venture capital, alliances, and holding firms. 2.4. Insurance Insurance packages are a special set of contracts. Examples include car insurance, life insurance, property insurance, medical insurance, and unemployment insurance. Car insurance, life insurance, and property insurance are typically provided by private companies. Medical insurance can be provided by the government, private companies, or both. Unemployment insurance is typically provided by the government. The government’s involvement in an insurance scheme is typically for the sake of low-income people.

3. The frameworks of contract theory Motivated by the widespread practice of sharecropping in traditional China, Cheung (1968) was the first to apply a contractual approach to economic trade. His work fell short of a contract theory, which was later developed by several researchers, including James Mirrlees. See Hart and Holmström (1987) for a detailed survey of the agency literature. There are two problems with Cheung’s work. First, it deals with a first-best problem, in which there are no moral hazards. This means that his solution is equivalent to a general equilibrium solution. Second, the admissible contracts are fixed-share contracts (with fixed income shares for participants). With this restriction, the optimal contract from this set of fixed-share admissible contracts may not be the optimal one in a larger set of admissible contracts. Two formulations of the agency problem were developed in the 1970s. In the state formulation, output is determined by a production function. In the distribution formulation, output is determined by a distribution function. 3.1. The state formulation The state formulation was initiated by Wilson (1969), Spence and Zeckhauser (1971), and Ross (1973). Consider a two-agent relationship in which the dominant

4 Complete contracts agent (the principal) tries to hire the other agent (the agent) to work on a project. Let A be the set of actions available to the agent and a be a generic element of A. Let ε~ represent the state of the economy. Output is determined by a function of the action and state: x~ ¼ xða; ε~ Þ. Output x is assumed to be verifiable, while action a is unverifiable, where the verifiability of output means that each value of output when produced can be observed by the court and is admissible as evidence to the court. Although output is produced ex post after uncertainty is realized, since it is verifiable, contractual terms can be conditional on its value and enforced by the court. On the contrary, since the agent’s action is not verifiable, contractual terms that are based on the action, if made, cannot be enforced by the court. Since a can typically take any positive real value, instead of just 0 or 1, we often call a the effort. That is, the value of a represents how much effort the agent has exerted on the project. Let v and u be the utility functions of the principal and the agent, respectively, and let c(a) be the private cost of effort, where private cost means that the agent covers the cost by himself. Let s(x) be the promised payment to the agent when output turns out to be x ex post. Then the payoffs for the principal and agent are respectively v[x  s(x)] and u[s(x)]  c(a). Let ϕðε Þ be the density function of ε~ : Then the agent’s expected utility is Z Uðs; aÞ ¼ ufs½xða; εÞgϕðεÞ dε  cðaÞ; and the principal’s expected utility is Z Vðs; aÞ ¼ fv½xða; ε Þ  s½xða; εÞgϕðε Þdε: The principal is to offer a contract s(⋅) to the agent in order to induce the agent to provide a desired amount of effort a.

3.2. The distribution formulation The distribution formulation was initiated by Mirrlees (1974, 1975, 1976) and was finalized by Holmström (1979). They propose a formulation that is equivalent to the state formulation, but with a more convenient setting. This formulation is now the basis of the standard agency model. Let f ðx ; aÞ be the density function of x~ ; dependent on a. Then the agent’s expected utility is Z Uðs; aÞ ¼ u½sðx Þ fðx ; aÞdx  cða Þ; and the principal’s expected utility is Z Vðs; aÞ ¼ v½x  sðx Þ f ðx ; aÞdx:

Complete contracts

5

Again, the principal is to offer a contract s(⋅) to the agent in order to induce the agent to provide a desired amount of effort a. This formulation is more convenient, since the endogenous function s(⋅) (to be determined in equilibrium) is a function of x where x is an independent variable. In contrast, in the state formulation, s(⋅) is a function of x where x is a function of other variables, which makes the task of solving for s(⋅) difficult.

4. The standard agency model In this section, we present the standard agency model developed by Mirrlees (1974) and Holmström (1979). This section is mainly based on Holmström (1979). See also Laffont (1995, pp. 180–198).1 The standard contract theory is about an output-sharing rule between a principal and an agent. The principal tries to hire the agent to work on a project. A problem is that the agent’s effort may be observable to the principal but is not verifiable to a third party such as a court, and is thus noncontractible. Hence, the principal must provide incentives in a contract for the agent to put in a sufficient effort voluntarily. The question is how to provide sufficient incentives in a contract. Let x~ be a random output from the firm. Given the agent’s effort a; the distribution function of x~ is Fðx; a Þ and the density function is fðx ; aÞ: The effort a is not verifiable or contractible; more precisely, it is uncontractible ex ante since it is nonverifiable ex post. In fact, only the output x is contractible (observable and verifiable ex post) in this model. The output is random ex ante, but it becomes known and verifiable ex post. Thus, a payment scheme s(⋅) can be based on the output x; the scheme pays s ¼ sðx Þ to the agent when output is x: The set of admissible contracts is fsðx Þjsðx Þis Lebesgue integrable and sðxÞ  0g: Here, condition sðx Þ ≥ 0 means limited liability to the agent. The agent’s utility function is uðs Þ  cða Þ;

with u0 40; u00 ≤ 0; c0 >0; c00 >0;

ð1:1Þ

where c(a) is the cost of effort which is private. The principal’s utility function is vðx  s Þ;

with v0 >0; v00 ≤ 0:

ð1:2Þ

In addition, assume that at least one of the parties is risk averse. This assumption is necessary for the standard agency theory. We will discuss the special case when both the principal and the agent are risk neutral in Section 5. Verifiable effort: the contractual approach As a benchmark, we first consider the problem in the Arrow–Debreu world, in which effort a is verifiable. In this case, the principal specifies both an effort and a payment scheme in a contract. That is, the principal can offer a contract of the form ½a; sð⋅Þ to the agent. Let u be the agent’s reservation utility, below which the agent

6 Complete contracts will turn down the contract. The principal is constrained by the participation constraint or individual rationality (IR) condition: Z u½sðx Þ fðx ; aÞdx  cða Þ  u : The principal’s problem is then Z max v½x  sðx Þ f ðx ; aÞdx a; sð⋅Þ Z s:t: IR: u½sðx Þ f ðx ; a Þdx ≥ cða Þ þ u :

ð1:3Þ

Its solution ða ∗∗ ; s ∗∗ Þ is said to be the first best, and hence the above problem is called the first-best problem. According to the first welfare theorem, it is Pareto optimal. Given each a; the support of the distribution function is the interval of x on which fðx ; aÞ > 0: This interval will generally be dependent on a. The following assumption is imposed in the standard agency model. Assumption 1.1. The support of the output distribution function is independent of a. The Lagrangian for (1.3) is Z  Z  L≡ v½x  sðx Þ f ðx ; aÞdx þ λ u½sðxÞfðx ; aÞdx  cða Þ  u ; where λ ≥ 0 is a Lagrange multiplier. The Hamiltonian for L can be defined as H ≡ vðx  sÞ þ λuðs Þ: Given Assumption 1.1 the first-order conditions (FOCs) for a and s(⋅) from the Lagrangian and Hamiltonian are respectively2 Z Z v½x  sðx Þ f a ðx ; a Þdx þ λ u½sðxÞ f a ðx; a Þdx ¼ λc0 ða Þ; ð1:4Þ v0 ½x−sðxÞ ¼ λ: u 0 ½sðx Þ

ð1:5Þ

Equation (1.5) indicates that the marginal rate of substitution MRS ¼ v0 ðx  s Þ=u0 ðsÞ is constant across all possible states x; implying the proportional sharing of risk in some sense. We call it the first-best risk sharing. The marginal rate of substitution (MRS) is constant because the Lagrangian is a linear social welfare function, where the weight λ is endogenous. This result is the same as that in the Arrow–Debreu world, in which the MRS is equal to the price ratio and thus constant in equilibrium. Equation (1.4) means that the marginal social utility of a equals the marginal social cost of a. By (1.5), we have λ40; implying a binding IR condition at the optimum. By differentiating (1.5) with respect to x; we have v00 ⋅½1  s0 ðx Þ ¼ λu00 ⋅s0 ðxÞ: Hence, if the principal is not risk neutral, we have 0 < s0 ðxÞ < 1; implying that both the agent’s income and the principal’s income increase strictly with x.

Complete contracts

7

If the principal is risk neutral, the contract will be a constant, i.e. the principal absorbs all the risk. Similarly, if the agent is risk neutral, the agent takes all the risk. This risk-sharing arrangement of a risk-neutral party taking all the risk makes sense. When a risk-averse party faces risk, it demands a risk premium (an extra payment to compensate for bearing the risk), while a risk-neutral party cares about expected income only. Hence, the risk-neutral principal can do better by taking all the risk. 4.1. Verifiable effort: the general equilibrium (GE) approach Alternately, we can apply the general equilibrium approach. We set up a pure exchange economy in which the agent and principal are the two consumers who exchange labor and commodity in a competitive market.3 Suppose that the principal pays a wage wðx Þ for the agent’s labor input a. Since output is random, this wage is state-contingent. The price for the commodity is pðx Þ: Let x1 be the agent’s consumption of the commodity and x2 be the principal’s consumption of the commodity. We will use superscripts d and s to denote demand and supply, respectively. Then the agent’s problem is Z   max u xd1 ðxÞ f ðx ; as Þdx  cðas Þ as ; xd1 ð1:6Þ Z Z s:t: pðx Þxd1 ðx Þdx ≤ wðxÞas dx: The principal’s problem is Z   max v xd2 ðxÞ f ðx ; ad Þdx d d a ; x2 Z Z Z d d s:t: pðx Þx2 ðx Þdx þ wðxÞa dx ≤ pðx Þxdx:

ð1:7Þ

The equilibrium conditions are x ¼ xd1 ðx Þ þ xd2 ðx Þ;

ð1:8Þ

as ¼ a d :

ð1:9Þ

The agent’s problem implies u0 ½x1 ðxÞ fðx ; a Þ ¼ λ1 pðx Þ; Z

Z u½x1 ðxÞ f a ðx; a Þdx þ λ1 wðxÞdx ¼ c′ðaÞ:

ð1:10Þ ð1:11Þ

The principal’s problem implies v0 ½x2 ðxÞ fðx ; a Þ ¼ λ2 pðx Þ; Z

Z v½x2 ðxÞ f a ðx; a Þdx ¼ λ2 wðx Þdx:

ð1:12Þ ð1:13Þ

8 Complete contracts The two budget conditions in the problems (1.6)–(1.7) must be binding. With the equilibrium conditions (1.8)–(1.9), one of the budget conditions must be redundant. Also, since a∗ is independent of x; wðx Þ can be arbitrary in equilibrium; hence, we can set wðxÞ ¼ fðx; a∗ Þ for any x: Hence, we have a total of six equations: the binding budget condition in (1.6), the equilibrium condition (1.8) for x; and four conditions in (1.10)–(1.13) These six conditions determine six variables/functions: x∗1 ðx Þ; x∗2 ðxÞ; pðx Þ; a∗ ; λ1 , and λ2 . To allow us to compare this solution with the solution from the contractual approach, we now reorganize the equations. By (1.10) and (1.12), we find v0 ½x2 ðx Þ λ2 ¼ : 0 u ½x1 ðx Þ λ1 By (1.11) and (1.13), we find Z Z λ2 λ2 u½x1 ðx Þ f a ðx ; aÞdx ¼ c0 ðaÞ: v½x2 ðx Þ f a ðx ; aÞdx þ λ1 λ1 Define sðx Þ ≡ x1 ðx Þ and u by Z u ≡ u½x1 ðx Þ f ðx ; aÞdx  cða Þ:

ð1:14Þ

ð1:15Þ

ð1:16Þ

Then the three equations (1.14)–(1.16) are the same as (1.4), (1.5) plus the binding IR condition in (1.3) Hence, the GE approach and the contractual approach imply the same solution. To solve for a solution, substituting (1.10) into the agent’s budget constraint yields Z a ¼ λ1 u0 ½x1 ðx Þx1 ðx Þ f ðx ; aÞdx: By (1.13), we find Z Z 1 λ2 v½x2 ðx Þ f a ðx ; a Þdx u0 ½x1 ðx Þx1 ðx Þf ðx ; aÞdx ¼ : a λ1 Substituting this into (1.14) and (1.15) to eliminate λ2 =λ1 and using the equilibrium condition (1.8) to eliminate x2 ðx Þ results in two equations for x1 ðx Þ and a. 4.2. Nonverifiable effort Suppose now that a is not verifiable. Since the principal cannot impose an effort level, she has to offer incentives in a contract to induce the agent to work hard. Given an income-sharing rule sð⋅Þ, the agent is free to choose an effort, and his problem is: Z ð1:17Þ max u½sðxÞ fðx; a Þdx  cðaÞ: a

Complete contracts

9

This is the incentive compatibility (IC) condition. The principal still decides the income-sharing rule s(⋅) and effort a. But, whereas before the principal had only to ensure that the agent accepts the contract (the IR condition) and the effort could be imposed upon the agent, now the principal has to provide not only an incentive for the agent to accept the contract but also an incentive to accept the effort (the IC condition). Hence the principal’s problem is Z max v½x  sðx Þ fðx ; aÞdx a; sð⋅Þ Z s:t: IR : u½sðx Þ fðx ; aÞdx ≥ cða Þ þ u ; Z a Þ: IC : a solves max u½sðx Þ f ðx; a^Þdxcð^ a^

The FOC of the IC condition (1.17) is Z u½sðx Þ f a ðx ; aÞdx ¼ c0 ðaÞ:

ð1:18Þ

The second-order condition (SOC) for the IC condition (1.17) is Z u½sðx Þ f aa ðx; a Þdx < c00 ðaÞ: The IC condition (1.17) is too difficult to handle in an optimization problem. As an alternative, the so-called first-order approach (FOA) is to substitute the original IC condition (1.17) by its FOC (1.18) in the principal’s problem. Rogerson (1985) and Jewitt (1988) find conditions under which the FOA is valid (i.e. by which the solution of the agency problem is indeed the right one). Rogerson (1985) limits distributions to have finite states. Under the FOA, we call (1.18) the IC condition. Thus, the principal’s problem becomes Z v½x  sðx Þ f ðx; a Þdx max a; sð⋅Þ Z ð1:19Þ s:t: IR: u½sðxÞ f ðx ; aÞdx ≥cða Þ þ u ; Z IC : u½sðxÞ f a ðx ; a Þdx ¼ c0 ðaÞ; Its solution ða∗ ; s∗ Þ is said to be the second best and hence the above problem is called the second-best problem. Due to the additional constraint, this solution is inferior to the first best. The IC and IR conditions provide different functions. An IR condition induces the agent to participate, but the agent may not work hard or invest an effort up to the principal’s desired amount. The principal has two ways to ensure the agent invests the desired amount of effort. One way is to write it in the contract, if this amount is verifiable. The other way is to provide incentives in the contract for the agent to invest the desired amount voluntarily.

10 Complete contracts The Lagrangian for (1.19) is Z  Z L  v½x  sðxÞ f ðx; aÞdx þ λ u½sðxÞ f ðx; aÞdx  cðaÞ  u ; Z þ

 u½sðxÞ fa ðx; aÞdx  c0 ðaÞ ;

where λ and μ are Lagrange multipliers, and λ  0: The Hamiltonian for L is H ≡ vðx  s Þ f ðx ; a Þ þ λuðs Þ f ðx ; a Þ þ μuðs Þ fa ðx ; a Þ: The FOCs for a and s(⋅) from the Lagrangian and Hamiltonian are respectively Z  Z ð1:20Þ v½x  sðx Þ f a ðx ; aÞdx þ μ u½sðxÞ faa ðx ; a Þdx  c 0 0 ðaÞ ¼ 0; v0 ½xsðx Þ f a ðx ; aÞ ¼λþμ : 0 u ½sðx Þ f ðx ; aÞ

ð1:21Þ

The second equation is called the Euler equation, which determines the optimal contract. Here, we can see that, if both parties are risk neutral, (1.21) cannot hold. This is the reason why one of the parties must be risk averse. Notice that the likelihood function is lnfðy; a Þ and ∂ ln f ðx ; aÞ f a ðx ; a Þ ¼ ; ∂a f ðx ; a Þ i.e.

f a ðx ;a Þ f ðx ;a Þ

∂2 ln f ðx ; aÞ ∂ f a ðx ; aÞ ¼ : ∂x∂a ∂y f ðx; a Þ

is the increase in ln fðx ; aÞ for a unit increase in a. And a and x are

complementary if

f a ðx ;a Þ f ðx ;a Þ

is increasing in x: The latter condition is called the

ðx ;a Þ monotone likelihood ratio property (MLRP). ∂ lnf measures a kind of correla∂x∂a tion between a and x: A higher value of lnfðx; a Þ means that a higher value of x is more likely to come from a higher value of a rather than by chance. Thus, ∂lnf∂að^x ;^a Þ is the marginal increase in the principal’s belief that the effort is a^ from the observation of x^ . We further assume that an increase in a generates an improvement in x in the sense of first-order stochastic dominance (FOSD), as stated in the following assumption. 2

Assumption 1.2. (FOSD). Fa ðx ; a Þ  0; for all ða; x Þ; with a strict inequality on a set of ða; x Þ that has a non-zero probability measure. Theorem 1.1 (Holmström 1979, Shavell 1979). Under Assumption 1.1 and Assumption 1.2, FOA and u00 < 0; we have μ > 0 for the optimal solution of (1.19). Proof. Suppose, on the contrary, that μ  0: Let rðx Þ  x  sðx Þ; and for the λ40 in (1.21), define r ðx Þ by

Complete contracts v0 ½r ðx Þ ¼ λ:  r ðxÞ

u0 ½x

11

ð1:22Þ

The advantage of r ðx Þ over rðx Þ is that we know ∫v½r ðx Þ f a ðx ; a∗ Þdx > 0: Then, by (1.21), for x satisfying f a ðx ; a∗ Þ  0; we have v0 ½rðx Þ v0 ½r ðx Þ ≤ λ ¼ : u0 ½x  rðx Þ u0 ½x  r ðx Þ Since v00 0 and u00 0: Also, by Assumption 1.2, Fa ðx; a Þ < 0 on a non-zero-probability-measure set. Therefore, Z v½r ðx Þ f a ðx ; a∗ Þdx > 0: However, if FOA is valid, the SOC for (1.17) must hold, which means Z u½sðy Þ f aa ðy ; a∗ Þdy  c00 ða∗ Þ  0:

ð1:23Þ

Then (1.20) cannot hold. This is a contradiction. We thus must have μ > 0: Q.E.D. Theorem 1.1 indicates that the risk sharing is no longer the first best. By deviating from the first-best risk sharing, the principal can provide incentives in the contract. To understand the result intuitively, assume that the principal is risk neutral. Then, the first-best risk sharing, as indicated by (1.5), requires full insurance (no risk) for the agent, i.e. s(x) is constant. However, if the agent’s action is not verifiable, under a fixed contract, the agent will choose zero effort. Thus, for the agent to have the incentive to work, the contract with uncontractible effort must relate a payment to output. This means that the agent must share some risk, implying inefficient risk sharing in the case of nonverifiable effort. Let us compare the first-best and second-best contracts. As shown in Figure 1.1, an increase in a causes the density curve to shift to the right. This implies that

12 Complete contracts

f a ðx ; aÞ > 0 for high x ;

f a ðx ; aÞ < 0 for low x:

Let s ∗ ∗ be the first-best contract determined by (1.5) and s∗ be the second-best contract determined by (1.21) Since the left-hand side of (1.21) is increasing in s; we have s∗ ðx Þ > s ∗∗ ðx Þ for high x ;

s∗ ðx Þ < s ∗∗ ðxÞ for low x:

Thus, the two contracts should have the relationship shown in Figure 1.2, which means that, in order to motivate the agent to work hard, the principal must use a steeper contract curve than the first-best contract to provide incentives. The secondbest contract rewards high output and punishes low output; at the same time, the second-best contract must also provide a proper risk-sharing scheme.

f (x,a)

f ( x,a ')

x

Figure 1.1 A shift in density function.

s s* s**

x

Figure 1.2 First- vs. second-best contracts.

Complete contracts

13

4

Example 1.4. The principal’s problem is Z   max ½x  sðxÞ f ðx; aÞdx s2S; a2A

Z

s:t: IC :

u½sðxÞ fa ðx; aÞdx ¼ c0 ðaÞ;

ð1:24Þ

Z u½sðxÞ f ðx; aÞdx  cðaÞ þ u;

IR :

where A ⊂ ℝþ is the effort space, and S is the contract space that consists of continuous contracts satisfying the limited liability condition: sðxÞ  0: Let pffiffiffi uðxÞ ¼ 2 x;

vðxÞ ¼ x;

3 u ¼ ; 4

cðaÞ ¼ a2 ;

1 x fðx; aÞ ¼ ea for x 2 ½0; þ1Þ: a The density function fðx ; a Þ states that the output follows the exponential distribution with mean Eðx Þ ¼ a and variance VarðxÞ ¼ a2 : We find the second-best solution: 1 a∗ ¼ ; 2



s∗ ðx Þ ¼ x þ

1 4

2

;

and the first-best solution: a ∗∗ ¼ 0:76;

s ∗∗ ðx Þ ¼ 0:44:

■

Example 1.2. Consider pffiffiffi uðxÞ ¼ 2 x;

vðxÞ ¼ x;

3 u ¼ ; 4

ðxaÞ2 1 fðx; aÞ ¼ pffiffiffiffiffiffi e 22 for x 2 R: 2

We have ðxaÞ2 x  a 1 fa ðx; aÞ ¼ pffiffiffiffiffiffi e 22 2 2

Hence, 1 xa ¼ λþμ 2 : ■ u0 ½sðy Þ σ

cðaÞ ¼ a2 ;

14 Complete contracts

5. The agency model with double moral hazard This section is based on the work of Kim and Wang (1998). They are the only authors to provide a thorough and rigorous theoretical foundation for the agency problem under double moral hazard. Their work can be viewed as an extension of that of Holmström (1979) from a single moral hazard setting to a double moral hazard setting. A risk-neutral principal hires a risk-averse agent with utility function u: ℝ ! ℝ to undertake a project. The utility function is increasing and concave. After the principal designs a wage contract s(⋅) for the agent, she provides her effort b∈ℝþ and the agent provides his effort a  0 noncooperatively. Let Cðb Þ and c(a) be the private costs of the principal’s effort and the agent’s effort, respectively. A joint effort is represented by z ¼ ϕða; bÞ: Let Fðxjz Þ be the output distribution function and fðxjz Þ be the density function. Since x is verifiable, the agent’s wage contract s(⋅) can be based on it, i.e. s(x) is the payment to the agent when output is x: The set of admissible contracts is S ≡ fs : ℝ→ ℝ js is Lebesgue measurableg: Assumption 1.3. a) c0 ðaÞ ≥ 0; c00 ðaÞ > 0; c0 ð0Þ ¼ 0; c0 ð∞Þ ¼ ∞. b) C0 ðb Þ ≥ 0; C00 ðbÞ > 0; C0 ð0Þ ¼ 0; C0 ð∞Þ ¼ ∞. Assumption 1.4. ϕa ða; bÞ > 0; ϕb ða; b Þ > 0; ϕaa ða; b Þ < 0; ϕbb ða; b Þ < 0; and ϕab ða; bÞ  0: Assumption 1.5. (MLRP). For any z  0; for x ∈ ℝ.

fx f

ðxjz Þ is strictly increasing in x;

Assumption 1.6. (CDFC: Convexity of the Distribution Function Condition). For any x ∈ ℝ; Fðxjz Þ is convex in z. Assumption 1.4 implies that the two parties’ efforts are productive and complementary. The complementarity condition is especially needed to characterize an optimal contract when the agent is risk averse (see Lemma B1 of Kim and Wang 1998). The assumption can be justified as a reason for the two parties’ undertaking of the joint project. Assumption 1.5 and Assumption 1.6 say that output is increasing in z with a decreasing rate in a stochastic sense. After the contract is accepted, assume that the two parties play a Nash game to determine their efforts. Hence, the IC conditions of the two parties are as follows: a solves Z ð1:25Þ max u½sðx Þ f ½xjϕða; b Þdx  cða Þ a≥0

and b solves Z max ½x  sðx Þ f ½xjϕða; bÞdx  CðbÞ: b≥0

ð1:26Þ

Complete contracts

15

By assuming that the first-order approach is valid for (1.25) and (1.26), the principal’s problem is: Z max ½x  sðxÞ f ½xjϕða; bÞdx  CðbÞ s2S; a; b0

Z þλ Z

s:t:

ϕa Z ϕb

 u½sðxÞ f ½xjϕða; bÞdx  cðaÞ

u½sðxÞ fz ½xjϕða; bÞdx ¼ c0 ðaÞ

ð1:27Þ

½x  sðxÞ fz ½xjϕða; bÞdx ¼ C0 ðbÞ

0  sðxÞ  l; 8 x; where λ40 is the weight placed on the agent’s utility in social welfare. The last constraint implies that the agent’s wage contract must exist in a given interval ½0; l : This constraint is needed to guarantee the existence of an optimal wage contract when the agent is risk averse. It is already well known that, in a single moral hazard setting, a lower bound is needed for the wage contract to guarantee the existence of an optimal contract. Otherwise, the principal can attain a result that is arbitrarily close to the complete information outcome by severely penalizing the agent with a small probability. In a double moral hazard setting, this issue arises not only from the agent’s side but also from the principal’s side. Kim and Wang show that, if there is no upper bound for the wage contract in the double moral hazard setting, then the double moral hazard situation arbitrarily approaches the single moral hazard situation. Actually, the principal can effectively provide her own incentives by designing a wage contract that penalizes herself severely when output is low and is otherwise the same as the wage contract which would be optimally designed in the single moral hazard situation. Since the principal can arbitrarily reduce the output range in which she is penalized by increasing the penalty amount, she can obtain a result that is arbitrarily close to that in the single moral hazard situation. Therefore, for an optimal contract to exist we need to impose not only a lower bound but also an upper bound on the wage contract. Let L be the Lagrange function for (1.27) without the last constraint. Then, Z  Z Lðs; a; b; 1 ; 2 Þ  ½x  sðxÞ f dx  CðbÞ þ λ uðsÞ f dx  cðaÞ  Z  þ 1 ϕa uðsÞfz dx  c0 ðaÞ  Z  þ 2 ϕe ½x  sðxÞ fz dx  C0 ðbÞ ; where μ1 and μ2 are Lagrange multipliers. The Hamilton function for L is

ð1:28Þ

16 Complete contracts H fx  sðxÞ  CðbÞ þ λ½uðsÞ  cðaÞ  1 c0 ðaÞ  2 C0 ðbÞgf þ ½1 ϕa uðsÞ þ 2 ϕb ðx  sÞfz :

ð1:29Þ

The Euler equation for s(⋅) implies the following equation for the optimal contract s∗ ð⋅Þ: λ þ 1 ϕa ffz ðxjϕ Þ 1 ; ¼ u0 ½s ðxÞ 1 þ 2 ϕb ffz ðxjϕ Þ

ð1:30Þ

where ϕ∗  ϕða∗ ; b∗ Þ; ϕ∗a  ϕa ða∗ ; b∗ Þ; ϕ∗b  ϕb ða∗ ; b∗ Þ; and ða∗ ; b∗ Þ are optimal efforts. Proposition 1.1. We have (i) μ1 > 0. (ii) μ2 ≥0. (iii) μ1 ϕ∗a > λμ2 ϕ∗b if l ≥l0 ; where l0 satisfies

l0 uðl0 Þ−uð0Þ

¼ λ.

The fact that μ1 > 0 implies that the moral hazard problem will always affect the agent, while the fact that μ2  0 implies that the moral hazard problem may (μ2 > 0) or may not (μ2 ¼ 0) affect the principal. Proposition 1.1 (iii) indicates that, if the upper bound l is large enough, the contract deals with the agent’s incentive problem more carefully, as implied by μ1 ϕ∗a > λμ2 ϕ∗b. By Proposition 1.1, we can now derive the optimal contract s∗ ðx Þ: Since the characteristics of an optimal contract when μ2 ¼ 0 are known from the standard agency model under single moral hazard, we hereafter exclusively focus on the case with μ2 > 0: Proposition 1.2. Assuming the FOA is valid, if l0  l  l1 ; where l0 is defined in ϕ∗ l1 Proposition 1.1(iii) and l1 satisfies uðl1 Þ−uð0Þ ¼ μ1 μ2 ϕa∗ ; then there is a unique, b

monotonically increasing, optimal contract s∗ ðx Þ defined by ( !) 8 1 þ 2 ϕb ffz ðxjϕ Þ > 1 > 0 > ; if x0  x; > < min l; ðu Þ λ þ 1 ϕa ffz ðxjϕ Þ  s ðxÞ ¼ > > > > : 0; if x5x0 ; where x0 satisfies

1 u0 ð0Þ

1 þ μ 2 ϕ∗

fz

ðx0 jϕ∗ Þ

¼ λ þ μ ϕ∗b ffz ðx jϕ∗ Þ (see Figure 1.3). 1 af

0

Proposition 1.3. Assuming the FOA is valid, if l > l1 ; where l1 is defined in Proposition 1.2 and ffz ðxjϕ∗ Þ is unbounded below, then there is a unique optimal contract s∗ ðx Þ defined by

Complete contracts

17

8 ! > 1 þ 2 ϕb ffz ðxjϕ Þ > 1 0 > > ; if x0  x; < ðu Þ λ þ 1 ϕa ffz ðxjϕ Þ  s ðxÞ ¼ > > 0; if x3  x5x0 ; > > : l; if x5x3 ; where x0 is defined in Proposition 1.2 and x3 satisfies

1 uðl Þuð0Þ

λþμ1 ϕ∗

fz

ðx3 jϕ∗ Þ

¼ 1þμ ϕ∗a ff z ðx 2 b f

3 jϕ

∗Þ

(see Figure 1.4). The fact that μ1 > 0 and μ2  0 illustrates that the optimal contract in (1.30) is less sensitive to output than that under single moral hazard. As a matter of fact, the key factor that makes the optimal contract under double moral hazard different from that under single moral hazard is μ2 ϕ∗b f z =f : Intuitively, in a single moral hazard setting, if there were changes in output x; they would be fully reflected in the agent’s rewards, because the agent is the only party applying effort. However, in a double moral hazard setting, changes in output will be reflected in the agent’s s*(x)

l (u′)–1

x0

μ 2φ*b μ 1φ*a

0

Figure 1.3 Optimal contract under double moral hazard, l0 ≤ l ≤ l. s*(x)

l

(u′)–1

x3

x2

x0

μ 2φ*b μ 1φ*a

0

Figure 1.4 Optimal contract under double moral hazard, l > l1 .

18 Complete contracts rewards with a discount, since the principal is also applying effort. In fact, the term μ2 ϕ∗b f z =f represents this discount. Proposition 1.2 and Proposition 1.3 also show that, unlike a single moral hazard setting, a double moral hazard setting requires not only MLRP and CDFC but also that the maximum reward for the agent must lie in a certain range, l < l1 ; which guarantees the monotonicity of the contract. This mainly comes from the fact that the incentives of both parties must be balanced in designing the contract under double moral hazard. If the maximum penalty against the principal is high (i.e. l > l1 ), then it is efficient to penalize the principal to the maximum extent possible (i.e. s∗ ðx Þ ¼ l ) and reward the agent also to the maximum extent possible when output is low (i.e. x < x3 ). Penalizing the principal and rewarding the agent when output is low will strengthen the principal’s incentive, but at the same time will weaken the agent’s incentive. However, the principal’s increased incentive can dominate the agent’s decreased incentive if the maximum penalty against the principal is sufficiently high, since the principal is risk neutral and the agent is risk averse. But this kind of incentive scheme will not work when the maximum penalty is not sufficiently high (i.e. l < l1 ).

6. The agency model with multiple agents This section is mainly based on Laffont (1995, 190–193). There are n agents i ¼ 1; 2; …; n; and one principal. Let x~i be the random output produced by agent i using effort ai : Let x ¼ ðx1 ; …; xn Þ and a ¼ ða1 ; …; an Þ: Let Ai be the effort space for ai : Let the joint distribution function be Fðx; a Þ and the density function be fðx ; aÞ: Given a system of payments si ðx Þ; the agent’s choices of actions are determined by a Nash equilibrium a∗ ¼ ða∗1 ; …; a∗n Þ; for which a∗i ∈ Arg max ui ½si ðx Þ fðx ; ai ; a∗−i Þdx  ci ðai Þ; ai ∈Ai

i ¼ 1; …; n:

Although there may be multiple equilibria, a unique solution is not necessary here. By the FOA, the principal’s problem is ) Z (X n v ½xi  si ðxÞ fðx; aÞdx max ai ; si ð Þ

Z

s:t: Z

i¼1

ui ½si ðxÞ f ðx; aÞdx  ci ðai Þ þ u; ui ½si ðxÞ fai ðx; aÞdx ¼ c0 i ðai Þ;

i ¼ 1; . . . ; n;

ð1:31Þ

i ¼ 1; . . . ; n:

The first issue is: under what circumstances does si ðx Þ depend only on xi ? We call s1 ðx1 Þ; …; sn ðxn Þ independent contracts. Following Holmström (1979), for n ¼ 2; given a2 ∈ A2 ; we say that x2 is uninformative about a1 if there is a function h such that f ðx1 ; x2 ; a1 ; a2 Þ ¼ hðx1 ; x2 ; a2 Þfðx1 ; a1 ; a2 Þ;

for all a1 ∈ A1 :

Complete contracts

19

That is, x1 is a sufficient statistic for a1 : The key here is that x2 and a1 are not related in a function. In this sense, observations in x2 cannot tell us anything about a1 :5 By this sufficiency condition, s1 ðx Þ does not depend on x2 : The basic intuition is simple enough. If x2 is not informative about a1 ; making the payment to agent 1 dependent on x2 will only increase his uncertainty without providing him with more incentives. But, if x2 is informative, then there is a trade-off between inducing effort and increasing uncertainty. Can the principal use the competition among the agents to her advantage? The literature examines a particular class of contracts called tournaments. Lazear and Rosen (1981) and Green and Stocky (1983) analyze a model incorporating multiple risk-averse agents, where each agent’s performance depends partly on his own effort, partly on his own luck and partly on a luck factor that affects all agents. Specifically, the output of agent i is xi ¼ ϕðai Þ þ εi þ ε; where ϕðai Þ is the expected output, εi is an individual-specific random factor, and ε is the common random factor. Assume that ε1 ; …; εn follow the same distribution and ε; ε1 ; …; εn are independent. The principal could offer each agent an independent contract si ðxi Þ in which each agent’s pay depends only on his own performance. However, Lazear and Rosen (1981) and Green and Stocky (1983) note that, in the real world, rewards are often based on relative performance – an employer may rank the performance of his workers and reward the better performer with a higher pay and promotion. Such a reward structure is called a tournament. Green and Stockey (1983) show that, when the number of agents is large, the order/ ranking of their outputs becomes a very precise estimator of the agents’ outputs net of common noises. Tournaments are thus approximately optimal. This leads to the second issue: under what conditions does a tournament yield better results than independent contracts? Lazear and Rosen (1981) and Green and Stocky (1983) show that the answer depends on the relative variability of the common luck factor versus those of the individual luck factors. If the individual luck factors have lower variances than the common luck factor, a tournament is better, and vice versa. This is intuitive. One weakness of a tournament is that it focuses on individuals’ average performance and ignores their luck factors. If the individual-specific factors have large variances, the tournament will often mistakenly offer some individuals too much and others too little, while independent contracts can take into account the variability of each individual properly and pay relatively reasonably. The third issue concerns the role of the principal when faced with a team of agents who work together for a single output. Their joint efforts affect output, and the contribution of each agent may be affected by other agents. In this case, Alchian and Demsetz (1972) argue that the role of the principal is to act as a monitor. Holmström (1982) develops a formal model, which demonstrates that the principal could play the role envisaged by Alchian and Demsetz (1972). Instead of a principal, as in (1.31), who designs and offers contracts to the agents, suppose that there is no principal and the n agents negotiate to reach an agreement for their joint production. Without a principal, there is a free-rider problem. To improve group performance, a penalty is

20 Complete contracts needed to guard against bad performance. But, without a principal, the threat of a penalty is not credible. The optimal penalty might be that they destroy some or all units for bad performance. But, once they have produced those units, they have no incentive to actually carry out the penalty themselves. If a principal is present, she can pay each individual based on his performance and pocket the rest. Faced with this credible threat, the team members would not slack. The principal has the incentive to monitor each individual closely and reward or punish him accordingly. Formally, suppose there are two agents, 1 and 2: If there is no principal, for verifiable output x; the two contracts s1 ðx Þ and s2 ðx Þ for the two agents need to satisfy s1 ðx Þ þ s2 ðxÞ ¼ x for any x: With a principal, the two contracts s1 ðx Þ and s2 ðx Þ are no longer constrained by s1 ðx Þ þ s2 ðxÞ ¼ x and the residual x  s1 ðx Þ  s2 ðx Þ goes to the principal. The pair of contracts s1 ðxÞ and s2 ðxÞ in the latter case can be designed more efficiently than in the former case. More specifically, with a principal, the optimization problem is: max

social welfare

s:t:

IC condtions;

a1 ;...;an s1 ;...;sn

ð1:32Þ

IR conditions: Without a principal, the optimization problem is: max

social welfare

s:t:

IC condtions;

a1 ;...;an s1 ;...;sn

ð1:33Þ

IR conditions; n X i¼1

si ðxÞ ¼

n X

xi :

i¼1

Obviously, problem (1.32) generally yields higher social welfare than (1.33).

7. Two-state agency models Two-state models are popular. For example, insurance models typically have only two states. With only two possible states, the agency problem becomes simpler. A good reference is the work of Mas-Colell et al. (1995, Chapter 14). Again, there is a principal who tries to hire an agent. The output can be either a high output xH or a low output xL : The probability of achieving xH is pðaÞ: The agent’s reservation utility is u : The private cost of effort is cðaÞ. 7.1. Verifiable effort Suppose that the principal pays wH for xH and wL for xL : The IR condition is pða ÞUðwH Þ þ ½1−pða ÞUðwL Þ  u þ cða Þ:

ð1:34Þ

Complete contracts Hence, if effort is verifiable, the principal’s problem is max pðaÞðxH  wH Þ þ ½1  pðaÞðxL  wL Þ a; wH ; wL

s:t: pðaÞUðwH Þ þ ½1  pðaÞUðwL Þ  u þ cðaÞ:

21

ð1:35Þ

The Lagrange function for (1.35) is L ¼ pðaÞðxH  wH Þ þ ½1  pðaÞðxL  wL Þ þ λ½pðaÞUðwH Þ þ ½1  pðaÞUðwL Þ  u  cðaÞ; where λ is a Lagrange multiplier and λ  0: The FOCs for ða; wH ; wL Þ are respectively p0 ðaÞðxH  wH  xL þ wL Þ þ λfp0 ðaÞ½UðwH Þ  UðwL Þ  c0 ðaÞg ¼ 0; pðaÞ ¼ λpðaÞU0 ðwH Þ; 1  pðaÞ ¼ λ½1  pðaÞU0 ðwL Þ; which imply λU0 ðwH Þ ¼ λU0 ðwL Þ ¼ 1;   c0 ðaÞ ¼ 0: xH −wH −xL þ wL þ λ UðwH Þ−UðwL Þ− 0 p ða Þ

ð1:36Þ

We immediately have λ ≠ 0 and   −1 1 : wH ¼ wL ¼ ðU0 Þ λ That is, the optimal salary w ∗ ∗ is independent of output and is w ∗ ∗ ¼ ðU0 Þ−1 ð1=λÞ: The institution is that, since the agent is risk averse and the principal is risk neutral, the principal takes all the risk. This is also indicated by the standard agency theory, as implied by (1.11). Since λ 6¼ 0; the IR condition (1.34) must be binding, implying Uðw ∗∗ Þ ¼ u þ cða ∗∗ Þ:

ð1:37Þ

Equation (1.36) then becomes xH  xL ¼ λ

c0 ða ∗∗ Þ p0 ða ∗∗ Þ

implying U0 ðw ∗ ∗ ÞðxH  xL Þ ¼

c0 ða ∗∗ Þ : p0 ða ∗∗ Þ

Equations (1.37)–(1.38) determine the first-best solution ða ∗∗ ; w ∗∗ Þ.

ð1:38Þ

22 Complete contracts 7.2. Nonverifiable effort With uncontractible effort, again suppose that the principal pays wH for xH and wL for xL : Given the wage contract, the agent considers his own problem: max pða ÞUðwH Þ þ ½1−pða ÞUðwL Þ  cða Þ; a

which implies the FOC: p0 ða Þ½UðwH Þ  UðwL Þ ¼ c0 ða Þ: The principal’s problem is to induce sufficient effort by considering the agent’s IC and IR conditions, i.e. max pðaÞðxH  wH Þ þ ½1  pðaÞðxL  wL Þ

a; wH ; wL

s:t: IC : p0 ðaÞ½UðwH Þ  UðwL Þ ¼ c0 ðaÞ;

ð1:39Þ

IR : pðaÞUðwH Þ þ ½1  pðaÞUðwL Þ  u þ cðaÞ: The Lagrange function for (1.39) is L ¼ pðaÞðxH wH Þþ ½1pðaÞðxL wL Þþfp0 ðaÞ½UðwH ÞUðwL Þc0 ðaÞg þ λ½pðaÞUðwH Þ þ ½1  pðaÞUðwL Þ  u  cðaÞ: where λ is a Lagrange multiplier and λ ≥ 0: The FOCs for ða; wH ; wL Þ are respectively p0 ðaÞðxH  wH  xL þ wL Þ þ fp00 ðaÞ½UðwH Þ  UðwL Þ  c00 ðaÞg ¼ 0; pðaÞ ¼ λpðaÞU0 ðwH Þ þ p0 ðaÞU0 ðwH Þ; 1  pðaÞ ¼ λ½1  pðaÞU0 ðwL Þ  p0 ðaÞU0 ðwL Þ; implying μ¼

p0 ðaÞðxH  wH  xL þ wL Þ 00 c ða Þ−p00 ða Þ½U ðwH Þ−U ðwL Þ ;

1 p0 ða Þ ; ¼ λ þ μ U0 ðwH Þ pða Þ 1 −p0 ða Þ : ¼ λ þ μ U0 ðwL Þ 1−pðaÞ The third equation implies λ ≠ 0; resulting in a binding IR condition. By Theorem 1.1, μ > 0: These three equations together with the IC and IR conditions determine ða∗ ; w∗H ; w∗L ; λ; μ Þ. By the IC condition, we have w∗H ≠ w∗L ; i.e. the agent shares some risks. The solution with contractible effort Pareto-dominates the solution with uncontractible effort, since the agent is indifferent but the principal is better off with contractible effort. In other words, problem (1.35) yields a higher profit than problem (1.39) since the former has fewer conditions.

Complete contracts

23

7.3. Example: insurance Consider a monopolistic insurance company with a single type of consumer. A consumer can influence his own probability of accident by putting in some preventive effort a; with private cost, which is increasing and convex. Assume that πða Þ is a decreasing function, indicating that a higher effort reduces the chance of an accident occurring. The company offers premium p for compensation z: Laffont (1995, 125–128) and Salanie (1999, 134–135) provide good references for this example. Insurance under Complete Information: The First Best If a is verifiable, the company can offer a contract of the form ða; p; z Þ and its problem is max p  ðaÞz p; z; a

s:t: ðaÞuðw  L þ z  pÞ þ ½1  ðaÞuðw  pÞ  u þ cðaÞ: The Lagrangian is L ¼ p  πðaÞz þ λfπðaÞuðw  L þ zpÞ þ ½1  πðaÞuðw  pÞ  ū  cða Þg;

where λ is a Lagrange multiplier and λ ≥ 0: The FOCs are 0¼

@L ¼ 1 þ λfðaÞu0 ðw  L þ z  pÞ  ½1  ðaÞu0 ðw  pÞg; @p

@L ¼ ðaÞ þ λðaÞu0 ðw  L þ z  pÞ; @z @L ¼ 0 ðaÞz þ λf0 ðaÞuðw  L þ z  pÞ  0 ðaÞuðw  pÞ  c0 ðaÞg; 0¼ @a

0¼

implying u0 ðw  L þ z  pÞ þ ðaÞ½u0 ðw  pÞ  u0 ðw  L þ z  pÞ ¼ u0 ðw  pÞ; zu0 ðw  L þ z  pÞ ¼ uðw  L þ z  pÞ  uðw  pÞ 

c0 ðaÞ : 0 ðaÞ

Then, if πðaÞ ≠ 1; we have u0 ðw  L þ z  pÞ ¼ u0 ðw  pÞ; implying full insurance: z ¼ L: That is, the company will offer full insurance under complete information. This is implied by the standard agency theory, in particular, by (1.5). The intuition is clear. Since the company is risk neutral and the consumer is risk averse, to avoid having to compensate the consumer for taking risk, the company takes all the risk. There is no incentive problem here, since the company can force the consumer to take action a ∗∗ : Then, u0 ðw  pÞ þ

1 c0 ðaÞ ¼ 0: L π 0 ða Þ

24 Complete contracts The IR condition must be binding, which implies p ¼ w  u − 1 ½ū þ cða Þ; and

1 c0 ðaÞ ¼ 0: u þ cðaÞ þ u0 u1 ½ L 0 ðaÞ This equation determines the optimal a ∗∗ under complete information. Insurance under Incomplete Information: The Second Best If effort a is not verifiable, so that the company cannot impose it in accident prevention, it can only offer incentives in a contract ða; p; z Þ to induce a desirable a. After accepting the contract, the consumer’s preferred effort a is determined by: max U  ðaÞuðw  L þ z  pÞ þ ½1  ðaÞuðw  pÞ  cðaÞ: a0

ð1:40Þ

The consumer’s marginal utility in accident prevention is: Ua ¼ π0 ðaÞ½uðw  L þ z  pÞ  uðw  pÞ  c0 ða Þ: If z ¼ L; we will have Ua < 0 for any a; implying that the consumer’s preferred effort is a ¼ 0: That is, with full insurance, the consumer would have no incentive to make any effort to prevent accidents from occurring. Obviously, the company cannot offer full insurance in this case. Hence, to provide incentives for accident prevention, the company ensures that the a in a contract must be the consumer’s preferred a. To do this, the company includes the FOC of (1.40) in its optimization problem [the SOC of (1.40) is automatically satisfied]. Hence, the company’s problem is max p  ðaÞz p; z; a

s:t: 0 ðaÞ½uðw  L þ z  pÞ  uðw  pÞ ¼ c0 ðaÞ; ðaÞuðw  L þ z  pÞ þ ½1  ðaÞuðw  pÞ  u þ cðaÞ: The IR condition must be binding. Thus the two constraints become uðw  L þ z  pÞ  uðw  pÞ ¼

c0 ða Þ ; π0 ðaÞ

πða Þ

c0 ðaÞ þ uðw  pÞ ¼ u þ cða Þ: π0 ðaÞ

Let x ≡ p  z: Then p  ðaÞz ¼ p  ðaÞðp  xÞ ¼ ½1  ðaÞp þ ðaÞx;   ðaÞ 0 1 p ¼ w  u u þ cðaÞ  0 c ðaÞ ;  ðaÞ   1  ðaÞ 0 1 c ðaÞ : x ¼ w  L  u u þ cðaÞ þ 0 ðaÞ

Complete contracts

25

Then the company’s problem becomes   ðaÞ max w  ðaÞL  ½1  ðaÞu1 u þ cðaÞ  0 c0 ðaÞ a  ðaÞ   1  ðaÞ 0 c ðaÞ :  ðaÞu1 u þ cðaÞ þ 0 ðaÞ This problem determines the optimal a* under incomplete information.

8. Linear contracts under risk neutrality Linear contracts are the simplest form of contracts, and they are very popular in applications. They offer a simple incentive mechanism. Examples of linear contracts are many: contractual joint ventures, equity joint ventures, crop-sharing contracts, fixed-price contracts, etc. Kim and Wang (1998) and Wang and Zhu (2005) provide some of the best references. A key theoretical question is: under what conditions is a linear contract optimal? When both parties in a contractual relationship are risk neutral, we say there is double risk neutrality; similarly, when both parties in a contractual relationship have moral hazard, we say there is double moral hazard. This section shows that, with moral hazard and uncertainty, a linear contract can be optimal only under double risk neutrality, and it can be the first best only under single moral hazard. From (1.21), we know that the optimal contract is generally nonlinear if one of the parties is risk averse. For example, if the principal is risk neutral with utility function uðz Þ ¼ z for all z; then (1.21) implies " 1 # f ðx; aÞ 1 a λþ : s ðxÞ ¼ ðu0 Þ fðx; aÞ For some popular output distributions, f a ðx ; a Þ=f ðx; a Þ is linear in x: For example, if output follows the normal distribution Nða; σ 2 Þ; then f a ðx ; a Þ=fðx; a Þ ¼ ðx  a Þ=σ 2 : If that’s the case, the optimal contract is nonlinear for a risk-averse agent. Hence, to find optimal linear contracts, we assume double risk neutrality in this section and specify utility functions uðzÞ ¼ z and vðz Þ ¼ z; for all z ∈ ℝ; for the agent and principal, respectively. 8.1. Single moral hazard Let a be the agent’s effort and f ðx; a Þ be the density function of output x: Let RðaÞ be the expected revenue RðaÞ ¼ ∫xf ðx; aÞdx. The first best With verifiable effort, the principal offers a contract of the form ða; sð⋅ÞÞ: Hence, the principal’s problem is

26 Complete contracts Z ½x  sðxÞ fðx; aÞdx  ¼ max s 2 S; a 2 A

Z sðxÞ fðx; aÞdx  cðaÞ þ u:

s:t:

Since the IR condition must be binding, this problem becomes a problem of social welfare maximization:  ¼ max RðaÞ  cðaÞ  u s2S; a2A

s:t:

Z sðxÞ f ðx; aÞdx  cðaÞ þ u:

Since the principal’s objective function is independent of the contract sðxÞ; this problem can be solved in two steps. First, by maximizing Rða Þ  cða Þ  ū; the first-best effort a ∗ ∗ is determined by R0 ða Þ ¼ c0 ða Þ: Second, given the optimal effort, an optimal contract will have to satisfy the IR condition only. To find an optimal contract, consider a fixed contract sðx Þ ¼ : By the binding IR condition, s ∗∗ ðx Þ ¼ cða ∗∗Þþu is such a contract. This is a first-best contract since it supports the first-best effort. The second best With unverifiable effort, the principal will still offer a contract of the form ða; sð⋅ÞÞ: But the principal has to provide incentives for the agent to accept this a. For this purpose, an IC is introduced. Hence, the principal’s problem is Z  ¼ max ½x  sðxÞ f ðx; aÞdx s 2 S; a2A

Z

s:t: IC :

sðxÞfa ðx; aÞdx ¼ c0 ðaÞ

Z sðxÞfðx; aÞdx  cðaÞ þ u:

IR :

Since the IR condition must be binding, this problem becomes a problem of social welfare maximization:  ¼ max RðaÞ  cðaÞ  u s2S; a 2A

Z

s:t: IC :

sðxÞfa ðx; aÞdx ¼ c0 ðaÞ

Z IR :

sðxÞfðx; aÞdx ¼ cðaÞ þ u:

ð1:41Þ

Complete contracts

27

Since the principal’s objective function is independent of sðxÞ; this problem can be solved in two steps. First, by maximizing RðaÞ−cða Þ−u ; the second-best effort a∗ is determined by R0 ða∗ Þ ¼ c0 ða∗ Þ; implying a∗ ¼ a ∗ ∗ : Second, given the optimal effort, an optimal contract will have to satisfy the IC and IR conditions only. We can find many contracts that can satisfy these conditions for a given a∗ : For a linear contract sðx Þ ¼ βx þ γ; the IC and IR conditions become Z Z β xfa ðx; a Þdx ¼ c0 ða Þ; β xfðx; a Þdx þ  ¼ cða Þ þ u; which imply β¼

c0 ða Þ ¼ 1; R0 ða Þ

 ¼ u þ cða Þ  βRða Þ ¼  :

This sðx Þ ¼ βx þ  is a second-best contract since it supports the second-best effort. Proposition 1.4. With single moral hazard and double risk neutrality, the linear contract s∗ ðx Þ ¼ x  π ∗ is optimal, implying the first-best effort a ∗∗ . Interestingly, this solution can be implemented by a change of ownership: the principal can simply sell the firm to the agent for payment π ∗ : When the agent is the owner, the incentive problem disappears. This solution leads to an important idea: incentive problems may be solved through an organizational approach. Coarse (1960) contributes precisely by proposing this idea. However, in this chapter, as we mentioned at the beginning, all contracts are treated as complete contracts. With complete contracts, an ownership transfer is not allowed. We will allow ownership transfers from the next chapter onwards when we use incomplete contracts. 8.2. Double moral hazard We have so far allowed the agent to invest only. In this subsection, we allow both parties in a joint venture to invest. Consider two agents, M1 and M2, engaged in a joint project. Efforts (investments) e1 and e2 respectively from M1 and M2 are private information. Let E be the effort space. Let c1(e1) and c2(e2) be the private costs of effort. Let h = (e1, e2) be the joint effort, and x~¼ Xð!; hÞ be the ex-post revenue depending on the state ! and joint effort h. Let x~follow a density function fðx; hÞ ex ante. Thus, the expected revenue is Z Rðe1 ; e2 Þ  xf ½x; hðe1 ; e2 Þdx: Assumption 1.7. h(e1,e2) is strictly increasing in e1 and . Assumption 1.8. c1 ðe1 Þ and c2 ðe2 Þ are convex and strictly increasing.

28 Complete contracts Assumption 1.9. Rðe1 ; e2 Þ is concave and strictly increasing in e1 and e2 . Given a contract ½s1 ðx Þ; s2 ðx Þ that specifies payments s1 ðx Þ and s2 ðxÞ to agents 1 and 2, respectively, the two agents play a Nash game to determine their efforts. In other words, given e2 ; agent 1 chooses his effort e1 by maximizing his own expected utility: Z s1 ðxÞ f ½x; hðe1 ; e2 Þdx  c1 ðe1 Þ; max e1 2E

which implies the FOC and SOC: Z h0 1 s1 ðxÞfh ½x; hðe1 ; e2 Þdx ¼ c0 1 ðe1 Þ; h00 1

Z

s1 ðxÞfh ½x; hðe1 ; e2 Þdx þ ðh0 1 Þ2

Z

s1 ðxÞfhh ½x; hðe1 ; e2 Þdx5c00 1 ðe1 Þ:

Similarly, given e1 ; agent 2 chooses his effort e2 by maximizing his own utility. Assume that contracting negotiation leads to social welfare maximum. This assumption is standard in the literature and is imposed on any negotiation outcome. Then the problem of maximizing social welfare can be written as V¼

max

si 2S; e1 ;e2 2E

s:t:

Rðe1 ; e2 Þ  c1 ðe1 Þ  c2 ðe2 Þ IC1 : h0 1 IC2 : h0 2

Z Z

s1 ðxÞfh ½x; hðe1 ; e2 Þdx ¼ c0 1 ðe1 Þ; s2 ðxÞfh ½x; hðe1 ; e2 Þdx ¼ c0 2 ðe2 Þ;

ð1:42Þ

SOC1 : for e1 ; SOC2 : for e2 ; RC : s1 ðxÞ þ s2 ðxÞ ¼ x; for all x 2 R þ ; where the last constraint is the resource constraint (RC). Since we allow a fixed transfer in the contract, IR conditions are unnecessary. The first-best problem is max Rðe1 ; e2 Þ  c1 ðe1 Þ  c2 ðe2 Þ

e1 ;e2 2E

s:t: RC : s1 ðxÞ þ s2 ðxÞ ¼ x; for all x 2 R þ ; where the contract needs to satisfy RC only. Proposition 1.5. Under Assumptions 1.7–1.9, with double moral hazard and double risk neutrality, there exists a linear output-sharing rule s∗i ðx Þ ¼ α∗i x;

Complete contracts where α∗i ¼

c i0ðe∗i Þ R i0ðe∗1 ;e∗2 Þ

29

; which induces the second-best efforts e∗i > 0 determined by

max Rðe1 ; e2 Þ  c1 ðe1 Þ  c2 ðe2 Þ

e1 ;e2 2E

s:t: R01 ðe1 ; e2 Þ ¼ c01 ðe1 Þ þ

h0 1 ðe1 ; e2 Þ 0 c 2 ðe2 Þ: h0 2 ðe1 ; e2 Þ

In addition, 0 < α∗i < 1; and the first-best outcome is not obtainable. Proof. Conditions IC1 and IC2 imply6 R01 ðe1 ; e2 Þ ¼ c01 ðe1 Þ þ

h0 1 ðe1 ; e2 Þ 0 c ðe2 Þ h0 2 ðe1 ; e2 Þ 2

ð1:43Þ

So the problem is equivalent to max

s1 2S; e1 ;e2 2E

s:t:

Rðe1 ; e2 Þ  c1 ðe1 Þ  c2 ðe2 Þ R0 1 ðe1 ; e2 Þ ¼ c0 1 ðe1 Þ þ

h0 1 ðe1 ; e2 Þ 0 c 2 ðe2 Þ; h0 2 ðe1 ; e2 Þ

ð1:44Þ

IC2 ; SOC1 ; SOC2 ; RC: This problem can be solved in two steps. First, we find a solution ðe∗1 ; e∗2 Þ from the following problem: max Rðe1 ; e2 Þ  c1 ðe1 Þ  c2 ðe2 Þ e1 ;e2 2E

s:t: R01 ðe1 ; e2 Þ ¼ c01 ðe1 Þ þ

h01 ðe1 ; e2 Þ 0 c 2 ðe2 Þ: h02 ðe1 ; e2 Þ

ð1:45Þ

This problem is not related to a contract. Second, given ðe∗1 ; e∗2 Þ; we look for a contract si ðx Þ that satisfies IC2 ; SOC1 ; SOC2 , and RC: There are many such contracts. Consider a simple sharing contract of the form si ðx Þ ¼ αi x for i ¼ 1; 2 with αi ¼

c 0i ðe∗i Þ : R 0i ðe∗1 ; e∗2 Þ

It is easy to verify that this contract satisfies IC2 . For RC, by (1.43), we have c 01 c 02 R 01 þ ¼ h 01 h0 2 h 01 Thus, α1 þ α2 ¼

c01 c02 c0 c0 R0 R01 R 1 R 2 R 1 þ ¼ ¼ 1: þ ¼ ¼ R01 R02 h01 xfh dx h02 xfh dx h01 xfh dx R01

Conditions IC2 and RC have now been verified. Since αi > 0 for both i ¼ 1 and 2; we must have 0 < αi < 1: Further, for contract si ðx Þ ¼ αi x; we have Z Z @2 @2 s xf ½x; hðe1 ; e2 Þdx ¼ α1 R001 ðe1 ; e2 Þ  0: ðxÞf ½x; hðe ; e Þdx¼ α 1 1 2 1 @e21 @e21

30 Complete contracts Hence, by Assumption 1.8 and Assumption 1.9, condition SOC1 is satisfied. Condition SOC2 can also be verified similarly. Finally, since the first-best solution ∗∗ ðe∗∗ 1 ; e2 Þ satisfies  0  0  R01 ðe 1 ; e2 Þ ¼ c1 ðe1 Þ ¼ c2 ðe2 Þ;

by condition (1.43), the solution ðe∗1 ; e∗2 Þ of problem (1.45) cannot be the first best. The proposition is thus proven. Q.E.D. The model in (1.42) is a model of two equal partners. An alternative setup is a principal–agent model. If one of the agents, say M1 ; is the principal and the other is the agent, we need to add an IR condition of the form Z s2 ðxÞf ½x; hðe1 ; e2 Þdx  c2 ðeÞ  u2 for the agent into problem (1.42). In this case, Preposition 1.5 still holds except that the optimal contract is a linear contract with the form s∗2 ðx Þ ¼ α∗2 x þ β2∗ ; where β2∗ is determined by the IR condition. In summary, an optimal linear contract exists under double risk neutrality; it is the first best under single moral hazard and it is the second best under double moral hazard. Example 1.3. Consider the following parametric case: hðe1 ; e2 Þ ¼ 1 e1 þ 2 e2 ;

~ ~ XðhÞ ¼ Ah;

ci ðei Þ ¼ e2i =2;

~ ¼ 1: The where μ1 ; μ2 > 0; A~ is a random variable with A~ > 0 and EðAÞ second-best solution is e1 ¼

31 ; 21 þ 22

e2 ¼

32 ; 21 þ 22

αi ¼

2i : 21 þ 22

The first-best solution is ∗∗ e∗∗ 1 ¼ μ 1 ; e 2 ¼ μ2 :

Q.E.D. There is no problem of risk sharing in a model under double risk neutrality, since both parties care about expected incomes only. Hence, under single moral hazard, when a single mechanism (a contract) is sufficient to handle the incentive problem, the optimal solution achieves the first best. However, under double moral hazard, when a single mechanism is not sufficient to handle the two incentive problems, the optimal solution cannot achieve the first best. For the optimal linear solution in Preposition 1.5, the value of α∗i reflects the relative importance of player i in the project. Example 1.3 indeed, shows this, where individual i’s marginal contribution to the project is represented by the parameter μi and the output share α∗i does indeed reflect his importance. Bhattacharyya and Lafontaine (1995) were the first to provide such a result in Preposition 1.5. Their result is for a special output process of the form x~ ¼ h þ ε~

Complete contracts

31

with a special distribution function Fðx ; h Þ ¼ Fε~ ðx  h Þ; where ε~ is a random shock with distribution function Fε~ : Kim and Wang (1998) were the first to provide this general theory on optimal linear contracts, with Wang and Zhu (2005) providing the proof for it. The optimality of linear contracts in this section is based on risk neutrality. Without double risk neutrality, linear contracts are generally not optimal. However, linear contracts are very popular in reality, and most parties involved are likely to be risk averse. This is puzzling within the framework of complete contracts. Our theory of incomplete contracts in Chapter 2 will provide a resolution to this puzzle. Linear contracts must be optimal incomplete contracts.

9. The one-step contract This section is based on Wang (2004). We now modify the standard agency model from yet another angle in an attempt to find a simple optimal contract. We have so far found a simple optimal contract, a linear contract, in a model under uncertainty only when there is no risk aversion. However, risk aversion is an important parameter in many applications. In this section, we find two conditions under which the agency model with risk aversion has a simple optimal contract known as a one-step contract. This contract is very convenient for applied agency problems. Real-world contracts between principals and agents are typically very simple and often have a multi-step bonus structure that specifies wage increases for certain minimum levels of performance. How can contracts be so simple in reality? We provide an answer by establishing the optimality of a one-step contract for the standard agency model under risk aversion. 9.1. The model We now abandon Assumption 1.1, and instead assume that the support of output is XðaÞ  ½AðaÞ; Bða Þ; where the boundaries of the support Aða Þ and BðaÞ are generally dependent on effort a: We allow the special cases with Aða Þ ¼ − ∞ and/or Bða Þ ¼ þ∞ for any a and those special cases with AðaÞ and/or Bða Þ being independent of a: We will also abandon the FOA. Given a utility function u : ℝþ ! ℝþ ; the agent’s expected utility is Z BðaÞ u½sðxÞ f ðx; aÞdx  cðaÞ: Uða; sÞ  AðaÞ

Hence, the principal’s problem is Z BðaÞ   max ½x  sðxÞfðx; aÞdx s2S; a2A

AðaÞ

s:t: IR :

Z

BðaÞ

u½sðxÞ f ðx; aÞdx  cðaÞ;

AðaÞ

IC : a 2 arg max Uð^ a; sÞ: a^

ð1:46Þ

32 Complete contracts Here, we assume that u ¼ 0: If u ≠ 0; we can replace cða Þ by Cða Þ ¼ cða Þ þ u and the model remains the same. This model is the same as the standard agency model except that we allow the boundaries of the domain to be dependent on effort. It turns out that this dependence is important for our alternative solution to the standard agency problem. Assumption 1.10. Utility function uðx Þ is concave, onto and strictly increasing. Bða Þ

Assumption 1.11. The expected revenue RðaÞ  ∫Aða Þ xfðx ; aÞdx is increasing and concave in a. Assumption 1.12 (FOSD). Fa ðx ; a Þ ≤ 0 for any x ∈ ℝ and a ∈ A. These three assumptions are natural requirements. In particular, u, being onto and strictly increasing, ensures that u − 1 ½cðaÞ is well defined, and Assumption 1.12 is required for any sensible contract theory. 9.2. The solution The following is the main result. Proposition 1.6. Let a∗ be the solution of the following equation: @u1 ½cða Þ ¼ R0 ða Þ; @a and suppose that the following two conditions are satisfied: @ ln u1 ½cða Þ @ ln cða Þ  ; @a @a  Fa ½Aða Þ; a  >

@ ln cða Þ : @a

ð1:47Þ

ð1:48Þ ð1:49Þ:

Then, under Assumptions 1.10–1.12, the optimal effort is and the optimal contract is  0 if x < Aða∗ Þ; ∗ s ðx Þ ¼ if x  Aða∗ Þ: u − 1 ½cða∗ Þ Furthermore, is the first-best solution. As shown in Figure 1.5, this solution looks distinctly different from the solution of the standard agency problem. In fact, it looks puzzling at first glance because of its simplicity. However, the intuition for the solution turns out to be quite simple. To induce the optimal effort a∗ ; the FOA suggests that the principal should associate each output level with a payment. This alternative approach suggests that the principal can induce a∗ by simply offering a bonus at a minimum level of performance. For the latter strategy to work, the dependence of the distribution function on effort at the left boundary of its domain is crucial. Condition (1.49) is precisely for establishing this dependence, which is, indeed, crucial for the solution.

Complete contracts

33

s*(x)

s

A(a*)

B(a*)

x

Figure 1.5 The optimal one-step contract.

We can immediately see some interesting features of the solution. First, the solution achieves the first best. This is impressive considering the simplicity of the contract. Second, it is a closed-form solution with a clear promotion component. In fact, the pay at the minimum level of performance is a jump, i.e. it could be in the form of a promotion, a bonus, or a change of nature of the contract. This solution looks very much like a typical employment contract, in which a fixed wage rate is given based on a certain level of education and working experience plus a potential bonus or promotion based on a minimum level of performance. Third, the solution has an interesting form with simple and intuitive expressions determining the optimal contract and effort. Based on the fact that s∗ ðx Þ ¼ u − 1 ½cða∗ Þ for x  Aða∗ Þ; (1.48) means that, for a percentage increase in a∗ ; the percentage increase in pay s∗ is greater than the percentage increase in cost. Condition (1.49) means that, for an increase in a∗ ; the increase in the probability of an output being larger than Aða∗ Þ is greater than the percentage increase in cost. We make some technical remarks about the conditions in Preposition 1.6. First, since −1 both cð⋅Þ and u − 1 ð⋅Þ are convex, u − 1 ½cðaÞ is convex in a; implying that ∂u ∂a½cða Þ is increasing in a: Also, since R0 ða Þ is decreasing in a; a∗ from (1.47) must be unique. Second, condition (1.48) means that the bonus elasticity of effort is larger than the cost elasticity of effort. Since u − 1 is convex, u − 1 ½cðaÞ is more convex than cðaÞ. This indicates that condition (1.48) can be easily satisfied for any a ∈ A: For example, for uðxÞ ¼ x1 − α and cða Þ ¼ aβ ; where α ∈ ½0; 1 is the relative risk aversion and β  1; (1.48) is satisfied for any a ∈ ℝþ. 9.3. Examples We consider two examples: one uses the exponential distribution and the other uses the uniform distribution. Example 1.4. Suppose uðxÞ ¼ x1α ;

cðaÞ ¼ aβ ;

1 xAðaÞ fðx; aÞ ¼ e a for x  AðaÞ; a

where Aða Þ is an arbitrary increasing function, α ∈ ½0; 1 is the relative risk aversion, and β  1: These functions include those in Example 1.1 as a special case. The key

34 Complete contracts difference between the two examples is that we allow the boundary Aða Þ to be dependent on effort. This dependence is crucial for our solution. We have β A0 ðaÞ RðaÞ ¼ AðaÞ þ a; Fa ½AðaÞ; a ¼  : u1 ½cðaÞ ¼ a1α ; a Equation (1.47) becomes β β ða Þ1α1 ¼ 1 þ A0 ða Þ: ð1:50Þ 1α Condition (1.48) is satisfied for any a  0: Condition (1.49) becomes β < A0 ða∗ Þ:

ð1:51Þ

Let AðaÞ ¼ γa for some γ41: Then (1.51) is satisfied for any a ∈ A and for β  γ; and (1.50) implies   1α   β ð1 þ Þð1  αÞ αþβ1 ð1 þ Þð1  αÞ αþβ1  ; s¼ : ■ a ¼ β β Example 1.5. Suppose 1 for AðaÞ  x  Aða Þ þ σ; σ where Aða Þ is an arbitrary increasing function, α ∈ ½0; 1 is the relative risk aversion, σ > 0; and β  1: We have β  1 u1 ½cðaÞ ¼ a1α ; RðaÞ ¼ AðaÞ þ ; Fa ½AðaÞ; a ¼  A0 ðaÞ: 2  Equation (1.47) becomes β β ða Þ1α1 ¼ A0 ða Þ: ð1:52Þ 1α Condition (1.48) is satisfied for any a  0: Condition (1.49) becomes 1 σ < a∗ A0 ða∗ Þ; β uðx Þ ¼ x1 − α ;

cðaÞ ¼ aβ ;

fðx; a Þ ¼

which is satisfied when σ is sufficiently small. Again, if we let AðaÞ ¼ a for some  41; then (1.52) implies   1α   β ð1  αÞ αþβ1 ð1  αÞ αþβ1  ; s ¼ : ■ a ¼ β β

10. A suboptimal linear contract In many published papers, researchers often use a limited set of admissible contracts for the standard agency model and find an optimal contract from this admissible set. In particular, researchers often limit admissible contracts to linear contracts and find an optimal linear contract from this set. This optimal contract is obviously inferior to the optimal contract in the standard agency model, and hence we call it a suboptimal linear contract or a third-best linear contract.

Complete contracts

35

Specifically, for the standard agency model in Section 4, let the set of admissible contracts be S ≡ fsðxÞjsðx Þis a linear function of the form sðxÞ ¼ α þ βx; where α; β ∈ ℝg: The principal’s problem can be written as Z v½ð1  βÞx  α f ðx; aÞdx max a; α; β2R

Z uðα þ βxÞ f ðx; aÞdx  cðaÞ þ u;

s:t: IR : Z

ð1:53Þ

uðα þ βxÞ fa ðx; aÞdx ¼ c0 ðaÞ:

IC :

The Lagrangian for (1.53) is Z  Z L  v½ð1  βÞx  α f ðx; aÞdx þ λ uðα þ βxÞ f ðx; aÞdx  cðaÞ  u Z þ

 uðα þ βxÞ fa ðx; aÞdx  c ðaÞ : 0

where λ and μ are Lagrange multipliers and λ  0: Under Assumption 1.1, the FOCs for problem (1.53) are Z  Z @L 00 ¼ v½ð1  βÞx  α fa ðx; aÞdxþ uðα þ βxÞfaa ðx; aÞdx  c ðaÞ ; 0¼ @a Z Z @L 0¼ ¼  v0 ½ð1  βÞx  α fðx; aÞdx þ λ u0 ðα þ βxÞfðx; aÞdx @α Z þ  u0 ðα þ βxÞ fa ðx; aÞdx; 0¼

@L ¼ @β

Z

þ

v0 ½ð1  βÞx  αxfðx; aÞdx þ λ Z

Z

u0 ðα þ βxÞxfðx; aÞdx

u0 ðα þ βxÞxfa ðx; aÞdx:

From these three equations, we can try to solve for ða∗ ; α∗ ; β∗ Þ: This solution is suboptimal and the suboptimal linear contract is s∗ ðx Þ ¼ α∗ þ β∗ x.

11. Linear contracts under certainty When there is no uncertainty, even in the presence of risk aversion, an optimal linear contract exists and it is the first best. We present this case since such a model or approach is popular in applications, even though it is trivial in theory.

36 Complete contracts Let the output space X ⊂ ℝ be a set of real numbers. Let A be the set of effort. Three functions are involved: the production function ϕ: A ! X ; the cost function c : A ! ℝ, and the utility function u : X ! ℝ: Assume ϕðaÞ > 0; ϕ0 ðaÞ > 0; ϕ00 ðaÞ  0; for all a 2 A; cðaÞ > 0; c0 ðaÞ > 0; c00 ðaÞ  0; for all a 2 A; uðxÞ > 0; u0 ðxÞ40; u00 ðxÞ50; for all x 2 X: Let S ¼ fs : X ! ℝþ js is Lebesgue integrableg: Notice that we have a concave utility function, which makes this model distinctly different from those of Bhattacharyya and Lafontaine (1995) and Kim and Wang (1998). Assume that a is unverifiable and output is not subject to uncertainty. Thus, output can be defined by a function x ¼ ϕðaÞ: Then the principal’s problem is 0  max ϕðaÞ  s½ϕðaÞ a2A; s2S

s:t: IR : ufs½ϕðaÞg  cðaÞ þ u; FOC : u0 fs½ϕðaÞgs0 ½ϕðaÞϕ0 ðaÞ ¼ c0 ðaÞ;

ð1:54Þ

SOC : ufs½ϕð^ aÞgis concave in a^: Lemma 1.1. The IR condition in (1.54) must be binding for the optimal solution. Proof. The proof turns out to be challenging to obtain. By the transformation of x≡ϕða Þ and letting Cðx Þ≡c ϕ − 1 ðx Þ; we find that problem (1.54) is equivalent to 0  max

x2X; s2S

x  sðxÞ

s:t: IR : u½sðxÞ  CðxÞ þ u; FOC : u0 ½sðxÞs0 ðxÞ ¼ C0 ðxÞ; SOC :

@ 2 u½sðxÞ 5C00 ðxÞ: @a2

Here, we have   c0 ϕ1 ðxÞ c0 ðaÞ C ðxÞ ¼ 0  1  ¼ 0 : ϕ ðaÞ ϕ ϕ ðxÞ 0

ð1:55Þ

Complete contracts

37

By letting fðx Þ ≡ u½sðxÞ; problem (1.55) is further equivalent to 0  max x  u1 ½fðxÞ x2X; f

s:t: IR : fðxÞ  CðxÞ þ u; FOC : f 0 ðxÞ ¼ C0 ðxÞ; SOC :

ð1:56Þ

@ 2 fðxÞ 5C00 ðxÞ: @x2

For problem (1.56), the IR condition must be binding. If not, given an optimal solution f ∗ ; we can find an ε > 0 such that f ðx Þ ¼ f ∗ ðx Þ  ε satisfies all the conditions but yields a higher payoff to the principal. This is a contradiction. Q.E.D. By Lemma 1.1, problem (1.54) becomes: max ϕðaÞ  u1 ½cðaÞ þ u

a2A; s2S

s:t: ufs½ϕðaÞg ¼ cðaÞ þ u; u0 fs½ϕðaÞgs0 ½ϕðaÞϕ0 ðaÞ ¼ c0 ðaÞ; ufs½ϕð^ aÞgis concave in a^: Then the problem can be solved in two steps. First, we find the optimal effort from max ϕðaÞ  u1 ½cðaÞ þ u: a2A

ð1:57Þ

This optimal effort is the first best. Second, given a; we find a contract that satisfies ufs½ϕðaÞg ¼ cðaÞ þ u;

ð1:58Þ

u0 fs½ϕðaÞgs0 ½ϕðaÞϕ0 ðaÞ ¼ c0 ðaÞ;

ð1:59Þ

ufs½ϕð^ aÞgis concave in a^:

ð1:60Þ

The FOC for (1.57) is

c0 ða0 Þ ; u0 u1 ½cða0 Þ þ u ¼ 0 ϕ ða0 Þ

ð1:61Þ

which is independent of the contract. This equation determines a0 : The following proposition determines the optimal linear contract. Proposition 1.7 (Optimal Linear Contract). (1) Given any level of effort a; there exists a unique linear contract ^ s ðx Þ ¼ α^ x þ β^ that satisfies the IR and FOC conditions (1.58)–(1.59), where α^ and β^ are constants defined by

38 Complete contracts

^¼ α

c0 ðaÞ ; u0 fu1 ½cðaÞ þ ugϕ0 ðaÞ

^ϕðaÞ: β^ ¼ u1 ½cðaÞ þ u  α

(2) If the level of effort is a0 defined by (1.61), then α^ ¼ 1 and the linear contract ^ s ðx Þ is optimal. Proof. For part (1), given contract s^ðx Þ ¼ α^ x þ β^ and effort a; (1.58)–(1.59) imply n o n o ^½ϕðaÞ þ β^ α ^ϕ0 ðaÞ ¼ c0 ðaÞ; ^½ϕðaÞ þ β^ ¼ cðaÞ þ u; u α u0 α implying ^ϕðaÞ; β^ ¼ u1 ½cðaÞ þ u  α

^¼ α

c0 ðaÞ : þ ugϕ0 ðaÞ

u0 fu1 ½cðaÞ

For part (2), by (1.61), for a ¼ a0 ; we immediately have α^ ¼ 1: Then, h i u½s^ðϕðaÞÞ ¼ u ϕðaÞ þ β^ ; which is obviously concave in a. Hence, the SOC (1.60) is satisfied. Hence, if the effort is optimal, the contract satisfies (1.58)–(1.60) and hence the contract is optimal. Q.E.D. Contracts generally involve two problems: incentive stimulation and risk sharing. In an uncertain environment with single moral hazard, if the agent is risk neutral, we can simply sell the firm to the agent, since the agent is willing to take all the risk at no extra cost, causing the moral hazard problem to disappear. This is stated in Preposition 1.4 which holds also in the standard agency model. If both parties have moral hazard problems, even though both are risk neutral, selling the firm will not solve the problem. In this case, although risk sharing is not an issue, we cannot hedge away the moral hazard problems simply by giving the firm to one of the parties involved. But a linear contract can be as good as any contract, as stated in Preposition 1.5. When the environment is certain, risk sharing is still not an issue. In this case, we can again sell the firm to the agent, even though he is risk averse, and the moral hazard problem will disappear, as stated in Preposition 1.7. Since linear contracts don’t allocate risk properly, only when risk sharing is not an issue can a linear contract be optimal.7 The first-best result is not surprising. Although the court cannot use the relationship x ¼ ϕðaÞ to infer a from x; the principal can. The principal can easily design a mechanism to punish the agent for not applying the desired amount of effort. It turns out that a linear contract can work.

Complete contracts

39

12. A recursive dynamic agency model This section is based on Wang (2003). A recursive dynamic agency model is developed for situations in which the state of nature follows a Markov process. The well-known repeated agency model developed by Spear and Srivastava (1987) is a special case of our model. It is found that the optimal effort depends not only on current performance but also on past performance, and the disparity between the two is a crucial determinant in the optimal contract. In a special case when both the principal and the agent are risk neutral, the first best can be obtained by a semilinear contract. In another special case of the repeated agency model, when the discount rate of time preferences converges to zero, the first best can again be obtained. For the general model, a computing algorithm that can be implemented in MathCAD is developed to find the solution numerically. Dynamic agency models have recently shown their power in explaining many economic phenomena and are increasingly being used in many fields, such as industrial organization, labor economics, health economics, insurance, and foreign trade; see, for example, Green (1987), Laffont (1995), and Wang (1995, 1997). However, existing infinite-period dynamic agency models are almost exclusively repeated agency models in which the underlying state of nature is an i.i.d. process. This type of model was first introduced by Spear and Srivastava (1987) and Green (1987), and the models have been subjected to many refinements and further developments in recent years; see, for example, Abreu et al. (1990), Thomas and Worrall (1990), Atkeson and Lucas (1992), and Wang (1995, 1997). In these models, although the agents do have concerns for future welfare, the nature of uncertainty simply repeats itself from period to period and the evolving nature of a typical dynamic model is missing. For example, the optimal effort stays the same throughout all contracting periods; it does not change according to the state of nature. This section develops a dynamic agency model that allows the underlying state of nature to be a Markov process. It includes the repeated agency model as a special case. By this, we have an infinite-period dynamic agency model that is suitable for a wide variety of agency problems. We call this type of dynamic agency model a recursive agency model, in contrast to a repeated agency model. The word “repeated” comes from repeated games. As we all know, the game of Prisoner’s Dilemma may yield the efficient solution if it is played repeatedly and has a small discount of future payoffs. Similarly, although the static principal-–agent model typically fails to reach the first best, it has been shown by Thomas and Worrall (1990), among others, that in a repeated agency model the efficiency of the optimal contract is improved and the contract can even reach the first best if there is no discounting of the future. We show that this result holds for the repeated version of our model. Our main interest in a recursive agency model has to do with the dynamic nature of the optimal contract. As we will show later, the optimal contract depends not only on current performance but also on past performance. Instead of contracting solely on current performance, the principal in a recursive dynamic setting may also use past information as a basis for contracting. In this way, the agent need not be rewarded handsomely for a good performance if the economy was in a good state during the last

40 Complete contracts period. The reason is that, with a serially correlated sequence of outputs, if the economy was in a good state during the last period, a good performance is likely due to a good state of the economy rather than a big effort. It is also found that the disparity between current and past performances is a crucial determinant in the optimal contract. The principal not only uses outputs to reward and punish, but also uses them to speculate on the effort inputs from the agent. In other words, the principal relies heavily on relative performance to improve efficiency in a dynamic contract. In repeated agency models, the state of nature is the best utility foregone for the agent. However, the best utility foregone is unobservable, which makes it difficult to test a theory empirically. In contrast, output and profit processes are observable. In this section, we assume that an agent’s utility value is not verifiable. Since outputs and profits are highly correlated over time in reality, especially for monthly data, by allowing the state of nature in an agency model to be Markovian, we can use an output or profit process to represent the state of nature and the results will be empirically testable. 12.1. The model Three agency models Consider a principal–agent relationship. A principal hires an agent to produce a product. The output x ∈ Y is random, where Y is an arbitrary Lebesgue-measurable set, Y⊂ℝ: For the static agency model, given the agent’s effort a ∈ A; where A is also an arbitrary Lebesgue-measurable set, A⊂ ℝ ; the output follows a process that can be described by a density function fðxjaÞ; defined for x ∈ Y: The principal’s preferences are described by a utility function vðx Þ of payoff x; and the agent’s preferences are described by a utility function uðx; aÞ of effort a and payoff x: We assume v0 > 0;

v00  0;

ux > 0;

uxx 50;

ua 50:

The principal can observe and verify the output x but cannot verify the effort a; she thus offers the agent a contract sðx Þ that is based on the output, where the contract is drawn from the following contract space: S 0 ¼ fs : Y ! ℝþ js is Lebesgue measurableg: Notice that we require sðx Þ≥0 and call it the limited liability condition. Finally, let Ū be the agent’s best alternative utility foregone if the agent accepts a contract from the principal. The following is the basic static agency model for the agency problem: Z v½x  sðxÞ fðxjaÞdx V  max a2A; s2S 0

Z

s:t: a 2 arg max 0 Z

a 2A

u½sðxÞ; a0 fðxja0 Þdx

 u½sðxÞ; afðxjaÞdx  U:

ð1:62Þ

Complete contracts

41

That is, the principal maximizes her own utility subject to the IC and IR conditions. This model has been extensively discussed in the literature; see, for example, Mirrless (1974), Holmstrom (1979), Grossman and Hart (1983), Holmstrom and Milgrom (1987), and Kim and Wang (1998). The solution of this basic model is generally the second best due to moral hazard on the agent’s part. To deal with the inefficiency of the second-best solution from (1.62), a repeated agency model is proposed. When the output process fyt g∞t ¼ 0 is i.i.d., Spear and Srivastava (1987) propose an infinite-period version of the agency problem:8 Z VðwÞ ¼ max fv½x  sðx; wÞ þ αV½Uðx; wÞg f ½xjaðwÞdx aðwÞ; Uð ;wÞ; sð ;wÞ

Z s:t: aðwÞ 2 arg max fu½sðx; wÞ; a þ βUðx; wÞgfðxjaÞdx Z

a2A

ð1:63Þ

fu½sðx; wÞ; aðwÞ þ βUðx; wÞg f ½xjaðwÞdx  w; for all w ∈ W ; where w is the best alternative utility foregone if the agent accepts a contract from the principal, and α and β are the principal’s and agent’s discount rates of time preferences, respectively. The solution will give four functions: að⋅Þ; Uð⋅;⋅Þ; sð⋅;⋅Þ, and Vð⋅Þ: There is also a large and recent literature on this model; see, for example, Rogerson (1985b), Abreu et al. (1990), Atkeson and Lucas (1992), and Wang (1995, 1997). In particular, Thomas and Worrall (1990) show that, in a repeated agency model, the efficiency of the optimal contract is improved and the contract can even reach the first best if there is no discounting of the future. It is natural to extend the dynamic agency model to a model that allows the state of nature to follow a Markov process. But what, then, is the proper formulation of the agency problem if the output process fyt g∞t¼0 is Markovian, i.e. yt þ 1 ~ f ð⋅jyt ; at Þ for a given density function f? Given a utility function Ū ðy Þ representing the best alternative foregone in the market for each output y; we will show that the following is a proper formulation of the agency problem: Z VðyÞ ¼ max fv½x  sðx; yÞ þ αVðxÞg f ½xjy; aðyÞdx aðyÞ; Uð Þ; sð ;yÞ

s:t:

aðyÞ 2 arg max Z

a2A

Z fu½sðx; yÞ; a þ βUðxÞg f ðxjy; aÞdx;

ð1:64Þ

fu½sðx; yÞ; aðyÞ þ βUðxÞg f ½xjy; aðyÞdx ¼ UðyÞ;  UðyÞ  UðyÞ; for all y ∈ Y: The above is a recursive agency model. The solution is a tuple of four functions ða; s; U; V Þ: In this case, the solution will obviously depend on the initial   is a given function, we do not need to express the utility w ¼ UðyÞ: But, since Uð⋅Þ  explicitly. That is, for the solution of (1.64), the contract’s dependence on U

42 Complete contracts  is implicit; instead, we use yt as the state of nature dependence of the solution on Uð⋅Þ at t and the solution will be specified as functions of the state. This treatment is consistent with the repeated model, in which w is treated as the state; in fact, (1.63) is  − 1 ðw Þ for a given w: This is due to the difference a special case of (1.64) with y ¼ U between the solutions of (1.63) and (1.64) – the solution of (1.63) is an open-loop solution while the solution of (1.64) is a closed-loop solution. The solution of our model (1.64) is time-consistent, due to the recursive nature of our model. Remark 1. We limit the output process to the first-order Markov process of the form yt þ 1 ~ f ð⋅jyt ; at Þ: It is straightforward to extend it to any finite-order Markov process. That is, our recursive model applies to any stationary random output process that depends on a fixed and finite number of periods of history. Remark 2. Recursive models cover a large variety of dynamic models. Economists almost never go beyond recursive models for theoretical analysis when an infiniteperiod model is used. Stokey and Lucas (1989) lay the foundation of recursive models for macroeconomics. Recursive models have two key advantages. First, the solutions are time-consistent. Second, they often provide tractable solutions with attractive characterizations. In contrast, nonrecursive models generally rely on numerical solutions. Remark 3. One problem with model (1.63) is that the principal uses Uðx; w Þ to represent the market condition in the next period. But the market condition can differ from Uðx; w Þ: For example, in a steady state, the market may still yield w in the next period, as opposed to Uðx; w Þ: The market condition is exogenously determined; the endogenously determined Uðx; w Þ cannot possibly reflect that. Such inconsistency does not exist in our model (1.64) where the market condition is represented by an exogenous process fyt g∞t ¼ 0 . The setup for the recursive model We now formally set up our model. At time t with a given output–action combination ðyt ; at Þ; denote yt þ 1 ~ fð⋅jyt ; at Þ to mean that yt þ 1 is conditionally distributed according to the density function f ð⋅jyt ; at Þ; i.e. fyt g∞t ¼ 0 is a first-order Markov process. We denote ℕ ¼ f0; 1; 2; …g as the set of all positive integers. Given a contract fst g∞t¼1 specifying a sequence of payments st to the agent, the utility function of the principal is v: ℝ ! ℝ; denoting the utility at t as vt ¼ vðyt −st Þ: The utility function of the agent is u: ℝþ A ! ℝ; denoting the utility at t as ut ¼ uðst ; at − 1 Þ: This is consistent with the standard treatment, for example, Spear and Srivastava (1987). The action at is aimed at the utility ut þ 1 in the next period. In addition, the action at is taken based on the realized state yt : Since the output process is a first-order Markov process, we will restrict the contract space to the following set:9 S ≡ s: Y 2 ! ℝþ js is Lebesgue measurableg:

Complete contracts

43

That is, since the output process is a first-order Markov process, we only need to consider feedback contracts that depend only on the current state of nature and the immediate past performance: st ¼ sðyt ; yt − 1 Þ: By this, we can transform a general dynamic agency problem into the recursive problem in (1.64). In this case, as we will show later, we must have at − 1 ¼ aðyt − 1 Þ: Then the action space is A ≡ fa: Y ! Aja is Lebesgue measurableg In summary, the sequences of variables fat ; st ; ut ; f t g are defined as at ¼ aðyt Þ; stþ1 ¼ sðytþ1 ; yt Þ; utþ1 ¼ uðstþ1 ; at Þ; ftþ1 ¼ fðytþ1 jyt ; at Þ: The expectation operator Et is defined by the Markov process fyt g∞t¼0 and the action function að⋅Þ; meaning that, for s  t and any Borel-measurable function : fY ! ℝ; we have Z Z Z ½ ðy Þ  ðy Þ f ½y jy ; aðy Þdy Eað Þ tþ1 tþ1 tþ1 t t tþ1 s að⋅Þ

f ½yt jyt1 ; aðyt1 Þdyt dysþ1 ; fa g

where ys is given. Similarly, the expectation operator Et τ is defined by the Markov process fyt g∞t ¼ 0 and the action series faτ g; meaning that, for s  t and any Borel-measurable function ψ : Y ! ℝ; we have Z Z Z fa g Es ½ ðytþ1 Þ  ðytþ1 Þfðytþ1 jyt ; at Þdytþ1 fðyt jyt1 ; at1 Þdyt dysþ1 : Also define

Z

Eay ½ ðxÞ 

ðxÞfðxjy; aÞdx:

Denote the history as ht ≡ ðy0 ; y1 ; …;yt Þ and L½Y  as L½Y ≡ fU : Y ! ℝjU is Lebesgue measurableg Furthermore, for convenience, we use the notations y ¼ yt and x ¼ yt þ 1 when there is no confusion. Consider a formal setting of the principal’s problem: 1 X faðht Þg max E0 αt1 v½yt  sðht Þ Vðy0 Þ ¼ fsðht Þg; faðht Þg

s:t:

t¼1

ytþ1  f ½ jyt ; aðht Þ;

8 t  0; fa g

faðht Þg 2 arg max1 E0 fa g2A

1 X

β1 u½sðh Þ; a 1 

¼1

s:t: yþ1  fð jy ; a Þ; faðht Þg

E0

1 X t¼1

   0 Þ; βt1 u sðht Þ; aðht1 Þ  Uðy

8 y0 2 Y:

44 Complete contracts Since we only consider feedback contracts in S with the form st ¼ sðyt ; yt − 1 Þ; the problem can be rewritten as 1 X faðht Þg max E0 αt1 v½yt  sðyt ; yt1 Þ Vðy0 Þ ¼ sð ; Þ2S; faðht Þg

t¼1

ytþ1  f ½ jyt ; aðht Þ;

s:t:

8 t  0; fa g

faðht Þg 2 arg max1 E0 fa g2A

1 X

β1 u½sðy ; y1 Þ; a1 

¼1

s:t: yþ1  fð jy ; a Þ; faðh Þg

E0

t

1 X

ð1:65Þ

 0 Þ; βt1 u½sðyt ; yt1 Þ; aðht1 Þ  Uðy

t¼1

8 y0 2 Y: The first objective is to prove that (1.64) and (1.65) are the same. The result is stated in Proposition 1, and its proof is provided in the appendix.  ∈ L½Y ; any solution of (1.65) must be a Proposition 1.8 (Recursiveness). Given U solution of (1.64), where for (1.64), a ∈ A; s ∈ S and V; U ∈ L½Y . Alternative setups There are two alternative setups for (1.64). In the first alternative setup, the IR condition is satisfied only for the initial market constraint. In this case, given a  of the best alternative opportunity at the start t ¼ 0 of the contract, utility level U the problem can be expressed as Z VðyÞ ¼

max

að Þ2A; Uð Þ2L½Y ; sð ; Þ2S

fv½x  sðx; yÞ þ αVðxÞgf½xjy; aðyÞdx

Z

s:t: aðyÞ 2 arg max Z

fu½sðx; yÞ; a þ βUðxÞgfðxjy; aÞdx;

a2A

ð1:66Þ

 fu½sðx; yÞ; aðyÞ þ βUðxÞgf½xjy; aðyÞdx ¼ UðyÞ; Uðy0 Þ  U:

In this case, the contract ensures acceptance at t ¼ 0; but it may need enforcement in  at a future date t; the agent would want to break the contract. the future. If Uðyt Þ < U In the second alternative setup, we ignore outside competition altogether and consider a simpler version: Z VðyÞ ¼ max fv½x  sðx; yÞ þ αVðxÞgf½xjy; aðyÞdx að Þ2A; Uð Þ2L½Y ; sð ; Þ2S

Z

s:t:

aðyÞ 2 arg max Z

a2A

fu½sðx; yÞ; a þ βUðxÞgfðxjy; aÞdx;

fu½sðx; yÞ; aðyÞ þ βUðxÞgf½xjy; aðyÞdx ¼ UðyÞ:

ð1:67Þ

Complete contracts

45

Our model (1.64) is the most restrictive version, (1.66) is less restrictive and (1.67) is the least restrictive. Our model (1.64) guarantees the acceptance of the contract by the agent at any state, (1.66) guarantees the acceptance of the contract at the initial state, while (1.67) does not guarantee anything. Consequently, the contract from (1.64) is time-consistent and thus needs no enforcement, but the contracts from (1.66) and (1.67) may need enforcement when the market condition changes. If (1.64) yields a solution, then (1.66) and (1.67) must yield solutions too. We will consider only model (1.64) in the rest of the section. 12.2. Analysis In this section, we investigate a few properties of model (1.64). Assumption 1.13 (A Separable Utility Function). uðs; aÞ ¼ uðs Þ−cða Þ. By Assumption 1.13, assuming that the FOA is valid, (1.64) becomes Z VðyÞ ¼ max fv½x  sðx; yÞ þ αVðxÞgfðxjy; aÞdx a2A; s2S

Z

s:t: Z

 gfa ðxjy; aÞdx ¼ c0 ðaÞ; fu½sðx; yÞ þ βUðxÞ

ð1:68Þ

 gfðxjy; aÞdx ¼ cðaÞ þ UðyÞ:  fu½sðx; yÞ þ βUðxÞ 0

0

Þ Let s ¼ hðx; z Þ be the unique solution of vuðx−s ¼ z for z > uv0 ð0ðxþÞÞ : h is well defined 0 ðs Þ and is strictly increasing in z (see the proof of Preposition 1.9). Let ( 0 hðx; zÞ; if z > uv0 ð0ðxÞþ Þ ; ϕðx; zÞ  0; otherwise:

Proposition 1.9 (The Optimal Contract). Under Assumption 1.13, assuming the FOA is valid, the optimal contract is  fa ð1:69Þ sðx; yÞ ¼ ϕ x; λðyÞ ½x j y; aðyÞ þ ðyÞ ; f where λðyÞ and μðy Þ are respectively the Lagrange multipliers of the IC and IR conditions in (1.62). Preposition 1.9 suggests that, given an immediate past performance y; the solution formula for the contract is exactly the same as that for the static model. In the special case when the principal is risk neutral, we have hðz Þ ¼ ðu0 Þ − 1 ð1=zÞ and ( hðzÞ; if z > u0 ð01 þ Þ ; ϕðzÞ ¼ 0; otherwise:

46 Complete contracts Then



fa sðx; yÞ ¼ ϕ λðyÞ ½xjy; aðyÞ þ ðyÞ : f

For example, if uðz Þ ¼

1 1−

1 

z1 −  ; where γ is the relative risk aversion (RRA), we

have hðzÞ ¼ z and u ∘ hðzÞ ¼

1 1−

1

z  − 1 for z > 0: For this special u to have concave

u ∘ h; we need   21 . The monotonicity of sðx; y Þ in current performance x; i.e. the positivity of λðy Þ; is guaranteed by the following proposition. Proposition 1.10 (Monotonicity). Under Assumption 1.13, assuming the FOA is valid and ffa ðxjy; aÞ is increasing in x; we must have λðyÞ > 0 and μðy Þ  0; for all y ∈ Y. Consider the repeated version of the recursive model, for which the output process fyt g∞t¼0 follows yt þ 1 ~ f ð⋅jat Þ; instead of yt þ 1 ~ f ð⋅jyt ; at Þ; for a given density function f : In this case, Ū ðy Þ; V ðy Þ and f ðxjy; aÞ are independent of y: Then (1.64) becomes Z 1 V ¼ max v½x  sðxÞ fðxjaÞdx a2A; s2S 0 1α Z ð1:70Þ s:t: a 2 arg max u½sðxÞ; a0  fðxja0 Þdx; 1 1β

Z

a0 2A

 u½sðxÞ; a fðxjaÞdx ¼ U:

Proposition 1.11 (First Best by Repetition). If α ¼ β; when α ! 1; the solution of  the repeated recursive model (1.70) converges to the first best with U¼0. 12.3. The recursive agency model under risk neutrality A semi-linear contract Linear contracts are very popular in applications of contract theory. An optimal secondbest linear contract was first found by Bhattacharyya and Lafontaine (1995) and a theory for optimal linear contracts was later established by Kim and Wang (1998). For the static model under risk neutrality and single moral hazard, a linear contract exists. And not only is this linear contract simple and intuitive, it is also the first best. Is there a similar contract for a dynamic model? The answer lies in the following proposition. Proposition 1.12 (A Semi-Linear Contract). If both the principal and the agent are risk neutral, assuming the FOA is valid, then there exists a semi-linear contract of the following form that induces the first-best effort a∗ ðy Þ:

Complete contracts sðx; yÞ ¼ ’ðyÞx þ ðyÞ;

47

ð1:71Þ

where ’ðy Þ and ðy Þ are defined by R   c0 ½a ðyÞ  β UðxÞf a ½xjy; a ðyÞdx R ’ðyÞ ¼ ;  xfa ½xjy; a ðyÞdx Z   ðyÞ ¼UðyÞ þ c½a ðyÞ  β UðxÞ f ½xjy; a ðyÞdx

ð1:72Þ

R  Z xf ½xjy; a ðyÞdx 0    R c ½a ðyÞ  β UðxÞfa ½xjy; a ðyÞdx ;  xfa ½xjy; a ðyÞdx where Vð⋅Þ and a∗ ð⋅Þ are determined by Z   fa ½xjy; a ðyÞdx ¼ c0 ½a ðyÞ; ½x þ αVðxÞ þ βUðxÞ Z

  f ½xjy; a ðyÞdx ¼ VðyÞ þ UðyÞ  ½x þ αVðxÞ þ βUðxÞ þ c½a ðyÞ:

ð1:73Þ

Given the solution in (1.71), under what conditions is the FOA valid? We find that, if ∫Ū ðx Þfaa ðxjy; a Þdx  0 and ∫xfaa ðxjy; aÞdx  0 for any ða; y Þ ∈ A Y; and ’ðyÞ  0 for any y ∈ Y; then the SOC for the IC condition of a∗ ðyÞ is satisfied for any y ∈ Y; i.e. the FOA is valid. Example 6. Let yt ¼ at þ yt1 þ "t ;

"t  Nð0; 2 Þ;

 UðyÞ ¼ by;

1 cðaÞ ¼ a2 ; 2

where b  0 and σ > 0 are constants. Then Z ðxayÞ2 1 Rðy; aÞ  xfðxjy; aÞdx ¼ a þ y; fðxjy; aÞ ¼ pffiffiffiffiffiffi e 22 ; 2 By Prepositon 1.12, the solution is 1 þ bðβ  αÞ ; a ¼ 1α   1  αbð1  βÞ 1  bð1  βÞ 1 1 þ bðβ  αÞ 2 sðx; yÞ ¼ ; x y 1α 1α 2 1α VðyÞ ¼

1  ð1  βÞb ½1 þ bðβ  αÞ2 : yþ 1α 2ð1  αÞ3

The FOA is valid in this case. If there is a discount on the future ðβ ≠ 1Þ; the two parties share outputs. ■ Many real-world contracts have simple income-sharing rules that evolve over time. For example, there are typically two types of joint ventures in foreign direct investment (FDI): equity joint ventures (EJVs) and contractual joint ventures (CJVs). In an EJV, the two firms, a local firm and a foreign firm, agree upon a specific division of equity ownership, which determines their profit shares. The equity division usually

48 Complete contracts doesn’t change during the whole cooperation period. On the other hand, in a CJV, profit shares for all the contracting years are specified on a yearly basis and the shares often change yearly. That is, EJVs typically have a constant sharing rule over the contracting years, while CJVs tend to have a variable sharing rule. The evolution of sharing rules over time is an interesting topic, although there is little research on it because of its complexity. One exception is the work of Tao and Wang (1998), who discuss the well-known, puzzling phenomenon of increasing the local firm’s share of revenue over the contracting years for CJVs in China. In a special case of our semi-linear contract when the output grows at a constant rate r; we find that the agent’s share of revenue sðx; y Þ=x is increasing in output, which is consistent with the observations of Tao and Wang (1998). In addition, this revenue share converges to a constant as output goes to infinity. The repeated model under risk neutrality If we replace the density function f ðxjy; a Þ by f ðxja Þ; then we have a repeated model, which is a special case of Preposition 1.12. Let Rða Þ  ∫xfðxja Þdx be the  Þ, and Vðy Þ are independent of y; and revenue function. In this case, f ðxjy; aÞ; Uðy thus Preposition 1.12 immediately gives the solutions:     þ cða ÞRða Þ; V ¼ Rða Þcða Þð1βÞU ; sðxÞ ¼ x þ ð1  βÞU 1α ð1:74Þ where a∗ is determined by R0 ða∗ Þ ¼ c0 ða∗ Þ: As expected, when α ¼ β ¼ 0; we have the well-known first-best solution for the static model. The Spear and Srivastava model is a repeated model, which can be compared with the repeated version of the recursive model. The Spear and Srivastava model under risk neutrality gives the following solution: Rða Þ  cða Þ  w; sðx; wÞ ¼ x þ w  β u þ cða Þ  Rða Þ; VðwÞ ¼ 1β ð1:75Þ where a∗ is again determined by R0 ða∗ Þ ¼ c0 ða∗ Þ; and u ≡Uðx; w Þ is a constant. Note that, to avoid inconsistency, we must impose α ¼ β for the Spear and Srivastava model under risk neutrality; also, u is undetermined and can be any arbitrary number. The solution in (1.74) of the repeated recursive model is the same as the solution  and α ¼ β: Our model is in (1.75) of the Spear and Srivastava model if w ¼ u ¼ U more general, since we allow α ≠ β. 12.4. Computation A computing algorithm By the analysis in Section 12.2, solving model (1.64) requires finding five functions ½sðx; yÞ; aðy Þ; VðyÞ; λðyÞ; μðy Þ; which are determined by the following five equations:

Complete contracts

49

Z VðyÞ ¼ Z Z

fv½x  sðx; yÞ þ αVðxÞgf ½xjy; aðyÞdx;

 gfa ½xjy; aðyÞdx ¼ c0 ½aðyÞ; fu½sðx; yÞ þ βUðxÞ  g f ½xjy; aðyÞdx ¼ UðyÞ  þ c½aðyÞ; fu½sðx; yÞ þ βUðxÞ 

sðx; yÞ ¼ ϕ x; λðyÞ R

λðyÞ ¼

ð1:76Þ



fa ½xjy; aðyÞ þ ðyÞ ; f

fv½x  sðx; yÞ þ αVðxÞgfa ½xjy; aðyÞdx R  gfaa ½xjy; aðyÞdx; c00 ½aðyÞ  fu½sðx; yÞ þ βUðxÞ

for all y ∈ Y: That is, the solution of (1.64) must satisfy these equations. Because there is generally no closed-form solution for (1.76), we now propose a computing algorithm that can be used to find a numerical solution. The computing procedure yields fast convergence of computation and is easy to implement. For simplicity, assume that the principal is risk neutral; a risk-averse principal will only slightly increase computational complexity. We also assume finite states; this assumption is needed for a numerical solution. Assumption 1.14 (A Risk-Neutral Principal). The principal is risk neutral. Assumption 1.15 (Finite States). Y is finite. Given these two assumptions, let Y ¼ ðy1 ; …;yn Þ and i  Uðy  i Þ; ai  aðyi Þ; sij  sðyj ; yi Þ; Vi  Vðyi Þ; U pij ðai Þ  fðyj jyi ; ai Þ;

i  ðyi Þ; λi  λðyi Þ:

Then the discrete time version of (1.76) is n  P Vi ¼ yj  sij þ αVj pij ðai Þ; j¼1

n   P j p0 ij ðai Þ ¼ c0 ðai Þ; uðsij Þ þ βU

j¼1

n   P j pij ðai Þ ¼ U i þ cðai Þ; uðsij Þ þ βU

j¼1

h 0 i p ða Þ sik ¼ ϕ λi pijij ðaii Þ þ i ; k ¼ 1; . . . ; n;  P yj  sij þ αVj p0ij ðai Þ j  P ; λi ¼ 00 j p00ij ðai Þ uðsij Þ þ βU c ðai Þ  j

50 Complete contracts

for i ¼ 1; …; n: These five equations determine five variables ai ; sij ; Vi ; λi ; μi Þ for i; j ¼ 1; …; n. Starting with a given s ∈ ℝn n ; we compute the solution in five simple steps based on the five equations; each step involves one equation. 1.

Given s, solve for a from X  j p0 ðai Þ ¼ c0 ðai Þ; uðsij Þ þ βU ij

i ¼ 1; . . . ; n:

ð1:77Þ

j

2.

Each ai is determined independently by one equation in (1.77). Given a and s; calculate using 0 B B V¼B B @

1  11 0 P v yj  s1j p1j ða1 Þ C B j C αp21 ða2 Þ 1  αp22 ða2 Þ αp2n ða2 Þ C C B C B . C B .. C: .. .. .. .. C B C A  . . . . A @P v yj  snj pnj ðan Þ αpn1 ðan Þ αpn2 ðan Þ 1  αpnn ðan Þ j

1  αp11 ða1 Þ αp12 ða1 Þ αp1n ða1 Þ

ð1:78Þ 3.

Given a; s and V; calculate λ by  P vðyj  sj Þ þ αVj p0ij ðai Þ j  P λi ¼ 00 j p00ij ðai Þ ; uðsij Þ þ βU c ðai Þ 

i ¼ 1; . . . ; n:

ð1:79Þ

j

4.

Given a and λ; solve for μ from  0  X p ij ðai Þ þ i pij ðai Þ u ϕ λi pij ðai Þ j X j ¼ U i þ cðai Þ; pij ðai ÞU þβ

i ¼ 1; . . . ; n:

ð1:80Þ

j

5.

Each μi is determined independently by one equation in (1.80). Finally, given a; λ and μ, calculate contract s by  0  p ij ðai Þ þ i ; i; j ¼ 1; . . . ; n: sij ¼ ϕ λi pij ðai Þ

ð1:81Þ

We can then go back to step 1 with the new s from step 5 for another round of calculations. We can start from an arbitrary contract sð0Þ ∈ ℝn n ; find a contract sð1Þ by following the five steps, find sð2Þ by following the five steps again, and so on until the contract stabilizes after N such cycles, i.e. until sðN Þ and sðN þ 1Þ are practically equal. The limiting s must satisfy the equations in (1.76). By this procedure, a computer can quickly find a solution at each step if one exists, and the sequence of

Complete contracts

51

solutions will typically converge very quickly. For the numerical presented next, the five-step system typically converges after about ten cycles. A numerical example We choose A ¼ ℝþ ; Y ¼ fy1 ; y2 g with y1 < y2 ; and fðy1 jy; aÞ ¼ eay ; fðy2 jy; aÞ ¼ 1  eay ; 1 ¼ 10; 2 ¼ 10:1; U U α ¼ 0:98; β ¼ 0:98;  ¼ 3; uðsÞ ¼ 1 ð1  es Þ; vðyÞ ¼ y; cðaÞ ¼ 12 a2 : That is, there are only two possible states, y1 and y2 ; the bad and good states respectively, and the principal is risk neutral. Such choices of functions for a numerical solution are typical in contract theory. With these choices, increases in effort a or past performance y will increase the chance of obtaining the higher output y2 : We also have fa ðy1 jy; aÞ ¼ −y; f

fa ye − ay ðy2 jy; aÞ ¼ ; f 1−e − ay

which yields ffa ðy2 jy; a Þ > ffa ðy1 jy; aÞ: We further have hðzÞ≡ 1 lnðz Þ; for z > 1. In the benchmark case, we choose y1 ¼ 1 and y2 ¼ 2; and the solution is       0:15 0:273 0:533 39:92 a¼ ; s¼ ; V¼ ; 0:17 0:610 0:786 39:68     0:38 2:65 λ¼ ; ¼ : 0:63 7:49 Generally, we choose either y1 ¼ 1 or y1 ¼ 1:5; and a continuum of y2 : y2 ∈ ð1:5; ∞ Þ: The solutions for ai and sij are depicted in the following six diagrams. The figures show those parts for which a solution exists. We make a few observations from the diagrams. First, no matter whether the economy is in a bad or a good state, the effort will always increase with an increase in the good-state output y2 ; unless the limited liability condition for when a good state is followed by a bad state ðy2 ! y1 Þ forces the principal to reduce payment for when a good state is followed by a good state ðy2 ! y2 Þ: There are two reasons for this result. A higher output when the economy is in a good state makes it worthwhile for the agent to work hard, since the pay increase will compensate for his cost of a higher effort. Furthermore, an increased performance gap between the two states will also make it easier for the principal to relate output to effort. Second, the contract payment to the agent is reduced when the output stays on the bad state for two consecutive periods. The reason may be that the two consecutive bad performances strengthen the principal’s belief that the agent is not working

52 Complete contracts a1

a2

0.3

0.35

y1 = 1.5

y1 = 1.5

0.3

0.25

0.25 0.2

y1 = 1

y1 = 1

0.2

0.15 0.15 y2 0.1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

y2 0.1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

s11

s12

0.3

1.3 1.2 1.1

y1 = 1

y1 = 1.5

1

0.25

0.9

y1 = 1

0.8 0.7 0.2

0.6 y1 = 1.5

0.5 0.4

y2 0.15 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 s21

y2 0.3 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 s22

0.7

1

0.6 0.9

0.5 0.4

y1 = 1

y1 = 1

0.8

0.3

0.7

y1 = 1.5

0.2 0.1

y1 = 1.5

y2 0 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

0.6 0.5 y 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 2

hard. But the reduction levels off when the good state is exceptionally good. A sharp difference in outputs for the two states leads the principal to believe that two consecutive bad performances can still be due to bad luck rather than low effort, since a small increase in effort with negligible cost to the agent could lead to a large performance improvement and a large reward – the agent should have realized this and should have chosen to work hard if the state were good. Symmetrically, the contract payment to the agent increases in the case of y2 ! y2 unless the limited liability condition limits the punishment of the agent in the case of y2 ! y1 . Third, the contract payment to the agent increases substantially if the output changes from being bad to good; conversely, the contract payment to the agent drops dramatically if the output changes from being good to bad. The better the good state, the larger are the reward and punishment. When limited liability limits

Complete contracts

53

the punishment of the agent, the principal reduces payment in the case of y2 ! y2 to compensate for the case of y2 ! y1 . Fourth, an increase in the bad-state output y1 will increase the effort in the bad state but reduce the effort in the good state. Since an increase in the bad-state output narrows the gap between the performances in the two states, the agent will distribute his effort more evenly between the two states. In essence, there is a substitution of efforts between the two states; efforts will be more evenly distributed if outputs are more evenly distributed. Finally, in contrast with the fourth point, when the output in the bad state is increased, the contract payment will be reduced for the cases of y1 ! y1 ; y2 ! y2 and y1 ! y2 ; but increased for the case of y2 ! y1 : This can be explained by the fact that an improvement of performance in the bad state reduces the performance gap between the two states, which leads to a more evenly distributed contract pay scheme. Notice that the increase in payment in the case of y2 ! y1 is marginal even if the improvement in the bad-state output is substantial. 12.5. Discussion This section proposes a dynamic agency model based on a Markov state process. The model is a generalization of the existing repeated agent model. We derive the solution equations and discuss some characteristics of the solution. Two special cases for which the first best can be achieved are discussed. The semi-linear contract is of particular interest. Judging from the popularity of the linear contract for the static model, we expect many applications for the semi-linear dynamic contract. However, a closed-form solution is unavailable for the general model, which is also true for the static standard agency model. We thus propose a computing algorithm using the well-known Mathcad software. Mathcad is powerful yet very user-friendly, and runs on the Microsoft Windows platform. A numerical example is given using the computing algorithm, from which we can observe a few characteristics of a dynamic contract. Many difficult technical problems remain to be tackled. First, we need an existence result for the solution of the equation set (1.76). This amounts to finding a domain of output y within which a solution of (1.76) is guaranteed. Second, we only know that any solution of problem (1.64) must be a solution of equation set (1.76); we need to establish the uniqueness of a solution for (1.76) to guarantee that any solution of (1.76) is in fact a solution of (1.64). Since problem (1.64) itself has already imposed quite stringent conditions, we expect equation set (1.76) to have a unique solution under mild conditions. Third, for the computing algorithm, a difficult issue is the convergence of the iterative process. This also amounts to finding a domain of output y within which a solution of the five-step algorithm converges. Fourth, in a dynamic model, since past performance may reveal a lot of information to the principal, as our numerical example clearly shows, there may exist many other types of first-best contracts, other than the one for the repeated model. Finally, renegotiation is not allowed in our model, just as in the typical infinite-horizon dynamic contract models in the literature.

54 Complete contracts

Appendix This appendix provides the proofs for all the propositions. Proof of Preposition 1.8 The objective is to prove that (1.64) and (1.65) are the same. Given a well-defined s ∈ S; let ( fas g1 s¼t

E Ut ðyÞ ¼ max 1 fas gs¼t

 )   βs  t1 u½sðys1 ; ys Þ; as1 yt ¼ y  s¼tþ1 1 X

ð1:82Þ

We need to show that Ut ðyÞ is time independent. We have  )   Ut ðyÞ ¼ max E β u½sðys1 ; ys Þ; as1 yt ¼ y  fas g1 s¼t s¼tþ1  ( )  1 X  st fas g1 ¼ max E s¼t1 β u½sðys1 ; ys Þ; as1 yt1 ¼ y  fas g1 s¼t1 s¼t (

fas g1 s¼t

1 X

st1

¼ Ut1 ðyÞ; where the second equality is obtained by replacing t by t−1: Here, the fact that fyt g∞t ¼ 0 is a first-order Markov process is critical. Notice that we have made use of the fact that the conditional density function f ð⋅jy; aÞ for yi þ 1 given yi ¼ y and ai ¼ a is the same as the conditional density function f ð⋅jy; aÞ for yi given yi − 1 ¼ y and ai − 1 ¼ a: Therefore, Ut − 1 ðyÞ is time independent. We can then rewrite (1.82) as 1 X fas g1 s¼t E βst1 u½sðys1 ; ys Þ; as1  Uðyt Þ ¼ max t 1 fas gs¼t

¼ max 1

fas gs¼t

¼ max at

s¼tþ1

( fa g1 Et s s¼t

u½sðyt ; ytþ1 Þ; at  þ

þ max 1

fas gs¼tþ1

) β

st1

u½sðys1 ; ys Þ; as1 

s¼tþ2

 Eat t

1 X

u½sðyt ; ytþ1 Þ; at  fa g1 Etþ1s s¼tþ1 β

1 X

β

st2

u½sðys1 ; ys Þ; as1 

s¼tþ2

¼ max Eat t fu½sðyt ; ytþ1 Þ; at  þ βUðytþ1 Þg: at

This first implies that a∗t ≡ aðht Þ ¼ aðyt Þ for any t: Second, we can then rewrite (1.65) as

Complete contracts

að Þ

Vðy0 Þ ¼ max

a2A; s2S

E0

1 X

αt1 v½yt  sðyt1 ; yt Þ

t¼1

s:t: ytþ1  f ½ jyt ; aðyt Þ; aðyÞ 2 arg max a2A

55

8 t  0;

Eay fu½sðx; yÞ; a

þ βUðxÞg s:t: x  fð jy; aÞ

ð1:83Þ

UðyÞ ¼ EaðyÞ y fu½sðx; yÞ; aðyÞ þ βUðxÞg; where x  fð jy; aÞ  Uðy0 Þ ¼ U: Similarly, let Vt ðyÞ ¼

( max

að Þ2A; sð Þ2S

s:t:

E

1 X

að Þ

s¼tþ1

α

st1

 )   v½ys  sðys1 ; ys Þyt ¼ y 

the constraints in ð1:88Þ:

We need to show that Vt ðyÞ is time independent. We have  ( )  1 X  að Þ st1 max E α v½ys  sðys1 ; ys Þyt ¼ y Vt ðyÞ ¼  að Þ2A; sð Þ2S s¼tþ1 s:t: ¼

max

að Þ2A; sð Þ2S

s:t:

the constraints in ð1:88Þ  ( )  1 X  að Þ st E α v½ys  sðys1 ; ys Þyt1 ¼ y  s¼t the constraints in ð1:88Þ

¼ Vt1 ðyÞ; where the second equality is obtained by replacing t by t−1: We have again made use of the fact that fyt g∞t¼0 is a first-order Markov process. Thus, Vt ðy Þ is time independent. Then, (1.83) becomes 1 X að Þ max E0 αt1 v½yt  sðyt1 ; yt Þ Vðy0 Þ ¼ að Þ2A; sð Þ2S

t¼1

( ¼

max að Þ2A; sð Þ2S

að Þ E0

v½y1  sðy0 ; y1 Þ þ

( ¼ ¼

max að Þ2A; sð Þ2S

max

að Þ2A; sð Þ2S

að Þ E0

v½y1 sðy0 ; y1 Þþ

1 X

) α

t1

v½yt  sðyt1 ; yt Þ

t¼2 að Þ αE1

1 X

) α

t2

t¼2 að Þ

E0 fv½y1  sðy0 ; y1 Þ þ αVðy1 Þg:

We have thus proved that (1.64) and (1.65) are the same.

v½yt sðyt1 ; yt Þ

56 Complete contracts Proof of Preposition 1.9 The Lagrange function of (1.68) is Z L ¼ fv½x  sðx; yÞ þ αVðxÞgf ½xjy; aðyÞdx Z   gfa ½xjy; aðyÞdx  c0 ½aðyÞ þ λðyÞ fu½sðx; yÞ þ βUðxÞ Z

ð1:84Þ

   fu½sðx; yÞ þ βUðxÞg f ½xjy; aðyÞdx  c½aðyÞ  UðyÞ :

þ ðyÞ

The Hamilton function for the Lagrange function is  fa 0  H ¼ vðx  sÞ þ αVðxÞ þ λðyÞ ½uðsÞ þ βUðxÞ ½xjy; aðyÞ  c ½aðyÞ f   þ ðyÞfuðsÞ þ βUðxÞ  c½aðyÞ  UðyÞg: Then the FOC for sðx; y Þ is Hs ¼ λðyÞu0 ðsÞ

fa ½xjy; aðyÞ þ ðyÞu0 ðsÞ  v0 ðx  sÞ ¼ 0; f

which implies v0 ð x  s Þ fa ¼ λðyÞ ½xjy; aðyÞ þ ðyÞ: 0 u ðsÞ f This equation gives a contract of the form s ¼ sðx; y Þ: By this, since v00 ðx−sÞ≤0 and u00 ðs Þ < 0; the SOC is satisfied: fa Hss ¼ v00 ðx  sÞ þ λðyÞu00 ðsÞ ½xjy; aðyÞ þ ðyÞu00 ðsÞ f 0 v ðx  sÞ 50: ¼ v00 ðx  sÞ þ u00 ðsÞ u0 ðsÞ Then, by Kim and Wang (1998), the FOC is sufficient for optimality. 0 v0 ðx−s Þ ¼ z for z > uv0 ð0ðxþÞÞ : h is well Let s ¼ hðx; z Þ be the unique solution of 0 ðs Þ u    defined on 0

v0 ðx Þ u0 ð0þ Þ

;

v0 ð0þ Þ u0 ðx Þ

: Since

v0 ðx−s Þ u0 ðs Þ

∂ ∂s

v ðx−s Þ u0 ðs Þ

¼− v0 0 u0 þu00 v0 ðu0 Þ2

> 0; the solution of

¼ z for s is unique; it also shows that h is strictly increasing in z: Let ( 0 hðx; zÞ; if z > uv0 ð0ðxÞþ Þ ; ϕðx; zÞ  0; otherwise: 0

Þ þ In the definition of ϕ; the case z≤ v ðx u0 ð0 Þ corresponds to the case with 0

λðyÞ ffa ½xjy; aðyÞ þ μðyÞ≤ v uðx0 Þ ð0þ Þ ; since v0 ð x−sÞ u0 ðs Þ

v0 ðx−sÞ u0 ðs Þ

is strictly increasing in s; we

½xjy; aðyÞ þ μðyÞ < for any s > 0 ; i.e. Hs < 0 for s > 0: have Thus, in this case, s ¼ 0 is the optimal solution. Preposition 1.9 is thus proven. λðyÞ ffa

Complete contracts

57

Proof of Proposition 1.10 Consider

Z

VðyÞ ¼ max Z s:t: Z

að Þ; sð Þ

fv½x  sðx; yÞ þ αVðxÞgf½xjy; aðyÞdx

 gfa ½xjy; aðyÞdx ¼ c0 ½aðyÞ; fu½sðx; yÞ þ βUðxÞ

ð1:85Þ

 gf½xjy; aðyÞdx  c½aðyÞ þ UðyÞ:  fu½sðx; yÞ þ βUðxÞ

We now prove that (1.68) and (1.85) are equivalent, i.e. they have the same solution (s). Obviously, any solution of (1.68) must be a solution of (1.85). Conversely, if a solution of (1.85) satisfies Z  g f ½xjy; aðyÞdx ¼ c½aðyÞ þ UðyÞ;  ð1:86Þ fu½sðx; yÞ þ βUðxÞ then it must also be a solution of (1.68). But suppose there is a solution ða0 ; s0 Þ of (1.85) that satisfies Z fu½s0 ðx; y Þ þ βŪ ðx Þgf ½xjy; a0 ðyÞdx > c½a0 ðy Þ þ Ū ðyÞ: Then we can find a positive number εðy Þ > 0 such that Z fu½s0 ðx; y Þ−εðy Þ þ βŪ ðxÞgf ½ xjy; a0 ðyÞdx  c½a0 ðy Þ þ Ū ðy Þ: We can then choose sðx; y Þ  u − 1 fu½s0 ðx; yÞ−εðyÞg for which Z fu½sðx; y Þ þ βŪ ðx Þgf ½xjy; a0 ðyÞdx−c½a0 ðyÞ  Ū ðyÞ; and

Z

 gfa ½xjy; a0 ðyÞdx fu½sðx; yÞ þ βUðxÞ Z

¼

 gfa ½xjy; a0 ðyÞdx ¼ c0 ½a0 ðyÞ: fu½s0 ðx; yÞ þ βUðxÞ

That is, ða0 ; sÞ satisfies the two constraints in (1.85) and yet pays the agent less, which must be strictly better off for the principal. This contradicts the fact that ða0 ; s0 Þ is a solution for (1.85). Thus, (1.86) must be satisfied. Therefore, (1.68) and (1.85) are equivalent. We can thus treat the Lagrange function defined in (1.84) as the Lagrange function for (1.85), in which case we must have μðy Þ ≥ 0 by the Lagrange Theorem.

58 Complete contracts Finally, by (1.69) and the increasingness of ffa ½xjy; aðy Þ in x; λðy Þ must be positive, otherwise the contract would be decreasing. Also, λðy Þ cannot be zero since a constant contract is not optimal. Preposition 1.10 is thus proved. Proof of Preposition 1.11 In this case, Ū ðyÞ; V ðy Þ and f ðxjy; a Þ are independent of y: Then (1.64) becomes Z V ¼ max fv½x  sðxÞ þ αVgfðxjaÞdx a; sð Þ

Z

s:t: a 2 arg max 0 Z

a 2A

gfðxjaÞdx ¼ U:  fu½sðxÞ; a þ βU

That is, V ¼ max a; sð Þ

gfðxja0 Þdx; fu½sðxÞ; a0  þ βU

1 1α

Z v½x  sðxÞ f ðxjaÞdx Z

s:t: a 2 arg max 0 1 1β

Z

a 2A

u½sðxÞ; a0  f ðxja0 Þdx;

 u½sðxÞ; a f ðxjaÞdx ¼ U:

For convenience, we assume that the FOA is valid and uðs; a Þ ¼ uðsÞ−cða Þ: Then the model becomes Z 1 v½x  sðxÞ fðxjaÞdx V ¼ max 1α a; sð Þ Z s:t: u½sðxÞ fa ðxja0 Þdx ¼ c0 ðaÞ; Z

 u½sðxÞ fðxjaÞdx ¼ cðaÞ þ ð1  βÞU:

The Lagrangian is Z  Z 1 L¼ v½x  sðxÞ f ðxjaÞdx þ λ u½sðxÞfa ðxja0 Þdx  c0 ðaÞ 1α   Z 1 1  þ u½sðxÞ f ðxjaÞdx  cðaÞ  U ; 1β 1β which is equivalent to Z  Z 0 0 0 u½sðxÞfa ðxja Þdx  c ðaÞ L ¼ v½x  sðxÞfðxjaÞdx þ λð1  αÞ þ

  Z 1α 1α  : u½sðxÞ f ðxjaÞdx  cðaÞ  ð1  αÞU 1β 1β

Complete contracts

59

Thus, if α ¼ β; when α ! 1; the limit of the solution is determined by the following Lagrangian: Z  Z u½sðxÞ fðxjaÞdx  cðaÞ ; Llim ¼ v½x  sðxÞ f ðxjaÞdx þ  which is the Lagrangian of the problem: Z v½x  sðxÞ f ðxjaÞdx max a; sð Þ

s:t:

Z u½sðxÞ f ðxjaÞdx ¼ cðaÞ:

The solution of this maximization problem is the first best, with Ū ¼ 0. Proof of Preposition 1.12 Assuming that the FOA is valid, problem (1.64) becomes Z ½x  sðx; yÞ þ αVðxÞ f ½xjy; aðyÞdx VðyÞ ¼ max aðyÞ; sð ;yÞ

Z

s:t: Z

  fa ½xjy; aðyÞdx ¼ c0 ½aðyÞ; ½sðx; yÞ þ βUðxÞ

ð1:87Þ

  f ½xjy; aðyÞdx ¼ UðyÞ  ½sðx; yÞ þ βUðxÞ þ c½aðyÞ:

Using the second condition in (1.87), we can eliminate sðx; y Þ from the objective function so that the problem becomes Z   f ½xjy; aðyÞdx  UðyÞ  ½x þ αVðxÞ þ βUðxÞ  c½aðyÞ VðyÞ ¼ max aðyÞ; sð ;yÞ

Z

s:t: Z

  fa ½xjy; aðyÞdx ¼ c0 ½aðyÞ; ½sðx; yÞ þ βUðxÞ

ð1:88Þ

  f½xjy; aðyÞdx ¼ UðyÞ  ½sðx; yÞ þ βUðxÞ þ c½aðyÞ:

Since the objective function is not affected by the contract, we can solve the problem in two steps: we first find the optimal effort from the following problem: Z   f ðxjy; aÞdx  UðyÞ  ½x þ αVðxÞ þ βUðxÞ  cðaÞ; ð1:89Þ VðyÞ ¼ max a2A

then, given the optimal effort a∗ ðy Þ from the above problem, we find a contract sðx; yÞ that satisfies the two conditions in (1.88): R   fa ½xjy; a ðyÞdx ¼ c0 ½a ðyÞ; ½sðx; yÞ þ βUðxÞ R ð1:90Þ   f½xjy; a ðyÞdx ¼ UðyÞ  ½sðx; yÞ þ βUðxÞ þ c½a ðyÞ:

60 Complete contracts Since a∗ is in fact the solution of the following problem without the IC condition: Z VðyÞ ¼ max ½x  sðx; yÞ þ αVðxÞ f ½xjy; aðyÞdx aðyÞ; sð ;yÞ

s:t:

Z

 f½xjy; aðyÞdx ¼ UðyÞ  ½sðx; yÞ þ βUðxÞ þ c½aðyÞ;

a∗ achieves the first best. Problem (1.89) implies the two equations in (1.73), which determine Vð⋅Þ and a∗ ð⋅Þ: For (1.90), consider a contract of the form: sðx; y Þ ¼ ’ðyÞx þ ψðy Þ:

ð1:91Þ

Substituting this into (1.90) yields ’ðyÞ and ψðy Þ in (1.72). Therefore, there exists a semi-linear contract of the form in (1.91) that can induce the first-best effort.

2

Incomplete contracts

1. The foundation of incomplete contracts 1.1. What is an incomplete contract? In the standard agency theory, contracts deal with income rights only. For a long time, economists have been perplexed by just how complex contracts are in theory when they are really quite straightforward. The reason for this discrepancy turns out to be simple: contracts are incomplete in reality. As we will explain in this subsection, a contract is an incomplete contract if it contains not only income rights but also control rights. Control rights are the rights to take an action or make a decision; income rights are the rights to claim income. In reality, contracts may include many other items and features, such as penalties, termination rights, promotions, bonuses, options, etc. The general view of a contract is that it is a blueprint that defines a trade, such as the exchange of labor, goods, or security for money. When many contingencies are involved in a trade, which is often the case, we expect highly complicated contracts. However, contracts in reality are typically very simple and are observed to be highly incomplete. Of course, by this we are referring only to the written part of the contract; the ensuing games that the participating parties may play could be quite complex indeed. Conceptually, a complete contract is a contract that specifies rights and obligations in every possible future state of the world; otherwise it would be an incomplete contract. For example, if there are only two possible states, good or bad, a contract that specifies payments for both states is a complete contract, while one that specifies a payment for the good state only is an incomplete contract. That is,
wg if good state; Complete contract : sc ðxÞ ¼ wb if bad state;
Incomplete contract :

si ðx Þ ¼

wg ?

if good state; if bad state:

In contract si ðxÞ; there is no mention of a payment for the bad state. However, under rationality, as long as the principal is aware that the world could evolve into a bad state, she will have a plan for it. For example, her plan may be that, if the world

62 Incomplete contracts really does evolve into a bad state, she decides an action at that time. She may not explicitly mention her plan of action in a contract. If by common practice the principal has the default right to decide if a state not described in a contract is realized, then she has no need to mention her plan in the contract. The contract in the latter case is:
s0 ðx Þ ¼

wg the principal decides ex post

if good state; if bad state:

This contract is complete. From this example, we can see that, if the trading parties are aware of all the states, their contract must be complete. That is, although realworld contracts may appear incomplete, by economic rationality and awareness,1 they must in fact all be complete. This explanation of incomplete contracts is consistent with Ben-Shahar’s (2004) argument that “by its legal definition a ‘contract’ cannot be incomplete” and Baker (2006)’s argument that “contracts are always obligationally complete.” In other words, on the one hand, since the contract si ðx Þ doesn’t specify payments for all the states, it is an incomplete contract. On the other hand, since the principal is fully aware that the world could evolve into a bad state, she must have a plan for it. In this sense, it is a complete contract. In other words, an “incomplete” contract is only incomplete in appearance; any contract under full awareness and rationality must be complete. The study of incomplete contracts is to uncover the missing items in an incomplete contract. An incomplete contract can be written in two ways: either it lists obvious items only, or it lists not only obvious items but also other items. An empty contract is an incomplete contract that lists obvious items only. By this argument, there are no incomplete contracts, only ones that appear to be incomplete but which are in fact complete in the sense that all possibilities would have been strategically taken care of by all parties. In theory, a description of incomplete contracts is tricky. Some contracts seem to have deliberately omitted terms (called unwritten terms) that could affect players’ behaviors. When this happens, a contract is considered incomplete. This is our general understanding of an incomplete contract. An incomplete contract is like a glass that is half full. The glass appears to be half full when in fact it is full, since the other half is filled with air. By this argument, all glasses are full, since they are all either completely filled with drinks or partially filled with something else. The difference is in whether a glass is filled with one matter or with multiple matters. By analogy, all contracts are complete, since they are all filled either with written statements or with a combination of written statements and other mechanisms. To put it another way, a contract is complete if the written part of the contract is the only mechanism in a contractual relationship; a contract is incomplete if its written part is only one of the mechanisms in a contractual relationship. Here, we emphasize the written part of a contract which could consist of a written part and an unwritten part. In the literature, the unwritten part is often neglected in discussions of incomplete contracts. We in this book, however, are interested in the unwritten part and sometimes specify it in a contract.

Incomplete contracts

63

In the presence of other mechanisms, an incomplete contract appears to be simple (simple in its written statements, but complicated in the underlying games the players are involved in since other mechanisms are involved), while a complete contract tends to be complicated since the written statements must deal with all problems. For example, an empty contract is incomplete since other mechanisms besides this contract must also play a role; in fact, in this case the contract is saying “since I am not doing anything, you guys (other mechanisms) have a bigger role to play.” However, in a model setting, we must clearly specify the unwritten part of a contract. A contract is considered incomplete if it omits the unwritten part; it is considered complete if it includes the unwritten part. In fact, when we say that all contracts are complete in reality, this is what we mean. Hence, in a world where economic agents are fully aware of everything (which is our assumption), a contract can be either complete or incomplete; the only difference is in the definition or appearance. Therefore, to clearly distinguish between these two concepts, a narrower definition is often used, as we do in this book: a contract is complete if it contains statements on income rights only; a contract is incomplete if it involves other rights. In this sense, we call the contracts in Chapter 1 complete contracts. We assume that income rights alone are enough to deal with all incentive problems, and, following Coase (1960), we recognize that other rights can also be used to deal with incentive problems. Then, what is the formal definition of incomplete contracts in this book? Welfare is the ultimate purpose of having a contractual relationship. In a model where welfare is derived solely from income, income is the ultimate purpose, while other rights are there to assist income rights. As explained above, if an income-sharing rule is incomplete, there must be other mechanisms to be introduced by the trading parties to deal with the incompleteness or to complete the contract. Hence, in this book, a contract is said to be incomplete if its income-sharing rule is considered incomplete. In other words, an incomplete contract is a contract that contains not only an income-sharing rule but also other mechanisms. By the same argument, a complete contract is a contract in which the income-sharing rule is capable of handling all possible contingencies so that additional mechanisms are unnecessary. To put it simply, a contract is complete if its income rights are complete and a contract is incomplete if its income rights are incomplete. Hence, we have the following formal definition of contracts: A complete contract ¼ fan income-sharing ruleg: An incomplete contract ¼ fan income-sharing rule þ other mechanismsg: In other words, a complete contract is a contract that contains an income-sharing rule only. If other mechanisms are present in a contract, we have an incomplete contract. For example, contract s0 ðx Þ defined above is an incomplete contract by our definition, although it is a complete contract by the conceptual description given earlier. Note that some incomplete contracts may not contain an income-sharing rule at all.

64 Incomplete contracts The income-sharing rules in Chapter 1 are called complete contracts, since each of those rules is supposed to take into account all possible contingencies without the assistance of any other mechanisms. Under economic rationality, since the principal does not employ any other mechanisms, the income-sharing rule must have taken into account all contingencies. Put differently, the contracts in Chapter 1 are called complete contracts because they are derived based on the assumption of a complete contract. That is, only an income-sharing agreement is allowed in each contract; any other mechanism is forbidden. For example, a change of ownership in the standard agency model is not allowed, or else there would not be an incentive problem. Why are contracts in reality so simple? Our definition of incomplete contracts can easily explain this. A real-world contract is viewed as simple since its incomesharing rule is simple. A real-world contract is simple because there are many mechanisms at work to assist the income-sharing rule. In our definition, a contract is said to be incomplete if there are multiple mechanisms at work. Therefore, realworld contracts are simple, since they are incomplete contracts. In our definition, an incomplete contract is an equilibrium outcome of a more complete setting. When the model setting is more complete (with many mechanisms at work simultaneously), the income-sharing rule can be very simple. In particular, if an allocation of control rights is a supplementary mechanism to an income-sharing rule, the income-sharing rule can be very simple indeed. And, with the assistance of other mechanisms, an optimal contract can be as simple as a linear sharing rule. For example, suppose that a project is financed in stages by a bond and efficiency is achieved. In this case, the efficient contract can be written as either C 1 ¼ fbondg

or C 2 ¼ fbond; staged financingg:

C 1 is obviously incomplete and is what the literature refers to as an incomplete contract, while C 2 is complete within the current setting. This difference in definition is why we say that incomplete contracts are actually complete. The fact that a simple contract such as a bond can achieve efficiency in this case is because of the existence of a second mechanism: staged financing. Contracts can be simple or they can look highly incomplete in the presence of other mechanisms. To be consistent with the tradition in the literature, we call C 2 an incomplete contract. The difference between our incomplete contract approach and that in the literature is that we explicitly identify the unwritten part of the contract while the literature tries to explain the incompleteness of C 1 mainly with transaction costs. The incompleteness is actually only in appearance. In other words, contracts may appear incomplete due to the presence of other mechanisms, rather than transaction costs as emphasized in the literature. Our definition of incomplete contract is consistent with the approach adopted by Wang and Zhou (2004), Qiu and Wang (2009), and Wang (2009), in which a simple income-sharing rule is optimal with the assistance of other mechanisms. According to transaction cost economics (Williamson 1975; Klein et al. 1978), institutions (firms in particular) are important since they are part of an incomplete contract. According to property rights theory (Grossman and Hart 1986; Hart and Moore

Incomplete contracts

65

1990), control rights (ownership and property rights in particular) are important since they are part of an incomplete contract. Hart argues that, if an incomesharing rule is very simple, control rights can play a role. In contrast, we argue that, since control rights are part of a contract, the income-sharing rule can be very simple. What is the mechanism-design approach? According to Mattesini and Monnet (2010), this approach “in general, begins by describing an economic environment, including preferences, technologies, and certain frictions – by which we mean spatial or temporal separation, information problems, commitment issues, and so on. One then studies the set of allocations that are attainable, respecting both resource and incentive feasibility constraints. Sometimes one also describes allocations that are optimal according to particular criteria. One then looks at these allocations and tries to interpret the outcomes in terms of institutions that can be observed in actual economies.” That is, the mechanism-design approach aims at explaining real institutions, focusing on the consistency between real institutions and theoretical solutions. Take banking institutions as an example: taking preferences, technologies, and frictions as primitives, the problem is whether or not banking institutions can arise endogenously. The incomplete contract approach is a mechanism-design approach that emphasizes contractual relationships, instead of the traditional problems in information revelation and transmission. In contrast, the general-equilibrium (GE) approach aims at building a theoretical foundation of the markets as opposed to institutions. Scholars tend to focus on two major issues with incomplete contracts: the reason for economic agents to choose an incomplete contract over a complete contract, and the optimal degree of incompleteness of a contract. Wang and Zhou (2004), Qiu and Wang (2009), and Wang (2009) tackle the first issue. Qiu and Wang (2009) explain a popular contract in private–public joint ventures using a contract theory by which contractual incompleteness is used to control product quality. The main point that they are trying to make is that contractual incompleteness allows options and flexibility in a contract so that the contract can handle contingencies optimally. As for the second issue, Wang (2009) argues that uncertainty in the external markets determines the optimal degree of incompleteness of a contract. For example, consider the law, which is a contract between the law enforcement authorities and potential violators. This contract is incomplete, just like a typical contract in reality. The argument is the same: the law itself must be complete, but the written law is incomplete. The law is complete in the sense that the court is able to make a judgement on any case. A judge would never say “I cannot make a judgement since the law is incomplete.” However, the written law, or the law in writing, is obviously incomplete. This incompleteness allows lawyers to play a role. One advantage of having an incomplete law is that it can serve as a deterrent against crimes. There is a common misconception that an incomplete contract must be simple and a complete contract must be more complicated. Consider a model setting in which a contract contains an income-sharing rule. If we say that the contract is complete, we are assuming that the income-sharing rule takes into account all

66 Incomplete contracts possible contingencies. If that’s the case, we only need to derive this incomesharing rule in equilibrium. But, if we say that the contract is incomplete, we are suggesting that the income-sharing rule is not supposed to take into account all possible contingencies. And if that’s the case, by rationality, we need to add some additional mechanisms into this contract or to assist the contract to deal with other contingencies. Suppose that control rights are allowed in the contract. That is, income rights (an income-sharing rule) are not enough and other rights, such as control rights, are considered. Then the equilibrium contract will contain not only an allocation of income rights (an income-sharing rule) but also an allocation of control rights. In terms of model setting, the complete contract is simpler than the incomplete contract, while in equilibrium the incomplete contract generally has a simpler income-sharing rule. 1.2. A foundation of organizations What are institutions? Institutions, according to North (1990), are “the rules of the game, and they comprise formal laws, informal constraints, shared norms, beliefs and self-imposed limits on behavior and their enforcement characteristics; i.e., courts, police, judgmental aunts, etc.” This broad definition bundles norms together with institutions and is also favored by Greif (2006), who argues that “an institution is a system of rules, beliefs, norms and organizations that together generate a regularity of social behavior.” The coexistence of organizations and markets hinges on a crucial trade-off: the price coordination versus planning coordination of economic activity. The key underlying factors determining this trade-off are information, incentives, risks, and externalities. Without these factors, markets would be perfectly efficient. Rights or control rights are particularly important among mechanisms. There are two sets of control rights. These two sets of rights are shown in Figure 2.1. A narrowly defined contract contains only a written part, called the written contract; a broadly defined contract contains both a written part and an unwritten part, with the latter being defined by mechanisms other than the written part. These mechanisms include an upfront ownership transfer (Coase), ex-post ownership transfer, conditional or contingent ownership transfer, ex-post renegotiation, investment strategies, control rights allocation, etc.

Rights

Written contracts

Contractual rights

Figure 2.1 The incomplete contract approach.

Mechanisms

Residual rights

Incomplete contracts

67

The standard agency model in Chapter 1 talks about an income-sharing contract only. Incomplete contracts are much more general. They include property rights (Coase, 1960) and ownership (Grossman and Hart, 1986). Incomplete contracts concern both income and control rights, where income rights refer to an incomesharing rule and control rights include property rights, ownership and various ex-post options. The allocation of income rights is an agreement on income sharing, while the allocation of control rights defines ownership and decision rights. Control rights may imply many unspecified rights for the holder. For example, if you are the owner of a copy machine, you have many implicit/unspecified rights, such as the right to lend it to someone, the right to smash it into pieces, and the right to use it anytime. Control rights also automatically imply responsibilities. So, as the owner of the copy machine, you are legally responsible for preventing anyone from using it for illegal copying. In theory, an organization or institution is a set of contracts plus a hierarchy system. This hierarchy system defines an allocation of control rights. Each contract is subject to this hierarchy system. For example, a contract may specify income rights plus a position in the hierarchy system for the contractor. Such a contract is an incomplete contract. Hence, incomplete contracts form the foundation of organization theory. Why are most human societies and organizations made up of a hierarchical system today? If relationships among all parties in an organization can be defined by a complete contract, there is no need for an organizational structure. Hence, the existence of an organization must be due to incomplete contracts. One special example is an army, which has many hierarchical layers. It seems that asymmetric information between higher-ranked and lower-ranked commanders and the need for a high degree of coordination are the reasons for the many hierarchical layers. In our opinion, one purpose of having an organization is to capture the benefit of central planning inside an organization and to shield individuals from outside competition. Hence, the boundaries of the firm must be a balance of benefits between central planning inside the firm and market competition outside the firm. If market competition were always better than central planning, we would have seen market competition inside the firm. Twenty years ago, nationwide central planning was finally abandoned by the communist-bloc countries. Today, all the economies in the world are becoming or are already market-based economies. At the same time, we also see central planning gaining popularity by way of expanding corporations. Sixty years ago, a nation’s total output mainly depended on millions of small firms. But in the last 60 years some firms have grown much bigger and many small firms have disappeared. Today, a small number of large corporations have a dominant influence on a country’s total output. For a typical Western country, listed companies play a major role in its gross national product (GNP). Within each company, central planning is the way of conducting production activity. Hence, in this sense, central planning is gaining popularity. In summary, in the last 60 years, nationwide central planning has been abandoned, while firm-level central planning has gained popularity.2

68 Incomplete contracts 1.3. The incomplete contract approach Within the world of incomplete contracts, the allocation of control rights and ownership is an important mechanism. There is a role for control rights to play if and only if contracts are incomplete. With incomplete contracts, ownership and control become relevant within the firm. In particular, as argued by Hart, when contractual incompleteness is coupled with asset specificity, a potential holdup problem ensues, which leads to the question of the optimal allocations of income and control rights in business relationships. After accepting a contract, the parties involved invest in their relationship, which creates assets for the firm. These assets may be relationship-specific in the sense that they are worth less outside their relationship. However, if the contract defining their relationship is incomplete, there are situations ex post in which the contract has no mention of their rights in some circumstances. They will have to negotiate the terms under these situations. Some parties may not have much bargaining power ex post since their assets are worth much less outside the relationship. Other parties take advantage of this and bargain for better terms. This is the so-called holdup problem in a relationship. Over the last two decades, contractual incompleteness, proposed by Grossman and Hart (1986), Hart (1988, 1995), and Hart and Moore (1990), has formed the foundation for organization theory, which is now playing a central role in the studies of the boundaries of the firm, corporate governance, and corporate law and regulation. Incomplete contracts have now become a paradigm for organization theory. For Hart, a contract is an allocation of rights, including income and control rights. Property rights and ownership are two forms of control rights. This stream of incomplete contract literature follows from Coase (1937, 1960) and focuses on the role of control rights (ownership and property rights). We will go beyond this stream of research in later chapters and consider all possible mechanisms, including the mechanism-design approach. In summary, we present organization theory in the following three steps: 1. 2. 3.

Income-sharing contracts, which focuses on income rights. This is our Chapter 1. The incomplete contract approach, which focuses on control rights (ownership and property rights) as an effective mechanism. This is our Chapter 2. The mechanism-design approach, which allows all possible mechanisms. We present various applications of this approach in the rest of this book.

1.4. Justifications for incomplete contracts The definition of incomplete contracts and their justifications in the literature are very different from ours. In the literature (see, for example, Hart and Moore 1999), contractual incompleteness is very much motivated by the following story. Imagine a buyer who requires a good from a seller. Suppose that the nature of the good is dependent on the state of the world. The state of the world is uncertain at the time of contracting. The two parties might write a contingent contract specifying exactly

Incomplete contracts

69

what kind of good is to be delivered in each state. However, if the number of states is very large, such a contract would be prohibitively costly to write. Hence, the two parties settle for an incomplete contract, which specifies trading deals for a few highly possible states only. In this story, contractual incompleteness is due to transaction costs. In this section, we describe a few popular justifications for contractual incompleteness, most of which are based on transaction costs. Contracts in reality appear to be highly incomplete. Hence, many researchers try to explain this “apparent incompleteness” or “incompleteness in appearance.” In the literature, a contract is considered incomplete if it appears to be incomplete. If so, one task for researchers is to find reasons for the incompleteness of a contract. Now we will describe a number of justifications for incomplete contracts in the literature, which can be classified along the four types of “transaction costs” identified by Tirole (1994): unforeseen contingencies, writing costs, enforcement costs, and renegotiation. Towards the end of this section, we add our own explanation of contractual incompleteness, which is not based on transaction costs. Our explanation applies only to our definition of incomplete contracts. Enforcement costs It is argued that contracts are incomplete since enforcing contracts is costly. An elaborate system consisting of courts, laws, and police forces needs to be set up. However, such a system is already in existence in every country. Since all countries have sufficient power to enforce any written item in a contract, why do we still have incomplete contracts? Observability When the parties’ valuations are mutually observable ex post, it may be optimal to negotiate the trade ex post after these valuations are observed, rather than rigidly contract the trade ex ante. However, this practice seems unrealistic in reality. The equilibrium solution of a trade based on ex-post negotiation is equivalent to an ex-ante renegotiation-proof contract. Since this contract is renegotiation-proof, verifiability of valuation is unnecessary. Such a contract is generally very simple. Hence, observability can ensure the existence of an ex-ante renegotiation-proof contract. Of course, this contract is generally less efficient than the optimal contract under verifiability. It is shown that, whenever the contracting parties’ valuations are mutually observable ex post, a contract can be made indirectly contingent on these valuations by allowing the parties to send messages and conditioning the trade on these messages (see Moore 1992 for a survey of the relevant implementation literature). In the context of the holdup problem, Rogerson (1992) shows that, using multistage message games and the concept of subgame-perfect implementation suggested by Moore and Repullo (1988), first-best investments can be implemented. Therefore, the cost of verifying publicly observable information may not be a serious constraint on contracting.

70 Incomplete contracts Unforeseen contingencies Oliver Hart’s well-known argument is that some aspects of future trades may not be foreseeable on the contracting date and therefore have to be left to future negotiation. For this reason, contracts are often incomplete. Maskin and Tirole (1999) show that, if the contracting parties can foresee their own payoffs (not necessarily valuations) in each future contingency, which is consistent with rationality, the parties can design a message game that effectively describes ex post all the trades that have not been described ex ante. They show that, when the parties can use such a game, inability to foresee future contingencies by itself does not constrain contracting. However, it is not clear what Hart means by unforeseen events. He could mean that the contracting parties are aware of the events but do not know the distribution function of the events; he could also mean that the contracting parties are not even aware of the events. If the parties know the distribution function, it is a problem of uncertainty; if the parties are aware of the events but do not know the distribution, they can apply subjective beliefs to the events (the Bayesian approach) so that these events can be considered in the contract; and if the trading parties are unaware of the events, the problem is beyond the scope of this book. Writing costs Some researchers argue that, if specifying contractual contingencies is costly, the optimal contract should trade the loss from contractual incompleteness off against the cost of adding more contractual clauses. A model of contractual incompleteness based on this idea has been offered by Dye (1985). However, many contracts with large stakes are highly incomplete; in fact, we don’t see a pattern of contracts with larger stakes being more complete. Writing costs cannot explain this. Consider a simple situation in which a household hires a babysitter. There is typically no written contract for this sort of hiring. However, an accident could happen that might cause the death of the child. Both parties are obviously aware of this possibility; yet, both agree to enter a contractual relationship without a written contract. Why? Renegotiation Any contract can be reopened for consensual renegotiation. This possibility reduces the benefits of complete contracts. Due to the possibility of renegotiation, contracting parities may not be willing to commit early. Hence, although renegotiation has the advantage of ensuring ex-post efficiency (by the assumption of Coasian bargaining), it has the disadvantage of causing ex-ante inefficiency. Consequently, in many cases, incomplete contracts are more efficient than complete contracts. However, in a contractual setting that allows renegotiation, the equilibrium contract is renegotiation-proof. Hence, the real question is: why is a renegotiationproof contract incomplete? The explanation is that the condition “renegotiation-proof” restricts the set of admissible contracts. If renegotiation-proofness rules out all complete contracts, then the optimal contract must be an incomplete contract.

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Complexity Segal (1999) argues that, when the environment is complicated, many IC conditions are needed to provide incentives. Each IC condition reduces efficiency further. In this sense, each IC condition represents a cost of complexity. When the environment is very complex, the total cost of complexity is high. When this cost is high enough, no contract (a special incomplete contract) is superior. Commitment Wang (2008) compares incomplete contracts and renegotiable complete contracts and argues for the cost of early commitment. With an incomplete contract, even if many items may be missing from the contract, the contracting parties know what is going to happen ex post and they make choices ex ante based on this knowledge. In a complete contract, even if all the items are in the contract, they may still need to negotiate a new contract ex post. So which contract is better? A contract represents an early commitment. Even though renegotiation is possible, the original agreement is an alternative to any negotiating party. Without a contract, there will be no such constraints – anything is up for negotiation. Hence, a less complete contract places fewer constraints on future negotiations.

Multiple mechanisms In our view, an optimal contract can be incomplete since there are other mechanisms in the model that can also provide incentives (see, for example, Wang and Zhou 2004, Qiu and Wang 2009, and Wang 2009). That is, an incomesharing rule may be a mechanism that can be used to provide incentives. Other mechanisms can also be deployed in the model to provide incentives. With the help of other mechanisms, a simple income-sharing rule may be optimal or efficient. This is our notion of incomplete contracts. Under this incomplete contract approach, it is even possible to achieve efficiency without an income-sharing rule. In some cases, having no formal agreement ex ante can also be an efficient solution.

2. Property rights as an economic instrument 2.1. The Coase Theorem This section discusses the Coase Theorem proposed by Coase (1960). The Coase Theorem. If there is a conflict of interest surrounding an asset and there are no transaction costs,  

granting property rights can ensure efficiency; the allocation of ownership does not matter.

72 Incomplete contracts Here, “no transaction costs” means no transaction costs for transferring ownership titles. Coase imposes two assumptions:  

Ownership is transferable. There are no transaction costs involved in ownership transfers.

Take a river or a forest as an example (see Figure 2.2). Without an assignment of property rights over the river or the forest, everyone will try to catch all the fish in the river or cut down all the trees in the forest. Coase argues that, to solve this problem, we only need to assign ownership of the river or the forest to someone, and a socially optimal solution will be assured. Coase further argues that to whom the ownership is assigned doesn’t matter – the ownership can be assigned to anyone and the socially efficient outcome will still be assured. We use an example to explain the theorem, as Coase (1960) does. Consider a situation in which a group of fishermen and a chemical factory are competing for a stream. The chemical factory wants to dump its waste into the stream and the fishermen want to fish in the stream. The factory is upstream and the fishermen are downstream. If the factory is allowed to operate, its product will yield a profit of πc dollars (without taking into account the cost of dumping its waste into the stream), but all the fish in the stream will be killed. If the factory doesn’t operate, the fishermen gain πf dollars from fishing. There is a conflict of interest. Coase argues that a simple solution to this problem is to grant ownership to someone, and, if there are no transaction costs, who has the ownership doesn’t matter. Suppose that the market value πc of the factory’s product is less than the

Figure 2.2 Chemical dumping vs. fishing.

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market value πf of all the fish in the stream at any one time. If the factory has the ownership of the stream (and presumably all the fish in it), the factory will sell the ownership to the fishermen and close down the factory, which leads to a socially efficient solution. If the fishermen have the ownership, the fishermen will go fishing, which is also a socially efficient solution. Even if a third party has the ownership, it will sell the ownership to the fishermen, which again leads to a socially efficient solution. That is, whoever has the ownership, as long as someone owns the stream, the outcome will be socially efficient.3 2.2. Discussion What is Coase’s contribution? There are two qualifications on Coase’s “efficient solution.” First, other things being equal, the solution yields the maximum social welfare achievable when ownership is costlessly transferrable. For example, if there is asymmetric information among trading parties, the solution achievable under the best ownership arrangement (best in terms of social welfare) is generally the second best. The first best may not be achievable under asymmetric information even if ownership is costlessly transferable. Second, Coase assumes that income distributions have no effect on economic efficiency. What is Coase’s contribution? On issues about firms, there are three approaches: the industrial organization (IO) approach, in which firms compete in the marketplace; the contractual approach, in which firms form joint ventures and alliances; and the organizational approach, in which many problems, especially information and incentive problems, can be solved through organizational means. It is Coase who proposed the organizational approach. Coase shows in a few examples that organizational means (ownership, property rights, and control rights) can potentially resolve conflicts of interest completely. When the contractual and organizational approaches are combined, we have the incomplete contract approach. Coase’s contribution is clear: an assignment of ownership or property rights can solve many economic problems. For example, let us apply Coase’s idea to the standard agency model in Chapter 1. Indeed, if the principal simply sells the firm to the agent, the first-best solution is obtained. Before Coase, economists viewed a competitive market as the only solution. Coase proposes an alternative solution: in some cases, an assignment of ownership can solve an economic problem. Karl Marx has the same idea of an ownership solution as Coase does (as pointed out by Chenggang Xu of the University of Hong Kong), but without a feasible solution. Marx points out two problems in the market economy: business cycles and income inequality. His proposed solution is to reallocate ownership of firms from capitalists to laborers. There are two problems with this proposal. First, there is no guarantee that laborers are better owners in any sense. Laborers may not have the skills and experience in managing firms; otherwise they would have already become managers and owners in the market economy. Second, how can this reallocation be realized? Coase argues that, if there are no transaction costs, rational

74 Incomplete contracts economic agents will voluntarily transfer property rights to the right person. Marx did not answer this question. Vladimir Lenin gave an answer to this question in the form of a revolution, which he led in 1917. The “laborers” were supposed to own all the firms in Russia after the revolution. The next question is: what kind of economic system should a country have if the laborers owned all the firms? Joseph Stalin proposed central planning.4 Both Lenin and Stalin were later proven wrong. Initial owner vs. final owner In technical terms, the Coase Theorem actually says something that is really trivial. Coase imposes three assumptions:   

Ownership is transferable. There are no transaction costs involved in ownership transfers. There are no liquidity constraints.

Under these three assumptions, obviously who has the ownership doesn’t matter. By economic rationality, the ownership will always be transferred to and exercised by the right person. Although Coase proposes a good idea, his theorem is trivial. How can ownership be allocated, or what is the best way to allocate ownership? Coase’s answer is simple: ownership can be arbitrarily assigned to anyone, as long as there are no costs involved in ownership transfers. The reason is simple: if there are no costs involved in ownership transfers, rational individuals will ensure that ownership ends up in the right hands even if it is initially allocated to the wrong person. There are two questions regarding ownership: (1) Who should the initial owner of the asset in question be? (2) Who should the final owner, who has the right to exercise the ownership rights, be? Coase answers the first question: the initial owner can be anyone as long as ownership is costlessly transferrable. We can easily understand the logic of this conclusion from our fishing example. The second question is the difficult one, which is what incomplete contracts are designed to address. In the fishing example, the latter question involves calculating πc and πf , which can be difficult. The payoffs of the two parties may be dependent on each other; for example, the factory may produce something that the fishermen want. The study of incomplete contracts is really about finding out who the final owner should be. Without a model with which to derive πc and πf , Coase avoided the difficult question. The difficult question is not about who the initial owner of an asset should be. Without transaction costs, obviously the most efficient user will eventually be the final owner, who exercises ownership rights. The difficult question is: who is the most efficient user conditional on the circumstances? Grossman and Hart (1986), Hart (1988, 1995), and Hart and Moore (1990) started the work of tackling this difficult task, which led to the foundation of organization theory. In the fishing example, the real question is who should be allowed to use the stream, rather than

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who should own the stream. To answer this question, we need to build a model for the factory–fishermen relationship and then determine who should be the final owner. The real difficulty is in finding πf and πf : The values of the stream to the two parties may be interlinked in equilibrium. That is, πc and πf may be conditional on ownership. For example, suppose that the only buyers of the factory’s product are the fishermen. If the factory produces the product and kills the fish in the process, the fishermen’s income will be reduced, which will lead to a lower profit for the factory; if the factory doesn’t produce and the fishermen get and sell all the fish, then the fishermen’s income will be higher, which will lead to a potentially higher profit for the factory. In this case, it is obvious who should be allowed to use the stream. Should investors be owners? However, due to liquidity constraints in practice, ownership transfers require investors and financiers. In general, economic activity requires the involvement of investors and financiers. This introduces a third party, investors and financiers, into the model, which leads to a further question: should an investor or financier be the owner? The second half of the book by Hart (1995) extensively discusses this issue, which is a focal point of corporate finance. Problems associated with ownership In practice, a transfer of ownership may solve one problem but create many others. First, ownership may create a monopoly problem. For example, if someone takes over an entire industry, the industry may change from being a competitive industry to being a monopolistic one, which causes inefficiency. As another example, can the pollution problem be solved by assigning air to someone? Obviously not, since the owner would then charge us for breathing in her air. Can overfishing be solved by assigning all oceans to someone? Obviously not, since the owner would then overcharge us for transporting goods across her oceans. Second, ownership is associated with risk. Although the owner can claim income from a property, she is responsible for its problems. For example, the sole owner of a firm is responsible for all its debts, and the owner of a firm is the first in line to be responsible for a fire in the firm’s premises. Third, ownership is associated with responsibility. Although the owner is the residual claimant for income, she has the residual responsibility for bad events that happen to the property. For example, the owner of a copy machine has the responsibility to prevent it from being used for illegal copying. Fourth, Coase analyzes the problem in isolation. He doesn’t consider the possibility that a change of ownership may cause a reaction from others, e.g. neighbors in the case of a change of ownership of a building, since it may have adverse effects on those others. Finally, ownership is often not assignable in reality, especially the ownership of land. For example, many countries do not allow private ownership of land, and in most countries rivers and forests can only be owned by the national government.

76 Incomplete contracts Land ownership disputes between nations are often next to impossible to settle without going to war.

3. Allocation of control rights This section presents an incomplete contract theory based on the work of Grossman and Hart (1986). It introduces a few innovative concepts essential to our understanding of incomplete contracts. Control rights are the rights to take an action or make a decision; income rights are the rights to claim income. Both rights can influence ex-ante investment decisions. In the paper, control rights are assignable, but income rights are not. The purpose of the paper is to determine the equilibrium allocation of control rights. The authors emphasize asset specificity and holdup problems in incomplete contracting. 3.1. The model Consider two firms, 1 and 2, engaged in a relationship that lasts two periods. The firms make relationship-specific investments a1 and a2 in period 1. In period 2, some further actions q1 and q2 are taken and the gains from trade are realized by the end of period 2. The controlling party of qi has the right to decide the value of qi : The surplus of firm i from the relationship is Bi ½ai ; ϕi ðq1 ; q2 Þ: Here, Bi is increasing in ai , and ϕi ; and ϕi is increasing in qi : B1 and B2 are realized at date 2. Renegotiation is allowed at t ¼ 1: Functions Bi ð:;:Þ and ϕi ð:;:Þ are common knowledge. Contracting 0 Ex ante

a1 , a2

ω realized renegotiation 1 Ex post

q1 , q2

Allocating revenue 2

For an action variable, there are two contracting possibilities. First, the value of the action variable may or may not be contractible; in this case, we say that the action variable is or is not contractible. Second, even if its value is not contractible, the control rights over it may or may not be contractible; in this case, we say that the action variable is or is not assignable. Contractibility of an action variable requires the value of the action variable to be verifiable, and the value of the action variable can be decided by the holder of the control rights when the time comes. Contractibility is stronger than assignability. Contractibility allows the two parties to jointly determine the value of the action variable, while assignability designates one player as the decision maker in this case, which is an extreme case of contracting. If an action variable is not contractible, it may still be assignable; but if it is nonassignable, it must be noncontractible. If an action variable is contractible, we will ignore its assignability, since it is more efficient to contract on its value than to assign its control rights in this case.

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For a value variable (such as income, profits, revenue, and output), there are also two contracting possibilities. First, it may or may not be contractible; in this case, we say that the value variable is or is not contractible. Second, even if its value is not contractible, the income rights over it may or may not be contractible; in this case, we say that the value variable is or is not assignable. Contractibility of a value variable requires the value of the value variable to be verifiable, and the value of the value variable can be claimed by the holder of the income rights when the time comes. Contractibility is stronger than assignability. Contractibility allows one player to receive a portion of the value, while assignability designates one player as the sole recipient, which is an extreme case of contracting. If a value variable is not contractible, it may still be assignable; but if it is nonassignable, it must be noncontractible. If a value variable is contractible, we will ignore its assignability, since it is more efficient to contract on its value than to assign its income rights in this case. Assumptions:    

The investments ai s are noncontractible and nonassignable ex ante, but they are observable ex post. The actions qi s are noncontractible ex ante but contractible ex post. The benefits Bi s are noncontractible ex ante. Functions Bi ½:;ϕi ð:;:Þ and ϕi ð:;:Þ are common knowledge.

Let us explain the assumptions. First, ai s need to be observable so that ex-post bargaining is feasible. Bargaining occurs after ai s have been made; since the two parties need to know the values of ai s in order to reach an agreement, ai s need to be observable. Second, since Bi s are noncontractible, an income-sharing contract is not possible. Third, Bi s and ϕi s need to be common knowledge, since the bargaining parties need the information to reach an agreement. Since Bi s are common knowledge, immediately after a bargaining agreement is reached, they can make a monetary transfer as required by the bargaining outcome. By this, B1 þ B2 need not be contractible ex post. Fourth, there are three possible assumptions on qi s, as shown in Table 2.1. Grossman and Hart have discussed case (1). Given that qi is ex-post contractible, firm i may be prepared to give up this right in exchange for a side payment from renegotiation at date 1: Who should have the control rights is an interesting issue in this case. The other two cases are less interesting. Case (2) is trivial, since the first best is achievable in this case, which will be shown later. Case (3) does not allow renegotiation, and hence control rights have no role to play; in this case, the controlling firm(s) decides qi s when the time comes. Table 2.1 Alternative assumptions on q1 and q2 Case

At t ¼ 0 (ex ante)

At t ¼ 1 (ex post)

(1) (2) (3)

Noncontractible Contractible Noncontractible

Contractible Contractible Noncontractible

78 Incomplete contracts In summary: 1.

2. 3.

4.

None of the variables ai , qi , and Bi is ex ante contractible. Thus, an initial contract can only be drawn for an allocation of control rights plus a possible payment for transferring the ownership of an asset. After the contract is signed, a1 and a2 are chosen in a Nash game. At date 1, the controlling firm of qi has the right to choose qi . Since qi s are contractible at date 1, the contract may be renegotiated and an ex-post contract may be signed. In equilibrium, renegotiation never happens and a transfer is made at time 0 to induce the controlling firm(s) to sign an agreement for supplying q1 ða Þ and q2 ða Þ; given inputs a ¼ ða1 ; a2 Þ in the first period, where q1 ða Þ and q2 ða Þ are defined in (1.1) in the next subsection. The income rights as defined in B1 and B2 are not assignable, meaning that firm i always has Bi , i ¼ 1; 2; under any condition. Instead, the control rights over ex-post actions q1 and q2 are assignable in a contract. The control rights over a1 and a2 are also not assignable.

With these four assumptions, the only possible contracts are: Ex-ante contract ¼ fallocation of control rights over q1 and q2 g: Ex-post contract ¼ fvalues of q1 and q2 g: After the initial contract is signed, a1 and a2 are chosen in a Nash game in period 1. But the contract may be renegotiated ex post to determine q1 and q2 . Finally, B1 and B2 are realized at date 2. 3.2. The first best The first-best solution consists of ða∗1 ; a∗2 ; q∗1 ; q∗2 Þ that maximizes the total surplus B1 þ B2 without any constraint. That is, the first-best problem is max

a1 ;a2 ;q1 ;q2

B1 ½a1 ; ϕ1 ðq1 ; q2 Þ þ B2 ½a2 ; ϕ2 ðq1 ; q2 Þ:

Given ða1 ; a2 Þ, denote qða Þ≡½q1 ða Þ; q2 ða Þ as the solution of the following ex-post problem: max B1 ½a1 ; ϕ1 ðq1 ; q2 Þ þ B2 ½a2 ; ϕ2 ðq1 ; q2 Þ: q1 ;q2

Obviously ða∗1 ; a∗2 Þ is the solution of the following problem: max maxfB1 ½a1 ; ϕ1 ðq1 ; q2 Þ þ B2 ½a2 ; ϕ2 ðq1 ; q2 Þg: a1 ;a2 q1 ;q2

ð2:1Þ

Incomplete contracts Then, by the envelope theorem, the FOCs for a∗i are ∂Bi ½ai ; ϕi ðqða ÞÞ ¼ 0; i ¼ 1; 2; ∂ai

79

ð2:2Þ

ÞÞ is the partial derivative with respect to ai when ϕi is taken as where ∂Bi ½ai ;ϕ∂ai ðqða i constant. Condition (1.2) implies that the first best can be achieved if qi s are ex ante contractible, even if ai s are not. By specifying qi ¼ q∗i in the ex-ante contract when qi s are ex ante contractible, firm i will choose ai ¼ a∗i to maximize its own surplus Bi ½ai ; ϕi ðq∗1 ; q∗2 Þ: Let us go back to the normal situation described in the last section, known as the second-best problem. Since qi s are ex post contractible, given a; ex-post efficiency can be guaranteed for the second-best problem through renegotiation. The qðaÞ ensures ex-post efficiency. It is in both parties’ interests to have this qða Þ: Hence, the contracts at time 0 and time 1 are respectively:

ex-ante contract ¼fcontrol rights over q1 and q2 ; alump-sum transferg; ex-post contract ¼fq1 ðaÞ; q2 ðaÞg: The ex-ante contract consists of an allocation of ownership rights and a lump-sum transfer of payment. There are four possible allocations of control rights: 1) 2) 3) 4)

Firm 1 controls q1 ; and firm 2 controls q2 : We call it “separate control.” Firm 1 controls both q1 and q2 : We call it “firm 1 in control.” Firm 2 controls both q1 and q2 : We call it “firm 2 in control.” Firm 1 controls q2 ; and firm 2 controls q1 : We call it “separate control.”

The determination of the optimal allocation of control rights is to find the best allocation among these four possibilities. We now proceed to analyze each of the four possible allocations of control rights. 3.3. Separate control Consider the case in which firm 1 has the right to choose q1 and firm 2 has the right to choose q2 at date 1. At date 1, they renegotiate a contract to determine q1 and q2 . If the bargaining process fails, a Nash game determines q1 and q2 . Let q^ ≡ ðq^1 ; q^2 Þ be the Nash outcome. Since B1 and B2 are increasing in ϕ1 and ϕ2 ; respectively, the solution is determined jointly by the following problems: max ϕ1 ðq1 ; q^2 Þ; q1

max ϕ2 ðq^1 ; q2 Þ q2

Assume ex-post efficiency for a bargaining solution, which is the standard treatment in the literature. Then the bargaining outcome must be qða Þ≡ ½q1 ða Þ; q2 ðaÞ, where a ≡ ða1 ; a2 Þ. Since the bargaining solution is ex-post efficient, it is generally

80 Incomplete contracts better than ð^ q 1 ; q^2 Þ. Then, according to the Nash bargaining theory, the payoffs of the two parties are respectively5 1 1 ða1 ; a2 ; q^Þ  B1 ½a1 ; ϕ1 ðq^Þ þ ðB1 fa1 ; ϕ1 ½qðaÞg þ B2 fa2 ; ϕ2 ½qðaÞg 2 B1 ½a1 ; ϕ1 ðq^Þ  B2 ½a2 ; ϕ2 ðq^ÞÞ; 1 2 ða1 ; a2 ; q^Þ  B2 ½a2 ; ϕ2 ðq^Þ þ ðB1 fa1 ; ϕ1 ½qðaÞg þ B2 fa2 ; ϕ2 ½qðaÞg 2

ð2:3Þ

B1 ½a1 ; ϕ1 ðq^Þ  B2 ½a2 ; ϕ2 ðq^ÞÞ: At t ¼ 0; the two firms know these expected payoffs after taking into account the possibility of renegotiation. They will thus choose their efforts ða1 ; a2 Þ based on these payoffs. Again, since ai s are not contractible, the firms will choose their efforts opportunistically. We assume that they play a Nash game to determine a1 and a2 . Let ða~ 1 ; a~ 2 Þ be the Nash solution, which is the solution of the following pair of problems: max 1 ða1 ; a~2 ; q^Þ; a1

max 2 ð~ a1 ; a2 ; q^Þ a2

Since 1 i ða1 ; a2 ; q^Þ  Bi ½ai ; ϕi ðq^Þ þ ðmaxfB1 ½a1 ; ϕ1 ðq1 ; q2 Þ þ B2 ½a2 ; ϕ2 ðq1 ; q2 Þg: 2 q1 ;q2 B1 ½a1 ; ϕ1 ðq^Þ  B2 ½a2 ; ϕ2 ðq^ÞÞ;

by the envelope theorem, the FOCs are 0¼

∂ξ i 1 ∂Bi ½ai ; ϕi ð^ q Þ 1 ∂Bi ½ai ; ϕi ðqða ÞÞ ¼ þ ; ∂ai 2 ∂ai 2 ∂ai

i ¼ 1; 2

ð2:4Þ

By comparing (1.4) with (1.2), we can see that ða~ 1 ; a~ 2 Þ generally cannot achieve the first best. This inefficiency arises because the firm puts a 50 percent weight on the noncooperative outcome q^i s, in spite of the fact that the noncooperative outcome would never occur. The reason for putting a 50 percent weight on the noncooperative outcome is because 50 percent of the controlling party’s income depends on the noncooperative outcome, and Bi ½ai ; ϕi ðq^ Þ serves as a bargaining chip for firm i. Hence, the firm has an incentive to increase the value of Bi ½ai ; ϕi ðq Þ: Finally, since the two firms receive incomes B1 fa~1 ; ϕ1 ½qða~Þg and B2 fa~ 2 ; ϕ2 ½qða~Þg at time 2, the transfer payment from firm 1 to firm 2 at time 0 is: aÞg  1 ð~ a1 ; a~2 ; q^Þ: B1 fa~1 ; ϕ1 ½qð~ This payment may be positive or negative. This payment can be made upfront since both parties know exactly what is going to happen throughout the whole cooperation period.

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3.4. Firm 1 in control Now, consider the case in which firm 1 has control over both q1 and q2 : If the firm is to exercise its control, its problem at date 1 is max ϕ1 ðq1 ; q2 Þ q1 ;q2

Let the solution be ðq1 ; q2 Þ. This solution is generally not ex post efficient. A bargaining process can improve efficiency at that point in time. Assuming ex-post efficiency for a bargaining solution, the bargaining solution must be qðaÞ as defined in (1.1). Then the Nash bargaining solution implies payoffs, ξ 1 ða1 ; a2 ; qÞ and ξ 2 ða1 ; a2 ; qÞ; for the two firms where ξ 1 and ξ 2 are as defined in (1.3) and q ≡ðq1 ; q2 Þ. In other words, the two firms can gain by signing an ex-post contract and the payoffs are as defined in (1.3) with ðq1 ; q2 Þ replacing ð^ q 1 ; q^2 Þ. Again, similar to the situation in (1.4), the Nash equilibrium solution ða1 ; a2 Þ from ξ 1 ða1 ; a2 ; qÞ and ξ 2 ða1 ; a2 ; qÞ is generally inefficient. Here, although firm 1 has full control of ðq1 ; q2 Þ ex post, it chooses to bargain with firm 2 in deciding ðq1 ; q2 Þ. If firm 1 were to choose qðaÞ by itself without signing an ex-post contract, firm 1 would get B1 fa1 ; ϕ1 ½qðaÞg instead of the higher B1 ½a1 ; ϕ1 ðqÞ, and firm 2 would get B2 fa2 ; ϕ2 ½qða Þg instead of the lower B2 ½a2 ; ϕ2 ðqÞ: Hence, firm 2 is willing to pay firm 1 to choose qða Þ: In other words, although firm 1 has the control rights over q1 and q2 , it has no income rights over B2 : To gain a share of B2 in exchange for choosing qða Þ instead of q; firm 1 has to bargain with firm 2. 3.5. Optimal control The case with firm 2 in control is exactly symmetric to the above case. Also, the case with firm 1 controlling q2 and firm 2 controlling q1 is symmetric to the case in subsection 3.3. We have now discussed all four cases. Which one is optimal? Intuitively, if only firm i cares about qi , then firm i should control qi : That is, if ϕi depends primarily on qi for both firms, then separate control should be optimal. Also, if firm 2 doesn’t care about q1 and q2 ; then firm 1 should be in control. That is, if ϕ2 hardly depends on ðq1 ; q2 Þ; having firm 1 in control should be optimal. Proposition 2.1. (Optimal Control) 1. 2. 3.

If ϕi ðq1 ; q2 Þ ¼ αi ðqi Þ þ εi βi ðq  i Þ for i ¼ 1; 2, where ε1 ; ε2 > 0 are small, then separate control is optimal. If ϕ2 ðq1 ; q2 Þ ¼ α2 þ ε2 δ2 ðq1 ; q2 Þ, where ε2 > 0 is small, then having firm 1 in control is optimal. If ϕ1 ðq1 ; q2 Þ ¼ α1 þ ε1 δ1 ðq1 ; q2 Þ, where ε1 > 0 is small, then having firm 2 in control is optimal.

The key message from this proposition is that a party should control a variable that has an important impact on its income.

82 Incomplete contracts 3.6. Discussion In the standard agency model, the agent has control rights over effort and income rights can be optimally allocated. In other words, control rights are not assignable but income rights are allocatable. In contrast, in the paper, control rights are assignable but income rights are not. Here, control rights refer to the control rights of q1 and q2 ; the control rights of a1 and a2 are not assignable. Nonassignable income rights mean that B1 and B2 are not contractible, but a lump-sum transfer is still feasible. Control rights yield bargaining power. Having control rights puts a firm in a good bargaining position and gives it better incentives. Hence, the allocation of control rights is important to economic efficiency. Coase argues that the allocation of ownership doesn’t matter. Why, then, does ownership matter so much in this study? The reason is that this study identifies the final owner. The initial ownership still doesn’t matter, as long as there are no ownership transfer fees. For example, when it is optimal for firm 1 to have full control, we can give control rights to firm 2. If so, firm 2 will be willing to sell the ownership for a profit, and the final owner will be firm 1, as long as there is no cost involved in this transfer of ownership. Therefore, there is no conflict between Coase and this study. The difference is that Coase talks about initial ownership whereas this study talks about final ownership. Since initial ownership is a trivial matter, in the rest of this book, ownership refers to final ownership. Hart’s contribution is that, when some variables are unverifiable, the allocation of control rights over those variables can be important. Before Grossman and Hart (1986), no one had ever mentioned this point. Note that the model has transferable utilities B1 and B2 ; meaning that one unit of B1 for firm 1 is equal to one unit of B2 for firm 2. Hence, we can treat B1 and B2 as money. If B1 and B2 are utility values, the problem becomes more difficult.

4. Allocation of income rights This section is based on the work of Hart (1988). The paper focuses on the allocation of income rights and ownership, where ownership implies an opportunity to gain private benefits. In contrast to the paper discussed in the last section, income rights are assignable in this paper, but control rights are not. Income rights are the rights to claim income, while control rights are the rights to decide an action. Ownership is also assignable, but ownership of an asset here is the right to cream off a fraction of profits from the asset. Ownership here doesn’t imply control rights. Hart emphasizes that a proper distribution of income is to provide incentives while a proper distribution of ownership is to avoid the use of assets for private benefits. The main conclusion is: since the owner of an asset can often use the asset for his own private benefit (unverifiable private benefit), it is often efficient to assign the ownership to just one person. We discuss three examples from the paper, which showcase the main ideas of the paper.

Incomplete contracts

83

Example 1: ownership of a single asset There is a machine that requires an input from a manager. Who should own the machine? Should it belong to an outsider who hires the manager to operate the machine? Consider a model in which the manager takes an unverifiable action x to yield a deterministic output Bðx Þ at date 1 with private cost x. Assume that the right to control the asset allows one to cream off a fraction 1  λ of the return Bðx Þ, with λ∈ð0; 1Þ. Assume that ð1  λ ÞBðx Þ is not verifiable, but the remaining return π ≡λBðxÞ is. Let a contract be sðπÞ if an outsider is the owner, where sðπÞ is a payment to the manager. Contracting

x

0

B(x) 1

The first-best problem is max BðxÞ  x x

which implies x∗ that satisfies B0 ðxÞ ¼ 1: If the manager is the owner, the manager can simply pay herself 100 percent of the profits, i.e. choose πðy Þ ¼ y: Then the manager’s return is R ≡ λBðx Þ þ ð1− λ ÞBðx Þ  x~ Hence, we have the first-best outcome in this case. If an outsider owns the machine, the manager still has control rights over x; even though the outsider owns the asset. With payment sðπÞ from the contract, the manager’s problem is max s½λBðxÞ  x: x

The FOC is λs0 ðπÞB0 ðxÞ ¼ 1: We assume s0 ðy Þ ≤ 1 for all y ≥ 0: If s0 ðyÞ > 1; by adding one dollar to y; the manager gets s0 ðy Þ > 1 dollars, i.e. the manager can gain by pumping cash into the project, which doesn’t make sense. By condition s0 ðπÞ ≤ 1; outsider ownership results in inefficiency. Hence, the manager should be the owner. Example 2: complementary activities Suppose that there is another asset, which complements the first. There are two managers. Specifically, given action x from manager 1 and action y from manager 2, let B1 ðx; y Þ and B2 ðyÞ be the returns from the two assets. Here, the presence of y in B1 ðx; y Þ represents an externality or complementarity.

84 Incomplete contracts Contracting

x,y

0

B1(x , y) , B2(y) 1

The first-best solution is ðx∗ ; y∗ Þ that solves: max B1 ðy Þ þ B2 ðy Þ  x  y: x; y

We have x∗ ¼ 0 and B01 ðy Þ þ B02 ðy Þ ¼ 1:

ð2:5Þ

Again, the owner of an asset can cream off an unverifiable fraction 1  λ of an asset’s return. Hence, the verifiable profits are π 1 ¼ λB1 ðx; yÞ;

π 2 ¼ λB2 ðyÞ:

A contract is a pair of payments s1 ðπ1 ; π2 Þ and s2 ðπ1 ; π2 Þ to the managers with s1 ðπ1 ; π2 Þ þ s2 ðπ1 ; π2 Þ ¼ π1 þ π 2: Suppose that B1 ¼ B1 ðyÞ: In the following, we discuss two alternative ownership arrangements. Under separate ownership, manager i owns Bi ; for both i ¼ 1; 2: Then the net returns of the two managers are R1 ¼ s1 ðπ1 ; π2 Þ þ ð1  λ ÞB1  x; R2 ¼ s2 ðπ1 ; π2 Þ þ ð1  λ ÞB2  y:

ð2:6Þ

In particular, manager 2’s problem is max s2 ½λB1 ðy Þ; λB2 ðyÞ þ ð1  λ ÞB2 ðyÞ  y: y

which implies the following FOC: λ

@s2 0 @s2 0 B1 ðyÞ þ λ B ðyÞ þ ð1  λÞB02 ðyÞ ¼ 1: @π 1 @π 2 2

ð2:7Þ

Assume that the managers can freely dispose of or boost profits, implying6 0≤

∂si ≤1: ∂πi

Since s1 þ s2 ¼ π1 þ π2 , we have ∂s2 ∂s1 ¼ 1 ≤1: ∂π1 ∂π1 Hence, λ

@s2 0 @s2 0 B ðyÞ þ λ B ðyÞ þ ð1  λÞB02 ðyÞ  λB01 ðyÞ þ B02 ðyÞ < B01 ðyÞ þ B02 ðyÞ: @π 1 1 @π 2 2

This means that a solution of (1.7) cannot possibly satisfy (1.5) too. Therefore, separate ownership cannot achieve the first best. The intuition for this result is clear.

Incomplete contracts

85

Since manager 2 is the only one who contributes to asset 1, she should own the asset to prevent manager 1 from creaming off a fraction of the asset’s output. Under single ownership with manager 2 owning both assets, the net returns of the two managers are R1 ¼ s1 ðπ1 ; π2 Þ  x; R2 ¼ s2 ðπ1 ; π2 Þ þ ð1  λ ÞB1 þ ð1  λ ÞB2  y:

ð2:8Þ

If we take s1 ¼ 0; then the optimal solution achieves the first best. Hence, single ownership is better than separate ownership. Of course, the same conclusion holds if manager 1’s contribution is small enough, with B1 ðx; y Þ ¼ γðyÞ þ ε1 δðx Þ;

B2 ðx; yÞ ¼ ηðyÞ þ ε2 ζ ðxÞ:

where ε1 and ε2 are small. This conclusion is the same as that in Example 1: the manager whose effort is important should own the asset. Since giving a manager ownership reduces the other manager’s incentives, the optimal ownership structure depends on the relative importance of the two managers’ efforts. Example 3: vertical relationship An upstream firm UF produces x as input for a downstream firm DF. The manager of UFis U; and the manager of DF is D: With investment y from D; the output of UF is Bðx; y Þ: The costs are Cðx Þ and y for U and D; respectively. Assume that the owner of UF has the ability to falsely report CðxÞ as cðx Þ ≡CðxÞ þ β, where β > 0: There is a social loss for this cost manipulation, so that the benefit to the owner is actually μβ, where μ ∈ð0; 1Þ; and the rest is wasted. This private benefit is unverifiable. We assume that Bðx; y Þ and cðxÞ are verifiable, but the cost y is private. C(x) UF U decides x D decides y

x DF

y

B (x , y)

A contract is a pair of payments sU ½Bðx; yÞ; cðx Þ and sD ½Bðx; y Þ; cðx Þ to the two managers, where sU þ sD ¼ Bðx; yÞ  cðx Þ. This means that the owner of UF will be reimbursed for the cost cðx Þ claimed and the two parties will share the joint profit Bðx; y Þ  cðx Þ: Without cost manipulation, the net returns of the two managers are RU ¼ sU ½Bðx; yÞ; CðxÞ;

RD ¼ sD ½Bðx; yÞ; Cðx Þ  y:

86 Incomplete contracts If U owns UF and D owns DF; then the profits of the two managers are RU ¼ sU ½Bðx; yÞ; CðxÞ þ  þ ;

RD ¼ sD ½Bðx; yÞ; CðxÞ þ   y:

ð2:9Þ Here, “ownership” is the right to gain from cost manipulation. Similarly, if D owns both firms, then the profits of the two firms are RU ¼ sU ½Bðx; yÞ; CðxÞ þ β;

RD ¼ sD ½Bðx; yÞ; Cðx Þ þ β  y þ μβ: ð2:10Þ

In the absence of a contract and cost manipulation, the net returns of the two managers are Bðx; y Þ  y and CðxÞ. Hence, the first-best solution ðx∗ ; y∗ Þ can be derived from: max Bðx; yÞ  CðxÞ  y x; y

which implies @Bðx ; y Þ ¼ C0 ðx Þ; @x

@Bðx ; y Þ ¼1 @y

ð2:11Þ

If D owns both firms and let sU ¼ 0 and sD ¼ Bðx; yÞ  cðx Þ: by (1.10), with net return RD ¼ Bðx; y Þ  Cðx Þ  y  ð1  μ Þβ and μ < 1, D won’t manipulate cost. Thus, D’s objective is max Bðx; y Þ  CðxÞ  y:

ð2:12Þ

y

Since RU ¼ 0; U is indifferent to any choice of x and is willing to choose x ¼ x∗ : Then, by (1.12), D will choose y∗ . Hence, if D owns both firms, the optimal solution is the first best. This arrangement allows the cost to be internalized. If U owns UF and D owns DF; by (1.9), the Nash equilibrium for ðx; yÞ implies @sU @B @sU 0 C ðxÞ ¼ 0; þ @B @x @c

@sD @B ¼ 1: @B @y

ð2:13Þ

If ðx∗ ; y∗ Þ were the solution, then (1.11) and (1.13) imply ∂sU ∂sD ¼1 ¼ 0: ∂B ∂B implying

∂sU ∂c

¼ 0. This means that

∂RU ∂sU ¼ þ μ > 0; ∂β ∂c Hence, U will manipulate cost, which leads to a social welfare loss of ð1  μ Þβ: Thus, the first-best solution is not achievable under separate ownership.

Incomplete contracts

87

In summary, if both firms are owned by a single owner, the solution is efficient; but if the two firms are owned separately, the solution is inefficient. 4.4. Discussion In all three examples, single ownership is optimal. Hart has some additional observations beyond the three examples: 1. 2. 3.

4.

If one individual is entirely responsible for the return of an asset, he should own it. If there are increasing returns to management, single ownership is optimal. If DF wishes to be supplied by UF; but DF’s business with UF is only a small fraction of both UF’s and DF’s total businesses, then we would expect a (long-term) contract between them rather than a merger. If an industry is declining, we would expect to see firms merge so as to save on overheads.

When control rights are applied to human assets instead of physical assets, the issue is about authority. Once an employee accepts a contract from a firm, he grants the firm authority to allocate his time for contingencies. The employee is willing to grant this authority because the firm is long-lived and wishes to maintain its reputation. A firm typically owns some assets and the firm can reallocate the assets to a new employee. But, when an employee leaves, he leaves with his human assets. Kreps considers a case in which the firm has control over intangible assets (good will and reputation) instead of physical assets. Williamson distinguishes between high-powered incentives provided by the market and low-powered incentives provided within a firm. Williamson argues that the different incentive arrangements inside and outside the firm are not coincidental. Ownership also has a “lock-in” effect, as argued by Williamson. In a buyer–seller relationship with an investment from the buyer, if the seller owns the investment, then the buyer cannot easily switch to an alternative seller. This reduces the incentive of the seller to provide a good service to the buyer, since the buyer has been “locked in.” In Hart’s model, liquidity constraints are not considered. Even if a manager has the funds, she may not want to bear the risks. External funding introduces interested parties, including creditors and equity holders, into a relationship. Then the situation becomes more complicated. Who should own the firm? Should the investor be an owner? How should control rights be allocated? Hart’s idea of optimal ownership in this paper is solely based on the avoidance of corruption. This seems rather simple-minded. Ownership likely involves more interesting matters. Hart (1988) refers to control rights that are not assignable and income rights that are, while Grossman and Hart (1986) refer to income rights that are not assignable and control rights that are. That is, Hart (1988) talks about the allocation of income rights, while Grossman and Hart (1986) talk about the allocation of control rights.

88 Incomplete contracts

5. Joint allocation of physical and human assets 5.1. Introduction This section is based on the work of Hart and Moore (1990). The paper focuses on a matching allocation of physical assets and human capital. A firm has a set of physical assets and each agent has human capital. Hence, the problem is to allocate the physical assets to match the human assets. This matching is considered to be crucial in an organization. In the paper, a physical asset is accessible to several people, some of whom have ownership rights to it. One major question is how the participants’ incentives depend on the ownership structure. Since control rights are not assignable in the paper, each agent can only decide whether or not to use his human capital (with private cost); he cannot decide how much to invest, and the right to make this decision cannot be reallocated to any other agent. Given a group of agents with a set of physical assets, the incomes from these assets are shared among the group members. Their shares are determined by the Shapley value. The basic theme is that, to produce something, physical assets and people are needed. But should the physical assets be owned by a single owner (integration) or should they be owned separately by several owners (separation)? And who should own the physical assets? 5.2. The model There are n individuals i ¼ 1; 2; …; n; and there are m pieces of physical assets a1 ; a2 ; …; am : These individuals form a group ℕ ≡ f1; 2; …ng and the available set of physical assets is A ≡ fa1 ; …; am g: There are two dates, 0 and 1: After a contract is accepted at date 0; each agent i takes an action xi : At date 1; a valuation v of output is generated and trade occurs. The cost of action is ci ðxi Þ: Allocate assets 0

x1 , . . . , xn

v 1

Valuation v is observable ex post but noncontractible ex ante. Hence, a contract specifying income rights is not possible. Investment xi s are not contractible either and their control rights are not assignable.7 Hence, a contract cannot specify values of xi s or their control rights. Consequently, there is no contract. The profit sharing will be determined by a multi-person bargaining process conducted under symmetric information. Since v is observable, a bargaining process can be conducted at t ¼ 1 to divide v: The Shapley value is used as the bargaining solution. The Shapley value is a cooperative bargaining solution under perfect information. Therefore, in fact, there is a contract, which says that the payoffs will be determined by the Shapley value. This contract also contains an allocation of physical assets.

Incomplete contracts

89

Consider a coalition S ⊂ℕ of agents who control a subset A ⊂ A of the available physical assets. Given an action profile x ¼ ðx1 ; …; xn Þ; with the set A of physical assets, the coalition S obtains value vðS; Ajx Þ; which is the maximum value that coalition S can obtain given A and x: A control structure α is a mapping from a coalition to a subset of assets. Then agent i’s income is given by his Shapley value: Bi ðαjx Þ ¼

∑

pðS Þ½vðS; αðS Þjx Þ  vðSnfig; αðSnfigÞjxÞ; ð2:14Þ

all S that contains i

where jSj denotes the number of players in coalition S and pðS Þ ¼

ðs  1Þ!ðn  sÞ! ; n!

s ¼ jSj:

That is, the Shapley value gives agent i his “expected” contribution to a coalition, where the expectation is taken over all the coalitions to which i might belong, assuming that the agent has an equal chance of joining any coalition. When we count these coalitions, the order of the members in a coalition matters. In particular, the number of coalitions that contains all the players is n!: Given the fact that agent i joins a coalition S with s ≡ jSj number of players, ðs  1Þ!ðn  sÞ! is the number of possible coalitions that the individual may join, where ðs  1Þ! is the number of possible ordered combinations of the coalition members in Snfig; ðn  s Þ! is the number of possible ordered combinations of noncoalition members in Sc ; and n! is the number of possible ordered combinations in f1; 2; …; ng: Hence, pðSÞ is the probability that this particular individual may join this particular coalition S; assuming that he has an equal chance of joining any coalition. Note that a contract contains an allocation of the physical assets only; there is no value-sharing agreement. The bargaining happens ex post at t ¼ 1; when α and x are given. Hence, Bi ðαjxÞ is dependent on x: Since output is already there and an agreement means an immediate allocation of cash on the table, we only need observability of the output. In many cases, the valuation vðS; Ajx Þ is very simple: if and only if certain players, named indispensable players, are in the coalition S will vðS; Ajx Þ yield a fixed number v; and any other player i is a valueless player, so that vðSnfig;αðS figÞjx Þ ¼ v: In this case, there are two ways to apply the Shapley value. In the first case, suppose that everyone in ℕ is indispensable, in the sense that if everyone is in a coalition then the coalition yields v; but if one of them is missing then the coalition yields nothing. If so, there is only one possible coalition, which is S ¼ ℕ; with s ¼ n and vðSnfig;αðS nfigÞjxÞ ¼ 0 for any x and i: Then a player’s Shapley value is 1 Bi ¼ v: n

ð2:15Þ

That is, the Shapley value in this case simply divides the total surplus equally without referring to an outside option. Then the profit of each agent is 1 πi ¼ v  ci ðxi Þ: n

90 Incomplete contracts In this case, this ℕ is supposed to be endogenously formed at t ¼ 0 and it includes only indispensable players. The value sharing is done at t ¼ 1:8 In the second case, ℕ is supposed to be exogenously given and it includes all the players. In this case, if each player is either indispensable or valueless, then each indispensable player’s Shapley value is given by (1.15) and each valueless player’s Shapley value is zero. Let us use one example to explain the above two alternative approaches to Shapley valuation. Suppose that there are three players i ¼ 1; 2 and 3; where players 1 and 2 are indispensable and player 3 is valueless. This means that, for any coalition S; if and only if both players 1 and 2 are in the coalition will the coalition yield value v; otherwise it yields nothing. What are the Shapley values of these three players? Using the first approach, we have ℕ ≡ f1; 2g at t ¼ 0: Then, by (1.15), players 1 and 2’s Shapley value at t ¼ 1 is Bi ¼ v=2: By the second approach, ℕ ≡ f1; 2; 3g: For player 1, he may join these coalitions: S1 ¼ f1g;

S2 ¼ f1; 2g;

S3 ¼ f1; 3g;

S4 ¼ f1; 2; 3g:

Coalitions S1 and S3 yield nothing, and in coalitions S2 and S4 player 1’s contribution is v: Hence, by (1.14), player 1’s Shapley value is B1 ¼

1 2! v vþ v¼ : 2 3! 3!

Similarly, B2 ¼ v=2: For player 3, he may join S1 ¼ f3g;

S2 ¼ f1; 3g;

S3 ¼ f2; 3g;

S4 ¼ f1; 2; 3g:

However, he contributes nothing in all the coalitions. Hence, B3 ¼ 0: The Shapley value is similar to the Nash bargaining solution that allocates equal shares of surplus to everyone, except that the status quo doesn’t have any implication for the Shapley value. More specifically, let B^ i be individual i’s benefit if Nash bargaining fails and let v be the total benefit for the whole society if they reach an agreement. Then the Nash bargaining solution is ! n X 1 B^j : v  Bi ¼ B^i þ n j¼1 This contrasts with the Shapley value in (1.15). 5.3. Examples Hart and Moore (1990) use three examples to illustrate their main ideas. In these examples, we will use (1.15) to calculate the Shapley value. Example 1: the owner should be indispensable Suppose that there are a chef, a skipper and a tycoon. There is one asset: a luxury yacht. The service (output) is gourmet seafare. By investing $100; the chef can produce a value of $240 to the tycoon. The skipper has no ability to produce any value. Assume there are no other customers and there is no other chef available.

Incomplete contracts

91

If the skipper owns the yacht, the chef will not invest, since only the group ℕ ≡ fskipper; tycoon; chefg yields a value and this value will be a three-way split. The value is $40 and each gets $80 (the Shapley value). However, this $ 80 cannot cover the chef’s investment of $100 Hence, the chef will not invest and the group yields nothing. If the tycoon owns the yacht, the chef will invest, since the group ℕ ≡ ftycoon; chefg yields a value and the value will be a two-way split. The value is $240 and each gets $120 which covers the chef’s investment. The skipper will not be invited. If the chef owns the yacht, the group ℕ ≡ ftycoon; chefg yields $240 The chef will invest, since the payoff will again be a two-way split and each gets $120 which covers the chef’s investment. Again, the skipper will not be invited. That is, when the chef or the tycoon owns the asset, the outcome is efficient; but if the skipper owns it, the outcome is inefficient. Therefore, those individuals who are indispensable or whose investments are crucial should own the asset. Example 2: the owner is whoever investments are specific to We now modify the above example so that the skipper can also invest. The skipper can invest $100 to create a value of $240 to the tycoon. If both the chef and the skipper invest, the total value is $480: If the skipper is the owner, he will form a group ℕ ≡ fskipper; tycoong and invest to produce $240. He and the tycoon will get $120 each. The skipper will not invite the chef, since the chef will choose not to invest. If the chef invests, the chef gets $160 out of a total of $480, resulting in a profit of $60. This is worse than not investing and getting $80 out of a total of $240. The situation is symmetric if the chef is the owner. If the tycoon is the owner, he will make deals with the skipper and the chef separately by forming two groups ℕ1 ≡ ftycoon; chefg and ℕ2 ≡ ftycoon; skipperg: In this case, both the skipper and the chef will invest and produce $240 each, and they will split the $240 evenly with the tycoon. In this case, social welfare is $480−100−100=$280, which is larger than in the earlier case. Therefore, when both the chef and the skipper take actions specific to the tycoon, it is strictly better when the tycoon owns the yacht than when either the chef or the skipper does. We see that it is most efficient to give ownership of assets to whomever the investments are specific to. Example 3: sole ownership is better for complementary assets Suppose now that the tycoon can also invest to create a value of 240: Further, suppose that the actions of the chef and the skipper are no longer specific to the tycoon. Instead, all of our three agents provide services to customers (who are outside our model setting).9 We continue to assume that the benefit of each agent’s action is $240; but the cost to agent i is ci instead of $100: Suppose also that the yacht consists of two pieces; neither piece is of any use without the other.

92 Incomplete contracts Would it ever be optimal for the chef to own one piece and the skipper to own the other? The answer is no. Under separate ownership, the chef, skipper, and tycoon would only act if, respectively, 240 > c1 ; 2

240 > c2 ; 2

240 > c3: 3

The chef needs the skipper to produce. Hence, the chef would only invest if 240=2 > c1 , since he has to pay half of the revenue to the skipper for using the asset. If the skipper considers offering his services (in addition to his deal with the chef before), he has to pay half of the revenue to the chef for using the yacht; so the skipper will act only if 240=2 > c2 . However, the tycoon needs both the chef and the skipper to produce. Hence, the tycoon only invests if 240=3 > c3 . In this case, the chef and the skipper will provide their own services if the relevant conditions are satisfied, just as before. This means that a service will be provided if one of the above conditions is satisfied. If two of the conditions are satisfied, two services will be provided; and if all conditions are satisfied, three services will be provided. Would it be better if the entire yacht were given to one individual, say the chef? In this case, the chef will act as long as 240 > c1 : If 240=2 > c2 , the skipper will provide his own service by paying the chef half of the revenue for using the yacht. Similarly, the tycoon will provide his service if 240=2 > c3 : Thus, a service will be provided if one of the following conditions holds: 240 > c1 ;

240 > c2 ; 2

240 > c3: 2

Again, a service will be provided if one of the above conditions is satisfied. If two of the conditions are satisfied, two services will be provided; and if all conditions are satisfied, three services will be provided. Clearly, giving both pieces of the asset to one agent leads to fewer holdups and greater efficiency. Therefore, assets that are highly complementary should be owned by one party only. 5.4. Discussion Hart and Moore (1990) are unique in that they view a firm as the outcome of a matched set of physical and human assets. The boundary of the firm is defined by an optimal allocation of assets, where physical assets are assigned to match human assets. Given an allocation of ownership over physical assets, humans gather to form coalitions. When a player joins a coalition, he not only brings in his own human assets; he also brings in the physical assets under his control. A firm ensures that each player is paid according to his contributions to the firm. Some economists view a firm as a set of contracts. Within the firm, there are contracts between the manager and his employees and between the owners and the manager. This firm may also have relationships with other firms defined by contracts. Hence, this set of contracts defines the firm and its boundary.

Incomplete contracts

93

6. Joint allocation of income and control rights This section is based on the work of Wang and Zhu (2005). In Grossman and Hart (1986), control rights are assignable, but income rights are not. In Hart (1988), income rights are assignable, but control rights are not. In Hart and Moore (1990), both control rights and income rights are assignable, but they must be bundled together, in the sense that control rights automatically imply full income rights. In Wang and Zhu (2005), both control rights and income rights are assignable, and they are not bundled together per se in the model setting. They show that, in equilibrium, control and income rights are bundled together in some circumstances, but not so in others. 6.1. The model Project Two risk-neutral parties, M1 and M2 ; are engaged in a joint venture. There are two periods (see Figure 2.3). The two agents contribute unverifiable investments, e1 and e2 ; in the first period. Specifically, in period 1 (ex ante), unverifiable investments (efforts) e1 and e2 are invested simultaneously with joint effort h ¼ hðe1 ; e2 Þ and private costs c1 ðe1 Þ and c2 ðe2 Þ: In period 2 (ex post), an action q ∈ ½0; 1 is taken by the controlling party or parties. Here, q is uncontractible ex ante, but its control right is assignable ex ante and q is contractible ex post. e1,e2

Contracting

ω realized renegotiation

q

Allocating revenue

1 Ex post

0 Ex ante

2

Figure 2.3 The timing of events.

There is uncertainty in the first period, and a random event ω ∈ Ω is realized at t ¼ 1; where Ω is the sample space. The two parties are allowed to renegotiate the initial contract at t ¼ 1: The output x is produced at t ¼ 2 and it is allocated according to the existing contract. Private benefit Let X∗ ðω; h Þ be the ex-post efficient revenue/output, dependent on random event ω and joint effort h: The controlling party can steal part of the total output. Specifically, the controlling party has the right to choose q ∈ ½0; 1 such that ð1  qÞX qX bi ð1  qÞX



is diverted for private use; is public revenue; is private benefit of the controlling party;

94 Incomplete contracts where bi ∈ ½0; 1Þ is Mi ’s ability to divert resources. That is, the controlling party may report only a fraction qX∗ of this X∗ : However, due to the possibility of renegotiation (which ensures ex-post efficiency), in equilibrium, the controlling party will always supply the efficient q∗ ¼ 1 to produce X∗ : That is, by the possibility of renegotiation, the ex-post efficient q∗ ¼ 1 will always be chosen. Contract Suppose that the two parties negotiate an ex-ante contract at t ¼ 0 and, if necessary, they renegotiate an ex-post contract at t ¼ 1: Given that the announced revenue qX∗ and the control rights of q are contractible ex ante, at the beginning of period 1, the two parties sign an ex-ante contract for:  

allocation of public revenue qX∗ (income rights) allocation of control rights over q (control rights)

This contract can be renegotiated ex post, after the investments are sunk, but before q is taken. Following the literature, we assume that renegotiation ensures ex-post efficiency. This means that the ex-post contract is ex post contract ¼ fq ¼ 1g This contract will be signed after a lump-sum monetary transfer between the two parties is made. We want to identify the ex-ante contract.

6.2. The solution We first consider the benchmark case in which action q is contractible ex ante. If so, contract negotiation implies the maximization of the joint surplus at t ¼ 0; which obviously implies the following solution:   ex ante contract ¼ α1 ¼ α∗1 ; α2 ¼ α∗2 ; q ¼ 1 ; where α∗1 and α∗2 are determined in the following proposition. Let f ðω Þ be the sample density function and Z Rðe1 ; e2 Þ ≡ X∗ ½ω; hðe1 ; e2 ÞfðωÞdω: Ω

The following proposition is from Chapter 1. Proposition 2.2. Under Assumptions 1.7–1.9, with double moral hazard and double risk neutrality, there exists a linear revenue-sharing rule s∗i ðxÞ ¼ α∗i x;

Incomplete contracts where α∗i ¼ max

e1 ;e2 2E

s:t:

c i0 ðe∗i Þ R 0i ðe∗1 ;e∗2 Þ

95

; which induces the second-best efforts e∗i > 0; determined by

Rðe1 ; e2 Þ  c1 ðe1 Þ  c2 ðe2 Þ R01 ðe1 ; e2 Þ ¼ c01 ðe1 Þ þ

h01 ðe1 ; e2 Þ 0 c 2 ðe2 Þ: h02 ðe1 ; e2 Þ

In addition, we have 0 < α∗i < 1; and the first-best outcome is not achievable. However, if action q is contractible ex post, but not ex ante, the controlling party will be in a position to use control rights for personal benefits. As will be shown later, to prevent corruption, we need a minimum output share for the controlling party i; such as αi ≥ k for some k ≥ 0: Without this condition αi ≥ k; we know that the optimal share is α∗i ; with this condition αi ≥ k; what is the optimal share? The following lemma gives the answer. Lemma 2.1. Suppose that firm i has the control and it must be given a share satisfying αi ≥k for some k ≥0: Then, if α∗i ≥ k; the optimal share for firm i is ^i ¼ k ðsee Figure 2:4Þ: α^ i ¼ α∗i ; but, if α∗i ≤ k; the optimal share for firm i is  b2 1

α*2

α1 0

α*1

k

1

b1

Figure 2.4 Lemma 2.1.

Proposition 2.1 is about the allocation of income rights, given the allocation of control rights. For the allocation of control rights, there are several possible choices: (1) M1 has control, (2) M2 has control, and (3) joint control. In the case of joint control, we assume that q is decided jointly by the two parties and they share output equally at αi ¼ 0:5: We now try to determine the control rights. At time t ¼ 1, given its revenue share αi in the initial contract, the controlling party considers its income:

96 Incomplete contracts i qX þ bi ð1  qÞX ¼ bi X þ ði  bi ÞqX :

ð2:16Þ

Hence, purely from this income, it will choose q ¼ 1 if αi ≥ bi ; but, if αi < bi , it will choose q ¼ 0. However, there is an alternative: the controlling party may demand renegotiation. Obviously, q ¼ 0 cannot imply an optimal solution; in fact, both parties can benefit from renegotiation in this case. In the case with αi < bi , if bargaining fails, the controlling party will choose q ¼ 0 so that the controlling party gets bi X∗ and the other party gets nothing; if bargaining leads to an agreement, the agreement must lead to q ¼ 1 so that social welfare is maximized ex post and is X∗ : Hence, by the Nash bargaining solution, from renegotiation, the controlling party gets 1 1 þ bi  X : bi X þ ðX  bX Þ ¼ 2 2 and the noncontrolling party gets 1 1 þ bi  0 þ ðX  bX Þ ¼ X : 2 2 i Since 1þb 2 > bi , by Lemma 2.1, this solution is worse than giving the controlling party the revenue share α^i ¼ bi . Thus, the controlling party should never be given a revenue share less than bi . This means that, if α∗i ≥ bi ; by (1.16), Mi will choose q ¼ 1 and both parties will choose the second-best effort, assuming Mi is the controlling party and is given αi ¼ α∗i : If Mi is assigned as the controlling party and α∗i < bi ; it should be given the share αi ¼ bi . The above analysis applies to each of the three possible allocations of control rights. We need to compare the social welfare of these three alternative allocations of control rights to identify the optimal one. After that, we can immediately derive the corresponding allocation of income rights from the above analysis. Given a set of specific functions, we can proceed to identify the optimal allocation of control rights. The following proposition identifies the optimal solution for a given set of simple parametric functions.

Proposition 2.3. Consider a parametric case with hðe1 ; e2 Þ ¼ 1 e1 þ 2 e2 ;

~ ~ XðhÞ ¼ Ah;

ci ðei Þ ¼ e2i =2;

ð2:17Þ

where μ1 ; μ2 > 0; A~ is a random variable with A~ > 0 and EðA~ Þ ¼ 1: Let ðα∗1 ; α∗2 Þ with α∗1 < α∗2 be the second-best sharing rule. Then, (1) If α∗i ≥ bi holds for both i ¼ 1 and i ¼ 2; an ex-ante contract that specifies αi ¼ α∗i and gives either party the control rights implements the second-best outcome. (2) If α∗i ≥ bi holds for either i ¼ 1 or i ¼ 2 but not both, an ex-ante contract that specifies αi ¼ α∗i and gives Mi the control rights to the party with α∗i ≥ bi is the unique optimal contract and implements the second-best outcome.

Incomplete contracts

97

(3) If α∗i < bi holds for both i ¼ 1 and i ¼ 2; the second-best outcome cannot be achieved. Depending on the parameters, either single-party control or joint control would be optimal. (a) If α∗1 < b1 < 1=2 and b2 > b1 þ α∗2  α∗1 ; having M1 in control is optimal. (b) If α∗2 < b2 < 2α∗2  21 and b2 < b1 þ α∗2  α∗1 ; having M2 in control is optimal. (c) If b1 > 1=2 and b2 > 2α∗2  21 ; joint control is optimal. Proposition 2.3 is graphically illustrated in Figure 2.5. The solution divides the possible combinations of ðb1 ; b2 Þ into six zones. The solution may be either the second or third best, where the third best is the optimal solution when the solution is inferior to the second best. In each zone, the optimal allocations of control and income rights are indicated, where SB stands for the second best and TB stands for the third best. b2 M1 control α1 = b1, TB

1 M1 control α1 = α*1, SB

2α2* – 0.5

Joint control, αi = 0.5, TB M2 control, α2 = b2, TB

α2* 0.5 Any single-party control, αi = α*i , SB

M2 control, α2 = α*2, SB

α1*

0

0.5

1

b1

Figure 2.5 Optimal control zones.

To show that our conclusion is not dependent crucially on the choice of functions in (1.17), we have chosen a set of more general functions in (1.18) and derived the control zones in Figure 2.6. By comparing Figure 2.5 with Figure 2.6, we can see that the choice of functions has little effect on our conclusions. Proposition 2.4. Consider a parametric case with μ

μ

hðe1 ; e2 Þ ¼ e11 e22 ;

X∗ ðω; h Þ ¼ A~ h; ~

ci ðeÞ ¼ ei = i ; ~

ð2:18Þ ~

where μ1 ; μ2 > 0;  i ≥ 1; A is a random variable, A > 0 and EðA Þ ¼ 1: Let ðα∗1 ; α∗2 Þ with α∗1 < 0:5 be the second-best sharing rule. Then, for β∗1 ≡ 0:5 and some constant β∗2 ; the corresponding control zones are indicated in Figure 2.6.

98 Incomplete contracts b2 B: M1 control α1 = b1, TB

1 A: M1 control, α1 = α1* , SB β *2

F: Joint control, αi = 0.5, TB

C: M2 control, α2 = b2, TB

α *2 0.5 E: Any single - party control, αi = α*i , SB

0

D: M2 control, α2 = α2* , SB

α1*

β *1 = 0.5

1

b1

Figure 2.6 Optimal control zones.

6.3. Discussion When we drew the above two figures, we assumed α∗1 < α∗2 : We first found that, although M1 is a minority holder (with an income share less than half), having M1 in control would be optimal when M1 ’s ability to acquire private benefits is weak. Specifically, in zones A and B, although M1 is a minority holder with income shares α1 ¼ α∗1 < 0:5 and α1 ¼ b1 ≤ 0:5; respectively, it is optimal for M1 to have the control right. Second, we found that income and control rights are often bundled together, in the sense that a majority holder tends to have control rights. Specifically, in zones C and D, M2 is a majority holder with income shares α2 ¼ b2 > 0:5 and α2 ¼ α∗2 > 0:5; respectively, and it is optimal for M2 to have the control rights. This bundling principle also holds in zone F, where the two parties have equal income shares and equal control rights. Zone E is special, in which, if the control right is given to M2 , then the bundling principle holds. Third, unbundling of control and income rights can sometimes be optimal. In zones A and B, although M1 is a minority holder, it is optimal for M1 to have the control rights. In zone E, if the control right is given to M1 ; then the minority holder has the control rights. When there is limited opportunity for private benefits, i.e. in zone E, the secondbest contract with income shares α∗1 and α∗2 and control rights by any single party is optimal. In contrast, when there is ample opportunity for private benefits, i.e. in zone F, joint control is optimal.

Incomplete contracts

99

Our results can explain different types of ownership structures: – – – –

EJVs, CJVs, 50:50 arrangements, 51:49 arrangements.

In an EJV, the party with a higher income share automatically has control rights. In a CJV, income and control rights are defined separately. There are no equity shares in a CJV; instead, annual income shares for both parties are specified, and control rights are independent of income rights. A key difference between EJVs and CJVs is that in EJVs income and control rights are bundled together, while in CJVs they are separated. The solution in zones C, D and E can be implemented by EJVs, while the solution in zones A and B can be implemented by CJVs, but not by EJVs. The solution in zone E can be implemented by either an EJV (if M2 has control) or a CJV (if M1 has control). Zone F indicates that a 50:50 arrangement can be optimal. Finally, our model can be modified to show that a 51:49 arrangement can also be optimal (see Wang and Zhu 2011). The Coase Theorem says that, in a world without transaction costs, ownership doesn’t matter. Why does ownership matter in our model, then? Even in zone E, the control rights are not supposed to be given to joint owners or a third party. Why? Coase only says that who the initial owner of control rights is doesn’t matter; as long as the control rights can be transferred without cost, the user of control rights will be the most suitable person to have those rights. For example, if M2 has the control rights initially in zone A, then he can do better by selling his control rights to M1 for a payment, which can benefit both parties. It doesn’t matter who has the initial control rights. As long as there is no transaction cost involved in a transfer of control rights, the control rights will end up in the right hands. The Coase Theorem can actually be applied to all the control areas in Figure 2.6 in that the control rights can be given to any party. Who the initial owner is has no effect on the final outcome. A more difficult question is: who should the final owner or the user of control rights be? Coase does not answer this question. Wang and Zhu (2005) do.

7. Contractual incompleteness for external risks So far in this chapter, we have assumed incomplete contracts and discussed the implications of control rights and income rights. In this section, we provide a justification for contractual incompleteness based on the argument of early commitment presented by Wang (2010). To justify incomplete contracts, Wang reconsiders the standard agency model by adding an exit option for the principal. The option is contractible, but the principal may choose not to write it into a contract. Wang identifies some interesting circumstances under which the option is not in the optimal contract. The main conclusion is that, to cope with external risks, a contract may be optimally

100 Incomplete contracts incomplete. In general, the degree of contractual incompleteness is positively dependent on the degree of external risks. We tend to have the impression that, when some variables are verifiable and if there are no transaction costs for including them in a contract, these variables must be utilized in an optimal contract. This is incorrect. One familiar example of our model is the tenure-track contract in academia. It seems feasible to offer more complete tenure-track contracts with clearly specified and verifiable conditions for tenure. Yet, tenure-track contracts are universally incomplete. The degree of incompleteness differs among universities and it is typically less complete for top-tier universities. At a top-tier university, a junior member’s performance is likely to be unsatisfactory at the time of the tenure decision. Wang’s study indicates that it is optimal for such a university to offer a highly incomplete tenure-track contract. 7.1. The model Consider an agency model in which a principal hires an agent to undertake a project. The project lasts two periods and yields a random output x at the end (date 2). In the first period, each party invests an effort, a from the agent and b from the principal, with private costs cða Þ and CðbÞ; respectively. The two efforts generate a joint effort hða; bÞ: The agent’s ability is represented by an index θ: The density function of the output is f ½x; θ; hða; b Þ: The timing of the events is shown in Figure 2.7. Ability θ is unknown to both parties at date 0 (ex ante); it becomes known to both parties at date 1 (ex post). The disinformation about θ is symmetric, meaning that the two parties have a common distribution function Φðθ Þ ex ante. As in the standard agency model, output x is verifiable and efforts a and b are observable ex post but not verifiable. An output-sharing rule is a real-valued function s:X ! ℝ; where X is the output space. In addition to the contractible output, we also allow the principal to have a contractible exit option. When the agent’s ability θ is revealed ex post, the principal has the option to exit from the output-sharing agreement sðx Þ and replace the agent by an outside agent from the ex-post labor market. The outside agent with ability θ0 is available ex post for a fixed wage rate s0 : The ex-post labor market is uncertain ex ante. That is, θ0 and s0 may be random ex ante and they only become known ex post. For our purpose, it suffices to allow only s0 to be random ex ante, and we will assume θ0 to be known ex ante. Assume that the distribution function of s0 is public knowledge ex ante and denote its mean Contracting, θ random

a,b

0 Ex ante

Figure 2.7 The timing of events.

Renegotiation, θ realized

Revenue distributed

1 Ex post

2 End

Incomplete contracts

101

as s 0 : Assume also that output x, ability θ; and s0 are independently distributed ex ante. Like the output, θ is random ex ante but is verifiable ex post. In other words, the principal can state explicitly in the contract that, when the agent’s ability falls below a certain level, he will be dismissed automatically. In this sense, we say that the exit option is contractible. The principal’s possession of this contractible exit option is a unique feature in the model. A key conclusion of Wang’s study is that the principal may choose not to include this exit option in a contract under certain conditions. For convenience, we call a contract a complete contract in this section if all contractible variables are employed in the contract; otherwise, we call it an incomplete contract. In this model, the principal will always include an output-sharing rule in a contract and, in addition, she has the option to include the exit option in the contract. If she explicitly includes the exit option in the contract, the contract is complete; otherwise, the contract is incomplete. As shown in Figure 2.7: 





At t ¼ 0; the principal offers a sharing rule sðxÞ; based on ex-post revenue x; to the agent. At the same time, depending on the type of contract, the principal may offer a commitment of continuous hiring in the contract. If the contract is accepted, the two parties simultaneously invest a and b and incur costs cða Þ and Cðb Þ; respectively. At t ¼ 1; the uncertainty surrounding θ is resolved. The principal considers the option to replace the agent with an outside agent, who has ability θ0, for a fixed salary s0 : A decision to hire a replacement will be based on the type of contract and the relative quality and cost of the outside agent. At t ¼ 2; the project is finished and payments are made based on the revenuesharing rule sðx Þ or s0 :

We denote the ex-post expected revenue as Z rða; b; θ Þ ≡ xf½x; θ; hða; bÞdx; X

and the ex-ante expected revenue without the exit option as Z ∞ Rða; bÞ ≡ rða; b; θ ÞdΦðθ Þ: 0

The ex-post expected welfare gain from exercising the option is gða; b; θ; s0 Þ ≡ rða; b; θ0 Þ  rða; b; θ Þ  s0 ; where rða; b; θ Þ is the loss of expected revenue if the agent is dismissed, rða; b; θ0 Þ is the expected revenue from an outside agent, and s0 is the cost of

102 Incomplete contracts hiring the new agent. Note that the costs of efforts have already been sunk at this point in time. At t ¼ 0; given an arbitrary random variable θ~; define Z  ~ Rða; b; θ~Þ≡Rða; bÞ þ Gða; b; θ~Þ; Gða; b; θÞ ≡ E gða; b; θ; s0 ÞdΦðθ Þ; 0

~ the gain function ~ and s0 : We call Gða; b; Þ where E is the expectation operator on , ~ and Rða; b; Þ the revenue function. Standard assumptions are imposed on the functions: ha ða; b Þ > 0; ra ða; b; θ Þ > 0; cða Þ > 0; CðbÞ > 0;

hb ða; b Þ > 0; rb ða; b; θ Þ > 0; c0 ða Þ > 0; C0 ðb Þ > 0;

haa ða; b Þ ≤ 0; rθ ða; b; θ Þ > 0; c ″ ða Þ > 0; C″ ðb Þ > 0:

hbb ða; b Þ ≤ 0;

7.2. Incomplete contracts Assume that the principal decides to offer an incomplete contract without any guarantee on continuous hiring. Given the agent’s type θ and efforts ða; bÞ; the option will be exercised if and only if gða; b; θ; s0 Þ > 0: Define θi ða; b; s0 Þ by the equation: rða; b; θ0 Þ ¼ rða; b; θi Þ þ s0 ;

ð2:19Þ

where the subscript i stands for incomplete contract. This θi is the threshold at which the principal decides to replace the agent ex post. The ex-ante expected gain from the option is G½a; b; θi ða; b; s0 Þ and the ex-ante expected revenue is R½a; b; θi ða; b; s0 Þ: The ex-ante utility values of the agent and the principal are, respectively, Z Z Uða; b; sÞ ≡ E

∞

θi ða;b;s0 Þ

Z

i ða;b;s0 Þ

Vða; b; sÞ  E

sðx Þf ½x; θ; hða; b Þϕðθ Þdxdθ; ½rða; b; 0 Þ  s0 ϕðÞd

0

Z

þE

1 i ða;b;s0 Þ

Z ½x  sðxÞ f ½x; ; hða; bÞϕðÞdxd

ð2:20Þ

¼ R½a; b; i ða; b; s0 Þ  Uða; b; sÞ: Suppose that, after the contract is signed, the two parties play a Nash game to determine efforts a and b: Then the principal’s problem is i 

max

s2S; a2A; b2B

s:t:

R½a; b; i ða; b; s0 Þ  Uða; b; sÞ  CðbÞ ICa : Ua ða; b; sÞ ¼ c0 ðaÞ; ICb : Vb ða; b; sÞ ¼ C0 ðbÞ; IR : Uða; b; sÞ  cðaÞ:

ð2:21Þ

Incomplete contracts

103

Proposition 2.5 (Incomplete Contracts). With incomplete contracts, the solution is determined by a linear contract sðx Þ ¼ α þ βx, and the efforts ðai ; bi Þ are determined by the following problem: i  max R½a; b; i ða; b; s0 Þ  cðaÞ  CðbÞ a2A; b2B

s:t: Ra ½a; b; i ða; b; s0 Þ ¼ c0 ðaÞ þ

ha ða; bÞ 0 C ðbÞ: hb ða; bÞ

ð2:22Þ

7.3. Complete contracts Assume now that the principal decides to offer a complete contract with a conditional guarantee on continuous hiring. Suppose now that the principal commits ex ante to replacing the agent when his ability is lower than a certain level θc ; where the subscript c stands for complete contract. Since θc is verifiable, it is contractible. The ex-ante expected gain from the option is Gða; b; θc Þ and the ex-ante expected revenue is Rða; b; θc Þ: The ex-ante utility values of the agent and the principal are, respectively, Z 1Z sðxÞf ½x; ; hða; bÞϕðÞdxd; Uða; b; c ; sÞ  c

Z Vða; b; c ; sÞ 

1

c

X

Z

½x  sðxÞ f ½x; ; hða; bÞϕðÞdxd Z

X

þE

c

ð2:23Þ

½rða; b; 0 Þ  s0 ϕðÞd

0

¼ Rða; b; c Þ  Uða; b; c ; sÞ: Suppose that, after the contract is signed, the two parties play a Nash game to determine efforts a and b Then the principal’s second-best problem is c 

max

s2S; a2A; b2B; c 0

s:t:

Rða; b; c Þ  Uða; b; c ; sÞ  CðbÞ ICa : Ua ða; b; c ; sÞ ¼ c0 ðaÞ; ICb : Vb ða; b; c ; sÞ ¼ C0 ðbÞ;

ð2:24Þ

IR : Uða; b; c ; sÞ  cðaÞ: Proposition 2.6 (Complete Contracts). With complete contracts, the solution is determined by a linear contract sðx Þ ¼ α þ βx and the efforts and threshold ðac ; bc ; θc Þ are determined by the following problem: c 

max

a2A; b2B; c 0

s:t:

Rða; b; c Þ  cðaÞ  CðbÞ IC : Ra ða; b; c Þ ¼ c0 ðaÞ þ

ha ða; bÞ 0 C ðbÞ: hb ða; bÞ

ð2:25Þ

104 Incomplete contracts 7.4. Superiority of the incomplete contract We now compare the two contracts and find conditions under which one contract is superior to the other. For this purpose, we choose the following simple parametric functions:   ~ cða Þ ¼ 1 a2 ; CðbÞ ¼ 2 b2 ; hða; b Þ ¼ μ1 a þ μ2 b; xðθ; h Þ ¼ Aθh; 2 2 ~ ~ where μi ≥ 0; A is random ex post with EðAÞ ¼ 1; θ takes two possible values θL and θH ex ante with θL ≤ θH and with probability pθ of taking θL ; and s0 also takes two possible values sL and sH ex ante with sL ≤ sH and with probability ps of taking sL : The means of s0 and θ are respectively s0  ps sL þ ð1  ps ÞsH ;   p L þ ð1  p ÞH : Then the problem under an incomplete contract can be written as   s0  cðaÞ  CðbÞ i ¼ max R a; b; 0  a2A; b2B hða; bÞ   s0 ha ða; bÞ 0 s:t: Ra a; b; 0  ¼ c0 ðaÞ þ C ðbÞ; hða; bÞ hb ða; bÞ and the problem under a complete contract can be written as    s0  cðaÞ  CðbÞ max R a; b; 0  c ¼ a2A; b2B; λ0 λ þ hða; bÞ    s0 ha ða; bÞ 0 s:t: Ra a; b; 0  ¼ c0 ðaÞ þ C ðbÞ: λ þ hða; bÞ hb ða; bÞ See Wang for the derivation. Suppose 0  L < 0 < H ;

L  0 

 s0 : hðac ; bc Þ

ð2:26Þ

Those cases beyond these two conditions are too trivial to discuss. The default restriction of sL ≤ sH and the second condition of (1.26) define the three areas A; B and C in Figure 2.8. We find   

In area C; sL and sH are too high, so that there is no gain from the exit option, implying that the incomplete contract is inferior to the complete contract. In area B; both sL and sH are low, so that the exit option will always be used by both contracts, implying that the two contracts are equally efficient. In area A; when sH is large enough, the incomplete contract is better; otherwise the complete contract is better. The derivations appear in the rest of this section.

We now find a condition on sH in area A by which the incomplete contract is strictly superior. In area A; the solutions to the two problems are

Incomplete contracts "

  # μ1 1 1 2  1 ac ¼ ½pθ ðθ0  θL Þ þ   þ ; γ1 μ1 μ21  22 "   # ; μ2 1 1 2  1  þ ; bc ¼ ½pθ ðθ0  θL Þ þ   2 μ2 μ21 μ22 and "   #

μ1 1 1 2  1  þ ; ai ¼ ps pθ ðθ0  θL Þ þ  γ1 μ1 μ21 μ22 "   # ;

μ2 1 1 2  1  þ ; bi ¼ ps p θ ð θ 0  θ L Þ þ   2 μ2 μ21 μ22

and "   # ½pθ ðθ0  θL Þ þ   2 μ21 μ22 1 2  1 Πc ¼ þ  þ  pθ s 0 ; 2 1 2 μ21 μ22

2 "   # ps pθ ð θ 0  θ L Þ þ  μ21 μ22 1 2  1 þ  þ  ps p θ s L : Πi ¼ 2 1 2 μ21 μ22

sL pssL + (1 − ps)sH = (θ0 − θL)h(ac , bc) 45o line

C

(θ0 − θL)h(ai , bi)

B

A

(θ0 − θL)h(ai , bi)

Figure 2.8 Dominant contracts in three areas.

(θ0 − θL)h(ac , bc) 1 − ps

sH

105

106 Incomplete contracts We find that Πi > Πc if and only if "   #   2 1 þ ps μ1 μ22 1 2  1 sH > pθ ð θ 0  θ L Þ þ  þ  þ ðθ0  θL Þ: 2 1 2 μ21 μ22 ð2:27Þ The existence of such an sH in area A satisfying condition (1.27) is guaranteed by the following condition:     pθ ðθ0  θL Þð1  p2s Þ þ 2ð1  ps Þ  2 μ 1 2  1 μ2 2 þ : ð2:28Þ <  1 μ2  2 μ1 pθ ðθ0  θL Þð1 þ p2s Þ þ 2ps  Then condition (1.27) defines a region within area A in which the incomplete contract is strictly superior. Notice that condition (1.28) has nothing to do with sL and sH : We make three interesting observations. First, when ps is close to 1; condition (1.28) holds. This makes sense; when the ex-post labor market is likely to be good (a low market wage rate), the incomplete contract is better. Second, since the left-hand side of (1.28) is decreasing in pθ ; the incomplete contract is likely to be better when pθ is large. The explanation for this is that, when the inside agent is likely to have low ability, the incomplete contract is superior. Third, since the left-hand side of (1.28) is decreasing in θ0 ; the incomplete contract is likely to be superior when θ0 is large. The explanation is symmetric to that for the second observation. Finally, we find that the right-hand side of (1.28) is minimum if and only if μ21 μ2 ¼ 2: 1 2

ð2:29Þ

In other words, when (1.29) is satisfied, the complete contract is likely to be better. Therefore, when the two parties’ skills are close substitutes in the sense defined by (1.29), the complete contract is likely to be superior. The above analysis is summarized in the following proposition. Proposition 2.7. The optimal incomplete contract dominates the optimal complete contract only under certain conditions. Generally speaking, the incomplete contract is superior when    

the future labor market is likely to be good, or the inside agent is likely to have low ability, or a highly capable outside agent is likely to be available, or the efforts from the two parties are complementary.

7.5. Discussion Without deviating from the basic setup of the standard agency model, Wang allows the principal to have an exit option in the standard agency model and

Incomplete contracts

107

investigate the circumstances under which the principal prefers to leave the option out of the contract. We find that a promising future, complementary efforts and an incompetent inside agent are important factors in the choice of an incomplete contract. In this model, the principal must balance incentives and risks in the choice of the type of contract: an incomplete contract has a negative effect on incentives, but it gives the principal the freedom to ensure ex-post efficiency; a complete contract is good for incentives, but it ties up the principal’s hands. In this model, as uncertainty of the future labor market increases, an incomplete contract may become optimal. From a different angle, within the class of renegotiable contracts, Segal (1999) shows that, when the trading environment grows more complex, an incomplete contract (in fact, no contract) becomes optimal in the limit. These two results are complementary. There are two critical differences between this model and existing models on incomplete contracts. First, there is a crucial difference between the existing notion of incompleteness in the literature and the notion in Wang’s (2010) paper. As defined by Maskin and Tirole (1999), incompleteness means that there is no explicit contractual statement on some of the states of nature. Wang (2010) defines incompleteness as there being no contractual agreement on some of the contractible options at disposal. Thus, his model is excluded from existing incomplete contract models, and in particular from Maskin and Tirole’s (1999) model on indescribable states of nature. Nevertheless, Wang (2010), Maskin and Tirole (1999), and Tirole (1999) all agree on the key point that transaction costs and bounded rationality are unnecessary for incomplete contracts to exist. Second, in existing models, the optimal complete contract can never be strictly inferior to an incomplete contract. In Wang’s (2010) model, we find conditions under which an incomplete contract is strictly superior or strictly inferior to a complete contract. There are also two approaches to incomplete contracts. One introduces transaction costs and bounded rationality, while the other does not. Following Williamson (1975, 1985), by invoking transaction costs and/or bounded rationality, an optimal contract can be reduced to something that looks like an incomplete contract.10 In this line of research, the trading parties involved know that a complete contract would serve them better if there are no transaction costs and if they are not restricted by bounded rationality. In contrast, Wang (2010) keeps unbounded rationality and imposes no transaction cost; instead, he identifies risks as a cause for incomplete contracts. A contract is offered ex ante, but many contingencies are realized ex post. The benefit of deciding on an option ex post is the reason why the principal may offer an incomplete contract, even though a complete contract provides better incentives and is costless to make in the model. Finally, in contrast to the popular argument about transaction costs being a cause of incompleteness in contracts, Wang (2010) shows that a lack of transaction costs can also be a cause of incompleteness in contracts. In our model, there is no verification cost on a contract to the contracting parties, and this can actually cause the principal to drop some items from a contract. Without any verification

108 Incomplete contracts cost, the agent may use the court at no private cost to defend his rights as defined in a contract. Therefore, to avoid being constrained by a complete contract, the principal may choose to leave some items out. Although this has a negative effect on incentives, the principal may still prefer an incomplete contract under some circumstances.

3

Corporate finance

In this chapter, we apply the incomplete contract approach to corporate finance. We investigate how financial instruments, especially convertibles, can be used to channel funds to corporate projects. The basic financial assets are bonds and equity shares. In terms of what is written down in the contract, a bond mainly specifies a linear income scheme, while an equity share mainly specifies a voting right. Most other financial assets allow other options. There are many possible options. These options are usually assignable ex ante and exercisable ex post, and hence are called ex-post options. In reality, bonds and equity shares are often combined to form various sophisticated assets. When an asset contains a conversion option, it is generally called a convertible. A contract or financial asset that includes an ex-post option is an incomplete contract. Hence, to discuss ex-post options, we need to apply the incomplete contract approach. Besides this purpose, the approach also allows us to discuss various rights in financial assets. For example, a bond holder typically has foreclosure rights and an equity holder has voting rights. For another example, paper money under the old system is a special option contract. It can be converted to gold at anytime and it has no fixed deadline for conversion. Under a fixed exchange rate system, the conversion ratio is predetermined. Under a flexible exchange rate system, however, the conversion ratio is determined by the market on the spot (a contingent conversion ratio). But nowadays, with the new system, paper money is debt. This debt has no interest and no maturity. By the incomplete contract approach, all financial assets can be treated as contracts so that we can discuss all kinds of features in financial assets within a single framework. A financial asset can be used for different purposes by different people. For example:




An option can be used to hedge against risks by farmers. An option can be used to diversify risks by portfolio managers. An option can be used to control moral hazards and deal with information problems in corporate finance by investors.

The first purpose is the original purpose of many option instruments. Farmers face large risks between the seeding time and the harvest time. A put option allows them

110 Corporate finance to hedge against the risks. After option contracts are issued, the characteristics of these options are valuable to fund managers. These valuable characteristics include the payment schedule and risk attributes. Fund managers can mix different option instruments to reduce the overall risk and increase expected return. Hence, option instruments can also serve the second purpose. And, when a company issues an option contract or negotiates with investors to raise funds, it will have to take into account risks, asymmetric information, and moral hazards. Option instruments can help in this regard too. The main theme of this chapter is to show that options and convertibles can indeed serve the last purpose well. Option contracts are important in practice and in theory. A typical financial asset is really a basic financial instrument plus a few options. Hence, an option theory is a key to our understanding of financial assets. We first present a theory on option contracts in the first section.

1. An option theory This section is based on the work of Demski and Sappington (1991). They show that option contracts can be used to deal with moral hazards effectively in financial investment. They consider an agency problem in which a principal hires an agent to work on a project. What is new in their model is that the principal has an option and, if the principal decides to exercise it ex post, the agent is required to purchase the firm at a predetermined price. They show that, even if efforts and output are unverifiable, if the agent’s effort is observable ex post to the principal, the double moral hazard problem can be resolved completely. In a standard principal–agent model, the firm is not for sale. In their paper, there is an option to sell the firm. If there is single moral hazard, as explained before, a sale of the firm resolves the agency problem completely; if there is double moral hazard, as will be shown later, a conditional sale of the firm resolves the agency problem completely. 1.1. The setup Consider a double moral hazard agency model in which the efforts from the two parties are unverifiable. Output is also unverifiable.1 But the agent’s effort is observable to the principal. Although a contract cannot include income rights, due to the unverifiability of output, it can contain an option conditional on the observable effort. This option can be considered as a control right. Since neither effort nor output is contractible, an initial contract can only contain the option and probably an upfront transfer. We call this an option contract, and there is no need to mention explicitly an initial fixed transfer in this contract. The principal owns the firm initially. An option contract is a pair of payments ðPa ; Pb Þ that states that, if the principal exercises her option, she sells the firm to the agent for payment Pa from the agent; if the principal keeps the firm by not

Corporate finance Contracting

Investment a

Investment b

0 Ex ante

Option

1 Ex post

111

Output 2 End

Figure 3.1 The timing of events.

exercising the option, she pays the agent Pb : That is, the worker pays Pa to buy the firm and the principal pays Pb as the wage. A decision on the option is made after the investments are made (see Figure 3.1). The timing of events is as follows: 1. 2. 3. 4. 5.

The worker decides whether or not to accept the option contract ðPa ; Pb Þ: If the worker accepts the contract, he decides his effort a. After observing a; the principal decides her effort b. Afterwards, the principal decides whether to exercise her option to sell the firm to the worker. The output is realized at t ¼ 2 and it all goes to the owner.

Note that no renegotiation is allowed at any time. Also, the players play a Stackelberg game in efforts. In fact, the decisions on b and the option are made at the same time in effect. 1.2. The model Suppose that the principal is risk neutral and the agent is risk averse.2 Let x be the revenue, fðxja; b Þ be the density function, and Rða; bÞ be the expected revenue: Z Rða; bÞ ≡ xfðxja; b Þdx: Given efforts, the expected revenue is Rða; b Þ: The costs of efforts are private and are a and b. Then the principal’s effort is decided by   b ða; Pa ; Pb Þ  max max½Rða; bÞ  b  Pb ; maxðPa  bÞ : ð3:1Þ option

b0

b0



 ^ Þ; 0 ; where b ¼ 0 if the principal exercises the option. Denote the solution as bða From (3.1), the principal will retain the firm if and only if Rða; bÞ − b − Pb ≥ Pa :

ð3:2Þ

Let the agent’s utility function be UðM − a Þ;3 where M is a monetary income. Then the agent’s payoff is ( R ^ ^  Pb < Pa ; Uðx  a  Pa Þfðxja; 0Þdx if R½a; bðaÞ  bðaÞ ^ bðaÞ ^  Pb  Pa : UðPb  aÞ if R½a; bðaÞ

112 Corporate finance Define the retention function as ( ^ ^  Pb  Pa ; 1 if R½a; bðaÞ  bðaÞ rða; Pa ; Pb Þ  0 if not: Then the agent’s effort is decided by

Z

a ðPa ; Pb Þ  maxf1  rða; Pa ; Pb Þg a0

^ Uðx  a  Pa Þf½xja; bðaÞdx

þ rða; Pa ; Pb ÞUðPb  aÞ:

ð3:3Þ  Assume that the agent’s reservation value is U: Then, at the initial stage, the agent will accept the contract if and only if  ð3:4Þ a ðPa ; Pb Þ  U: At t ¼ 0, the principal’s problem is to determine a contract offer ðPa ; Pb Þ; taking into account (3.1), (3.3) and (3.4). 1.3. Result Let ða∗ ; b∗ Þ be the first-best efforts, which are determined by max Rða; bÞ − a − b: a; b

The following is the only result in the paper. Proposition 3.1. The first-best outcome can be achieved by an option contract ðPa ; Pb Þ defined by  UðPb  a Þ ¼ U:

Pa ¼ Rða ; b Þ  b  Pb ;

ð3:5Þ

Proof. The ex-post valuation of the principal is ( Pa ; if exercising the option; P ¼ max Rða; bÞ  b  P ; if not: b b

If the agent supplies a∗ ; since max Rða∗ ; b Þ − Pb − b ¼ Rða∗ ; b∗ Þ − Pb − b∗ ¼ Pa ; b

the principal will supply b∗ and retain ownership. If the agent supplies a > a∗ ; since maxfRða; bÞ  b  Pb g  maxfRða ; bÞ  b  Pb g ¼ Pa : b0

b0

the principal will retain ownership. Finally, if the agent supplies a < a∗ ; since maxfRða; bÞ  bg  Pb  maxfRða ; bÞ  bg  Pb ¼ Pa ; b0

b0

the principal will give up ownership. Therefore, the principal will not exercise the option if and only if a ≥ a∗ : Thus the ex-ante valuation of the agent is

Corporate finance  a ¼

UðPb  aÞ; R Uðx  a  Pa Þfðxja; 0Þdx;

113

if a  a ; if a < a :

From this, we can see that the agent will never choose a > a∗ ; since it is definitely worse than choosing a ¼ a∗ : If the agent chooses a < a∗ ; then he will own the firm. In this case, the principal will not make any investment, i.e. b ¼ 0; and the agent will get Z R  a ðPa ; Pb Þ ¼ Uðx  a  Pa Þfðxja; 0Þdx  U ðx  a  Pa Þfðxja; 0Þdx ¼ U½Rða; 0Þ  a  Pa  ¼ U½Rða; 0Þ  a  Rða ; b Þ þ b þ Pb  ¼ Uf½ðRða; 0Þ  0  aÞ  ½Rða ; b Þ  a  b Þ þ ðPb  a Þg  < UðPb  a Þ ¼ U: Again, this is worse than picking a ¼ a∗ : Thus, the optimal choice of the agent is a ¼ a∗ : If so, the principal will respond by supplying b∗ and retaining ownership. This outcome is the first best. Q.E.D. 1.4. Discussion Since efforts and output are not verifiable, only a simple contract is possible. It turns out that the double moral hazard problem can be resolved completely when the principal, who can observe the agent’s action, has the option of requiring the agent to purchase the firm at a predetermined price. The threat of becoming a full residual claimant makes the agent want to work hard. As long as the agent has incentives to work hard, the principal will have incentives to keep the ownership. It is the incomplete contract approach that results in efficiency. The simplicity of the solution is amazing. The key is an ex-post conditional ownership transfer. Although the principal doesn’t face a holdup problem in equilibrium, it is not a single moral hazard problem, and hence an upfront ownership transfer cannot lead to efficiency. Demski and Sappington (1991) were the first to propose an ex-post conditional ownership transfer. One key assumption is that efforts are sequential. The principal’s effort comes after the agent’s effort, which allows the principal to make her effort conditional on the agent’s effort. This offers the principal an additional instrument, her investment, to deal with the agent’s incentive problem. In contrast, simultaneous investments, without this second instrument, may not achieve efficiency. As shown by Edlin and Hermalin (2000), simultaneous investments can achieve efficiency under the same setting only if the investments are cooperative. We will discuss Edlin and Hermalin’s paper (2000) in Section 3. In addition, the option is exercisable after the agent’s effort. Since the principal can exercise the option after she has observed the agent’s effort, she can punish the agent for any deviation. Both the principal’s effort and the option will put pressure on the agent to perform.

114 Corporate finance This option contract is not renegotiable. What will happen if it is renegotiable? A number of studies, such as those of Nöldeke and Schmidt (1995) and Edlin and Hermalin (2000), have extended the work of Demski and Sappington (1991) by allowing ex-post renegotiation. The basic message is that efficiency may not be achievable if renegotiation is allowed. From (3.2), if Rða; b Þ − b − Pb < Pa , the principal will sell the firm. However, both parties can benefit ex post from renegotiation in this case. Since the principal’s investment is made afterwards, without an agreement, the principal will not invest at all. A better solution is to induce the principal to keep the firm. The trouble is, if renegotiation is possible, both players become opportunistic ex ante, which may lead to ex-ante inefficiency. The next two sections will discuss option contracts when renegotiation is allowed. In the current setting, the principal’s decisions on her effort input and the option can be made at the same time in effect. Hence, the principal doesn’t face a holdup/ commitment problem. If efforts are made simultaneously, efficiency may not be achievable.4 Mann and Wissink (1988) ask whether a money-back guarantee on a product can induce efficiency. They assume the two parties act simultaneously; consequently, their solution is inefficient. Threatening the workers about having to buy out the firm is not a realistic scheme, and a simple liquidity constraint will rule out this buying possibility. One way to address this constraint is to have a bonus component in a contract, instead of a transfer of ownership. A bonus is an option for the principal. She may or may not exercise the option ex post. If this bonus is related to output, it can be an effective tool for inducing good incentives. In the current setting, actions are completely unobservable to a third party. Hermalin and Katz (1991) examine a one-sided moral hazard problem in which renegotiation is possible and the agent’s action is imperfectly observable to a third party but perfectly observable to the principal. They show that the threat of having the court resolve a dispute generally enables the principal to employ her private information to discipline the agent. If the principal can only observe the agent’s effort partially or randomly, say with probability α∈ð0; 1Þ, it can be shown that the first best can still be achievable in equilibrium, provided that shirking is observable to the principal with a high probability. This result requires a risk-averse agent, who is averse to the risk of being punished for shirking. Proposition 3.2 does not impose a limited liability condition. That is, we assume that Rða; 0Þ ≥ Pa : What will happen if we impose limited liability? If Rða; 0Þ < Pa , the agent knows that the principal will pay vða; 0Þ instead of Pa . With this knowledge, the agent’s behavior will change, which in turn will affect the equilibrium outcome. We will discuss this issue in Section 4. Finally, the solution in Proposition 3.1 is not unique. In this option contract, if the option is not exercised, the agent is paid his total cost a∗ þ U − 1 ðŪ Þ in the relationship; if it is exercised, the agent pays the principal the maximum profit or the total surplus. Schmidt (2003) provides other optional contracts, as stated in the following proposition, which also induce the first-best effects but allow the two sides to share the surplus.

Corporate finance

115

Proposition 3.2. Assume Ū ¼ 0 and let I be the initial investment of the principal. LetðPa ; Pb Þ satisfy I ≤ Pa ≤ Rða∗ ; b∗ Þ − a∗ − b∗ ;

Pb ¼ Rða∗ ; b∗ Þ − b∗ − Pa :

ð3:6Þ

Then the option contract ðPa ; Pb Þ induces both parties to take the first-best actions a∗ and b∗ :The payoffs of the agent and the principal are respectively U ¼ Pb − a∗ and V ¼ Pa − I; which satisfy the two IR conditions U ≥ 0 and V ≥ 0:

2. Option contracts with renegotiation This section is based on the work of Che and Hausch (1999). They show that option contracts may not be efficient if renegotiation is allowed. 2.1. The model Consider a two-stage game between a buyer and a seller. Both are risk neutral. In the first stage, the seller and the buyer make investments s ≥ 0 and b ≥ 0; respectively. In the second stage, the level of trade q ≥ 0 is determined. The value of the trade for the buyer is vðq; ε; b; sÞ and the cost for the seller is cðq; ε; b; s Þ; where ε is a random shock drawn from ½0; 1 with distribution Fðε Þ: Given state θ ≡ ðε; b; sÞ; if q can be freely chosen, the maximum net joint surplus is ϕðθ Þ ≡ max vðq; θ Þ − cðq; θ Þ; q≥0

ð3:7Þ

R and the expected surplus is Φðb; s Þ ≡ ϕðε; b; sÞdFðε Þ: Both parties can observe θ at t ¼ 2; but it is not verifiable. The quantity of trade q; transfer payments between the two parties, and their reports on θ are verifiable. The verifiability of θ means that a contract can specify a menu of quantity-transfer pairs ðqðθ Þ; tðθ ÞÞ; where qðθ Þ is the quantity traded and tðθ Þ is the transfer made at state θ: Although θ is not verifiable, the true θ will be reported by the two parties under the so-called “shoot the liar” mechanism: both parties are asked   to report their ^ ^ ^ observed state θ; and if their reports match then qðθÞ; tðθÞ for that state is enforced; otherwise, both parties are penalized by zero trade and zero transfer. By this mechanism, neither party has anything to gain from lying, assuming renegotiation is not allowed. Alternatively, we can also apply the revelation principle to the direct mechanism and impose two IC conditions to induce truthful reporting from the two parties. This alternative approach yields the same results. The first-best outcome Social welfare is W∗ ðb; sÞ ≡ Φðb; sÞ − b − s: The Nash equilibrium solution ðb∗ ; s∗ Þ is determined jointly by the following two problems: max Φ ðb; s Þ − b; b≥0

max Φ ðb; sÞ − s: s≥0

116 Corporate finance This pair ðb∗ ; s∗ Þ of investments maximizes social welfare, and they are hence the first-best (efficient) investments. No contract outcome Suppose the seller has bargaining power α ∈ ½0; 1: When there is no initial contract, the utility functions of the two parties are U0s ðb; sÞ  ðb; sÞ  s;

U0B ðb; sÞ  ð1  Þðb; sÞ  b:

Let ðb0 ; s0 Þ be the optimal investments from the following problems: max ð1 − αÞΦ ðb; s Þ − b; b≥0

max α Φ ðb; sÞ − s: s≥0

This pair of problems is the solution under ex-post negotiation when there is no contract. Obviously, this solution is not efficient. Contracting without renegotiation Let a contract be a pair ðqðθ Þ; tðθ ÞÞ; where q is the quantity traded and t is the transfer from the buyer to the seller. Proposition 3.3. Under some technical conditions on vðq; θ Þ and cðq; θ Þ; when renegotiation is not allowed, the first best can be achieved and the first-best contract is defined by qðθ Þ ¼ q∗ ðθ Þ;

tðθ Þ ¼ v½q∗ ðθ Þ; θ − Φ ðb; sÞ þ T;

where q∗ ðθ Þ solves (3.7) and T is an arbitrary constant that can be used to induce both parties to participate. Contracting with renegotiation However, efficiency may not be achievable if renegotiation is possible. Cooperative investments have a positive externality on the other party’s value and cost such that vs ðq; ε; b; sÞ > 0 and cb ðq; ε; b; s Þ < 0; while selfish investments benefit the investing party herself such that vb ðq; ε; b; sÞ > 0 and cs ðq; ε; b; sÞ < 0: The literature, such as the work of Nöldeke and Schmidt (1995), shows that option contracts can achieve efficiency in a bilateral trade, even in the face of specific investments and incomplete contracting. However, the literature typically considers selfish investments that benefit the investor only. Che and Hausch (1999) find a very different result for cooperative investments that benefit the investor’s partner. This result is stated in the following: Proposition 3.4. Under certain technical conditions on vðq; θ Þ and cðq; θ Þ; if investments are sufficiently cooperative and renegotiation is possible, then there exists an intermediate range of bargaining share for which contracting has no value, i.e. contracting is less efficient than ex-post negotiation.

Corporate finance

117

This result is inefficient. In fact, even if the cooperative nature of the investments is sufficiently weak so that contracting has a value, an efficient outcome may not be possible. 2.2. An example To understand the above theoretical results, we present an example. This example shows how renegotiation causes inefficiency. Suppose a buyer purchases one unit of a good from a seller. With investment e; let vðeÞ be the value of the good to the buyer. The investment e is selfish if e is from the buyer and it is cooperative if e is from the seller. The first-best effort e∗ is from max vðe Þ − e: e

If there is no initial contract and the parties split the surplus ex post equally, then each party receives vðeÞ=2 and the investing party chooses e0 from max e

1 vðe Þ − e: 2

The solution is not efficient. If there is an initial contract that specifies a fixed price p for the trade and if the buyer makes the investment e (selfish investment), the optimal effort is from max vðe Þ − p − e; e

which implies an efficient e∗ : However, if the seller makes the investment e (cooperative investment), the optimal e is from max p − e; e

which is zero. This solution is not efficient either. The above contract is too simple. The first-best outcome can actually be achieved with a cooperative investment and a more sophisticated contract if there is no renegotiation. Consider an option contract for the buyer: the buyer will exercise the option to buy the product at price p ¼ vðe∗ Þ if vðeÞ ≥ vðe∗ Þ and reject the good if vðe Þ < vðe∗ Þ: Then, if renegotiation is not possible, the buyer will choose e ¼ e∗ : However, if renegotiation is possible, given a price p specified in the option contract, if the buyer rejects the good, then the parties will renegotiate and split the surplus from trade, giving the buyer a payoff of vðeÞ=2: Thus, knowing this, the buyer will reject the good if and only if p > vðe Þ=2: Given this response, the seller’s payoff is 8 1 1 > < vðe Þ if p > vðeÞ; 2 2 1 > : p if p ≤ vðe Þ: 2

118 Corporate finance Hence, the seller’s problem is   1 min vðeÞ; p  e: e 2 Clearly, the optimal investment in this case cannot exceed e0 : In fact, the optimal contract must specify p ≥ vðe0 Þ=2; which implies e0 is the optimal investment. That is, with renegotiation, the outcome is the same as not having any contract. This example clearly shows that, with cooperative investments, renegotiation can undermine an option contract completely.

3. Option contracts with renegotiation revisited This section is based on the work of Edlin and Hermalin (2000). Renegotiation is allowed and it has two effects. On the one hand, the agent’s investment is sunk at the time of renegotiation (the holdup effect). On the other hand, the agent’s investment may give him an advantage in bargaining (the threat-point effect). That is, with renegotiation, the agent may have an incentive to overinvest so as to bargain for a good deal; however, as the principal may exercise the option to obtain the asset if it is of high value after the agent’s investment, the agent may have an incentive to underinvest. This study shows that, when renegotiation is allowed, an option contract can still achieve the first best if the threat-point effect dominates the holdup effect. Also, the second-best solution turns out to be to not have any contract. 3.1. The model As in Section 1, the model has double moral hazard, with a risk-neutral principal and a risk-averse agent. Output x is not verifiable. An option contract is the only possible contractual solution. An option contract is a pair ðPa ; Pb Þ of payments. Upon acceptance of the contract, the principal transfers ownership of the asset to the agent with payment Pa from the agent. After the agent has chosen his effort, the principal has the option to buy back the improved asset at price Pb : If the principal declines to exercise her option, the agent retains ownership. After the principal has decided whether or not to exercise her option, she chooses her effort. The final owner of the asset receives output x at the end. Contracting 0 Ex ante

Investment a

Option 1 Ex post

Investment b

Output 2 End

The efforts from the agent and the principal are respectively a and b: The costs of those efforts are also a and b: Given ða; b Þ; the distribution and density functions of x are Rrespectively Fðxja; bÞ and fðxja; bÞ: Then the expected revenue is Rða; b Þ ≡ xf ðxja; bÞdx: If the principal decides to buy back the asset, the value of the asset to the principal is

Corporate finance

119

VðaÞ ¼ max Rða; b Þ − b: b≥0

Let uðx Þ be the agent’s utility function.5 If the principal is not going to buy back the asset, the value of the asset to the agent is Z UðaÞ ≡ uðx Þfðxja; 0Þdx: The first-best effort a∗ is the solution of the following first-best problem: max VðaÞ − u − 1 ðaÞ:

ð3:8Þ

a

Assumption 3.1 (FOSD). Fðxja; bÞ is decreasing in ða; b Þ: Proposition 3.5 (Che–Hausch 1999). If renegotiation is infeasible, then the option contract ðPa ; Pb Þ defined below achieves the first best: Pa ¼ Vða∗ Þ − u − 1 ða∗ Þ;

Pb ¼ Vða∗ Þ:

After the contract is signed, the expected value of the asset to the agent if the principal doesn’t exercise the option is Z uðx − Pa Þfðxja; 0Þdx; which implies that the certainty equivalence of the asset to the agent at t ¼ 1 is Z uðx  Pa Þfðxja; 0Þdx þ Pa : CEða; Pa Þ  u1 The total gain from renegotiation is Sða; Pa Þ ≡ Vða Þ − CEða; Pa Þ: Assume that the bargaining outcome is determined by an efficient monotonic sharing rule: the two agents split the surplus according to σ A : ℝ ! ℝþ and σ P : ℝ ! ℝþ , satisfying 1. 2. 3.

(efficiency): σ A ðs Þ þ σ P ðs Þ ¼ s; and either (weak monotonicity): σ 0A ðsÞ ≥ 0 and σ 0 P ðs Þ ≥ 0; or (strict monotonicity): σ 0A ðs Þ > 0 and σ 0 P ðsÞ > 0:

The Nash bargaining solution is a strictly monotonic sharing rule. The principal’s payment to the agent after renegotiation becomes pða; Pa Þ ≡ CEða; Pa Þ þ σ A ½Sða; Pa Þ: The principal will therefore exercise her option if Pb ≤ pða; Pa Þ: Otherwise she will let it expire and then renegotiate. We say that the threat-point effect dominates if ∂CEða∗ ; Pa Þ ≥ V0 ða∗ Þ ∂a

ð3:9Þ

120 Corporate finance for Pa satisfying ð3:10Þ u − 1 ða∗ Þ ¼ pða∗ ; Pa Þ − Pa : That is, the threat-point effect dominates if the marginal impact of the agent’s action increases the value of the asset to him more than it increases the value of the asset to the principal. Proposition 3.6. If the renegotiation sharing rule is strictly monotonic and condition (3.9) fails for the Pa satisfying (3.10), then there isn’t a single option contract that can achieve the first best. Proposition 3.7. If the renegotiation sharing rule is monotonic and either condition (3.9) holds for the Pa satisfying (3.10) or the agent has full bargaining power at a∗ and Pa ; then an efficient option contract exists provided that u½ pða; Pa Þ − Pa  − a is ideally quasi-concave in a: If the agent is risk neutral, then CEða; Pa Þ ¼ Rða; 0Þ: Hence, condition (3.9) holds if ∂2 Rða∗ ; 0Þ ≤ 0; ∂a∂b meaning that the two efforts are substitutable. On the other hand, condition (3.9) fails if the two efforts are complementary such that ∂2 Rða∗ ; 0Þ > 0: ∂a∂b Hence, substitutability of efforts is crucial to the existence of an efficient option contract if renegotiation is possible. 3.2. Discussion Demski and Sappington (1991) show that option contracts can be efficient. Che and Hausch (1999) point out the failure of option contracts to achieve efficiency when renegotiation is allowed. Edlin and Hermalin (2000) look closely at the situation described by Che and Hausch (1999) and find that the substitutability of efforts can restore efficiency. When the first best is not achievable with an option contract, it may still be achievable with a more sophisticated contract. Edlin and Hermalin (2000, Section 4) discuss this possibility for a risk-neutral agent. Since the principal’s effort is made after renegotiation, her effort does not really play a role in the model. Consequently, only Vða Þ is used to represent the principal’s valuation and b is completely missing from discussions. Che and Hausch (1999, Section I) discuss the idea presented by Edlin and Hermalin (2000); the critical difference between the two papers is that Edlin and Hermalin allow the possibility that the agent’s effort is valuable (the threat point) even if the principal exercises the option to penalize the agent. This feature creates the threat-point effect, which can support the efficient outcome under certain conditions even if renegotiation is allowed.

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When the two parties’ investments are substitutes at the margin, the threat-point effect tends to dominate. However, Nöldeke and Schmidt (1998) show that, even if investments are complements at the margin, the first best can be achieved. The crucial difference between Nöldeke and Schmidt’s (1998) work and that of Edlin and Hermalin (2000) is in the settings, with Edlin and Hermalin assuming that the exercise date of the option comes before the principal’s investment date. Holdup means that the agent cannot capture the full marginal contribution of his effort to the principal’s value Vða Þ; which reduces the agent’s investment incentives. It turns out, however, that, even though holdup is unavoidable, it does not necessarily cause underinvestment. Offsetting the holdup effect is the threat-point effect: the agent’s investment strengthens his bargaining position by increasing the value of the asset if the agent retains it. If this threat-point effect is sufficiently large, it will dominate the holdup effect, and an efficient option contract can exist when renegotiation is allowed. Proposition 3.5 is similar to Demski and Sappington’s (1991) result, but the firstbest option contracts are different. The agent’s payoff function given by Demski and Sappington (1991) is UðM − a Þ; the one given by Edlin and Hermalin (2000) is UðMÞ − a with u ¼ 0: With this new payoff function, the first-best option contract ðPa ; Pb Þ is redefined as Pa ¼ Rða∗ ; b∗ Þ − b∗ − Pb ;

Pb ¼ u − 1 ða∗ Þ:

The option contract ðPa ; Pb Þ in Proposition 3.5 can be rewritten as Pa ¼ Rða∗ ; b∗ Þ − b∗ − u − 1 ða∗ Þ ¼ Rða∗ ; b∗ Þ − b∗ − Pb ;

Pb ¼ Vða∗ Þ:

Hence, the two option contracts are different. This means that first-best option contracts are not unique. In fact, as shown by Schmit (2003), an option contract ðPa ; Pb Þ achieves the first best if   Pb ∈ Vða∗ Þ; u − 1 ða∗ Þ : Pa ¼ Rða∗ ; b∗ Þ − b∗ − Pb ; From the proof of Proposition 3.7, the principal must be indifferent between exercising and not exercising the option in equilibrium. On the one hand, the option has a value; but, on the other hand, the option cannot have a positive value or else the agent would have invested too much and the principal would be inclined to exercise the option. Hence, in equilibrium, the option must have zero value. But, even then, it can still be used to guard against undesirable out-ofequilibrium behavior. With renegotiation, the agent has the incentive to overinvest so as to bargain for a good deal. But the principal may exercise the option to obtain the asset at a low price. Due to this, the agent will not invest more than a∗ : In other words, the option can prevent a > a∗ :

4. Option contracts with limited liability This section is based on the work of Wang (2004). Wang investigates another issue in relation to option contracts: limited liability. We find that option contracts are

122 Corporate finance generally not able to achieve efficiency (the first best) under limited liability. We identify the optimal option contract in this case. Interestingly, our solution shows that limited liability may cause either overinvestment by the agent or overcompensation to the agent, both of which are in sharp contrast to the standard outcome of a principal–agent model. 4.1. The model Consider a principal–agent relationship between an entrepreneur (EN) and a venture capitalist (VC), in which the VC is the principal and the EN is the agent. The project needs an initial investment of I dollars from the VC to get started. The VC may put in a second installment b in the second stage if the project survives through the first stage. The monetary value of the project is vða; bÞ by the end of the project. The inputs a and b are unverifiable, but the input a is observable ex post. Contracting

Investment a

0 Ex ante

Investment b

Option 1 Ex post

Output 2 End

A convertible security is a pair ðR; P Þ of payments that allows the principal to either receive payment R at date 1 (not to convert the debt into equity) or purchase the firm by making an additional payment P (to convert the debt into equity). Given ðR; P Þ and the EN’s utility function uðx Þ; the payoffs are ( EN

U

¼ (

UVC ¼

u½vða; bÞ  a  R

if VC does not convert

uðP  aÞ

if VC converts

RbI

if VC does not convert

vða; bÞ  b  P  I

if VC converts

4.2. Convertible security with unlimited liability Let ða ∗∗ ; b ∗∗ Þ be the first-best investments from the two parties. They are determined by max vða; b Þ − a − b:

a; b ≥ 0

With unlimited liability, the following result is well known. Proposition 3.8 (First Best). Under unlimited liability, the first-best outcome can be achieved by a convertible security ðR ∗∗ ; P ∗∗ Þ defined by uðP ∗∗ − a ∗∗ Þ ≥ u ;

ð3:11Þ

R ∗∗ ¼ vða ∗∗ ; b ∗∗ Þ − b ∗∗ − P ∗∗ :

ð3:12Þ

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123

This proportion indicates that there are many first-best solutions. However, the optimal solution that maximizes the principal’s payoff is unique, which is stated in the following. Proposition 3.9 (Optimum). Under unlimited liability, the optimal convertible security ðR ∗∗ ; P ∗∗ Þ is defined by uðP ∗∗ − a ∗∗ Þ ¼ u ;

ð3:13Þ

R ∗∗ ¼ vða ∗∗ ; b ∗∗ Þ − b ∗∗ − P ∗∗ :

ð3:14Þ

This optimal convertible security is unique and it achieves the first best. 4.3. Convertible security with limited liability In this subsection, assume that EN has only limited liability, in the sense that the maximum amount that he has to pay at time 1 is what he can get from the firm at that moment. At that time, if VC decides to cash in, the firm’s value is only vða; 0Þ; which is the liquidation value. Denote LðaÞ ≡ vða; 0Þ: For the convertible security ðR ∗∗ ; P ∗∗ Þ defined in Proposition 3.9, if Lða ∗∗ Þ < R ∗∗ ; the security is no longer feasible since the EN cannot pay R ∗∗ if the VC decides to cash in. Although the equilibrium outcomes in Proposition 3.8 and Proposition 3.9 never violate the limited liability condition in equilibrium, limited liability can be crucial to an equilibrium outcome. Given a; we define Pða Þ by the following condition: u½Pða Þ − a ¼ u

or PðaÞ ≡ a þ u − 1 ðu Þ:

We call a convertible security ðR; P Þ with P ¼ Pða Þ a binding convertible security for a; otherwise it is called a non-binding convertible security. Define the value function RðaÞ as RðaÞ  max fvða; bÞ  b  PðaÞg: b0

ð3:15Þ

Rða Þ is the ex-post profit for the VC if she retains ownership. Denote also bða Þ as the optimal investment from (3.15). Proposition 3.10. Under limited liability,



If Lða ∗∗ Þ ≥ Rða ∗∗ Þ; the first-best outcome can be achieved with the binding convertible security ðRða ∗∗ Þ; Pða ∗∗ ÞÞ: If Lða ∗∗ Þ < Rða ∗∗ Þ; the first-best outcome cannot be achieved with any binding convertible security.

Proposition 3.10 indicates that, since limited liability limits the return on debt, it provides natural protection for the EN and limits the ability of a convertible security

124 Corporate finance R(a) L(a) L(a)

R(a) 0

al

0

a

a**ah

al

a**

a

L(a)

R(a) 0

a**

a

Figure 3.2 Optimal investment.

to achieve efficiency. If Lða ∗∗ Þ < Rða ∗∗ Þ; there are three possibilities, as shown in the following figures. The optimal solutions for the three cases are stated in the following proposition. Proposition 3.11. Under limited liability,






(Efficiency). If Lða ∗∗ Þ ≥ Rða ∗∗ Þ; the unique optimal convertible security is ½Rða ∗∗ Þ; Pða∗∗ Þ; which is the first best. (Overinvestment). If Lða ∗∗ Þ < Rða ∗∗ Þ and ah defined in Figure 3.2 exists, the unique optimal convertible security is ½Lðah Þ; Pðah Þ; which induces the second-best investment ah : This solution does not achieve the first best. In this case, the VC’s surplus is Lðah Þ − I and the EN has no surplus. (Overcompensation). If Lða ∗∗ Þ < Rða ∗∗ Þ and ah does not exist, the unique optimal convertible security is ½Lða ∗∗ Þ; Pða ∗∗ Þ; which induces the first-best investment a ∗∗ : This solution achieves the first best. In this case, the VC’s surplus is Lða ∗∗ Þ − I and the EN’s surplus is Rða ∗∗ Þ − Lða ∗∗ Þ:

With limited liability, the VC is less likely to convert the debt into equity. Hence, the EN has more incentive to invest. In case (c), limited liability serves as a limit on how much surplus the VC can squeeze out of the project. As a result, the EN may have a positive surplus, in contrast to the standard outcome of a principal–agent model.

5. A signaling equilibrium of financial instruments This section is based on the work of Stein (1992). Stein argues that corporations may use convertible bonds as an indirect way to get equity into their capital

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125

structures when adverse-selection problems make a conventional stock issue unattractive. The call option is critical – it is the only way to actually force investors to exercise their conversion option early. By this, the cost of excessive debt is reduced. Stein’s theory is based on three key points: 1. 2. 3.

Asymmetric information makes conventional equity issues unattractive, which is shown by Myers and Majluf (1984). There is a cost of excess debt, called financial distress. Convertibles in reality mostly have a call feature so that companies can force early conversion to reduce debt.

If the firm’s value is less than its debt, the firm is in financial distress. There is a lump-sum cost for this financial distress, which is c > 0: For a discounted bond, the face value (or par value) is the amount paid to a bond holder at maturity. The call price (or redemption price) of a convertible is the amount of money the issuer has to pay to redeem the convertible when it is called; this price is set at the time when the convertible is issued. The exercise price (or strike price) is the price at which an option or warrant holder can buy or sell the underlying asset. The conversion ratio is the number of shares for each convertible when it is converted. The conversion value (or parity value) is the value of a convertible when it is to be converted. A callable convertible consists of three features: face value, conversion ratio, and call price. Without a call, the holder can either cash in at the face value at maturity or convert the investment to equity according to the conversion ratio. Upon a call, the holder can either convert the investment according to the conversion ratio or cash in at the call price. The holder will convert if the conversion value is above the call price; otherwise he will redeem it at the call price (see Figure 3.3).

.

Investor Don’t convert

Convert

.

Equity

Firm Call

No call

.

Face value F Convert Equity

Figure 3.3 A callable convertible.

Investor Don’t convert

Call price K

126 Corporate finance 5.1. The model Stein considers a separating equilibrium in financing strategies for three possible types of firms: good, medium, and bad. In a separating equilibrium, the good firm issues debt, the bad firm issues equity, and the medium firm issues convertibles. There are three time points: 0, 1, and 2. There is no time discount. All agents are risk neutral. The firm needs an initial investment of I at time 0, which must be raised from external sources. Output x is realized at time 2. Contracting

0 Ex ante

Option

Shock 1

1 Ex post

Shock 2

Output

2 End

Each firm has output XH or XL ; with XH > I > XL : The only differences among the firms are the probabilities of producing XH : The type of firm, i.e. the probability of producing XH ; is private information. The random outputs at t ¼ 0 as represented by lotteries are good firm :

x~ ¼ ðXH ; 1Þ;

medium firm : x~ ¼ ðXL ; p ; XH ; 1 − p Þ; bad firm : x~ ¼ ðXL ; z; ðXL ; p; XH ; 1 − p Þ; 1 − zÞ ¼ ðXL ; q; XH ; 1 − q Þ: There is a shock to the bad firm in period 1. The shock doesn’t affect the other two firms. The bad firm has the same risk as the medium firm if it doesn’t encounter the shock in period 1; but the bad firm has probability z of experiencing a bad shock in period 1 such that it will produce XL for sure. Let q ≡ z þ pð1 − z Þ: Then the bad firm has probability q of producing XL : The bad firm is defined by a two-stage lottery. Using the Axiom of Compound Lotteries, we reduce the lottery to a one-stage lottery with probability q of a poor performance. Assume qXL þ ð1 − q ÞXH ≥ I; which means that it is worthwhile for the bad firm to produce. And, if that’s the case, it is worthwhile for all the firms to produce. Proposition 3.12. If the cost of financial distress is high enough, so that c > ðI − XL Þ; then the following choices ensure a separating Bayesian equilibrium:






The good firm issues debt with a face value of I: The medium firm issues a convertible bond. The convertible has face value F > XL ; call price K satisfying XL < K < I; and conversion ratio I pXL þð1 − p ÞXH . The bad firm issues a fraction qXL þð1I− q ÞXH of equity.

Proof. Step 1. The good firm will choose debt.

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127

If the good firm issues debt, since XH > I; it will never be in financial distress. The cost of debt is I: If the good firm chooses to issue the convertible, no matter whether an investor is forced to convert or chooses to convert, he still gets I XH > I: pXL þ ð1 − p ÞXH

ð3:16Þ

Hence, if F ≤ I; the investors will convert voluntarily; if F > I; the investor may not choose to do so. Either way, it costs the good firm more to issue the convertible. If the good firm chooses to issue equity, the value of the equity share is I XH > I: qXL þ ð1 − q ÞXH This cost is again higher than the straight debt. Hence, the best choice for the good firm is debt. Step 2. The medium firm will choose the convertible. If the medium firm offers the convertible, it can always force conversion at t ¼ 1: Since K < I; this conversion is always successful, so that the firm faces no financial distress. The expected cost of this convertible is I: If the medium firm offers equity by offering a fraction qXL þð1I− q ÞXH of equity, the expected value of this offer is I ½pXL þ ð1 − p ÞXH  > I: qXL þ ð1 − q ÞXH This cost is too high. If the medium firm offers debt with face value I; there is a probability p that the firm has a value of XL that is below the debt, in which case the firm suffers financial distress. Hence, the expected cost of issuing the debt is pc: On the other hand, the value of the debt is pXL þ ð1 − pÞI; which is sold for I: Hence, the expected gain from issuing the debt is pðI − XL Þ: With c > I − XL ; the firm will choose not to issue the debt. Step 3. The bad firm will choose equity. If the bad firm issues equity with a fraction qXL þð1I− q ÞXH of equity, it has no financial distress. The expected cost is I: If the bad firm issues the convertible, with probability z; the firm will experience a bad shock. Then the conversion value of forced conversion is I XL ; pXL þ ð1 − p ÞXH which is less than the call price K: In this case, the firm will not be able to force conversion, since the investor will choose to receive the call price instead of converting voluntarily. Since K > XL ; the firm cannot afford K in this case. Hence, the firm

128 Corporate finance will not force conversion. Consequently, the firm is left with a debt burden of F: Since XL < F; there is financial distress. Hence, the expected cost of financial distress is zc: On the other hand, there is a gain from issuing the convertible. Since the convertible will remain a debt claim worth XL with probability z and will be converted into equity worth6 I with probability 1 − z; the value of the convertible is zXL þ ð1 − zÞI: But, if the bad firm sells it for I; the gain from this sale will be zðI − XL Þ: Given condition c > ðI − XL Þ; the bad firm will choose not to sell the convertible. If the bad firm chooses to issue debt, this debt has a true value of qXL þ ð1 − qÞI:7 The bad firm sells it for I: The gain from this sale is qðI − XL Þ: However, the cost of financial distress is qc: Again, by condition c > ðI − XL Þ; the bad firm will not choose to sell the debt. In summary, we have shown that, given the three assets, no firm will try to hide its identity by offering a different asset. Q.E.D. 5.2. Discussion In Stein’s work, the three assets provide the same expected return but have different risks. Hence, the equilibrium is not stable: with a tiny bit of risk aversion, there will be no demand for equity and convertibles. Renegotiation is not allowed. So the parties involved are not allowed to renegotiate at time 1 after the firm types become known to the public. The result relies crucially on a large cost of financial distress. This cost is vague. It is not a risk premium. It is a kind of psychological cost. There is no theory for it. Each firm issues only one type of asset. In reality, a typical firm issues all three types of asset. In fact, a firm will be willing to issue any asset if there is demand for it. Stein looks at the issue from the firm’s point of view only (the supply side of financial assets). But we should also look from investors’ point of view (the demand side). For example, most venture capitalists demand convertibles, but some demand debt. As a result, a firm may issue debt as well as convertibles. In fact, 95 percent of financing by venture capitalists is done with convertibles, and these convertibles are almost exclusively convertible preferred shares. Auto conversion is popular. In reality, staged financing and convertibles go hand in hand. Stein explains the existence of convertibles, but he doesn’t explain why staged financing is popular and why convertibles are used in staged financing.

6. Convertibles as an efficient financing solution under asymmetric information This section is based on the work of Chakraborty and Yilmaz (2004). The key point they try to bring out is that there is an efficient pooling equilibrium to an asymmetric information problem. This result relies crucially on a public signal. This signal can be partial, which reveals partial information only, but it needs to be sufficiently reliable. Such a model can easily find efficient separating equilibria, but a separating solution is not in the interest of Chakraborty and Yilmaz.

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129

Following Myers and Majluf (1984), a large number of scholars focus on different modes of financing that allow the management to signal its type and thus solve the underinvestment problem (a separating equilibrium), including Stein (1992). However, separation by signaling is typically costly and creates another source of inefficiency. Chakraborty and Yilmaz show that an efficient pooling solution is possible, if there is a partial resolution of information by a public signal. This solution does not imply a complete resolution of information in equilibrium. Convertibles have received considerable attention in the literature due to the importance of the convertibles market. These securities are quite common. The global convertible market is worth about $610 billion and growing. In 2001, for example, there were around 400 new issues in the US, which raised a total of $106.8 billion. About 95 percent of convertible securities are callable, and an overwhelming majority of these have restrictive call provisions. About 95 percent of convertibles are callable (see Lewis, Rogalski et al. 1998) and an overwhelming majority of these have restrictive call provisions, such as provisional calls. This study shows that convertibles with restrictive provisions can result in an efficient pooling solution. 6.1. The model The model is the same as Myers and Majluf’s (1984). A firm may have either type θ1 managers or type θ2 managers, where θi is the manager’s private information and θ2 is a better type with θ2 > θ1 : Let λi ¼ Prðθ ¼ θi Þ with λ1 þ λ2 ¼ 1: Initially, the firm consists of equity only, with the number of shares outstanding being M ¼ 1: The firm is to acquire an investment of $1 to invest in a new project. Without this investment, the firm’s expected value is Ai for θ ¼ θi : The new investment yields a random output of x~ with distribution function Gðxjθ Þ: Then the expected xjθi Þ: Assume value is Vi ¼ Eð~ FOSD : Gðxjθ2 Þ ≤ Gðxjθ1 Þ:

ð3:17Þ

By (3.17), we have V2 ≥ V1 : We assume a strict version: V2 > V1 : We also assume Vi − 1 > Ai for both i so that both types of managers will invest under complete information. Also, to rule out riskless debt financing, assume Gð1jθ1 Þ > 0; i.e. there is a possibility that output x is less than the investment $1: Let Sðx Þ be the payoff from a security S when output is x: An admissible security is an increasing Sðx Þ satisfying 0 ≤ Sðx Þ ≤ x;

for all x:

ð3:18Þ

By FOSD, the admissibility condition (3.18) implies that xÞjθ1 ; E½Sð~ xÞjθ2  > E½Sð~

for any x~:

For example, equity and debt are admissible securities. Given a share α; an equity security is Sðx Þ ¼ αx; which satisfies (3.18). The expected value of this equity security for manager i is αVi : For any discount bond with face value F ≥ 0;8 we have

130 Corporate finance Sðx Þ ¼ minðx; FÞ; which also satisfies (3.18). The expected value of this debt security with manager i is x; F Þjθi : Di ðF Þ ≡ E½min ð~ Investors are assumed to be competitive, so that at each date they value all securities according to their own expected values, given publicly available information. Contracting

Information revelation options

Output: y

0 Ex ante

1 Ex post

2 End

Figure 3.4 The timeline of events.

The model has three dates, 0, 1, and 2 (see Figure 3.4). At date 0, the manager decides whether to engage in a project and what securities to issue to acquire the required investment. The market is uninformed about θ and the securities are valued by the market. At date 1, some of the asymmetric information is resolved. Specifically, the market publicly observes a signal m 2 fm1 ; m2 g of θ; with Prfm ¼ mi j ¼ i g ¼  2 ð0:5; 1;

for all i:

The case β ¼ 1 corresponds to the case of perfect resolution of asymmetric information. On the other hand, the case β ¼ 0:5 corresponds to the case when there is no additional useful information from the signal. For the time being, we assume that the signal m is exogenous. The manager is allowed to issue a security at date 0 that is conditional on the signal m: At date 2, output is realized and distributed according to the existing securities. The equilibrium is a perfect Bayesian equilibrium. We assume that the manager cares only about the existing shareholders, which is a common assumption in the literature. 6.2. An example Suppose V1 ¼ 4 and V2 ¼ 8 with λi ¼ 0:5; and A1 ¼ 2:5 and A2 ¼ 6:5: Under complete information, both types θi of managers will engage in the project. Suppose that the total number of shares is 1. Since the total value of the firm is Vi ; the share price is Vi : With an investment of $1, the investor buys 1=Vi shares. Then the expected payoff to the existing shareholders is Vi − 1; which is the new firm value after the investment. It is easy to find a separating equilibrium in which manager 1 issues an equity share and manager 2 foregoes the investment opportunity. Consider α1 ¼ 1=V1 ¼ 0:25 and α2 ¼ 0:25 − ε; where ε > 0 is sufficiently small. The security S1 ðx Þ ¼ α1 x is just enough to raise the required investment of $1: The expected payoff for the existing shareholders with manager 1 is ð1 − α1 ÞV1 ¼ 3 > A1 ; but the expected payoff for the existing shareholders with manager 2 is ð1 − α2 ÞV2 ≈ 6 < A2 :

Corporate finance

131

Hence, manager 1 will engage in the project while manager 2 will forego the opportunity. Here, investors can identify the managers’ types by their equity share offers. We can easily find a simple belief system to support this solution as a separating Bayesian equilibrium. This solution is inefficient. We can also find a pooling equilibrium. Let V be the expected value of the firm: V ¼ λ1 V1 þ λ2 V2 :  Consider an equity security Sðx Þ ¼ αx with α ¼ 1=V¼1=6 issued by both managers. In this case, both managers will sell the security. Depending on the managers’ types, the security yields the following payoffs to the existing shareholders: ð1 − αÞV1 ¼

10 > A1 ; 3

ð1 − α ÞV2 ¼

20 > A2 : 3

Even in this case, the payoff 20=3 with manager 2 under asymmetric information is less than the payoff V2 − 1 ¼ 7 with manager 2 under complete information. The payoff of the firm with manager 2 is diluted by issuing the equity security. Similarly, dilution occurs for other simple securities like debt. But Chakraborty and Yilmaz show that an optimal security will avoid such dilution. We present this result in the next subsection. 6.3. An optimal callable convertible Perfect revelation of information ex post Assume β ¼ 1; in which case the information asymmetry is perfectly resolved at date 1. Denote a callable convertible by ðF; α; k; TÞ; where T is the exercise date of the convertibility option and the call option, k is the call price, α is the conversion ratio (the share of the firm the holder can have), and F is the face value. We now design an efficient callable convertible ðF ∗ ; α∗ ; k∗ ; T ∗ Þ: First, suppose that F ∗ is determined by D1 ðF ∗ Þ ¼ 1:

ð3:19Þ

This means that the debt part of the convertible is a fair valued security for the bad manager. Second, suppose that the equity part of the convertible is also a fair valued security for the good manager, which means that α∗ is determined by α∗ V2 ¼ 1:

ð3:20Þ

Third, suppose that T ∗ ¼ 1; meaning that all the options expire on date 1. Fourth, we choose k∗ properly so that each type of manager decides to finance the project efficiently (without any dilution). At date 1, if m ¼ m2 ; everyone knows θ ¼ θ2 : Since D2 ðF ∗ Þ ≥ D1 ðF ∗ Þ ¼ 1 ¼ α∗ V2 ;

132 Corporate finance no one will voluntarily convert to equity. If k∗ ≤ α∗ V2 ¼ 1;

ð3:21Þ

the convertible holder will convert if a call is made. We choose k∗ satisfying (3.21). Then since ð1 − α∗ ÞV2 ≥ V2 − D2 ðF ∗ Þ; the manager will call. At date 1, if m ¼ m1 ; everyone knows θ ¼ θ1 : Since D1 ðF ∗ Þ ¼ 1 ¼ α∗ V2 > α∗ V1 ; no one will voluntarily convert to equity. If k∗ ≥ D1 ðF ∗ Þ ¼ 1;

ð3:22Þ

the manager will not call. Therefore, by choosing k∗ ¼ 1 satisfying (3.21) and (3.22), the manager will call if θ ¼ θ2 and conversion happens, but the manager will not call if θ ¼ θ1 and no conversion happens. Last, suppose that the investors believe θ ¼ θ1 whenever the manager issues any other security at date 0. Then neither type of manager has an incentive to deviate. By this, we have a Bayesian equilibrium. The following proposition is a summary. Proposition 3.13. Suppose β ¼ 1:Then it is a Bayesian equilibrium for both types of managers to issue the callable convertible ðF ∗ ; α∗ ; k∗ ; T ∗ Þ; defined by (3.19)–(3.22). The good manager will call and a conversion will take place, but the bad manager will not call and no conversion will take place. In equilibrium, the expected value of the convertible is 1 regardless of θ; the expected payoff to the firm (initial shareholders) with a type θi manager is Vi − 1; which is the first-best solution. Note that the outcome can also be implemented by short-term debt maturing at date 1 and then refinancing at date 1. Specifically, a manager can issue short-term risk-free debt and then issue any other security to retire the debt at date 1. The problem in this case with β ¼ 1 is trivial, since the message m can fully reveal the manager’s type. Imperfect revelation of information ex post We now assume β ∈ ð0:5; 1Þ: In this case, to achieve efficiency, a call restriction is to be imposed on the bad manager.9 Denote a callable convertible by ðF; α; k; p; TÞ; where T is the exercise date of the options, k is the call price, α is the conversion ratio (the equity share of the firm the bond holder can have), p is the trigger price for the call restriction, and F is the face value. Here, the call restriction serves as a commitment device by which the firm commits not to call if the stock price is below the trigger price. We now design an efficient callable convertible ðF; α; k; p; TÞ: First, suppose T ¼ 1: Second, suppose that F and α are determined by βD1 ðF Þ þ ð1 − β ÞαV1 ¼ 1;

ð3:23Þ

Corporate finance ð1 − β ÞD2 ðF Þ þ βαV2 ¼ 1:

133 ð3:24Þ

That is, if the manager thinks that the investors will hold onto debt if she is viewed as bad and the investors will get equity if she is viewed as good, the equation (3.23) means that the expected cost of the convertible is $1 for a bad manager and the equation (3.24) means that the expected cost of the convertible is $1 for a good manager. When β is large enough, equations (3.23) and (3.24) have a solution ðα; FÞ satisfying αVi < Di ðF Þ;

for

i ¼ 1; 2:

ð3:25Þ

That is, the expected value of the equity claim is always less than the expected value of the debt. There are two implications. First, by (3.23) and (3.24), we know that Di ðF Þ must be higher than the investment of $1 for both types of managers. That is, to compensate the investors, the debt is sweetened so that it has a higher value than it would obtain under complete information. Second, since the investors know (3.23) and (3.24), they won’t voluntarily convert and the manager will always want to force conversion. Thus, if there is no restriction on conversion, the convertible would be worth less than $1: Consequently, a restriction on the call provision is necessary, as it enables the manager to commit to not calling the convertible under certain conditions. With this guarantee, an investor may be willing to buy the convertible. Third, let i ðmj Þ ¼ Prð ¼ i jm ¼ mj Þ: Choose k satisfying k < μ1 ðm2 ÞαV1 þ μ2 ðm2 ÞαV2 :

ð3:26Þ

By this, if m ¼ m2 ; a convertible holder will convert if the convertible is called. Finally, choose p satisfying 2

2

i¼1

i¼1

∑ μi ðm1 Þ½Vi − Di ðF Þ < p < ∑ μi ðm2 Þð1 − α ÞVi :

ð3:27Þ

Such a p exists by (3.25). By this, the manager will be able to call if m ¼ m2 ; but the manager will not be able to call if m ¼ m1 : Proposition 3.14. There exists β∗ ∈ ð0:5; 1Þ such that for β > β∗ ; there is an equilibrium callable convertible ðF; α; k; p; TÞ; for which T ¼ 1; α and F satisfy (3.23) and (3.24), k satisfies (3.26), and p satisfies (3.27). In this pooling equilibrium, the manager calls only when m ¼ m2 ; regardless of what the value of θ is. The convertible cannot be called and is not converted when m ¼ m1 : At date 0, the expected value of the convertible is $1 to the investors and Vi − 1 to the firm. This outcome is the first best. In Proposition 3.14, we show that efficiency can be achieved in a pooling equilibrium. There exist other equilibria, involving similar securities, that also achieve

134 Corporate finance the same outcome. In particular, there exists a separating equilibrium in which the good manager issues a callable convertible similar to the one above and the bad manager issues any other security, say equity. In such a separating equilibrium, the signal m has no role to play. The key to Proposition 3.14 is that, even when the initial information asymmetry is never perfectly resolved, there exists at least one efficient equilibrium. 6.4. Discussion Mikkelson (1981) reports that an announcement of a call results in statistically significant negative returns to stockholders. Dann and Mikkelson (1984) and Eckbo (1986) also report that an announcement of an issuance of convertibles results in statistically significant negative returns to stockholders. These empirical findings are inconsistent with Chakraborty and Yilmaz’s (2004) conclusion that a call doesn’t necessarily imply the firm is bad in a pooling equilibrium, although it does imply that the firm is good in a separating equilibrium. In contrast, we will show in the next section that only a bad firm (a firm with a bad performance ex post) will call, which is consistent with the empirical findings.

7. Sequential financing using convertibles This section is based on the work of Wang (2009). Wang emphasizes three popular features in real-world corporate finance: milestones, sequential financing, and convertibles. Sequential financing is a very popular financing strategy in corporate finance. However, sequential financing may cause many incentive problems. Also, many types of financial instruments can be deployed to implement the financing strategy. Why convertibles? Wang shows that convertibles can be deployed to resolve the incentive problems completely. This may explain why convertibles are often used to implement sequential financing in reality. 7.1. The model The project Consider a firm that relies on an investor for investment. The project lasts two periods. The manager provides his effort x after accepting the contract, with private cost cðx Þ: The investor provides the necessary funding K in two stages, with an initial installment k in the first period and a second installment K − k in the second period. Output y is produced at the end of the second period, which is random ex ante with distribution function Φðy ; x; k Þ: Sequential financing is to allocate the total investment K between the two installments. Specifically, the investor provides a total of k in funds at the beginning of the first period. After the uncertainty is realized at the end of the first period, the

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135

investor considers providing the rest of the funds K − k. The investor has the option of not providing the second installment without penalty. If the project is abandoned in the middle by either the manager or the investor, the firm is liquidated for a fraction of the initial investment θk; where θ ∈ ½0; 1Þ :The manager and investor share output at the end of the project based on the existing contract. We have two mechanisms: a financial instrument and a sequential financing strategy. These two mechanisms form an incomplete contract. The financial instrument is either debt or a convertible. Timing The uncertainty of output is realized and publicly revealed at t ¼ 1: The manager’s effort is observable at time t ¼ 1 but not verifiable. The investor’s investments k and K − k are verifiable, but her option of whether or not to continue her investment at t ¼ 1 is not ex-ante contractible.

Contract

x,k

0 Ex ante

Information revelation options, renegotiation 1 Ex post

K−k

Output: y 2 End

Figure 3.5 The timeline of events.

At t ¼ 0; the two parties negotiate a contract. If the contract is accepted, the investor invests k and the manager applies effort x and incurs cost cðx Þ: At t ¼ 1; the investor considers the options to quit and to renegotiate. If the project is bad, the investor abandons the project without investing K − k ; if the project is mediocre, she may demand to negotiate a new contract; if the project is good, she continues to invest in the project and provides the necessary investment K −k: The manager may also demand renegotiation or decide an option ex post (see Figure 3.5). Financial instruments We consider two financial instruments, convertibles and bonds, for implementing the sequential financing strategy. A convertible pays a guaranteed rate of return r; it also provides an option for the investor to convert his investment into equity at anytime (either at t ¼ 1 or t ¼ 2) at a guaranteed conversion ratio τ: The investor can decide to purchase convertibles at t ¼ 0 and t ¼ 1 and convert his investments at t ¼ 1 and t ¼ 2: A convertible may also contain options for the manager. A bond pays a guaranteed rate of return r: The investor can decide to purchase bonds at t ¼ 0 and at t ¼ 1. Each financial instrument may also carry other rights.

136 Corporate finance Assumptions Denote Φðy ; x; k Þ as the distribution function of output y conditional on investments ðx; k Þ: Assumption 3.2 (FOSD). Φx ðy; x; k Þ ≤ 0 and Φk ðy; x; k Þ ≤ 0 for all ðy; x; kÞ: Assumption 3.3. The support of Φðy; x; k Þ of output y is independent of investments ðx; k Þ: Model setup Both the manager and investor are risk neutral in income. Let I and M be the ex-ante payoffs of the investor and the manager, respectively. Assume that the two parties negotiate and bargain over the terms of a contract ex ante and ex post. The problem is max M þ I

x; k;

s:t:

ð3:28Þ

@M ¼ 0: @x

Besides x and k; there may be other choice variables. IR conditions are ignored.10 There are several incentive problems. First, there is the traditional incentive problem, resulting from an unverifiable input from the manager. Second, the investor has an incentive problem, since her second investment decision comes after the manager’s investment. Third, the possibility of the second installment not coming affects the manager’s incentives. This can work in two ways: the manager may be discouraged by the risk of not having the second installment; the manager may also work harder to boost performance in order to secure the second installment. Finally, some special rights from the financial instrument used in the financing strategy will affect the incentives of both parties. 7.2. Benchmark: the first best Consider the first-best problem. Let y1 ≡ K − ð1 − θÞk: Then if and only if y ≥ y1 will it be ex-post efficient to continue the operation by committing the second investment. Hence, social welfare is Z 1 M þ I ¼ kðy1 ; x; kÞ þ ðy  K þ kÞdðy; x; kÞ  k  cðxÞ : y1

The first-best problem is max M þ I :

x; k0

Proposition 3.15 (First Best). The first-best solution ðx∗ ; k∗ Þ is determined by two equations:

Corporate finance Z

∞

ð1 − θ ÞΦðy1 ; x; k Þ þ Z

Φk ðy ; x; k Þ dy ¼ 0;

137 ð3:29Þ

y1 1

x ðy; x; kÞ dy þ c0 ðxÞ ¼ 0:

ð3:30Þ

y1

7.3. Debt Suppose that the investor invests in stages in the form of debt (see Figure 3.6). Assumption 3.4 (Foreclosure). If the firm is expected to go bankrupt, a debt holder has the right to foreclose the firm at t ¼ 1 by selling off the firm for the liquidation value. The two parties negotiate a contract t=0 The investor provides k

The manager provides x

t = 1, uncertainty realized, renegotiation allowed The investor provides K − k

Foreclosure

t=2

0 θk

Bankrupt

Normal y − (1 + r ) K (1 + r ) K

0 y

Figure 3.6 The game tree for debt financing.

Let y2 ≡ ð1 þ r ÞK: The firm will not go bankrupt if y > y2 ðsee Figure 3:7Þ: Hence, the payoffs are Z 1 ½y  ð1 þ rÞKdðy; x; kÞ  cðxÞ ; M  y2

Z

I  kðy1 ; x; kÞ þ Z þ

1

y2

½y  ðK  kÞdðy; x; kÞ

y1

½ð1 þ rÞK  ðK  kÞdðy; x; kÞ  k :

y2

The contractual problem is max M þ I x; k

s:t:

@M ¼ 0: @x

ð3:31Þ

138 Corporate finance default

bankrupt y1

investor receives debt repayment y2

y

Figure 3.7 Decision intervals for debt.

Proposition 3.16. Debt financing is generally inefficient. Let the solution ^ The solution is efficient if and only if ^ and let y^1 ≡ K − ð1 − θÞk: of (3.31) be ð^ x; kÞ Z y2 ^ x ðy; x^; kÞdy ¼ 0: ð3:32Þ y^1

Furthermore, if (3.32) fails, both the manager and investor will underinvest. R y2 Φðy;x;k Þ dy is a risk measure of bankruptcy, which is used in the definition of y 1

second-order stochastic dominance (SOCD). Hence, condition (3.32) means that the risk of bankruptcy is minimal. This is consistent with Stein (1992) and casual observations of the VentureXpert database. The explanation is: the manager is the sole residual claimant; the investor, with a fixed income, cares only about bankruptcy. 7.4. Straight convertible Suppose that the investor invests in stages in the form of convertibles. Assumption 3.5. A convertable is renegotiable at any time. If renegotiation leads to default, a convertible holder shares the proceeds with the manager based on their share holdings ðτ; 1 − τ Þ: Let y3 ≡ ð1 þ r ÞK=τ; where y1 is the threshold for default, y2 is the face value of debt, and y3 is the threshold for conversion (see Figure 3.8). The ex-ante payoff functions are Z y3 ½y  ð1 þ rÞKdðy; x; kÞ M  ð1  Þkðy1 ; x; kÞ þ Z þ

y2 1

ð1  Þydðy; x; kÞ  cðxÞ;

y3

Z

I  kðy1 ; x; kÞ þ Z þ

y2

½y  ðK  kÞdðy; x; kÞ

y1 y3

½ð1 þ rÞk  ðK  kÞdðy; x; kÞ

y2

Z þ

1

½y  ðK  kÞdðy; x; kÞ  k :

y3

default

bankrupt y1

debt y2

equity y3

Figure 3.8 Decision intervals for a straight convertible.

y

Corporate finance

139

The contractual problem is max M þ I x; k; 

s:t:

@M ¼ 0: @x

Proposition 3.17 (Inefficiency). Straight convertibles are inefficient. The intuition is that, although the manager gains a fraction of the liquidation value in the case of y < y1 when the firm defaults, he is not the sole residual claimant in the case of y > y3 when the firm does very well. A straight convertible fails to provide enough incentives to the manager. 7.5. Callable convertible Suppose that the investor invests in stages in the form of callable convertibles. The callable convertible works in this way. The investor has the option to convert her investment into τ equity shares or to demand payment ð1 þ rÞK at maturity. But, if the investor doesn’t convert her investment, the manager can call. If the manager calls, the investor can either convert her investment into τ equity shares or receive the call price q for her investment. Assumption 3.6. In period 2, the manager has the option to call with call price q. Without a call, the choices are as follows: EN’s payoff : (1 − τ ) θk default

0

y − (1 + r) K

(1 − τ) y

bankruptcy

debt

equity

y1

y2

y3

y

Figure 3.9 Decision intervals before a call.

With a call, the choices are as follows:

EN’s payoff : (1 − τ ) θk default y1

0

y−q

(1 − τ ) y

bankruptcy

Failed conversion

Forced conversion

q

Figure 3.10 Decision intervals after a call.

q/τ

y

140 Corporate finance

. .

The two parties negotiate for an agreement t=0 VC provides k VC

EN provides x t = 1, uncertainty realized, renegotiation allowed VC provides K − k

Default

.

EN

(1 − τ)θk τθk

t=1 EN calls

No call

VC

Bankrupt 0 y

.

t=2

VC Convert

No convert y − (1 + r) K (1 + r) K

Bankrupt

(1 − τ) y τy

.

t=2

No convert Convert y−q q

0 y

(1 − τ) y τy

Figure 3.11 The game tree for callable convertible financing.

With q ≤ y2 ; the payoffs are

Z

q=

EN  ð1  Þkðy1 ; x; kÞ þ Z þ

1

ð1   Þydðy; x; kÞ  cðxÞ;

q=

Z

VC  kðy1 ; x; kÞ þ Z þ

ðy  qÞdðy; x; kÞ

q

q

½y  ðK  kÞdðy; x; kÞ

y1 q=

½q  ðK  kÞdðy; x; kÞ

q

Z þ

1

½y  ðK  kÞdðy; x; kÞ  k :

q=

The contractual problem is max EN þ VC

x; k; ; q

s:t:

@EN ¼ 0: @x

Proposition 3.18 (Callable Convertible). With a call option, for the first-best investments ðx∗ ; k∗ Þ with y∗1 ≡ K − ð1 − θÞk∗ ;



Any callable convertible with a call price q ≥ y1 is inefficient. A callable convertible can achieve efficiency if

Corporate finance Z

y∗1 0

Φx ðy; x∗ ; k∗ Þdy < θk∗ Φx ðy∗1 ; x∗ ; k∗ Þ:

141 ð3:33Þ

That is, we find an efficient callable convertible ðK; r; τ ∗ ; q∗ Þ with principal K; interest rate r; conversion ratio τ ∗ ∈ð0; 1Þ; call price q∗ ∈ 0; y∗1 ; convertible and callable after t ¼ 1; and maturity at t ¼ 2; where K > 0 and r > 0 can take arbitrary values. Condition (3.33) requires the increasing effort from the manager to reduce the risk of default sufficiently quickly. By (3.33), the investor may not worry about the negative effect of the call option on her, since the tendency to call when the output is low is compensated by a smaller chance of default. That is, with a call option, the manager’s incentive is improved sufficiently, and by (3.33) the net effect of the call option on the investor’s incentive is neutral. Therefore, efficiency is achievable under (3.33). However, part (a) is disappointing. The call price q∗ for an efficient callable convertible is too low, which is unrealistic. A restriction on call allows a large q: 7.6. Callable convertibles with call protection Suppose that the investor invests in stages in the form of callable convertibles. These callable convertibles contain certain call protection. A provisional call means that the convertible cannot be called unless the stock trades above a stated level. The spot stock price in our model is y and the strike price or conversion price is K=τ: Let y be the trigger price. We should have y ∈½y2 ; y3 . Assumption 3.7. In period 2, if y ≥ y; the manager has the option to call with call price q and conversion ratio τ: Given a trigger price y ∈½y2 ; y3  and a call price q ≤ τ y; the ex-post decisions are as follows: Default

Bankruptcy

Debt

y2

y1

Call & Convert

y

Equity

y3

y

Figure 3.12 Decision intervals for a callable convertible with protection.

Proposition 3.19 (Call Protection). With a provisional call option, for the first-best investments ðx∗ ; k∗ Þ with y∗1 ≡ K − ð1 − θÞk∗ ; and any y ∈½y2 ; y3 ; if x ðy; x ; k Þ  0;

for

y1 < y < y;

ð3:34Þ

then staged financing using convertibles can achieve efficiency. That is, we find an efficient callable convertible ðK; r; τ ∗ ; q∗ ; yÞ with principal K; interest rate r; conversion ratio τ ∗ ∈ð0; 1Þ; call price q∗ ≤τ ∗ y; trigger price y∈½y2 ; y3 ; convertible after t ¼ 1; callable after t ¼ 1 if y ≥ y; and maturity at t ¼ 2; where K > 0; r > 0;

142 Corporate finance q∗ ≤τ ∗ y and y ∈½y2 ; y3  can take arbitrary values. In particular, if we take y ¼y2 ; condition (3.34) is unnecessary. Condition (3.34) means that an increase

 in the manager’s investment reduces the chance of producing an output in y∗1 ; y : Hence, condition (3.34) encourages the manager to expend more effort and the investor to invest. Condition (3.34) is easy to satisfy. For example, consider output process: y~ ¼ μðx; k Þ þ ε~;

ð3:35Þ

where μðx; k Þ is the mean output with μx ðx; k Þ > 0 and μk ðx; k Þ > 0 (which is the FOSD), and ε~ is a random shock with zero mean. Condition (3.34) is satisfied if the trigger price y is less than the mean output. The intuition for the efficiency result is as follows. When condition (3.33) fails, the call option has a negative net effect on the investor’s incentive, even though it has a positive effect on the manager’s incentive. One way to rectify this problem is to impose some restrictions on calls. A trigger price serves the purpose. It prevents the manager from forcing conversion when the firm is doing badly. With this trigger price, a proper balance of incentives between the manager and investor is achieved for many output distribution functions satisfying some minor conditions such as (3.34). 7.7. Discussion Chakraborty and Yilmaz (2004) and Wang (2009) both emphasize callability and call protection. They both find a provisional call to be important. Chakraborty and Yilmaz (2004) use the asymmetric-information approach, while Wang (2009) uses the incomplete contract approach. The conclusions are very different. For example, Chakraborty and Yilmaz (2004) conclude that a call doesn’t necessarily imply that the firm is bad in a pooling equilibrium, although it does imply that the firm is good in a separating equilibrium. In contrast, Wang (2009) shows that only a bad firm (a firm with a bad performance ex post) will call, which is consistent with empirical evidence. In Wang (2009), a firm’s type is determined ex post by bad incentives or bad luck, while in Chakraborty and Yilmaz (2004) a firm’s type is exogenously given. Here we would like to reiterate our incomplete contract approach. The fact that a simple contract such as a convertible can achieve efficiency in this case is due to a second mechanism: sequential financing. This study illustrates our central idea in this book: contracts can be simple or can look highly incomplete in the presence of other mechanisms. The efficient contract in Wang (2009) can be written as either C1 ¼ fcallable convertibleg or C2 ¼ fcallable convertible; sequential financingg: C1 is obviously incomplete and is what the literature refers to as an incomplete contract, while C2 is complete within the current model. This example shows why we say that incomplete contracts are actually complete in our definition of incomplete contracts. To be consistent with the tradition in the literature, we call C2 an incomplete contract. The difference between our incomplete contract approach and that in

Corporate finance

143

the literature is that we explicitly identify the “hidden part” of the contract while the literature tries to explain the incompleteness of C1 mainly by transaction costs. The incompleteness is actually only in appearance and is not due to transaction costs. In other words, contracts may seem incomplete due to the presence of other mechanisms, rather than transaction costs as emphasized in the literature.

8. Investment syndication with endogenous networks This section is based on the work of Wang and Wang (2011). As an effective investment strategy, investors often invest jointly in a company by forming an investment syndicate, which is typically led by a lead investor. Investment syndicates are very popular. For example, 64 percent of US, 60 percent of Canadian, and 30 percent of European venture capital investments are syndicated. Wang and Wang provide a theory on the endogenous formation of networks in investment syndication and analyze how risk aversion, productivity, risk, and cost affect incentives and investment. 8.1. The model Consider a project in which the entrepreneur (EN) provides unverifiable effort a: The EN (she) is risk neutral. Funding is provided through a syndicate. Investors are risk averse. The syndicate (SN) is organized by a lead investor (he). The syndicate is defined by ðn; α1 ; …; αn Þ; where n is the number of investors in the syndicate, called the lead VC’s network capital, and αi is investor i’s risk aversion. One crucial determinant of a lead investor’s ability to organize a syndicate is his business connections, known as the lead investor’s network capital. When choosing a lead investor, the EN needs to strike a balance between a syndicate’s ability to provide funding and its potential to bargain for company profits. Each investor has utility function: ui ðx Þ ¼

1 x 1 − αi : 1 − αi

ð3:36Þ

A project is a lottery x~ defined by x~ ≡ ðxðIÞ; pða Þ; 0; 1 − pðaÞÞ; where I is the total investment, xðI Þ is the output when the project is successful, and pðaÞ is the probability of success. xðIÞ is increasing and pðaÞ is increasing and concave. The problem is solved backwards in three steps as follows:

EN chooses n

Investors jointly provide I EN applies effort a

0

Figure 3.13 The timeline of events.

Bargain for output λ for SN , 1 − λ for EN 1

144 Corporate finance We take an incomplete contract approach. In our model setting, there is no contract ex ante at t ¼ 0 and the two parties bargain for output shares ex post at t ¼ 1: However, the equilibrium solution is implementable with an equity-sharing agreement ex ante. Given n∗ ; an upfront equity-sharing agreement that gives the syndicate an equity share of λðn∗ Þ implements the solution. Step 1: the EN’s effort After output is produced, the two parties bargain for a share. If no agreement is made, both parties get nothing. If there is an agreement, they share output x: Assuming that the lead investor behaves as a risk-neutral agent when bargaining with the risk-neutral EN, the Nash bargaining solution implies ex-post payoffs: x; EN ¼ ½1  λðnÞ~

SN ¼ λðnÞ~ x;

where λðn Þ is the syndicate’s output share from Nash bargaining. With private cost cðaÞ and risk neutrality in income, the EN’s ex-ante payoff is UEN ¼ ½1 − λðnÞpðaÞxðI Þ − cðaÞ; Then the EN chooses a by the first-order condition (FOC): ½1  λðnÞp0 ðaÞxðIÞ ¼ c0 ðaÞ:

ð3:37Þ

The SOC is satisfied since pðaÞ is concave. Step 2: the investors’ investment Investor i with investment Ii receives a share Ii =I from the syndicate’s total income λðn Þ~ x: With the coordination of the lead investor, the investors form a cooperative group. Then, assuming ui ð0Þ ¼ 0; the syndicate’s investment decision is determined through welfare maximization:    n

 Ii λðn Þx ∑j Ij − Ii USN ≡ max ∑ pða Þui ∑ j Ij fIi g i ¼ 1 This problem implies the total investment I: Based on the existing literature, we assume homogeneity among the investors, i.e. all investors have the same risk aversion α: Hence, we have Ii ¼ I=n for all i: Then the FOC is:  λðnÞxðIÞ 1 x0 ðIÞ ; ð3:38Þ 1 ¼ pðaÞn n xðIÞ ^ n Þ: which determines the optimal investment Iða; Step 3: the EN’s choice of network capital The EN selects a lead investor with a certain network capital n: Given equations (3.37) and (3.38), the EN’s problem is

Corporate finance

145

max ½1  λðnÞpðaÞxðIÞ  cðaÞ n; a; I

s:t: IC1 : ½1  λðnÞp0 ðaÞxðIÞ ¼ c0 ðaÞ;  λðnÞxðIÞ 1 x0 ðIÞ ¼ 1: IC2 : pðaÞn n xðIÞ

ð3:39Þ

We do not include IR conditions in problem (3.39), since we implicitly allow the bargaining process to include a monetary transfer between the two parties to ensure both IR conditions for the EN and the syndicate. It turns out that such a transfer is unnecessary, since the solution of (3.39) automatically satisfies the two IR conditions when we choose a set of simple functions in the following analysis. 8.2. Analysis To analyze the solution, we use parametric functions: ui ðx Þ ¼

1 1−α x ; 1−α

pða Þ ¼ aγ ;

cða Þ ¼ a β ;

xðI Þ ¼ Iδ ;

λðn Þ ¼ ρn;

where α; γ; δ; ρ ∈ ð0; 1Þ; β ≥ 1: Denote θ ≡ γ=β: We have






α is risk aversion, γ represents project riskiness, β represents easiness of effort, δ represents productivity (effective use of investment), ρ is an investor’s income share.

Since the equilibrium solution a∗ is less than 1; an increase in β reduces the cost of effort cða∗ Þ and chance of success pða∗ Þ: Hence, β represents easiness of effort and γ represents project riskiness. Given this set of parametric functions, we have a closed-form solution: δ ; ð1 þ αδ Þρ

n∗ ¼

a ¼ I ¼ λ∗ ¼



1 þ  

1 þ   1 þ  

1 þ  

δ : 1 þ αδ

ð3:40Þ

1 þ ð Þð1þ Þ



ð Þð1þ Þ

 

2 1 þ  2

1 þ 



ð Þð1þ Þ



ð Þð1þ Þ

; ð3:41Þ ; ð3:42Þ

In contrast, if the network capital is exogenous (i.e. n is a given constant), with a corresponding change to problem (3.39), the solution is

146 Corporate finance 

1 þ

ð1 þ Þ ð1þ Þ 1 ð1 þ Þ ð1þ Þ a^ ¼ ð1  nÞ  n ; 

 

ð1 þ Þ ð1þ Þ



1 n ð1 þ Þ ð1þ Þ : I^ ¼ ð1  nÞ 

ð3:43Þ

Syndication versus solo Is it possible to have a syndicated investment that is smaller than solo investment (i.e. an underinvestment)? Is it possible that incentive is worse under syndication? By denoting I^ and a^ in (3.43) respectively as functions I^ðn Þ and a^ ðnÞ of n; we can ^ ∗ Þ; and compare compare solo investment I^ð1Þ with syndicated investment I∗ ¼ Iðn ∗ ∗ effort a^ ð1Þ under solo investment with effort a ¼ a^ ðn Þ under syndication. The ^ ∗ Þ < Ið1Þ ^ question is, is it possible to have Iðn and a^ ðn∗ Þ < a^ ð1Þ when n∗ ≥ 1? It turns out that it is. Proposition 3.20 (Syndication vs. Solo). Given condition n∗ ≥ 1; i.e. ρ ≤


;

Solo investment is larger than syndicated investment if and only if ρ≤1− θ≤




δ 1þαδ

δ : 1 þ αδ

ð3:44Þ

Incentive under solo investment is better if and only if 1

: 1 þ 

ð3:45Þ

A solo investor has good incentives, since she is the sole recipient of the income share for investors and hence may invest a lot. With syndication, there are multiple sources of funding, but each investor’s willingness to invest is low, since the investors have to share the benefits. Both the EN and the investors care about output shares, but, while the EN welcomes more investment, the investors prefer to invest less. Depending on project risk, risk aversion, productivity, and cost of effort, each party may have a tendency towards one objective or the other. In cases when the EN does not care so much about the investment amount, for example, if productivity is high, solo investment may be larger than syndicated investment. Wang and Wang’s theory does indeed suggest this. Condition (3.44) is more likely to hold if δ is large, in which case solo investment would be larger than syndicated investment. One important function of the syndicate is to share risk. If risk aversion is low, the syndicate’s role of risk sharing is reduced, while output sharing within the syndicate will still have a negative effect on investment. Hence, if risk aversion is low, solo investment may be larger than syndicated investment. Wang and Wang’s theory does indeed confirm this. Condition (3.44) is more likely to hold if α is small, in which case solo investment would be larger than syndicated investment. On the other hand, if risk aversion is high, a solo investor will

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invest conservatively, while syndication allows risk sharing so that investors will be more willing to invest. In this case, we expect syndicated investment to be larger than solo investment. Indeed, when α is large, condition (3.44) is more likely to fail, in which case syndicated investment would be larger, according to Wang and Wang. When the project is highly risky, we expect a preference for syndication on both sides. Indeed, when the project is risky, as represented by a large γ; condition ρ ≤ 1 − θ in (3.44) tends to fail, implying a larger syndicated investment. If effort is costly, the EN will rely more on investment than her own effort, in which case she is likely to go for syndication. Indeed, if β is small, condition ρ ≤ 1 − θ in (3.44) tends to fail, in which case syndicated investment would be larger. Network capital Proposition 3.21 (Network Capital).


The syndicate size or network capital is diminishing in risk aversion, and this negative effect of risk aversion on syndicate size is also diminishing in risk aversion: ∂n∗ < 0; ∂α




∂2 n ∗ > 0: ∂α2

ð3:46Þ

The syndicate size or network capital is increasing in productivity, and this positive effect of productivity on syndicate size is also diminishing in productivity: ∂n∗ > 0; ∂δ

∂2 n ∗ < 0: ∂δ2

ð3:47Þ

Proposition 3.21 suggests that more risk aversion implies a smaller syndicate. This result makes sense, since highly risk-averse investors demand high compensation for their investment. Consequently, the EN prefers a small syndicate to reduce the investors’ bargaining power for compensation. Proposition 3.21 also indicates that, for a project with high return potential, the EN will pick a lead investor to organize a large syndicate. Investment and incentives We now investigate the effects of risk aversion, productivity, and an investor’s output share on incentive and investment. While λ is the output share of the syndicate, ρ is the output share of each investor. The effect of ρ is clear from (3.40) and (3.41): an increase in the investor share ρ reduces the syndicate size, effort and investment. The reason is that a larger investor share reduces the EN’s incentive, which in turn reduces investment; consequently the EN will choose a smaller syndicate size.

148 Corporate finance Proposition 3.22 (The Effect of Risk Aversion on Investment and Incentive).


Investment generally increases with risk aversion. Specifically, investment increases with risk aversion if and only if 2 ð1þ  Þ 1  1 1 ð1þ  Þ1  1 1 2 1 ð 1þ Þ:  ð1 þ  Þ ½ð1 þ   Þ e ð3:48Þ 




Incentive generally improves with risk aversion. Specifically, incentive improves with risk aversion if and only if 1



 1  2 ð1 þ  Þ1 ½ð1 þ   Þ1 : 1

1

ð3:49Þ

Typically, θ ≡ γ=β is small, since β is larger than 1: Hence, we will generally have ∗ δ < 1 − θ: If so, Proposition 3.22(a) can be expressed as ∂I ∂α ≥ 0 if and only if ( ) 11 2 ð1þ 1 Þ 1 1 1    2   ð1 þ  Þ1 ½ð1 þ   Þ 1 e ð1þ  Þð1þ Þ : With reasonable parameter values (where α assumes values between 0 and 1Þ; the right-hand side of the above inequality is above 70. Hence, condition (3.48) generally holds. This means that investment generally increases with risk aversion. By the same reasoning, incentive generally improves with risk aversion. The explanation is that risk aversion puts pressure on performance, which improves incentive and in turn induces more investment. Higher risk aversion also reduces the syndicate size and hence results in a larger output share for the EN, which also improves incentive. It is interesting to see that more risk-averse investors end up investing more in equilibrium. Proposition 3.23 (The Effect of Productivity on Investment and Incentive).


Investment increases with productivity if and only if ρ is small. Specifically, investment increases with productivity if and only if

 2 1 2þ 1  1 þ   ð1Þ  eð1Þð1þÞðð1Þ 1þ  Þð11þ Þ : ð3:50Þ  1 þ  1 þ 




Incentive is generally decreasing in productivity. Specifically, incentive is decreasing in productivity if and only if 

2 1 þ 

1

 1 þ   ð1Þ 1 þ ð1Þ e :  1 þ 

ð3:51Þ

For reasonable parameter values, the right-hand side of (3.50) is between 0 and 10; and the right-hand side of (3.51) is less than 0:17: Hence, investment is increasing in productivity if and only if ρ is small, and incentive is generally decreasing in productivity. There are two reasons for these results. On the one hand, since a larger investor output share ρ has a negative effect on incentive, ρ cannot be too large for

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the EN to have enough incentive. If the investor share is small, with sufficient incentive, the EN welcomes higher productivity and applies more effort, which in turn induces investors to invest more. On the other hand, a large investor share sufficiently dampens the EN’s incentive; with insufficient incentive, higher productivity puts less pressure on the EN to perform, which implies less incentive and investment. Cost of effort and project risk Proposition 3.24 (The Effects of Easiness of Effort and Project Riskiness).


Investment is generally positively associated with easiness of effort and negatively associated with project riskiness. Specifically, an increase in easiness of effort β or a decrease in project riskiness γ raises investment if and only if 2 1

1þ   ð1Þð1þ Þ 1 þ    e  :  1 þ  1 þ 




Incentive is generally negatively associated with project riskiness. Specifically, a decrease in project riskiness γ improves incentive if and only if 2 1

1þ   1þ  ð1Þð1þ Þ 1 þ   1þ  e   :  1 þ  1 þ 




ð3:52Þ

ð3:53Þ

Incentive is generally positively associated with easiness of effort. Specifically, an increase in easiness of effort β improves incentive if and only if (3.52) holds.

For reasonable parameter values, the right-hand sides of (3.52) and (3.53) are less than 0:02 and 0:06; respectively. Hence, these two conditions generally hold. Endogenous vs. exogenous networks One natural question is whether the type of networks, either endogenous or exogenous, is important to our results. In Figure 3.14, we graph investment by 0.04

I*

δ = 0.5

0.03

δ = 0.3

0.02 0.01 0.24

0.26

0.28

0.3

0.32

α

Figure 3.14 Investment with endogenous network.

150 Corporate finance Iˆ 0.11

δ = 0.3

0.1 0.09

δ = 0.5

0.08 0.07 0.06 0.24

0.26

0.28

0.3

0.32

α

Figure 3.15 Investment with exogenous network.

arbitrarily setting ρ ¼ 0:2; γ ¼ 0:5; β ¼ 2 and allowing risk aversion to change; in Figure 3.15, we regraph investment with the same parameter values except for a fixed n0 at an equilibrium value. Comparing these two figures, we find that productivity has opposite effects on investment. Hence, the endogeneity of network is important to our conclusions. 8.3. Discussion Existing studies (mostly empirical works) on investment syndication have mostly focused on the role of the lead investor. Wang and Wang focus instead on strategic interactions between managers and investors. In their model, investment decisions together with syndication are determined in three stages. Most of the existing studies have focused only on the second stage. Wang and Wang demonstrate that network endogeneity is crucial; opposite effects are found if the network is treated as exogenous instead of endogenous.

4

Corporate governance

This chapter presents a few applications of organization theory in corporate governance.

1. Contractual joint ventures in FDI This section is based on the work of Tao and Wang (1998). A long-standing deterrent to FDI in developing countries is weak enforcement of binding contracts. A local firm may learn business skills from a cooperating multinational firm before ditching that firm and doing business on its own with the acquired skills. This may deter foreign firms from forming joint ventures with local firms. Joint ventures in FDI have two popular forms: EJVs and CJVs. In an EJV, participating firms have equity shares and profits are divided based on equity shares. The shares are typically constant throughout the cooperation period. In a CJV, there are no equity shares; instead, yearly revenue shares and responsibilities are specified in the contract. Table 4.1 presents a few real-world examples of CJVs. The fact that much of the FDI in the early days of China’s open-door era was in the form of CJVs, despite weak contract enforcement, is puzzling to economists. Tao and Wang’s (1998) theory can explain the puzzle. 1.1. The model Consider a foreign firm and a local firm which jointly undertake a project through a contractual arrangement (see Figure 4.1). The foreign firm has technology and the local firm has some complementary inputs. The project lasts two periods. Ei and ei, i =1, 2, are efforts/investments from the foreign and local firms, respectively, where Ei is 0 or 1 and ei 2 [0, 1]. These efforts are unverifiable. The success probability of the project in the first period is p1 = e1E1. The foreign firm’s effort is interpreted as providing either good technology, Ei = 1 with cost Ci > 0, or bad technology, Ei = 0 with zero cost. The local firm’s effort is interpreted as developing the foreign firm’s 1þβ

β ei β ; where β > 0. Once technology into a product with private cost cðei Þ ¼ 1þβ successful, the project generates verifiable revenue Ri > 0; otherwise it has zero revenue. The model is a true two-period model, with a payment in every period (see Figure 4.1).

152 Corporate governance Table 4.1 Contractual joint venturesa Company

Duration

Revenue shareb

Technology

Stone cutting

10 years from 1986 5 years from 1985 6 years from 1983 10 years from 1985 unspecified from 1984 8 years from 1985

50/50

Equipment from West Germany, some from Italy Basically traditional technology Packaging machinery, imported from Japan Equipment from Japan

Dried duck processing Confectionery manufacturing Footwear manufacturing Restaurant Kitchen equipment for restaurant Jewellery manufacturing

10 years from 1985

Camera manufacturing

10 years from 1985

55/45 50/50 for first 3 years 55/45 for last 3 years 50/50 10/90 0/100 for first 3 years 40/60 for next 2 years 50/50 for next 1 year 60/40 for last 2 years 56/44 over and above capital repayment to Hong Kong side 30/70 of net profit

Hong Kong style restaurant supervision NA

Hong Kong designs

NA

a

This table is adopted from Thoburn et al. (1990) Chinese revenue share versus foreign revenue share

b

Contract (X1,X2)

e1, E1

0

Output R1 1

e2, E2

Output R2 2

Figure 4.1 The timeline of events.

Through cooperation in the first period, the local firm may learn the foreign firm’s technology. The probability of doing so is given by ϕ = ke1E1, where k 2 (0, 1]. If the local firm learns the technology in the first period, the success probability in the second period is p2 = e2. If the local firm does not learn the technology in the first period, the success probability in the second period is p2 = e2E2. Assume that the foreign firm designs the contract. There are two types of contracts. First, the foreign firm can write a nonbinding contract (incomplete contract) that allows the local firm to dismiss the foreign partner at time 1. In the second period, the local firm will work on its own if it has learned the technology; otherwise it will be in both firms’ interests to continue their cooperation. Second, the foreign firm can write a binding contract (complete contract) that prevents the local firm from doing business on its own upon learning the technology. A nonbinding contract is a revenue-sharing rule plus

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153

an option for the local firm to leave anytime. A binding contract doesn’t contain the option. Binding contracts require contract enforcement whereas nonbinding contracts do not. There are four classes of contracts in total: nonbinding two-period contracts, binding two-period contracts, one-period contracts in the first period, and one-period contracts in the second period. One-period contracts can be considered as either nonbinding or binding contracts. We call the two-period contracts long-term contracts and the one-period contracts short-term contracts. We first derive the optimal contract in each class. We then find the best contract among the four optimal contracts. The foreign firm (the contract designer) will propose this best contract to the local firm. If this best contract turns out to be the most preferred contract by both parties, it becomes what we call the dominant contract. 1.2. Long-term contracts The optimal nonbinding long-term contract We solve for the optimal contract backwards. For a given contract (X1, X2), in the second period, if the local firm has learned the technology, it will work on its own and its problem is: ∗ ≡ max e2 R2 −cðe2 Þ; π2;F e2 ≥0

which yields e2;F ¼ Rβ2 ;

π 2;F ¼

1þβ 1 ðe2;F Þ β : 1þβ

If the local firm has not learned the technology, it maintains the contractual relationship with the foreign firm and the local firm’s problem is: π^2  max e2 E2 ðR2  X2 Þ  cðe2 Þ;

ð4:1Þ

e2 0

which yields ( e^2 ¼

½E2 ðR2  X2 Þβ ; 0;

if X2  R2 ; otherwise;

1þβ

π^2 ¼

1 e^ β : 1þβ 2

If the local firm has not learned the technology, the foreign firm’s expected profit is Π2 ¼ E2 ðe2 X2 − C2 Þ; implying E^ 2 ¼ 1 if and only if e2 X2 − C2 ≥ 0: Thus, there are two possible Nash equilibria. If X2  R2

ðR2  X2 Þβ X2  C2  0;

and

ð4:2Þ

the Nash equilibrium is E^2 ¼ 1;

e^2 ¼ ðR2  X2 Þβ

ð4:3Þ

154 Corporate governance with 1þβ

^ 2 ¼ e^2 X2  C2 ; Π

π^2 ¼

1 e^ β ; 1þβ 2

^ 2 ¼ 0 and π^2 ¼ 0: otherwise the Nash equilibrium is E^2 ¼ 0 and e^2 ¼ 0; with Π In the first period, the local firm’s problem is: h i π 2 ; ð4:4Þ π^N  max e1 E1 ðR1  X1 Þ  cðe1 Þ þ  ke1 E1 π 2;F þ ð1  ke1 E1 Þ^ e1 0

which implies 8n h ioβ < E1 R1  X1 þ kðπ 2;F  π^2 Þ ; e^1 ¼ : 0

if

X1  R1 þ kðπ 2;F  π^2 Þ;

otherwise;

and 1þβ

π2 ; π^N ¼ π^1 þ ^

where

π^1 

1 e^ β : 1þβ 1

The foreign firm’s expected total profit is ^ 2; ΠN ¼ E1 ðe1 X1  C1 Þ þ ð1  ke1 E1 Þ Π ^ 2 Þ  C1  0: implying E^1 ¼ 1 if and only if e1 ðX1  kΠ Again, there are two possible Nash equilibria in the first period. If h iβ R1  X1 þ kðπ 2;F  π^2 Þ X1  R1 þ kðπ 2;F  π^2 Þ; and ^ 2 Þ  C1  0; ðX1  kΠ

ð4:5Þ

then the Nash equilibrium is

h iβ e^1 ¼ R1  X1 þ kðπ 2;F  π^2 Þ ;

E^1 ¼ 1;

ð4:6Þ

with ^ N ¼ e^1 X1  C1 þ ð1  k^ ^ 2; Π e1 ÞΠ

π^N ¼ π^1 þ ^ π2 ;

^ N ¼ Π ^ 2 and otherwise the Nash equilibrium is E^1 ¼ 0 and e^1 ¼ 0; with Π π2 : π^N ¼ ^ Finally, the foreign firm designs a contract. Under conditions (4.2) and (4.5), which imply the efforts in (4.3) in the second period and the efforts in (4.6) in the first period, the foreign firm solves: ΠN 

max

X1 ; X2 2ð1; 1Þ

^ 2; e^1 X1  C1 þ ð1  k^ e1 ÞΠ

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155

which yields: i 1 h R1 þ kðπ 2;F  π 2 þ βΠ2 Þ ; 1þβ  β h iβ β ¼ R1 þ kðπ 2;F  π 2  Π2 Þ ; 1þβ

X1;N ¼ e1;N

E1 ¼ 1;

R2 ; 1þβ   βR2 β e2 ¼ ; 1þβ

X2 ¼

E2 ¼ 1;

π N ¼ π 1;N þ π 2 ; ΠN ¼ Π1;N þ Π2 ; where 1þβ 1þβ 1 1 ðe1;N Þ β ; π 2  ðe2 Þ β ; 1þβ 1þβ 1þβ 1þβ 1 1  ðe1;N Þ β  C1 ; Π2  ðe2 Þ β  C2 : β β

π 1;N  Π1;N

For the optimal X1 and X2 ; the first constraints in (4.2) and (4.5) are automatically satisfied. The second constraints in (4.2) and (4.5) are equivalent to Π∗1;N ≥ 0 and Π∗2 ≥ 0; respectively. This solution is summarized in the following lemma. Lemma 2 (Nonbinding Contract). If Π∗1;N ≥ 0 and Π∗2 ≥ 0; there exists an equilibrium nonbinding two-period contract ðX∗1;N ; X∗2 Þ with efforts e∗1;N ; e∗2 ; E∗1 ; E∗2 and profits πN∗ and Π∗N . ∗ We may regard π1;N ; π2∗ ; Π∗1;N , and Π∗2 as profits without learning. The foreign firm’s expected first-period profit is e∗1;N X∗1;N − C1 ¼ Π∗1;N þ δke∗1;N Π∗2 ; and its expected discounted second-period profit is δΠ∗2 − δke∗1;N Π∗2 : Hence, under the nonbinding contract, the foreign firm gets a larger revenue share than Π∗1;N in the first period to compensate for the expected loss from learning.

The optimal binding long-term contract In this case, the local firm does not have the option to leave even if it can do better on its own. In the second period, if the local firm has learned the technology, its problem is: π^20 ¼ max e2 ðR2  X2 Þ  cðe2 Þ; e2 0

which gives ( e^02 ¼

ðR2  X2 Þβ ;

if X2  R2 ;

0;

otherwise;

π^20 ¼

1þβ 1 ð^ e02 Þ β : 1þβ

^20 ¼ e^02 X2 : The rest is similar to the derivation of the nonbinding For the foreign firm, Π contract.

156 Corporate governance Lemma 3 (Binding Contract). If Π∗1;B ≥ 0 and Π∗2 ≥ 0; there exists an equilibrium binding contract ðX∗1;B ; X∗2 Þ with efforts e∗1;B ; e∗2 ; E∗1 ; E∗2 and profits πB∗ and Π∗B . The foreign firm’s expected first-period profit is e∗1;B X∗1;B − C1 ¼ Π∗1;B − δke∗1;B C2 ; and its expected second-period profit is δΠ∗2 þ δke∗1;B C2 : Hence, under the binding contract, the foreign firm gets a smaller first-period revenue share than Π∗1;B due to the expected gain from learning. 1.3. Optimal short-term contracts The firms may also have one-period contracts. The optimal one-period contracts are stated in the following lemma. Since it takes one period to learn, contract enforcement is unnecessary for one-period contracts. Lemma 4 (One-Period Contracts). 

If Π∗1st ≥ 0; there exists an equilibrium one-period contract X∗1;1st in the first ∗ and Π∗1st ; where period with efforts e∗1;1st and E∗1;1st and profits π1st R1 k  þ π ; 1 þ β 1 þ β 2;F   βR1 kβ  β þ π 2;F ; ¼ 1þβ 1þβ

X1;1st ¼ e1;1st

π 1st ¼ 

1þβ 1 ðe1;1st Þ β ; 1þβ

E1;1st ¼ 1;

1þβ 1 Π1st ¼ ðe1;1st Þ β  C1 : β

If Π2  0, there exists an equilibrium one-period contract X∗2 in the second period with efforts e∗2 and E∗2 and profits π2∗ and Π∗2 ; where X∗2 ; e∗2 ; E∗2 ; π2∗ and Π∗2 are as in Lemma 2.

1.4. The dominant contract We call a contract the dominant contract if it is preferred by both firms to all other contracts. Proposition 4.1 (Dominant Contracts).    

If Π∗1;N ≥ 0 and Π∗2 ≥ 0; the dominant contract is the nonbinding contract in Lemma 1. If Π∗1;N < 0 but Π∗2 ≥ 0; the dominant contract is the one-period contract in the second period in Lemma 3. If Π∗1st ≥ 0 but Π∗2 < 0; the dominant contract is the one-period contract in the first period in Lemma 3. If Π∗1st < 0 and Π∗2 < 0; there is no feasible contract.

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157

Proposition 4.1 establishes the firms’ preferences for all possible cases. It implies that contract enforcement is unnecessary in all possible cases. 1.5. Characteristics of CJVs Characteristic 1 (Patterns of Revenue Shares). 

The foreign firm’s revenue shares in the second period are always the same under both the binding and nonbinding contracts: X∗2 1 : ¼ R2 1þβ



Across periods, under the nonbinding contract, the foreign firm demands a larger revenue share in the first period: X∗1;N X∗ > 2: R1 R2



ð4:7Þ

ð4:8Þ

However, under the binding contract, the foreign firm demands a smaller revenue share in the first period: X∗1;B X∗ < 2: R1 R2

ð4:9Þ

The first-period revenue shares reflect gains or losses from learning. Under the nonbinding contract, the revenue share for the foreign firm declines over time, which is consistent with empirical evidence. This empirical evidence supports our theory that, if a long-term contract is to be chosen, it must be the nonbinding contract. The following result shows that the key reason for the preference for the nonbinding contract is that it provides better incentives to the local firm. Characteristic 2 (Contractual Preference). In the first period, the local firm always chooses more effort under the nonbinding contract than under the binding contract, i.e. e∗1;N > e∗1;B : The following result shows, further, that the profits of both firms increase with faster learning under the nonbinding contract. Characteristic 3 (Payoffs under Faster Learning). The profits of both firms, πN∗ and Π∗N ; increase as k increases under the nonbinding contract. While poor protection of intellectual property rights (IPRs) in developing countries is thought to deter inward FDI, empirical evidence has been inconclusive. In fact, Latin American and East Asian countries were the most active places for FDI in the 1980s, even though these countries are known to be weak in contract enforcement and protection of IPRs. Also, there is little variation among the Visegrad countries

158 Corporate governance (i.e. Poland, Hungary, the Czech Republic, and Slovakia) in contract enforcement, but much variation in FDI (Peitsch 1995). In our theory, with endogenous learning by doing, the nonbinding contract makes both parties better off, which casts doubt on the popular view that attributes a negative incentive for incoming FDI to an unsettled legal environment. 1.6. Discussion Tao and Wang’s (1998) conclusion relies heavily on a key feature in their model, i.e. a time lag for learning. The time lag is enough for the foreign firm to compensate for its potential loss of IPRs. Tao and Wang’s (1998) model is suitable only for the early stages of an economic transition. At those stages, learning by doing is an important factor affecting the local firm’s incentives. Strict contract enforcement may be necessary for the later stages of an economic transition.

2. Endogenous governance transparency This section is based on the work of Hidalgo (2009), which endogenizes the choice of governance transparency. Consider a company whose manager raises capital in financial markets that are subject to imperfections arising from the nonobservability of returns for investors. The investors can observe a signal correlated with returns. The manager optimally decides the quality of the signal (the quality of governance transparency) based on the trade-off between the possibility of expropriating profits and the opportunity of raising more capital. Hidalgo shows that one important driving force of governance transparency is the competition in the product market: tougher market competition translates into fewer frictions in the capital market, so that investors have greater possibilities for portfolio diversification. The manager reacts to market competition by increasing disclosure/transparency. 2.1. The model There are two periods. There is one investor, but there are n managers competing for the investor’s fund. The investor is endowed with capital and each manager is endowed with a risky project. Managers No manager has any financial capital endowment. The managers are risk neutral and the investor is risk averse. All the projects are the same, with the same returns and risk characteristics. A project has a random gross return R~ 2 fR; 0g:That is, for investment k in the first period, the output is either Rk or in the second period. The   success probability of the project is Pr R~ ¼ R ¼ 1=n. Here n is the number of firms/managers on the market. It is a measure of the degree of competition among firms. Only one firm can succeed.

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159

Information structure Assume that realizations of output are private information to the manager. The investor receives only a binary signal with two possible values: high and low. The correlations between the signal and the realized return of a project are listed in Table 4.2. Table 4.2 Correlations between the signal and the realized return signal output

h

l

R

p

1−p

0

1−p

p

This table implies that the signal is symmetric so that Prðsignal ¼ hjR~ ¼ RÞ ¼ p;

Prðsignal ¼ ljR~ ¼ 0Þ ¼ p:

This probability p represents the quality of the signal. If p ¼ 0:5; the signal conveys no information. Hence, we assume p ≥ 0:5: Timing At time t ¼ 0; manager i proposes her project Ci ¼ fR; 0; 1=n; pi g to the investor.   Then the investor offers a contract wli ; whi to the manager, where wsi specifies the income the manager will receive in reported state s: This contract is based on the manager’s report on return. Contracting

Transparency

Investment k

∼

Output Rk

Report

0

Signal

1

Figure 4.2 The timeline of events.

The investor The investor has an endowment of capital K: His utility function is  U ¼ c1 1 c2 ;

where ct is consumption in period t. The budget constraint is n

c1 þ ∑ ki ¼ K; i¼1

where ki is the investor’s investment in manager i’s project. The investor will observe the signal and receive the manager’s report. If the signal and the manager’s report are not consistent, the investor may go to a court to verify the information. We assume that the court is able to verify the information. If the court finds the

160 Corporate governance manager guilty of false reporting, the manager pays penalty Fki ; where F is an exogenous constant. 2.2. Solution We solve the problem backward. Contract A contract is based on the manager’s report on output. Given investment k; in order for the manager to report the truth when return is low, the following IC condition is necessary:1       p wl þ ð0  0Þk þ ð1  pÞ wl þ ð0  0Þk  p wh þ ð0  RÞk   þ ð1  pÞ wh þ ð0  RÞk ; where the left-hand side is the expected payoff when she tells the truth and the righthand side is the expected payoff when she lies. Similarly, for the manager to report the truth when return is high, the following IC condition is necessary:     p wh þ ðR  RÞk þ ð1  pÞ wh þ ðR  RÞk      p wl þ ðR  0Þk  Fk þ ð1  pÞ wl þ ðR  0Þk ; where the left-hand side is the expected payoff when she tells the truth and the righthand side is the expected payoff when she lies. These two IC conditions can be summarized as follows: ðR  pFÞk  wh  wl  Rk:

ð4:10Þ

Since the investor only needs to ensure minimum conditions for truthful reporting, he tries to pay as little as possible under the conditions in (4.10). Hence the first inequality in (4.10) must be binding. If so, the second inequality is automatically satisfied and it must be strict. We thus have the following proposition. Proposition 4.2. For a manager with pi ; the investor will offer the following contract: wl ∗ ¼ 0;

wh ∗ ¼ ðR − pi F Þk:

which induces truthful reporting. Investment decision We now consider the investor’s decision in his investment portfolio. His problem is    max E c1 1 c2 c1 ; fki g; c2

s:t:

c1 þ

n X

ki  K;

i¼1

c2 ¼ Rki  wh i if manager i succeeds; i ¼ 1; . . . ; n:

ð4:11Þ

Corporate governance

161

where wh∗ i is defined in Proposition 4.2. Since only one of the managers can succeed, each of the managers has the probability 1=n of successfully producing the high output. Proposition 4.3. The solution of (4.11) is 

ki

ðpi = pi Þ1

¼



n  1 þ ðpi = pi Þ1

K;

ð4:12Þ

where pi is the average governance transparency of other firms: P 1 pi  n1 j6¼i pj : Optimal transparency The following problem gives the manager’s choice of transparency:

pi

1 ðR  pi FÞki n

s:t:

ki

max



¼

ðpi = pi Þ1



n  1 þ ðpi = pi Þ1

K:

This problem can be easily solved to yield the following result. Proposition 4.4. The quality of governance transparency is pi ¼

R ðn  1Þ ; F ðn  1Þ þ ð1  Þn

ð4:13Þ

where we need to restrict the parameters so that p∗i ∈½0:5; 1 for a sensible p∗i . By (4.13), we find that p∗ is strictly increasing in n and R, but strictly decreasing in F: The optimal transparency’s increasingness in n indicates that market competition induces better governance transparency. As investment becomes more risky, the investor demands higher transparency. Transparency reduces information risk (risk resulting from imperfect information). For the investor, there is a trade-off between investment risk 1−1=n and information risk 1− p: In the work of Hidalgo (2009), the investment risk is exogenous and the information risk is endogenous. The investor demands lower information risk (larger pÞ to compensate for high investment risk (larger n Þ: The optimal transparency’s increasingness in R indicates that more profitable firms are more transparent. A more profitable firm is eager to attract investment and is hence willing to offer more transparency. The optimal transparency’s decreasingness in F indicates that a higher penalty reduces transparency. It means that a stronger legal institution is a substitute for governance transparency.

162 Corporate governance Discussion Hidalgo’s (2009) paper is based on an interesting idea and a nice model. The results are interesting and empirically testable.

3. Endogenous governance structure This section is based on the work of Chen and Wang (2009). Using the franchise industry as a case for study, we investigate how firms compete using contractual terms and organizational structures. Business franchise is popular in many industries, especially in the retail industry. Regarding organizational choices, a franchisor has two decisions to make regarding each outlet. The first decision is whether the outlet should be company-owned or franchised. If it is to be franchised, the second decision is the design of a franchise contract. A franchise contract is a linear contract in practice: payment ¼ α þ β : profits; where the payment is paid by the franchisee to the franchisor, α is the initial lump-sum payment, β is the royalty rate, and the profits are the outlet’s annual profits. Although strategic interactions among firms within an industry and across industries have been viewed as being of major importance in economic theory, empirical analysis has only recently become feasible by the recent development of spatial econometrics. Chen and Wang analyze whether and how franchise chains interact with their neighbors in choosing their corporate governance strategies, such as organizational structures and contractual terms. A number of theoretical models and empirical studies suggest that decisions of self-interested economic agents are interdependent spatially in three different patterns: preference interactions, constraint interactions, and expectation interactions. Indeed, by properly choosing a spatial weights matrix, we can identify the impacts of such strategic interactions. Using cross-sectional data of US franchise chains, Chen and Wang find that organizational structures among franchise chains at different distances are positively correlated, implying that organizational structures are a corporate strategy for dealing with head-on competition. Similarly, contractual terms among franchise chains at different distances are also positively correlated, implying that contractual terms are also a corporate strategy for dealing with head-on competition. This is economic Darwinism. However, the two corporate strategies, organizational structures and contractual terms, adopted by franchise chains at different distances are negatively correlated. In other words, franchise chains at different distances choose to diversify between their organizational structures and contractual terms. Hence, those surrounded by chains with relatively high proportions of franchised outlets (PFOs) tend to choose high royalty rates (RRs), and those surrounded by chains with relatively low RRs tend to choose low PFOs. This indicates that competition drives diversification. One implication of this is that the popular oligopoly structure with a few big

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163

franchise chains and a large number of medium-sized franchise chains may be a result of this diversification strategy. Besides the above spatial factors, Chen and Wang have also examined many traditional non-spatial factors. They find some substantially different results from the literature after taking into account spatial factors. First, Lafontaine (1992) argues that a high RR leads to less franchising (fewer PFOs). In contrast, Chen and Wang find that most variables have same-direction impacts on both RR and PFOs. Second, as pointed out by Rubin (1978) and Lal (1990), the more important a franchisor’s inputs are, the larger the RR is. However, Chen and Wang show that such a relationship does not always hold. A franchisor’s inputs can be divided into lump-sum inputs and ongoing inputs. A lump-sum input, such as publicly shared business experience and initial training, is not closely related to the franchisor’s incentives for providing such a service; hence, in this case, a low royalty rate is sufficient. An ongoing input, such as national advertising and ongoing support, is very much dependent on the franchisor’s incentives for providing such a service; in this case, a high royalty rate is necessary. Third, as in the work of Caves and Murphy (1976), the capital-market-imperfection argument states that a high PFO is associated with increasing needs for financial capital by the franchisor. However, Chen and Wang find that the initial investment α is negatively correlated with PFO. In other words, the data indicates that franchising is not a way of obtaining capital for a franchisor.

4. Staged financing in venture capital This section is based on the work of Wang and Zhou (2004). Wang and Zhou investigate staged financing in an environment where an investor faces moral hazard and uncertainty. Staged financing plays two roles in this model: to control risk and to mitigate moral hazard. New startups in high-tech industries tend to have high expected returns and high risk. Instead of providing all the necessary capital upfront, venture capitalists invest in stages to keep the project under control. Staged investment allows venture capitalists to monitor the progress before making refinancing decisions. It reduces losses from inefficient continuation and creates an exit option for venture capitalists. By threatening termination, venture capitalists also have some control over potential moral hazards. With the flexibility of staged financing, many projects which might otherwise have been abandoned under upfront financing become profitable. Wang and Zhou show that, when used together with a fixed-share contract (fixed income shares for contracting parties), staged financing acts as an effective complementary mechanism to contracting in controlling agency problems. The efficiency of staged financing approaches the first best for highly promising firms. 4.1. The model The project Consider an entrepreneur (EN) who relies on a venture capitalist (VC) for investment. The project lasts two periods. The EN’s effort x is applied throughout the two

164 Corporate governance periods, and the VC provides a total investment k in the two periods. The cost of effort for the EN is cðx Þ: The expected output is fðx; k Þ: The output faces a random shock μ such that the realized output is y ¼ μf ðx; k Þ: The VC knows the distribution function of μ at the beginning of the first period (ex ante), and the shock is realized and publicly revealed at the beginning of the second period (ex post). The VC offers to provide a total of k1 in funds at the beginning of the first period. After the uncertainty is realized at the beginning of the second period, the VC considers providing a total of k2 in funds. To realize the output by the end of the second period, the necessary amount, k; of investment must be made, i.e. k1 þ k2 ≥ k: If the project is abandoned in the middle by either the EN or the VC, the project fails without any output and the initial investment k1 is lost. Contracting As in a typical agency model, assume that only the output is contractible ex ante. In particular, the EN’s effort x is unverifiable and the VC’s investment strategy is noncontractible. Assume that the VC considers equity financing only. In other words, the contract proposed by the VC is a fixed-share contract ðs; 1− sÞ of output, with output share s for the EN and 1−s for the VC. As some information becomes available when μ is realized ex post, the two parties may want to renegotiate the contract. Renegotiation ensures ex-post efficiency, but it causes ex-ante opportunistic behaviors, which may result in ex-ante inefficiency. In other words, there is a trade-off: disallowing renegotiation may result in ex-post inefficiency, but allowing renegotiation may result in ex-ante inefficiency. Consequently, economic agents in reality sometimes find ways to commit themselves to “no-renegotiation.” Timing The timing of the events is illustrated in Figure 4.3. Contract: s Effort: x Investment: k1

0 Ex ante

Realization: µ Investment: k2

Output: μ f (x,k)

1 Ex post

2 End

Figure 4.3 The timeline of events.

 

At t = 0, the VC offers a fixed-share contract ðs; 1  sÞ to the EN and decides on an investment plan ðk1 ; k2 Þ; if the contract is accepted, the VC invests k1 and the EN applies effort x and incurs cost cðx Þ: At t = 1, the uncertainty is resolved. The VC considers the options to quit or to renegotiate. If the project is bad, she abandons the project without investing k2 ;

Corporate governance



165

if the project is mediocre, she demands to negotiate a new contract; and if the project is good, she continues to invest in the project and provides the necessary investment k2 : The EN, on the other hand, would never want to renegotiate, since, once the project starts, his effort is sunk. At t=2, the project is finished, and the VC and EN divide the output based on the existing contract.

Assumptions A few assumptions are needed for a tractable model. There is a linear production function fðx Þ ¼ αx; and the output is y ¼ μf ðx Þ with μ being random ex ante and observable ex post. In order to have a closed-form solution, we will also use the following specific parametric functions: μ ∼ U ½0; 2;

cðx Þ ¼

1 β x ; with β

β ≥ 1;

where U½0; 2 denotes the uniform distribution on interval ½0; 2. These two functional forms lead to a unique closed-form solution, which allows us to arrive at a few clear-cut conclusions quickly. Here a value of μ < 1 (or μ > 1Þ means a contraction (or expansion) of output, and the costs in our equilibrium solutions converge to zero quickly as β ! ∞: Further, assume that both the EN and the VC are risk neutral in income. For simplicity, also assume no discount of time preferences and no interest rate across periods. 4.2. Decision problems We examine several variations of the model in this subsection. In the first case, we do not allow ex-post renegotiation; in the second case, we do not allow staged financing; and in the last case, we allow both. Staged financing without renegotiation We examine staged financing without renegotiation in this case. At the beginning of the second period, given the investment k1 ; the EN’s effort level x and the resolved uncertainty μ; the VC decides whether to continue investing in the project or to k2 , the VC will provide the refinancing if   ; abandon ship. Letting   xð1sÞ otherwise the VC will stop investing in the project. Expecting the possibility of termination ex post, the EN’s expected profit from the project is Z 1 sxgðÞd  cðxÞ: ΠEN ¼ 

Let X be the effort space. Assuming that the EN’s reservation profit is zero, the VC’s ex-ante second-best problem is

166 Corporate governance ΠVC

Z ¼

max

s2½0; 1; k1 ;k2 0; x2X

1 k2 xð1sÞ

½ð1  sÞx  k2 gðÞd  k1

Z

s:t:

IR : IC :

1 k2 xð1sÞ

@ @x

Z

sxgðÞd  cðxÞ; 1

k2 xð1sÞ

ð4:14Þ

sxgðÞd ¼ c0 ðxÞ;

k1 þ k2  k: That is, the VC maximizes her ex-ante profit subject to her ex-post IR condition    the EN’s ex-ante IR condition, the EN’s IC condition, and the capital requirement condition. Note that, without renegotiation, the VC’s ex-post profit may not be maximized. This restriction eliminates ex-post opportunistic behavior from the VC and avoids an ex-ante incentive reaction from the EN. This restriction will be dropped later. If the EN’s effort is verifiable, the VC can demand an effort level in the contract without providing sufficient inducement, and staged financing is used only to control risk. The VC’s problem in this case is (4.14) without the IC condition. We call the solution in this case the first best. Upfront financing without renegotiation Now, consider the case in which the VC is required to invest all the required capital k upfront. In this case, the VC loses the option to quit. The effort is unverifiable and the fixed-share contract is now the only instrument that can be used to control moral hazard. By dropping the capital requirement condition and the VC’s ex-post IR condition    in (4.14), the VC’s problem becomes  VC ¼ Π

Z max

s2½0; 1; x2X

s:t:

1

0

IR:

ð1  sÞxgðÞd  k Z

1

0

@ IC : @x

sxgðÞd  cðxÞ;

Z

1 0

ð4:15Þ

sxgðÞd ¼ c0 ðxÞ:

Unlike in staged financing, the EN’s IR condition in upfront financing is not binding. The reason why the VC cannot expropriate the entire surplus is that she only has one instrument, in the case of upfront financing, with which to control moral hazard and risks. Staged financing gives the VC additional power, which allows her to expropriate the entire ex-ante surplus.

Corporate governance

167

Staged financing with renegotiation We now allow the possibility of renegotiation ex post. When the uncertainty is resolved ex post, the project may turn out to be bad or mediocre, so that the VC is unwilling to continue investing given the existing contract. However, the project may still be socially viable. In this case, the EN has no alternative but to renegotiate with the VC for a new contract. We assume that the VC has the control rights so that she gets all the efficiency gains from renegotiation. There are three possible cases at t ¼ 1:  



When    the project continues without renegotiation. Renegotiation cannot increase ex-post efficiency in this case and thus one of the parties will refuse to renegotiate. k2  the project is still socially viable and the VC needs a new  5; When x contract to continue financing. The VC must be provided with extra compensation and the EN has no alternative but to yield to the VC’s demands. Having the control rights, the VC takes over the firm and captures all the efficiency gain αxμ − k2 : k2 , the project is no longer socially viable and it will be terminated. When μ < αx

With renegotiation, the VC’s second-best problem under staged financing becomes Z 1  ^ max ½ð1  sÞx  k2 gðÞd ΠVC  k s2½0; 1; k2 0; x2X

2 xð1sÞ

Z þ

k2 xð1sÞ

Z

s:t:

IR:

ðx  k2 ÞgðÞd  k1

k2 x

1

k2 xð1sÞ

@ IC : @x

Z

sxgðÞd  cðxÞ; 1

k2 xð1sÞ

sxgðÞd ¼ c0 ðxÞ:

If we drop the IC condition, we have the VC’s first-best problem. 4.3. Analysis To understand the role of staged financing in alleviating moral hazard and in sharing risks, we listed three cases above plus two first-best cases. Their solutions are summarized in Table 4.3, where social welfare is defined by SW ¼ ΠVC þ ΠEN and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi







2 3 2 3 1 q q p q q p þ 3   ; þ þ s^ ¼ þ 3  þ 3 2 2 3 2 2 3 4β þ 1 14β  52 ; q : p 3ð2β  1Þ 27ð2β  1Þ

qffiffiffiffiffiffiffiffiffiffi 1  β2





ðβ  1Þxβ  k

ΠVC

ΠVC

SW

pffiffiffiffiffiffiffiffiffiffiffiffiffi β2  1

β1 β β x

ΠVC

ðβ1Þ2 β β x

k

ðβ  1Þxβ

1 2 β1 β

1 β

1 β

1 2 β1 1þβ

Second best

First best

Second best ðβ1Þð2β1Þ 12 β β3

1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Þβ1 1þβ ðβ1Þð2β1Þ 1þ 12 β3

ΠVC

2βðβ 1Þ ð1þβÞð2β1Þ x ð2

 sÞ  k

qffiffiffiffiffiffi 2xð1  sÞ β1 βþ1

@ 4

0

1þ

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

With renegotiation

Without renegotiation

k2

x

s

Staged finance

Staged finance

Table 4.3 The equilibrium solutions

1 2βs2 β1 1þs

2

Þð1sÞ k xð2þs1þs ΠVC

qffiffiffiffiffiffi 2xð1  sÞ 1s 1þs



s^

∗∗

First best

1  β1 β

β2 1 β β x

k

ðβ  1Þxβ  k

0



1 β

Second best

Upfront finance

Corporate governance

169

Proposition 4.5 (Efficiency). With or without renegotiation, the solution under staged financing approaches the first best when β goes to infinity. Proposition 4.5 suggests that staged financing, coupled with a fixed-share contract, can approximately achieve the first best. This may explain why many joint ventures use the simple form of equity sharing. The comparison of the three social welfare curves is shown in Figure 4.4. SF, UF, SB and FB stand for staged financing, upfront financing, the second best and the first best, respectively. For example, the top line is the social welfare for the first best under staged financing; other lines are similarly indicated. SW

1 SWFB,SF (β)

0.8

SWSB,SF (β)

0.6 SWSB,UF (β)

0.4

0.2 β

0 2

2.5

3

3.5

4

Figure 4.4 Efficiency.

The intuition is clear. The β coefficient measures the strength of moral hazard, with higher values of β signifying a weaker entrepreneurial moral hazard. This explains why, as β ! ∞; the difference in efficiency between the second best and the first best diminishes. 

Proposition 4.6 (Social Welfare). Without renegotiation, there exists a β > 1 such  that if and only if β > β will we have SWSB ; SF > SWSB ; UF : With renegotiation, for any β > 1; we have SWSB; SF > SWSB; UF . Figure 4.5 presents a graphic illustration of Proposition 4.6 when renegotiation is not allowed. It suggests that when moral hazard is serious (with a small β Þ; upfront financing is better. The reason is that, while the VC prefers staged financing, the EN prefers upfront financing. Upfront financing may offer better incentives to the EN, especially when moral hazard is serious. On the other hand, when ex-post renegotiation is allowed, it gives the VC an extra mechanism to control moral hazard. With an improvement in ex-post efficiency, the VC can offer a larger share to the EN to induce more effort.

170 Corporate governance SW

0.4 SWSB−SF (β) 0.3

SWSB−UF (β)

0.2

0.1

β

0 1.1

1.4

1.7

2

Figure 4.5 Welfare comparison: SF vs. UF

This mechanism becomes particularly important when moral hazard is serious (with a small β Þ; and, as a consequence, it results in a sufficient improvement of ex-ante efficiency so that staged financing continues to be better than upfront financing. Proposition 4.7 (Incentives). With or without renegotiation, the effort level under staged financing is higher than that under upfront financing. A comparison of the three effort levels is shown in Figure 4.6. A higher effort level in staged financing can be easily understood. Staged financing introduces a threat: the VC may possibly abandon the project. This induces the EN to work extra hard Effort 1

xFB−SF (β) 0.8

xSB−SF (β)

0.6

β

0.4 2

4

Figure 4.6 Effort comparison.

6

8

10

Corporate governance

171

in an attempt to provide the VC with incentives for ex-post refinancing. In other words, moral hazard on the part of the VC under staged financing will have an extra impact on the EN’s incentives, which induces more effort. Proposition 4.8 (Contract). Without renegotiation, the optimal shares in all three cases are the same. With renegotiation, however, the second-best share for the EN is higher than the first-best share. Proposition 4.8 suggests that, without the option of ex-post renegotiation, the contract plays only a marginal role in controlling moral hazard because the contract remains the same in very different situations. Thus, as a complementary mechanism to contracting, staged financing must play a crucial role in controlling moral hazard. This is further verified by the fact that the VC commits less initial investment under the second best than she does under the first best. A larger amount of late investment imposes pressure on the EN to input more effort. On the other hand, with the option of ex-post renegotiation, since the VC is in control, she is willing to offer a higher share to the EN, which induces more effort from the EN. Proposition 4.8 can be understood from a cost-benefit analysis of the two different mechanisms. There are costs to the VC for using staged financing and a sharing contract. The cost of staged financing is the risk associated with the early investment; the cost of a sharing contract is a lower share for the VC. When the marginal cost of using staged financing is always lower than the marginal cost of using a sharing contract, the VC will rely heavily on her financing strategy and leave her contracting option unchanged. Only when the VC has the option to renegotiate will she use the contracting option as well. Staged financing has a few other more intuitive properties. First, with the flexibility of staged financing, a project will generally generate a higher expected value to the VC. Thus, projects that are not profitable under upfront financing may become profitable under staged financing. Second, when β is larger, since the incentive problem becomes less important, the income share for the EN is smaller. Finally, the VC is inclined to provide less early investment under moral hazard. Since late investment imposes pressure on the EN, the VC uses her investment patterns to control moral hazard.

5. Contracting vs. integration This section is based on the work of Wang and Zhu (2004). Wang and Zhu investigate the boundaries of the firm based on Williamson’s transaction cost economics (TCE) and Grossman, Hart and Moore’s property rights theory (GHM). By TCE, while interfirm transactions may be governed by contracts, intrafirm transactions are governed by the implicit law of forbearance. By GHM, some observable but unverifiable information can be useful in the internal governance of the firm, but not in a contractual relationship.

172 Corporate governance 5.1. The model Consider a situation in which a buyer needs a specific intermediate good that can be produced by a seller. The value of the good depends on both the quality of the good and the buyer’s value-enhancing effort. The production cost of the good depends also on quality as well as on the seller’s cost-reducing effort. The buyer can choose either to contract out the production to an independent seller or to integrate with a selling firm. Following GHM, the trade itself and the accounting cost are contractible, so that an explicit procurement contract, including the commonly observed fixed-price and cost-based contract, between two independent firms can be used. Moreover, we assume that the accounting cost can be manipulated and hence the true production cost is not verifiable to a third party; but accounting manipulation can be observed by the buyer at a cost of auditing. The model has two parties, a buyer and a seller, who are risk neutral in income. The buyer needs one unit of an intermediate good, a widget. The widget can be either specific or general. A general widget can be bought from a market. The production cost of a general widget is Cg ðq;δÞ; where q  q  0 is quality and     0 is the seller’s cost-reducing investment. The value of a general widget to the buyer is Vg ðq; βÞ; where β  β  0 is the buyer’s value-enhancing investment. Assume that the buyer is a monopsonist and that the market for a general widget is competitive. Because of competition, the price of a general widget is Pg ðqÞ  min  ½Cg ðq; Þ þ : If the buyer uses a general widget, he will choose qg and β to maximize his profit: π g ðq; βÞ  Vg ðq; βÞ  Pg ðqÞ  β: A specific widget is not readily available on the market. The value of the specific widget to the buyer is Vðq; β Þ. The specific widget can be sold on the competitive market for a value Vm ðq Þ if it is not used by this buyer. The production cost to the e  Cðq; Þ þ ; where θ is a random variable with mean zero. seller is C We make two conventional assumptions about the cost and value functions. Assumption 4.1. Cðq; δ Þ is increasing and convex in q and is decreasing and convex in δ: Assumption 4.2. Vðq; β Þ is increasing and concave in q and β respectively. We also make two assumptions regarding product specificity. Assumption 4.3. Vðq; βÞ  ½Cðq; Þ þ   β  π g ðqg ; βÞ  0 for all q; β and δ: Assumption 4.4. Vm ðqÞ−½Cðq;δÞ þ δ  ≤ 0 for all q and δ. Assumption 4.3 says that it is always more efficient for the buyer to use a specific widget. Assumption 4.4 says that a seller cannot expect to make a profit by selling the specific widget to any other buyer. It follows from Assumptions 4.3 and 4.4 that Vðq; βÞ  Vm ðqÞ for all q and β;

ð4:16Þ

Corporate governance Vðq; βÞ  π g ðqg ; βÞ þ β þ Vm ðqÞ for all q and β;

173 ð4:17Þ

implying that it is always ex post efficient for the buyer to use the specific widget after it is produced and all the investments (β and δ Þ are sunk. To simplify the analysis, assume both Vðq; β Þ and Cðq; δ Þ are separable in quality and investment so that Vq ðq; β Þ ¼ Vq ðq Þ and Cq ðq; β Þ ¼ Cq ðq Þ: Our qualitative results are not affected by this simplification. Assumption 4.5. Vgβ ðqg ; βÞ < Vβ ðβÞ for all β: Assumption 4.6. Vm q ðqÞ < Vq ðq Þ for all q: Assumption 4.5 says that the buyer’s investment is more productive with a specific widget than with a general widget, and Assumption 4.6 says that quality improvement has more value to the buyer than to any outside party. The first-best investments and quality maximize the total expected surplus: Vðq; βÞ  Cðq; Þ  β  :

ð4:18Þ ∗

∗

∗

The FOCs for the first-best (efficient) solution ðq ; β ; δ Þ are Vq ðq Þ ¼ Cq ðq Þ;

Vβ ðβ Þ ¼ 1;

C ð Þ ¼ 1:

ð4:19Þ

At the beginning, both parties have symmetric but imperfect information about θ: Neither β nor δ is observable to the other party. V and q are observable but not verifiable. Moreover, the actual production cost C is not contractible. The two firms can write a contract contingent on the reported accounting cost A ¼ ð1 þ z ÞC; where z 2 ½0; z is the portion of artificially boosted cost. The deadweight loss of accounting manipulation is κðzÞC; where κðzÞ is convex and increasing, zÞ ¼ 1: κz ð0Þ ¼ κð0Þ ¼ 0 and κz ð The buyer has an incentive to secure the specific widget while the seller needs assurance of compensation for it. In the case of contracting, the two firms can sign a procurement contract for the specific widget. The contract can be made conditional on the nature of the widget (specific or general), the trade itself, and the accounting cost. In the case of integration, the buyer acquires the seller and hires the seller as a manager. 5.2. Contracting We consider three types of procurement contracts: fixed-price, accounting costbased, and open-term. Fixed-price contract The simplest contract is a fixed-price contract, under which the seller agrees to produce and deliver a specific widget and the buyer agrees to pay the seller a price P: Since the payment does not depend on the cost, the seller has no incentive to

174 Corporate governance manipulate accounting cost, and the buyer has no need to audit the true cost. Thus, the seller’s ex ante expected payoff is P  Cðq; Þ  :

ð4:20Þ

Clearly, she will choose qf ¼ q and f ¼  ; where superscript ‘f ’ stands for a fixedprice contract. On the other hand, the buyer will choose βf ¼ β to maximize Vðq; βf Þ  P  βf :

ð4:21Þ

The price will be P ¼ Cðq;  Þ þ  so that the seller just breaks even. Proposition 4.9. Under a fixed-price contract, both the buyer and the seller invest efficiently, but the seller has no incentive to provide quality. Moreover, since the seller has no incentive for cost manipulation, there is no need for auditing. Accounting cost-based contract Since quality is not directly contractible, providing incentives for quality requires the seller’s payoff to be at least indirectly contingent on the quality of the widget. Given that the accounting cost partly reflects the quality of the good, the two parties can sign a contract that relates the price P to the accounting cost A: We consider a general form of contract PðA Þ; called a cost-based contract. Given a contract PðA Þ; the seller’s problem is max π S  P½ð1 þ zÞCðq; Þ  ½1 þ κðzÞCðq; Þ  :

ð4:22Þ

q;;z

The FOC for qc implies PA ½ð1 þ zÞCðq; Þ ¼

1 þ κðzÞ : 1þz

ð4:23Þ

Then the buyer’s problem is max π B ¼ Vðq; βÞ  P½ð1 þ zc ÞCðq; Þ  β

q; β; Pð Þ

ð4:24Þ

s:t: IC : PA ½ð1 þ zc ÞCðq; Þð1 þ zc Þ ¼ 1 þ κðzc Þ; IR : P½ð1 þ zc ÞCðq; Þ  ½1 þ κðzc ÞCðq; Þ    0:

The IC condition is from (4.23). The choice of investment β is independent of other choices and hence βc ¼ β∗ : We can solve the buyer’s optimization problem in two steps. First, we choose q to maximize the objective function without the constraints. The FOC with respect to qc is: Vq ðqc Þ ¼ PA ½ð1 þ zc ÞCðq; Þð1 þ zc ÞCq ðqc Þ:

ð4:25Þ

Then, using the IC condition, we have the solution qc ; which satisfies Vq ðqc Þ ¼ ½1 þ κðzc ÞCq ðqc Þ:

ð4:26Þ

By the first expression in (4.19) and Assumptions 4.1 and 4.2, we have q < q∗ : c

Corporate governance

175

The second step is to find a contract that satisfies the IC and IR conditions. Consider a linear contract of the form P ¼ P0 þ αA; where P0 is a lump-sum transfer payment and α is a cost-sharing parameter. The IC and IR conditions are satisfied if α and P0 are chosen as ¼

1 þ κðzc Þ ¼ κz ðzc Þ; 1 þ zc

P0 ¼ :

ð4:27Þ

Thus, a linear cost-sharing contract P ¼ P0 þ αA is optimal. We now show that α^ ≠ 1: By (4.23), π S ¼ 150; implying c ¼ : Further, we have  1 þ κðzÞ  κz ðzÞ Cðq; Þ: ð4:28Þ π Sz ¼ 1þz If πzS < 0 for all z; then zc ¼ 0; which, when plugged back into (4.28), leads to a contradiction, πzS ¼ Cðq; δÞ > 0. If πzS > 0 for all z; then zc ¼ z; which also leads to a contradiction because κz ð zÞ ¼ ∞: Therefore, πzS ¼ 0; and zc satisfies κz ðzc Þ ¼

1 þ κðzc Þ 1 þ zc

and zc > 0. Let ψðzÞ ¼ αz −κðzÞ: Since ψ 0 ðzc Þ ¼ 0 and ψ″ðz Þ ¼ −κzz ðz Þ < 0 for all z; we have ψ 0 ðzÞ > 0 for z ∈½0; zc Þ: Then, since ψð0Þ ¼ 0, we must have ψðzc Þ ¼ αzc −κðzc Þ > 0. From (4.27), the above implies that zc > κðzc Þ: Hence, α < 1. This means that a cost-plus contract PðA Þ ¼ P0 þ A, where P0 is a constant, is not optimal. Proposition 4.10. Under the optimal cost-sharing contract, the seller chooses to manipulate the accounting cost, has no incentive to invest, and provides inefficient quality, whereas the buyer invests efficiently. Moreover, a cost-plus contract is not an optimal cost-based contract. Open-term contract Consider the following open-term contract: the buyer pays the seller a fixed amount of a down payment T and in exchange the seller promises to produce a specific widget; but the trading price is left open for negotiation after the widget is produced. Given the status quo values Vm ðq Þ and Vg ðqg ;β Þ−pg for the seller and the buyer respectively, social welfare is Vðq; βÞ  Vm ðqÞ  Vg ðqg ; βÞ þ pg : The Nash bargaining solution implies the negotiated price: 1 P ¼ Vm ðq Þ þ ½Vðq; β Þ−V m ðqÞ−V g ðqg ; β Þ þ pg : 2 Hence, the ex ante expected payoff for the seller is 1 T þ Vm ðqÞ þ ½Vðq; βÞ  Vm ðqÞ  Vg ðqg ; βÞ þ pg   Cðq; Þ  : 2

ð4:30Þ

ð4:31Þ

ð4:32Þ

176 Corporate governance Thus, the seller will choose qo and δo ; where the superscript o stands for an openterm contract, to satisfy the following conditions: C ðo Þ ¼ 1;

ð4:33Þ

which implies δo ¼ δ∗ ; and 1 o o ½Vq ðqo Þ þ Vm q ðq Þ ¼ Cq ðq Þ: 2

ð4:34Þ

By Assumptions 4.1, 4.2 and 4.6, (4.34) implies qo < q∗ : The ex ante expected payoff for the buyer is 1 Vg ðqg ; βÞ  pg  T þ ½Vðq; βÞ  Vm ðqÞ  Vg ðqg ; βÞ þ pg   β: 2

ð4:35Þ

The buyer’s investment choice, βo ; solves 1 ½Vβ ðβÞ þ Vgβ ðqg ; βÞ ¼ 1; 2

ð4:36Þ

which, by Assumptions 4.2 and 4.5; implies that βo< β∗ : Finally, the fixed transfer payment T will be chosen to allow the seller to break even. Proposition 4.11. Under an open-term contract with a fixed transfer payment, the seller invests efficiently but provides inefficient quality, whereas the buyer’s investment is inefficient. Moreover, the seller has no incentive for accounting manipulation and hence there is no need for auditing. 5.3. Vertical integration One difference between contracting and integration is that, due to the law of forbearance, no contract other than a wage contract can be effectively enforced by a third party under integration, and the employee’s (the seller’s) income is based only on a fixed wage rate and the duration of employment. Another difference is that, under integration, the employer (the buyer) can fire the employee, whereas as separate entities the buyer can only fire the entire firm owned by the seller. This means that it is much easier to replace an employee than to replace a contractor. If the seller is fired, we assume that the only alternative is to buy a general widget. It can be shown that integration without auditing is less efficient than contracting with cost sharing. That is, if integration is desirable, auditing is necessary. This explains the assumed informational difference between markets and hierarchies in TCE. Under integration, the cost of production must be fully paid by the employer. In the production stage, the buyer can simply order the seller to produce the specific widget at a quality level q and, because all the production expenses are paid for, the seller has no incentive not to follow the order. Moreover, being paid a fixed wage w, the seller has no incentive to make any cost-reducing investment, i.e.  ¼ : Thus, the employer’s expected payoff is

Corporate governance Vðq; βÞ  Cðq; Þ    β  D:

177 ð4:37Þ

where D is the cost of internal auditing. We immediately find that the employer will choose β ¼ β∗ and q ¼ q∗ : Proposition 4.12. In the case where the buyer acquires the selling firm and hires the seller as an employee, the buyer invests efficiently and chooses to audit the cost, whereas the seller has no incentive to invest but provides efficient quality. Comparison of the four different modes of governance The Table 4.4 summarizes the results from the four different modes of governance discussed in this section. As shown, each mode of governance has its comparative advantage and none of them has an absolute advantage over the rest. Integration has the advantage when quality is very important but cost-reducing investments are not. If quality is not important but cost-reducing investments are, then a fixed-price contract is the best. When the degree of specificity in value-enhancing investments and quality provision is low, then open-term contracting is a good choice. Costsharing contracting has the advantage over integration when accounting manipulation is not a serious problem. Table 4.4 Summary of results Mode of governance

Incentive for quality

Incentive for cost reduction

Incentive for value Accounting enhancement manipulation

Auditing

Fixed-price contract Open-term contract Costsharing contract Integration

No

Efficient

Efficient

No

No

Some

Efficient

Some

No

No

Some

No

Efficient

Yes

No

Efficient

No

Efficient

No

Yes

5.4. Discussion Based on the idea of TCE, different modes of governance work for different contract law regimes; i.e., whereas interfirm transactions may be governed by court-enforceable contracts, intrafirm transactions are governed by the implicit law of forbearance. The law of forbearance limits the use of incentive contracts within the firm. Wang and Zhu (2004) show advantages and disadvantages of different modes of governance. Based on the idea of GHM that it is easier to replace an employee than an independent contractor, they show that observable but unverifiable cost information may be effectively used in internal governance of the firm but not in a contractual interfirm relation.

5

Public governance

The incomplete contract approach is also popular for dealing with issues of public governance. In this chapter, we show a few such applications.

1. Private–public partnerships via ownership and regulation This section is based on the work of Qiu and Wang (2011). Governments, and especially those in developing countries, are often involved in large infrastructure projects. Due to the requirements of funding, management, and expertise, governments often invite private firms to join them in such projects. These joint projects are often defined by contractual agreements. The most popular form of such a contractual agreement in large infrastructure projects is a Build-Operate-Transfer (BOT) contract. Using an incomplete contract approach, we show that a BOT contract with an ownership mechanism under regulation is capable of achieving full efficiency. 1.1. The model Suppose that a government decides to implement a project, such as building a railway, and invites a private firm to join in. The government will grant a firm to build the project. The project is built at t = 0 (see Figure 5.1). The firm makes investment k(q) for quality q, with k(0)=0 and k′(q)>0. The service life of a project is two periods, with typically 30 years in each period in practice. The firm has the right to operate the project and keep all the profits earned in the first period (firm ownership). After that, the project is to be transferred to the government unconditionally (government ownership). Such a deal is called a BOT contract. However, the government may decide to extend the firm’s ownership ex post. Such an extension is often granted in reality. After the project is built at t = 0, let xi (pi, q) be the demand in period i for the service offered by the project, where pi is the price. At t=0, x1(p1, q) is known, but x~2 ðp2 ; q Þ is random. At t = 1, x2(p2, q) becomes known. Assume xi, q(p, q)>0 and xi, p(p, q) 0 and c00i ðx Þ ≥ 0. Then single-period operating profits are: 1 ðp1 ; qÞ  p1 x1 ðp1 ; qÞ  c1 ½x1 ðp1 ; qÞ; x2 ðp2 ; qÞ: ~2 ðp2 ; qÞ  p2 x~2 ðp2 ; qÞ  c2 ½~

ð5:1Þ

Assume that quality q is unverifiable but observable ex post. The government offers a BOT contract to the firm before the firm makes its investment to build the project. Firm ownership 0 Ex ante contracting

Period 1

1 λ Ex post Ownership extension

Government ownership

Period 2

2 End

Figure 5.1 The timeline of events.

There are two policy instruments: price control and ownership extension. Specifically, the government has the right to decide prices p1 and p2 and to extend the firm’s ownership ex post by length λðq Þ, where p1 is decided at t ¼ 0 and p2 and λ are decided at t ¼ 1. Three problems must be dealt with: quality, risk sharing, and monopoly power. The effects of price control are twofold: to limit monopoly power and to provide the incentive to invest. The effects of ownership extension are also twofold: to provide incentive for investment and to share risks.

1.2. Solutions The first-best solution If the government can finance, build, and operate the project completely on its own as efficiently as private firms can, the government’s problem is to maximize social welfare: W ¼ max s1 ðp1 ; qÞ þ 1 ðp1 ; qÞ þ E½s~2 ðp2 ; qÞ þ ~2 ðp2 ; qÞ  kðqÞ: ð5:2Þ p1 ; p2 ; q

The FOCs are 0   p 1 ¼ c1 ½x1 ðp1 ; q Þ ;

ð5:3Þ

0  x2 ðp p 2 ¼ c2 ½~ 2 ; q Þ ;

ð5:4Þ

   0  ~2;q ðp s1;q ðp 1 ;q Þþ E s 2 ; qÞ ¼ k ðq Þ :

ð5:5Þ

∗∗ ∗∗ They determine the first-best solution C ≡ ðp∗∗ 1 ; p2 ; q Þ.

180 Public governance The second-best solution Now, we return to our BOT model, in which the government has to rely on a private firm to finance, build, and operate the project in the first period. Three issues are involved: risk sharing, incentives, and monopoly power. The government has two instruments at its disposal: price control and ownership mechanism. Given the contract, the firm’s problem is to choose q to maximize its expected profit: 2 ðp2 ; qÞ  kðqÞ: ðp1 ; p2 ; qÞ  1 ðp1 ; qÞ þ E½λðqÞ~

ð5:6Þ

The government’s problem is W ¼

max

p1 ; p2 ; p0 2 ; q; ~λðÞ

s1 ðp1 ; qÞ þ 1 ðp1 ; qÞ þ EfλðqÞ½~s2 ðp2 ; qÞ þ ~2 ðp2 ; qÞg þ Ef½1  λðqÞ½~ s2 ðp0 2 ; qÞ þ ~2 ðp0 2 ; qÞg  kðqÞ

s:t:

IC :

ð5:7Þ

ðp1 ; p2 ; qÞ  ðp1 ; p2 ; qÞ for all q 2 Q:

Note that, in a social welfare maximization problem, IR conditions are unnecessary, since the government will take into account the firm’s interest, and, when necessary, offer a lump-sum upfront subsidy. Proposition 5.1. An optimal solution to problem (5.7) is 8  p1 ¼ p 1 ; > > >  0   > p ¼ p > 2 ¼ p2 ; < 2 q ¼ q ; >  >   > kðqÞ=~ 2 ðp >  2 ; q Þ; for 0  q5q ; > : λ ðqÞ ¼     kðq Þ=~ 2 ðp2 ; q Þ; for q  q :

ð5:8Þ

We have 0 ≤ λ∗ ðqÞ < 1. This BOT contract is efficient, which induces the first-best quality q ∗∗ (see Figure 5.2). The optimal contract completely solves the problems. It induces the firm to invest efficiently in quality, offers a special way of sharing risk, and effectively tackles the monopoly problem under the firm’s ownership.

λ(q)

λ(q∗∗)

0

q∗∗

Figure 5.2 The optimal ownership mechanism.

q

Public governance

181

Notice that λ∗ ≠1. Since monopoly in the second period is a source of inefficiency, the government optimally chooses an appropriate length of period as a reward for high quality. ~  at all, ~  ¼ kðq Þ: There is no randomness in  The firm’s ex-post profit is  2 2 implying that the government takes all the risks. This makes sense, since the government cares about social welfare.

1.3. Comparison of alternative solutions Besides the efficient solution, there are a few alternative solutions that are also seen in reality under some circumstances. To conveniently discuss and compare these cases, we choose a set of parametric functions. Let 1 1 x1 ðp1 ; qÞ ¼ qαp1 ; x2 ðp2 ; qÞ ¼ A~ðqαp2 Þ; c1 ðxÞ ¼ c2 ðxÞ ¼ x2 ; kðqÞ ¼ q ; 2  where α > 0 and θ > 1 : The distribution function for the random constant A~ is arbitrary. Extendable BOT: ownership mechanism + price control The efficient BOT solution described in Proposition 5.1 under the ownership mechanism and price control is: " !# 1 2 ~ 1 1 A A~ q ; ; p1 ¼ þE q ; p2 ¼ q ¼ ~ αð1 þ αÞ 1þα αð1 þ αAÞ 1 þ αA~ 8  2 > 2 >   < 2 α þ A1~ ðq Þ q ; for 0  q5q ; 1 1   W  ðq Þ : ¼ λ ðqÞ ¼  2 > 2  > : 2 α þ 1 ðq Þ2 ; for q  q ;  A~ We call this an extendable BOT since the firm’s ownership is extendable. Regulated BOT: price control only Suppose now that the government controls prices but does not apply the ownership mechanism (i.e. no possibility of an ownership extension). In this case, given a contract, the firm’s problem is to choose quality to maximize its expected profit: ðp1 ; qÞ  1 ðp1 ; qÞ  kðqÞ: The government’s problem is  b ¼ max s1 ðp1 ; qÞ þ 1 ðp1 ; qÞ þ E½s2 ðp02 ; qÞ þ 2 ðp02 ; qÞ  kðqÞ W p1 ; p02 ; q

s:t: IC : ðp1 ; qÞ  ðp1 ; qÞ for all q 2 Q:

ð5:9Þ

ð5:10Þ

182 Public governance The solution is 2

1 32   A~ þ E 1þαA~ 62 7 7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qb ¼6 h  i5 ; 4α 4ð1Þ A~ 1 1 þ 1 þ 1þα 1þα þ E 1þαA~

1 1þα

qb þ qb1 A~ ; p2b ¼ qb ; 1þα 1 þ αA~

 ~  α qb22 q2b 1 1 A    qb : þE Wb ¼ ~ 2α 1 þ α 2ð1 þ αÞ  1 þ αA p1b ¼

Unregulated BOT: Ownership Mechanism only Suppose now that the government applies the ownership mechanism but leaves prices to be freely decided by the firm. The firm’s problem is to choose fp1 ; p2 ; qg to maximize its expected profit: 2 ðp2 ; qÞ  kðqÞ: ðp1 ; p2 ; qÞ ¼ 1 ðp1 ; qÞ þ E½λðqÞ~

ð5:11Þ

The government’s problem is ^b ¼ W

max

p1 ; p2 ; p02 ; q; λðÞ

s1 ðp1 ; qÞ þ 1 ðp1 ; qÞ þ EfλðqÞ½~ s2 ðp2 ; qÞ þ ~2 ðp2 ; qÞg þ Ef½1  λðqÞ½~ s2 ðp02 ; qÞ þ ~2 ðp02 ; qÞg  kðqÞ

s:t: IC : ðp1 ; p2 ; qÞ  ð p1 ; p2 ; qÞ; for all q 2 Q; p1 ; p2  0: The solution is 1

2 1 ; q^b ¼ αð2 þ αÞ ^λb ðqÞ ¼ 0;

1þα A~ p^02b ¼ q^b ; q^b ; αð2 þ αÞ 1 þ αA~ "  ~ # 2   1 1 1 1  A ^b ¼ q^2b þ W þE  q^b : 2α 2 þ α 2  1 þ αA~ p^1b ¼

Surprisingly, ownership has no role to play in this case. Ownership and monopoly power are both incentive instruments. However, without price regulation, granting monopoly power is socially very costly, since the monopoly price would be prohibitively high. Granting monopoly power in the first period to the firm provides enough incentives as it is; granting monopoly power in the second period as well would be over the top, implying no ownership extension in the optimal solution. Regulated privatization Instead of BOT, privatization is also a popular solution in some sectors of an economy, although rare in large infrastructure projects. Consider a situation in

Public governance

183

which the project is completely run by a private firm. We call this privatization. In this case a private firm builds the project and then operates it in both periods. Suppose that the government controls prices. Then, given prices ðp1 ; p2 Þ, the firm’s problem is to choose quality to maximize its profit: r ðp1 ; p2 ; qÞ  1 ðp1 ; qÞ þ E½~2 ðp2 ; qÞ  kðqÞ: Then the government’s problem is  p ¼ max s1 ðp1 ; qÞ þ 1 ðp1 ; qÞ þ E½~ s2 ðp2 ; qÞ þ ~2 ðp2 ; qÞ  kðqÞ W p1 ; p2 ; q

s:t: IC : ðp1 ; p2 ; qÞ  ðp1 ; p2 ; qÞ; for all q 2 Q: The optimal solution is denoted as ð p1p ; p2p ; qp Þ: Unregulated privatization Suppose now that the government does not even control prices in the case of privatization. Then the firm’s problem is to choose prices and quality to maximize its expected profit: ^ p ¼ max 1 ðp1 ; qÞ þ E½~2 ðp2 ; qÞ  kðqÞ:  p1 ; p2 ; q

The optimal solution ð^ p1p ; p^2p ; q^p Þ is: 1

 ~ 2 1 1 A þ E ; q^p ¼ αð2 þ αÞ α 2 þ αA~ 1þα 1 þ αA~ q^ ; q^p ; p^2p ¼ ~ p αð2 þ αÞ αð2 þ αAÞ ( "   2 #) 2   1 1 1 1 1  2 ~ ^  q^ : Wp ¼ þE A q^p þ 2α 2þα 2  p 2 þ αA~ p^1p ¼

Comparison In the above, we find three BOT contracts: an extendable BOT, a regulated BOT, and an unregulated BOT, plus two forms of privatization. We compare these five solutions in terms of quality, price, and welfare. QUALITY

The choice of quality is driven by the cost of investment and the effect of quality on demand. We find that under price control, to induce a greater demand, the firm tends to opt for a higher quality. Our first observation from Figure 5.3 is that the extendable BOT has the highest quality. That is, the highest quality is ensured only when both the ownership mechanism and price control are employed.

184 Public governance 5

q∗

4

qp

3

2

1 qˆb qb

qˆp

α

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Figure 5.3 Quality curves.

Our second observation is that the worst two cases are the nonextendable BOTs. In particular, a nonextendable BOT is always worse than privatization with or without regulation. Privatization implies better quality, and this ownership arrangement turns out to be more effective than price control or price incentive (without price control). The third observation is that, although privatization provides the firm with better incentive for quality, it results in substantially lower quality than the extendable BOT. The explanation is that, when the length of ownership is not dependent on the firm’s investment in quality, due to the increasing marginal cost of quality, the firm does not invest in high quality. The fourth observation is that regulated privatization (or BOT) is better than unregulated privatization (or BOT) when it comes to quality. The reason is that the government can provide some incentive and bear some of the risk involved through price control. The reasoning is as follows. Without the ownership mechanism, price control is the only way for the government to provide incentive and bear risk. With the firm’s understanding that quality has a positive effect on demand, in an unregulated environment, if the quality is high, the demand is strong and the firm can charge high prices; in a regulated environment, however, in order to justify a higher price, the firm needs to provide higher quality. A social welfare consideration tends to mean higher quality, while a profit consideration takes into account both the cost of quality and the effect of quality on demand. Through price control, the government can force the firm to take into account social welfare rather than profits alone. It is this forced commitment to quality that results in higher quality.

Public governance

185

5

p1b

4

p1p pˆ 1p

3

2

p1*

pˆ 1b

1

α

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.4 Price curves in the first period. PRICE

First, one most noticeable phenomenon in Figure 5.4 is that the first-period price under the regulated BOT is quite high; in fact, higher than all others. As we know, the regulated price p1b is committed at t ¼ 0 when quality is chosen. Hence, to encourage quality, the government sets a high p1b With such a committed price, the firm is willing to provide high quality. Since the first period is the only time period in which the firm has ownership, price control is exercised fully to encourage investment in quality. On the other hand, the second-period price p2b is committed at t ¼ 1 after quality is chosen. Hence, p2b cannot be used as an incentive tool; instead, it is determined by the chosen quality. Second, as shown in Figure 5.5, except for the extendable BOT, each secondperiod price is a reflection of the chosen quality. On the one hand, in an unregulated environment, an unregulated/monopoly price is associated with demand, and demand is in turn associated with quality, implying a correlation between the price and quality. On the other hand, while the firm sets a monopoly price, the government sets a price based on the P=MC rule (the price equals the marginal cost). Hence, given the same quality, a regulated price will be lower than an unregulated price. However, since regulation can help induce incentive for quality and higher quality implies higher demand, a regulated price tends to be closely correlated with quality. Third, two forces determine the prices: incentive for quality and monopoly power. It turns out that, in both BOT and privatization cases, a regulated price is always higher than the corresponding unregulated price. This indicates that, when the government exercises price control, it focuses more on providing incentives

186 Public governance 5

p2p pˆ 2p

4

p∗ 2

3

p2b 2

1

pˆ 2b

α

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5.5 Price curves in the second period.

for the firm to achieve quality than on keeping the firm’s monopoly power in check. Finally, only the prices for the regulated BOT differ substantially between the two periods. This is consistent with our first observation that the first-period price is the key incentive tool under BOT. WELFARE

First, from Figure 5.6, we can see that only the extendable BOT is efficient. In an unregulated environment, one-period ownership does not provide enough incentive under a nonextendable BOT, on the one hand, and monopoly power is unchecked under privatization, on the other hand. In a regulated environment, price control does not provide enough incentive under a nonextendable BOT, on the one hand, and price control is not enough to keep monopoly power in check under privatization, on the other hand. Second, social welfare is closely associated with quality. By comparing Figure 5.6 with Figure 5.3, we can see that the ranking order of the welfare curves is the same as that of the quality curves. This means that, although we have other issues, such as risk sharing and monopoly power, the issue of quality is the key. Third, in terms of social welfare, the extendable BOT is better than privatization, which is in turn better than the nonextendable BOT under either environment. With the ownership mechanism, BOT is the best solution in a regulated environment; without the ownership mechanism, BOT is the

Public governance

187

10 9 8 W∗

7 6

Wp

5 4 3 2 ˆ W b

1

Wb

ˆ W p

α

0 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 5.6 Welfare curves.

worst solution in either environment. In fact, privatization is substantially better than BOT in either environment, indicating the importance of the ownership mechanism. Finally, under privatization, regulation is not important at all, and it can only slightly improve social welfare, although it can noticeably improve quality. In contrast, under BOT, regulation is crucial and it can substantially improve social welfare. Due to a more severe incentive problem under a nonextendable BOT, price control as an incentive tool is more important under BOT than under privatization. 1.4. Discussion Qiu and Wang (2011) built a model that aimed at using a positive agency theory to explain a unique approach to BOT projects. They allowed the government the option of extending a firm’s ownership ex post. This ownership mechanism had never before been discussed in the literature. Since some information is unknown ex ante but becomes known ex post, a decision on ownership partition is made ex post only after the information becomes known. The ownership mechanism and price regulation are known to exist in almost every real-world BOT project. We investigate how these mechanisms deal with the issues of risk sharing, incentives, and monopoly power. Our conclusion is consistent with the real-world popularity of the two mechanisms. We also show the necessity of coupling the two instruments. That is, the ownership mechanism can work effectively, but it works well only in a regulated environment. The monopoly

188 Public governance wants to reduce quantity to push up the price, and the best way to do this is to reduce quality. Under regulation, the price should not be high; otherwise, the monopoly would have the incentive to increase quantity (by increasing quality). Empirical studies on this mechanism are almost nonexistent. One rare example is a recent paper by Brickley et al. (2006) studying the impact of contract extension on the incentives of franchisees in business franchising. They find that “contract duration is positively and significantly related to the franchisee’s physical and human capital investments.” This empirical finding is consistent with our theoretical finding.

2. On the hierarchy of government This section is based on the work of Wang and Xiao (2007). Recently, there has been a growing interest in public decision making. Some policy decisions, such as monetary and trade policies, tend to move to a higher level of government, while others, such as health care and education policies, tend to move to a lower level. Should a social program be run at a higher or lower level of government? Wang and Xiao consider a government-run program supplying a private good to consumers. We focus on a trade-off between the advantage of having a high level of government in resource allocation and its disadvantage in consumer information. We look into the effect of differences in income levels, marginal costs, and preferences on the structure of public governance. One general conclusion is that, when regional differences are large, the central government should be in charge; otherwise, local governments should be in charge. Wang and Xiao define a notion of bureaucratic inefficiency as lacking detailed information on the people the government serves. A lower office tends to have better information on the local people it serves, while a higher office typically has a wider range of resources at its disposal and is hence more efficient in resource allocation. 2.1. The model Consider a country with two provinces i ¼ 1; 2 and a central government. Each province has a local government. Assume that a local government knows each consumer’s type θ in its own province, but the central government knows only the distribution of consumer type. The density function of consumer type in province i is f i ðθ Þ, i.e. the population size of type θ is f i ðθ Þ: These density functions are the true density functions and are public knowledge. Assume that the support of each density function is ½0; 1: A government is to provide an amount xi ðθ Þ of a service from a certain program to the citizens of type θ in province i and charges a user fee ti ðθ Þ. A typical consumer’s utility function is Ui ¼ θvi ðxi Þ−ti ; where i is the province, θ is the consumer type, xi is the amount of the program that the consumer consumes, and ti is the consumer’s payment for the program. The

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189

value function vi ðx Þ is increasing and concave. The cost to the program for consumption xi is ci ðxi Þ: The cost function ci ðxi Þ is increasing and convex. With xi ðθ Þ amount being consumed by consumer θ in province i for θ ∈½0; 1, the cost of the program for province i is Z 1 ci ½xi ðÞ fi ðÞd: 0

Since an individual typically spends a small portion of his/her total budget on a government program, there should not be a hard budget constraint for each individual. Hence, we introduce a variable Ii , called the income level or resource level for province i, to represent a resource constraint of the provincial government. With this, we impose a cost restriction on each province: Z 1 ti ðÞ fi ðÞd  Ii : 0

2.2. Local governments in charge If a local government is in charge of the program, the local government’s problem is Z 1 Vi ¼ max fvi ½xi ðÞ  ci ½xi ðÞgfi ðÞd xi ðÞ; ti ðÞ

0

s:t: IRi : vi ½xi ðÞ  ti ðÞ  0; Z 1 ti ðÞfi ðÞd  Ii ; RCi : Z BCi :

ð5:12Þ

0

1

0

fti ðÞ  ci ½xi ðÞgfi ðÞd ¼ 0;

where RCi is the resource constraint in province i, and BCi is the local government’s budget constraint. The solution is determined by the following proposition. Proposition 5.2 (The LG Problem). The local government’s problem is to solve for x∗i ðθ Þ from: Z 1 Vi  max fvi ½xi ðÞ  ci ½xi ðÞgfi ðÞd xi ðÞ

0

s:t: RCi :

Z 0

1

ð5:13Þ

ci ½xi ðÞ fi ðÞd  Ii ;

and to find a payment scheme t∗i ðθ Þ satisfying IRi and BCi . 2.3. The central government in charge Since the central government does not know each consumer’s type θ, an IC condition is needed. Given a package f½xi ðθ Þ; ti ðθ Þjθ ∈½0; 1g designated for

190 Public governance

h i province i, a consumer of type θ in province i chooses xi ðθ^ Þ; ti ðθ^ Þ if θ^ solves the following problem: ^  ti ðÞ: ^ max vi ½xi ðÞ ^

The IC condition ensures that the consumer chooses θ^ ¼ θ after the package is announced. Hence, given the total income I of the whole country, the central government’s problem is 2 Z 1 X V¼ max fvi ½xi ðÞ  ci ½xi ðÞgfi ðÞd xi ðÞ; ti ðÞ; i¼1;2

i¼1

0

^  ti ðÞ; ^ i ¼ 1; 2; IC :  2 arg max vi ½xi ðÞ

s:t:

^

IR : vi ½xi ðÞ  ti ðÞ  0; i ¼ 1; 2; 2 Z 1 X ti ðÞfi ðÞd  I; RC : i¼1

BC :

0

2 Z X i¼1

ð5:14Þ

0

1

fti ðÞ  ci ½xi ðÞgfi ðÞd ¼ 0:

i ðÞ 0 Let Ji ðÞ    1F fi ðÞ : As in standard auction theories, assume J i ðθ Þ≥0.

Proposition 5.3 (The CG Problem). The central government’s problem is to solve for x^ i ðθ Þ, i ¼ 1; 2, from: 2 Z 1 X V  max fvi ½xi ðÞ  ci ½xi ðÞgfi ðÞd: xi ðÞ; i¼1;2

s:t:

i¼1

0

SOC : x0i ðÞ  0; i ¼ 1; 2; 2 Z 1 X RC : ci ½xi ðÞ fi ðÞd  I; i¼1

BC :

0

2 Z X i¼1

0

1

fvi ½xi ðÞJi ðÞ  ci ½xi ðÞgfi ðÞd ¼ 0;

and set the user fee t^i ðÞ; i ¼ 1; 2, as Z  xi ðÞ  vi ½^ xi ðτÞ dτ: t^i ðÞ ¼ vi ½^ 0

ð5:15Þ

ð5:16Þ

xi ðÞ is how much the consumer θ benefits from the program In (5.16), the term vi ½^ R and the discount term 0 vi ½^ xi ðτÞdτ eliminates the consumer’s incentive to choose a policy designed for a lower type (with a smaller θ Þ, since a consumer of lower type receives less discount.

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191

The BC condition in problem (5.15) is interesting. The budget needs to be balanced. xi ðÞ, but he However, for an individual θ in province i, his cost of consumption is ci ½^ xi ðÞ and t^i ðÞ means that the central governpays t^i ðÞ The difference between ci ½^ ment overcharges those consumers who benefit more from the program and subsidizes those who benefit less from the program. The BC condition causes a reallocation of income, which reduces income inequality to some extent. Hence, a reduction of income inequality plays a role in the central government’s problem.

2.4. Theoretical analysis Differences in income levels Income is typically geographically concentrated in any country, due to economies of scale, endowments of natural resources, educational traditions, and infrastructural backgrounds. Hence, the role of differences in income levels is important to the issue of which level of government should be in charge. Denote the value of the local governments’ problem as Vi ðIi Þ and the value of the central government’s problem as VðI Þ: If the local governments are in charge of the program, social welfare is V1 ðI1 Þ þ V2 ðI2 Þ; and if the central government is in charge, social welfare is VðI1 þ I2 Þ: Assumption 5.1. For any I > 0, we have Vi ðIÞ < VðIÞ for both i: That is, it is always worse for the central government to spend all the resources in one province only. Proposition 5.4. Under Assumption 5.1, the space ℝ2þ of income levels can be divided into four regions, which suggests that, when the two provinces have sufficiently unbalanced income levels, the central government should be in charge; otherwise the local governments should be in charge (see Figure 5.7). I2

Indifference region

CG region

I2 CG region LG region

I1

Figure 5.7 The control regions depending on income levels.

I1

192 Public governance For example, on the issue of fiscal federalism, a general taxation program is mainly run by the central government in a country. One explanation is that income is typically geographically unbalanced.

Differences in costs In this case, the costs are assumed to have fixed marginal costs: c1 ðxÞ ¼ c1 x;

c2 ðx Þ ¼ c2 x;

ð5:17Þ

where c1 ≥ 0 and c2 ≥ 0 are fixed constants. Also, assume vi ðxi Þ ¼

1 x1α ; 1α i

where α 2 ð0; 1Þ:

ð5:18Þ

Here, α is the relative risk aversion. Further, to focus on the differences in costs in this case, assume the same density function in the two provinces: f 1 ðθ Þ ¼ f 2 ðθ Þ ≡ f ðθ Þ:

ð5:19Þ

Given income levels I1 and I2 , with I ¼ I1 þ I2 , denote the value of the local governments’ problem as Ui ðci Þ and the value of the central government’s problem as Uðc1 ; c2 Þ: If the local governments are in charge of the program, social welfare is U1 ðc1 Þ þ U2 ðc2 Þ; and if the central government is in charge, social welfare is Uðc1 ; c2 Þ: Proposition 5.5. Under Assumption 5.1, the space ℝ2þ of marginal costs can be divided into four regions, which suggests that, when the two provinces have sufficiently unbalanced marginal costs, the central government should be in charge; otherwise the local governments should be in charge (see Figure 5.8). c2

Indifference region

CG region

c2 CG region LG region

c1

Figure 5.8 The control regions depending on marginal costs.

c1

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193

Differences in preferences The role of differences in consumer preferences has been an important issue in the literature. For example, Oates’ (1972) Decentralization Theorem states that, with heterogeneous consumers and without spillover effects, decentralization is preferable. In this case, the utility functions are assumed to have fixed marginal utilities: v1 ðxÞ ¼ v1 x; v2 ðx Þ ¼ v2 x; where v1 ≥ 0 and v2 ≥ 0 are two constants. Also, assume ci ðx Þ ¼

1 β x ; β

where

β > 1:

ð5:20Þ

Further, to focus on the differences in preferences in this case, assume the same density function: f ðθ Þ ≡ f 1 ðθ Þ ¼ f 2 ðθ Þ: Given income levels I1 and I2 , with I ≡ I1 þ I2 , denote the value of the local governments’ problem as Wi ðvi Þ and the value of the central government’s problem as Wðv1 ; v2 Þ: If the local governments are in charge of the program, social welfare is W1 ðv1 Þ þ W2 ðv2 Þ; and if the central government is in charge, social welfare is Wðv1 ; v2 Þ: Proposition 5.6. Under Assumption 5.1, the space ℝ2þ of marginal utilities can be divided into four regions, which suggests that, when the two provinces have sufficiently unbalanced preferences, the central government should be in charge; otherwise the local governments should be in charge (see Figure 5.9).

v2

CG region LG region

v2 CG region Indifference region

v1

v1

Figure 5.9 The control regions depending on marginal utilities.

194 Public governance 2.5. Discussion Local governments are closer to the people they serve, and they serve only those people within the province under their control, which is typically a small percentage of the total population of a country. A local government has an advantage in local consumer information. However, the central government has an advantage in resource allocation. Hence, when preferences, costs, or income diverge greatly across localities, this advantage in resource allocation may dominate the informational weakness of centralization. This trade-off implies the necessity of a hierarchy system in which a set of social programs is optimally managed at various levels of government depending on their requirements for information and resources. Wang and Xiao’s (2007) analysis offers some insights about governments:



Different levels of government have different comparative advantages in serving people. A higher level of government is endowed with more power in allocating resources across regions; a lower level of government is physically located closer to the local population and has better information on it. The government has to manage many social programs. Different social programs have different requirements for information knowledge and resource allocation. To achieve overall efficiency, the government needs to have a hierarchy system to efficiently utilize the comparative advantages at various levels of control.

3. The agency problem of litigation This section is based on the work of Wang (2008b). The legal system forms a part of public governance, especially in developing countries, where a government should put more effort into establishing a sound legal system. A fixed-share contract allocates fixed shares of total income to trading parties, regardless of the size of total income. Fixed-share contracts are very popular in reality. In the USA, plaintiffs in 96 percent of tort litigations pay their lawyers on a fixed-share basis. Lawyers’ fees are typically a fixed share of the recoveries across all lawsuits of the same type. For example, the lawyer’s fee is typically 29 percent for auto accidents, 32 percent for medical malpractice, 34 percent for asbestos injuries, and 17 percent for aviation accidents. We present a simple agency model with a risk-averse principal and a risk-neutral agent and show that the observed fixed-share contracts in litigation are optimal behaviors. 3.1. The model The plaintiff and his lawyer first assess the validity of the case and the potential recovery R and then decide their strategy. If the case is valid, they can either sue or seek an out-of-court settlement. If an early out-of-court settlement fails and the lawyer and plaintiff decide to bring the case to court, the lawyer and plaintiff enter into a contractual relationship for the lawsuit. During the process of the lawsuit, the lawyer applies effort which affects the winning chance. A court ruling finally arrives and the lawyer is then paid according to the contract (see Figure 5.9).

Public governance File a lawsuit Contract offer

195

Verdict Apply effort

Figure 5.10 The sequence of events.

Given an estimated recovery R and a winning probability p, the outcome is  R; with probability p; R~ ¼ 0; with probability 1  p: We write this lottery in a simple form: R~ ≡ ðR; p; 0; 1−pÞ, where R is fixed and known. Suppose that the lawyer can exert effort a to achieve a certain winning probability pðaÞ, where pða Þ is increasing in a: For convenience, instead of dealing with an effort variable, we will deal with the probability variable p directly. We call p either the winning probability or effort. Let Cðp; R Þ be the cost associated with the effort for achieving a winning probability p for a recovery R: Assume that Cðp; R Þ is increasing in p and R: In fact, we will use a simple form for the cost function with a constant marginal cost: Cðp; R Þ ≡ cpR, where c ∈ ð0; 1Þ is a fixed constant. Here, the condition c < 1 ensures the plaintiff’s IR condition. Assume that the recovery R is observable and verifiable, but the winning chance p is not verifiable. The pay scheme is:  αw ; if the case is won; lawyer fee : s ¼ αl ; if the case is lost; where αl ; αw ∈ ½0; R : Here, conditions αl ≥ 0 and αw ≥ 0 are limited liability conditions for the lawyer and αl ≤ R and αw ≤ R are limited liability conditions for the plaintiff. The plaintiff’s utility function is an arbitrarily increasing and concave function v: ℝ ! ℝ; assume vð0Þ ¼ 0 for convenience. Hence, the plaintiff/principal is risk averse. Assume also that the lawyer/agent is risk neutral. This is in sharp contrast to the standard agency model. Then, given the potential lawyer fees ðαl ; αw Þ and an assessment of the recovery R, if R is a known constant, the lawyer’s problem is max pαw þ ð1−pÞαl −Cðp; R Þ:

ð5:21Þ

p∈½0;1

Hence, the plaintiff’s problem is V ¼

max

αl ; αw 2½0; R; p2½0;1

s:t:

pvðR  αw Þ þ ð1  pÞvðαl Þ IR : pαw þ ð1  pÞαl  Cðp; RÞ;

ð5:22Þ

IC : p 2 arg max p0 αw þ ð1  p0 Þαl  Cðp0 ; RÞ: p0 2½0;1

196 Public governance 3.2. The solution Problem (5.22) has the following solution. Proposition 5.7. There is a unique optimal contract and it is in the form of contingent fees. Specifically,

~ ¼ cR, ~ where R~ ≡ ðR; p; 0; 1−pÞ. The optimal contract is s∗ ðRÞ ∗ The optimal winning chance is p ¼ 1. This solution is the first best.

Hence, the plaintiff pays the lawyer nothing if the case is lost and pays him a fixed proportion of the recovery if the case is won; this fixed proportion equals the marginal cost of effort. This proportion is fixed regardless of the plaintiff’s preferences and the potential size of the recovery. Only in the case of different types of lawsuits may marginal costs, and thus the shares of recovery for the lawyer, differ. One puzzling phenomenon in reality is that, although the income share is fixed across different states and provinces within one country, it generally varies across different countries under similar law systems. Some argue for differences in tradition and culture. Our solution suggests that the fixed-share rule is determined by the marginal cost of a lawyer’s effort. The marginal cost of a lawyer is typically fixed within one country due to labor mobility, while it varies across countries due to labor immobility. Having a zero constant term in our linear contract is interesting; it agrees with realworld practices. Optimal linear contracts are found in certain agency models, but this is the first time a pure share rule has been found to be optimal. Not having a constant term in the contract means that the lawyer’s income is completely contingent on outcome. Having a fixed share that is independent of preferences and output is also interesting. Linear contracts are not only popular in the real world; they tend to have a stable structure within the same class of contractual relationships. Our solution represents the first time that this possibility has been shown in theory. This result may shed some light on our understanding of similar observations in many other types of business relationships. 3.3. Discussion Optimality of contingent fees Although contingent fees are a widely adopted form of pay scheme in plaintiff– lawyer relationships, its optimality has never been shown in theory. We find an optimal linear contract that is strikingly simple and is consistent with the popular pay scheme of contingent fees in use today. Our solution explains a few key features of contingent fees. First, the widely adopted contingent fees in plaintiff–lawyer relationships are optimal behaviors.

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197

Second, contingent fees are an efficient solution to the agency problem of litigation. Third, within the same type of lawsuits, the pay ratio of contingent fees is fixed. Imposing limited liability conditions Among the limited liability conditions αl ; αw ∈ ½0; R  in our model, only the condition αl ≥ 0 is binding in the solutions. Hence, only the condition αl ≥ 0 is necessary in our model. Interestingly, a legal provision in American law explicitly imposes the limited liability condition αl ≥ 0. What is the implication of this condition for our solution and why is this condition necessary in practice? To understand the implication of the law, we derive the following result for the case without the limited liability conditions. Corollary 5.1. Without the limited liability conditions,

The second-best solution achieves efficiency as before. The first-best contract is ðα∗l ; α∗w Þ ¼ ð−ð1−cÞR; cRÞ and the first-best winning chance is p ∗∗ ¼ 1.

With the limited liability constraint αl ≥ 0, as shown in Figure 5.11, for a given p, the contract is at point A: Without the limited liability constraint αl ≥ 0, the contract is at point B: At point B, the plaintiff effectively sells the case to the lawyer for a lumpsum payment ð1−cÞpR: With this solution, the plaintiff faces no risk, while the riskneutral lawyer has no problem with taking more risks. The effect of the legal provision in American law is to prevent such a trade. αw

Indifference curves

.

B

cR

.

A

IR line cpR 1−p

αl

Figure 5.11 The first- and second-best solutions.

However, since the winning chance is p ∗∗ ¼ 1 in equilibrium, the total surplus in either case is V ∗∗ ¼ v½ð1−cÞR : This means that the legal provision has no effect on social welfare.

198 Public governance Why is this ruling necessary in practice? One view in the literature argues that this ruling may serve to reduce the number of lawsuits in the United States. In other words, although we show p∗ ¼ 1 in equilibrium, if the plaintiff is not too confident about winning the case, he may opt for out-of-court settlement, which may improve efficiency when the welfare of defendants or the whole society is taken into account. For example, if the plaintiff is not sure about the lawyer’s ability, he’s not sure about winning the case. Alternatively, we argue that the requirement of αl ≥ 0 may be due to the problem of asymmetric information. Before the lawyer decides to take the case, she may not have all the information. If the lawyer is to buy the case before she sees evidence from the defendant in the court, the plaintiff has the incentive to hide information from the lawyer. The problem of adverse selection by plaintiffs may lead to inefficiency. In this case, government intervention, by imposing the limited liability condition, may improve efficiency. Banning of contingent fees Interestingly, although we’ve shown the optimality and efficiency of contingent fees, many European countries, such as Great Britain, ban this sharing rule and lawyers are paid by the hour instead. Other countries, like Canada, are talking about imposing a ban on the sharing rule as well. This ban may be consistent with our results. Precisely because the contractual solution between plaintiffs and their lawyers is so efficient, the governments may want to ban it. For example, since a plaintiff is sure of winning a lawsuit, he may not have much incentive to take precautionary measures to prevent accidents; and, if an accident does happen, he may not have much incentive to agree to an out-of-court settlement. In a more complete model that includes a pre-trial strategic game, an efficient solution for the plaintiff may not be most desirable for the whole society. The 100% winning probability With p∗ ¼ 1, the plaintiff–lawyer team in our model will always win in equilibrium. If so, why is there litigation? There are several reasons for it. First, the vast majority of legal disputes in practice are settled out of court. This fact is consistent with our 100% winning probability. When a defendant realizes that the simple fixed-share contract between the plaintiff and a lawyer can be so efficient, he is likely to want to settle out of court. Second, as shown by experimental economics, it usually takes quite a while for players in a game to gain enough experiences before they finally arrive at an equilibrium. Most defendants and plaintiffs, in actual fact, are inexperienced in legal matters. Hence, they may occasionally end up in court, even though an out-of-court settlement is better for all parties. Third, a defendant and a plaintiff may not agree on the compensation. In this case, they may need a court to decide. In fact, much of a pre-trial negotiation in practice is over the size of compensation, not about whether the plaintiff has a case.

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199

Potential applications Similar contractual relationships exist in a wide range of scenarios, including the share-cropping rule in land leasing between landlords and tenants, CJVs and EJVs in FDI between foreign and local firms, franchising in retailing between franchisors and franchisees, and book publishing contracts between authors and publishers. This section provides an understanding of the linear contractual relationship in its simplest form.

Notes

1 Complete contracts 1 For more details of the standard principal–agent model, see Ross (1973), Stiglitz (1974), Harris and Raviv (1979), and Shavell (1979). 2 There are no boundary conditions such as sð1Þ ¼ s and sð1Þ ¼ s
for s(x) for some given constants s and s Thus, we will have two transversality conditions for the optimal contract. If we impose sð−∞ Þ ≥ 0 and sð∞ Þ ≥ 0; then the conditions are sðtÞHs_ jt¼1 ¼ 0;

3 4 5 6 7 8

sðtÞHs_ jt¼1 ¼ 0;

Hs_ jt¼1  0;

Hs_ jt¼1  0:

_ these conditions are all satisfied. Note also that by However, since H doesn’t contain s; assumption the optimal solution must be a continuous function if the admissible set is assumed to contain continuous contracts only. In a general equilibrium setting, the firm is typically risk neutral. In our model, the principal has a utility function; hence, she cannot be treated as a firm. This example is from Holmström (1979, p. 79), but his solution seems to be erroneous. The explanation is that, although we do not observe a density function directly, we can statistically deduce the density function from many observations of the same relationship and then figure out ða1 ; a2 Þ by0 observing ðx1 ; x2 Þ. 1 ;e2 Þ 0 Also: R02 ðe1 ; e2 Þ ¼ c02 ðe2 Þ þ hh0 21 ðe ðe1 ;e2 Þ c1 ðe1 Þ: Cheung (1968) limits his contracts to linear contracts only when deriving a suboptimal contract. Our result shows that his linear contract is optimal, since there is no uncertainty in his model. Since the second constraint in problem (1.63) holds for any w; it must also hold for Uðw; yÞ: Thus, we must have Z Uðw; yÞ ¼ fuðs½Uðw; yÞ; y0 ; a½Uðw; yÞÞ þ βU½Uðw; yÞ; y0 gfðy0 ja½Uðw; yÞÞdy0 ;

for all w 2 W and y 2 Y: Spear and Srivastava (1987) list this condition as one of three constraints, but it is actually redundant. 9 For simplicity, we do not impose the condition st ≤ yt for all t ∈ ℕ in our presentation. However, the numerical example in Section 5.2 does satisfy this condition.

2 Incomplete contracts 1 It is certainly possible that some contracts in reality are incomplete due to unawareness. However, unawareness is beyond the Bayesian paradigm. Within the Bayesian paradigm, full awareness is assumed, in the sense that each player is fully aware of all possible

Notes

2

3 4 5 6

7 8

9 10

201

states, even though he/she may not know when a specific state may be realized or what the chance of it being realized is. In this book, we assume full awareness. Just like the organization of firms, shareholding (ownership) has also been gradually becoming organized (institutionalized) in the last 60 years. For a typical Western country today, only about 15 percent of the shares traded on the stock exchanges of that country are held by domestic individuals; 50 percent are held by domestic institutions and about 30 percent by foreigners. This is dramatically different from the situation in the 1960s, when the majority of shares were held by domestic individuals. See, for example, Jones (2004). Note that central planning doesn’t necessarily mean the elimination of a price system to allocate resources. All centrally planned economies and some large corporations use a price system to allocate resources in their system. To avoid the issue of externality, assume that π c and π f include all externalities. Lenin proposed to adopt the European solution: the market solution. But he passed away before he could implement his plan. See Wang (2012) for a few popular cooperative solutions, including the Nash bargaining solution and the Shapley value. @s1 < 0, the manager can earn a higher income by disposing of some output. If If @π 1 @s2 > 1, the manager can earn a higher income by borrowing money elsewhere and @π 2 claiming it to be a part of the output. xi can be observable ex post. Here, we don’t allow option contracts. The story will be very different if the owner can take an action conditional on her observation of xi . Of course, this can also be equivalently done at t = 0 if the value sharing is based on the Shapley value at t = 1 Hence, besides an allocation of physical assets and an agreement to share value based on the Shapley value, the contract can include an allocation of income rights, as long as it is based on the Shapley value. As in the previous two examples, the yacht can be used to provide several services simultaneously. We can alternatively assume that the yacht can only provide one service at a time, but the conclusions are the same. See, for example, Spier (1992), Anderlini and Felli (1994, 1996), Segal (1995, 1999), MacLeod (1996), Aghion and Tirole (1997), Hart and Moore (1988, 1999), Bernheim and Whinston (1998), Che and Hausch (1999). See Tirole (1994) for a survey of the relevant literature.

3 Corporate finance 1 This assumption is unnecessary. If efficiency is achievable when output is unverifiable, it will most certainly be achievable by the same solution when output is verifiable. 2 The results are no different if the principal is risk averse. In fact, risk aversion in this model does not play any role, and neither does the randomness of revenue. 3 We could use the more typical utility function of the form uðMÞ  cðaÞ: The results would remain the same. 4 Demski and Sappington (1991, Section 4) suggest that in this case efficiency may be achievable with a random payment scheme. Since the risk-averse agent may not want to take the risk, when the variance of Pb is large enough, the first best can be achieved. 5 This is equivalent to assuming a risk-neutral agent (risk neutral in income) and letting the cost of a be an arbitrary cost function cðaÞ: This cðaÞ: replaces u1 ðaÞ in this section. 6 With probability 1  z: this bad firm is the same as the medium firm, in which case the convertible is worth I. 7 Again, with probability q, this bad firm is the same as the good firm, and hence its debt is worth I: 8 The price is assumed to be $1, implying that F is the gross interest rate. 9 However, in reality, most convertibles have call restrictions; but we do not expect most issuers to be bad.

202 Notes 10 In bargaining models, as opposed to principal–agent models, IR conditions can be ignored. In a principal–agent model, an IR condition for the principal is usually ignored, but an IR condition for the agent must be included. 4 Corporate governance 1 By assumption, when the signal is low but the manager claims that it is high, there is no lawsuit; only when the signal is high but the manager claims that it is low will there be a lawsuit.

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Index

action variable, 76 admissible contracts, 3, 5 assignable, 75, 76 bond, 2, 125 Build-Operate-Transfer (BOT), 178 bureaucratic inefficiency, 188 call price, 125 callable convertible, 2, 125 Convexity of the Distribution Function Condition (CDFC), 14 Coase Theorem, 71 complete contract, 1, 27, 61 contract, 1 contractible, 5 contractual approach, 5 contractual joint ventures (CJVs), 47 control rights, 61 conversion ratio, 109, 125 conversion value, 125 convertible, 2, 109, 122 cooperative investments, 116 cost plus contract, 175 distribution formulation of contract, 4 dominant contract, 153 double moral hazard, 27, 110, 118 double risk neutrality, 25

first best, 3, 6 first-best risk sharing, 6 first-order approach (FOA), 9 first-order conditions (FOCs), 6 first-order stochastic dominance (FOSD), 10 fixed-share contract, 3, 163 foreclosure, 137 foreign direct investment (FDI), 47 franchise contract, 162 franchised outlets, 162 general equilibrium (GE) approach, 11 governance transparency, 158 Grossman, Hart and Moore’s property rights theory (GHM), 171 holdup effect, 118 holdup problem, 68 incentive compatibility (IC) condition, 1, 9 incentive problem, 1 income level, 189 income rights, 61 incomplete contract, 1, 61, 63 independent contracts, 18 individual rationality (IR) condition, 6 information risk, 161 initial owner, 74 institutions, 64, 66 investment risk, 161

equity joint ventures (EJVs), 47 Euler equation, 10, 16 exercise price, 125 ex-post options, 109 external risk, 99

labor contract, 2 lead investor, 143 limited liability condition, 5, 40 linear contracts, 25

face value, 125 financial assets, 2, 109 financial distress, 125

marginal rate of substitution, 6 Markov process, 41 mechanism-design approach, 65

Index mechanisms, 1, 66 monotone likelihood ratio property (MLRP), 10 Nash bargaining solution, 81 network capital, 143 option contract, 109, 110 organizational approach, 27, 73 par value, 125 parity value, 125 policy, 1 provisional call, 141 recursive agency model, 39 redemption price, 125 relational contracts, 3 relative risk aversion (RRA), 46 renegotiation, 66, 70 repeated agency models, 39, 40 royalty rate, 162

second-order condition (SOC), 9 selfish investments, 116 semi-linear contract, 46 Shapley value, 89; privatization, 182 standard agency model, 5 state formulation of contract, 3 static agency model, 40 strike price, 125 suboptimal linear contract, 34 substitutable inputs, 120 support of a distribution function, 6 syndicate, 143 third-best linear contract, 34 threat-point effect, 118 tournament, 19 transaction cost economics (TCE), 171 transaction costs, 68, 69 trigger price, 132 two-state agency model, 20 unforeseen contingencies, 70

second best, 9 second-best problem, 9

value variable, 77

209