MANAGERIAL ECONOMICS INSTRUCTOR'S MANUAL [FULL, 8 ed.] 0393920499

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INSTRUCTOR’S MANUAL

Managerial Economics EIGHTH EDITION

INSTRUCTOR’S MANUAL

Managerial Economics EIGHTH EDITION

W. Bruce Allen • Neil A. Doherty • Keith Weigelt • Edwin Mansfield Jean Cupidon TEXAS TECH UNIVERSITY

B

W • W • NORTON & COMPANY • NEW YORK • LONDON

W. W. Norton & Company has been independent since its founding in 1923, when William Warder Norton and Mary D. Herter Norton first published lectures delivered at the People’s Institute, the adult education division of New York City’s Cooper Union. The fi rm soon expanded its program beyond the Institute, publishing books by celebrated academics from America and abroad. By midcentury, the two major pillars of Norton’s publishing program—trade books and college texts— were firmly established. In the 1950s, the Norton family transferred control of the company to its employees, and today—with a staff of four hundred and a comparable number of trade, college, and professional titles published each year—W. W. Norton & Company stands as the largest and oldest publishing house owned wholly by its employees.

Copyright © 2013 by W. W. Norton & Company, Inc. All rights reserved. Printed in the United States of America. Associate media editor: Nicole Sawa Production manager: Ben Reynolds Composition by Westchester Publishing Ser vices Manufacturing by Sterling Pierce ISBN 978- 0-393-92049-9 W. W. Norton & Company, Inc. 500 Fifth Avenue, New York, N.Y. 10110- 0017 wwnorton.com W. W. Norton & Company Ltd. Castle House, 75/76 Wells Street, London W1T 3QT 1 2 3 4 5 6 7 8 9 0

CONTENTS

PART 1: THE NEED FOR A GUIDE Chapter 1 | Introduction

1

PART 2: THE NATURE OF MARKETS Chapter 2 | Demand Theory

13

Chapter 3 | Consumer Behavior and Rational Choice

31

Chapter 4 | Estimating Demand Functions

51

PART 3: PRODUCTION AND COST Chapter 5 | Production Theory

70

Chapter 6 | The Analysis of Costs

86

PART 4: MARKET STRUCTURE AND SIMPLE PRICING STRATEGIES Chapter 7 | Perfect Competition

103

Chapter 8 | Monopoly and Monopolistic Competition

115

PART 5: SOPHISTICATED MARKET PRICING Chapter 9 | Managerial Use of Price Discrimination Chapter 10 | Bundling and Intrafirm Pricing

133 153

v

vi | Contents

PART 6: THE STRATEGIC WORLD OF MANAGERS Chapter 11 | Oligopoly

169

Chapter 12 | Game Theory

188

Chapter 13 | Auctions

206

PART 7: RISK, UNCERTAINTY, AND INCENTIVES Chapter 14 | Risk Analysis

220

Chapter 15 | Principal–Agent Issues and Managerial Compensation 239 Chapter 16 | Adverse Selection

258

PART 8: GOVERNMENT ACTIONS AND MANAGERIAL BEHAVIOR Chapter 17 | Government and Business

271

Chapter 18 | Optimization Techniques

293

CHAPTER 1

Introduction

Lecture Notes 1. Introduction •

•

Objectives ÿ To provide a guide to making good managerial decisions ÿ To use formal models to analyze the effects of managerial decisions on measures of a firm’s success Managerial Economics versus Microeconomics ÿ Managerial economics differs from microeconomics in that microeconomics focuses on description and prediction while managerial economics is prescriptive. ÿ Managerial economics prescribes behavior, whereas microeconomics describes the environment. ÿ Managerial economics is an integrative course that brings the various functional areas of business together in a single analytical framework. ÿ Managerial economics exhibits economies of scope by integrating material from other disciplines and thereby reinforcing and enhancing understanding of those subjects.

2. The Theory of the Firm •

Managerial Objective ÿ To make choices that will increase the value of the firm ÿ Managers in profit-oriented organizations try to increase the net present value of expected future cash flows. ÿ The value of the firm is defined as the present value of future profits:

1

2 | Chapter 1 n  2 ÿ Present value of = 1 + +$+ 2 expected future profits 1 + i (1 + i ) (1 + i ) n ÿ More compactly, we write: n t ÿ Present value of = (1 expected future profits t =1 + i)t ÿ Given that profit  total revenue  total cost, then we write: n TRt − TCt ÿ Present value of = expected future profits t =1 (1 + i)t ÿ Notation Profit in time t  Total Revenue in time t  Total Cost in * pt time t Interest rate * i Number of time periods * n * TRt Total Revenue in time t * TCt Total Cost in time t Managerial Choices ÿ Influence total revenue by managing demand ÿ Influence total cost by managing production ÿ Influence the relevant interest rate by managing finances and risk Managerial Constraints ÿ Environmental and antitrust laws ÿ Resource scarcity ÿ Legal or contractual limitations

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∑

•

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STRATEGY SESSION: Bono Sees Red, and Corporate Profits See Black Discussion Questions 1. How can a firm assess the benefits and costs of cause marketing? 2. What other examples of cause marketing can you identify? 3. What Is Profit? •

Two Measures of Profit ÿ Accounting profit * Historical costs, legal compliance, reporting requirements * The accountant is concerned with controlling the firm’s day-today operations, detecting fraud and embezzlement, satisfying tax and other laws, and producing records for various interested groups ÿ Economic profit * Market value; opportunity, or implicit cost

Introduction | 3 * *

The economist is concerned with decision making, rational choice among strategies A more useful measure for managerial decision making

4. Reasons for the Existence of Profit •

•

Profit ÿ Measures the quality of managers’ decision-making skills ÿ Encourages good management decisions by linkage with incentives Sources of Profit: Three profit-generating areas ÿ Innovation: Producing products that are better than existing products in terms of functionality, technology, and style ÿ Risk taking: Future outcomes and their likelihoods are unknown, as are the reactions of rivals. ÿ Market power: Managers also earn profit by exploiting market inefficiencies. Common tactics include * building barriers to entry * employing sophisticated pricing strategies * diversification efforts * making good strategic production decisions

5. Managerial Interests and the Principal–Agent Problem •

Principal–Agent Problem ÿ The interests of a firm’s owners and those of its managers may differ, unless the manager is the owner. ÿ Separation of ownership and control * The principals are the owners. They want managers to maximize the value of the firm. * The agents are the managers. They want more compensation and less accountability. Because the firm’s owners find it difficult to adequately distinguish between actions that maximize profits and those that do not, managers have incentives to enrich themselves. * The divergence in goals is the principal–agent problem. * To deal with this problem, owners (the principals) often use  contracts to converge their preferences and those of their agents. * Moral hazard exists when a person behaves differently when he or she is not subject to the risks associated with his or her behavior. * Managers who do not maximize the value of the firm may do so because they do not suffer as a result of their behavior.

4 | Chapter 1 ÿ Solutions * Devise methods that lead to convergence of the interests of the firm’s owners and its managers. * Examples: Stock option plans or bonuses linked to profits 6. Demand and Supply: A First Look •

Market ÿ A group of firms and individuals that interact with each other to buy or sell a product ÿ Part of an economy’s infrastructure ÿ A social institution that exists to facilitate economic exchange ÿ Relies on binding, enforceable contracts

STRATEGY SESSION: Baseball Discovers the Law of Supply and Demand Discussion Questions 1. Do you see a relationship between variable pricing of baseball game tickets and odds-making on horse races? 2. How do you think real-time variable pricing would influence the practice of ticket scalping? 7. The Demand Side of a Market •

•

Demand Curve ÿ It shows managers how many units they sell at a given price, holding other possible influences constant. ÿ It is negatively sloped. ÿ It pertains to a par ticular period of time. Other influences on demand decisions include * consumer income * prices of substitutes and complements * advertising expenditures * product quality * government fiat Total Revenue Function ÿ A firm’s total revenue (TR) for a given time period is equal to the price charged (P) times the quantity sold (Q) during that time period. ÿ TR  P  Q ÿ The demand function reflects the effect of changes in P on quantity demanded (Q) per time period and, hence, the effect of changes in P on TR.

Introduction | 5 8. The Supply Side of a Market •

Supply side is represented by a market supply curve. ÿ The market supply curve shows how many units of a commodity sellers will offer at any price. ÿ It is positively sloped. ÿ It pertains to a par ticular period of time. ÿ Decreases in the cost of inputs (labor, capital, land) or technological progress cause supply curves to shift to the right.

9. Equilibrium Price •

•

Disequilibrium ÿ Price is too high. * Excess supply or surplus * Causes price to fall ÿ Price is too low. * Excess demand or shortage * Causes price to rise Equilibrium Price ÿ A situation in which quantity demanded is equal to quantity supplied ÿ Price is sustainable. ÿ The market is in balance because everyone who wants to purchase the good can, and every seller who wants to sell the good can.

10. Actual Price • • • • •

The price that is of interest to the manager Invisible hand: When no governmental agency is needed to induce producers to drop or increase their prices If the actual price is above the equilibrium price, there will be a surplus that will put downward pressure on the actual price. If the actual price is below the equilibrium price, there will be a shortage that will put upward pressure on the actual price. If the actual price is equal to the equilibrium price, then there will be neither a shortage nor a surplus and the market is said to be in equilibrium.

11. What If the Demand Curve Shifts? • •

Demand and supply curves are not static. They shift in reaction to changes in the environment. Increase in Demand ÿ Represented by a rightward or upward shift in the demand curve

6 | Chapter 1

•

ÿ Result of a change that makes buyers willing to purchase a larger quantity of a good at the current price and/or to pay a higher price for the current quantity ÿ Will create a shortage and cause the equilibrium price to increase Decrease in Demand ÿ Represented by a leftward or downward shift in the demand curve ÿ Result of a change that makes buyers purchase a smaller quantity of a good at the current price and/or continue to buy the current quantity only if the price is reduced ÿ Will create a surplus and cause the equilibrium price to decrease

12. What If the Supply Curve Shifts? •

•

Increase in Supply ÿ May be caused by technological advances ÿ Represented by a rightward or downward shift in the supply curve ÿ Result of a change that makes sellers willing to offer a larger quantity of a good at the current price and/or to offer the current quantity at a lower price ÿ Will create a surplus and cause the equilibrium price to decrease Decrease in Supply ÿ Represented by a leftward or upward shift in the supply curve ÿ Result of a change that makes sellers willing to offer a smaller quantity of a good at the current price and/or to offer the current quantity at a higher price ÿ Will create a shortage and cause the equilibrium price to increase

STRATEGY SESSION: Life During a Market Movement Discussion Questions 1. Several factors are mentioned as contributing to disequilibrium in global food markets. Among them are emotions (panic), government restrictions on trade, the Malthusian specter of population growth outpacing food production, slowing productivity growth in the agricultural sector, rising incomes, and the production of ethanol. Which of these are supply factors and which are demand factors? How does each influence market price? 2. The market price for crude oil fluctuated widely during 2008. What supply and demand factors contributed to these fluctuations? Is the petroleum market subject to any of the same factors cited as influencing agricultural markets?

Introduction | 7 Chapter 1: Problem Solutions 1. A book is to be written by Britney Spears. Batman Books agrees to pay Britney $6 million for the rights to this not-yet-written memoir. According to one leading publisher, Batman Books could earn a profit of roughly $1.2 million if it sold 625,000 copies in hardcover. On the other hand, if it sold 375,000 copies, managers would lose about $1.3 million. Publishing executives stated that it was hard to sell more than 500,000 copies of a nonfiction hardcover book, and very exceptional to sell 1 million copies. Were Batman managers taking a substantial risk in publishing this book? Solution: There was a substantial risk of loss. On the other hand, there was substantial opportunity for gain. Risk is often unavoidable. The appropriate balance between risk and return is what should determine managers’ decisions. Successful decisions in circumstances of risk are a source of profit. 2. Some say that any self-respecting top manager joining a company does so with a front-end signing bonus. In many cases this bonus is in the seven figures. At the same time the entering manager may be given a bonus guarantee. No matter what happens to firm profit, he or she gets at least a percentage of that bonus. Do long-term bonus guarantees help to solve the principal– agent problem, or do they exacerbate it? Why? Solution: An executive who spends a lifetime working for a single company or in a single industry has a poorly diversified human capital portfolio. Such an executive also often has a significant, undiversified financial investment in the form of stock options and pension plans that are used in partial substitution for current salary to align the long-term wealth of the executive with that of the shareholders. As an executive climbs the corporate ladder, the value of his or her human capital becomes more closely tied to the fortunes of the firm and industry. This lack of diversification requires a compensating risk premium. A large signing bonus may allow a risk-averse executive to make an investment, which increases the value of the firm but which the executive would otherwise avoid because of concern for his or her own personal wealth; thus the bonus may reduce the principal–agent conflict. Of course the benefits of reduced risk to the executive come at the potential cost of indifference to the wealth of the shareholders. Although a large signing bonus may help solve the incentive alignment problem, compensation that is too great and too insensitive to the fortunes of the shareholders makes the principal–agent problem worse.

8 | Chapter 1 3. If the interest rate is 10%, what is the present value of the Monroe Corporation’s profit in the next 10 years? Number of Years in the Future

Profit (millions of dollars)

1 2 3 4 5 6 7 8 9 10

8 10 12 14 15 16 17 15 13 10

Solution: Use formula (1.1) for t  1, 2, . . . , 10 to obtain the following table: Number of Years in the Future

Profit (millions of dollars)

(1  i)t

1 2 3 4 5 6 7 8 9 10

8 10 12 14 15 16 17 15 13 10

0.90909 0.82645 0.75131 0.68301 0.62092 0.56447 0.51316 0.46651 0.42410 0.38554

Present Value (millions of dollars)

Total

7.27272 8.26450 9.01572 9.56214 9.31380 9.03152 8.72372 6.99765 5.51330 3.85540 77.55047

The answer is $77.55047 million. 4. Managers at Du Pont de Nemours and Company expect a profit of $2.9 billion in 2012. Does this mean that Du Pont’s expected economic profit will equal $2.9 billion? Why or why not? Solution: Economic profits differ from accounting profits because of differences in the way depreciation is measured, differences in the way revenues and costs are recognized in terms of timing, and the inclusion of the opportunity cost of owner-supplied inputs in the calculation of economic profits.

Introduction | 9 Du Pont’s economic profits might well be negative if accounting profits do not exceed the risk-adjusted rate of return multiplied by the firm’s equity value. 5. William Howe must decide whether to start a business renting beach umbrellas at an ocean resort during June, July, and August of next summer. He believes he can rent each umbrella to vacationers at $5 a day, and he intends to lease 50 umbrellas for the three-month period for $3,000. To operate this business, he does not have to hire anyone (but himself), and he has no expenses other than the leasing costs and a fee of $3,000 per month to rent the business location. Howe is a college student, and if he did not operate this business, he could earn $4,000 for the three-month period doing construction work. a. If there are 80 days during the summer when beach umbrellas are demanded and Howe rents all 50 of his umbrellas on each of these days, what will be his accounting profit for the summer? b. What will be his economic profit for the summer? Solution: a. TR  (80 days)  (50 umbrellas)  ($5 per day)  $20,000 TC  (3 months)  ($3,000 per month rent)  ($3,000 umbrella lease)  $12,000 Accounting Profit  TRTC  $8,000 b. Economic Profit  Accounting Profit  Opportunity Cost Economic Profit  $8,000  $4,000  $4,000 6. On March 3, 2008, a revival of Gypsy, the Stephen Sondheim musical, opened at the St. James Theater in New York. Ticket prices ranged from $117 to $42 per seat. The show’s weekly gross revenues, operating costs, and profit were estimated as follows, depending on whether the average ticket price was $75 or $65:

Gross revenues Operating costs Profit

Average Price of $75

Average Price of $65

$765,000 600,000 165,000

$680,000 600,000 80,000

a. With a cast of 71 people, a 30-piece orchestra, and more than 500 costumes, Gypsy cost more than $10 million to stage. This investment was in addition to the operating costs (such as salaries and theater rent). How many weeks would it take before the investors got their money back, according to these estimates, if the average price was $65? If it was $75?

10 | Chapter 1 b. George Wachtel, director of research for the League of American Theaters and Producers, has said that about one in three shows opening on Broadway in recent years has at least broken even. Were the investors in Gypsy taking a substantial risk? c. According to one Broadway producer, “Broadway isn’t where you make the money any more. It’s where you establish the project so you can make the money. When you mount a show now, you really have to think about where it’s going to play later.” If so, should the profit figures here be interpreted with caution? d. If the investors in this revival of Gypsy make a profit, will this profit be, at least in part, a reward for bearing risk? Solution: a. Given a price of $75, the weekly operating profit of $165,000 would pay off the $10 million investment in 10,000/165  60.6 or 61 weeks. If the price is $65, it would take 10,000/80  125 weeks to pay off the investment. This does not provide for any return on investment, however. b. The investors in Gypsy were indeed taking a substantial risk. If only one in three shows breaks even, two out of three make losses. c. The profit figures should be interpreted with caution because they do not take into account the likelihood of profits when, and if, the show goes on the road. d. Yes. 7. If the demand curve for wheat in the United States is P  12.4  QD where P is the farm price of wheat (in dollars per bushel) and QD is the quantity of wheat demanded (in billions of bushels), and the supply curve for wheat in the United States is P  2.6  2QS where QS is the quantity of wheat supplied (in billions of bushels), what is  the equilibrium price of wheat? What is the equilibrium quantity of wheat sold? Must the actual price equal the equilibrium price? Why or why not? Solution: Setting demand equal to supply in equilibrium, that is, QD  QS  QE, yields 12.4  QD  2.6  2QS QE  15/3  5 PE  12.4  5  2.6  (2)(5)  $7.40

Introduction | 11 The actual price need not be equal to equilibrium price, although it will generally tend to move toward it because of the equilibrating effects of shortage and surplus. Factors that might prevent the actual price from equaling the equilibrium price include the cost and availability of information, transportation costs, and lack of opportunities for price-equalizing arbitrage. 8. The lumber industry was hit hard by the downturn in housing starts in 2010 and 2011. Prices plunged from $290 per thousand board feet to less than $200 per thousand board feet. Many observers believed this price decrease was caused by the slowing of new home construction because of the glut of unsold homes on the market. Was this price decrease caused by a shift in the supply or demand curve? Solution: Because the demand for lumber is derived in large part from the demand for new housing construction, a decline in construction would be likely to cause the demand for lumber to fall, leading to lower lumber prices since the demand curve shifts to the left. Supply would not be affected by changes in housing construction. 9. From November 2010 to March 2011 the price of gold increased from $1,200 per ounce to over $1,800 per ounce. Newspaper articles during this period said there was little increased demand from the jewelry industry but significantly more demand from investors who were purchasing gold because of the falling dollar. a. Was this price increase due to a shift in the demand curve for gold, a shift in the supply curve for gold, or both? b. Did this price increase affect the supply curve for gold jewelry? If so, how? Solution: a. A change in the value of the dollar causes the dollar price of globally traded commodities to change. If the value of the dollar falls, the dollar price of commodities will rise. In this case, a decline in the value of the dollar can be expected to cause the market for gold (with price measured in dollars) to experience an increase in demand and a decrease in supply and thus an increase in price. There may also have been an additional increase in demand due to expectations by investors that the dollar price of gold will continue to rise. Finally, there may have been a further supply decrease if producers, speculating that prices would rise further, withheld gold from the market.

12 | Chapter 1 b. Gold is an input to the production of jewelry. An increase in the price of gold would therefore be expected to reduce the supply of jewelry, resulting in higher jewelry prices.

CHAPTER 2

Demand Theory

Lecture Notes 1. Introduction •

Objectives ÿ To explain the importance of market demand in managerial decision making ÿ To understand the many factors that influence the demand for a product ÿ To measure and analyze the sensitivity of demand to changes in factors affecting demand. The tool used for this type of sensitivity analysis is demand elasticity. * Elasticity: Measures the percentage change in one factor given a small (marginal) percentage change in another factor * Elasticity: Measures the sensitivity of one factor to another * Demand elasticity: Measures the percentage change in quantity demanded of a product given a small (marginal) percentage change in another factor that affects the demand for the product ÿ Explain the role of managers in influencing and predicting market demand. * Managers can influence demand by controlling, among other things, advertising, product quality, and distribution strategies. * Managers cannot control, but need to understand, elements of the competitive environment that influence demand, including the availability of substitute or complement goods, their pricing, and the advertising strategies employed by their sellers. * Managers cannot control, but need to understand how, the macroeconomic environment influences demand, including interest rates, taxes, and both local and global levels of economic activity such as the level of income in the economy.

13

14 | Chapter 2 2. The Market Demand Curve • • • •

•

•

Market Demand Schedule: A table showing the total quantity of the good purchased at each price during a given time period Market Demand Curve: A plot of the market demand schedule on a graph Example (Table 2.1): Demand schedule for tablets It shows the total quantity of tablets demanded at each price, not the quantity demanded from a par ticular firm. ÿ Convention: Price is on the vertical axis and quantity is on the horizontal axis. ÿ Example (Figure 2.1): Demand curve for tablets Characteristics of the Market Demand Curve ÿ Quantity demanded is for output of the entire market or the industry, not of a single firm. ÿ For most products and services, the market demand curve slopes downward and to the right. ÿ Example: The quantity of tablets demanded increases as the price of tablets falls. ÿ Quantity demanded is defined with regard to a par ticular time period. Determinants of the position and shape of the market demand curve— Some of the important factors include ÿ Consumer tastes or preferences * An increase in consumer tastes shifts the demand curve to the right. * A decrease in consumer tastes shifts the demand curve to the left. ÿ Consumer income (or more specifically per capita disposable income) * Normal and inferior goods * Example (Figure 2.3): Increase in income causes an increase in demand for tablets; that is, tablets are a normal good. ÿ Population size in the market

STRATEGY SESSION: The Customer Is Always Right—Wrong! Discussion Questions 1. Like retail technology stores, clothing stores have their angels and devils. How do you think the devils prey on clothing stores, and how could their behavior be discouraged? How do you think angels could be encouraged to shop at a par ticular clothing store? Answer: Devils buy clothes, wear them, and then return them for a refund. Stores can refuse to provide refunds on returns and, instead, provide a credit for future purchases or only allow exchanges. Angels buy lots of clothes on impulse. Stores could offer quantity discounts or a “shoppers club” with special notification of sales.

Demand Theory | 15 2. Some electronics stores refuse to allow customers to return or exchange products, instead requiring them to deal directly with the manufacturer. What are the pros and cons of this approach with regard to the stores’ objective of encouraging angels and discouraging devils? 3. Industry and Firm Demand Functions •

Market Demand Function: The relationship between the quantity demanded of a product and the various factors that influence this quantity ÿ Quantity demanded of good X: Q  f(factors)  f(P, P_r, I, T, N, A, …) ÿ Factors include * price of X: P * incomes of consumers: I * tastes of consumers: T * prices of related goods in consumption: P_r * population size: N * advertising expenditures: A * general demand function: Q  f(factors)  f(P, P_r,I,T,N,A, other factors) ÿ Example (equation 2.1): A linear demand function: Q  b1P  b2 I  b3S  b4 A Assumes that population is constant P  price of tablets I  per capita disposable income S  average price of software A  amount spent on advertising b1, b2, b3, and b4 are parameters that are estimated using statistical methods, namely, regression analysis. Parameters: Constant or variable terms used in the function that helps managers determine the specific form of the function but not its general nature * Example: Q  2,000P  70I  375S  0.0001A Relationship between the market demand function and the market demand curve * The market demand curve shows the relationship between Q and P when all other variables are held constant at specific values. * The market demand function does not explicitly hold any values constant. Example (equation 2.3): Suppose I  13,000, S  400, and A  50 million. Then Q  700P  200(13,000)  500(400)  0.01(50,000,000) That is * * * * * *

ÿ

ÿ

ÿ ÿ

16 | Chapter 2 *

Q  2,900,000  700P (direct demand function)

Solving for P in terms of Q gives the inverse demand function: P  4,143  0.001429Q (graphed in Figure 2.4) ÿ Example: Shifting the demand curve Suppose the price of software falls from $400 to $200. Then Q  700P  200(13,000)  500(200)  0.01(50,000,000) Q  3,000,000  700P *

Solving for P gives

P  4,286  0.001429Q (graphed in Figure 2.4) •

Note: Same slope; only the vertical intercept changes. The Firm’s Demand Curve ÿ Negative slope with regard to own price * Slope may not be the same as that of the market demand curve ÿ Represents a portion of market demand * Market share * Responds to same market and macroeconomic factors as the market demand curve

4. The Own-Price Elasticity of Demand •

Own-Price Elasticity of Demand: More simply referred to as the price elasticity of demand, it is the concept that managers use to measure the percentage change in quantity demanded of their firm’s products resulting from a 1% change in the products’ own prices. ÿ The elasticity of a function is the percentage change in the dependent (Y) variable in response to a 1% change in the independent (X) variable. ÿ The price elasticity of a demand function is the percentage change in quantity demanded in response to a 1% change in price. ÿ Formula: ⎛ P ⎞ Q =⎜ ⎟ ⎝ Q ⎠ P ÿ Along a demand function, it is given by ⎛ P ⎞ ∂Q =⎜ ⎟ ⎝ Q ⎠ ∂P ÿ Price elasticity of demand generally is different at different prices and on different markets.

Demand Theory | 17 •

Terminology ÿ Price elasticity of demand is symbolized by the Greek letter eta (h) (not a general convention). ÿ Fact: 0 ≥ h ≥ ∞. That is, price elasticity of demand is always nonpositive. ÿ When |h| 1, demand is elastic. ÿ When |h| 1, demand is inelastic. ÿ When |h|  1 or h  1, demand is unitary. Limiting cases: ÿ When h  0, demand is perfectly inelastic and the demand curve is vertical. * Quantity demanded is the same at all prices. ÿ When h  ∞, demand is perfectly elastic and the demand curve is horizontal. * Price is the same for all quantities demanded. * If price rises, quantity demanded falls to zero. * If price falls, quantity demanded increases without limit. • Linear Demand Curves ÿ The slope of a linear demand curve is constant. ÿ If the demand curve is neither vertical nor horizontal, the price elasticity will vary depending on price level. * At the midpoint of a linear demand curve, h  1, with h approaching zero as price approaches the vertical intercept. * At prices above the midpoint, demand is elastic, with h approaching negative infinity as quantity approaches zero or as price approaches the vertical intercept. * At prices below the midpoint, demand is inelastic, with h approaching 0 as price approaching 0. ÿ Given a demand curve defined as P  abQ, the price elasticity of −1 a − bQ = − P / (a − P ) demand is:  = ⎛⎜ ⎞⎟ ⎝ b⎠ Q

5. Point and Arc Elasticities •

The point price elasticity formula should be used when working with an estimated demand curve or when the change in price is very small. It is written as ⎛ P ⎞ ∂Q =⎜ ⎟ ⎝ Q ⎠ ∂P ÿ Calculated value of price elasticity for small changes in prices will differ depending on whether P and Q are the starting values or the ending values after the price change. The change will be small if the change is small.

18 | Chapter 2

•

* Example: P1  99.95, P2  100.00, Q1  20,002, and Q2  20,000 * h  [(20,002  20,000)/(99.95  100)][99.95/20,002]  0.1999 * h  [(20,000  20,002)/(100  99.95][100/200,000]  0.22 ÿ If the price change is large, then the direction of change will influence the calculated elasticity. * Example: P1  5, P2  4, Q1  3, and Q2  40 * h  [(40  3)/(4  5)][5/3]   61.67 * h  [(3  40)/(5  4)][4/40]   3.70 ÿ This problem is corrected by using the arc or midpoint formula. Recommendation: The midpoint or arc elasticity formula should be used to estimate the price elasticity of demand from a demand schedule where price differences are not very small. It is given by ⎛ Q ⎞ ⎛ P1 + P2 ⎞ =⎜ ⎝ P ⎟⎠ ⎜⎝ Q1 + Q2 ⎟⎠

ÿ Example: P1  5, P2  4, Q1  3, and Q2  40 ÿ h  [(40  3)/(4  5)][(5  4)/(3  40)]  7.74 6. Using the Demand Function to Calculate the Price Elasticity of Demand Whenever the demand function is specified, one usually uses the point price elasticity of demand. • Example: Given ÿ Q   700P  200I  500S  0.01A ÿ Q  Quantity demanded of tablet computers ÿ Price  P  3,000 ÿ Income  I  13,000 ÿ Software  S  400 ÿ Advertising  A  50,000,000 • Derive the demand curve ÿ Q   700P  (200)(13,000)  (500)(400)  (0.01)(50,000,000) ÿ Q  2,900,000  700P • Determine Q ÿ Q  2,900,000  (700)(3,000)  800,000. ∂Q We have = −700. Thus ∂P • h  (700)(3,000/800,000)  2.62 • For P  2,000, Q  2,900,000  (700)(2,000)  1,500,000, so h  (700)(2,000/1,500,000)  0.93

Demand Theory | 19 7. The Effect of Price Elasticity on the Firm’s Revenue •

Derivation of relationship between marginal revenue (TR/Q or dTR/ dQ) and the price elasticity of demand ÿ Total revenue: TR  PQ ÿ Consider quantity as a function of price: Q  f(P). Then, differentiating TR with respect to P, using the product rule for derivative, gives dTR/dP  Q(dP/dP)  P(dQ/dP) Dividing by Q gives (1/Q)(dTR/dP)  (dP/dP)  (P/Q)(dQ/dP)  1  h

• •

•

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1/Q is positive. Implications: ÿ Case 1: If h  1, that is, unitary elastic demand, then 1  h  0, and dTR/dP  0, so total revenue is at a maximum and a change in P will have no effect on total revenue. ÿ Case 2: If h 1 (inelastic demand ), then 1  h 0 and dTR/dP 0, so an increase in P (and consequent decrease in Q) will increase total revenue. ÿ Case 3: If h 1 (elastic demand ), then 1  h 0, and dTR/dP 0, so an increase in P (and consequent decrease in Q) will reduce total revenue. ÿ To summarize: If the price elasticity is unitary, any price change will cause an equal and opposite percentage change in quantity demanded. Total revenue will remain constant. If the price elasticity is in the inelastic range, then a 1% change in P will cause less than a 1% change in quantity in the opposite direction. Therefore, total revenue will change in the same direction as price. If the price elasticity is in the inelastic range, then a 1% change in P will cause more than a 1% change in quantity in the opposite direction. Therefore, total revenue will change in the opposite direction from price.

PROBLEM SOLVED: Price Elasticity of Demand: Philip Morris Discussion Questions 1. The decline in total revenue from cigarette sales in 1993 is attributed to Philip Morris’s cut in the price of cigarettes. Are there other factors that might have contributed to this decline in revenue? Answer: The price elasticity of demand assumes that “all other things” are held constant. Changes in taxes, consumer income, or attitudes toward

20 | Chapter 2 tobacco during this period might have reduced demand, while the price cut increased quantity demanded. If this was the case, then the true price elasticity would likely be closer to 1. 8. Funding Public Transit •

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Given ÿ Price (fare) elasticity of demand for public transit in the United States is estimated to be about 0.3. Facts: ÿ All public transit systems in the United States lose money. ÿ Public transit systems are funded by federal, state, and local governments, all of which have budget issues. Which transit systems have the most difficult time getting public funding? ÿ Revenue from sales will increase if fares are increased because demand is inelastic. ÿ Costs will likely decrease if fares are increased because quantity demanded (ridership) will fall. ÿ Implication: Managers of public transit will therefore increase fares if they do not receive enough public funds to balance their budgets.

9. Determinants of the Own-Price Elasticity of Demand • •

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Number and Similarity of Available Substitutes Fact: A product with many close substitutes generally has elastic demand. Product’s Price Relative to a Consumer’s Total Budget ÿ Facts: * Products for which the typical consumer spends only a very small fraction of her income are quite elastic. ° Examples: Thimbles, rubber bands, salt * Products that command a larger percentage of the consumer’s total budget tend to be more price elastic. ° Examples: Kitchen appliances, automobiles Time Period Available for Adjustment to a Price Change ÿ Fact: For nondurable goods, demand is likely to be more elastic over a long period relative to a short period. ÿ Rationale: The longer the time period, the easier it is for consumers to substitute one good for another.

10. The Strategic Use of the Price Elasticity of Demand • •

Managers can change the price elasticity of demand for their products. A Useful Tool: Product differentiation

Demand Theory | 21 • • •

Differentiation strategies convince consumers the product is unique; hence it has fewer substitutes. Caution: Differentiation is not effective if consumers do not perceive it. Example: Strategic pricing of first-class (h  0.45), regular economy (h  1.30), and excursion (h  1.83) airline tickets between the United States and Europe ÿ First-class prices should be relatively high because demand is inelastic. ÿ Regular economy and excursion prices should be relatively low because demand is elastic. ÿ If consumers perceive that a product has fewer substitutes, then their price elasticity of demand for the product will decrease (become less elastic) in absolute value. ÿ Differentiation strategies do not require actual differences in products, only a perceived difference.

STRATEGY SESSION: Elasticity in Use Discussion Questions 1. Suppose that a manufacturer sells a product through an upscale boutique and, with a different brand name, through a discount retailer. The elasticity of demand at the boutique is 1.2, and at the discount retailer it is 2.6. If the optimal price at the boutique is $85, what price (PD) should be charged at the discount retailer? Answer: 85(1  1/1.2)  PD (1  1/2.6) so PD  $23.02 2. A consulting firm charges $250 per hour to Fortune 500 companies. The estimated elasticity of demand for consulting services is 3.1. The firm is planning to spin off a subsidiary firm that will work with smaller businesses. The estimated elasticity of demand for these firms is 7.3. What price per hour (PS), to the nearest dollar, should be charged by the subsidiary? Answer: 250(1  1/3.1)  PS (1  1/7.3) so PS  $200 11. Total Revenue, Marginal Revenue, and Price Elasticity •

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A firm’s total revenue (TR) is equal to the total amount of money consumers spend on the product in a given time period. ÿ Linear demand curve: P  a– bQ ÿ Corresponding total revenue curve: TR  PQ  aQ– bQ2 Marginal Revenue: The incremental revenue earned from selling an additional unit of output ÿ MR  dTR/dQ  d( aQ– bQ2)/dQ  a  2bQ * h  (1/b)[(a–bQ)/Q]  a/(bQ)  1

22 | Chapter 2 * Case 1: If Q  a/2b, then h  1, unitary elastic demand. * Case 2: If Q a/2b, then h 1, and demand is inelastic. * Case 3: If Q a/2b, then h 1, and demand is elastic. ÿ MR  dTR/dQ  d(PQ)/dQ  P(dQ/dQ)  Q(dP/dQ)  P[1  (Q/P) (dP/dQ). Thus MR  P(1  1/h) * Case 1: |h| 1 (elastic) implies MR 0. * Case 2: |h| 1 (inelastic) implies MR 0. * Case 3: |h| 1 (unitary elastic) implies MR  0. 12. The Income Elasticity of Demand •

Income Elasticity of Demand (hI): Measures the percentage change in quantity demanded (Q) resulting from a 1% change in consumer income (I) ÿ Income can be defined as aggregate consumer income or as per capita disposable income, depending on circumstances. ⎛ Q ⎞ ⎛ I ⎞ ÿ I = ⎜ ⎝ I ⎟⎠ ⎜⎝ Q ⎟⎠ ÿ For a demand function, it is given by ⎛ I ⎞ ∂Q =⎜ ⎟ ⎝ Q ⎠ ∂I ∂Q > 0 for normal goods. ∂I * For most products, hI 0; that is, most goods are normal, since increases in aggregate income are associated with increases in aggregate consumer spending. ∂Q ÿ hI 0 for inferior goods because < 0 for inferior goods. ∂I * Examples: Hamburgers and public transportation * Fact: When the economy is expanding, products with high-income elasticities will enjoy a significant increase in sales and managers must prepare for probable significant increase in sales. Strategic Management and the Income Elasticity of Demand ÿ The demand for a product that has an income elasticity of demand that is large in absolute value will vary widely with changes in income caused by economic growth and recessions. ÿ Managers can lessen the impact of economic changes on such products by limiting fixed costs so that changes in production capacity can be made quickly. ÿ Managers can forecast demand for products using the income elasticity of demand combined with forecasts of aggregate income. ÿ hI 0 for normal goods because

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Demand Theory | 23 PROBLEM SOLVED: Income Elasticity of Demand Discussion Questions 1. Suppose that a market demand function is defined as Q  20,000  8P  0.1I and suppose that P  2,000 and I  20,000. What is the income elasticity of demand at this point? Answer: Here, we have ∂Q  0.1 (a normal good). Then we have ∂I

Q  6,000, so that hI  0.1(20,000/6,000)  1/3 2. If the income elasticity of demand for a product is unitary, then a 1% change in income will change demand in the same direction by 1%. If price remains constant, then spending on the product will change by 1% and, consequently, spending on the product will be the same percentage of income after the income change as it was before. If the income elasticity of demand is greater than one, then spending will increase as a percentage of income as income increases. If it is less than one, spending will decrease as a percentage of income as income increases. How do you think the percentage of income spent on jewelry, food, clothing, housing, and automobiles responds to a 1% increase in income? 13. Cross-Price Elasticities of Demand •

Cross-Price Elasticity of Demand (hXY): The percentage change in quantity demanded of one good (QX) resulting from a 1% change in the price of a related good (PY) ÿ Income can be defined as aggregate consumer income or as per capita disposable income, depending on circumstances. ⎛ QX ⎞ ⎛ PY ⎞ ÿ  XY = ⎜ ⎟ ⎟⎜ ⎝ PY ⎠ ⎝ QX ⎠ ÿ hXY 0 if the two products are substitutes. * Example: Wheat and corn ÿ hXY 0 if the two products are complements. * Example: Computers and computer software ÿ hXY  0 if the two products are independent or unrelated. * Example: Butter and airline tickets ÿ Example: A linear demand function * Given: QX  1,000  0.2PX  0.5PY  0.04I, QX  2,000, and PY  500 * hXY  0.5(500/2,000)  0.125 so the two products are substitutes.

24 | Chapter 2 •

Strategic management and the cross-price elasticity of demand ÿ Managers can use information about the cross-price elasticity of demand to predict the effect of competitors’ pricing strategies on the demand for their product. ÿ Antitrust authorities use the cross-price elasticity of demand to determine the likely effect of mergers on the degree of competition in an industry. * A high positive cross-price elasticity, indicating that the two goods are strong substitutes, suggests that a merger would significantly reduce competition in the industry. * A high negative cross-price elasticity of demand, indicating that the two goods are strong complements, suggests that a merger might give the merged firm excessive control over the supply chain. The merged firm may refuse to sell the intermediate product to other producers.

14. The Advertising Elasticity of Demand •

Advertising Elasticity of Demand (hA): The percentage change in quantity demanded (Q) resulting from a 1% change in advertising expenditure (A) ⎛ Q ⎞ ⎛ A ⎞ ÿ A = ⎜ ⎝  A ⎟⎠ ⎜⎝ Q ⎟⎠ ÿ Example calculation: A linear demand function * Suppose: Q  500  0.5P  0.01I  0.82A and suppose, A/Q, the amount of advertising per unit of output, is $2. Then * hA  0.82(2)  1.64, indicating that a 1% increase in income results in a 1.64% increase in the quantity demanded.

15. The Constant-Elasticity and Unitary Demand Function •

Constant-Elasticity Demand Function: Mathematical form of a demand function that always yields that same elasticity, regardless of the product’s price and consumers’ income and other factors that influence demand ÿ Example: A multiplicative demand function Q  200P 0.3I 2 ÿ Price elasticity of demand   0.3 ÿ Income elasticity of demand  2.0 ÿ More generally, an example of a constant elasticity demand function is Q aP  bIc where a, b, c are parameters to be estimated. ÿ Price elasticity of demand  –b ÿ Income elasticity of demand  c

Demand Theory | 25

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ÿ Fact: For a multiplicative demand function, the exponents represent elasticities. Unitary elastic demand function and total revenue (TR) ÿ TR  PQ so if TR is constant, Q  (TR)(P  1) ÿ Price elasticity of demand   1 ÿ More generally, a unitary elastic demand curve is given by the demand function Q  mP  1 or Q  m/P ÿ Its graph: A rectangular hyperbola

Chapter 2: Problem Solutions 1. The Dolan Corporation, a maker of small engines, determines that in 2012 the demand curve for its product is P  2,000  50Q where P is the price (in dollars) of an engine and Q is the number of engines sold per month. a. To sell 20 engines per month, what price would Dolan have to charge? b. If managers set a price of $500, how many engines will Dolan sell per month? c. What is the price elasticity of demand if price equals $500? d. At what price, if any, will the demand for Dolan’s engines be of unitary elasticity? Solution: a. For Q  20, P  2,000  50(20)  $1,000. b. For P  500, we have 500  2,000  50Q. Solving for Q gives 50Q  1,500, that is, Q  30. c. h  (≠Q /≠P)(P/Q)  ( 1/50)(500/30)  1/3. d. For P  $1,000, h  (≠Q/≠P)(P/Q)  ( 1/50)(1,000/20)  1. Alternatively, set (1/50)(2,000  50Q)/Q   1. Solve for Q and then P. 2. The Johnson Robot Company’s marketing managers estimate that the demand curve for the company’s robots in 2012 is P  3,000  40Q where P is the price of a robot and Q is the number sold per month. a. Derive the marginal revenue curve for the firm. b. At what prices is the demand for the firm’s product price elastic? c. If the firm wants to maximize its dollar sales volume, what price should it charge?

26 | Chapter 2 Solution: a. TR  PQ (3,000  40Q)Q  3,000Q  40Q2. Hence, MR  dTR/dQ  3,000  80Q. b. Recall: For a linear demand curve P  a  bQ, demand is elastic if P a/2. Hence, we have P ≥ $1,500. c. Revenue is maximized at MR  0. That is, 3,000  80Q  0. Solving for Q gives Q  3,000/80  37.5 and therefore P  3,000  40(37.5), that is P  $1,500. Alternatively, one can use the fact that for a linear demand curve P  a  bQ, revenue is maximized at Q  a/2b or P  a/2. 3. After a careful statistical analysis, the Chidester Company concludes the demand function for its product is Q  500  3P  2Pr  0.1I where Q is the quantity demanded of its product, P is the price of its product, Pr is the price of its rival’s product, and I is per capita disposable income (in dollars). At present, P  $10, Pr  $20, and I  $6,000. a. What is the price elasticity of demand for the firm’s product? b. What is the income elasticity of demand for the firm’s product? c. What is the cross-price elasticity of demand between its product and its rival’s product? d. What is the implicit assumption regarding the population in the market? Solution: a. h  (≠Q/≠P)(P/Q)  ( 3)(10/1,110)   3/111 b. hI  (≠Q/≠I)(I/Q)  (0.1)(6,000/1,110)  60/111 c. hcross  (≠Q/≠Pr)(Pr /Q)  (2)(20/1,110)  4/111 d. The model for market demand implicitly assumes that the population is constant. 4. The Haas Corporation’s executive vice president circulates a memo to the firm’s top management in which he argues for a reduction in the price of the firm’s product. He says such a price cut will increase the firm’s sales and profits. a. The firm’s marketing manager responds with a memo pointing out that the price elasticity of demand for the firm’s product is about 0.5. Why is this fact relevant? b. The firm’s president concurs with the opinion of the executive vice president. Is she correct?

Demand Theory | 27 Solution: a. Whether total revenue will go up or down when the product price is lowered and more units are sold depends on whether the quantity of units sold increases by a greater percentage than the price is reduced by. That is, it depends on whether the demand is elastic or inelastic. The nature of the price elasticity of demand can be used as a test to determine the effect of a price change on total revenue. This is the so-called total revenue test. b. Assuming that the marketing manager is correct that the demand elasticity is 0.5, then demand is price inelastic and therefore a price reduction will cause the number of units sold to increase by a smaller percentage than price has fallen, and both the president and executive vice president will have egg on their faces when total revenues decline after the price is reduced. 5. Managers of the Hanover Manufacturing Company believe the demand curve for its product is P5Q where P is the price of its product (in dollars) and Q is the number of millions of units of its product sold per day. It is currently charging $1 per unit for its product. a. Evaluate the wisdom of the firm’s pricing policy. b. A marketing specialist says that the price elasticity of demand for the firm’s product is 1.0. Is this correct? Solution: a. We have TR  (5  Q)Q  5Q  Q2 . Therefore, MR  5  2Q. At P  1, Q  4 and MR  3 and total revenue decreases. The price is too low; increasing the price and selling fewer units would increase revenues. b. No, while dQ/dP 1, (dQ/dP)(P/Q)  1/4 at P  1. 6. On the basis of historical data, Richard Tennant has concluded, “The consumption of cigarettes is . . . [relatively] insensitive to changes in price. . . . In contrast, the demand for individual brands is highly elastic in its response to price. . . . In 1918, for example, Lucky Strike was sold for a short time at a higher retail price than Camel or Chesterfield and rapidly lost half its business.” a. Explain why the demand for a par ticular brand is more elastic than the demand for all cigarettes. If Lucky Strike raised its price by 1% in 1918, was the price elasticity of demand for its product greater than 2?

28 | Chapter 2 b. Do you think that the demand curve for cigarettes is the same now as it was in 1918? If not, describe in detail the factors that have shifted the demand curve and whether each has shifted it to the left or right. Solution: a. As we define a product more narrowly, consumers have better substitutes (whose prices are held constant) as the price of the good under consideration varies. This makes the demand for a good more elastic the more narrowly the good is defined. We are not told how much Lucky Strike was priced above Camel and Chesterfield, but assuming that the margin was less than 25%, we can conclude that the cross-price elasticity was greater than 2. This isn’t exactly right; we must also assume that Lucky Strike’s fall in sales resulted from a reduction in the price of Camels and Chesterfields for the cross-price elasticities as explained to students. b. Population, per capita income, and subsidized health care have all increased; this probably caused the demand curve for cigarettes to shift out, or to the right. Public health education and general education have increased; this probably shifted the demand curve in, or to the left. 7. According to S. Sackrin of the U.S. Department of Agriculture, the price elasticity of demand for cigarettes is between 0.3 and 0.4, and the income elasticity of demand is about 0.5. a. Suppose the federal government, influenced by findings that link cigarettes and cancer, were to impose a tax on cigarettes that increased their price by 15%. What effect would this have on cigarette consumption? b. Suppose a brokerage house advised you to buy cigarette stocks because if incomes were to rise by 50% in the next decade, cigarette sales would be bound to spurt enormously. What would be your reaction to this advice? Solution: a. Cigarette consumption would fall by between 4.5 and 6.0%. b. Assuming that the prices of cigarettes were to remain constant, a 50% increase in income would cause sales of cigarettes to increase 25%. The weighted average of all income elasticities equals 1, so consumption of noncigarette items would increase by more than 50% and certainly more than the 25% performance of cigarettes. I would not follow the broker’s advice. 8. A survey of major U.S. firms estimates on average, the advertising elasticity of demand was only about 0.003. Doesn’t this indicate that firms spend too much on advertising?

Demand Theory | 29 Solution: No. The fact that the elasticity of demand with respect to advertising is relatively small (0.003) does not necessarily mean that an additional dollar spent on advertising would not be profitable or that the last dollar spent was not profitable. This number indicates that a 1% increase in advertising expenditure results in a 0.003% increase in the quantity demanded of the product. 9. The McCauley Company hires a marketing consultant to estimate the demand function for its product. The consultant concludes that this demand function is Q  100P3.1I 2.3A0.1 where Q is the quantity demanded per capita per month, P is the product’s price (in dollars), I is per capita disposable income (in dollars), and A is the firm’s advertising expenditures (in thousands of dollars). a. What is the price elasticity of demand? b. Will price increases result in increases or decreases in the amount spent on McCauley’s product? c. What is the income elasticity of demand? d. What is the advertising elasticity of demand? e. If the population in the market increases by 10%, what is the effect on the quantity demanded if P, I, and A are held constant? Solution: Observation: This is a multiplicative demand function. Therefore, the exponents represent elasticities. a. The price elasticity of demand is 3.1. b. An increase in price will cause revenues to fall because the demand is elastic. c. The income elasticity of demand is 2.3. d. The advertising elasticity of demand is 0.1. e. If P, I, and A are held constant, per capita consumption, Q, is constant. Notice that if N represents the population size, aggregate market demand is given QT  100NP3.1I 2.3A0.1 indicating that the population elasticity is 1. Therefore, a 10% increase in population gives rise to a 10% increase in the quantity demanded. 10. The Schmidt Corporation estimates that its demand function is Q  400  3P  4I  0.6A

30 | Chapter 2 where Q is the quantity demanded per month, P is the product’s price (in dollars), I is per capita disposable income (in thousands of dollars), and A is the firm’s advertising expenditures (in thousands of dollars per month). Population is assumed to be constant. a. During the next decade, per capita disposable income is expected to increase by $5,000. What effect will this have on the firm’s sales? b. If Schmidt wants to raise its price enough to offset the effect of the increase in per capita disposable income, by how much must it raise its price? c. If Schmidt raises its price by this amount, will it increase or decrease the price elasticity of demand? Explain. Make sure your answers reflect the fact that elasticity is a negative number. Solution: Observation: This is a linear demand function. ∂Q a. We have  = 4 , meaning that a $1,000 increase in per capita dispos∂I able income results in a 4-unit increase in the quantity demanded of the product. Therefore, if per capita disposable income is expected to increase by $5,000, then the firm’s sales will increase by 20 units per month. b. Here, several variables are changing, not just one. Write Q  3 P  4 I  0.6 A  3 P  4 I, assuming A is constant in the analysis. Now, offsetting the effect of the increase in per capita disposable income means that the combined effects of a price increase and a per capita disposable income increase leave quantity demanded unchanged. Hence, 3 P  4 I  0. Solving for P gives P  20/3  $6.67. That is, price should be raised by $6.67 per unit. c. There are two possible interpretations, both leading to more elastic demand. You could assume that the question is asking if demand is ∂Q more elastic after both the income and price have increased. Since is ∂P unchanged and P/Q has increased, the demand will be more elastic. Alternatively, you might assume that the question is asking, as we increase the price to choke off the anticipated increase in the quantity demanded after income has gone up, does the demand become more or less elastic? This is, of course, just moving up a linear demand curve, which implies an increasingly elastic demand.

CHAPTER 3

Consumer Behavior and Rational Choice

Lecture Notes 1. Introduction •

Objectives ÿ To explain the determinants of individual consumer demand and market demand * Internal classification scheme— Consumer preference orderings or ranking * Need to allocate limited budgets— Consumer constraints * Rational behavior in economic terms— Consumers wish to maximize their well-being given their budget constraints. * Effect of risk is discussed in Chapter 14. The idea is to show how consumers make choices under risk or uncertainty. * Effect of asymmetric information is discussed in Chapters 15 and 16. * Understand Problem 11 which defines a state’s budget constraint and its priorities and requires determination of the optimal choice. ÿ To show how market demand is a horizontal summation of consumer demands ÿ To come up with a quantitative measure of consumer well-being or satisfaction or happiness or the amount of pleasure from consumption ÿ To formulate a theory or model of consumer choice (under certainty) ÿ Model’s tools: * Utility functions * Indifference curves * Budget lines ÿ To explain how managers can exert some control over demand by controlling some of its determinants

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32 | Chapter 3 * * *

Pricing Advertising Product quality

2. Indifference Curves •

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Ms. Popovich—A Running Example ÿ She lives in South Pasadena, California. We will get to know about her preferences and her responses to market incentives. ÿ She has a limited budget that she must allocate between food and clothing. * If her available budget increases, she can buy more of both goods. * If the price of one good increases, she can buy less of that good. ÿ Her utility-maximizing combination of food and clothing is found where her indifference curve is tangent to her budget line. * If she receives a coupon, how will her budget constraint change? * If she receives a quantity discount, how will her budget constraint change? * Why is she generally better off with a cash gift than a gift or gift card of equal value? * How does she balance work and leisure? * If the price of a good changes, how will her consumption bundle change? * How is her demand curve derived? Indifference Curve: A graph containing points representing market bundles among which the consumer is indifferent Key Assumption: Nonsatiation axiom: Consumers are insatiable; that is, more of a good is preferred to less of it. Utility refers to the amount of happiness or satisfaction or pleasure that a person derives from consumption of a good or service. ÿ If two combinations of two goods, that is, two bundles, yield the same level of utility, and hence are equally desirable, then they are on the same indifference curve. Characteristics of Indifference Curves (Figure 3.1) Three main characteristics: ÿ A consumer has many indifference curves. * More is preferred to less; hence indifference curves farther from the origin represent higher levels of utility. That is, market bundles on higher indifference curves are preferred to bundles on lower indifference curves. ÿ Every indifference curve must slope downward and to the right, as long as the consumer prefers more of each commodity to less.

Consumer Behavior and Rational Choice | 33 If a consumer gets more of one commodity, then in order to maintain a constant level of utility, the consumer must get less of the other commodity. * If a consumer loses some of one commodity, then in order to maintain a constant level of utility, the consumer must get more of the other commodity. ÿ Noncrossing property of indifference curves: Indifference curves cannot intersect (Figure 3.2). * This property is easily proved by contradiction. The main idea of the proof is that if they were to cross, then the nonsatiation axiom would be violated. *

3. The Marginal Rate of Substitution (MRS) •

Definition: The marginal rate of substitution of product X (measured on the horizontal axis) for product Y (measured on the vertical axis) is equal to the number of units of product Y that must be given up in order to gain an extra unit of product X and still maintain a constant level of utility. ÿ The marginal rate of substitution (at a point) is equal to the negative of the slope of an indifference curve ( at that point). MRS(x for y)   slope of IC   dy/dx ÿ Claim: The marginal rate of substitution is equal to the ratio of the marginal utility of product X to the marginal utility of product Y. MRS(x for y)  MU X / MUY To see this, we observe that the indifference curve, at level c, is given by the set of all bundles (x, y) such that U(x, y)  c, where c is a given constant. Then totally differentiating this identity gives (MU X)dx  (MUY )dy  dc  0 (because c is a constant). Then, solving for the slope gives dy/dx  MU X /MUY But, by definition, the MRS is the negative of the slope of the indifference curve. It follows that MRS(x for y)  MU X /MUY as we previously claimed. Note: The shape of an indifference curve depends on the assumptions about consumer preferences. ÿ The curved shape of an indifference curve implies that the marginal rate of substitution decreases as X increases.

34 | Chapter 3 •

Implications (Figure 3.3) ÿ If the marginal rate of substitution is large, then indifference curves are steep, and the consumer will sacrifice a large amount of product Y in order to get one more unit of product X. ÿ If the marginal rate of substitution is small, then indifference curves are flat, and the consumer will sacrifice only a small amount of product Y in order to get one more unit of product X.

4. The Concept of Utility •

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Utility: Indicates the level of enjoyment or preference attached by a consumer to a par ticular market bundle ÿ Ranks alternative market bundles ÿ The higher the level of utility assigned to a bundle, the higher the level of satisfaction the consumer realizes from it. ÿ Market bundles on the same indifference curves have the same utility. ÿ Given two market bundles, A and B, there are three possibilities: * A is preferred to B. * B is preferred to A. * Indifference: A and B are equally preferred. This is commonly called the completeness axiom or consistency axiom of consumer preferences. This axiom (or property) is required for a theory of consumer behavior. ÿ Transitivity property of preferences. It is assumed that consumer preferences are transitive. It means that: Given any three bundles of goods A, B, and C, then we have the following: * If A is preferred to B and B is preferred to C, then A must be preferred to C. Indifference curves are also known as iso-utility curves. ÿ For product Y on the vertical axis and product X on the horizontal axis ÿ Recall that: * Marginal rate of substitution   Slope of IC *  Slope of IC  Y/X   (U/X)/(U/Y)  MU X /MUY * Ms. Popovich consumes food (f, on the vertical axis) and clothes (c, on the horizontal axis), so her marginal rate of substitution is  f/c.

5. The Budget Line •

Equation ÿ For product Y (with price PY) measured on the vertical axis and product X (with price PX) measured on the horizontal axis, and income I, the budget line is: * I  YPY  XPX

Consumer Behavior and Rational Choice | 35 It indicates that the consumer spends all of her income on the two goods X and Y (nothing is saved). * Graphed as: Y  I/PY  (PX/PY)X ÿ Illustration (Figure 3.4): For product Y  food with price Pf measured on the vertical axis and product X  clothing with price Pc measured on the horizontal axis, and budget I, the budget line is: * I  YPf  XPc * Graphed as: Y  I/Pf  (Pc/Pf)X Shifting the Budget Line ÿ An increase in the price of product X, or a decrease in the price of product Y, will make the budget line pivot (or rotate) and become steeper because the slope is affected. ÿ A decrease in the price of product X, or an increase in the price of product Y, will make the budget line pivot and become flatter. ÿ An increase in budget will move the budget line further from the origin but will not change the slope (parallel shift). ÿ A decrease in budget will move the budget line closer to the origin but will not change the slope. Example (Figure 3.5): Change in budget: For product Y with price Pf measured on the vertical axis and product X with price Pc measured on the horizontal axis, and budget I. ÿ Pf  3, Pc  60, I  300 * 300  3Y  60X * Y  100  20X * Intercepts: Y  100, X  5 ÿ Pf  3, Pc  60, I  600 * 600  3Y  60X * Y  200  20X * Intercepts: Y  200, X  10 ÿ Pf  3, Pc  60, I  900 * 900  3Y  60X * Y  300  20X * Intercepts: Y  300, X  15 Example (Figure 3.5): Change in price: For product Y with price Pf measured on the vertical axis and product X with price Pc measured on the horizontal axis, and budget I. ÿ Pf  3, Pc  60, I  600 * 600  3Y  60X * Y  200  20X * Intercepts: Y  200, X  10 ÿ Pf  6, Pc  60, I  600 * 600  6Y  60X * Y  100  10X * Intercepts: Y  100, X  10 *

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36 | Chapter 3 6. The Equilibrium Market Bundle • •

Given a consumer’s indifference curves and budget line, the consumer’s equilibrium market bundle can be derived. Equilibrium Market Bundle: The market bundle that, among all the items the consumer can afford to purchase, yields the maximum utility ÿ Located on the budget line at a point that corresponds to the highest feasible indifference curve ÿ Two aspects of the consumer’s market bundle: * It can be determined by the tangency condition: It is given by the point on the budget line at which MRS  Price ratio  PX /PY, for any two goods X and Y. Illustration (Figure 3.7) * By corner solutions Illustration (Figure 3.8)

STRATEGY SESSION: The Effect of a Time Constraint on Consumer Behavior Discussion Questions 1. Suppose that Mildred Evans has a high marginal rate of substitution. Is she more likely to choose an equilibrium bundle that is located on line segment AE or EC? Answer: Line segment EC. 2. Suppose that Mildred Evans has a low marginal rate of substitution. Is she more likely to choose an equilibrium bundle that is located on line segment AE or EC? Answer: Line segment AE. 3. What would happen to Mildred’s time constraint if a new football stadium was built near her home and the time required to attend a football game fell to 4 hours? How would this be likely to affect her equilibrium bundle? Answer: The vertical intercept of the time constraint would increase from 4 to 6. The time constraint would become steeper. Line segment AE would be longer and line segment EC would be shorter. If her equilibrium bundle was on segment AE, but not at point E, then it would be unchanged. If her equilibrium bundle was on line segment EC, including point E, then her new equilibrium would include more baseball games and could include more or fewer football games.

Consumer Behavior and Rational Choice | 37 7. Maximizing Utility: A Closer Look •

The equilibrium bundle in Figure 3.7 is found where an indifference curve is tangent to the budget line. ÿ Marginal rate of substitution  MRS   Pc /Pf , for two goods, Clothing and Food. ÿ MRS is the rate at which the consumer is willing to trade one good for the other while holding utility constant. Example: MRS  4 means that the consumer is willing to give up 4 pounds of food for an extra piece of clothing. ÿ The slope of the budget line (Pc/Pf) is the rate at which the consumer is able to trade one good for the other at market prices. For example, if Pc /Pf  4, this means that the consumer must give up 4 pounds of food to obtain one more piece of clothing.

STRATEGY SESSION: A Manager’s Trade-Off Between Output and Profit Discussion Questions 1. Do you think that the type of behavior described in this problem could be detected by shareholders? Why or why not? 2. Do you think that the owner of a company who is also its manager would be likely to engage in this type of behavior? Why or why not? 8. Corner Solutions •

Occur when a consumer chooses to consume zero units of a good (Figure 3.8) ÿ It implies that the price of the good exceeds the value to the consumer. ÿ Indifference curve intersects, but is not tangent to, the budget line at either its vertical or horizontal intercept. ÿ The marginal rate of substitution is not equal to the slope of the budget line.

9. How Managers Can Strategically Influence Consumer Choices •

•

Managers can change consumer choices by influencing preferences through advertising strategies and by influencing the budget line through pricing strategies. Example: Coupon that offers a discount of $18 to food customers who spend at least $180 in one store ÿ Figure 3.9 shows the equilibrium bundle in the absence of the coupon on a graph that assumes that the budget is $200 and prices of clothing (Y) and of food (X) are both equal to $1.

38 | Chapter 3

Total Cost of Donuts 4.5 4 3.5 Total Expenditure

•

ÿ Figure 3.10 shows a budget line that represents the effect of the coupon offer. * The budget constraint is shifted upward by 18 at X  180. * The equilibrium bundle that is determined by indifference curve U0' is not influenced by the coupon. * The equilibrium bundle that is determined by indifference curve U0 is influenced by the coupon, and the new equilibrium bundle is located on the higher indifference curve, U1. Application: Quantity discount (Figure 3.11) ÿ If Ms. Popovich buys from one to three donuts, they cost $0.50 each. Six donuts cost a total of $2.00. Seven donuts cost $2.00 plus $0.50. A dozen donuts cost $3.00. Ms. Popovich has $4.00 to spend on donuts or on “all other things.” The relationship between total expenditure and number of donuts consumed is shown below.

3 2.5 2 1.5 1 0.5 0 0

2

4

6

8

10

12

14

16

Donuts

The budget line under quantity discount pricing, as shown in the figure, is a step function. * The budget line under constant pricing, with a price of $0.50 per donut, has a vertical intercept of 4 and a horizontal intercept of 8. * If Ms. Popovich has indifference curve U0', then a change from constant pricing to quantity discount pricing will have no effect on the equilibrium bundle. * If Ms. Popovich has indifference curve U0, then a change from constant pricing to quantity discount pricing will cause a change *

Consumer Behavior and Rational Choice | 39

•

•

in the equilibrium bundle from 4 donuts to 7 donuts, which is on higher indifference curve U1. Application: Why most individuals prefer cash to a specific gift (Figure 3.12) ÿ Ms. Popovich is maximizing utility at the tangency between her budget line (I0) and indifference curve U0. The vertical axis represents all other goods and the horizontal axis “gift” goods. ÿ Budget line I0  PG represents the effect of receiving one unit of the gift good in the form of cash. ÿ If Ms. Popovich’s preferences include indifference curve U1', then the gift will provide her with exactly the same amount of utility as a cash gift of equal value. ÿ If Ms. Popovich’s preferences include indifference curve U2, then one unit of the gift good would leave her utility unchanged, while a cash gift would increase her utility. ÿ In the case described above, a cash gift with value A1  A0 would provide more utility (increase from U0 to U1) than a gift valued at A2  A0, which would leave utility unchanged. ÿ Love: This analysis does not account either for the sentimental value of a gift to the recipient or for the time and effort expended by the giver in acquiring it. Example: Gift cards instead of cash ÿ At best, they are not universally accepted. ÿ At worst, they tie the recipient to a par ticular store. ÿ They’re like money but not as useful because they are not as fungible.

PROBLEM SOLVED: Do I Stay or Do I Go? Use of Indifference Curves *

* * * *

Describes the trade-off between hours of leisure (HL) and income (I), defined as the wage rate (W) times hours of work (H W), in terms of the budget constraint I  24*WW*HL , with slope equal to W. The slope of an indifference curve is equal to MUL/MUI. Equilibrium is located where MUL/MUI  W. The figure in the text shows the effect of different values of W on hours worked. Because leisure is a normal good, more of it will be desired relative to income as income rises. If income rises far enough, as shown in the figure (shift from U3 to U4), then additional income (a higher W) can lead to an increase in the consumption of leisure and a reduction in hours worked.

40 | Chapter 3 Discussion Questions 1. Identify the two corner solutions to this problem. Discuss the circumstances under which each might be reasonable. Answer: Retirement or pulling an all-nighter, but the latter is not sustainable. 2. Suppose that compensation increases from W to 1.5 * W if Ms. Popovich works more than 8 hours. What does the new budget constraint look like? Answer: It has a kink at Leisure  16. To the right of the kink, the wage is W. To the left of the kink, the wage is 1.5 * W. The vertical intercept is I  8 * W  16 * W * 1.5. See below. Budget Constraint: W = 10 350 300

Income

250 200 150 100 50 0 0

5

10

15

20

Leisure Hours

10. Deriving the Individual Demand Curve •

•

•

A consumer’s demand curve shows how much of a product that a person will purchase during a given time period at various prices while all other variables (income, other prices, etc.) are held constant. Figure 3.14 shows the effect of a change in the price of food on Ms. Popovich’s equilibrium market bundle in terms of indifference curves. The results shown keep clothing consumption constant, which is not what would typically be expected. Figure 3.15 shows the same results in the form of a demand curve for food.

Consumer Behavior and Rational Choice | 41 11. Deriving the Market Demand Curve •

• • •

• • •

The market demand curve is a summary of all of the demand curves of individual consumers for a good in the market. It is obtained by horizontal summation. Horizontal summation adds the quantities demanded by all consumers at each market price. A market is generally composed of buyers with different tastes and preferences (nonhomogeneous or heterogeneous buyers). Managers can strategically exploit this heterogeneity by identifying submarkets and charging each submarket a different price. This is called price discrimination. An increase in the number of consumers in a market will cause market demand to increase. The market demand curve is the same as a firm’s demand curve only in the case of monopoly because there is only one seller on the market. The market demand curve is not the same as a firm’s demand curve if the firm is not a monopolist.

12. Consumer Surplus • •

• • •

•

Consumer’s Reservation Price: The price at which the consumer values each number of units demanded. Also known as willingness to pay (WTP). Consumer Surplus: The difference between what an individual is willing to pay (the reservation price) and what that individual has to pay (the market price) for a product. Marginal buyers, who pay a price equal to their reservation price, do not capture any consumer surplus. Consumer surplus is, along with producer surplus, a measure of the efficiency of markets and social benefits of market transactions. Managers can develop pricing strategies that allow them to capture consumer surplus by charging different prices to individuals or groups of individuals that are closer to their reservation prices. Figure 3.16 shows a demand curve with consumer surplus labeled as area A.

STRATEGY SESSION: The Trade-Off between Risk and Return Discussion Questions 1. Most people are described as risk averse. Do risk-averse people make risky investments? 2. Interpret the marginal rate of substitution for the risk-return indifference curves.

42 | Chapter 3 Chapter 3: Problem Solutions 1. The market for sports per for mance drinks experienced a big shift in 2012 as sales of low-calorie sports drinks grew by over 25%. Many attributed this shift to greater use by women who wanted a sports drink without many calories. a. If a woman desires two containers of low-calorie sports drink as much as one container of high-calorie sports drink, what do her indifference curves (between low- and high-calories sports drinks) look like? b. Do they have the typical shape of indifference curves? Why or why not? Solution: a. It is linear with a slope  (Low-Calorie)/(High-Calorie)  2. The reason is that the MRS is 2 anywhere on an indifference curve, and therefore the slope of the indifference curves is 2. An example of her indifference curve is shown below. Indifference Curve 12

Low-Calorie

10 8 6 4 2 0 0

1

2

3

4

5

6

High-Calorie

b. The shape is not typical. Indifference curves do not generally have a constant marginal rate of substitution. Often, for theoretical convenience, it is assumed that preferences are strictly convex, to convey the idea that averages are preferred to extremes. But this type of indifference curve indicates a legitimate type of preference. Note: Diminishing marginal utility alone is not, in general, sufficient to ensure diminishing marginal rate of substitution.

Consumer Behavior and Rational Choice | 43 2. In recent years fresh bagel sales have been growing at about 30% per year. Once considered an ethnic food to be eaten with cream cheese and lox, bagels now “have become the new donut to bring to the office,” according to Michael Goldstein of Goldstein’s Bagel Bakery in Pasadena, California. But one problem with bagels is that they get stale fast. In the words of Ray Lahvic, editor emeritus of Bakery Production and Marketing, “the worst thing in the world is a day-old bagel.” If a market researcher asserts that the slope of the typical consumer’s indifference curves between fresh bagels and day-old bagels is 1, would you agree with this assertion? Why or why not? Solution: No. If the slope of the consumer’s indifference curve is 1, and therefore the MRS is 1, then the consumer is willing to trade one fresh bagel for one dayold bagel, and the goods would be perfect substitutes for the consumer. Rather, we would expect a consumer would be willing to trade several dayold bagels for a single fresh bagel, in which case the slope of the indifference curve would not be 1. 3. On a piece of graph paper, plot the quantity of lamb consumed by Ms. Turner along the vertical axis and the quantity of rice she consumes along  the horizontal axis. Draw the indifference curve that includes the following market bundles. Each of these market bundles gives equal satisfaction: Market Bundle

Lamb (Pounds)

Rice (Pounds)

1 2 3 4 5 6 7 8

2 3 4 5 6 7 8 9

8 7 6 5 4 3 2 1

44 | Chapter 3 Solution:

4. In the previous question, what is the marginal rate of substitution of rice for lamb? How does the marginal rate of substitution vary as Ms. Turner consumes more lamb and less rice? Is this realistic? Solution: Observe that the indifference curve is basically a straight line. Let Y  amount of lamb (in pounds) an X  amount of Rice (in pounds). Take the points (or bundles): (8, 2) and (2, 8) (or any other 2 points on the line). Slope  (8  2)/(2  8)  1. Since the slope of the IC is constant at any point on the indifference curve, then MRS  1 anywhere on the IC. That is, the marginal rate of substitution of rice for lamb is 1 and is constant. This means that Ms. Turner’s indifference curve is linear. This is not realistic. It is more likely that, because of diminishing marginal utility, the marginal rate of substitution will not be constant. 5. Suppose Richard has an after-tax income of $500 per week and must spend it all on food or clothing. If food is $5 per pound and clothing is $10 per piece, draw his budget line on a piece of graph paper, where the amount of food is measured along the vertical axis and the amount of clothing is measured along the horizontal axis.

Consumer Behavior and Rational Choice | 45 Solution: The equation of the budget line is given by 5F  10C  500, where C  amount of clothing and F  amount of food purchased. If F  0, then C  50. Similarly, if C  0, then F  100. The budget line is obtained by connecting these two points.

6. In the previous problem, what is the budget line if Richard’s weekly income increases to $600? What is his budget line if his income is $500, but the price of food increases to $10 per pound? What is his budget line if his income is $500, but the price of clothing increases to $20 per piece? Draw each of these budget lines on the piece of graph paper used in the previous problem. Solution: If income increases to $600, the equation of the budget line is given by 5F  10C  600. That indicates a parallel shift of the budget line to the right given the slope remains the same. If income is $500 but the price of food increases to $10 per pound, then the equation of the budget line is given by 10F  10C  500. Slope  1. The previous budget line will rotate inward or counterclockwise. If the price of clothing increases to $20 per piece, then the equation of the budget line is given by 10F  20C  500. Equivalently, the equation is F  50  2C. Slope  2. The original budget line rotates clockwise. In general, we observe that when income increases, the budget line shifts out from the origin with constant slope. When the price of one good changes, the budget line pivots, with the intercept of the good with the changing price changing.

46 | Chapter 3

7. Maria has budgeted a total of $9 to spend on two goods: chips and salsa. She likes to consume a unit of chips in combination with a unit of salsa. Any unit of chips that she cannot consume in combination with a unit of salsa is useless. Similarly, any unit of salsa that she cannot consume in combination with a unit of chips is useless. If the price of a unit of chips is $0.50 and the price of a unit of salsa is $0.10, how many units of each good does she purchase? Solution: Maria will maximize utility at point A, where she purchases 15 units of both chips and salsa. Note that her indifference curves are 90-degree angles.

Consumer Behavior and Rational Choice | 47 8. In the following diagram, we show one of Jane’s indifference curves and her budget line. a. If the price of good X is $100, what is her income? b. What is the equation for her budget line? c. What is the slope of her budget line? d. What is the price of good Y? e. What is Jane’s marginal rate of substitution in equilibrium?

Solution: Notice that for two goods X and Y, the budget line is given by the equation PX X  PY Y  I , or Y  I/PY  (PX/PY)X. Now, if we set X  0, then the vertical intercept is given by Y  I/PY. Similarly, setting Y  0 gives the horizontal intercept X  I/PX . Given the above information, we have that I/PX  40 and I/PY  80. Now, a. If PX  $100, then we have I/100  40, that is, I  $4,000. b. Now, 4,000/PY  80, PY  50. The equation is given by X  40  (PY /PX)Y  40  (0.5)Y. c. The slope of the budget line is PY/PX  0.5. d. PY  $50  I/Y  4000/80. Recommendation: Put good X on the horizontal axis and good Y on the vertical axis. e. The MRS is equal to the price ratio in equilibrium. Therefore, we have MRS  PY/PX  $60/$100  0.5. Note: If we have good X on the horizontal axis and good Y on the vertical axis, then, in equilibrium, the MRS is given by MRS  PX/PY. 9. Sarah has $300 to allocate between opera tickets and movie tickets. The price of each opera ticket is $60, and the price of each movie ticket is $6.

48 | Chapter 3 Her marginal rate of substitution of opera tickets for movie tickets equals 5, regardless of what market bundle she chooses. How many opera tickets does she purchase? Solution: Let X  number of opera tickets and Y  number of movie tickets. The equation of the budget line is given by 60X  6Y  300, or, equivalently, Y  50  10X. The slope of the budget line is 10. Now, given the MRS is constant at 5, then the slope of the indifference curves is 5 at any point on the indifference curves. It follows that the indifference curves are straight lines with slope 5. Therefore, the budget line is steeper than the indifference curves. A graphical solution is given below. It shows that the corner solution is on the vertical axis. Thus, we see that the equilibrium solution is a corner solution, meaning that she will spend the entire $300 on movie tickets and none on opera tickets. This indicates that she will buy 50 movie tickets and no opera tickets. Y

50

BL IC6 IC5

IC1

IC4

5

X

10. Suppose Milton has $50 to be divided between corn and beans and that the price of beans is $0.50 per pound. What will be the relationship between the price of corn and the amount of corn he will buy if U  log Qc  4 log Qb, where U is his utility, Qc is the quantity of corn he consumes (in pounds), and Qb is the quantity of beans he consumes (in pounds)? Solution: The tangency condition is given by MRS  Pc /Pb. We have MRS  MUc /MUb. But we have

Consumer Behavior and Rational Choice | 49 MUc  1/Qc and MUb  4/Qb. Therefore, we obtain MRS  Qb/(4Qc). Thus, the tangency condition implies that PbQb  4PcQc. Moreover, the budget line is given by I  PbQb  PcQc Substituting for PbQb and taking into account that I  50 gives 50  4PcQc  PcQc so that Qc  10/Pc  10P1 . This is actually the c demand function or demand curve for corn. The price elasticity of demand is 1, and the graph is a rectangular hyperbola. 11. The state of New York received $3 billion (from federal sources and a state petroleum tax) to be spent on highways and/or mass transit (subways, buses, and urban rail lines), both of which could be used to meet the transportation needs of the state’s population. a. If each mile of mass transit costs $20 million, what is the maximum number of miles of mass transit that these funds would have enabled the state to construct? b. If each mile of highway costs $10 million, what is the maximum number of miles of highways that these funds would have enabled the state to construct? c. If the number of miles of mass transit constructed is put on the vertical axis of a graph and the number of miles of highways constructed is put on the horizontal axis, can a budget line (showing the maximum number of miles of mass transit that can be constructed, given each number of miles of highways constructed) be drawn for the state? If so, what is the slope of this budget line? (Assume that the $3 billion is the only source of funds for mass transit or highway construction.) d. If the public and the state government agree that every extra mile of mass transit adds three times as much to the state’s transportation capability as an extra mile of highways, how much of the $3 billion should be spent on mass transit if the objective is to maximize transportation capability? Solution: One can see the $3 billion as the main budget or the main source of income for the state and highways and mass transit as two goods or commodities. a. 3,000/20  150 b. 3,000/10  300 c. See the diagram below.

50 | Chapter 3 d. The state of New York should spend all of its money on mass transit (unless there are diminishing returns to investing in it) to maximize transportation capability. Mass Transit

150

0

150

300

Highways

CHAPTER 4

Estimating Demand Functions

Lecture Notes 1. Introduction •

Objectives ÿ To investigate techniques to estimate a product’s demand function Three Techniques of Interest: * Consumer surveys * Market experiments * Regression analysis ÿ To investigate a demand curve for a par ticular product ÿ To offer an introduction to complex questions and some of the statistical techniques designed to provide firms with information about markets ÿ To discuss some of the important problems in regression analysis such as * multicollinearity * serial correlation (or autocorrelation)

2. The Identification Problem • • •

It is not always easy to obtain estimates. In a competitive market, price is determined by both the demand and supply curves for a product. The demand and supply curves may change from one period to the next. ÿ The idea: Because a manager does not necessarily hold constant a variety of nonprice variables such as consumer tastes, income, the price of other goods, and advertising, she cannot be sure that the demand curve was fixed during the period when the measurements were made, and therefore the ceteris paribus assumption is violated. Unless all variables plausibly believed to affect the quantity demanded are

51

52 | Chapter 4

•

included in the demand estimation, the true demand curve may be significantly different from the demand curve one estimates. ÿ An important question: How can managers estimate a demand curve if it has not remained fixed in the past? Main tools of analysis: ÿ Consumer interviews or surveys ÿ Market experiments ÿ Econometric techniques

3. Consumer Interviews •

•

The idea: ÿ To obtain information concerning the demand function for a particular product, managers frequently interview consumers and administer questionnaires concerning their buying habits, motives, and intentions. ÿ Managers may also run focus groups in an attempt to discern consumers’ tastes. Drawback: Directly asking people how much they would buy of a par ticular product at par ticular prices does not in practice work very well, given that the answers provided by consumers are not very accurate. ÿ Consumer surveys and focus groups may have an impact on presidential politics. Former president George H. W. Bush’s reverence for the flag and the Pledge of Allegiance was used in the 1988 campaign after his advisers found that focus groups responded positively to those “hot buttons.” ÿ Despite its limitations, many managers believe that the technique can reveal a great deal of information about consumer preferences.

4. Market Experiments •

• • •

Involve selective variations in prices or other marketing techniques in otherwise similar test markets to try to ascertain how variations in prices, advertising, or packaging might affect sales ÿ The idea: To vary the price of the product while attempting to keep other market conditions fairly stable or to take changes in other market conditions into account Controlled laboratory experiments can sometimes be carried out: Consumers are given money and told to shop in a simulated store. Drawback: Consumers participating in such an experiment know that their actions are being monitored, and this can affect their behavior. Decision on carrying market experiment must be based on cost-benefit analysis. ÿ The idea: Direct experimentation can be expensive or risky because consumers may be lost and profits cut by the experiment.

Estimating Demand Functions | 53 ÿ Example: Raising the price of a product as part of a market experimentation: Potential buyers may be driven away. 5. Regression Analysis •

•

•

Regression analysis is concerned primarily with the study of the relationship between one variable called the explained (or dependent) variable and one or more other variables called independent or explanatory variables. ÿ Example: Relationship between the level of sales of a par ticular product and advertising expenditure incurred on that product Object of interest: Estimating a demand curve or demand function ÿ Note: Regression analysis is not limited just to estimating economic relationships. A Regression Model ÿ Suppose a firm’s demand function is given by Y  A  B1X  B2P  B3I  B4Pr

•

•

(4.1)

where Y  Quantity demanded of the firm’s product (the dependent variable) X  Marketing expense (such as advertising) P  Price I  Consumer disposable income Pr  Price of rival brands are the independent variables. ÿ Managers need estimates of the model’s parameters A, B1, B2, B3, B4. ÿ Need data on Y, X, P, I, and Pr. Two aspects of regression analysis: ÿ 1. Simple linear regression analysis ÿ 2. Multiple linear regression analysis ÿ Two special cases of interest * Case 1: Marketing expense is the only factor to significantly affect demand (simple linear regression model). * Case 2 (more realistic): Quantity demanded is affected by two factors: advertising expense and price (multiple linear regression model). Simple Linear Regression Model ÿ Model: A simplified or idealized representation of reality. ÿ Simple linear regression analysis assumes that the mean value of Y, given the value of X, is a linear function of X: Yi  A  BXi  ei where Yi  the ith observation on the dependent variable Y

(4.2)

54 | Chapter 4

•

Xi  the ith observed value of the independent variable ei  error term Equation 4.2 is called the population regression line or true regression line. ÿ Assumptions: * The values of ei are independent and have zero mean. * The error terms ei have the same variance. This is called homoscedasticity. ÿ Goal: To obtain estimates of the parameters A and B. Call them a and b ÿ Estimation method: Method of least squares   a  bX. ÿ Estimates a and b lead to sample regression equation Y  ÿ Define residuals ei  Yi  Y I  observed–predicted. ÿ Sum of squared residuals ∑in (Yi − Yi )2 = ∑in (Yi − a − bX i )2 ÿ Goal: To choose a and b to minimize the sum of squared residuals. The idea is to differentiate the above sum of squared residuals with respect to a and b and set the partial derivatives equal to 0. These are called first-order conditions in unconstrained optimization theory. ÿ Formulas for a and b are given in equations 4.6 and 4.7 or the alternative formula. ÿ a: intercept coefficient and b: slope coefficient. Example: ÿ Data: From Table 4.1   2.536  1.504X ÿ Sample regression line or regression of Y on X: Y Predicted value of sales given that $4 million are devoted to market  2.536  1.504(4)  8.55 million units. ing expense: Y

6. Coefficient of Determination •

Question: How can managers measure how well a regression line fits the data? ÿ Most commonly used measure of goodness of fit of a regression line: the coefficient of determination, R-Sq. * Formulas exist for the coefficient of determination, but it is usually reported in computer printouts. ÿ Facts: * R-Sq is always between 0 and 1. 2 * The closer R-Sq is to 1, the better the fit. That is, an R near 1 means that the data are bunched around the sample regression line. * The closer R-Sq is to 0, the poorer the fit. * R-Sq  1, the fit is perfect.

Estimating Demand Functions | 55 7. Multiple Regression • •

•

•

Includes two or more independent variables Usually carried out with the help of statistical software packages such SAS and Minitab (and a lot more) or EXCEL ÿ SAS and Minitab are expensive software, and therefore not all students have access to them. But the good news is that there is some free online available software that is even more friendly to use. One example is GRETL. It can be easily downloaded online. Two important steps in using multiple regression analysis: ÿ To identify the dependent and independent variables ÿ To specify the mathematical form of the equation relating the (conditional) mean of the dependent variable to the independent variables Illustration: The Miller Pharmaceutical Company ÿ Suppose the firm’s manager feels that its sales depend on its price, as well as on its own marketing expense. That is, assume the following model: Yi  A  B1Xi  B2Pi  ei

(4.10)

where Xi  marketing expense during the ith year (in millions of dollars) Pi  price (in dollars) of the product during the ith year. Yi  firm’s sales in the ith year. ÿ Goal: Use multiple linear regression analysis to estimate the unknown constants A, B1, and B2. ÿ Assuming estimated parameter values are: a, b1, b2, respectively, obtain the estimated regression line

•

i  a  b X  b P Y 1 i 2 i  )2 = ∑ n (Y − a − b X − b P )2 ÿ Sum of squared residuals: ∑in (Yi − Y i 1 i 2 i i i ÿ Goal: To choose a, b1, and b2 that minimize the sum of squared residuals. This is the essence of the method of least squares. This is done by taking the first partial derivatives of the above expression with respect to the parameters and setting them equal to zero. That leads to solving a system of three equations with three unknowns. ÿ Fact: The estimated parameters or least-squares estimators are usually given by software packages. Example: Using data from Table 4.3 ÿ Computer printout indicates that b2  0.352 a  2.529 b1  1.758   2.529  1.758X  0.352P (4.12) ÿ Estimated regression equation: Y i i i ÿ The slope coefficient b1  1.758 indicates that a $1 increase in marketing expense results in an increase in estimated sales of 1.758 million units.

56 | Chapter 4 ÿ Goodness of fit can be measured using the “multiple correlation coefficient,” denoted by R squared: R2 , also reports by computer printouts. Analyzing Overall Fit: The F Statistic • The idea: To know whether the independent variables simultaneously influence the dependent variable • Application: The marketing director may ask whether the data indicate that either marketing expense or price really influences the firm’s sales. ÿ Tool used: The F statistic ÿ Computer printout gives F  525.72. • Facts: ÿ Large values of F tend to imply that at least one of the independent variables has an effect on the dependent variable. Testing Hypothesis about Individual Regression Coefficients: t Statistic • The rationale: The least-squares estimator of the parameter B1, for example, varies from one sample to another and by chance may be positive even if the true value of B1 is zero. • The interest lies in whether a par ticular independent variable influences the dependent variable. • Question: Does the amount allocated to marketing expense affect the firm’s sales? In fact, the analyst wants to test whether the true value of B1 is zero. • Tool used: The t statistic • Computer printouts report the t statistic. • All other things being equal, the higher the value of the t statistic (in absolute value), the smaller the probability that the true value of the regression coefficient in question really is zero. 8. Multicollinearity •

Question: What happens if independent variables are correlated? ÿ Multicollinearity: A situation in which two or more independent variables are very highly correlated. This is called the multicollinearity problem. ÿ Perfect multicollinearity: Exact linear relationships among the independent variables * Rarely encountered in practice. But near or very high multicollinearity where independent variables are approximately linearly related frequently arises in applications. ÿ If multicollinearity exists in the data, it is impossible to estimate the parameters with accuracy. ÿ Causes the t statistic to report that the individual coefficients are not significantly different from 0, whereas the F statistic reports that the right-hand-side variables taken as a group are statistically significant. ÿ There are remedies when multicollinearity is present.

Estimating Demand Functions | 57 9. Serial Correlation • • •

•

• •

Classical linear regression analysis assumes that the error terms are independent. The point: The error terms may not be independent but are serially correlated. Illustration: ÿ If the error term in one period is positive, the error term in the next period is almost always positive. ÿ If the error term in one period is negative, the error term in the next period is almost always negative. To see whether serial correlation is present, the test statistic used is the Durbin-Watson test statistic (see equation 4.15). It is based on the residuals. One way to deal with serial correlation is to take first differences of all the independent and the dependent variables. Two remedies: ÿ Weighted least squares ÿ Make a transformation on the dependent variable, such as a log transformation.

10. Further Analysis of the Residuals • •

Residuals, ei can be used to test whether some of the assumptions underlying classical linear regression analysis are met. Residual plots are frequently used as a detecting device.

Chapter 4: Problem Solutions 1. The Klein Corporation’s marketing department, using regression analysis, estimates the firm’s demand function, the result being Q  104  2.1P  3.2I  1.5A  1.6Z R2  0.89 Standard error of estimate  108 where Q is the quantity demanded of the firm’s product (in tons), P is the price of the firm’s product (in dollars per ton), I is per capita income (in dollars), A is the firm’s advertising expenditure (in thousands of dollars), and Z is the price (in dollars) of a competing product. The regression is based on 200 observations. a. According to the statistical software, the probability is 0.005 that the t statistic for the regression coefficient of A would be as large (in absolute

58 | Chapter 4 terms) as it is in this case if in fact A has no effect on Q. Interpret this result. b. If I  5,000, A  20, and Z  1,000, what is the Klein Corporation’s demand curve? c. If P  500 (and the conditions in part b hold), estimate the quantity demanded of the Klein Corporation’s product. d. How well does this regression equation fit the data? Solution: a. Advertising almost certainly has a nonzero effect on Q. There is only a 1 in 200 chance of such a high t statistic if in fact Q and A are not related. If Klein can be sure that Q is not causing A, then it should conclude that increases in A will, other things being equal, increase Q. b. Q  104  2.1P  3.2(5,000)  1.5(20)  1.6(1,000)  17,536  2.1P c. P  500 → Q  17,526  1,050  16,476 d. R2  .89; the regression explains 89% of the variation in Q. Therefore the regression seems to fit the data well enough. But we have no information on whether or not this is the best fit. 2. Since all the Hawkins Company’s costs (other than advertising) are essentially fixed costs, it wants to maximize its total revenue (net of advertising expenses). According to a regression analysis (based on 124 observations) carried out by a consultant hired by the Hawkins Company, Q  23  4.1P  4.2I  3.1A where Q is the quantity demanded of the firm’s product (in dozens), P is the price of the firm’s product (in dollars per dozen), I is per capita income (in dollars), and A is advertising expenditure (in dollars). a. If the price of the product is $10 per dozen, should the firm increase its advertising? b. If the advertising budget is fixed at $10,000 and per capita income equals $8,000, what is the firm’s marginal revenue curve? c. If the advertising budget is fixed at $10,000 and per capita income equals $8,000, what price should managers charge? Solution: The goal here is, given the estimated demand function, that the manager wants to maximize the firm’s total revenue, net of advertising expense. a. Total revenue less advertising expenditures is TR  A  PQ  A  23P  4.1P2  4.2IP  3.1AP  A  23P  4.1P2  4.2IP  (3.1P  1)A  640  42I  30A if P  10

Estimating Demand Functions | 59 (TR  A)/A  30 0 if P  10. It indicates that an increase in advertising expenditures will cause net revenue to increase, given that P  $10 per dozen. More importantly, it says that a $1 increase in advertising expenses will lead to a $30 increase in net revenues, assuming the price is fixed at $10 per dozen. Therefore, the firm should increase advertising. b. We have Q  23  4.1P  4.2(8,000)  3.1(10,000)  64577  4.1P or equivalently P  (1/4.1)(23  Q  33,600  31,000)  15,750.4878  0.2439Q Therefore, total revenue is given by TR  PQ  15,750.4878Q  0.2439Q2 and marginal revenue is given by MR  15,750.49  0.4878Q c. Assume the manager wants to maximize total revenue. Total revenue is  maximized at the point where marginal revenue equals zero. Given the  levels of advertising and per capita income, set MR  0; that is, 15,750.48  0.4878Q  0. Solving for Q gives Q  15,750.4878/0.4878  32,289; this implies P  15,750.4878  0.2439(32,289)  7,875. 3. The 2012 sales and profits of seven clothes companies were as follows: Firm

Sales ($ billions)

Profit ($ billions)

5.7 6.7 0.2 0.6 3.8 12.5 0.5

0.27 0.12 0.00 0.04 0.05 0.46 0.00

Maxx Bleu Goden Triex Chateau L&T Eastview

a. Calculate the sample regression line, where profit is the dependent variable and sales is the independent variable. b. Estimate the 2012 average profit of a clothing firm with 2012 sales of $0.2 billion. c. Can this regression line be used to predict a clothing firm’s profit in 2026? Explain. Solution: a. Let Y denote profit and X denote sales. The regression line is given by Y  a  bX , where

60 | Chapter 4 b=

∑( X i − X mean )(Yi − Ymean )

( Xi − X mean )

2

=

4.278429 = 0.035609 120.1487

a  Ymean  bXmean  0.134286  0.035609(4.285714)   0.1833. Of course, one can use the alternative formula for b. For that we have X  30.0

Y  0.94

One obtains b =

X 2  248.72

XY  8.307

n7

29.949 = 0.1356 941.04

b. X  2 → Y  0.052893. That is, about 0.05 billion dollars expected profit. c. The data sample is a cross section of firms in 2012. It might be used to predict the profits of another steel firm in the same year but not to predict the profits of even one of the firms from this sample in a later time period, since the industry structure, technology, costs, and prices of important inputs have likely changed over the years. 4. The Cherry Manufacturing Company’s chief engineer examines a random sample of 10 spot welds of steel. In each case, the shear strength of the weld and the diameter of the weld are determined, the results being as follows: Shear strength (pounds)

Weld diameter (thousandths of an inch)

680 800 780 885 975 1,025 1,100 1,030 1,175 1,300

190 200 209 215 215 215 230 250 265 250

a. Does the relationship between these two variables seem to be direct or inverse? Does this accord with common sense? Why or why not? Does the relationship seem to be linear? b. Calculate the least-squares regression of shear strength on weld diameter. c. Plot the regression line. Use this regression line to predict the average shear strength of a weld 1/5 inch in diameter. Use the regression line to predict the average shear strength of a weld 1/4 inch in diameter.

Estimating Demand Functions | 61 Solution: a. The best way to answer the question is to use a scatterplot diagram. However, consistent with commonsense expectations, larger tested weld diameters with a few exceptions have greater strength. b. One can easily use Excel to obtain XY  2,219,370 X 2  506581

X  2239 Y 2  95062500

Y  9750

Therefore, the slope parameter, b, is given by b=

10(2219370) − (2239)(9750) 363450 = = 6.898 10(506581) − (2239)(2239) 52689

The intercept is given by a  Ymean  bXmean  975  6.898(223.9)  569.468 Also, we have R-Sq  0.758459 Regression equation: Y  569.468  6.898X c. Actual and fitted y versus x 1300 1200

actual fitted

1100

y

1000 900 800

700 600

190

200

210

220

230 x

We have X  200 → Y  810.14. Also, when X  250 → Y  1,155.04.

240

250

260

62 | Chapter 4 5. The Kramer Corporation’s marketing manager calculates a regression, where the quantity demanded of the firm’s product (designated as “C1”) is the dependent variable and the price of the product (designated as “C2”) and consumers’ disposable income (designated as “C3”) are independent variables. The Minitab printout for this regression follows: MTB > regress c1 on 2 predictors in c2 and c3 The regression equation is C1  40.8  1.02 C2  0.00667 C3 Predictor Coef Constant 40.833 C2 1.02500 C3 0.006667

Stdev 1.112 0.06807 0.005558

s  1.361 R-sq  91.62% Analysis of Variance SOURCE Regression Error Total

DF 2 21 23

SS 422.92 38.92 461.83

SOURCE C2 C3

DF 1 1

SEQ SS 420.25 2.67

a. b. c. d. e. f.

t-ratio 36.74 15.06 1.20

p 0.000 0.000 0.244

R-sq(adj)  90.8% MS F p 211.46 114.11 0.000 1.85

What is the intercept of the regression? What is the estimated regression coefficient of the product’s price? What is the estimated regression coefficient of disposable income? What is the multiple coefficient of determination? What is the standard error of estimate? What is the probability that the observed value of the F statistic could arise by chance, given that neither of the independent variables has any effect on the dependent variable? g. What is the probability, if the true value of the regression coefficient of price is zero, that the t statistic is as large (in absolute terms) as we observe? h. What is the probability, if the true value of the regression coefficient of disposable income is zero, that the t statistic is as large (in absolute terms) as we observe? i. Describe briefly what this regression means. Solution: a. 40.833 b. 1.025

Estimating Demand Functions | 63 c. d. e. f. g. h. i.

0.006667 91.6% 1.361 Virtually zero: 0.000 (as indicated by the corresponding p) Virtually zero: 0.000 (as indicated by the corresponding p) 24.4% or 0.244 (as indicated by the corresponding p) The regression results say that if the true relationship is linear and if we haven’t left out important variables, then the quantity demanded is statistically significantly reduced by increases in its price and less statistically significantly increased by increases in disposable income.

6. Railroad executives must understand how the costs incurred in a freight yard are related to the output of the yard. The two most important services performed by a yard are switching and delivery, and it seems reasonable to use the number of cuts switched and the number of cars delivered during a particular period as a measure of output. (A cut is a group of cars that rolls as a unit onto the same classification track; it is often used as a unit of switching output.) A study of one of the nation’s largest railroads assumed that Ci  A  B1Si  B2Di  ei where Ci is the cost incurred in this freight yard on the ith day, Si is the number of cuts switched in this yard on the ith day, Di is the number of cars delivered in this yard on the ith day, and ei is an error term. Data were obtained regarding Ci, Si, and Di for 61 days. On the basis of the procedures described in this chapter, these data were used to obtain estimates of A, B1, and B2. The resulting regression equation was Cˆ i  4,914 + 0.42Si + 2.44Di where Cˆ i is the cost (in dollars) predicted by the regression equation for the ith day. a. If you were asked to evaluate this study, what steps would you take to determine whether the principal assumptions underlying regression analysis were met? b. If you were satisfied that the underlying assumptions were met, of what use might this regression equation be to the railroad? Be specific. c. Before using the study’s regression equation, what additional statistics would you like to have? Why? d. If the Durbin-Watson statistic equals 2.11, is there evidence of serial correlation in the residuals? Solution: a. One tool: Residual plot. Plot the residuals against values of Si and Di to see whether there is some pattern in them that would suggest a nonlinear

64 | Chapter 4 relationship between the variables, whether there is serial correlation, or whether the variance in the errors is not constant across observations. b. If the regression estimates are statistically significant and economically important and if the railroad company has good forecasts of Si and Di, it can predict its future costs and net cash flows with some confidence; this is important in its financing operations. c. We need the t statistics on the parameter estimates and the F statistic to determine whether there is any explanatory power in the regression; the model could meet the assumptions needed for regression analysis but not explain enough of the variation in costs to make it economically useful. The R2 and RMSE would also shed light on this issue. The DurbinWatson statistic would alert us to any serial correlation, although we should have caught that in the residual plots done in part a. d. The Durbin-Watson is between du and 4  du for the critical values of du and d1 given in an appendix; this tells us that there is no serial correlation in the error terms. 7. Mary Palmquist, a Wall Street securities analyst, wants to determine the relationship between Chile’s gross domestic product (GDP) and the profits (after taxes) of the Carlton Company. She obtains the following data concerning each variable: Year

Gross domestic product (Billions of dollars)

Carlton’s profits (Millions of dollars)

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012

688 753 796 868 936 982 1,063 1,171 1,306 1,407 1,529 1,706

355 339 361 357 278 363 510 573 661 705 688 931

a. What are the least-squares estimates of the intercept and slope of the true regression line, where Carlton’s profits are the dependent variable and GDP is the independent variable? b. On the average, what effect does a $1 increase in gross domestic product seem to have on the profits of Carlton? c. If Ms. Palmquist feels that next year’s GDP will be $2 trillion, what forecast of Carlton’s profits will she make on the basis of the regression?

Estimating Demand Functions | 65 d. What is the coefficient of determination between the nation’s gross domestic product and GE’s profits? e. Do the results obtained in previous parts of this problem prove that changes in Carlton’s profits are caused by changes in the gross domestic product? Can we be sure that Carlton’s profit is a linear function of the GDP? What other kinds of functions might be as good or better? f. If you were the financial analyst, would you feel that this regression line was an adequate model to forecast Carlton’s profits? Why or why not? Solution: a. A computer printout gives a  134.542, b  0.585801 Suggestion: I suggest that the reader use either EXCEL or GRETL to perform the regression. SAS and Minitab are expensive software, and therefore not all students have access to them. But the good news is that there is some free online available software that is even more friendly to use. One example is GRETL. It can be easily downloaded online. Regression equation: y  134.542  0.585801x, where Y is profits and X is GDP. The regression line is plotted below. Actual and fitted y versus x 1000 900

actual fitted

800

y

700 600 500 400 300 200

800

1000

1200 x

1400

1600

b. Note that dy/dx   0.585801. This indicates that a $1 increase in GDP seems to increase GE’s profit by $0.0005858, on average.

66 | Chapter 4 c. GDP  2,000,000,000 → Carlton’s Profit  y  0.5858(2,000)  134.542  1,037.058 millions. d. R2  .897361. Therefore, about 89.74% of the total variation in y is explained by the regression. e. All that we have shown is that GDP and Carlton’s profits are correlated. This could result simply from inflation since both figures seem to be in nominal terms. There might not be any correlation between their real values. Even if their real values are correlated, it could be that Carlton’s profits are affected by something that also affects GDP or even that changes in Carlton’s profits cause changes in GDP instead of the other way around. The criticism that the authors are driving at in this question is an important one that could be leveled against every regression interpretation given in the chapter. Without looking at the residuals in detail, we cannot know whether or not the linear regression assumption is good. We might be led to try a quadratic, log, or log-linear regression equation. f. I would not be comfortable with this estimation since there must be many important omitted variables. For instance, increases in the price of oil might affect Carlton’s profits by increasing the demand for electricity and electrical devices. The regression is far too simple to be useful. 8. In the manufacture of cloth, the weft packages should not disintegrate unduly during weaving. A direct measure of the tendency to disintegrate exists, but it is laborious and uneconomical to carry out. In addition, there are indirect measures based on laboratory tests. Managers of the Brockway Textile Company want to determine the extent to which one of these indirect measures is correlated with the direct measure. If the correlation is high enough, they believe that they may be able to use the indirect measure instead of the direct measure. An experiment was carried out in which both the direct and indirect measures of the tendency to disintegrate were calculated for 18 lots of packages. The results follow: Measure Lot

Direct

Indirect

1 2 3 4 5 6 7 8 9

31 31 21 21 57 80 35 10 0

6.2 6.2 10.1 8.4 2.9 2.9 7.4 7.3 11.1

Estimating Demand Functions | 67 Measure Lot

Direct

Indirect

10 11 12 13 14 15 16 17 18

0 35 63 10 51 24 15 80 90

10.7 4.1 3.5 5.0 4.5 9.5 8.5 2.6 2.9

a. What is the coefficient of determination between the two measures? b. What linear regression line would you use to predict the value of the direct measure on the basis of knowledge of the indirect measure? c. On the basis of your findings, write a brief report indicating the factors to be weighed in deciding whether to substitute the indirect measure for the direct measure. Solution: a. R2  .710189 b. Direct  87.2735  8.05732 (indirect) c. There appears to be an inverse relationship between the indirect measure and the more accurate direct measure. Variation in the indirect measure explains 71% of the total variation in the dependent variable. When these two observations are eliminated, the coefficient estimates don’t change much, but the R2 drops from .71 to .627. Depending on the importance of having a good measure of strength and the cost of gathering the direct measure, Brockway may or may not want to accept the indirect measure. 9. The Kingston Company hires a consultant to estimate the demand function for its product. Using regression analysis, the consultant estimates the demand function to be log Q  2.01  0.148 log P  0.258 log Z where Q is the quantity demanded (in tons) of Kingston’s product, P is the price (in dollars per ton) of Kingston’s product, and Z is the price (in dollars per ton) of a rival product. a. Calculate the price elasticity of demand for Kingston’s product. b. Calculate the cross elasticity of demand between Kingston’s product and the rival product. c. According to the consultant, R2  0.98 and the standard error of estimate is 0.001. If the number of observations is 94, comment on the goodness of fit of the regression.

68 | Chapter 4 Solution: a. This is the so-called log-linear demand function. Fact: For a log-linear demand function, the coefficients represent elasticities. Recommendation: I recommend that instructors teach this type of material in their managerial courses. Another way to see this is to rewrite the demand function in multiplicative form and realize that the exponents represent elasticities. We have then Price elasticity of demand: EP  0.148. Alternatively, one can use the definition of elasticity to obtain EP  d log Q/d log P  0.148 b. We have EZ  0.258. Alternatively, EZ  d log Q/d log Z  0.258. c. The R2  0.98 is extremely high. Except in time-series regressions using nominal data, we would almost never observe such a high R2, so the goodness of fit of this regression is tremendous. 10. Managers of the New Hope and Ivyland Short Line Railroad conducted an experiment in which they reduced fares by about 28% for approximately a year to estimate the price elasticity of demand. This large fare reduction resulted in essentially no change in the railroad’s revenues. a. What problems exist in carrying out an experiment of this sort? b. Taken at face value, what seemed to be the price elasticity of demand? Solution: a. Many important determinants of ridership other than the fare, such as the price of gasoline or personal disposable income, may have changed over the year, and so the result does not identify a demand curve but rather points on shifting demand curves. A change in rail fare that lasts for 1 year will be viewed by consumers as permanent and therefore may be difficult to reverse. b. The price elasticity appears to be equal to 1. 11. Because of a shift in consumer tastes, the market demand curve for highquality red wine has shifted steadily to the right. If the market supply curve has remained fixed (and is upward sloping to the right), there has been an increase over time in both the price of such wine and in the quantity sold. a. If one were to plot price against quantity sold, would the resulting relationship approximate the market demand curve? b. If not, what would this relationship approximate?

Estimating Demand Functions | 69 Solution: a. No, the data points are each on different demand curves. b. The data points trace out the fixed upward-sloping supply curve for red wine. 12. Managers of the Brennan Company used regression analysis to obtain the following estimate of the demand function for its product: log Q  2  1.2 log P  1.5 log I where Q is quantity demanded, P is price, and I is consumers’ disposable income. a. Brennan’s president is considering a 5% price reduction. He argues that these results indicate that such action will result in a 6% increase in the number of units sold by the firm. Do you agree? Why or why not? b. The firm’s treasurer calculates that the probability that the t statistic of log P is as large (in absolute value) as it is, given that log P has no real effect on log Q, is about 0.5. He says that the estimate of the price elasticity is unreliable. Do you agree? Why or why not? c. How can the managers obtain a more accurate estimate of the price elasticity of demand? Solution: a. This is a log-linear demand function and therefore the coefficients represent elasticities. The price elasticity of demand for the good is given by EP  1.2, meaning that a 1% reduction in price leads to a 1.2% increase in the quantity demanded of the product. Therefore, a 5% reduction in price results in a 6% increase in the number of units sold by the firm. Hence, the firm’s president is correct. b. The treasurer is also right. If there is a 50% chance that we could have observed the 1.2 coefficient on log P if the true coefficient is zero, we can’t have much faith in the estimate. The probability of rejecting the null hypothesis is too high. c. The firm should analyze the residuals from the regression to see whether a specification other than log-linear is suggested or whether there is a serial correlation or heteroskedasticity problem. Per capita income and price will be highly correlated if both are expressed in nominal terms; in that case respecifying them in real terms should increase the t statistic on price. Finally, if more data can be gathered economically, then increasing the amount of data could improve the statistics.

CHAPTER 5

Production Theory

Lecture Notes 1. Introduction •

•

•

• • • • • •

70

Objectives ÿ To explain how managers should determine the optimal method of production by applying an efficient production process ÿ To understand the linkages between production processes and costs Production processes include all activities associated with providing goods and services, including: * employment practices * acquisition of capital resources * product distribution * managing intellectual resources Production processes define the relationships between resources used and goods and services produced per time period. ÿ Managers exert control over production costs by understanding and managing production technology. To examine the concept of production function including production function with one variable input and those with two variable inputs To understand the concept of isoquant To find the optimal combination of inputs, including interior solution and corner solution To understand the concept of returns to scale To explore the concept of output elasticity To briefly discuss the problem of estimating production functions

Production Theory | 71 STRATEGY SESSION: The Yankees’ Deal for Alex Rodriguez Discussion Questions 1. A good economic decision is one in which the benefits of an action exceed the costs of the action by the maximum possible amount. In general, if the cost of resources used to increase production is less than the revenue generated, profits will increase. Alex Rodriguez is a rather costly resource. What sources of benefit are cited in the Strategy Session? Are you convinced that the Yankees made a good decision by acquiring Rodriguez? Why or why not? 2. Baseball, like other types of entertainment, often provides massive compensation packages to star employees. The justification cited is that, even though an employee who is “almost as good” can be hired for a fraction of the amount paid for a star, the value of having the best can make the expenditure worthwhile. Does this seem fair? Would it make sense to cap the compensation that can be paid to star employees at some multiple of the lowest paid employee? What effect do you think this would have on entertainment revenues? 2. The Production Function with One Variable Input •

•

•

• •

A production function shows the maximum amount that can be produced per time period with the best available technology from any given combination of inputs. It can be represented as ÿ a table ÿ a graph ÿ an equation * A production function summarizes the characteristics of existing technology at a given time. * Production is dynamic: Methods, designs, and factor costs change. Example: Production function with one variable input: ÿ Q  30L  20L2  L3 (an equation) (equation 5.1) where L is amount of labor used in the production process. * A production function as table: Table 5.1 and Figure 5.1 * A graph as a production function: Figure 5.1 * Q  Hundreds of parts produced per year * L  Number of machinists hired * Fixed capital  Five machine tools The idea: For a one variable production function, only one input is variable and the other inputs are fixed during the period under consideration (the short run). Measures of efficiency Question: How does output change as one input varies?

72 | Chapter 5 •

Several measures: ÿ Average Product of Labor  APL  Q/L * Common measuring device for determining the units of output, on average, per worker ÿ Marginal Product of Labor  MPL  Q/L (for a table) * Product of Labor  MPL  dQ/dL (for an equation) assumes that labor can be varied continuously. * Metric for analyzing the efficiency of each input in which the input’s MP is equal to the incremental change in output created by a small increase in the variable input. ÿ Example: Production function: Q  30L  20L2  L3 * Table 5.2 and Figure 5.2 2 * APL  Q/L  30  20LL 2 * Using calculus: MPL  30  40L  3L * APL is at a maximum, and MPL  APL , at L  10 and MPL  APL  130. * MPL is at a maximum at L  6.67 and MPL  163.33. ÿ Why does MPL  APL when APL is at a maximum? * If MPL APL , then APL must be increasing. * If MPL APL , then APL must be decreasing.

3. The Law of Diminishing Marginal Returns •

•

Law of Diminishing Returns: A well-known occurrence where when managers add equal increments of an input while holding other input levels constant, the incremental gains to output eventually get smaller The empirical observation that as the number of one input increases, holding other inputs constant, a point will be reached beyond which the marginal product of the variable input decreases * It is not a theorem that can be proved or disproved.

4. The Production Function with Two Variable Inputs •

•

Q  f(X1, X2) , where both X1 and X2 are variable inputs. ÿ Q  Output rate ÿ X1  Level of first input ÿ X2  Level of second input ÿ AP1  Q/X1 and MP1  Q/X1 (for a table) or ≠Q/≠X1 (for an equation) ÿ AP2  Q/X2 and MP2  Q/X2 (for a table) or ≠Q/≠X2 (for an equation of the production function) Example: ÿ Table 5.3 and Figure 5.3 (for a graphical representation of a production function with two variable inputs) ÿ The graph of Q  f(X1, X2) is a surface in a three-dimensional space.

Production Theory | 73 5. Isoquants •

•

Isoquant: Curve showing all possible (efficient) input bundles capable of producing a given output level. ÿ Graphically constructed by cutting horizontally through the production surface (Figure 5.3) at a given output level ÿ Isoquants representing different output levels are shown in Figure 5.4. ÿ An isoquant shows all combinations of X1 and X2 such that f(X1, X2)  c, where c is a fixed amount of output. Properties of Isoquants ÿ Isoquants farther from the origin represent higher output levels. ÿ Given a continuous production function, every possible input bundle is on an isoquant and there are an infinite number of possible input combinations. ÿ Isoquants slope downward to the left and are convex to the origin.

STRATEGY SESSION: How Nucor Stays on the Production Function Discussion Questions 1. What actions has Nucor taken to control and reduce principalagent problems within its organization? Answer: Compensation plan is performance-based; flat organizational structure reduces information asymmetry; equal treatment of all employees encourages active participation in company operations; ISO 9000 certification encourages accountability and performance. 2. Steel production is an “old” smokestack manufacturing technology. How has Nucor succeeded in competing effectively against foreign competition while other domestic steel producers have been less successful? How do you think the isocost curves and isoquants for Nucor differ from those of its domestic competitors? Answer: Assuming that the markets for resources are competitive, Nucor’s isocost curves are the same as those of its competitors. Its isoquants, however, are closer to the origin, which means that a given level of output can be attained with a lower resource utilization by Nucor than by its competitors, at least over some range of output rates. 6. The Marginal Rate of Technical Substitution • •

Marginal Rate of Technical Substitution (MRTS): Shows the rate at which one input is substituted for another (with output remaining constant) Suppose the production function is given by Q  f(X1, X2).

74 | Chapter 5

•

•

ÿ MRTS  Negative of the slope of an isoquant  X2/X1 with Q held constant and X2 on the vertical axis ÿ Alternative formula for MRTS: MRTS  MP1/MP2 ÿ MRTS  Absolute value of the slope of an isoquant MRTS and Isoquants (with X2 on the vertical axis) ÿ If the MRTS is large, it takes a lot of X2 to substitute for one unit of X1, and isoquants will be steep. ÿ If the MRTS is small, it takes little X2 to substitute for one unit of X1, and isoquants will be flat. ÿ If X1 and X2 are perfect substitutes, MRTS is constant, and isoquants will be straight lines connecting the two axes. * Example: Linear production function ° Suppose the production function is given by f(X1, X2)  2X1  4X2. Isoquants are given by the set of all input bundles (X1, X2 ) such that 2X1  4X2  c, for some constant c. These form a family of straight lines with slope 1/2. ÿ If X1 and X2 are perfect complements, no substitution is possible, MRTS is undefined, and isoquants will be right angles. * Examples of such isoquants are given by the so-called Leontief Production Function. * Example shown in Figure 5.5 Ridge Lines ÿ Ridge lines: The lines that profit-maximizing firms operate within, because outside of them, marginal products of inputs are negative ÿ Economic region of production is located within the ridge lines. ÿ Example shown in Figure 5.6 ÿ The idea: Managers cannot use input bundles outside the ridge lines if they want to maximize profit.

7. The Optimal Combination of Inputs •

Isocost Curve: Curve showing all the input bundles that can be purchased at a specified cost * L  Level of labor * PL  Price of labor * K  Level of capital * PK  Price of capital * M  Total outlay or personal disposable income * Assuming all income is spent on the two goods, the budget constraint is given by PLL  PKK  M Alternatively, solving for K gives us ÿ K  M/PK  (PL/PK)L

(5.4) (5.5)

Production Theory | 75

•

ÿ This is a straight line, called the consumer’s budget line or isocost curve, with * Vertical intercept: Found by setting L  0 in equation 5.4 and is equal to M/PK * Horizontal intercept: Found by setting K  0 and is equal to M/PL * Slope   PL/PK * Graph: Figure 5.7 * It shows all the input bundles that can be purchased at a specified cost M. Optimal Combination of Inputs ÿ Can be found either through cost minimization or output maximization ÿ Maximize output for given cost: * Manager’s problem: Choose the input bundle that maximizes the output derived with a given level of income, given factor market prices. * Here the constraint is predetermined. It is the budget constraint. * For convex isoquants, the optimal input bundle can be found by the tangency condition. ÿ The idea: An efficient manager should choose the point on the isocost curve that is tangent to the highest valued isoquant. ÿ Tangency between isocost and isoquant * MRTS  MPL/MPK  PL/PK or, alternatively, * MPL/PL  MPK /PK * Marginal product per dollar spent should be the same for all inputs. * Example: Figure 5.8 * In the case or more than two inputs, the tangency condition says that the manager chooses the input bundle such that * MPa /Pa  MPb/Pb  …  MPn /Pn ÿ Minimize cost for a given output: Figure 5.9

PROBLEM SOLVED: What Skills Do We Need? Discussion Questions 1. Suppose that the monthly wage of a technician decreases to $1,000 per month. How many engineers and technicians will be employed, and how many units of output will be produced? Answer: E  4 2/3, T  9 1/3, and Q  140 2. Suppose that the wages of technicians and engineer are $2,000 and $4,000, respectively, and that total spending on production increases to $52,000. How many engineers and technicians will be employed, and how many units of output will be produced? Answer: E  8, T  10, and Q  166

76 | Chapter 5 PROBLEM SOLVED: The Efficient Minds of Managers Discussion Questions 1. Suppose that the wage rate increases to $100 per hour and the price of a machine increases to $25 per hour. How many workers and machines should be used to produce 80 units of output? Answer: L  40 and K  160. The cost of both inputs increased by 25%, so the relative prices, and hence the slope of the isocost curve, remain the same. Consequently, the optimal employment of resources does not change, and total cost increases by 25%. 2. Suppose that a new process changes the production function to Q  20L 0.4K0.6. If input prices remain the same, how many workers and machines should be used to produce 80 units of output? Answer: L  27.3 and K  163.8. The productivity of K has increased, so employment of K increases and employment of L decreases. 8. Corner Solutions • •

•

•

•

Optimal input bundles with just one input deployed Optimal input combination does not occur at a point of tangency between isocost and isoquant curves. ÿ In a two-input case, one of the inputs will not be used at all in production. ÿ Illustration: Figure 5.10 * Case 1: Just capital is used: MPL/PL MPK /PK * Case 2: Just labor is used: MPL/PL MPK /PK If two inputs are perfect complements (isoquants are right angles), then both inputs will be used, but the optimal combination will not occur at a point of tangency between isocost and isoquant curves. If two inputs are perfect substitutes (isoquants are straight lines), then only one input will be used, but the optimal combination will not occur at a point of tangency between isocost and isoquant curves. There will be a corner solution. Illustration: Figure 5.10

9. Returns to Scale •

Long-Run Effect of an Equal Proportional Increase in All Inputs ÿ Increasing returns to scale: When output increases by a larger proportion than inputs ÿ Decreasing returns to scale: When output increases by a smaller proportion than inputs

Production Theory | 77

•

•

ÿ Constant returns to scale: When output increases by the same proportion as inputs Sources of Increasing Returns to Scale ÿ Indivisibilities: Some technologies can only be implemented at a large scale of production. ÿ Subdivision of tasks: Larger scale allows increased division of tasks and increases specialization. ÿ Probabilistic efficiencies: Law of large numbers may reduce risk as scale increases. ÿ Geometric relationships: Doubling the size of a box from 1  1  1 to 2  2  2 multiplies the surface area by four times (from 3 to 12) but increases the volume by eight times (from 1 to 8). This applies to storage devices, transportation devices, and the like. Sources of Decreasing Returns to Scale ÿ Coordination inefficiencies: Larger organizations are more difficult to manage. ÿ Incentive problems: Designing efficient compensation systems in large organizations is difficult.

STRATEGY SESSION: Economies in Oil Tankers Discussion Questions 1. In the Strategy Session, an oil tanker is likened to a geometric shape—a cylinder. Do other three-dimensional geometric shapes have similar relationships between surface area and volume? Answer: Yes, there are many examples. For example, the surface area of a sphere is 4 •  • r 2 and the volume is 4/3 •  • r 3, where r is the radius. A doubling of the radius from r  1 to r  2 will cause the surface area of a sphere to triple from 12.6 to 50.3. The same change will cause the volume to increase seven times from 4.2 to 33.5. 2. If such impressive economies of scale exist with tanker ships, why aren’t they larger than they are? What factors limit the practical scale of these ships? 10. The Output Elasticity •

Output Elasticity: The percentage change in output resulting from a 1% increase in all inputs ÿ Note: A more common measure of output elasticity is the percentage change in output resulting from a 1% increase in a single input. This is referred to as the output elasticity of a certain input.

78 | Chapter 5

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ÿ Fact: For a Cobb-Douglas production function, the exponents represent output elasticities. ÿ Accordingly, the coefficients 0.3 and 0.8 in the Cobb-Douglas function below would be referred to as the output elasticities of labor and capital, respectively. Cobb-Douglas Production Function Example: Q  0.8L 0.3K0.8 ÿ Q  Parts produced by the Lone Star Company per year ÿ L  Number of workers ÿ K  Amount of capital ÿ Output elasticity  1.1 for infinitesimal changes in inputs ÿ Example calculation for 1% increase in both inputs 0.3 0.8 * Q'  0.8(1.01L) (1.01K)  1.011005484Q Therefore, if managers increase the use of both inputs by 1%, output increases by slightly more than 1.1% and output elasticity is approximately 1.1.

11. Estimations of Production Functions • • •

Managers need to estimate production functions. One of the first steps in estimating a production function is to specify a mathematical form. Managers commonly use the Cobb-Douglas production function. Cobb-Douglas Mathematical Form: Q  aLbKc ÿ MPL  Q/L  b(Q/L)  b(APL) ÿ Log-linear estimation: log Q  log a  b log L  c log K, obtained by taking the log of both sides of the above equation. ÿ Returns to scale * b  c 1 → increasing returns to scale * b  c  1 → constant returns to scale * b  c 1 → decreasing returns to scale

PROBLEM SOLVED: Finding the Optimal Mix Discussion Questions 1. A production function for broiler chicken growth can be represented by the following Cobb-Douglas function: G  1.4(CS)0.3. If this function is solved with G set equal to 1, it yields (approximately) C  0.3/S. Plot this isoquant over the range of S from 0.25 to 2.5. Assume that the price of corn is the same as that of soybean oilmeal. What combination of the two inputs should be used to produce a one-pound increase in broiler weight? Answer: C  0.55 and S  0.55 yields G  0.98. See the graph below, which shows the isoquant, the isocost, and the optimum combination.

Production Theory | 79

Indifference Curve for G = 1 3 2.5

Corn

2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Soybean Oilmeal

2. What will happen to the optimal combination of feed if the price of corn increases? If the price of soybean oilmeal increases? Show the effects on your graph. Answer: If corn is more expensive, the isocost will be flatter, C will decrease, and S will increase. If soybean oilmeal is more expensive, the isocost will be steeper, C will increase, and S will decrease. 12. Appendix: Lagrangian Multipliers and Optimal Input Combinations •

• • •

Given ÿ Production function: Q  f(X1, X2) ÿ Firm’s expenditure (E*) on inputs: E*  X1P1  X2P2 * Implicit form: E*  X1P1  X2P2  0 Lagrangian Solution to Output Maximization ÿ Lagrangian function: L1  f(X1, X2)  (E*  X1P1  X2P2) Lagrangian Solution to Output Maximization ÿ Lagrangian function: L2  X1P1  X2P2  (Q*  f(X1, X2)) Conditions for Optimum ÿ MP1/P1  MP2/P2 ÿ MP1/MP2  P1/P2

80 | Chapter 5 Chapter 5: Problem Solutions 1. In the Elwyn Company, the relationship between output (Q) and the number of hours of skilled labor (S) and unskilled labor (U) is Q  300S  200U  0.2S2  0.3U2 The hourly wage of skilled labor is $10, and the hourly wage of unskilled labor is $5. The firm can hire as much labor as it wants at these wage rates. a. Elwyn’s chief engineer recommends that the firm hire 400 hours of skilled labor and 100 hours of unskilled labor. Evaluate this recommendation. b. If the Elwyn Company decides to spend a total of $5,000 on skilled and unskilled labor, how many hours of each type of labor should it hire? c. If the price of a unit of output is $10 (and does not vary with output level), how many hours of unskilled labor should the company hire? Solution: a. Q  300S  200U  0.2S2  0.3U2 MPU  Q/U  200  0.6U. MPS  Q/S  300  0.4S An optimal combination of skilled and unskilled labor must satisfy the tangency condition: MPU/PU  (200  0.6U)/5  (300  0/4S)/10  MPS /PS → S  3U  250. (1) At S  400 and U  100, this efficiency condition doesn’t hold, and so the proposed hiring levels are not efficient. b. Combining equation (1) with the budget constraint 5,000  10S  5U yields 5,000  30U  2,500  5U. That is, 7,500  35U, so that U*  214.286 and S*  392.86. c. Profit maximization requires that the value of marginal product be equal to the wage rates. Hence, for the unskilled labor we have VMPU  PU . Given that MPU  200  0.6U, we have then 10(200  0.6U)  5, that is, U*  332.5. 2. A consulting firm specializing in agriculture determines that the following combinations of hay and grain consumption per lamb will result in a 25pound gain for a lamb:

Production Theory | 81 Pounds of hay

Pounds of grain

40 50 60 70 80 90 110 130 150

130.9 125.1 120.1 115.7 111.8 108.3 102.3 97.4 93.8

a. The firm’s president wants to estimate the marginal product of a pound of grain in producing lamb. Can he do so on the basis of these data? b. The firm’s president is convinced that constant returns to scale prevail in lamb production. If this is true and hay and grain consumption per lamb are the only inputs, how much gain accrues if the hay consumption per lamb is 100 pounds and the grain consumption per lamb is 250.2 pounds? c. What is the marginal rate of technical substitution of hay for grain when between 40 and 50 pounds of hay (and between 130.9 and 125.1 pounds of grain) are consumed per lamb? d. A major advance in technology occurs that allows farmers to produce a 25-pound gain per lamb with less hay and grain than the preceding table indicates. If the marginal rate of technical substitution (at each rate of consumption of each input) is the same after the technological advance as before, can you draw the new isoquant corresponding to a 25-pound gain per lamb? Solution: a. No. Every possible pairwise comparison involves changes in both pounds of hay and pounds of grain, so the marginal product of grain cannot be isolated. b. CRS → f(H0, G 0)  25 ↔ f(2H0, 2G 0)  50. 50 pounds will accrue. c. H/G  10/5.8  1/0.58. That is, 1 pound of hay is required to compensate for the reduction of 0.58 pound of grain on average over the interval in question. d. No. We know that the new 25-pound isoquant will have the same shape and will be shifted in toward the origin, but we don’t know how far. 3. The Ascot Corporation, which produces stationery, hires a consultant to estimate its production function. The consultant concludes that Q  0.9P  0.06L

82 | Chapter 5 where Q is the number of pounds of stationery produced by Ascot per year, L is the number of hours of labor per year, and P is the number of pounds of paper used per year. a. Does this production function seem to include all the relevant inputs? Explain. b. Does this production function seem reasonable if it is applied to all possible values of L? Explain. c. Does this production function exhibit diminishing marginal returns? Solution: a. As a long-run production function, this seems to be lacking a variable for capital equipment. b. The production function says that without any labor, paper can transform itself into stationery. This production option may be specified over the range of P and L observed, but it doesn’t make sense in reality. c. No. The marginal returns to both P and to L are constant. We have MPP  0.9 and MPL  0.06. 4. A Cobb-Douglas production function was estimated for six types of farms. There were five inputs in the production function: (1) land, (2) labor, (3) equipment, (4) livestock and feed, and (5) other resource services. The exponent of each input was as follows: Exponent Farm type Crop farms Hog farms Dairy farms General farms Large farms Small farms

Land

Labor

Equipment

Livestock and feed

Other resource services

0.24 0.07 0.1 0.17 0.28 0.21

0.07 0.02 0.01 0.12 0.01 0.05

0.08 0.1 0.06 0.16 0.11 0.08

0.53 0.74 0.63 0.46 0.53 0.43

0.02 0.03 0.02 0.03 0.03 0.03

a. Do there appear to be increasing returns to scale in any of these six types of farms? b. In what type of farm does a 1% increase in labor have the largest percentage effect on output? c. Based on these results, would you expect output to increase if many of the farms included in this sample were merged?

Production Theory | 83 Solution: a. No. The sum of the coefficients from the Cobb-Douglas production function for each farm type is less than 1. b. The elasticity of output with respect to labor is the highest for general farms. It is 0.12. c. Unclear. Although there doesn’t appear to be any economies of scale for any farm type, there might be economies of scope from combining two farms of different types. On the basis of the data provided, we cannot conclude whether such economies of scope exist. 5. According to the chief engineer at the Zodiac Company, Q  AL  K , where Q is the output rate, L is the rate of labor input, and K is the rate of capital input. Statistical analysis indicates that   0.8 and   0.3. The firm’s owner claims the plant has increasing returns to scale. a. Is the owner correct? b. If ß were 0.2 rather than 0.3, would she be correct? c. Does output per unit of labor depend only on  and ? Why or why not? Solution: a. Yes. It can be seen that     1.1, which is greater than 1; therefore the production function exhibits increasing returns to scale. Thus, increasing both L and K by 1% causes output to go up by more than 1%. b. If ß  0.2 instead of 0.3, then we would have     1, indicating that there would be constant returns to scale. c. Q/L  APL  AKß/L1  , which also depends on A, K, and L. The average product of labor is a function of L because the marginal product of L is not constant. The average product depends on K because as K increases, Q increases, so holding L constant, Q/L increases. Finally, A scales the Q, and so if A goes up, Q/L goes up proportionately. 6. According to data obtained by the U.S. Department of Agriculture, the relationship between a cow’s total output of milk and the amount of grain it is fed is as follows: Amount of Grain (Pounds)

Amount of Milk (Pounds)

1,200 1,800 2,400 3,000

5,917 7,250 8,379 9,371

(This relationship assumes that forage input is fi xed at 6,500 pounds of hay.)

84 | Chapter 5 a. Calculate the average product of grain when each amount is used. b. Estimate the marginal product of grain when between 1,200 and 1,800 pounds are fed, when between 1,800 and 2,400 pounds are fed, and when between 2,400 and 3,000 pounds are fed. c. Does this production function exhibit diminishing marginal returns? Solution: a. and b. are given in the following table. G 1,200 1,800 2,400 3,000

Average product Q/G

Marginal product Q/G

5,917/1,200  4.93 7,250/1,800  4.03 8,379/2,400  3.49 9,371/3,000  3.12

1,333/600  2.22 1,129/600  1.88 992/600  1.65

c. Yes, we can see that succeeding 600-pound increments of grain increase milk production by successively smaller amounts. That is, marginal product is decreasing. 7. An electronics plant’s production function is Q  5LK, where Q is its output rate, L is the amount of labor it uses per period, and K is the amount of capital it uses per period. The price of labor is $1 per unit of labor, and the price of capital is $2 per unit of capital. The firm’s vice president for manufacturing hires you to determine which combination of inputs the plant should use to produce 20 units of output per period. a. What advice would you give him? b. Suppose the price of labor increases to $2 per unit. What effect will this have on output per unit of labor? c. Is this plant subject to decreasing returns to scale? Why or why not? Solution: a. Tangency condition: MPL/MPK  PL/PK But, we have MPL/MPK  5K/5L  K/L  PL/PK  1/ 2. Hence, the tangency condition is K/L  1/2; that is, L  2K. Furthermore, the output constraint is such that 5LK  20. Substituting L into this equation gives us 10K2  20. Thus, K *  21/2. L*  81/2. This is the optimal input mix. b. As before, the optimal input mix is given by the tangency condition, where the new price ratio is 1. Hence, the tangency condition gives K/L  1, or L  K. Substituting this into the output constraint gives us 5K 2  20. That is, K * 2 and L*  2. Output per unit of labor increases from 20/81/2 to 20/2  10.

Production Theory | 85 c. No. Given that the production function is of Cobb-Douglas type, we have     1  1  2. Therefore, the plant exhibits increasing returns to scale. 8. Volvo A.B., the Swedish auto firm, operated a car assembly plant at Uddevalla in 1988. The idea was to have a small team of highly skilled workers build an entire car. According to the proponents, this would reduce the tedium associated with the conventional assembly line and cut absenteeism and turnover among workers. In 1991 there were reports that it took 50 hours of labor to assemble a car at Uddevalla, in contrast to 25 hours at Volvo’s conventional assembly plant at Ghent, Belgium. If you were Volvo’s chief executive officer, what questions would you ask Uddevalla’s managers, and what steps would you take? Solution: That it requires more labor to produce a car in Uddevalla than in Ghent is a concern if there are not compensating reductions in the amount of other inputs required to produce a car in Uddevalla. I would want to know how much is spent on capital equipment at each plant and the difference in the wage rates of the skilled workers at Uddevalla and the presumably less skilled workers at Ghent. Without this additional information we cannot tell whether the technique chosen at Uddevalla is efficient.

CHAPTER 6

The Analysis of Costs

Lecture Notes 1. Introduction •

Objectives ÿ To explain how managers can use their knowledge of the relationship between costs and output to make decisions that maximize the value of the firm ÿ To answer the following key question: How are costs related to output? ÿ To explain the basics of cost analysis and describe models that help managers create competitive advantages using cost analysis ÿ To understand the difference between short-run and long-run costs and the importance of this distinction for managerial decisions ÿ To analyze two important issues in cost analysis: Economies of scope and economies of scale ÿ To examine the managerial issue of break-even analysis

2. Opportunity Costs •

• • •

86

The opportunity cost of producing a par ticular product is the revenue a manager could have received if she had used her resources to produce the next best alternative product or service. Opportunity costs are the revenues foregone if resources or inputs are not optimally used. The point: Managers want to reduce opportunity costs. Some Key Concepts: ÿ Opportunity cost doctrine: The inputs’ values (when used in their most productive way) together with production costs (the accounting costs of producing a product) determine the economic cost of production.

The Analysis of Costs | 87 ÿ Historical cost: The money that managers actually paid for an input * Fact: Historical costs can be misleading. ÿ Explicit costs: The ordinary items accountants include as the firm’s expenses * Examples: Firm’s payroll and payment for raw material ÿ Implicit costs: The foregone value of resources that managers did not put to their best use ÿ Doctrine of sunk costs: Sunk costs are resources that are spent and cannot be recovered. * Fact: Rational managers must ignore sunk costs and choose between possible strategies by evaluating only future costs and benefits. 3. Short-Run Cost Functions •

•

Definitions ÿ Cost function: Function showing various relationships between input costs and output rate ÿ Short run: * The time span between the one where the quantity of no input is variable and one where the quantities of all inputs are variable * A period so short that manager cannot alter the quantity of some inputs ÿ Fixed inputs: When the quantities of plant and equipment cannot be altered ÿ Scale of plant: This scale is determined by fixed inputs. ÿ Variable inputs: Inputs that a manager can vary in quantity in the short run ÿ Three short-run cost concepts: * Total fi xed cost (TFC): The total cost per period of time incurred for fixed inputs. It does not vary with the amount of output. Example: Property taxes * Total variable cost (TVC): The total cost incurred by managers for variable inputs. These costs increase as output rises. * Total cost (TC): The sum of total fixed and total variable costs, that is, TC  TFC  TVC Example (Table 6.1 and Figure 6.1): TC  100  50Q  110Q2  Q3

4. Average and Marginal Costs •

Definitions ÿ Average fi xed cost (AFC): The total fixed cost divided by output AFC  TFC/Q

88 | Chapter 6

•

•

ÿ Average variable cost (AVC): The total variable cost divided by output AVC  TVC/Q ÿ Average total cost (ATC): The total cost divided by output ATC  TC/Q ÿ Marginal cost (MC): The incremental cost of producing an additional unit of output MC  TC/Q * Note: Marginal cost is defined as TC/Q for discrete changes in Q and as dTC/dQ for continuous changes in Q. Cost Relationships ÿ The behavior of cost is largely determined by the production function. ÿ Let U be the number of inputs used and W the cost per unit of input. We have: ÿ AVC  TVC/Q  W(U/Q)  W(1/AP) where AP is the average product of U. ÿ Recall the behavior of AP: * Initially increases with output * Reaches a maximum * Then begins to decrease Fact: AVC mirrors the behavior of AP. Thus, * when AP increases, AVC decreases; * when AP decreases, AVC increases. Hence, AVC will: * initially increase * hit a minimum * then increase ÿ Marginal cost (MC): The incremental cost of producing an additional unit of output ÿ MC  TVC/Q  W(U/Q)  W(1/MP) where MP is the marginal product of U. ÿ MC  AVC when AVC is at a minimum. ÿ MC  ATC when ATC is at a minimum. Example (Table 6.2 and Figure 6.2) Suppose total cost function is given by TC  100  50Q 11Q2  Q3 We have ÿ AFC  100/Q ÿ AVC  50  11Q  Q2 ÿ ATC  AFC  AVC  100/Q  50  11Q  Q2 ÿ MC  dTC/dQ  dTVC/dQ  50  22Q  3Q2 ÿ AVC is at a minimum where dAVC/dQ  –11  2Q  0; that is, when Q  5.5 MC  AVC  19.75.

The Analysis of Costs | 89 ÿ ATC is at a minimum where dATC/dQ  –(100/Q2) 11  2Q  0; that is when Q  6.64. We have then MC  ATC  36.11 STRATEGY SESSION: The Effects of Output on the Cost of Producing Aircraft Discussion Questions Assume that the relationship between the unit cost of production as a percentage of the cost of producing 400 units (C) and the number of units produced in a single production run (Q) is determined by the following function: C  2000Q0.5. 1. If 325 units are produced, how much higher is the cost of production compared to 400 units? Answer: Q  110.94, so cost is 10.94% higher. 2. How many units will have to be produced in a single run in order to reduce unit cost by 10% below that of producing 400 units? Answer: Q  (2000/C)2  (2000/90)2  494 5. Long-Run Cost Functions •

•

Definitions ÿ Long-run total cost function: The relationship between long-run total cost and output ÿ In the long run, all inputs are variable, and managers can build any scale or type of plant. ÿ Managers need to produce at the minimum average cost. ÿ Long-run average cost function: Function showing the minimum cost per unit of all output levels when any desired size plant is built * It is the envelope of the short-run functions. ÿ Long-run marginal cost function: Function representing how varying output affects the cost of producing the last unit if the manager has chosen the most efficient input bundle Examples (Figures 6.3– 6.5)

PROBLEM SOLVED: Managerial Use of Cost Functions Discussion Questions 1. Suppose that the cost of capital increases to $4 per unit. What is the costminimizing level of K as a function of Q? Answer: TC  8L  4K  Q2/2K  4K dTC/dK  −(Q2)/(2K2)  4  0 → K  (Q)(21.5)  Q/2.83

90 | Chapter 6 2. Suppose that the cost of capital decreases to $1 per unit. What is the costminimizing level of K as a function of Q? Answer: TC  8L  K  Q2/2K  K dTC/dK  (Q2)/(2K2)  1  0 → K  (Q)(22)  Q/4 6. Managerial Use of Scale Economies •

•

Economies of Scale: Occur when the firm’s average unit cost decreases as output increases ÿ Example: Figure 6.6—Economies of scale at Texas nursing homes ÿ Economies of scale can result from larger plant size and/or an increase in the number of plants. ÿ Economies of scale may be exploited by changes in production, distribution, raising capital, advertising, and other business processes. ÿ Managers must understand their cost relationships to recognize where to best exploit scale economies. Diseconomies of Scale: When the average costs per unit of output increase.

7. Managerial Use of Scope Economies •

Economies of Scope: Exist when the cost of jointly producing two (or more) products is less than the cost of producing each one alone. That is, C(Q1  Q2) C(Q1)  C(Q2)

•

A simple way for managers to estimate the extent of their scope economies is to use the following measure: S= ÿ ÿ ÿ ÿ

•

C (Q1 ) + C (Q2 ) − C (Q1 + Q2 ) C (Q1 + Q2 )

S measures the degree of economies of scope. C(Q1)  Cost of producing Q1 units of product 1 C(Q2)  Cost of producing Q2 units of product 2 C(Q1  Q2)  Cost of producing Q1 units of product 1 and Q2 units of product 2 ÿ If there are economies of scope, S 0. ÿ S measures the percentage of saving that results from producing them jointly rather than individually. Example $12 million + $6 million − $15 million ÿ S= = 0.20 $15 million

The Analysis of Costs | 91

ÿ ÿ ÿ ÿ

That is, economies of scope have lowered costs by an estimated 20%. C(Q1)  $12 million C(Q2)  $6 million C(Q1  Q2)  $15 million Implies that scope economies have resulted in a 20% cost advantage

STRATEGY SESSION: Economies of Scope in Advertising Agencies Discussion Questions 1. What are the different “goods” that are produced by an advertising agency, and how might they yield economies of scope in practice? Answer: The goods are different media outlets used for advertising. For example, TV advertising and newspaper advertising are treated as different goods. Economies of scope might result when a single agency conducts an advertising campaign using multiple types of media. Economies would result from the reuse of themes, images, and logos in the different media outlets. 2. Can you think of a type of business where economies of scope are unlikely to exist? If you can’t, explain why you think all types of businesses experience economies of scope. If you can, explain why you think your example does not have access to economies of scope. 8. Managerial Use of Break-Even Analysis • •

Break-Even Point (QB): The output level that must be reached if managers are to avoid losses Given that TR  PQ and TC  TFC  TVC  TFC  AVC (Q), then at the break-even output level, we must have: PQB  TFC  AVC (QB). Solving for QB gives Q B  TFC/(P – AVC).

•

ÿ TFC  Total fixed cost ÿ P  Price ÿ AVC  Average variable cost Example (Figure 6.7) ÿ TFC  $600,000 ÿ P  $3 ÿ AVC  $2 ÿ QB  600,000

92 | Chapter 6 STRATEGY SESSION: Mr. Martin Gets Chewed Out by the Boss Discussion Questions 1. Economic modeling involves simplification. In virtually every case, it is impossible to precisely represent every complex detail that is relevant to a managerial decision. Mr. Martin is suffering because he oversimplified the problem. Under what circumstances would his model have been appropriate and his analysis been relatively accurate? 2. How would you redo the break-even analysis model in order to make it a more realistic representation of the company’s potential profit? 9. Profit Contribution Analysis • • •

Profit contribution analysis: A break-even analysis to understand the relationship between price and profit QB'  (TFC  Profit target)/(P– AVC) ÿ QB'  Minimum output level that will attain the profit target Example ÿ QB'  ($600,000  $1,000,000)/($3  $2)  1,600,000 ÿ TFC  $600,000 ÿ Profit target  $1,000,000 ÿ Price  $3 ÿ AVC  $2

Chapter 6: Problem Solutions 1. An MIT study has estimated costs for producing steel with three different technologies: (1) coke, blast furnace, basic oxygen furnace, ingots, and finishing mills; (2) coke, blast furnace, basic oxygen furnace, continuous casting, and finishing mills; and (3) steel scrap, electric arc furnace, continuous casting, and finishing mills. Under reasonable assumptions concerning input prices, the estimated average costs per ton are as follows:

The Analysis of Costs | 93

Cost category Process materials Energy Direct Capital Other Total

Coke, blast furnaces, basic oxygen furnace, ingots, finishing mills

Coke, blast furnaces, basic oxygen furnace, continuous casting, finishing mills

Steel scrap, electric arc furnace, continuous casting, finishing mills

$148.34 21.15 83.43 102.06 46.74 $401.72

$136.19 15.98 75.09 99.93 41.67 $368.86

$122.78 41.58 67.43 54.08 24.47 $310.34

a. The MIT report concludes that “unless significant changes occur in other technologies, the electric-furnace continuous-casting route will dominate domestic production.” Why? b. At the same time, the report notes that the price of scrap (which is used in this route) “could increase as electric furnace production expands because of the increased demand.” Why is this relevant? c. The report also concludes that regardless of which of these technologies is used, cost per ton is about 25 to 30% higher if wages are $26 per hour rather than $2 per hour. What does this imply about the competitiveness of U.S. steel producers relative to producers in other countries that pay wages far below U.S. levels? d. If these cost figures are long-run average costs, under what circumstances would they also equal long-run marginal costs? Solution: a. Recovering steel from steel scrap is less costly than either of the other techniques for producing steel. Recovery enjoys an advantage both in its variable costs and its fixed costs. Over time, as blast furnaces wear out, they will be replaced with electric arc furnaces, which have much lower average total costs. In the short run, if the capital cost component in the study represents sunk costs, it remains profitable to run the two types of blast furnaces because their average variable costs are lower than the average total cost of a state-of-the-art electric arc furnace. b. Steel scrap is an important input for the electric arc recovery mills. As the output of recycling mills expands, the price of steel scrap will be driven up. This will cause the average costs of the recovery mills to rise and so provide a classic example of an increasing cost industry. Whether recovery mills ever completely replace primary production blast furnaces depends on how quickly the price of scrap steel increases as the recycling industry expands.

94 | Chapter 6 c. If Korean firms have all the same other costs but their workers are paid $2 rather than $26 as paid to workers in the United States, then U.S. manufacturers would have a 25 to 30% competitive disadvantage. d. Long-run average cost curves equal long-run marginal cost curves when there are constant returns to scale. 2. The Haverford Company is considering three types of plants to make a particular electronic device. Plant A is much more highly automated than plant B, which in turn is more highly automated than plant C. For each type of plant, average variable cost is constant so long as output is less than capacity, which is the maximum output of the plant. The cost structure for each type of plant is as follows:

Average Variable Costs Labor Materials Other Total Total fixed costs Annual capacity

Plant A

Plant B

Plant C

$1.10 0.90 0.50 $2.50 $300,000 200,000

$2.40 1.20 2.40 $6.00 $75,000 100,000

$3.70 1.80 2.00 $7.50 $25,000 $50,000

a. Derive the average costs of producing 100,000, 200,000, 300,000, and 400,000 devices per year with plant A. (For output exceeding the capacity of a single plant, assume that more than one plant of this type is built.) b. Derive the average costs of producing 100,000, 200,000, 300,000, and 400,000 devices per year with plant B. c. Derive the average costs of producing 100,000, 200,000, 300,000, and 400,000 devices per year with plant C. d. Using the results of parts (a) through (c), plot the points on the long-run average cost curve for the production of these electronic devices for outputs of 100,000, 200,000, 300,000 and 400,000 devices per year. Solution: a. At Q  100: ACA  AFCA  AVCA  $300,000/100,000  $2.50  $5.50. At Q  200: ACA  AFCA  AVCA  $300,000/200,000  $2.50  $4.00. At Q  300: ACA  AFCA  AVCA  $600,000/300,000  $2.50  $4.50. At Q  400: ACA  AFCA  AVCA  $600,000/400,000  $2.50  $4.00. b. At Q  100: ACB  AFCB  AVCB  $75,000/100,000  $6.00  $6.75. At Q  200: ACB  AFCB  AVCB  $150,000/200,000  $6.00  $6.75. At Q  300: ACB  AFCB  AVCB  $225,000/300,000  $6.00  $6.75. At Q  400: ACB  AFCB  AVCB  $300,000/400,000  $6.00  $6.75. c. At Q  100: ACC  AFCC  AVCC  $50,000/100,000  $7.50  $8.00. At Q  200: ACC  AFCC  AVCC  $100,000/200,000  $7.50  $8.00.

The Analysis of Costs | 95 At Q  300: ACC  AFCC  AVCC  $150,000/300,000  $7.50  $8.00. At Q  400: ACC  AFCC  AVCC  $200,000/400,000  $7.50  $8.00. d. See the diagram below. Average 6 Total Cost 5

4

3

2

1

0 100

150

200

250

300

350

400

Quantity (thousands per year)

3. The Abner Corporation, a retail seller of television sets, wants to determine how many television sets it must sell to earn a profit of $10,000 per month. The price of each television set is $300, and the average variable cost is $100. a. What is the required sales volume if the Abner Corporation’s monthly fixed costs are $5,000 per month? b. If the firm sells each television set at a price of $350 rather than $300, what is the required sales volume? c. If the price is $350, and if average variable cost is $85 rather than $100, what is the required sales volume? Solution: Here we are given the following information: Product’s price: P  $300; AVC  100  TVC/Q. Hence, TVC  100Q. a. We have TC  TFC  TVC  5,000  100Q Since total revenue is given by TR  300Q, then profit is given by Profit  TR – TC  300Q  5,000  100Q  200Q  5,000 Now set profit equal to 10,000, that is, 200Q  5,000  10,000. Solving for Q gives us Q  75. b. The procedure is the same as before. Now, total revenue is given by

96 | Chapter 6 TR  350Q and profit is given by Profit  TR – TC  350Q  5000  100Q  250Q  5,000 Now set profit equal to 10,000, that is, 250Q  5,000  10,000. Solving for Q gives us Q  60. c. We have TC  5,000  85Q and Profit  TR – TC  350Q  5000  85Q  265Q  5,000 Now set profit equal to 10,000, that is, 265Q  5,000  10,000. Solving for Q gives us Q  56.60377. 4. According to a statistical study, the following relationship exists between an electric power plant’s fuel costs (C) and its eight-hour output as a percentage of capacity (Q): C  16.68  0.125Q  0.00439Q2 a. When Q increases from 50 to 51, what is the increase in the cost of fuel for this electric plant? b. Of what use might the result in part (a) be to the plant’s managers? c. Derive the marginal (fuel) cost curve for this plant, and indicate how it might be used by the plant’s managers. Solution: a. TC(Q  51)  TC(Q  50)  $34.47339  $33.905  $0.56839. Another way to answer the question is to calculate the marginal cost at the output level of 50. b. Marginal cost is useful for determining the profit-maximizing level of output. This information will be useful if the manager finds that it is possible to expand sales at the current price and this price exceeds $0.57. c. The marginal cost function is given by MC  dC/dQ; that is, MC  0.125  0.00878Q → MC(Q  50)  $0.564, MC(Q  51)  $0.57278. Clearly, the MC curve is a straight line with a slope of 0.00878 and a vertical intercept of 0.125. Notice that TC  TC(Q  51)  TC(Q  50) is the average of the marginal cost evaluated at Q  50 and Q  51. Marginal fuel cost might be used to choose among alternative techniques or among alternative output levels.

The Analysis of Costs | 97 5. The following table pertains to the Lincoln Company. Fill in the blanks:

Output 0 1 2 3 4 5 6 7 Solution:

Total Cost

Total Fixed Cost

Total Variable Cost

Average Fixed Cost

Average Variable Cost

$50 75 100 120 135 150 190 260

_______ _______ _______ _______ _______ _______ _______ _______

_______ _______ _______ _______ _______ _______ _______ _______

_______ _______ _______ _______ _______ _______ _______ _______

_______ _______ _______ _______ _______ _______ _______ _______

Output

Total Cost

Total Fixed Cost

Total Variable Cost

Average Fixed Cost

Average Variable Cost

0 1 2 3 4 5 6 7

$50 75 100 120 135 150 190 260

$50 50 50 50 50 50 50 50

$0 25 50 70 85 100 140 210

— $50.00 25 16.7 12.5 10 8.33 7.14

— $25.00 25 23.33 21.25 20 23.33 30

Notice that the cost of producing 0 unit, as given in the table, is 50. Therefore it is a fixed cost. Average fixed and variable costs are not defined when the firm is producing nothing. 6. The Deering Manufacturing Company’s short-run average cost function in 2012 was AC  3  4Q where AC is the firm’s average cost (in dollars per pound of the product), and Q is its output rate. a. Obtain an equation for the firm’s short-run total cost function. b. Does the firm have any fixed costs? Explain. c. If the price of the Deering Manufacturing Company’s product (per pound) is $3, is the firm making profit or loss? Explain. d. Derive an equation for the firm’s marginal cost function.

98 | Chapter 6 Solution: a. We know that AC  TC/Q . Therefore, we have TC  QAC. That is, TC  3Q  4Q2. b. No, the firm’s total costs are equal to zero when Q  0. c. If P  $3, then total revenue is given by TR  3Q and profit is given by profit  3Q  ( 3Q  4Q2)  – 4Q2. Hence, at this price the firm will always face losses no matter how much output it produces. It could be optimal for the firm to produce nothing in the short run at this market price. d. MC  dTC/dQ  3  8Q 7. The president of the Tacke Corporation believes that statistical research by his staff shows that the firm’s long-run total cost curve can be represented as TC  0 Q1PL2PK3 where TC is the firm’s total cost, Q is its output, PL is the price of labor, and PK is the price of capital. a. Tacke’s president says that 1 measures the elasticity of cost with respect to output—that is, the percentage change in total cost resulting from a 1% change in output. Is he correct? Why or why not? b. He also says that if 1 1, economies of scale are indicated, whereas if 1 1, diseconomies of scale are indicated. Is he correct? Why or why not? c. According to Tacke’s president, the value of 3 can be estimated by regressing log (TC/PK) on log Q and log (PL/PK). Is he correct? Why or why not? Solution: a. The president is correct that a1 is the elasticity of cost with respect to output. This can be proven by writing out the elasticity as dTC/dQ(Q/TC) or by noting that d log TC/d log Q  1. Note: Observe that the long-run cost function is a multiplicative function; therefore, as in the case of multiplicative demand functions or production functions, the exponents represent elasticities. No calculation is required once this observation is made. b. Diseconomies of scale exist when the average cost per unit of output decreases as output increases. We have d(TC/Q)/dQ  (1  1)TC/Q. Thus, we see that if 1 1, the above expression is negative and therefore there are economies of scale, and if 1 1, then the expression is positive and there are diseconomies of scale. Therefore, we conclude that the firm’s president is correct.

The Analysis of Costs | 99 c. The president is correct as long as total costs are homogeneous of degree 1 in input prices; this is a common assumption in economics. TC  0 Q1(PL)2(PK)3 TC/PK  0 Q1(PL)2(PK)31 log (TC/PK)  log 0  1 log Q  2log PL  (3  1) log PK If 2  3  1, log (TC/PK)  log 0  1 log Q  2 log (PL/PK) Therefore, from this, we observe that the president is correct only if the condition 2  3  1 is satisfied. Otherwise, he is not correct. Notice also that if the above condition holds, then estimating 2 provides an estimate for 3 . Now, under the above condition, 0, 1, 2,and 3 can be estimated, or derived from estimated coefficients, using multiple linear regression techniques from Chapter 5. 8. Engineers sometimes rely on the “0.6 rule,” which states that the increase in cost is given by the increase in capacity raised to the 0.6 power; that is, C2  C1(X2/X1)0.6 where C1 and C2 are the costs of two pieces of equipment, and X1 and X2 are their respective capacities. a. Does the 0.6 rule suggest economies of scale? b. Some experts have stated that in the chemical and metal industries, the 0.6 rule can be applied to entire plants rather than individual pieces of equipment. If so, will the long-run average cost curve in these industries tend to be negatively sloped? c. Can you think of a way to test whether this rule is correct? Solution: a. C2/C1  (X2/X1)0.6 C2  (X2/X1)0.6 C1 log C2  0.6 log X2  0.6 log X1  log C1 d log C2/d log X2  0.6 → economies of scale b. Yes, economies of scale at the plant level would lead to downwardsloping long-run average cost curves unless there were diseconomies of multiplant management. c. If cost and output data could be gathered and differences in other variables such as input prices could be controlled for, regression analysis might identify a long-run average cost curve for the industry.

100 | Chapter 6 9. The Dijon Company’s total variable cost function is TVC  50Q  10Q2  Q3 where Q is the number of units of output produced. a. What is the output level where marginal cost is a minimum? b. What is the output level where average variable cost is a minimum? c. What is the value of average variable cost and marginal cost at the output specified in the answer to part (b)? Solution: a. MC  50  20Q  3Q2. dMC/dQ  –20  6Q  0 → Q  20/6 or 10/3. Of course, the second derivative of MC is positive, indicating that, indeed, the above quantity is a minimizer for MC. b. AVC  50  10Q  Q2. Average variable cost is at a minimum at the quantity where marginal cost equals average variable cost. We can find this quantity by setting MC  AVC and solving for Q, or we can set dAVC/dQ  0 and solve for Q. Either way Q*  5. c. MC(Q  5)  AVC(Q  5)  25 10. The Berwyn Company is considering the addition of a new product to its product line. The firm has plenty of excess manufacturing capacity to produce the new product, and its total fixed costs would be unaffected if the new product were added to its line. Nonetheless, the firm’s accountants decide that a reasonable share of the firm’s present fixed costs should be allocated to the new product. Specifically, they decide that a $300,000 fixed charge will be absorbed by the new product. The variable cost per unit of making and selling the new product is $14, which is composed of the following: Direct labor Direct materials Other Total

$8.20 1.90 3.90 $14.00

a. Should the Berwyn Company add the new product to its line if it can sell about 10,000 units of this product at a price of $25? b. Should it add the new product if it can sell about 10,000 units at a price of $20? c. Should it add the new product if it can sell about 10,000 units at a price of $15? d. What is the minimum price for the new product that will make it worthwhile for Berwyn to add the new product to its line?

The Analysis of Costs | 101 Solution: a. AVC  $14. AFC(Q  10,000)  $300,000/10,000  $30. Therefore, ATC(Q  10,000)  $44. If the accountants prevail, the product will not be introduced at a price of $25 since this will not cover the ATC at the level of sales achievable at this price. Nonetheless, the price of $25 exceeds the average variable cost of $14, so the product should be introduced. b. As in part a, the price does not cover the average total cost but does cover the average variable cost, so the product should be introduced. c. As in part a, the price does not cover the average total cost but does cover the average variable cost, so the product should be introduced. d. $14. 11. The Jolson Corporation produces 1,000 wood cabinets and 500 wood desks per year, the total cost being $30,000. If the firm produced 1,000 wood cabinets only, the cost would be $23,000. If the firm produced 500 wood desks only, the cost would be $11,000. a. Calculate the degree of economies of scope. b. Why do economies of scope exist? Solution: TC(Q1  1,000  Q2  500)  $30,000 TC(Q1  1,000)  $23,000 TC(Q2  500)  $11,000 Since the joint cost, $30,000, is smaller than the sum of the separate costs, $34,000, then economies of scope exist. a. The extent to which economies of scope exist is measured by the ratio TC (Q1 = 1,000) + TC (Q2 = 500) − TC (Q1 = 1,000, Q2 = 500) TC (Q1 = 1,000 + Q2 = 500) $23,000 + $11,000 − $30,000 = = 0.13333. $30,000

S=

b. Economies of scope occur because of excess capacity that can be utilized more fully when additional products are produced or because of byproducts from one process that can be economically captured only if another product is produced in close proximity. 12. The Smith Company made and sold 10,000 metal tables last year. When output was between 5,000 and 10,000 tables, its average variable cost was $24. In this output range, each table contributed 60% of its revenue to fixed costs and profit.

102 | Chapter 6 a. What was the price per table? b. If the Smith Company increases its price by 10%, how many tables will it have to sell next year to obtain the same profit as last year? c. If the Smith Company increases its price by 10%, and if its average variable cost increases by 8% as a result of wage increases, how many tables will it have to sell next year to obtain the same profit as last year? Solution: AVC  $24 for 5,000 Q 10,000 a. Variable costs  0.4PQ AVC  0.4P → P  $60 b. Fixed costs  profits  0.6[$60(10,000)]  $360,000 At P  $66, P  AVC  $42. $360,000/$42  8,571.43 c. AVC  1.08($24)  $25.92 At P  $66, P – AVC  $40.08. $360,000/$40.08  8,982.036

CHAPTER 7

Perfect Competition

Lecture Notes 1. Introduction •

Objectives ÿ To explain how managers should respond to different competitive environments (or market structures) in terms of pricing and output decisions ÿ To examine pricing behavior in perfectly competitive markets ÿ To briefly investigate four broad categories of markets: * Perfect competition * Monopoly * Monopolistic competition * Oligopoly ÿ Market power * A firm’s pricing market power depends on its competitive environment. * In perfectly competitive markets, firms have no market power. They are “price takers.” They make decisions based on the market price, which they are powerless to influence. * In markets that are not perfectly competitive (which describes most markets), most firms have some degree of market power. ÿ Strategy in the absence of market power ÿ Firms cannot influence price and, because products are not unique, they cannot influence demand by advertising or product differentiation. ÿ Managers in this environment maximize profit by minimizing cost, through the efficient use of resources, and by determining the quantity to produce.

103

104 | Chapter 7 2. Market Structure •

•

•

•

•

Perfect Competition: When there are many firms that are small relative to the entire market and produce similar products Characteristics: ÿ Managers are price takers: They accept the decision of the aggregate market. They are ruled by the “invisible hand.” ÿ Products are standardized (identical). ÿ There are no barriers to entry or exit. ÿ There is no nonprice competition. * Example: Some sectors of agriculture Imperfect Competition ÿ Firms have some degree of market power and can determine prices strategically. ÿ Products may not be standardized. ÿ Firms employ nonprice competition. * Product differentiation * Advertising * Branding * Public relations Monopolistic Competition: When there are many firms and consumers, just as in perfect competition; however, each firm produces a product that is slightly different from the products produced by the other fi rms. ÿ There are no barriers to entry. ÿ Example: Retail trade Monopoly: Markets with a single seller ÿ Barriers to entry prevent competitors from entering the market. ÿ A monopolist is likely to have strong control over price. ÿ Example: Public utilities Oligopoly: Markets with a few sellers ÿ There are significant barriers to entry. ÿ It is the most prevalent category in present-day business. ÿ Examples: Automobile manufacturing and oil refining ÿ Barriers to Entry: Barriers that determine how easily firms can enter an industry, depending on the market structure * In perfect competition and monopolistic competition, barriers to entry are low. * There are considerable barriers to entry in oligopoly. * In a monopoly, entry is blocked.

Perfect Competition | 105 3. Market Price in Perfect Competition • •

Market price is determined by the intersection of the market demand curve and the market supply curve. Example ÿ Demand: P  22  0.5QD ÿ Supply: P  4  0.25QS ÿ Equilibrium: Obtained by setting demand equal to supply; that is, ÿ 22  0.5Q  4  0.25Q and solving for Q gives Q  24 thousand units and P  $10. If there are 1,000 firms in the market, then each produces 24 units. If one firm alters output, there will be virtually no effect on market price, so each firm faces a nearly horizontal demand curve. Fact: The demand curve for the output of the entire industry is downward sloping, but the demand curve for the output of a single firm is regarded as horizontal.

4. Shifts in Supply and Demand Curves • • •

It is important for managers to understand the factors that cause supply and demand curves to shift. They have significant consequences for firm performance, and managers must try to anticipate them and respond as best as they can. Two of the most important factors causing shifts in supply curves: ÿ Technological advances * Advances in technology tend to shift a product’s supply curve to the right. ÿ Changes in input prices * Increases in input prices tend to shift a product’s supply curve to the left because they push up the firm’s costs.

5. The Output Decision of a Perfectly Competitive Firm • • • •

Question: How much output should firms operating in a perfectly competitive environment produce? Suppose the firm chooses the output level that maximizes profit. Profit is maximized at the quantity of output (Q) where marginal revenue equals marginal cost and marginal cost is increasing. Firm’s profit:   TR – TC  PQ – TC, where P is fixed. Now, ÿ /Q  TR/Q  TC/Q  0 ÿ Marginal revenue (MR)  TR/Q  P (since P does not depend on Q) ÿ Marginal cost (MC)  TC/Q.

106 | Chapter 7

•

Hence, the first-order condition for profit maximization entails that P  MC, where MR  P. * There is no above-normal economic profit. * The nature of competition is to grind the price down to marginal cost. Profit Maximization Example ÿ Market price (P)  $10 ÿ Total revenue (TR)  PQ  10Q ÿ Total cost (TC)  1  2Q Q2 ÿ Profit ()  PQ–TC  10Q  (1  2Q  Q2)  8Q  1  Q2 ÿ MR  10 ÿ MC  2  2Q ÿ MR  MC → Q  4 ÿ Table 7.2: Cost and Revenues of a Perfectly Competitive Firm ÿ Figure 7.2: Relationship between Total Cost and Total Revenue of a Perfectly Competitive Firm ÿ Figure 7.3: Relationship of Profit and Output of a Perfectly Competitive Firm ÿ Table 7.3: Marginal Revenue and Marginal Cost: Perfectly Competitive Firm ÿ Figure 7.4: Marginal Revenue and Marginal Cost of a Perfectly Competitive Firm

6. Setting the Marginal Cost Equal to the Price • • • • •

Managers in perfectly competitive markets often accrue negative profits (or loss), even if they satisfy the preceding rules. All the manager can do in the short run is to produce at a loss or discontinue production. The decision to close a plant should answer one question: Does the product price cover the average variable cost? Case: Figure 7.5 Shutdown Point: When the price equals the minimum average variable cost ÿ If price is greater than average variable cost, produce a level of output where marginal cost is equal to price, even if this results in negative profit. Profit will exceed that which would result from shutting down. ÿ If price is less than average variable cost, shut down and produce no output. Negative profit will be equal to total fixed costs. ÿ Figure 7.5: Short-Run Average and Marginal Cost Curves

Perfect Competition | 107 STRATEGY SESSION: Forecasting the Price of Salmon Discussion Questions 1. Does it seem reasonable to assume that the supply of salmon will be unaffected by a 10% decrease in price? If it is affected, how will it influence the predicted price change? Answer: If quantity supplied declines as price falls, as would seem likely, then a smaller decrease in price will be required to bring the market to equilibrium. 2. Suppose that the demand for salmon is projected to exceed supply by 15%. What do you predict will happen to price? Answer: Price will rise by up to 10% to equate supply and demand. However, because of the response of quantity supplied to the higher price, it is unlikely that a 10% price increase will be necessary to return the market to equilibrium. PROBLEM SOLVED: The Kadda Company Discussion Questions 1. Suppose that, due to an increase in fixed costs, the firm’s new total cost curve is: TC  1,000  6Q  2Q2. How many units should be produced if P  30, and what is AVC at the profit-maximizing rate of production? Answer: Use the rule: P  MC. We have then dTC/dQ  MC  6  4Q  30  P → Q  6. Also, AVQ  TVC/Q  (6Q  2Q2)/Q  6  2Q. If Q  6, then AVC  $18. 2. Suppose that, due to an increase in variable costs, the firm’s new total cost curve is: TC  800  10Q  2Q2. How many units should be produced if P  30, and what is AVC at the profit-maximizing rate of production? Answer: dTC/dQ  MC  10  4Q  30  P → Q  5 and AVC  20 •

Summary: If the manager maximizes profit or minimizes loss, the output is set so that the short-run marginal cost equals the price and the marginal cost is rising.

108 | Chapter 7 7. Another Way of Viewing the Price Equals Marginal Cost Profit-Maximizing Rule • •

•

•

Question: If a firm has one fixed input (say capital) and one variable input (say labor, L), how much of its variable input should it utilize? Marginal Revenue Product (MRP): The amount an additional unit of the variable input adds to the firm’s total revenue ÿ Marginal revenue product of labor  MRPL ÿ MRPL  TR/L  (TR/Q)(Q/L)  (MR)(MPL) Marginal Expenditure on Labor (MEL): The amount an additional unit of labor adds to the firm’s total costs ÿ MEL  TC/L  (TC/Q)(Q/L)  (MC)(MPL) Profit is maximized where the employment of the variable input is such that marginal revenue product is equal to marginal expenditure. ÿ Equivalent to MR  MC in terms of output ÿ Managers should stop expanding output when marginal revenue equals marginal cost.

8. Producer Surplus in the Short Run •

•

Producer Surplus: The difference between the market price and the price the producer is willing to receive for a good or service (the producer’s reservation price) ÿ A firm’s reservation price is the marginal cost of production above the shutdown point. ÿ Producer surplus is a firm’s variable cost profit: TR – TVC. ÿ Producer surplus is the difference between a firm’s supply curve and the market price under perfect competition. ÿ Social Welfare: At a given price, it is measured by the sum of consumer surplus and producer surplus. Examples * Figure 7.6: Producer Surplus and Variable-Cost Profit * Figure 7.7: Market Social Welfare (A  B) of a Perfectly Competitive Price Policy, P*

9. Long-Run Equilibrium of the Firm • •

Question: In the long run, how much will managers in a competitive firm produce? Conditions ÿ Quantity produced is such that profit is equal to zero and price is equal to: * the lowest point on the long-run average (total) cost (LAC) curve and the relevant short-run total cost curve. * long-run marginal cost and short-run marginal cost.

Perfect Competition | 109

•

•

ÿ Adjustment to equilibrium ÿ If P LAC, then economic profit is earned and new firms enter the industry. This increases supply, thereby driving down price and hence profit. ÿ If P LAC, the firm will exit the industry. As firms exit, supply falls, causing price and profit to rise. Only when economic profit is zero (which means that long-run average cost curve equals price ) is a firm in long-run equilibrium. ÿ If firms are earning negative profits, then firms will exit the industry, market supply will decrease, and price will rise to the long-run equilibrium level. ÿ If firms are earning positive profits, then firms will enter the industry, market supply will increase, and price will fall to the long-run equilibrium level. Illustration ÿ Figure 7.8: Long-Run Equilibrium of a Perfectly Competitive Firm

PROBLEM SOLVED: Output at the Bergey Company Discussion Questions 1. Suppose that long-run average cost changes to AC  200  4Q  0.05Q2. What is the quantity at which AC is at a minimum? What is the value of AC and MC at this rate of production? What is the long-run equilibrium price? Answer: dAC/dQ 4  0.1Q  0 → Q  40 units per day and AC(Q  40)  MC(Q  40)  120. This means that the long-run equilibrium price is $120. 2. Suppose that long-run average cost changes to AC  200  3Q  0.05Q2. What is the quantity at which AC is at a minimum? What is the value of AC and MC at this rate of production? What is the long-run equilibrium price? Answer: dAC/dQ  –3  0.1Q  0 → Q  30 and AC  MC  155, and the long-run equilibrium price is $155. 10. The Long-Run Adjustment Process: A Constant-Cost Industry •

•

Constant- Cost Industry: An industry in which an increase in output does not lead to an increase in input prices ÿ Horizontal long-run supply curve Figure 7.9: Long-Run Equilibrium in a Constant-Cost Industry

110 | Chapter 7 11. The Long-Run Adjustment Process: An Increasing-Cost Industry •

• •

Increasing- Cost Industry: An industry in which an increase in output leads to an increase in input prices ÿ Upward-sloping long-run supply curve Some industries are decreasing-cost industries. ÿ Downward-sloping long-run supply curves Figure 7.10: Long-Run Equilibrium in an Increasing-Cost Industry

12. How a Perfectly Competitive Economy Allocates Resources •

Example: Demand for corn increases and demand for rice decreases. ÿ Short-run equilibrium * The price of corn will increase, the quantity of corn produced will increase, and corn producers will earn positive economic profits. * The price of rice will decrease, the quantity of rice produced will decrease, and rice producers will earn negative economic profits. ÿ Long-run equilibrium * Firms will reallocate resources away from the production of rice and toward the production of corn. * The price of corn will decrease from its high, the quantity of corn produced will increase further, and corn producers will find that their economic profits decline to zero. The price of rice will increase from its low, the quantity of rice * produced will increase from its low, and rice producers will find that their economic profits rise to zero.

Chapter 7: Problem Solutions 1. The Hamilton Company is a member of a perfectly competitive industry. Like all members of the industry, its total cost function is TC  25,000  150Q  3Q2 where TC is the firm’s monthly total cost (in dollars) and Q is the firm’s monthly output. a. If the industry is in long-run equilibrium, what is the price of the Hamilton Company’s product? b. What is the firm’s monthly output? Solution: a. Since Hamilton is a representative firm, we can determine the competitive long-run equilibrium price by finding Hamilton’s minimum average total cost.

Perfect Competition | 111 ATC  TC/Q  25,000/Q  150  3Q dATC/dQ  –25,000/Q2  3  0 → Q*  (25,000/3)1/2  91.287 ATC(Q  Q*)  25,000/91.287  150  3(91.287)  697.72 Alternatively, we can find the long-run equilibrium by setting LAC  LMC, where LAC and LMC are the long-run average and marginal cost functions. b. Hamilton will produce where marginal cost equals the price, $697.72. Marginal cost equals average total cost where average cost is minimized, so Hamilton will produce 91.287 per month. 2. In 2012, the box industry was perfectly competitive. The lowest point on the long-run average cost curve of each of the identical box producers was $4, and this minimum point occurred at an output of 1,000 boxes per month. The market demand curve for boxes was QD  140,000  10,000P where P was the price of a box (in dollars per box) and QD was the quantity of boxes demanded per month. The market supply curve for boxes was QS  80,000  5,000P where QS was the quantity of boxes supplied per month. a. What was the equilibrium price of a box? Is this the long-run equilibrium price? b. How many firms are in this industry when it is in long-run equilibrium? Solution: a. Since the industry is perfectly in equilibrium, each firm takes the market price as given and merely reacts to it. This equilibrium price is given by setting demand equal to supply. That is, 80,000  5,000P  140,000  10,000P This gives P  $4, and the equilibrium quantity is 100,000. The long-run equilibrium price must be $4 since this is the minimum long-run average total cost of all the firms. Thus the short-run equilibrium price is equal to the long-run equilibrium price. b. Each firm in the industry is producing 1,000 boxes at the price of $4 per box because the firm is operating in equilibrium. We have found that the industry total output is 100,000 boxes. Therefore, the number of firms in the industry is 100,000/1,000  100. 3. The Burr Corporation’s total cost function (where TC is the total cost in dollars and Q is quantity) is TC  200  4Q  2Q2

112 | Chapter 7 a. If the firm is perfectly competitive and the price of its product is $24, what is its optimal output rate? b. At this output rate, what is its profit? Solution: a. Given that the firm is perfectly competitive, use the P  MC rule. That is, set MC  4  4Q  24 P and get Q  5. Also, we observe that the MC function is increasing given that it is a straight line with a positive slope of 4. Hence the second-order condition for profit maximization is also satisfied. Therefore 5 units is the optimal output rate. b. Profit  24(5)  [200  4(5)  2(5)2]  120  270  –150, which is a loss. 4. The supply and demand curves for pears are QS  10,000P QD  25,000  15,000P where QS is the quantity (tons) supplied, QD is the quantity (tons) demanded, and P is the price per pear (in hundreds of dollars per ton). a. Plot the supply and demand curves. b. What is the equilibrium price? c. What is the equilibrium quantity? Solution: a. Price 2.5

2

1.5

1

Demand Supply 0.5

0 0

5

10

15

Quantity in Thousands

20

25

Perfect Competition | 113 b. Setting QS  QD, we get 25,000  15,000P  10,000P → P  1. c. Q  10,000. 5. The White Company is a member of the lamp industry, which is perfectly competitive. The price of a lamp is $50. The firm’s total cost function is TC  1,000  20Q  5Q2 where TC is total cost (in dollars) and Q is hourly output. a. What output maximizes profit? b. What is the firm’s economic profit at this output? c. What is the firm’s average cost at this output? d. If other firms in the lamp industry have the same cost function as this firm, is the industry in equilibrium? Why or why not? Solution: a. MC  20  10Q. Given that the firm is perfectly competitive, then use the P  MC rule to find the equilibrium quantity. That is, 20  10Q  50. Thus Q  3. Moreover, we observe that the MC curve is an increasing function of Q since it is a straight line with a positive slope of 10. Thus, the quantity Q  3 lamps per hour maximizes profit. b. Profit  TR(Q  3)  TC(Q  3)  50(3)  [1,000  20(3)  5(3)2]  –$955 c. ATC  [1,000  20(3)  5(3)2]/3  $368.33 d. None of the firms are covering their fixed costs of $1,000. In fact, the firms are experiencing big losses relative to revenues, so the industry cannot be in equilibrium. Firms will exit until the price increases to the minimum average cost of the remaining firms in the industry (to about $161.42). 6. The long-run supply curve for a particular type of kitchen knife is a horizontal line at a price of $3 per knife. The demand curve for such a kitchen knife is QD  50  2P where QD is the quantity of knives demanded (in millions per year) and P is the price per knife (in dollars). a. What is the equilibrium output of such knives? b. If a tax of $1 is imposed on each knife, what is the equilibrium output of such knives? (Assume the tax is collected by the government from the suppliers of knives.) c. After the tax is imposed, you buy such a knife for $3.75. Is this the longrun equilibrium price? Solution: a. The demand curve can be rewritten as PD  25 0.5QD, and the supply curve is given as PS  3. Setting PD  PS and solving for QD, we get Q  44.

114 | Chapter 7 b. The price will now have to cover the $1 tax. Setting PD  PS  1 and solving for QD, we get Q  42. c. No, the long-run equilibrium price must be $4. If the price initially is $3.75, the existing firms are only receiving $2.75 per knife after they pay the $1 tax and so must be losing money. Exit from the industry will occur until the net price received by the firms increases back to $3.

CHAPTER 8

Monopoly and Monopolistic Competition

Lecture Notes 1. Introduction •

Objectives ÿ To investigate how managers set price and output when they have market power ÿ To investigate how managers act when they have market power ÿ To examine issues the monopolist manager must consider in choosing the optimal price and quantity combination ÿ To show the profit-maximizing rule for managers in monopolistic competitive markets. Managers still have market power, but they must deal with intramarket rivals. ÿ To analyze a common pricing strategy: Cost-plus pricing ÿ To examine the multiproduct firm: Demand interrelationships ÿ To analyze monopsony ÿ To investigate advertising and promotion policies ÿ To discuss a simple rule with respect to advertising expenditures ÿ To investigate the empirical evidence on the usefulness of advertising, brand equity, and the price elasticity of demand with respect to managerial behavior ÿ Monopolistically competitive firms have market power based on product differentiation, but barriers to entry are modest or absent.

2. Pricing and Output Decisions in Monopoly •

The demand curve faced by managers of monopolies is downward sloping. ÿ With monopoly power, the firm’s demand curve is the market demand curve. A monopolist is the only seller of a product for which there are no close substitutes, and is protected by barriers to entry.

115

116 | Chapter 8 •

Managers with market power must decide both price and quantity. They are no longer passive price takers. They have more strategy power and are rewarded with higher economic profit. • Managers with monopoly power do not have to consider the actions of market rivals because there are none. They still have to understand: * product competition * spatial competition * temporal competition * Fact: An unregulated monopolist maximizes profit by choosing the price and output where the difference between total revenue and total cost is the highest. Profit:   TR  TC. Profit maximization requires that d/dQ  0 . That is, MR  MC. Fact: Under monopoly, as under perfect competition, managers maximize profit if they set output at the point where MR  MC. Also, to fulfill the second managerial rule of profit maximization, we must have PM AVC. Example: Consider a monopolist with a demand curve given by the demand function P  10  Q. ÿ Total revenue: TR  PQ  10Q  Q2 ÿ Marginal revenue: MR  10  2Q ÿ Total cost: TC  1  Q  0.5Q2 ÿ Marginal cost: MC  1  Q ÿ For profit maximization, set MR  10  2Q  1  Q  MC → Q  3 and P  10  3  7. Also, AVC(Q  3)  1  0.5(3)  2.5 and therefore the second managerial condition for profit maximization is satisfied. ÿ Profit  Q(P  ATC) ÿ Table 8.1: Cost, Revenue, and Profit of a Monopolist ÿ Figure 8.1: Total Revenue, Total Cost, and Total Profit of a Monopolist ÿ Figure 8.2: Profit and Output of a Monopolist ÿ Table 8.2: Marginal Cost and Marginal Revenue of a Monopolist ÿ Figure 8.3: Marginal Revenue and Marginal Cost of a Monopolist ÿ Observation: Relative to managers in perfectly competitive markets, monopolists choose a higher price and a lower output. ÿ Suppose, in the previous example, managers were forced to behave as  perfectly competitive. Then they would set P  MC. That is, P  10  Q  1  Q; that is, Q  4.5 and P  $5.5. • Marginal Revenue, Elasticity, and Price ÿ Unlike perfect competition, MR is less than price and depends on Q. ÿ Formula: MR  P[1  (1/h)]  P[1  (1/|h|)]  P  P/|h| P. * A profit-maximizing monopolist will not produce where demand is inelastic; that is, where |h| 1, because MR 0.

Monopoly and Monopolistic Competition | 117 *

In equilibrium, MC  MR  P[1  (1/|h|)], so the profit-maximizing price is P 

MC ⎡ ⎛ 1 ⎞⎤ ⎢1 − ⎜ ⎟ ⎥ ⎢⎣ ⎝ | | ⎠ ⎥⎦

* Clearly, P MC. ÿ Figure 8.4: Output and Price Decisions of a Monopolist PROBLEM SOLVED: The McComb Company Discussion Questions Consider a monopolist producing and selling a product with the demand curve: P  30  6Q (8.4) where P is price (in thousands of dollars) and Q is the firm’s output (in thousands of units). The firm’s total cost function is TC  14  3Q  3Q2 (8.5) where TC is total cost (in millions of dollars). We have TR  PQ  (30  6Q)Q  30Q  6Q2 MR  30  12Q and MC  3  6Q Set MR  MC; that is, 30  12Q  3  6Q; that is, Q  1.5 and P  $21. Thus, to maximize profit, managers should set a price of $21,000 and produce and sell 1,500 units, and the resulting profit is $6.25 million. Also, managers in both markets must choose a price higher than AVC. • Two Additional Examples 1. A monopolist has a demand curve defined as Q  28  2P. The firm’s total cost curve is TC  10  2Q  Q2. What price and output will maximize the firm’s profit, and how much profit will be earned? Answer: Set MR  14  Q  2  2Q  MC → Q  4, P  12,   48  34  14 2. A monopolist has a demand curve defined as Q  28  4P. The firm’s total cost curve is TC  10  2Q  Q2. What price and output will maximize the firm’s profit, and how much profit will be earned? Answer: MR  7  0.5Q  2  2Q  MC → Q  2, P  6.5,   13  18  5 Also, notice that AVC(Q  2)  2  2  $4 and therefore P AVC; that is, the second managerial condition for profit maximization is satisfied.

118 | Chapter 8 STRATEGY SESSION: Franchiser versus Franchisee? Discussion Question 1. In the motion picture industry, top performers and other artists are paid, at least in part, with “points,” or a percentage of gross revenue. How do you think this influences their DVD pricing preferences? Answer: They prefer that DVD prices be lower than the profit-maximizing price. 3. Cost-Plus Pricing • • •

• •

Many managers use the simple heuristic of cost-plus pricing. Many managers act as if cost is the primary driver of price. Cost-Plus Pricing: Simplistic strategy that guarantees that price is higher than the estimated average cost ÿ Studies of pricing behavior suggest that many managers use cost-plus pricing and do not price optimally. ÿ Percentage markup of this strategy is expressed as Markup  (Price  Cost)/Cost (8.6) ÿ Price  (Cost)(1  Markup) ÿ Example: Price  6, Cost  4, Markup  0.50 Profit margin: The price of a product minus its cost ÿ Profit margin  Price  Cost Target Return: What managers hope to earn and what determines the markup ÿ P  L  M  K  (F/Q)  (A/Q) * L  unit labor cost * M  unit material cost * K  unit marketing cost * F  total fixed costs * Q  units to produce * A  gross operating assets *   desired profit rate (%) ÿ Example * L2 * M1 * K3 * F  10,000 * Q  1,000 * A  100,000 *   15% * P  2  1  3  (10,000/1,000)  0.15(100,000,1,000)  $31

Monopoly and Monopolistic Competition | 119 •

•

•

•

Allocation of Indirect Cost among Products ÿ Often done on the basis of average variable costs ÿ Example: * Indirect costs  $3 million * Variable costs  $2 million * Indirect cost allocation  3/2  150% of variable cost * If a product’s variable cost is $10, then the allocated indirect cost is ($10)(1.5)  $15. If there is a 40% markup, then the product price is ($10  $15)(1.4)  $25. Cost-Plus Pricing at Therma-Stent ÿ Factory cost  $2,300 ÿ Markup  40%  $920 ÿ Price  $3,220 Cost-Plus Pricing at Internet Companies and Government-Regulated Industries ÿ Cost-plus pricing is often used by Internet companies and governmentregulated industries. Question: Can cost-plus pricing maximize profit? * This pricing technique does not explicitly account for important factors on both the demand and supply sides. For example, it does not consider the pricing behavior of rivals. * Fact: If applied well, cost-plus pricing may result in managers almost maximizing profit. ÿ Optimal markup  |h|/(|h|  1) (8.9) ÿ Fact: A manager can maximize profit using cost-plus pricing with a markup equal to the value specified in equation (8.9). ÿ Optimal markup is higher if demand is less elastic. ÿ Table 8.3: Relationship between Optimal Markup and Price Elasticity of Demand ÿ Observation: There is a negative association between elasticity and markup.

PROBLEM SOLVED: The Humphrey Corporation Discussion Questions 1. Automobile prices are notoriously negotiable, but the limits to negotiation are set by market conditions. How would the negotiating position of a seller, and consequently the markup, change if a car was rare and in high demand? What does this imply about the price elasticity of demand for the car? Answer: The seller would have more market power, and the markup would consequently be higher. The price elasticity of demand is relatively inelastic.

120 | Chapter 8 2. In bazaars around the world, haggling over price is a way of life. Do you think a seller’s reservation price (the lowest price that will be accepted) is influenced by demand elasticities? Explain. 4. The Multiple-Product Firm: Demand Interrelationships • •

•

Multiple-Product Firm (Good X and Good Y) Managers at monopolies need to recognize that a change in the price of quantity sold of one product may influence the demand for other products. ÿ Total revenue  TR  TR X  TRY * MR X  TR/QX  TR X /QX  TRY /QX * MRY  TR/QY  TR X /QY  TRY /QY ÿ If the two goods are substitutes, then TR X/QY and TRY /QX are negative. ÿ If the two goods are complements, then TR X/QY and TRY /QX are positive. Pricing of Joint Products: Fixed Proportions ÿ Total marginal revenue curve: The vertical summation of the tow marginal revenue curves for individual products ÿ Figure 8.5: Optimal Pricing for Joint Products Produced in Fixed Proportions (Case 1) * Marginal revenue of both products is positive at the optimal level of output. ÿ Figure 8.6: Optimal Pricing for Joint Products Produced in Fixed Proportions (Case 2) * The marginal revenue of one product is negative at the optimal level of output. * If a product’s marginal revenue is negative, then the firm will dispose of a quantity sufficient to bring marginal revenue to zero and thereby maximize revenue on that product.

PROBLEM SOLVED: Profit Maximizing at Humphrey Discussion Questions 1. Managers face the total cost function TC  100 Q  2Q2 (8.12) The demand curve for the firm’s two products are PA  200  QA (8.13) PB  150  2QB (8.14) Humphrey managers want to know how many units of each leg they should produce to maximize profit. Answer: Q  QA  QB  34.9 and PA  $165.10 and PB  $80.20.

Monopoly and Monopolistic Competition | 121 •

An Additional Example: Suppose that the demand for product B changes to PB  100  2QB. What is the new equilibrium price and output configuration? Answer: PA  170.1, PB  50, QA  29.9, and QB  25 (4.9 units are destroyed).

2. Suppose that the demand for product B changes to PB  100  2QB. What is the new equilibrium price and output configuration? Answer: PA  140.1, PB  280.2, QA  59.9, and QB  59.9. •

Output of Joint Products: Variable Proportions ÿ Isocost curve: Curve showing the amounts of goods produced at the same total cost ÿ Isorevenue lines: Lines showing the combinations of output of products that yield the same total revenue ÿ Optimal combinations of goods are found where isocost and isorevenue lines are tangent. ÿ Optimal total production is found where profit is maximized, which occurs at a point of tangency where the difference between cost and revenue is maximized.

5. Monopsony •

•

Monopsony: Markets that consist of a single buyer ÿ Contrast with monopoly markets that consist of a single seller. ÿ Buyers on a competitive market face a horizontal supply curve; they are price takers. ÿ There is only one buyer on a monopsony market, and this buyer faces the upward-sloping market supply curve, which means that marginal cost is above the supply price. ÿ The monopsonist controls price. ÿ Under monopsony, to maximize profit, managers will equate marginal benefit to marginal expenditure (or marginal cost). ÿ The optimal number of workers hired under a monopsony is less than the optimal number of workers hired under perfect competition. ÿ The optimal wage under monopsony is less than the optimal wage under perfect competition. Example: Monopsony labor market ÿ Labor supply: P  c  eQ ÿ Total cost: C  PQ  (c  eQ)Q ÿ Marginal cost: C/Q  c  2eQ  MC ÿ Figure 8.8: Optimal Monopsony Pricing ÿ The wage (P) and quantity hired (Q) are both less than at the competitive equilibrium.

122 | Chapter 8 6. Monopolistic Competition •

Characteristics of Monopolistic Competition ÿ Product differentiation—products are not perceived as identical by consumers. ÿ Managers have some pricing discretion, but because products are similar, price differences are relatively small. ÿ Competition takes place within a product group. • Product Group: Group of firms that produce similar products • Conditions that must be met, in addition to product differentiation, to define a product group as monopolistically competitive 1. There must be many firms in the product group. 2. The number of firms in the product group must be large enough that each firm expects its actions to go unheeded by its rivals and unimpeded by possible retaliatory moves on their part. 3. Entry into the product group must be relatively easy, and there must be no collusion, such as price fixing or market sharing, among managers in the product group. • Price and Output Decisions under Monopolistic Competition ÿ Figure 8.9: Short-Run Equilibrium in Monopolistic Competition. * Identical to short-run equilibrium under monopoly ÿ Figure 8.10: Long-Run Equilibrium in Monopolistic Competition * Entry and exit of firms from the product group shift individual firms’ demand curves. * Long-run equilibrium occurs where profit is equal to zero. 7. Advertising Expenditures: A Simple Rule •

Question: How much should a profit-maximizing manager spend on advertising? ÿ Assume diminishing returns to advertising expenditures ÿ Assume that quantity demanded depends only on price and advertising expenditures ÿ Table 8.4: Relationship between Advertising Expenditures and Quantity * Illustrates diminishing marginal returns (see below) between advertising expenditures (A) and quantity demanded (Q) A

Q

dQ/dA

0.80 0.90 1.00 1.10 1.20

15.00 17.00 18.50 19.50 20.00

20 15 10 5

Monopoly and Monopolistic Competition | 123 •

•

Derivation ÿ Net profit  P  MC (omitting advertising expenditures) ÿ Advertising expenditures are optimal if the increase in net profit from an additional dollar spent on advertising is equal to one dollar. ÿ If Q is defined as the number of extra units sold as a result of an additional dollar of advertising expenditures, then advertising expenditures are optimal when Q(P  MC)  1. ÿ The above implies that the marginal revenue from an extra dollar of advertising  |h| when advertising expenditures are optimal. Managers should therefore increase advertising expenditures until this condition is reached. Using Graphs to Help Determine Advertising Expenditure ÿ Figure 8.11: Optimal Advertising Expenditure * Curve A: Relationship between advertising expenditures and the absolute value of the price elasticity of demand—higher expenditures cause demand to become less elastic. * Curve B: Relationship between marginal revenue from an extra dollar of advertising expenditures and total advertising expenditures * The intersection between Curve A and Curve B defines the optimal level of advertising expenditures. * Curve B' represents a shift in Curve B due to increasing advertising effectiveness. It results in an increase in advertising expenditures.

8. Advertising, Price Elasticity, and Brand Equity: Evidence on Managerial Behavior •

•

•

Promotions ÿ Appeal to price sensitivity ÿ Price-oriented message ÿ Attempt to erode brand loyalty ÿ Attempt to increase price elasticity and limit the premiums consumers are willing to pay for brand-name products Advertising ÿ Attempts to build brand loyalty ÿ Loyalty is measured as the frequency of repeat purchases ÿ Product-quality-oriented message Evidence ÿ Promotions do increase the price elasticities of consumers. ÿ Promotions have less effect on brand loyalists. ÿ The effects of promotions decay over time ÿ Price elasticity of nonloyalists was found to be four times that of loyalists in one study. ÿ The effects of advertising on brand loyalty erode over time, and price becomes more important to consumers.

124 | Chapter 8 Chapter 8: Problem Solutions

1. Harry Smith owns a metal-producing firm that is an unregulated monopoly. After considerable experimentation and research, he finds that the firm’s marginal cost curve can be approximated by a straight line, MC  60  2Q, where MC is marginal cost (in dollars) and Q is output. The demand curve for the product is P  100  Q, where P is the product price (in dollars) and Q is output. a. If Smith wants to maximize profit, what output should he choose? b. What price should he charge? Solution: a. Harry should produce where marginal revenue equals marginal cost. Setting MR  dTR/dQ  d(100Q  Q2)/dQ  100  2Q and therefore, the condition yields 100  2Q  60  2Q → Q  10. b. Harry’s market clearing price at Q  10 is P  100  10  $90. 2. The Wilson Company’s marketing manager has determined that the price elasticity of demand for its product equals 2.2. According to studies she carried out, the relationship between the amount spent by the firm on advertising and its sales is as follows: Advertising Expenditure

Sales

$100,000 200,000 300,000 400,000

$1.0 million 1.3 million 1.5 million 1.6 million

a. If the Wilson Company spends $200,000 on advertising, what is the marginal revenue from an extra dollar of advertising? b. Is $200,000 the optimal amount for the firm to spend on advertising? c. If $200,000 is not the optimal amount, would you recommend that the firm spend more or less on advertising? Solution: a. Marginal revenue of advertising tells us about the increase in total revenue resulting from a $1 increase in the amount of advertising on the product. If the company is spending $200,000 on advertising, then over the range $200,000 and $300,000 for advertising, marginal revenue is given by 0.2/0.1  2. Therefore, a $1 increase in the amount of advertising results in a $2 increase in sales. Thus, the marginal revenue is 2. b. The optimal level of advertising occurs where the marginal revenue from an additional dollar spent on advertising equals the price elasticity of the

Monopoly and Monopolistic Competition | 125 demand curve. In this case if the elasticity is 2.2 and the advertising marginal revenue is 2, then advertising is not at its profit-maximizing level. c. The firm should spend more (but not $100,000 more) on advertising. 3. The Coolidge Corporation is the only producer of a par ticular type of laser. The demand curve for its product is Q  8,300  2.1P and its total cost function is TC  2,200  480Q  20Q2 where P is price (in dollars), TC is total cost (in dollars), and Q is monthly output. a. Derive an expression for the firm’s marginal revenue curve. b. To maximize profit, how many lasers should the firm produce and sell per month? c. If this number were produced and sold, what would be the firm’s monthly profit? Solution: a. We have TR  PQ. Let’s solve for P in terms of Q. We have 2.1P  8,300  Q; that is, P  3,952.381  0.4762Q. TR  PQ  (3,952.381  0.4762Q)Q  3,952.381Q  0.4762Q2 MR  dTR/dQ  3,952.381  0.9524Q b. Set MR  3,952.381  0.9524Q  MC  dTC/dQ  480  40Q 3,472.381  40.9524Q Q  84.8 c. Profit  [3,952.381  0.4762(84.8)](84.8)  [2,200  480(84.8)  20(84.8)2] Profit  331,737.54  186,724.80 Profit  $145,012.74 4. The Madison Corporation, a monopolist, receives a report from a consulting firm concluding that the demand function for its product is Q  78  1.1P  2.3Y  0.9A where Q is the number of units sold, P is the price of its product (in dollars), Y is per capita income (in thousands of dollars), and A is the firm’s advertising expenditure (in thousands of dollars). The firm’s average variable cost function is AVC  42  8Q  1.5Q2 where AVC is average variable cost (in dollars).

126 | Chapter 8 a Can we determine the firm’s marginal cost curve? b. Can we determine the firm’s marginal revenue curve? c. If per capita income is $4,000 and advertising expenditure is $200,000, can we determine the price and output where marginal revenue equals marginal cost? If so, what are they? Solution: a. Yes! We only need the TVC to calculate marginal cost. Given the AVC, we have TVC  42Q  8Q2  1.5Q3. MC  dTC/dQ  dTVC/dQ  42  16Q  4.5Q2 b. Solving for P gives P  (78  2.3Y  0.9A)/1.1  Q/1.1. Hence, TR  PQ  Q(78  2.3Y  0.9A)/1.1  Q2/1.1 MR  dTR/dQ  (78  2.3Y  0.9A)/1.1  2Q/1.1 We need information on per capita income and on advertising expenditures to determine marginal revenue as a function of Q alone. c. Set MR  MC. MR  242.9  1.82Q  42  16Q  4.5Q2  MC → 4.5Q2  14.18Q  200.9  0 Solving this using the quadratic formula, we get Q  8.44 and P  235.24. 5. The Wilcox Company has two plants with the marginal cost functions MC1  20  2Q1 MC2  10  5Q2 where MC1 is marginal cost in the first plant, MC2 is marginal cost in the second plant, Q1 is output in the first plant, and Q2 is output in the second plant. a. If the Wilcox Company minimizes its costs and produces five units of output in the first plant, how many units of output does it produce in the second plant? Explain. b. What is the marginal cost function for the firm as a whole? c. Can we determine from these data the average cost function for each plant? Why or why not? Solution: a. Wilcox is minimizing its costs if it is using both plants at outputs that equalize their marginal costs. That is set MC1  MC2. If Wilcox is producing 5 units in the first plant, its marginal cost there is MC1  20  2(5)  30. For MC2 to equal 30, we set 10  5Q2  30; that is, Q2  4. Wilcox must produce 4 units at the second plant.

Monopoly and Monopolistic Competition | 127 b. Setting MC1  MC2 → Q1  2.5Q2  5, so Q  Q1  Q2  3.5Q2  5. Q2  (Q  5)/3.5 MC  MC1  MC2  10  5(Q  5)/3.5  (10/7)(Q  12) Note: Here is the rationale behind the above methodology for this type of cost-minimization procedure for a multiproduct plant. Suppose the total cost function is given by C(Q1, Q2)  TC(Q). Now, differentiating this with respect to Q1, with the proviso that Q  Q1  Q2, the total output produced and using the chain rule, we have MC1  ∂C(Q1,Q2)/∂Q1  (∂TC/∂Q)(∂Q/∂Q1)  MC(Q), because ∂Q/∂Q1  1. Similarly, MC2  ∂C(Q1,Q2)/∂Q2  (∂TC/∂Q)(∂Q/∂Q2)  MC(Q), because ∂Q/∂Q2  1. Hence, we see that MC  MC1  MC2. This also indicates that cost minimization requires that MC  MC1  MC2. c. We can determine average variable cost for each plant including marginal cost for producing at both plants, but because we don’t have information on fixed costs, we cannot determine average total cost for each plant. 6. If the Rhine Company ignores the possibility that other firms may enter its market, it should set a price of $10,000 for its product, which is a power tool. But if it does so, other firms will begin to enter the market. During the next two years it will earn $4 million per year, but in the following two years it will earn $1 million per year. On the other hand, if it sets a price of $7,000, it will earn $2.5 million in each of the next four years because no entrants will appear. a. If the interest rate is 10%, should the Rhine Company set a price of $7,000 or $10,000? Why? (Consider only the next four years.) b. If the interest rate is 8%, should the Rhine Company set a price of $7,000 or $10,000? Why? (Consider only the next four years.) c. The results in parts (a) and (b) pertain to only the next four years. How can the firm’s managers extend the planning horizon? Solution: a. If a price of $10,000 is chosen, the present discounted value of the profit stream is given by 4/1.1  4/(1.1)2  1/(1.1)3  1/(1.1)4  $8.376477 million If the price is set at $7,000, then the present discounted value of the profit stream is given by 2.5/1.1  2.5/(1.1)2  2.5/(1.1)3  2.5/(1.1)4  $7.924663 million Therefore, we see that a price of $10,000 should be set.

128 | Chapter 8 b. At an 8% interest rate, the present discounted value of profits is given by 4/1.08  4/(1.08)2  1/(1.08)3  1/(1.08)4  $ 8.661921 million (for a $10,000 price) and 2.5/1.08  2.5/(1.08)2  2.5/(1.08)3  2.5/(1.08)4  $8.28 million. If the interest rate were zero, then the two profit streams would have equal value of 4  4  1 1  $10 million (for a $10,000 price) and 2.5(4) $10 million (for a $7,000 price). For any positive rate of interest the $10,000 price yields a higher present discounted value because it has the same undiscounted total cash flow and has more of it sooner than the $7,000 price does. c. The firm would need to estimate profits for these additional years and include them in the present discounted value calculations. 7. During recessions and economic hard times, many people—particularly those who have difficulty getting bank loans—turn to pawnshops to raise cash. But even during boom years, pawnshops can be profitable. Because the collateral that customers put up (such as jewelry, guns, or electric guitars) is generally worth at least double what is lent, it generally can be sold at a profit. And because usury laws allow higher interest ceilings for pawnshops than for other lending institutions, pawnshops often charge spectacularly high rates of interest. For example, Florida’s pawnshops charge interest rates of 20% or more per month. According to Steven Kent, an analyst at Goldman, Sachs, pawnshops make 20% gross profit on defaulted loans and 205% interest on loans repaid. a. In 2012 there were about 15,000 pawnshops in the United States. This was much higher than in 2007, when the number was about 12,000. Why did the number increase? b. In a particular small city, do the pawnshops constitute a perfectly competitive industry? If not, what is the market structure of the industry? c. Are there considerable barriers to entry in the pawnshop industry? (Note: A pawnshop can be opened for less than $250,000, but a number of states have tightened licensing requirements for pawnshops.) Solution: a. The pawnshop business appears to be highly profitable on the basis of the problem. We should expect that entry into this industry would follow such high profits unless there were strong barriers to entry. If the current entry in the industry were not enough to eliminate all the supranormal profits being earned, we should expect continued entry until profits were eliminated. b. The market for pawnshop services is likely to be quite local, as local perhaps as neighborhoods. In this case even though there might be 100 pawnshops in a large city, the market for any individual shop is likely to be oligopolistic or monopolistic.

Monopoly and Monopolistic Competition | 129 c. Barriers to entry appear low in this industry given that a pawnshop can be opened for less than $250,000. Pawnshops don’t appear to be any more difficult to start than restaurants or hardware stores. 8. In 1996 dairy farmers, hurt by a decade of low milk prices, began reducing their herds. Subsequently Kenneth Hein, a Wisconsin farmer, said he was getting $16 per 100 pounds of milk, rather than $12, which he had gotten earlier. a. Why did the price increase? b. Dairy cattle are often fed corn. When Hein got $16 per 100 pounds of milk, he paid $5 a bushel for corn; but when he got $12 per 100 pounds of milk, he paid $2.50 a bushel for corn. Does this mean that Hein made less money when the price of milk was $16 than when it was $12? Solution: a. The price increased because the supply of milk shifted to the left. As dairy farmers began reducing the size of their herds, less milk was supplied to the market at every price. b. Not necessarily. Changes in the relative price of an input should cause farmers to change their optimal mix of inputs. This may mean that the farmers changed their mix of feed to include less corn (whose relative price is rising) and more of other inputs. 9. The demand for diamonds is given by PZ  980  2QZ where QZ is the number of diamonds demanded if the price is PZ per diamond. The total cost (TCZ) of the De Beers Company (a monopolist) is given by TCZ  100  50QZ  0.5QZ2 where QZ is the number of diamonds produced and put on the market by the De Beers Company. Suppose the government could force De Beers to behave as if it were a perfect competitor—that is, via regulation, force the firm to price diamonds at marginal cost. a. What is social welfare when De Beers acts as a single-price monopolist? b. What is social welfare when De Beers acts as a perfect competitor? c. How much does social welfare increase when De Beers moves from monopoly to competition? Solution: a. The firm, being a single monopolist, produces where marginal revenue  marginal cost. So, given that TR  PQ  980Q  2Q2 and MC  50  Q, we have 980  4Q  50  Q, and the profit-maximizing output and price for the firm is 186 diamonds at a price of $608. To find social welfare,

130 | Chapter 8 determine the sum of consumer and producer surplus. At this point, consumer surplus is 0.5(186)(980  608)  Area above the equilibrium piece and underneath the demand curve  $34,596. TR  806(186)  113,088 and total variable cost  50(186)  0.5(186)2  26,598. Therefore, producer surplus  TR  TVC  113,088  26,598  $86,490. Therefore, total surplus  social welfare  34,596  86,490  $ 121,086. b. When De Beers acts as a perfect competitor, the market will produce where price  marginal cost. Now, 980  2Q  50  Q, and the industry output is 310 diamonds at a price of $360. Consumer surplus is 0.5(310) (620)  $96,100. Producer surplus is 0.5(310)(310)  $48,050. Total surplus is $144,150. c. Eliminating De Beers’s monopoly power would create an additional $23,064 of surplus. 10. The Hassman Company produces two joint products, X and Y. The isocost curve corresponding to a total cost of $500,000 is QY  1,000  10QX  5Q 2X where QY is the quantity of product Y produced by the firm and QX is the quantity of product X produced. The price of product X is 50 times that of product Y. a. If the optimal output combination lies on this isocost curve, what is the optimal output of product X? b. What is the optimal output of product Y? c. Can you be sure that the optimal output combination lies on this isocost curve? Why or why not? Solution: a. Hassman maximizes its profits, given that it is spending $500,000 on inputs, by choosing quantities of X and Y so that the slope of the isorevenue function equals the slope of the isocost function. Total revenue is given by TR  PXQX  PYQY. Isorevenue curves are obtained by setting PXQX  PYQY  c  constant. This implies that PXQX  PYQY  c  0. Therefore the slope of the isorevenue curves is given by QY/QX  PX/PY. Given that PX  50PY, we have PX/PY  50 and the slope of the isocost function is dY/dX  10  10QX. Solving 50  10  10QX → QX  4. b. QY  1,000  10(4)  5(4)2  880 c. We have determined the profit-maximizing combination of outputs given the $500,000 expenditure on inputs. There is no certainty that this is the profit-maximizing level of input expenditures.

Monopoly and Monopolistic Competition | 131 11. The McDermott Company estimates its average total cost to be $10 per unit of output when it produces 10,000 units, which it regards as 80% of capacity. Its goal is to earn 20% on its total investment, which is $250,000. a. If the company uses cost-plus pricing, what price should it set? b. Can it be sure of selling 10,000 units if it sets this price? c. What are the arguments for and against a pricing policy of this sort? Solution: a. $15  [0.20($250,000)  $10(10,000)]/10,000 b. There is no assurance that 10,000 units can be sold at this price. c. Markup pricing is not sensible if the markup is not based on the elasticity of demand. If the decision maker knows the elasticity of demand and uses it, markup pricing is profit maximizing. If the decision maker doesn’t, then it can be silly. 12. The Morrison Company produces tennis rackets, the marginal cost of a racket being $20. Because there are many substitutes for the firm’s rackets, the price elasticity of demand for its rackets equals about 2. In the relevant range of output, average variable cost is very close to marginal cost. a. The president of the Morrison Company feels that cost-plus pricing is appropriate for his firm. He marks up average variable cost by 100% to set price. Comment on this procedure. b. Because of heightened competition, the price elasticity of demand for the firm’s rackets increases to 3. The president continues to use the same cost-plus pricing formula. Comment on its adequacy. Solution: a. The profit-maximizing markup from marginal cost is |h|/(|h|  1), which if the elasticity is 2, equals 200%. If marginal cost equals average variable cost, then the 100% markup from average variable cost is NOT profit maximizing. b. If the elasticity changes to 3, then the profit-maximizing markup is 150%. 13. The Backus Corporation makes two products, X and Y. For every unit of good X that the firm produces, it produces two units of good Y. Backus’s total cost function is TC  500  3Q  9Q2 where Q is the number of units of output (where each unit contains one unit of good X and two units of good Y) and TC is total cost (in dollars). The demand curves for the firm’s two products are PX  400  QX PY  300  3QY

132 | Chapter 8 where PX and QX are the price and output of product X and PY and QY are the price and output of product Y. a. How much of each product should the Backus Corporation produce and sell per period? b. What price should it charge for each product? Solution: a. We are given QY  2QX. We have PX  400  QX, PY  300  3QY, MC  3  18Q. Total revenue  PXQX  PYQY  (400  QX)QX  (300  3QY)QY  (400  Q)Q  (300  6Q)2Q  1,000Q  13Q2, where we set Q  QX. The firm’s total profit is given by   TR  TC  500  997Q  22Q2. The first-order condition for profit maximization entails 997  44Q  0. Thus Q  QX  22.66 and QY  45.32. Alternatively, MR X  400  2QX  400  2Q, MRY  300  6QY  600  24Q. Setting MR  MR X  MRY  1,000  26Q  3  18Q  MC Q  997/44  22.66, QX  22.66, and QY  45.32.

yields

We also need to check that each product’s individual marginal revenue is positive at the proposed solution values, which in fact each is. b. PX  400  22.66  377.33, PY  300  3(45.32)  164.04

CHAPTER 9

Managerial Use of Price Discrimination

Lecture Notes 1. Introduction •

Objectives ÿ To discuss the rationale behind price discrimination ÿ To investigate usefulness of price discrimination as a legitimate pricing strategy ÿ To analyze the use of coupons and rebates as price discrimination strategy ÿ To discuss the three types of price discrimination techniques * First-degree price discrimination * Second-degree price discrimination * Third-degree price discrimination ÿ To discuss peak and load pricing ÿ To discuss two-part tariff ÿ To explain how managers use price discrimination to increase profits * Identify submarkets with different price elasticities of demand * Segment the market and charge different prices to consumers in each submarket

2. Motivation for Price Discrimination •

Figure 9.1: Single-Price Monopolist Profit-Maximizing Outcome ÿ This figure is a very important illustration of the idea of price discrimination. ÿ As indicated in the figure, by charging PM, the monopolist sells the output QM. Customers in area AB of the demand curve value the good at a higher price than PM, but they are asked to pay PM for it. Then consumer surplus  CS  area given by V.

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134 | Chapter 9 ÿ The idea: If the manager is going to capture some (or all) of region V and some (or all) of region X  Z, she cannot do it with a single-price strategy. ÿ Surplus can be captured from either area with a strategy that involves two or more prices. ÿ Managers’ motivation is to capture the additional profit area in V and areas X  Z by comparing the benefit and the cost of doing so. ÿ Consumers in region BC are willing to pay a price as high as PM but will pay a price higher than it costs the producer to make the good. * Variable-cost profit  W  Y * Total revenue  PMQM  W  Y  U * If P PM, area X  Z is bigger. * If P PM, V becomes bigger. The manager cannot increase profit from PM because it is the profitmaximizing price for the single-price monopolist. * Observation: Sophisticated pricing is more costly to implement than simple single pricing. ÿ Single-price monopoly equilibrium fails to capture all consumer surplus and also results in a deadweight loss. 3. Price Discrimination •

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Price Discrimination: Occurs when the same product is sold at more than one price ÿ Examples: * An airline may sell tickets on a par ticular flight at a higher price to business travelers than to college students. * An automobile dealer: Selling same type of car to different buyers ÿ Differences in price among similar products are not evidence of price discrimination unless these price differences are not based on cost differences. ÿ Even if the products are not precisely the same, price discrimination is said to occur if similar products are sold at prices that are in different ratios to their marginal costs. ÿ Managers need to master three basic types of price discrimination: * First-degree price discrimination * Second-degree price discrimination * Third-degree price discrimination First-Degree Price Discrimination (or Perfect Discrimination) Illustration: Figure 9.1: A Simple Monopoly (Single-Price) ProfitMaximizer. ÿ The simple monopolist makes a variable-cost profit of W  Y and leaves the consumer surplus (CS) of V with the consumers of segment AB.

Managerial Use of Price Discrimination | 135 ÿ If managers could perfectly price discriminate, they would charge the consumers in segment AB their reservation prices, capturing all the consumer surplus and turning it into producer surplus of V  W  Y. This increases variable cost by X  Z. ÿ By perfectly discriminating in both the AB and the BC segments, managers increase the firm’s variable cost profit by V  X  Z: the area the simple monopolist was not exploiting in Figure 9.1. The idea: If managers can capture all of V  X  Z, we say that they are practicing discrimination of the first degree. * In essence, the strategy allows the managers to charge each consumer his or her reservation price. By doing so, the managers guarantee that consumer surplus is zero. * Managers sell to a consumer as long as the reservation price (which the manager can charge and the consumer is willing to pay) exceeds the marginal cost of production. Key idea: In perfect discrimination the firm’s demand curve becomes the firm’s marginal revenue curve. ÿ Key fact: The perfectly discriminating manager maximizes profit by producing until marginal revenue (represented by the demand curve) is equal to the output’s marginal cost. ÿ All customers are charged a price equal to their reservation price. ÿ The firm captures 100% of the consumer surplus. ÿ Using first-degree price discrimination, CS  0 (it has all been captured) and producer surplus PS  V  W  X  Y  Z. But, under perfect competition, PS  Y  Z and CS  V  W  X and social welfare  V  W  X  Y  Z, indicating that the welfare is the same under both pricing mechanisms. But consumers benefit under perfect competition, and producers get all the benefit of first-degree price discrimination. ÿ Equilibrium output and marginal cost is the same as under perfect competition. ÿ There is no deadweight loss. ÿ First-degree price discrimination requires that firms have a relatively small number of buyers and that they are able to estimate buyers’ reservations prices. ÿ The pricing strategy may be operationalized by means of a two-part tariff. STRATEGY SESSION: When Can You Haggle? Discussion Questions 1. Have you had the experience of haggling over price in the United States? Did you enjoy the experience or did you hate it? Why?

136 | Chapter 9 2. Have you had the experience of haggling over price in a foreign country? How was it different from the United States? Did you enjoy the experience or did you hate it? Why? PROBLEM SOLVED: Honest Sanjay’s Use of First-Degree Price Discrimination Discussion Questions 1. Suppose that Sanjay moves his business to a larger city where demand is P  12  Q, where P is the unit price (in thousand). Marginal cost conditions are the same, MC  $2 (thousand). Sanjay faces fixed costs of $5 (thousand). What price should Sanjay charge under simple monopoly pricing, and how much monthly profit will be earned? Answer: Set MR  MC, under simple monopoly, where TR  PQ  (12  Q)Q  12Q  Q2 MR  12  2Q  2  MC so Q  5 and price of cars is P  12  5  7 or $7,000. Sanjay’s total revenue per month  (12  5)5  35 or $35,000. Variable costs  2(5)  10 or $10,000 and fixed cost is $5,000. Hence, profit  35  15  20 or $20,000. 2. Suppose that Sanjay moves his business to a larger city where demand is P  12  Q. Suppose Sanjay can hire a slick sales force. The salespeople are paid strictly a commission of $1 (thousand) per car. What quantity will Sanjay sell under first-degree price discrimination, and how much monthly profit will be earned? Answer: Under the model of sales, Sanjay’s marginal cost is $3 (thousand) per car. Under perfect discrimination, profit maximization requires that MR  P  MC. Thus, set P  12  Q  3  MC so Q  9

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Total revenue  $67.5 (thousand) and fixed costs  $5 (thousand), a profit of $35.5 (thousand) Second-Degree Price Discrimination ÿ Most commonly used by utilities ÿ Example: Public utilities: Gas, electric, water, etc. ÿ Illustration: Figure 9.2: Second-Degree Price Discrimination ÿ The idea: By charging different prices, managers increase profit relative to a single-price strategy. ÿ Different prices are charged for different quantities of a good. ÿ Unlike first-degree price discrimination, managers leave a consumer surplus of A  B  C.

Managerial Use of Price Discrimination | 137 •

Third-Degree Price Discrimination ÿ The most common form of price discrimination ÿ Three conditions must hold true for this pricing strategy to succeed. 1. Demand must be heterogeneous; that is, different demand segments must have different price elasticities of demand. 2. Managers must be able to identify and segregate the different segments. 3. Markets must be successfully sealed so that customers in one segment cannot transfer the goods to another segment. * The idea: Managers identify individuals with similar traits and group them together. Managers then appeal to the group. ÿ Example of third-degree price discrimination: Students are price sensitive. * Limited income makes students more responsive to price differences. * Students’ price elasticity of demand is thus likely to be more elastic than that of other segments. * Students can be readily identified by their student IDs, aiding in segmentation. ÿ Other conditions * Segments must differ significantly in their price elasticities. * Managers must be able to identify and target the segments at moderate cost. * Buyers must be unable to transfer a product from one segment to another. * These two conditions are referred to as the ability to “segment and seal” the market. ÿ Optimal strategy (allocation) * Allocate total output so that marginal revenue in all segments is equal to the firm’s marginal cost. * Optimal price ratios ⎡ ⎛ 1 ⎞⎤ ⎢1 − ⎜ ⎟⎥ P1 ⎢ ⎝ | 2 | ⎠ ⎥  ⎢ P2 ⎞⎥ ⎛ ⎢1− ⎜ 1 ⎟ ⎥ ⎢⎣ ⎝ |1 | ⎠ ⎥⎦

(9.1)

Note: It does not pay to discriminate if the two price elasticities are the same, implying P1  P2. * Segments with a lower (in absolute value) price elasticity are charged a higher price. ÿ Illustration: Figure 9.3: Third-Degree Price Discrimination ÿ Strategy implementation: A two-group case *

138 | Chapter 9 * *

* *

* *

Managers choose total output. They must look at costs as well as demand in the two groups. The manager will then maximize profit when the MC of the entire output is equal to the common value of the MR in the two groups. Firm’s profit  TR1  TR2  TC  TR1(Q1)  TR2(Q2)  TC(Q) The monopolist has two output choices, so profit is maximized from the first-order conditions, resulting from an unconstrained optimization of a function of two variables. Optimality condition: MR1  MR2  MC (the two-group case) For n classes of demanders, the profit-maximizing rule would be: MR1  MR2  …  MRn  MC

STRATEGY SESSION: That Darling Little Mouse Is Really a Price Discriminator Discussion Questions 1. In the example, a driver’s license with state of residence information provides the basis for sealing and segmenting. What other information on a driver’s license could be used to seal and segment? 2. Student identification cards and AARP membership cards are two additional methods used to seal and segment. Can you think of any other methods of sealing and segmenting that are or might be used? • Managerial Use of Third-Degree Price Discrimination ÿ Airline tickets * Demand for business travel is much less elastic than for leisure travel. * Business travelers pay more for tickets. * Conditions imposed by airlines yield other benefits, such as the ability to schedule more accurately well in advance and to charge ticket-holders to reschedule. ÿ New business models have reduced the ability of airlines to engage in price discrimination. * Internet ticket sales provide more information and flexibility to buyers. * There is a time cost to searching the Internet for low prices, and online information may not always be accurate. * Priceline.com, an Internet company that lets customers bid on tickets, facilitates first-degree price discrimination.

Managerial Use of Price Discrimination | 139 STRATEGY SESSION: Mickey Mouse Pricing at Amusement Parks Discussion Question 1. Describe why this might be a revenue-enhancing third-degree price discrimination strategy in which customers self-select among the three market segments. How do you think the price elasticities of demand differ among the segments? STRATEGY SESSION: Yield Management and Airline Performance Discussion Questions 1. Summarize, in your own words, the key components of yield management. 2. Is yield management a viable pricing strategy under perfect competition? Explain. PROBLEM SOLVED: Third-Degree Price Discrimination Discussion Questions 1. Suppose that, because of a decline in the value of the dollar relative to the euro, the demand curve for the drug in Europe shifts to PE  20  0.5Q. How many units should the drug manufacturer sell in Europe and the United States, and what prices should be charged? Answer: MRE  20  QE  2  MC → QE  18 and PE  11. QU and PU are unchanged. 2. Suppose that, because of a decline in the value of the euro relative to the dollar, the demand curve for the drug in the United States shifts to PU  40  QU. How many units should the drug manufacturer sell in Europe and the United States, and what prices should be charged? Answer: MRU  40  2QU  2  MC → QU  19 and PU  21. QE and PE are unchanged. 4. Using Coupons and Rebates for Price Discrimination •

Coupons and rebates are used to segment a market. ÿ People who use coupons or send in rebates are likely to have more elastic demand than those who do not. ÿ Coupons and rebates lead people to self-select their market segment. ÿ These devices reduce the price of products. ÿ Not all consumers use coupons.

140 | Chapter 9

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ÿ Managers use coupons and rebates to price discriminate because other consumers are willing to pay more—that is, to buy the good without a coupon. Example: Pricing strategy Managers at the fish company choose a posted price (P), but then issue a coupon for $X off in the newspaper local to the consumer types. Question: What should the value of P and X be? To maximize profit, the MR in each market should be equal, and they should be, in turn, equal to the MC of the company. ÿ P(1  1/|hR|)  (P  X)(1  1/|hS |)  MC ÿ P  market price ÿ X  discount from coupon or rebate ÿ hR  price elasticity of demand by those who don’t use coupons or rebates ÿ hS  price elasticity of demand by those who do use coupons or rebates Example: Barnegat Light Fish Company price crab cakes ÿ MC  2 ÿ hR   2 ÿ MR  MC → P  4 ÿ hS   5 ÿ MR  (4  X)[1  (1/|5|)]  2  MCX  1.5 ÿ Those who are more price elastic use coupons, the less price elastic people do not. Therefore, by using coupons (or rebates), managers can price discriminate and increase their profits.

5. Peak Load Pricing •

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Issues in Pricing Strategy ÿ The demand for some goods is time sensitive or seasonal. ÿ Plant capacity is constant. ÿ Examples * Electricity generation * Roadways * Resort and hotel rooms * Intertemporal pricing of intellectual property— early release charges peak pricing, and later release charges trough pricing— books released first as hard-bound with higher price followed by paperback at a lower price—leaders and followers in markets Strategic Response ÿ During peak time periods, when demand is high, managers should charge a higher price (PP). ÿ During trough time periods, when demand is low, managers should charge a lower price (PT).

Managerial Use of Price Discrimination | 141 ÿ Marginal cost often follows a cyclical pattern in which MC is high during peak periods and low during trough time periods. ÿ Firms should equate marginal cost and marginal revenue separately in the two time periods to determine the appropriate prices. ÿ Figure 9.4: Determination of Peak and Trough Prices ÿ Rule to follow: Set MR  MC ÿ Not a third-degree price discrimination STRATEGY SESSION: The Future Is Now: The Futures Market for Super Bowl Tickets Discussion Questions 1. More conventional futures markets, for example, the foreign exchange market for dollars and euros, guarantee that you can exchange a certain  number of dollars for a certain number of euros at a future date. Under what circumstances might this type of “insurance” be useful to a firm? Under what circumstances would this type of futures contract be worthless? 2. More conventional futures markets, for example the commodities markets, guarantee that you can purchase a certain commodity at a certain price at a future date. Under what circumstances might this type of “insurance” be useful to a firm? Under what circumstances would this type of futures contract be worthless? STRATEGY SESSION: Why Do Your Laundry at 3 A.M.? Discussion Questions 1. How responsive do you think energy demand is to temporal changes in pricing? 2. With peak load pricing, consumers can choose to adjust their consumption or to leave it unchanged. How do you think the price elasticity of demand of those who adjust their behavior differs from those who do not? STRATEGY SESSION: A Change from Markup Pricing to Sophisticated Pricing Discussion Questions 1. What are the pros and cons of markup pricing? Under what conditions might it be appropriate? 2. What are the pros and cons of sophisticated pricing? Under what conditions might it not be appropriate?

142 | Chapter 9 6. Two-Part Tariffs • • •

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•

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Two-Part Tariff: When managers set prices so that consumers pay an entry fee and then a use fee for each unit of the product they consume Two-part tariffs are common in the business world. Examples ÿ Clubs (golf, health, discount, etc.) that charge a membership fee and a per use fee. ÿ Wireless phone plans that charge a fixed fee and then additional fees per minute. ÿ Personal seat licenses (PSL) for sports stadiums—a fixed cost that gives the purchaser the right to buy tickets to games. Strategy when all demanders are the same ÿ Model * Assume that all consumers have the same preferences, defined by the demand curve P  a  bQ. * Assume that the firm’s marginal cost is constant. * Entry fee is equal to consumer surplus. * Use fee is equal to marginal cost. * Total revenue is the same as under first-degree price discrimination. ÿ Figure 9.5: Optimal Two-Part Tariff When All Demanders Are the Same A Two-Part Tariff with a Rising Marginal Cost ÿ Strategy is the same as when marginal cost is constant. ÿ Variable cost profit is positive when marginal cost has a positive slope. ÿ Figure 9.6: Optimal Two-Part Tariff When Marginal Cost Is Rising A Two-Part Tariff with Different Demand Curves ÿ Model * Market consists of strong demanders and weak demanders. ÿ Pricing strategies * When strong demand is much stronger than weak demand: Set use fee equal to marginal cost and entry fee equal to the strong demanders’ consumer surplus. Weak demanders will be excluded from the market. * When strong demand is not much stronger than weak demand: Set use fee equal to marginal cost and entry fee equal to the weak demanders’ consumer surplus. Weak demanders will not be excluded from the market. * When strong demand is not much stronger than weak demand: Set use above marginal cost at a price that maximizes variable cost profit and entry fee equal to the weak demanders’ consumer surplus. Weak demanders will not be excluded from the market. * Optimal strategy when strong demand is not much stronger than weak demand is found by comparing total average cost profit from the two strategies. ÿ Figure 9.7: Optimal Two-Part Tariff with Two Demand Types

Managerial Use of Price Discrimination | 143 STRATEGY SESSION: Making Them Pay Twice: Personal Seat Licenses for Sports Teams Discussion Questions 1. This is an example of a two-part tariff in which the entry fee is the cost of the personal seat license and the entry fee is the price of a ticket. It differs from other two-part tariffs in that there is an active market for the licenses. If there is an increase in the price of licenses over time, what does it imply about the demand for tickets? Answer: It would imply that the demand for tickets is increasing. 2. This is an example of a two-part tariff in which the entry fee is the cost of the personal seat license and the entry fee is the price of a ticket. It differs from other two-part tariffs in that there is an active market for the licenses. If there is an increase in the price of tickets, how will it affect the price of licenses? Answer: An increase in ticket price would reduce the area of consumer surplus and therefore reduce the price of licenses. STRATEGY SESSION: Costco and the Two-Part Tariff Discussion Question 1. The simple theory of a two-part tariff implies that price will be equal to marginal cost. In the case of Costco, price is above marginal cost. How do you explain this? Answer: In the case of multiple consumer segments with different demand curves, it may be optimal to price above marginal cost. STRATEGY SESSION: Verizon Local Calling Plans Discussion Questions 1. How many message units does a consumer need to use per month, on average, in order to be indifferent between the low-use and moderate ser vice? Answer: 20  (7.4  5.2)/(0.10)  42 2. How many message units does a consumer need to use per month, on average, in order to be indifferent between the moderate and flat-rate service? Answer: 75  (8.95  7.4)/(0.065)  99

144 | Chapter 9 PROBLEM SOLVED: Two-Part Tariff Pricing Discussion Questions 1. Suppose that the demand for C-Pal’s product increases to P  10  Q. Assuming a use fee of $4, how many units will be sold, and what is the optimal entry fee? Answer: Write Q  10  P. For P  $4 and revenue from the use fee is 4(6)  $24 and profit is $13,000. 2. Suppose that the demand for C-Pal’s product increases to P  40  2Q and its marginal cost changes to MC  1  Q. What is the appropriate use fee, how many units will be sold, and what is the optimal entry fee? Answer: MC  1  Q  40  2Q  P → Q  13, so P  14 and the entry fee is (13)(40  14)/2  169. STRATEGY SESSION: Scientific Pricing—Even for Great Art? Discussion Question 1. Does it seem reasonable to you that great artists would produce their most valuable works either early in their careers or late in their careers, but not both? Why or why not? PROBLEM SOLVED: A Two-Part Tariff with Different Demands Discussion Questions 1. Suppose the strong demand increases to PS  10  QS, and weak demand increases to PW  8  QW. What is the total revenue from use fees and the variable cost profit if the entry fee is set based on strong demand? Answer: QS  8 and QW  0 so total revenue from use fees is $16. Variable cost profit  (10  2)(8)/2  32 2. What is the total revenue from use fees and the variable cost profit if the entry fee is set based on weak demand? Answer: QS  8 and QW  6 so total revenue from use fees is $28. Variable cost profit  (2)(8  2)(6)/2  36 3. Suppose the strong demand increases to PS  10  QS and weak demand increases to PW  8  QW. What is the price that maximizes variable cost profit? What are total use fees, the total revenue from use fees, and the variable cost profit?

Managerial Use of Price Discrimination | 145 Answer: P*  3, QS  7, and QW  5 so total revenue from use fees is $36. Variable cost profit  (2)[(8  3)(5)/2]  (7  5)(3  2)  37

Chapter 9: Problem Solutions 1. Managers at the Ridgeway Corporation produce a medical device that they sell in Japan, Europe, and the United States. Transportation costs are a negligible proportion of the product’s total costs. The price elasticity of demand for the product is 4.0 in Japan, 2.0 in the United States, and 1.33 in Europe. Because of legal limitations, this medical device, once sold to a customer in one country, cannot be resold to a buyer in another country. a. The firm’s vice president for marketing circulates a memo recommending that the price of the device be $1,000 in Japan, $2,000 in the United States, and $3,000 in Europe. Comment on his recommendations. b. His recommendations are accepted. Sales managers send reports to corporate headquarters saying that the quantity of the devices being sold in the United States is lower than expected. Comment on their reports. c. After considerable argument, the U.S. sales manager agrees to lower the price in the United States to $1,500. Is this a wise decision? Why or why not? d. Can you be sure that managers are maximizing profit? Why or why not? Solution: a. The recommendation is not correct. Profit maximization requires the marginal revenue (MR) in each market be the same and equal to marginal cost. MRJ  PJ(1  1/|hJ|) so MRJ  PJ(1  1/|4|)  0.75PJ MRUS  PUS (1  1/|2|)  0.50PUS MRE  PE (1  1/|4/3|)  0.25PE Profit maximization requires that MR be equal for each market: MRJ  MRUS  MRE or 0.75PJ  0.50PUS  0.25PE Since (0.75)($1,000) ≠ (0.50)($2,000) ≠ (0.25)($3,000), profit is not maximized. b. Since the U.S. price is too high, we should not be surprised if U.S. sales are below expectations. c. If the U.S. price falls to $1,500, then MRJ  MRUS  MRE  $750. Whether or not this is a good decision depends on whether or not Ridgeway’s MC  750. d. We don’t know if Ridgeway is maximizing profit because we don’t know its MC.

146 | Chapter 9 2. Ann McCutcheon is hired as a consultant to a firm producing ball bearings. This firm sells in two distinct markets, each of which is completely sealed off from the other. The demand curve for the firm’s output in the first market is P1  160  8Q1, where P1 is the price of the product and Q1 is the amount sold in the first market. The demand curve for the firm’s output in the second market is P2  80  2Q2, where P2 is the price of the product and Q2 is the amount sold in the second market. The firm’s marginal cost curve is 5  Q, where Q is the firm’s entire output (destined for either market). Managers ask Ann McCutcheon to suggest a pricing policy. a. How many units of output should she tell managers to sell in the second market? b. How many units of output should she tell managers to sell in the first market? c. What price should managers charge in each market? Solution: Optimal strategy: Set MR  MC in each market. We have TR1  (160  8Q1)Q1 and TR2  (80  2Q2)Q2. Hence, MR1  160  16Q1 and MR2  80  4Q2. From the optimality conditions, we have 160  16Q1  5  (Q1  Q2) and 80  4Q2  5  (Q1  Q2), 155  17Q1  Q2 and 75  Q1  5Q2. Solving this system of equations with two unknowns (either by elimination or substitution) gives 75  Q1  5(155  17Q1) → Q1  8.33 and Q2  13.33. a. Q2  13.33 b. Q1  8.33 c. P1  160  8(8.33)  93.33 P2  80  2(13.33)  53.33 3. The Lone Star Transportation Company hauls coal and manufactured goods. The demand curve for its services by the coal producers is PC  495  5QC where PC is the price (in dollars) per ton-mile of coal hauled and QC is the number of ton-miles of coal hauled (in thousands). The demand curve for its services by the producers of manufactured goods is PM  750  10QM where PM is the price (in dollars) per ton-mile of manufactured goods hauled, and QM is the number of ton-miles of manufactured goods hauled (in thousands). The firm’s total cost function is TC  410  8(QC  QM)

Managerial Use of Price Discrimination | 147 where TC is total cost (in thousands of dollars). a. What price should managers charge to haul coal? b. What price should managers charge to haul manufactured goods? c. If a regulatory agency were to require managers to charge the same price to haul both coal and manufactured goods, would this reduce the firm’s profit? If so, by how much? Solution: TC  410  8(QC  QM), MCC  MCM  8 MRC  495  10QC, MR M  750  20QM a. Setting MCC  MRC, we get 8  495  10QC → QC  48.7 → PC  251.5. b. Setting MCM  MR M, we get 8  750  20QM → QM  37.1 → PM  379. c. Rearranging the demand functions, we get QC  (495  P)/5 and QM  (750  P)/10 QC  QM  (1,740  3P)/10  174  0.3P P  580  (10/3)Q Setting MR  580  (20/3)Q  8  MC yields Q  85.8. p  [580  (10/3)85.8](85.8)  410  8(85.8)  $24,129, whereas when price discriminating, the monopolist earns p  [495  5(48.7)](48.7)  [750  10(37.1)](37.1)  410  8(48.7  37.1)  $25,213. This implies that the monopolist earns $1,084 less when not allowed to price discriminate. 4. Electric companies typically have 5–10 different rate schedules for their main customer groups. The average price charged to large industrial users may differ substantially from that charged to residences. Moreover, many consumers pay a price for electricity based on the time of day they use it. For example, the prices charged by Consolidated Edison, a large New York electric utility, and Pacific Gas and Electric, a major California electric utility, are as follows:

148 | Chapter 9 Price Company and Time of Day of Electricity Use (Cents per Kilowatt-Hour) Consolidated Edison 8 a.m.–10 p.m. (peak hours) 10 p.m.–8 a.m.(off-peak hours) Pacific Gas and Electric Summer Noon–6 p.m. (peak hours) 6 p.m.–noon (off-peak hours) Winter Noon–6 p.m. (peak hours) 6 p.m.–noon (off-peak hours)

27.0 4a

28.3 9.2 11.3 8.0

a

Approximate figure

Electric utilities use their cheapest generators continuously and start up their more costly ones as demand goes up. Consequently, at 3 a.m., a utility might meet its requirements from a hydroelectric dam that produces electricity for $0.02 per kilowatt-hour. However, on a hot day in August, when air conditioners are running full blast, demand would be so great that the utility would be forced to use its most costly generators—perhaps an oilfired plant where electricity costs $0.07 per kilowatt-hour. a. Does price discrimination occur in the market for electricity? b. Why have some state regulatory commissions, including the Public Service Commission of New York, ordered that time-of-day rates be phased in for residential consumers? c. In many areas, both residential and industrial consumers tend to pay a lower price per kilowatt-hour if they use more rather than less electricity. Is this price discrimination? If so, what kind of price discrimination is it? d. Explain why price discrimination is used by managers of electric companies. Solution: a. When a utility charges different prices to consumers and large industrial users, it is probably practicing price discrimination. This assumes that consumers and industrial users have different demand elasticities and that the difference in price is not entirely due to differences in the marginal cost of delivering (as opposed to generating) power. Similarly, the difference between peak and off-peak pricing to consumers represents the results of a combination of price discrimination and differences in the marginal cost of generating electricity at peak and off-peak levels of demand. b. Time-of-day pricing has the effect of reducing peak demand for electricity. This reduces the average cost of electricity generation overall, con-

Managerial Use of Price Discrimination | 149 serves resources, and reduces the need to add additional generating capacity to the power grid. c. This quantity discount is an example of second-degree price discrimination. d. Price discrimination is used by electric companies to manage demand in order to reduce costs and enhance revenues. 5. In the town of Oz, there are two types of tennis players: wizards and imps. Wizards and imps do not socialize, so it would be impossible to start a tennis club that both types would join. Imps have access to credit but a weak demand for tennis as follows: PI  30  QI where QI refers to the number of games they would play if the price of a game were PI. Because of their access to credit, they would be willing to pay an upfront fee to join the club. Wizards live from paycheck to paycheck and would be willing to pay for each tennis game as they go along. Their demand is PW  40  QW where QW refers to the number of games they would play if the price of a game were PW. There are an equal number of wizards and imps (for simplicity, assume one of each). The marginal cost of one game of tennis is a constant 2. You can design your tennis facility to attract either wizards or imps (but not both). Which clientele would you like to attract and what would be your profit per “person”? Solution: Applying a two-part tariff to the imps and setting the use fee equal to marginal cost results in QI  28. The optimal entry fee, and the variable cost profit, is (30  2)(28)/2  392. Applying simple monopoly pricing to the wizards yields: MR  40  2QW  2  MC → QW  19, profit  361.

PW  21,

and

variable

cost

The facility should cater to imps, which will yield a profit of $392 per person. 6. The managers of Roosevelt’s (a local yet upscale bar) are considering charging an admission fee on Thursday nights. They contemplate how to charge. Should they: Option 1. Use just a beverage charge per beverage ordered or Option 2. Use an admission charge (a fee to enter the establishment) and a beverage charge per beverage ordered?

150 | Chapter 9 There are two types of people who frequent Roosevelt’s: Over 21 Students (S) and Over 21 Student Wannabees (W). Each Student has a demand for beverages of P  8  QS where QS is the quantity of beverages demanded if the price of a beverage is P. Each Wannabee has a demand for beverages of P  8  2QW where QW is the quantity of beverages demanded if the price of a beverage is P. The marginal cost of serving a beverage is a constant $2. For simplicity, assume there is one demander of each type. Roosevelt’s must (by law) charge all customers the same admission charge and the same per beverage charge. Beverages do not have to be sold in integer amounts, and prices do not have to be in integer amounts. a. Under option 1, what is the profit-maximizing price per beverage? b. Under option 2, what is the profit-maximizing two-part tariff? c. What is Roosevelt’s profit under Roosevelt’s best choice? Solution: a. Under option 1, market demand is the horizontal summation of the demand curves: Q  8  P  4  P/2  12  (3/2)P. So P  8  (2/3)Q and MR  8  (4/3)Q. MR  8  (4/3)Q  2  MC → Q  4.5, P  5, QS  3, QW  1.5, and the variable cost profit  13.5. b. Under option 2, there are three alternative strategies. Strategy 1: Charge a price equal to marginal cost and charge an entry fee based on the strong demand. In this case, P  2, Q  6, and the entry fee, and the variable cost profit, is (8  2)(6)/2  18. Strategy 2: Charge a price equal to marginal cost and charge an entry fee based on the weak demand. In this case, P  2, Q  9, the entry fee is (8  2)(3)/2  9, and the variable cost profit  18. Strategy 3: Charge a price greater than MC that maximized variable cost profit and charge an entry fee based on weak demand. In this case, P  3.5, Q  6.75, the entry fee is (8  3.5)(6.75)/2  5.0625, and the variable cost profit  20.25. c. Profit is highest under option 2 with Strategy 3: 20.25. 7. The demand for a strong demander for a round of golf is PS  6  QS

Managerial Use of Price Discrimination | 151 where QS is the number of rounds demanded by a strong demander when the price of a round of golf is PS. The demand for a weak demander for a round of golf is PW  4  QW where QW is the number of rounds demanded by a weak demander when the price of a round of golf is PW. The cost of providing an additional round of golf to either type of golfer is a constant 2. There is one golfer of each type. The club has decided that the best pricing policy is a two-part tariff. However, it’s your job to tell the club the optimal entry fee and the optimal use fee to maximize the club’s profit. The club cannot price discriminate on either the use or the entry fee. The club’s fixed cost is 1. What are the club’s optimal use fee and the optimal entry fee? Solution: There are three alternative strategies. Strategy 1: Charge a price equal to marginal cost and charge an entry fee based on the strong demand. In this case, P  2, Q  4, and the entry fee, and the variable cost profit, is (6  2)(4)/2  8. Strategy 2: Charge a price equal to marginal cost and charge an entry fee based on the weak demand. In this case, P  2, Q  6, the entry fee is (4  2)(2)/2  2, and the variable cost profit  4. Strategy 3: Charge a price greater than MC that maximized variable cost profit and charge an entry fee based on weak demand. In this case, P  3, Q  4, the entry fee is (4  3)(1)/2  1/2, and the variable cost profit  5. Strategy 1 is the best alternative. 8. The university museum has two types of visitors. One type is university employees; and the other type is people nonaffiliated with the university. All university employees have identical annual demands for museum visits, given by PP  30  QP (for each university employee) where QP is the number of visits demanded if the price is PP per visit. Nonaffiliated people all have identical annual demands for museum visits, but differ from university employees: PN  100  QN (for each nonaffiliated person)

152 | Chapter 9 where QN is the number of visits demanded if the price is PN per visit. The museum can identify university employees by their university ID card, while a nonaffiliated person does not possess a university ID. The university’s profit-maximizing museum is contemplating two different pricing policies: Policy 1 • For university employees: An annual membership fee and an additional price-per-visit. (Only university employees are eligible for this membership plan.) • For nonaffiliated visitors: A single price-per-visit, with no membership fee. (This price per visit is not necessarily the same as the university employee price per visit.) Policy 2 • This policy would offer a different price-per-visit for each type of visitor, but no membership fees at all. The museum has a constant marginal cost of $6 per visit, regardless of the visitor’s type. For simplicity, assume that there is one university employee and one nonaffiliated person in the target population. How much more profit does the best policy yield than the other policy? Solution: Policy 1 uses a two-part tariff for employees and simple monopoly pricing for nonemployees. For employees, PP  MC  6, so QP  24, the membership fee, and variable cost profit, is (30  6)(24)/2  288. For nonemployees, MR  100  2QN  6  MC → QN  47, PN  53, and variable cost profit is 2,209. Total profit for Policy 1 is 2,497. Policy 2 uses price discrimination. Nonemployee results are the same as under Policy 1. For employees, MR  30  2QP  6  MC → QP  12, PP  18, and variable cost profit is (12)(18  6)  144. Total profit for Policy 2 is 2,353. Policy 1 yields a profit that is 144 greater than Policy 2.

CHAPTER 10

Bundling and Intrafirm Pricing

Lecture Notes 1. Introduction •

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Objectives ÿ To examine the mechanics of bundling ÿ To explain how managers can use bundling and tying strategies to increase profit when customers have heterogeneous tastes ÿ To look at bundling as a preemptive entry strategy ÿ To analyze transfer pricing ÿ To explain how firms use transfer pricing to provide incentives to wholly-owned subsidiaries and divisions and to shelter profit from taxes in a global environment ÿ To examine transfer pricing in a perfectly competitive market for upstream product ÿ To look at the global use of transfer pricing ÿ To explain tying at IBM, Xerox, and Microsoft Bundling: Another sophisticated pricing strategy used by managers Two Aspects of Bundling: * Simple bundling * Mixed bundling ÿ Simple (pure) bundling: Occurs when managers offer several products or services as one package so consumers do not have an option to purchase package components separately * The goods are offered only in a package. * Example: Inclusion of a service contract with a product ÿ Mixed bundling: Allows consumers to purchase package components either as a single unit or separately * The goods have a package price as well as a stand-alone price.

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154 | Chapter 10 The bundle price is generally less than the sum of the price of the individual components. * Examples: ° Season tickets to sporting events or value meals at McDonald’s ° Opera companies: Offering tickets to individual performances as well as season tickets ÿ Bundling is best used when there is a wide variance in consumers’ price sensitivity of demand and when market conditions make it difficult to price discriminately. ÿ Managers prefer to form bundles so as to create negative correlations across consumers. ÿ Negative correlation: Exists when some customers have higher reservation prices for one item in the bundle but lower reservation prices for another item in the bundle, whereas another group of customers has the reverse preferences *

STRATEGY SESSION: Bundling Carbon Credits with Gas Sales Discussion Questions 1. The goal of cap-and-trade programs is to reduce the amount of carbon (primarily carbon dioxide) that is emitted into the atmosphere in the most efficient fashion. Do you think that the actions taken by Gazprom are consistent with this goal? 2. The United States refused to participate in the Kyoto Protocols. Do you think that it should? Why or why not? 2. The Mechanics of Bundling • •

Question: Why do managers commonly use bundling? Advantages of Bundling ÿ Bundling can increase the seller’s profit if customers have varied tastes. ÿ Bundling can emulate perfect price discrimination when perfect price discrimination is not possible (because knowing individual reservation prices is either too difficult or expensive to pursue) or it is not legal to charge multiple prices for the same product. ÿ Fact: Bundling does not require knowledge of individual consumers’ reservation prices, but only the distribution of consumers’ reservation prices over the goods. ÿ Three pricing mechanisms: * Pricing separately * Pure bundling * Mixed bundling

Bundling and Intrafirm Pricing | 155 •

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Strategies * Are the goods independent or complementary? * Independent goods: Worth of the bundle to consumers  sum of the separate goods in the bundle * Complementary goods: The goods as a bundle have a value greater than the sum of the separate reservation prices. Example: Software and hardware ÿ Assumption: Goods are independent, so the value of a bundle is equal to the sum of the reservation prices of the goods in the bundle. ÿ Separate pricing: Goods are not bundled. * Prices are set equal to profit-maximizing monopoly prices. ÿ Pure bundling * Bundle price is set to maximize profit. ÿ Mixed bundling * Bundle price and individual good prices are set to maximize profit. ÿ Optimal strategy is that which maximizes profit. Example Figures ÿ Notation * ri  Reservation price of good i * * pi  Price charged for good i * * PB  Price of bundle ÿ Figure 10.1: Price Separately * * * If r1 p1 and r 2 p2 , then consumer buys neither good. * * * If r1 p1 and r 2 p2 , then consumer buys only good 1. * * * If r1 p1 and r 2 p2 , then consumer buys only good 2. * * * If r1 p1 and r 2 p2 , then consumer buys both goods. ÿ Figure 10.2: Pure Bundling * * If (r1  r 2) PB , then consumer does not buy the bundle. * * If (r1  r 2) PB , then consumer buys the bundle. ÿ Figure 10.3: Mixed Bundling * * * * Buy neither good nor bundle: (r1  r2) PB , r1 p1, and r 2 p2. * * Buy bundle: (r1  r 2) PB . * * * * * Buy good 1 only: r1 p1, r 2 p2, and r 2 (PB  p1). * * * * * Buy good 2 only: r 2 p2, r1 p1, and r1 (PB  p2). Example 1 ÿ Assumptions * Perfect negative correlation among consumer reservation prices (Figure 10.4) * No variation in total bundle valuation; all value the bundle at $100 * Unit cost of production for each good  $1 ÿ Table 10.1: Shows the separate price strategy * The idea: Whether consumers purchase goods separately depends on their reservation price for the good relative to the prices charged by sellers.

156 | Chapter 10 Managers choose the optimal simple monopoly prices for good 1 and good 2 (the ones that maximize profit). ÿ Figure 10.2: Pure Bundling The idea: Whether consumers purchase the bundle depends on the sum of their reservation prices for the good relative to the bundled price charged by the seller. * Optimal separate prices for good 1 and good 2: Profit  $264 ÿ Figure 10.3: Mixed Bundling * The idea: Whether the consumer purchases the goods separately or as a bundle depends on the consumer surplus. * Critical to mixed bundling is creating a credible mixed bundle. * Credibility of the bundle means that managers correctly anticipate which consumers will purchase the bundle or the goods separately. * Optimal pure bundle price for consumers A, B, C, and D: Profit  $392 ÿ Figure 10.4: Example of Perfect Negative Correlation of Consumers’ Reservation Prices * Managers can always come up with a mixed bundle by pricing the individual goods at prices at which no consumer purchases the goods. * Mixed bundling always weakly dominates pure bundling. * Optimal mixed bundle prices: Profit  $392 ÿ Table 10.5: Optimal Mixed Bundle Prices When Consumers Buy Bundle and at Least One of the Separately Priced Goods: Profit  $373.98 Definitions ÿ Recall: Credibility of the bundle: When managers correctly anticipate which customers will purchase the bundle or the goods separately ÿ Extraction: When the manager extracts the entire consumer surplus from each customer ÿ Exclusion: When the manager does not sell a good to a customer who values the good at less than the cost of producing it ÿ Inclusion: When a manager sells a good to a consumer who values the good at greater than the seller’s cost of producing the good ÿ Fact: Extraction, exclusion, and inclusion are all satisfied by perfect price discrimination. That is, perfect price discrimination extracts all available consumer surplus, does not sell to anyone for less than cost, and sells to anyone who values the good more than cost. ÿ Pricing separately will satisfy exclusion but will not result in complete extraction or inclusion. ÿ Pure bundling can result in complete extraction, but if consumer reservation prices do not have a perfect negative correlation, extraction will be less than complete. It is also possible for pure bundling to fail to attain full inclusion and exclusion. *

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•

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ÿ The profit from mixed bundling is always equal to or better than that of pricing separately or pure bundling. Example 2 ÿ Assumptions * Perfect negative correlation among consumer reservation prices (Figure 10.4) * No variation in total bundle valuation; all value the bundle at $100 * Unit cost of production for each good  $11 ÿ Table 10.6: Optimal Pure Bundle Price: Optimal Separate Prices for Good 1 and Good 2: Profit  $204 ÿ Table 10.7: Optimal Pure Bundle Price for Consumers A, B, C, and D: Profit  $312 ÿ Table 10.8: Optimal Mixed Bundle Prices: Profit  $312 ÿ Table 10.9: Optimal Mixed Bundle Prices When Consumers Buy Bundle and at Least One of the Separately Priced Goods: Profit  $313.98 ÿ The idea: In general, optimal pricing solutions among these three methods entail a trade-off among the concepts of extraction, inclusion, and exclusion. ÿ Negative correlation of reservation prices enables a manager to fully extract all consumer surplus with a pure bundle when the cost of production is low. ÿ Negative correlation is not required to make bundling the best choice (see Figure 5). Example 3 ÿ Assumptions * Perfect negative correlation among consumer reservation prices (Figure 10.4) * No variation in total bundle valuation; all value the bundle at $100 * Unit cost of production for each good  $55 ÿ Table 10.10: Optimal Separate Prices for Good 1 and Good 2: Profit  $70 ÿ Table 10.11: Optimal Pure Bundle Price for Consumers A, B, C, and D: Profit  $0 ÿ Table 10.12: Optimal Mixed Bundle Prices at Any Pure Bundle Price over $100 (So No Bundle Is Purchased): Profit  $70 Conclusions from Examples 1–3: When reservation prices are negatively correlated ÿ When production cost is low, pure bundling will extract all consumer surplus. ÿ When production cost rises, mixed bundling is best. ÿ When production cost rises further, separate pricing is best. ÿ The optimal separate prices are always equal to consumers’ reservation prices.

158 | Chapter 10

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•

•

ÿ The optimal pure bundle price is always equal to the sum of consumers’ reservation prices. ÿ The optimal mixed bundle prices are not necessarily equal to reservation prices or their sum. Example 4 ÿ Assumptions * Distribution of reservation prices is uniform over the range $0 to $100 for each good. * Correlation is zero. * There are 10,000 customers. * Production cost is zero. ÿ Figure 10.5: Optimal Separate Prices in the Case of Uniformly Distributed Consumer Reservation Prices: Profit  $500,000 ÿ Figure 10.6: Optimal Pure Bundle Price in the Case of Uniformly Distributed Reservation Prices: Profit  $544,331.10 ÿ Figure 10.7: Optimal Mixed Bundle Pricing in the Case of Uniformly Distributed Reservation Prices: Profit  $549.201. Mixed bundling is even better than pure bundling. Example 5 ÿ Assumptions * Quantity discounting is a form of mixed bundling. * Unit cost of production for each good  $1 ÿ Table 10.13: Reservation Prices for the First and Second Units of a Good by Consumers A and B ÿ Table 10.14: Optimal Separate Prices for the Good: Profit  $6 ÿ Table 10.15: Optimal Pure Bundle Price for Two Units of the Good: Profit  $7 ÿ Table 10.16: Optimal Mixed Bundling Prices for the Case of a Single Good: Profit  $7.99 Example 6 ÿ Assumptions * Three consumers with negatively correlated reservation prices * Each consumer wants no more than one unit of each of two goods * Cost of production is $4 ÿ Table 10.17: Consumer Reservation Prices for Good X and Good Y (in Dollars) ÿ Table 10.18: Best Separate Price Strategy: Profit  $16.00 ÿ Table 10.19: Best Pure Bundling Strategy: Profit  $15.99 ÿ Table 10.20: Best Mixed Bundling Strategy: Profit  $17.97

3. When to Unbundle •

Example ÿ Assumptions

Bundling and Intrafirm Pricing | 159 * Three consumers with negatively correlated reservation prices * Each consumer wants no more than one unit of each of two goods. ÿ Table 10.21: The Reservation Prices for Consumers A, B, and C for Good X, Good Y, and a Bundle of Good X and Good Y ÿ Figure 10.8: Depiction of Bundling Problem in Table 10.21 * Pure bundling gives the lowest profit. * Mixed bundling gives the highest profit. 4. Bundling as a Preemptive Entry Strategy • •

•

Bundling can be used to deter entry. Assumptions ÿ The bundle offered by Alpha Company is made up of W and S, has a cost of 4, and will be priced at $X. ÿ The Beta Company is developing C that has a cost of 2 and is a close substitute for W. ÿ The Gamma Company is developing N that has a cost of 2 and is a close substitute for S. ÿ Only Alpha has the assets to produce a bundle. ÿ Alpha’s entry cost to the market with a bundle is 30. Entry cost for each product individually is 15. ÿ Beta’s entry cost to the market is 17. ÿ Alpha’s entry cost to the market is 17. ÿ Demand for the goods is perfectly negatively correlated. Example ÿ Table 10.22: The Reservation Prices for Consumers A, B, and C for Good W or C, Good S or N, and a Bundle of Good W and Good S or a Bundle of Good C and Good N

STRATEGY SESSION: How the New Yorker Used Bundling Discussion Questions 1. What does this strategy’s success imply about the reservation prices of advertisers?

•

Answer: The reservation price for advertising in The New Yorker is less  than the price of advertising in The New Yorker. The reservation price of advertising in the The New Yorker plus the reservation price of advertising in either of the other two magazines is less than the bundle price. Some Guidelines to Help Managers Construct More Effective Bundling Policies:

160 | Chapter 10 1. If goods’ reservation prices are positively correlated, pure bundling can do no better than separate pricing (but mixed bundling might). 2. If the marginal cost of producing a good exceeds its reservation price, in general you should think carefully about selling it. 3. If goods’ reservation prices are correlated perfectly negatively and the marginal cost of production of the goods is zero, pure bundling is best. 4. If goods’ reservation prices are negatively correlated, as the marginal cost of production increases, mixed bundling is likely to be better than pure bundling. 5. Tying at IBM, Xerox, and Microsoft •

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Tying: A pricing technique in which managers sell a product that needs a complementary product ÿ This is a form of bundling that applies to complementary products. ÿ Tying requires market power. ÿ Tying may be used to protect a monopoly. ÿ Tying may be used to ensure that a firm’s product works properly and that its brand name is protected. ÿ Tying can lower transaction costs. ÿ Tying is also used to protect product integrity. * Examples: Maintenance contracts and franchise requirements Examples ÿ Xerox required customers who leased copy machines to use Xerox paper. ÿ IBM required customers who leased computers to use IBM punch cards. ÿ Microsoft forced customers to use Internet Explorer with its operating system in order to exclude Netscape and protect its monopoly. ÿ Notice to wholesale liquor dealers from the Department of the Treasury (2003) prohibited the following tying practices: * Requiring a retailer to purchase a regular case of distilled spirits to be able to purchase the spirits in a special holiday container * Requiring a retailer to purchase 10 cases of a winery’s Chardonnay with 10 cases of the winery’s Merlot * Requiring a retailer to purchase a two-bottle package of a winery’s Merlot and Chardonnay to purchase cases of the winery’s Merlot

6. Transfer Pricing • •

Transfer Price: Payment that simulates a market where no formal market exists Transfer pricing is prevalent.

Bundling and Intrafirm Pricing | 161

•

ÿ Refers to intrafirm pricing among wholly-owned subsidiaries or divisions ÿ The purpose of transfer prices: * To encourage profit-maximizing or cost-minimizing behavior by providing an incentive * To measure the performance of semiautonomous divisions Illustration: The transfer pricing issue facing managers. * Demand curve for the downstream product: PD  PD (QD) where PD is the price of the downstream product and QD is in units of the downstream product Upstream Producer Quantity Produced = QU TCU = TCU (QU)

Multidivisional Firm π = TRD –TCU –TCD

Transfer Price PU

Downstream Producer Quantity Produced = QD QD = f (LD , KD|QU) TCD = TCD(QD|QU)

Market Demand PD = PD(QD)

⎛ TRD ⎞ ⎛ QD ⎞ ⎛ TC D ⎞ ⎛ QD ⎞ ⎛ TCU ⎞  =⎜ ⎟ –⎜ ⎟ –⎜ ⎟ =0 ⎟⎜ ⎟⎜ QU ⎝ QD ⎠ ⎝ QU ⎠ ⎝ QD ⎠ ⎝ QU ⎠ ⎝ QU ⎠ ⎡⎛ TRD ⎞ ⎛ TC D ⎞ ⎤ ⎛ QD ⎞ ⎛ TCU ⎞ ⎢⎜ ⎟ =⎜ ⎟ ⎟ –⎜ ⎟ ⎥⎜ ⎢⎣⎝ QD ⎠ ⎝ QD ⎠ ⎥⎦ ⎝ QU ⎠ ⎝ QU ⎠

or (MR D 2 MCD)MPU  MCU (condition for profit maximization) •

In the absence of an external market, the optimal transfer price is the marginal cost of the upstream product (MCU) when the optimal quantity (QU) is produced.

•

Example: Figure 10.9: Determination of the Transfer Price, Given No External Market for the Transferred Good

7. Transfer Pricing: A Perfectly Competitive Market for the Upstream Product • •

The optimal transfer price is the competitive market price. Example: Figure 10.10: Determination of the Transfer Price, Given a Perfectly Competitive External Market for the Transferred Product

162 | Chapter 10 8. The Global Use of Transfer Pricing • •

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•

•

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Transfer pricing is widespread. For international transfers, the most common methods of determining transfer prices are market-based transfer prices and full productions costs plus a markup. ÿ A comparison with the results of an earlier study indicates that the market-based transfer prices are increasingly being used. Managers can use transfer pricing to shift profits between divisions in order to minimize tax liability. ÿ Increase profit in low-tax countries and decrease profit in high-tax countries. Notation and Implication ÿ Assume that there is no external market for the upstream product and that all profits are expressed in the same currency. ÿ a  Tax rate in a downstream country ÿ b  Tax rate in an upstream country, where a b ÿ After-tax profit in the downstream country  (1  a)(TR D 2 TCD 2 PUQU) ÿ After-tax profit in the upstream country  (1  b)(PUQU 2 TCU) ÿ Suppose all profits are expressed in the same currency and that PU was used to maximize before-tax profit. ÿ Overall conglomerate’s after-tax profit  (1  a)(TRD 2 TCD)  (1  b) (TCU)  (a 2 b)(PUQU) ÿ Increasing the transfer price (PU) will increase after-tax profit. Question: Why have transfer prices become so important on the international level? Four basic reasons for the importance of global transfer prices: ÿ Increased globalization ÿ Different level of taxation in various countries ÿ Greater scrutiny by tax authorities ÿ Inconsistent rules and laws in different tax jurisdictions Transfer Price Policies that Cause the Fewest Problems ÿ Comparable uncontrolled price (arms-length price) ÿ Cost-plus prices, using the arms-length markup ÿ Resale price

Chapter 10: Problem Solutions 1. The Locust Corporation is composed of a marketing division and a production division. The marginal cost of producing a unit of the firm’s product is $10 per unit, and the marginal cost of marketing it is $4 per unit. The demand curve for the firm’s product is

Bundling and Intrafirm Pricing | 163 P  100  0.01Q where P is the price per unit (in dollars) and Q is output (in units). There is no external market for the good made by the production division. a. How should managers set the optimal output? b. What price should managers charge? c. How much should the production division manager charge his counterpart in marketing for each unit of the product? Solution: a. The marginal cost for the firm is MC  MCP  MCM  10  4  14. Also, the firm’s total revenue is TR  PQ  (100  0.01Q)Q  100Q  0.01Q2 and marginal revenue is given by MR  100  0.02Q. Now, set the firstorder condition for profit maximization: MR  MC. That is, 100  0.02Q  10  4 → 86  0.02Q → Q  4,300. That is, managers should set the optimal output at 4,300 units. b. P  100  0.01(4,300)  $57 c. The transfer price should be set at the selling division’s marginal cost, $10 in the case of the producing division. 2. The Xerxes Company is composed of a marketing division and a production division. The marketing division packages and distributes a plastic item made by the production division. The demand curve for the finished product sold by the marketing division is P0  200  3Q0 where P0 is the price (in dollars per pound) of the finished product and Q 0 is the quantity sold (in thousands of pounds). Excluding the production cost of the basic plastic item, the marketing division’s total cost function is TC0  100  15Q 0 where TC0 is the marketing division’s total cost (in thousands of dollars). The production division’s total cost function is TC1  5  3Q1  0.4Q12 where TC1 is total production cost (in thousands of dollars) and Q1 is the total quantity produced of the basic plastic item (in thousands of pounds). There is a perfectly competitive market for the basic plastic item, the price being $20 per pound. a. What is the optimal output for the production division? b. What is the optimal output for the marketing division? c. What is the optimal transfer price for the basic plastic item? d. At what price should the marketing division sell its product?

164 | Chapter 10 Solution: a. The producing division should produce to the point where its marginal cost equals the competitive price for the basic plastic item. MC1  3  0.8Q1  20 → Q1  17/0.8  21.25 pounds b. The marketing division should market to the point where its net marginal revenue from marketing equals its marginal marketing cost: MR  MC. Use equation (10.4) to obtain MR  MR0  MC0  200  6Q 0  15  20 → Q 0  165/6  27.5 pounds. c. The basic plastic item should be transferred at the competitive intermediate price of $20. d. P1  200  3(27.5)  $117.50. 3. Knox Chemical Corporation is one of the largest producers of isopropyl alcohol, or isopropanol, as it frequently is called. Isopropanol is used to produce acetone, an important industrial chemical; it is also used to make various chemical intermediate products. Because Knox Chemical produces both acetone and these chemical intermediates, it uses much of the isopropanol it makes. One of the many tasks of Knox’s product manager for isopropanol is to set transfer prices for isopropanol within the company. a. Knox’s product manager for isopropanol generally sets the transfer price equal to the prevailing market price. Is this a sensible procedure? b. When the production of phenol expands rapidly, a great deal of acetone is produced because it is a by-product of the process leading to phenol. What effect do you think this has on the market price of isopropanol? c. In producing a pound of phenol, 0.6 pound of acetone is produced. Are phenol and acetone joint products? d. Are they produced in fixed proportions? Solution: a. This is a sensible procedure. If Knox can sell its isopropanol in a competitive market, the appropriate transfer price is the market price. b. The market and transfer prices of isopropanol will fall. c. Yes d. Yes

Bundling and Intrafirm Pricing | 165 4. The reservation prices (in dollars) of three classes of demanders (classes A, B, and C) for Ricky Parton’s (a Latin country-western singer) compact disks are given in the table that follows: Class

CD 1

CD 2

A B C

11 8 9

5 9 10

It costs $4 to produce and distribute each compact disk. The company can sell each CD separately, can put them together as a boxed set (that is, as a pure bundle), or can sell them in a mixed bundling format (offer the CDs both separately and as a boxed set). Assume that each demander wants only one of each of the CDs at the reservation price (or at any lower price) and that there are an equal number of demanders in each class. For simplicity, assume that the only costs are those mentioned here. a. What pricing method would you advise Ricky’s company to use? b. How much better (profitwise) is the best pricing method than the second most profitable pricing method? Solution: a. Pricing the two-CD bundle at $16 will yield revenue  $48. This is the best pricing method. b. Separate pricing, with the price of CD 1  8 and the price of CD 2  9 will yield revenue  42. The second-best pricing method is $6 worse than the best method. 5. Bob and Ron’s Stereo sells televisions and DVD players. They have estimated the demand for these items and have determined that there are three consumer types (A, B, and C) of equal number (assume one for simplicity) that have the following reservation prices for the two products. Bob and Ron’s cost for a TV is 9 and for a DVD player is 9. It will cost Bob and Ron 18 to produce a bundle of one TV and one DVD player. Consumer

TV

DVD Player

A B C

28 29 30

12 4 10

Any consumer’s reservation price for a bundle of one TV and one DVD player is the sum of their reservation prices for each item. Consumers will demand (at most) one TV and one DVD player. a. If Bob and Ron only consider pricing each item separately, pricing a pure bundle, or pricing a mixed bundle as their pricing policy, what price(s) would maximize their profit and what would be their profit?

166 | Chapter 10 b. If Bob and Ron were able to perfectly price discriminate (that is, charge different prices to different consumers), how much would their profit increase over their optimal profit in part a? Solution: a. Under separate pricing, TVs should be priced at $28 and DVD players should be priced at $10. This yields net revenue  $104  (5)(9)  $59. With a pure bundle price of $33, net revenue  $99  (6)(9)  $45. With a mixed bundle, the bundle should be priced at $40, TVs should be priced at $29, and the DVD player should be priced at $12. Bundles will be sold to consumers A and D. Consumer B will buy a TV but not a DVD player. Net revenue  $109  (5)(9)  $64, so the mixed bundle is best. b. There would be no additional net revenue. The mixed bundle extracts all consumer surplus. 6. The University of Pennsylvania basketball team will play both the University of Kansas and Nowhere University this year on Penn’s campus. Kansas is a nationally ranked team, while Nowhere is just plain terrible. The athletic director traditionally prices each game separately. You approach him and point out that two other pricing options exist. One possibility is to offer a pure bundle, that is, a ticket package containing one Kansas ticket and one Nowhere ticket. The second possibility is a mixed bundle. In this situation, a pure bundle is offered, but admissions to the games can also be sold separately. It costs Penn a constant 5 per spectator to produce a game. It would cost Penn 10 to produce a bundle of a Kansas game and a Nowhere game. Three types of potential spectators exist (A, B, and C). There are an equal number of types (for simplicity, assume one of each type). Their reservation prices for each game are shown below: Spectator

Kansas

Nowhere

A B C

40 49 3

13 3 30

Penn’s policy is not to price discriminate. A spectator’s reservation price for a bundle of the two games is the sum of their reservation prices for each game. A spectator wants (at most) one admission to each game. a. What’s your pricing advice to the athletic director (so that the director maximizes Penn’s profit)? b. Given the current pricing policy of Penn, what’s your advice worth to the athletic director?

Bundling and Intrafirm Pricing | 167 Solution: a. Under separate pricing, Kansas tickets should be priced at $40 and Nowhere tickets should be priced at $30. This yields net revenue  $110  (3)(5)  $95. With a pure bundle price of $52, net revenue  $104  (4)(5)  $84. With a mixed bundle, the bundle should be priced at $53, Kansas tickets should be priced at $49, and the Nowhere tickets should be priced at $30. The bundle will be sold to consumer A, a Kansas ticket will be sold to consumer B, and a Nowhere ticket will be sold to consumer C. Net revenue  $132  (4)(5)  $112, so the mixed bundle is best. b. The mixed bundle increases net revenue by $17 over the separate pricing strategy. 7. GeeM has a sporty wheel package and a luxury interior package that it is considering offering to its auto buyers. GeeM has estimated that there are three consumer types (A, B, and C—all of equal magnitude—for simplicity, consider it one of each type). Consumers want (at most) one of each package. It costs GeeM 5 to produce a sporty wheel package and 10 to produce a luxury interior package. It will cost GeeM 15 to produce a bundle consisting of both packages. The following are the consumer reservation prices for each package: Consumer

Wheels

Interior

A B C

11 35 18

24 12 28

A consumer’s reservation price for a bundle consisting of sporty wheels and a luxury interior is the sum of the individual component reservation prices. GeeM does not price discriminate. GeeM has solicited your help in pricing the wheel and interior package. You know that they could sell the packages separately, as a pure bundle, or as a mixed bundle. Of those three pricing strategies, which one would maximize GeeM’s profit? What are the prices (what is the price) that you suggest? How much better is the best pricing strategy than the second best pricing strategy? Solution: Under separate pricing, the Wheels package should be priced at $35 and the Interior package should be priced at $24. This yields net revenue  $83  (1) (5)  (2)(10)  $58. With a pure bundle price of $46, net revenue  $92  (2) (15)  $62. With a mixed bundle, the bundle should be priced at $46, the Wheels package should be priced at $22, and the Interior package should be

168 | Chapter 10 priced at $24. The bundle will be sold to consumers B and C and the Wheels package will be sold to consumer A. Net revenue  $116  (2)(15)  (1) (10)  $76, so the mixed bundle is best by $14 over the pure bundle. 8. Food For Life makes health foods for active, outdoor people. Their three basic products are whey powder, a high-protein strength bar, and a meal additive that has the taste and consistency of sawdust. Research shows that consumers fall into two types (A and B), and these are described in the table below by their reservation prices for the products. Each consumer will demand no more than one unit of any product at their reservation price. The consumers will value a bundle of the products at the sum of the constituent reservation prices. Each product costs $3 to produce. A bundle of all three products costs $9 to produce. Food For Life does not price discriminate. Consumer

Whey

Strength

Sawdust

A B

10 3

16 10

2 13

There is an equal number of each consumer type (for simplicity, one of each type). What pricing (profit-maximizing) strategy (among pricing separately, pure bundling, and mixed bundling) would you recommend to Food For Life? Why? Only bundles of all three products need to be considered. Solution: Under separate pricing, Whey should be priced at $10, Strength should be  priced at $10, and Sawdust should be priced at $13. This yields net revenue  $43  (4)(3)  $31. With a pure bundle price of $26, net revenue  $52  (2)(9)  $34. With a mixed bundle, the bundle should be priced at $28, Whey should be priced at $10, Strength should be priced at $10, and Sawdust should be priced at $13. The bundle will be sold to consumer A, and Strength and Sawdust will be sold to consumer B. Net revenue  $51  (9)  (2)(3)  $36, so the mixed bundle is better by $2 over separate pricing.

CHAPTER 11

Oligopoly

Lecture Notes 1. Introduction •

Objectives ÿ To study four main types of oligopoly: * Cournot Oligopoly * Stackelberg Oligopoly * Bertrand Oligopoly * The Kinked Demand Curve Oligopoly Model ÿ To explain how managers of fi rms that operate in an oligopoly market use strategic decision making to maintain relatively high profits ÿ To understand how the reactions of market rivals influence the effectiveness of decisions in an oligopoly market ÿ To examine cooperative behavior ÿ To discuss collusive agreements and their breakdowns ÿ To examine collusive behavior of managers

2. Cooperative Behavior • Recall that: ÿ Oligopoly is a market with a small number of firms. It is characterized by interdependence and the need for managers to explicitly consider the reactions of rivals in formulating pricing policy. ÿ It is protected by barriers to entry that result from government fiat, economies of scale, or control of strategically important resources. * Examples:

169

170 | Chapter 11 The U.S. petroleum industry: A few firms account for most of the industry’s refining capacity. ° Market for commercial aircraft: Dominated by Boeing and Airbus * Question: Why do oligopolies exist? ° A high entry barrier ° Government fiat ° Economies of scale: Costs decrease as output expands. * Oligopolists often realize relatively high profits. Oligopolies are global phenomena. The small number of firms in an oligopoly market tends to encourage cooperative behavior (collusion). This can * increase profits; * decrease uncertainty; * raise barriers to entry. There are usually incentives for cooperative parties to cheat. Cartel: When a collusive arrangement is made openly and formally ÿ Cartels and collusion in general are illegal in the United States. ÿ Sherman Antitrust Act (1890): Outlawed most collusive agreements ÿ Cartels maximize profit by restricting the output of member firms to a level where the marginal cost of production of every firm in the cartel is equal to the market’s marginal revenue and then charging the market-clearing price. ÿ Managers need to estimate the marginal cost curve for the cartel as a whole. ÿ If input prices do not increase as the cartel expands, then the marginal cost curve is the horizontal summation of the individual firms’ marginal cost curves. ÿ The need to allocate output among member firms results in an incentive for the firms to cheat by overproducing and thereby increase profit. ÿ Figure 11.1: Price and Output Determination by a Cartel ÿ Cartel managers also determine the distribution of sales across members. ÿ If the purpose of the managers is to maximize the profits of the corporate entity, they should allocate sales to cartel members so that the marginal cost of all members is equal and in turn equal to the cartel’s marginal revenue. °

• •

• •

3. The Breakdown of Collusive Agreements • •

Cartels are not stable. By producing a quantity of output that exceeds the quota established by a cartel, a firm can generally increase profits. ÿ Figure 11.2: Instability of Cartels

Oligopoly | 171 ÿ There is a constant threat to the existence of a cartel. ÿ Trust is vital to a cartel’s existence. 4. Price Leadership •

•

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Price Leadership: In oligopolistic industries, managers at one firm have significant market power and can set their price and rivals then follow their lead. Assumptions ÿ There is a dominant firm, the price leader, that sets price in the market. ÿ There are also a number of small firms, follower firms, which behave as price takers, producing a quantity where marginal cost is equal to price. Their supply curve is the horizontal summation of their marginal cost curves. ÿ The price leader faces a residual demand curve that is the horizontal difference between the market demand curve and the followers’ supply curve. ÿ The price leader produces a quantity where the residual marginal revenue is equal marginal cost. Price is then set to clear the market. Figure 11.3: Price Leadership by a Dominant Firm

STRATEGY SESSION: Cranberries: Where 30% of the Market Are Price Takers Discussion Questions 1. Does a market that operates according to the price leadership model provide more or less consumer surplus than a market that is perfectly competitive? Answer: Less 2. Does a market that operates according to the price leadership model provide more or less consumer surplus than a market that is controlled by a monopolist? Answer: More, because a portion of market supply is sold at a price equal to marginal cost. PROBLEM SOLVED: Ghoshal, Inc.: A Numerical Example Discussion Questions Suppose that the industry demand curve is Q  100  5P. The followers’ supply curve is QS  10  P, where QS is the total amount supplied by all the small firms combined. The price leader’s marginal cost is

172 | Chapter 11 MC  (8/3)QA. (11.1) where QA is the leader’s output (i.e., Goshal’s output). 1. What is the price leader’s residual demand curve? 2. What is the price leader’s marginal revenue function? 3. What price will the price leader charge, and how much will it produce? 4. What quantity will be produced by the followers? Solution 1. To derive the demand curve for the leader, we subtract QS from Q to obtain QA  Q 2 QS  100  5P  10  P, That is, QA  90  6P or alternatively, P  15  (1/6)QA. 2. Gosha’s total revenue is given by TR A  PQA  15  (1/6)QA)QA  15QA  (1/6)(QA)2. Marginal revenue is given by MR A  15  (1/3)QA. 3. Profit maximization implies MR A  MCA, that is 15  (1/3)QA  (8/3)QA. That is, QA  5 and the price is P  $14.167. Therefore, if the managers at Goshal want to maximize profit, they should set their price at $14.167. 4. The followers take the price as given as set by the leader. Hence, they will produce QS  10  14.167  $24.167. 5. Possible Behavior in Markets with Few Rivals • • • •

•

Managers need to anticipate the behavior of others. Consider the two-firm case: Duopoly. Duopoly: A market in which there are only two sellers Assume that: ÿ Firms produce identical products in a simultaneous move scenario. ÿ Rival managers make decisions simultaneously (a simultaneous move game). * When rivals make decisions without knowing the decisions of others, we say decision making is simultaneous (as opposed to the sequential move game in which managers know the decisions of others before making their decisions). * Managers who take actions before others are called first movers or market leaders. When Rivals Are Few: Price Competition ÿ Price competition is a common strategy used by managers.

Oligopoly | 173 ÿ Key idea: Managers should try to avoid this behavior. ÿ Price competition tends to drive prices down to marginal cost and so should be avoided by managers. ÿ Example: * Consider two firms A and B with identical total cost functions: TCi  500  4qi 0.5qi2 *

(11.4),

where I  A, B and qi is firm i’s output. Market demand as seen by managers of both firms is: P  100  Q  100  qAqB (11.5)

where P is the product’s price per unit. * Marginal cost for firm I is: MCi  4  qi * Firm i’s total revenue: TRi  Pqi  (100  qA2qB)qi * Marginal revenue of firm A and B is obtained as MR A  100  2qA2qB

and

MR B  100  2qB2qA.

If managers at both firms want to compete on price, the competition will drive price down to the level of their marginal cost. * Set MCA  P to get firm A’s reaction function: 4  qA  100  qA2 qB → qA  48  0.5qB (11.7) * Set MCB  P to get firm B’s reaction function: 4  qB  100  qA2 qB → qB  48 - 0.5qB (11.8) * Pricing game outcome: Substitute (11.8) into (11.7) to get *

qA  48  0.5(48  0.5qB)  24  0.25 qA

•

Hence qA  32 and qB  48  0.5(32)  32. Note that each firm has identical cost functions. * Industry’s output: Q  32  32  64 units * Price: P  100  64  $36 * Each firm has a total revenue of $1,152 and a total cost of $1,140; hence each firm earns a profit of $12. When Rivals Are Few: Collusion ÿ Question: What if the two managers both realized the dangers of a price war and instead chose to act cooperatively? ÿ Example: The two-firm case Assume they can legally form a cartel. Under these circumstances, the market demand curve is the cartel’s demand curve and the cartel’s marginal cost curve is the horizontal summation of each firm’s marginal cost curve. * The two firms have identical total cost functions: TCi  500  4qi  0.5qi2 * Market demand: P  100  Q  100  qA2 qB * Marginal revenue: 100  2Q

174 | Chapter 11 * *

* * * * *

Marginal cost: MCi  4  qi and qi  4  MCi Hence qA  4  MCA and qB 4  MCB Summing the marginal cost horizontally by adding up the quantities produced at any given MC gives: Q  qA  qB  8  2MC. → MC  4  0.5Q, the cartel’s marginal cost. The cartel behaves as a monopolist and sets marginal revenue equals marginal cost. Total revenue: TR  PQ  (100  Q)Q  100QQ2, so that MR  100  2Q. Set MC  MR: 4  0.5Q  100  2Q → Q  38.4 (qi  19.2) and P  100  38.4  $61.6 Cartel’s total revenue: $2,365.44 Total profit is $843.20, or $421.60 for each firm.

STRATEGY SESSION: Cartels Come in Many Shapes and Sizes Discussion Questions 1. Why do you think that collusion among shipping firms is exempt from U.S. antitrust laws? 2. Do you think that other U.S. industries, for example, tobacco producers or distilleries, should be exempt from antitrust laws? What effect would it likely have on the markets for these products? STRATEGY SESSION: How Seagoing Chemical Haulers May Have Tried to Share the Market Discussion Question 1. Based on the ease with which firms can identify the profits available from collusive behavior, do you think it likely that many firms engage in this behavior, even though it is illegal? • When Rivals Are Few: Quantity (Capacity) Competition ÿ Strictly competing on price is often a lose–lose strategy. ÿ Managers could compete on something other than price. ÿ Idea: Managers should try to compete on any metric that affects profit and gives them a higher profit relative to competing on price. ÿ One metric: Quantity or production capacity This type of competition is called Cournot. ÿ The Cournot model makes the following assumptions: Rivals * make simultaneous decisions, * have the same estimate of market demand, * have an estimate of the other’s cost function, and * choose their profit-maximization output conditional on their rivals’ outputs.

Oligopoly | 175

•

•

ÿ Example 1: Monopoly by firm A * Market demand: P  100  Q  100  qA * Marginal revenue: 100  2Q * Marginal cost: MCA  4  Q * MC  MR: 4  Q  100  2Q → Q  32 and P  68 ÿ Example 2: Firm B produces qB  96. * Residual market demand to firm A: P  4  qA * Optimal output is qA  0. ÿ Example 3: Firm B produces qB  50. * Residual market demand to firm A: P  50  qA * Optimal output is qA  15.33. ÿ Example 4: General solution * Market demand: P  100  Q  100  qB * Marginal revenue for firm A: MR A  100  2 qB * Marginal cost for firm A: MCA  4  qA * Managers at firm A maximize profits by setting MC  MR, yielding firm A’s reaction function: 4  qA  100  2qA  qB → qA  32  (1/3)qB (11.10) This is firm A’s reaction function or best-response function. * By symmetry, we have Firm B’s reaction function: qB  32  (1/3)qA * Cournot equilibrium: Solving the two reaction functions simultaneously yields qA  qB  24, and each firm earns a profit of $364. * Figure 11.4: Cournot Reaction Functions for Firms A and B The Cournot Scenario with More than Two Firms ÿ Example * Market demand: P  a  b Qi * Marginal revenue: MR  a  (n  1)bQi * MC  c  eQi * MC  MR: c  eQi  a  (n  1)bQi → Qi  (a  c)/[(n  1)b  e] * Table 11.2: Price, Output, and Profits with Multiple Cournot Competitors (with a  100, b  1, c  4, and e  1) * The addition of a small number of entrants in a Cournot situation can result in significant price competition and erosion of profits. When Managers Move First: Stackelberg Oligopoly * Aim: To consider a situation in which managers at one firm are able to implement actions prior to those of rival managers * Managers at firm A choose and credibly commit to an output decision. * Managers at firm B know the decision when they choose their own output. The oligopoly model was proposed in 1934 by Heinrich Von Stackelberg.

176 | Chapter 11 In the Stackelberg model, * Firms set output; * One firm (the leader) acts before the others. ÿ When, in a Cournot environment, one firm moves first and optimizes production based on knowledge of its rival’s reaction function, there is a first-mover advantage. ÿ Example * Market demand: P  100  Q  100  qA  qB * Marginal revenue for firm A: MR  100  2qAqB * Marginal cost for firm A: MCA  4  qA * Solution method: Solve backward. * Given QB, firm B chooses its output by maximizing its profit. * Firm B’s reaction function: qB  32  (1/3)qA * Now, given firm B’s reaction, the leader (firm B) chooses its output by setting MCA  MRB : 4  qA  100  2qA  [32  (1/3)qA] → qA  27.43, qB  22.86, firm A’s profit is $377.71, and firm B’s profit is $283.67. The first-mover advantage (additional profit) over the Cournot solution for firm A is $13.71. ÿ Example: When firm A has a lower cost, its first-mover advantage is increased. 2 * Firm A’s cost function: TCA  500  4qA  0.5qA 2 * Firm B’s cost function: TCB  500  10qB  0.5qB * Firm A’s reaction function: qA  32  (1/3)qB * Firm B’s reaction function: qB  30  (1/3)qA * If firm A goes first: P  $51.143, qA  28.286, pA  $433.429, qB  20.571, and pB  $134.776. * If firm B goes first: P  $51.429, qA  23.714, pA  $343.551, qB  24.857, and pB  $220.857. * A general managerial rule: If you have the market strength so that market rivals cede you the power to move first, use it. *

6. Duopolists and Price Competition with Differentiated Products • •

The Bertrand Model or Bertrand Oligopoly Model Example: Two producers who sell differentiated but highly substitutable products ÿ Assume MC  0 for both firms ÿ Demand for firm 1’s product: Q1  100  3P1  2P2 ÿ Demand for firm 2’s product: Q2  100  3P2  2P1 ÿ Total revenue for firm 1: TR1  P1(100  3P1  2P2)  100P1  3P12  2P1P2 TR1  TR11  TR12 where TR11  100P1  3P12 and TR12  2P1P2

Oligopoly | 177

•

ÿ Marginal revenue for firm 1: MR1  TR1/P1  (TR11/P1)  (TR11/P1) MR1  100  6P1  2P2 ÿ Bertrand reaction function for firm 1: MR  MC1  0: 100  6P1  2P2  0 → P1  (50/3)  (1/3)P2 ÿ Bertrand reaction function for firm 2: MR  MC2  0: 100  6P2  2P1  0 → P2  (50/3)  (1/3)P1 ÿ Solving the two reaction functions simultaneously yields: P1  P2  $25, q1  q2  75, p1  p2  $1,875. ÿ Figure 11.5: Bertrand Reaction Functions and Equilibrium for Firms 1 and 2 Example: Collusion Two producers who sell differentiated but highly substitutable products and collude or merge ÿ TR  TR11  TR22  TR12  100P1  3P12  100P2  3P22  4P1P2 ÿ MR1  100  6P1  4P2 ÿ MR2  100  6P2  4P1 ÿ Reaction function for firm 1: (MR1  0): P1  (50/3)  (2/3)P2 ÿ Reaction function for firm 2: (MR2  0): P2  (50/3)  (2/3)P1 ÿ Solving the two reaction functions simultaneously yields: P1  P2  $50, q1  q2  50, p1  p2  $1,875.

7. The Sticky Pricing of Managers •

•

Asymmetrical Responses to Price Changes ÿ If a firm increases price, other firms do not follow, so the firm’s demand is relatively elastic. ÿ If a firm reduces price, other firms follow, so the firm’s demand is relatively inelastic. ÿ Result: The firm’s demand curve has a “kink” at the current price, and the firm’s marginal revenue curve is vertical at the quantity that corresponds to the kink. ÿ Implication: Changes in marginal cost that do not move above or below the vertical section of the marginal revenue curve do not cause the optimal level of output or price to change. Figure 11.6: The Situation of the Kinked Demand Curve

Chapter 11: Problem Solutions 1. The Bergen Company and the Gutenberg Company are the only two firms that produce and sell a par ticular kind of machinery. The demand curve for their product is P  580  3Q

178 | Chapter 11 where P is the price (in dollars) of the product, and Q is the total amount demanded. The total cost function of the Bergen Company is TCB  410QB where TCB is its total cost (in dollars) and QB is its output. The total cost function of the Gutenberg Company is TCG  460QG where TCG is its total cost (in dollars) and QG is its output. a. If these two firms collude and they want to maximize their combined profit, how much will the Bergen Company produce? b. How much will the Gutenberg Company produce? c. Will the Gutenberg Company agree to such an arrangement? Why or why not? Solution: a. We observe that MCB  410

and

MCG  460.

Hence, it is seen that MCB < MCG. If the two firms collude, the firm with the lowest marginal cost will lead the production process. Since Bergen’s marginal cost is always less than Gutenberg’s marginal cost, then Bergen would produce all the combination’s output. Bergen Company’s total revenue is given by TR  PQ  (580  3P)Q. Therefore, marginal revenue is given by MR  580  6Q Profit maximization implies MR  MC. Hence, we have 410  580  6Q → QB  170/6

and

QG  0

b. As discussed in part (a), Gutenberg’s marginal cost is always greater than Bergen’s. If Gutenberg were to produce 1 unit and Bergen 1 unit less, it would reduce their combined profits by the difference in their marginal costs. If Gutenberg were to produce 1 unit without any reduction in Bergen’s output, it would reduce their combined profits by the same amount. Therefore, the Gutenberg Company will produce nothing. c. If direct payments for output restrictions between the fi rms were legal, Gutenberg would accept the zero output quota. But if competition were to break out, Gutenberg would make zero profits and Bergen would earn $2.00. Thus the most Bergen would pay for Gutenberg’s cooperation is $408.33 and the least Gutenberg would accept to not produce is $0.01.

Oligopoly | 179 2. The can industry is composed of two firms. Suppose that the demand curve for cans is P  100  Q where P is the price (in cents) of a can and Q is the quantity demanded (in millions per month) of cans. Suppose the total cost function of each firm is TC  2  15q where TC is total cost (in tens of thousands of dollars) per month and q is the quantity produced (in millions) per month by the firm. a. What are the price and output if managers set price equal to marginal cost? b. What are the profit-maximizing price and output if the managers collude and act like a monopolist? c. Do the managers make a higher combined profit if they collude than if they set price equal to marginal cost? If so, how much higher is their combined profit? Solution: a. Since each firm has a constant marginal cost of $0.15, the price must also equal $0.15 for price to equal marginal cost. Setting price equal to MC gives 100  Q  0.15; that is, Q  99.85 million per month. b. If they collude, they will produce where marginal revenue equals marginal cost. Marginal revenue is given by MR  100  2Q. Setting marginal revenue equal to marginal cost, where MC  15 (to comply with the units of measurement for P and Q), same for both firms, we have 100  2Q  15, that is, 2Q  85, that is, Q  42.5. Therefore, the joint profit-maximizing combined output is Q  42.5. Since the firms have constant marginal costs, only one firm should operate; thereby they avoid the fixed costs of the other firm. Their combined profits would be p  57.5(42.5)  [2  15(42.5)]  1,804.25, or $18,042,500. If they cannot avoid the fixed costs of one of the firms by shutting it down, their combined profits would be $18,022,500. c. Since they lose $40,000 if they compete and earn $18,022,500 if they collude, they earn $18,062,500 more if they collude than if they compete. 3. An oligopolistic industry selling a par ticular type of machine tool is composed of two firms. Managers at the two firms set the same price and share the total market equally. The demand curve confronting each firm (assuming that the other firm sets the same price) follows, as well as each firm’s total cost function.

180 | Chapter 11 Price (Thousands of Dollars)

Quantity Demanded per Day

Daily Output

Total Cost (Thousands of Dollars)

10 9 8 7 6

5 6 7 8 9

5 6 7 8 9

45 47 50 55 65

a. Assuming that each manager is correct in believing that managers at the other firm will charge the same price as they do, what price should each charge? b. Under the assumptions in part (a), what daily output rate should managers at each firm set? Solution: a. Each should charge a price of $9,000. b. Each firm should produce 6 units, given the unit price charged of $9,000. 4. James Pizzo is president of a firm that is the industry price leader; that is, it sets the price and the other firms sell all they want at that price. In other words, the other firms act as perfect competitors. The demand curve for this industry’s product is P  300  Q, where P is the price of the product and Q is the total quantity demanded. The total amount supplied by the other firms is equal to Qr, where Qr  49P. (P is measured in dollars per barrel; Q, Qr, and Qb are measured in millions of barrels per week.) a. If Pizzo’s firm’s marginal cost curve is 2.96Qb, where Qb is the output of his firm, at what output level should he operate to maximize profit? b. What price should he charge? c. How much does the industry as a whole produce at this price? d. Is Pizzo’s firm the dominant firm in the industry? Solution: a. The residual demand faced by Pizzo is Qb  (Q  Qr )  300  P  49P  300  50P, which can be written as P  6  0.02Qb. Setting his marginal revenue equal to his marginal cost gives 6  0.04Qb  2.96Qb → Qb  2. Pizzo determines that the profit-maximizing quantity is 2. b. P  6  0.02(2)  $5.96 c. Q  300  P  294.04 units , Qr  49(596)  292.04 units d. No. The elasticity of Pizzo’s residual demand at his chosen level of output is nearly 150. Notice that his marginal cost is $5.92, and yet his market power only allows him to charge a price of $5.96. He produces less than 1% of the industry’s output.

Oligopoly | 181 5. The International Air Transport Association (IATA) has been composed of 108 U.S. and European airlines that fly transatlantic routes. For many years, IATA acted as a cartel: It fixed and enforced uniform prices. a. If IATA wanted to maximize the total profit of all member airlines, what uniform price would it charge? b. How would the total amount of traffic be allocated among the member airlines? c. Would IATA set price equal to marginal cost? Why or why not? Solution: a. The IATA would charge the price that clears the market at the level of output where the marginal revenue equals the horizontally summed marginal cost curves of each operator in the market. b. The traffic should be allocated so that the marginal costs of all the members operating in the market would be equal and that no member not currently in the market would have a lower marginal cost. c. No! Operating as a cartel allows IATA to behave as a monopolist. Therefore, the IATA would set a price equal to the marginal cost multiplied by 1/(1  1/E), where E is the price elasticity of demand in the market in question. 6. In late 1991 two firms, Delta Airlines and the Trump Shuttle, provided air shuttle service between New York and Boston or Washington. The one-way price charged by both firms was $142 on weekdays and $92 on weekends, with lower off-peak advance purchase fares. In September 1991 Delta increased the per-trip shuttle mileage given to members of the Delta frequentflier program from 1,000 to 2,000 miles, even though actual mileage from New York to either Boston or Washington is about 200 miles. Moreover, Delta also offered an extra 1,000 miles to frequent fliers who made a roundtrip on the same day, raising a possible day’s total to 5,000 miles. Almost simultaneously, Trump changed the frequent-flier mileage it gave shuttle passengers. (It participated in the One Pass frequent-flier program with Continental Airlines and some foreign carriers.) What sorts of changes do you think Trump made? Why? Solution: If Delta’s promotion were viewed by Trump as a cost effective method of wooing customers, it would have to respond with similar frequent-flier bonuses. Alternatively, Trump might try to initiate a different promotion, but to do nothing would cause it to lose customers to Delta. 7. Two firms, the Alliance Company and the Bangor Corporation, produce vision systems. The demand curve for vision systems is P  200,000  6(Q1  Q2)

182 | Chapter 11 where P is the price (in dollars) of a vision system, Q1 is the number of vision systems produced and sold per month by Alliance, and Q2 is the number of vision systems produced and sold per month by Bangor. Alliance’s total cost (in dollars) is TC1  8,000Q1 Bangor’s total cost (in dollars) is TC2  12,000Q2 a. If managers at these two firms set their own output levels to maximize profit, assuming that managers at the other firm hold constant their output, what is the equilibrium price? b. What is the output of each firm? c. How much profit do managers at each firm earn? Solution: a. Let us calculate the profits of each firm. Alliance and Bangor’s profits can be written as p1  [200,000  6(Q1  Q2)]Q1  8,000Q1  192,000Q1  6Q12  6Q1Q2 p2  [200,000  6(Q1  Q2)]Q2  8,000Q2  188,000Q2  6Q22  6Q1Q2 Profit-maximizing levels of output are determined by setting the first derivative of each firm’s profit function equal to zero and solving for Q1 and Q 2. p1/Q1  192,000  12Q1  6Q2  0 p2/Q2  188,000  6Q1  12Q2  0 Rewrite the above equations as 12Q1  6Q2  192,000 6Q1  12Q2  188,000 Multiply the first equation by 2 and add the resulting two equations to eliminate Q2. We obtain then 18Q1  196,000. Hence, we have Q1  10,888.89. Plugging this value into any of the above equations gives 12(10,888.89)  6Q2  192,000 That is, Q2  10,222.22. Price: P  200,000  6(10,888.89  10,222.22)  $73,333.34 b. Alliance’s output is obtained above and is equal to 10,888.89, and Bangor’s output is 10,222.22.

Oligopoly | 183 c. Profits will be p1  192,000Q1  6Q21  6Q1Q2  $711,407,550 p2  188,000Q2  6Q22  6Q1Q2  $626,962,890 8. In Britain price competition among bookshops has been suppressed for over 100 years by the Net Book Agreement (of 1900), which was aimed at preventing price wars. However, in October 1991 Waterstone and Company began cutting book prices at its 85 British shops. According to Richard Barker, Waterstone’s operations director, the decision to reduce the price of about 40 titles by about 25% was due to price cuts by Dillons, Waterstone’s principal rival. a. According to the president of Britain’s Publishers Association, the pricecutting was “an enormous pity” that will “damage many booksellers who operate on very slim margins.” Does this mean that price-cutting of this sort is contrary to the public interest? b. Why would Dillons want to cut prices? Under what circumstances would this be a good strategy? Under what circumstances would it be a mistake? Solution: a. No. A price war, although always an “enormous pity” for producers, usually is very good news for consumers. b. If demand is elastic at high prices and if Dillons has a comparative advantage with respect to its rivals at high volumes, it may prefer a low competitive price to a high collusive one. If demand is inelastic or if Dillons does not have a comparative advantage over its rivals at high volumes, it may be a mistake to cut prices. 9. In the 1960s Procter & Gamble recognized that disposable diapers could be made a mass-market product and developed techniques to produce diapers at high speed and correspondingly low cost. The result was that it dominated the market. According to Harvard’s Michael Porter, who made a careful study of this industry, the following were some ways in which Procter & Gamble might have signaled other firms to deter entry.

184 | Chapter 11

Tactic 1. Signal a commitment to defend position in diapers through public statements, comments to retailers, etc. 2. File a patent suit.

Cost to Procter & Gamble

Cost to an Entrant

None

Raises expected cost of entry by increasing probability and extent of retaliation.

Legal fees

Incurs legal fees plus the probability that P & G wins the suit with subsequent cost to the competitor. Raises expected risk of price-cutting and the probability of P & G’s retaliation to entry. Raises the expected cost of entry by forcing entrant to bear possible product development and changeover costs contingent on the ultimate configuration of the new generation.

3. Announce planned capacity expansion.

None

4. Announce a new generation of diapers to be introduced in the future.

None

a. In considering these possible tactics, why should managers at Procter & Gamble be concerned about their costs? b. Why should managers be concerned with the costs to an entrant? c. By the 1990s Procter & Gamble had to compete with high-quality, private-label diapers (as well as with Kimberly-Clark, which successfully entered the market in the 1970s). In March 1993 its Pampers brand had about 30% of the market, and its Luvs brand had about 10%. The price of Luvs and Pampers exceeded that of discount brands by over 30%. Should Procter & Gamble have cut its prices? d. In 1993 Procter & Gamble sued Paragon Trade Brands, a private-label producer, alleging infringement of two patents. Are lawsuits of this kind part of the process of oligopolistic rivalry and struggle? Solution: a. Obviously, Procter & Gamble must be concerned with its own costs. If it adopts a tactic that is far more costly to itself than to a potential entrant, it may cost more than it is worth. b. The point of these tactics is to raise the cost to a potential entrant and thus discourage entry.

Oligopoly | 185 c. Whether Procter & Gamble should have cut its price depends on whether the discount brands (and Kimberly-Clark, which has become a major rival) would cut their prices in response and by how much. In fact, Procter & Gamble did reduce its price substantially (by 16% in the case of Luvs). According to Edwin Artzt, chairman of Procter & Gamble, “We believe our profits are going to grow, because we’re going to get volume back.” d. Yes. Procter & Gamble wanted to reduce what it regarded as improper imitation of its technology. On the other hand, firms that are sued often regard such suits as attempts to intimidate them. 10. Under which circumstances do managers find it profitable to increase the quality of their products? Do the benefits always exceed the costs? Why or why not? Solution: Firms only find it profitable to raise the quality of their products if consumers appreciate quality enough to pay for it with higher product prices. Firms benefit if their market share increases; however, they must weigh these benefits against the costs of producing and marketing a higher-quality product. Firms tend to gain advantage from this type of product development if they are able to innovate substantially, and then follow these innovations with new production processes that allow for cost advantages. 11. The West Chester Corporation believes that the demand curve for its product is P  28  0.14Q where P is price (in dollars) and Q is output (in thousands of units). The firm’s board of directors, after a lengthy meeting, concludes that the firm should attempt, at least for a while, to increase its total revenue, even if this means lower profit. a. Why might managers adopt such a policy? b. What price should managers set if they want to maximize total revenue? c. If the firm’s marginal cost equals $14, do managers produce a larger or smaller output than they would to maximize profit? How much larger or smaller? Solution: a. Higher market shares established by sacrificing current profits might allow the firm to charge higher prices and earn higher profits in the future because of having established a brand name. Also there might be a learning curve effect; this means that future cost savings from increased current output should be added to current marginal revenues when determining the profit-maximizing level of current output and price.

186 | Chapter 11 b. To maximize sales, the firm should produce where marginal revenue equals zero. MR  28  0.28Q  0 → Q  100. c. If the firm’s marginal cost equals 14, the firm should produce the quantity where profit is maximized. This implies setting MR  MC, that is 28  0.28Q  14, that is, Q  50,000. Hence, it should produce 50,000 units, which is 50,000 units less than if it maximizes sales. 12. Steve Win has purchased land from the city of Atlantic City in the Marina section. There are stories of a new casino building boom in Atlantic City (MGeeM is also talking about entering, and Gump is opening his fourth casino). Some talk is circulating that Win will subdivide his new land purchase and perhaps three casinos will be built on the site. Suppose Win subdivides his land into two parcels. He builds on one site and sells the other to another gambling entrepreneur. Win estimates that the demand for gambling in the Marina area of Atlantic City (after accounting for the presence of two existing casinos in the Marina and adjusting for the rest of the casinos in Atlantic City) is P  750  5Q where P is the price associated with gambling and Q is the quantity of gambling (think of P as the average amount that a typical patron will net the casino, an amount paid for the entertainment of gambling, and Q as the number of gamblers). Win, of course, does not sell the other parcel until his casino is built (or is significantly far along); thus he has a first-mover advantage. Win’s total cost (TCW) of producing gambling is TCW  20  40QW  15.5QW2 where QW is the number of gamblers in Win’s casino and the total cost (TCR) of producing gambling for Win’s rival is TCR  10  50QR  20QR2 where QR is the number of gamblers in the rival’s casino and QW  QR  Q Would Atlantic City have done better to sell the land as two separate parcels rather than as a single parcel to Win (given that Win was going to subdivide, Win and his rival could not collude, and Win did not have the ability to produce as a monopolist)? You may assume that Win and his rival could have been Cournot duopolists. If Atlantic City could do better, show why and by how much. Carry all calculations to the thousandths decimal point.

Oligopoly | 187 Solution: The monopoly solution for Win is to produce where MR  MC. That is, setting 750 10Q  40  31Q. We obtain Q  17.317 at the price P  $663.42. However, if Win and the rival produce as simultaneous moving duopolists, each chooses output to maximize profit. Win’s profit is pw  [750  5QW  5QR]QW  20  40QW  15.5QW2 Profit maximization, given QR, implies setting the partial derivative with respect to QW equal to zero: 750  10QW  5QR  40  31QW  0 We obtain Win’s reaction function: QW  17.317  0.1219QR Similarly, we have pR  750Q R  5QWQR 0 5QR2  10  50QR 2 20QR2 Profit maximization, given QW, requires 750  5QW  10QR  50  40QR  0 or 700  50QR  5QW  0 Therefore the reaction function is QR  14  0.1QW substituting QR into QW to obtain QW  17.317  0.1219(14  0.1QW) Hence we have QW  15.803

and QR  12.42.

If Win can act as a first mover, then solving backward gives the rival’s reaction function as found above. Now, given this reaction function, Win chooses output to maximize profit. Plugging this reaction function into Win’s profit above and setting the firstorder condition gives 640  40QW  0. That is, QW  16 units and QR  14  1.6 12.4 for a total of Q  28.4. Price is P  750  5(28.4)  $608. The first-mover advantage is small in this case.

CHAPTER 12

Game Theory

Lecture Notes 1. Introduction •

Objectives ÿ To discuss the tools of making strategy through game theory ÿ To examine the concept of equilibrium from a game-theoretic standpoint ÿ To examine the concept of a dominant strategy ÿ To discuss the concept of a Nash equilibrium (NE) ÿ To solve games using backward induction ÿ To examine repeated games ÿ To study the concept of reputation building ÿ To study strictly competitive games

2. Making Strategy and Game Theory •

•

•

•

188

Strategic Managerial Decisions: Characterized by interactive payoffs in which managers must explicitly consider the actions likely to be taken by their rivals in response to their decisions Nonstrategic Managerial Decisions: Do not involve other decisions makers, so the reactions of other decision makers do not have to be considered Interactive: When the consequence of a manager’s decision depends on both the manager’s own action and the actions of others ÿ There are no unconditional optimal strategies in game theory; the optimality of a strategy depends on the situation in which it is implemented. Game theory can help managers better understand others strategically.

Game Theory | 189 • •

Using game theory, managers can increase their ability to anticipate the actions of others. Game theory provides an assessment framework to help managers.

3. Strategy Basics • •

Before a game is played, potential players have to understand the rules, which define the game. All game-theoretic models are defined by five parameters. ÿ 1. The players: A player is an entity that makes decisions; models describe the number and identities of players. * Changes in either the identities of the players or the number of players can alter the game. ÿ 2. The feasible strategy set: Actions with a nonzero probability of occurring comprise the feasible strategy set. * Any strategy not considered by one manager, but is then played by others, puts managers at a strategic disadvantage. * Each player has a feasible strategy set. ÿ 3. The outcomes or consequences: The feasible strategies of all players intersect to define an outcome matrix. * A par ticular outcome is defined by the strategy choice of each player. ÿ 4. The payoffs: Every outcome has a defined payoff for every player. Players are assumed to be rational, that is, to prefer a higher payoff to a lower one. ÿ 5. The order of play: Play may be simultaneous or nonsimultaneous, that is, sequential.

4. Visual Representation •

•

Payoff representation can take one of two forms: ÿ Matrix form: Form that summarizes all possible outcomes ÿ Extensive form: Form that provides a road map of player decisions * Game trees: Game trees are another name for extensive form games and are akin to decision trees. The fundamental difference between a game tree and decision tree is one of strategy. ÿ A game tree is strategic; a decision tree is not. ÿ Decision trees have no interactive payoffs. ÿ Extensive form games represent the revelation of a player strategy with decision nodes. ÿ The nodes specify: * the player’s identity and * the feasible strategy set.

190 | Chapter 12 •

Examples ÿ Figure 12.1: A Two-Person Simultaneous Game 1. Players: There are two players: Allied and Barkley. * Barkley: The row player * Allied: The column player 2. Order of play: Simultaneous— each must reach a decision without knowing that of the other. 3. Feasible strategy set: Each player can either maintain the current spending or increase it. 4. Outcomes: Because there are only two players and each has two strategic options, there are four possible outcomes. 5. Consequences: The payoffs are listed for each player within every possible outcome. ÿ Figure 12.2: Allied–Barkley Pricing: Sequential * All simultaneous games are played with imperfect information; that is, when revealing his strategy, a player does not know the strategies of all the others. ÿ Figure 12.3: Allied–Barkley Pricing: Simultaneous * Uses information sets to use the extensive form to represent simultaneous decisions * These models let managers anticipate the future by predicting the actions of others and those of the managers themselves.

5. Solution Concepts •

Question: How does game theory let managers see the future?

•

Answer: It anticipates (correctly) the behavior of others because it believes individuals act according to prescribed rules. Hence: ÿ The key to the solution of game theory problems is the anticipation of the behavior of others.

6. Equilibria •

Equilibrium: When no player has an incentive to unilaterally change his or her strategy * Once there, no one moves unilaterally. * Players cannot unilaterally change behavior because they cannot increase their payoff. Hence: No player is able to improve his or her payoff by unilaterally changing strategy.

Game Theory | 191 Assumption: Behavior is directed by preferences. Players are rational: Players do not want to hurt themselves and accept a lower payoff. An equilibrium is: * rational * optimal * stable Key idea: Each player tries to obtain the highest payoff possible, given the actions of other players in the game. * Once an equilibrium is reached, no player has an incentive to unilaterally change behavior. * *

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7. Dominant Strategies •

Question: What if managers act without regard to the actions of others?

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Dominant Strategy: A strategy whose payout in any outcome is higher relative to all other feasible strategies ÿ A strategy that is optimal regardless of the strategies selected by rivals. ÿ Weakly dominated strategy: A strategy that does at least as well as any other strategy for some outcomes and better than any strategy for the remaining outcomes ÿ Managers choose dominant strategies to optimize their expected return. ÿ Fact: Managers should always choose a dominant strategy if it is available. ÿ Example: Figure 12.1: A Two-Person Simultaneous Game * Barkley has a dominant strategy, which is to increase spending. * Allied has a dominant strategy, which is to maintain the current spending ÿ Prediction: We have a dominant strategy equilibrium: Allied managers will maintain their current spending, and those at Barkley will increase it. Figure 12.4: Matrix Form Representation of Figure 12.2 * Barkley has a dominated strategy, which is to charge $1.00: $1.35 dominates $1.00. There is no circumstance under which this strategy would yield a payoff greater than the other feasible strategies. * Allied has two dominated strategies, given the elimination of a Barkley price of $1.00. These are to price at $0.95 and to price at $1.30. * No player should ever play a dominated strategy because he or she is always better off playing the dominant one. * The existence of dominated strategies reduces the set of possible outcomes.

192 | Chapter 12 A rational player always chooses his or her dominant strategy. In matrix form games, the degree of rationality is measured by the rounds of iterated dominance needed to reach the equilibrium. * Fact: Few games have a dominant strategy equilibrium. ÿ Figure 12.5: Iterative Dominance * Shows the elimination of dominated strategies * *

8. The Nash Equilibrium • • •

• •

John Nash developed guidelines for behavior that are rational, optimal, and stable. Player’s objective: To maximize the payoff conditional on what he or she thinks of others Assumption: All players are rational, and therefore, every player should choose the best strategy conditional on all other players doing the same. The Model ÿ There are N players in the game. ÿ Each of N players chooses a strategy s*, where i  1, 2, 3, ... , N. i ÿ An outcome of the game is represented as an array of strategies s*  (s1*, s2*, ... , sN*). ÿ The payoff to player i when s* is selected is Bi(s*), where I is any player, i  1,2,3, ... , N. ÿ Definition: A Nash equilibrium is an array of strategies such that Bi(s1*, s2*, ... , sN*) ≥ Bi(s1' , s2*, ... , sN*) for all outcomes. * There is no array of strategies better than s* for any player. * This equilibrium is rational, optimal, and stable. * The NE is optimal in the sense that all players try to maximize their payoffs. * It is stable in the sense that no player can unilaterally change strategy and realize a higher payoff. Fact: A Nash equilibrium exists for all games with a finite number of players and outcomes. Nash’s recommendation: Maximize your payoff conditional on all others doing the same. Figure 12.6: New Product Introduction * The numbers represent profits. * Decision rule: Look for dominated strategies. * Observation: Neither player has a dominated strategy. * Algorithm: For each strategy, indicate the behavior of others. ÿ Nash equilibrium (NE) is where the best responses intersect and it is obtained by the group. Nash equilibrium for the game: Barkley introduces product sigma, and Allied introduces product alpha.

Game Theory | 193 9. Strategic Foresight: The Use of Backward Induction •

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Definitions ÿ Strategic Foresight: A manager’s ability to make decisions today that are rational given what is anticipated in the future ÿ Game theory models strategic foresight through: ÿ Backward Induction: Used in game theory to solve games by looking to the future, determining what strategy players will choose (anticipation), and then choosing an action that is rational, based on those beliefs * Backward induction is most easily seen in extensive form games because of the ability to map the choices of the players. * In sequential games, backward induction involves starting with the last decisions in the sequence and then working backward to the first decisions, identifying all optimal decisions. In other words, * To use backward induction, we must come back from the future. We anticipate the future actions of others and then choose actions that are rational, conditional on our expected behavior of others. Example ÿ Figure 12.10: Allied–Barkley Expansion Decision ÿ Solution:

Example: Backward Induction and the Centipede Game ÿ Centipede game: A sequential game involving a series of six decisions that shows the usefulness of backward induction in strategic thinking ÿ Solving the game: Use backward induction. ÿ We need to go to the end of the game and work backward from the future. ÿ The optimal solution is for the first player to end the game immediately.

194 | Chapter 12 •

The Credibility of Commitments ÿ Credible: When the costs of falsely making a commitment are greater than the associated benefits ÿ We want to know whether we should believe others. ÿ A warranty increases the commitment cost. ÿ Example * Figure 12.12: Does Barkley Have a Credible Threat? * It is not in Barkley’s interest to drop price in response to Allied’s price cut. The threat to do so is not credible. * Solution:

ÿ Subgame: A segment of a larger game * Figure 12.12 has three subgames, and all are in equilibrium at the optimal solution, which implies that the equilibrium is a subgame perfect equilibrium. * Result: In repeated games, all subgame perfect equilibria are also Nash equilibria, although not all Nash equilibria are subgame perfect equilibria. * Nash equilibria that are based on noncredible threats are not subgame perfect. 10. Repeated Games • • •

The business world is characterized by repeated interactions. Question: How can a perception of future interaction affect behavior? Illustration: Use a class of games known as: Prisoner’s dilemma

Game Theory | 195 Example: ÿ Allied and Barkley produce an identical product and have similar cost structures. ÿ Each player must decide whether to price high or low. ÿ Figure 12.13: Pricing as a Prisoner’s Dilemma * Solution is for both to price low. * Both would be better off if both priced high. ÿ Repeated play can lead to cooperative behavior in a prisoner’s dilemma game * Trust, reputation, promises, threats, and reciprocity are relevant only if there is repeated play. * Cooperative behavior is more likely if there is an infinite time horizon than if there is a finite time horizon. * If there is a finite time horizon, then the value of cooperation, and hence its likelihood, diminishes as the time horizon is approached. Backward induction implies that cooperation will not take place in this case. * Folk theorem: This theorem states that any type of behavior can be supported by an equilibrium (as long as the players believe there is a high probability that future interaction will occur). * In finite horizon games, the Nash equilibrium is identical to that of a one-shot game. 11. Incomplete Information Games •

Incomplete Information Games (IIG): A branch of game theory that loosens the restrictive assumption that all players have the same information ÿ The introduction of incomplete information makes it possible to derive cooperation as an equilibrium behavior. ÿ One strategy many players use to experiment is “tit for tat”: Strategy that allows players to cooperate in the first period and in all succeeding periods the players mimic the preceding period’s strategy of the other player ÿ In IIG models, players possess asymmetric information. ÿ Asymmetric information is summarized in terms of player types. A type has characteristics unknown to other players that have different preference (payoff) functions. * Low-cost type and high-cost type * Tough type and soft type Example ÿ Figure 12.14: Tough or Soft Barkley Managers * If Barkley managers are tough, they will fight, and Allied will not enter the market. * If Barkley managers are soft, they will not fight and will enter the market.

196 | Chapter 12 Recall: Tit for tat: Strategy in which players cooperate in the first period, and in all succeeding periods the players mimic the preceding period’s strategy of the other player. 12. Reputation Building •

The presence of a future and incomplete information are the necessary ingredients for building reputations. In their presence, a reputation is a rent-generating asset. ÿ Reputation requires a time horizon and incomplete information. ÿ Reputation is based on a player’s history of behavior and involves inferring future behavior based on past behavior.

13. Coordination Games • • • •

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It is often profitable for managers to coordinate actions with others. Game theory models represent coordination games as containing more than one Nash equilibrium. The issue is: Which one to choose? Coordination games have more than one Nash equilibrium, and the players’ problem is which one to select. Matching Games ÿ Two Nash equilibria ÿ Problems in coordination arise from players’ inability to communicate, players with different strategic models, and asymmetric information. ÿ Figure 12.15: Product Coordination Game * Nash equilibrium is for one firm to produce for the industrial market and the other to produce for the consumer market. * Both firms would prefer the equilibrium with the higher payoff. Battle of the Sexes ÿ In this coordination game, players still want to coordinate, but they prefer different outcomes. Because of the different preferences, each player prefers a payoff not favored by the other. ÿ Two Nash equilibria ÿ Players prefer different equilibria. ÿ Figure 12.16: Battle of the Sexes * Nash equilibrium is for one to produce the high end and the other to produce the low end. * Both players prefer to produce the high-end product. Assurance Games ÿ Figure 12.17: Stag Hunt or Assurance Game * Outcome 12, 12 is Pareto dominant, since both players are better off, but it is risk dominated because if one firm decides to shift and the other does not, then the player that shifts receives a payoff of zero.

Game Theory | 197 Achieving the Pareto dominant solution requires cooperation and trust because of the risk of reneging. First-Mover Games ÿ Two Nash equilibria ÿ Players prefer different equilibria. ÿ Figure 12.18: First-Mover Advantage * Nash equilibrium is for one firm to produce the superior product and the other to produce the inferior product. * Both firms want to produce the superior product, which yields the higher payoff, by moving first. * Barkley is predicted to move first because the payoff is higher for Barkley and therefore Barkley can afford to spend more to speed up development. Hawks and Doves ÿ Two Nash equilibria ÿ Players prefer different equilibria. ÿ Figure 12.19: Hawks and Doves * Nash equilibrium is for one player to behave as a hawk and the other to behave as a dove. * Both players want to behave as a hawk, which yields the higher payoff. * If both players act like hawks, conflict ensues. *

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14. Strictly Competitive Games • • •

Some games are strictly competitive: Any gain by one player means a loss by another player. These games are called: Zero-Sum Games: A competitive game in which any gain by one player means a loss by another player. Figure 12.20: Advertising Campaigns ÿ Nash equilibrium is for Barkley to choose campaign 2 and for Allied to choose campaign A.

Chapter 12: Problem Solutions 1. Two soap producers, the Fortnum Company and the Maison Company, can stress either newspapers or magazines in their forthcoming advertising campaigns. The payoff matrix is as follows:

198 | Chapter 12 Maison Company Stress newspapers

Stress magazines

Stress newspapers

$8 mm, $9 mm

$7 mm, $8 mm

Stress magazines

$9 mm, $8 mm

$8 mm, $7 mm

Fortnum Company

a. Is there a dominant strategy for each firm? If so, what is it? b. What will be the profit of each firm? c. Is this game an example of the prisoner’s dilemma? Solution: a. Maison has a dominant strategy: Stress newspapers. Fortnum Company has a dominant strategy: Stress magazines. b. Eliminating the dominated strategies from the game leads to the dominant strategy equilibrium: (Fortnum plays “Stress magazines,” Maison plays “ Stress newspapers”). Therefore, Maison’s profit will be $8 million. Fortnum’s profit will be $9 million. c. This is not a prisoner’s dilemma game. The players do not end up with an outcome from which both would be better off if they cooperated. 2. The Ulysses Corporation and the Xenophon Company are the only producers of a sophisticated type of camera. They each can engage in either a high or a low level of advertising in trade journals. The payoff matrix is as follows: Xenophon Company Low level

High level

Low level

$12 mm, $13 mm

$11 mm, $12 mm

High level

$13 mm, $12 mm

$12 mm, $11 mm

Ulysses Corporation

Game Theory | 199 a. Will Ulysses engage in a high or a low level of advertising in trade journals? b. Will Xenophon engage in a high or a low level of advertising in trade journals? c. Is there a dominant strategy for each firm? Solution: a. Ulysses corporation has a dominant strategy: Engage in a high level of advertising because Ulysses’s profits are higher if it advertises at a high rather than at a low level; regardless of what Xenophon does, it will advertise at the high level. In other words, Ulysses will play its dominant strategy. b. Xenophon has a dominant strategy: Engage in a low level of advertising given that Xenophon’s profits are higher if it advertises at a low rather than at a high level; regardless of what Ulysses does, it will advertise at the low level. In other words, Xenophon company will play its dominant strategy, as it should. c. As mentioned in parts (a) and (b), both players have a dominant strategy. 3. The New York Times reports that Wal-Mart has decided to challenge Netflix and enter the online DVD-by-mail market. Because of economies of scale, Wal-Mart has a slight cost advantage relative to Netflix. Wal-Mart is considering the use of a limit pricing strategy. It can enter the market by matching Netflix on price. If it does, and Netflix maintains its price, then both firms would earn $5 million. But if Netflix drops its price in response, Wal-Mart would have to follow and would earn $2 million; Netflix would earn $3 million. Or Wal-Mart could enter the market with a price that is below Netflix’s current price but above its marginal cost. If it does, Netflix would make one of two moves. It could reduce its price to below that of Wal-Mart. If it does, Wal-Mart will earn a profit of $0, and Netflix will earn a profit of $2 million. Or Netflix could keep its present price. If Netflix keeps its present price, Wal-Mart can keep its present price and earn $6 million (while Netflix earns $4 million). Or Wal-Mart can increase its price and earn $2 million while Netflix earns $6 million. a. Draw the extensive form of this game and solve it. b. Draw the game’s matrix form and identify any Nash equilibria.

200 | Chapter 12 Solution: a. Extensive form: Note Wal-Mart payoffs appear first.

The game can be solved using backward induction from the extensive form, starting from the last nodes. Wal-Mart should enter and drop its price. Netflix will not drop its price because it would only get 2. If Netflix maintains its price, it will either get 4 or 6, both of which are better than 2. If Netflix maintains its original price (after Wal-Mart has entered with a dropped price), Wal-Mart will maintain that dropped price because it yields them 6 (as opposed to 2 if Wal-Mart increased its price). If Wal-Mart initially matches Netflix’s price, Netflix will maintain its initial price and earn 5. If Netflix drops its initial price, Netflix will only earn 3. Thus, Wal-Mart will earn 5. Since Wal-Mart earns 6 by initially lowering its price, it is better for Wal-Mart to drop its price when entering and not match Netflix’s price when entering. b. The solution to this part is obtained in the same way as in the previous exercises. The main point here is to list all the strategy sets for each player. The solution is the same as in part (a) using backward induction.

Game Theory | 201 4. Two rival PE firms are interested in finding the same two start-ups. Each would prefer not to get into a bidding war with the other regarding either of the start-ups. Payoffs are given in the following table: PE Firm A Invest Start-up 1

Invest Start-up 2

Invest 1

10, 10

60, 40

Invest 2

25, 55

20, 20

PE Firm B

What are the reservation prices of managers at the two firms? What will bidding look like? Solution: It can be easily seen that no player has a dominant strategy in this game. Hence, no firm has a strong incentive to bid. Also, there are two Nash equilibria: (Firm B playing “Invest 2,” Firm A playing “ Invest Start-up 1”) and (Firm B playing “Invest 1,” Firm A playing “Invest Start-up 2”). For the first NE, given the payoffs received by the two players, the reservation price for firm B is 25 and the reservation price for firm A is 55. Similarly, for the second NE, the reservation price for firm B is 60 while the reservation price for firm A is 40. 5. Two soft-drink producers, York Cola and Reno Cola, secretly collude to fix prices. Each firm must decide whether to abide by the agreement or to cheat on it. The payoff matrix is as follows: York Cola Abide by agreement

Cheat

Abide by agreement

$29 mm, $29 mm

$26 mm, $30 mm

Cheat

$30 mm, $26 mm

$28 mm, $28 mm

Reno Cola

202 | Chapter 12 a. What strategy will each firm choose, and what will be each firm’s profit? b. Does it matter whether this agreement is for one period or for three periods? c. Is this game an example of the prisoner’s dilemma? Solution: a. Reno and York each have Cheat as their dominant strategy, so they will choose their dominant strategy and each will earn $28 million. It can also be seen that the game has Nash equilibrium (“Cheat,” “Cheat”); that is, each player is choosing to cheat. b. Since abiding by the agreement would raise their profits to $29 million each if this game were to be played out an infinite number of times, the dominant strategy would be for both to abide if they thought that a defection would be met with cheating by their opponents in all future rounds. c. Yes, this is a prisoner’s dilemma since the firms are stuck in an outcome from which both could be made better off by cooperation. 6. Part 1: Firm A currently monopolizes its market and earns a profit of $10 million. Firm B is a potential entrant that is thinking about entering the market. If B does not enter the market, it earns a profit of $0, while A continues to earn a profit of $10 million. If B enters, then A must choose between accommodating entry and fighting it. If A accommodates, then A earns $5 million and B earns $5 million. If A fights, then both firms lose $5 million. Draw the game in extensive form and predict the outcome. Part 2: Again, consider the above game. Now, suppose the decision of B to enter is reversible in the following way. After B enters the market and A has decided either to fight or accommodate, B can choose to remain in the market or exit. All payoffs from the above game remain the same. However, if B decides to exit the market, then B suffers a loss of $1 million, while A regains its old profits of $10 million. Draw the game in extensive form and predict the outcome.

Game Theory | 203 Solution: Part 1: $5m, $5m Accommodate

Enter

Firm A Fight

–$5m, –$5m

Firm B Do not enter Firm A $0, $10

Outcome of the game: Using backward induction, we see that firm B will enter the market and firm A will choose to accommodate. Part 2: –$1m, $10m Exit Firm B Accommodate

Stay

$5m, $5m

Firm A Exit

Enter

Fight

–$1m, $10

Firm B

Firm B

Stay Do not enter

–$5, –$5

Firm A $0, $10

The outcome of the game using backward induction is that firm B will not enter the market. However, if it is assumed that B has perfect foresight, then if B enters the market and A accommodates, then A will not exit the market

204 | Chapter 12 given its payoff is $5 if staying while the loss is $1 if exiting. Hence, the branch tree with payoff $1, $10 would not be part of the picture at all. But the outcome remains the same. 7. The Rose Corporation is one of two sellers of paint. It pursues a tit-for-tat strategy. However, it has great difficulty in telling whether its rival is secretly cutting its price. What problems is this likely to cause? Solution: With a bit of tit-for-tat strategy, each player makes a move that mirrors his opponent’s previous move. If a player cannot observe the rival’s moves, then tit for tat would be difficult to play. 8. Consider a father who is trying to discipline his child. The father insists that the child must go with the rest of the family to visit their grandmother. The child prefers to go to the movies with a friend. The father threatens to punish the child if the child doesn’t visit the grandmother. If the child goes with the family to visit the grandmother, both the child and the father receive one unit of utility. If the child refuses to go to the grandmother’s house, and the father punishes the child, the child receives one unit of utility, and the father receives one unit of utility. If the child refuses to go and the father relents (does not punish), the child receives two units of utility, and the father receives none. a. Draw this game in matrix form. b. Draw this game in extensive form. c. Solve this game via backward induction. Solution: a.

b. Assume that the child moves first. Then we have the following tree diagram;

Game Theory | 205

We can draw a similar tree assuming that the father moves first. c. Via backward induction, the child maximizes by not going. Given that the child doesn’t go, the father maximizes (in this single-period game) by not punishing. 9. The Boca Raton Company announces that if it reduces its price subsequent to a purchase, the early customer will get a rebate so that he or she will pay no more than those buying after the price reduction. a. If the Boca Raton Company has only one rival, and if its rival too makes such an announcement, does this change the payoff matrix? If so, in what way? b. Do such announcements tend to discourage price cutting? Why or why not? Solution: a. Yes, the payoff matrix changes. However, if both players make the same offer, the payoffs are altered in the same way relative to one another; price reductions reduce payoffs for both firms. b. These announcements tend to discourage price cutting as long as the threats are credible. Since both firms are going to lose in the instance of price cutting, there is no incentive for either to pursue such a strategy.

CHAPTER 13

Auctions

Lecture Notes 1. Introduction •

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Objectives ÿ To explain how managers can apply game theory to the analysis of auctions ÿ To examine auction mechanisms and revenue generation ÿ To describe some bidding strategies ÿ To examine the effect of risk aversion ÿ To discuss the notion of “Winner’s curse” ÿ To describe the importance of auction mechanisms and their use in strategic decisions related to negotiations and monopoly markets Focus: Auction mechanism Managers design auctions to generate higher profits or they bid more efficiently to capture greater surplus.

2. A Short History of Auctions •

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206

First auction recorded: The annual Babylonian marriage market described by Herodotus (circa 300 BCE) ÿ Most beautiful women were auctioned first at the highest prices. ÿ Less attractive women were auctioned for negative bids. The man with the highest negative bid got the woman and also got paid for it from the positive bids tendered for the beautiful women. Ancient Greece: Auctions were used to award mineral rights. Roman authorities: Auctions were used to collect debts owned by individuals. By the 1980s auctions were used to sell over $5 billion of goods yearly.

Auctions | 207 •

Internet auction: Auction use expanded rapidly with the development of the Internet and e-commerce in the 1990s.

3. Types of Auction Mechanisms • • •

• 1. 2. 3. 4. •

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All auctions involve a bidding process. In most auctions, a service or product is made available, and buyers bid for the ownership rights. Reverse auctions (as opposed to normal auctions) occur when a buyer announces the need for a product or service and sellers bid for the right to sell the buyer the good or service and the low bidder wins the auction. Managers use one of four auction mechanisms: English or ascending-bid auction Dutch or descending-bid auction First-price, sealed-bid auction Second-price, sealed-bid auction English or Ascending-Bid Auction ÿ Reserve price: The lowest price at which a seller is willing to sell a product; also called a reservation price ÿ In English auctions, the initial price is set at the seller’s reserve price. Buyers then bid against each other with a succession of higher prices until one bidder remains, who receives the good or service at his or her final stated price. ÿ Two special types of ascending-bid auctions: * Japanese auction: Auction in which bidders bid until the price exceeds their reservation price * Ascending-bid time auction: Auction in which the bidding continues for a specified time The high bidder at the end of this period receives the good or service (assuming the bid is greater than the seller’s reservation price). It is used extensively by Internet auction sites such as eBay and uBid. ÿ Sniping: When bidders use programs to ensure that they submit the last-second best bid Dutch or Descending-Bid Auction ÿ Dutch auction: A descending-bid system in which initial prices are set very high and are lowered at set intervals until accepted by bidders ÿ The price is lowered until one bidder accepts the announced price. ÿ Used in the Netherlands to sell flowers First-Price, Sealed-Bid Auction ÿ Bidders submit price bids known only to themselves. ÿ Bidders do not know the valuations of others. ÿ The highest bidder receives the item at his or her stated price.

208 | Chapter 13 •

Second-Price, Sealed-Bid Auction ÿ The highest bidder receives the good at the bid price of the secondhighest bidder. These auctions are also called Vickrey auctions, named after William Vickrey. ÿ Vickrey auctions: Second-price, sealed-bid auctions in which the highest bidder receives the good or service at the bid price of the second-highest bidder ÿ Fact: Second-price auctions have characteristics conducive to bidders truthfully revealing their valuations.

4. Auction Mechanism and Revenue Generation • • •

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In auctions, managers want to maximize profits. Auction rules explain variance in generating revenues. An Advantage of Auction Use: Relative to markets, it is easier and less costly for managers to influence structural parameters such as product differentiation, entry barriers, and location through the use of auction rules. A Model: assumptions ÿ Bidders are symmetric. * Bidders with identical reservation prices who observe the same signal will submit the same bids. * Bidders have similar distributions of valuation that are known to all. * Reservation prices may differ across bidders. ÿ Bidders are risk-neutral. * Bidders maximize expected value and not risk-adjusted utility. * Most managers assume that individuals are risk-averse. ÿ Signals are independent. Signals are independent in one of two ways. In: * Private-value auctions: Auctions in which reservation prices are a function of information and utility * Common-value auctions: Auctions in which all bidders value the good similarly In private-value auctions, signals are independent if reservation prices depend on utility (experience) rather than information (about the value of the good at auction). In common-value auctions, signals are independent if reservations prices are the same for all bidders, but the true value of the good at auction is unknown. If signals are independent, then what one manager estimates does not depend on the estimates of others, although the distribution of signals is commonly known.

Auctions | 209 ÿ Notation * b = bid * p = price paid by winning bidder * PrW = probability of winning the auction * Expected profit = PrW(b − p) * Expected profit is the same for the four types of auctions. * b − p = surplus ÿ Fact: Auction formats with identical surplus functions offer the same probability of winning at any given bid and hence recommend the same optimal strategy. ÿ Implication: * Revenue equivalence theorem: Theorem showing that whether a manager chooses an English, Dutch, sealed-bid, or second-price, sealed-bid auction, the choice does not affect the auction’s expected total surplus and hence does not affect the expected revenues. 5. Bidding Strategies • • • •

•

•

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Question: What are the optimal bidding strategies across the four auction formats? Optimal behavior is defined by a dominant strategy. Dominant Strategy in an Ascending Auction: Bidders should always be willing to bid up to their reservation prices. A Strategic Rule: ÿ If the current bid is higher than or equal to your reservation price, don’t bid. ÿ If the current bid is lower than your reservation price, increase the bid. English or Ascending Bid Auction ÿ The rules allow managers to publicly bid multiple times. ÿ Bidders gain information about rivals’ reservation prices, but it is irrelevant to the optimal bid. ÿ The difference between the reservation price of the second-highest bidder and that of the highest bidder, minus the minimum necessary incremental bid, is the potential surplus. ÿ The optimal winning bid is equal to the bid made by the bidder with the second-highest reservation price plus the minimum necessary additional increment to win the auction. Vickrey Auction ÿ Winning bidder pays the amount bid by the second-highest bidder, so the bidder’s surplus is equal to the difference between the secondhighest bidder’s reservation price and the winner’s reservation price. Descending Auctions or First-Price Sealed-Bid Auctions ÿ No dominant strategy ÿ Bidders must consider what others will bid.

210 | Chapter 13

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ÿ Bidders should bid their estimates of the reservation price of the second-highest bidder and, assuming that all bidders bid less than their actual reservation price, the bidder with the highest reservation price will win the auction. ÿ Managerial beliefs about the reservation of the second-highest bidder are influenced by many factors. One important one is the number of bidders. ÿ The more bidders there are, the closer a person’s bid should be to their reservation price. ÿ Assume the distribution of bids is uniform. Then the optimal bidding strategy is given by * b = v − [(v − L)/n] * b = optimal bid * v = bidder’s reservation price * L = lowest possible bid * n = number of bidders Example: Suppose our own evaluation of an item is $3, and we believe the bids will eventually be distributed between $0 and $15. What is the optimal strategy? Solution: * v=3 * L=0 * b = 3 − [(3 − 0)/2] = $1.50 for n = 2 * b = 3 − [(3 − 0)/3] = $2.00 for n = 3 The rules of second-priced, sealed-bid auctions encourage all bidders to tell the truth. Bidders are better off if they tell the truth. This leads to the notion of: Incentive-compatible: Type of rule that encourages managers to reveal their true preferences

6. Strategies for Sellers • • •

Many managers use auctions to sell goods, services, and assets. Rule: Maximize profits with MR = MC. No quantity decision. What determines the optimal pricing point is the distribution of reservation prices across bidders. That is: Optimal pricing strategy depends on the distribution of reservation prices across bidders. ÿ Figure 13.1: Relationship between the Seller’s Expected Revenue and the Winning Bidder’s Expected Marginal Revenue ÿ The manager maximizes profit by selling to the bidder with the highest reservation price.

Auctions | 211 ÿ Example * Distribution of reservation prices = F(b) * Probability that bid b will be the winning bid = 1 − F(b) * Expected revenue = b[1 − F(b)] = Area under the MR curve for any b. * Under general conditions, managers should consider the following decision rule in setting a reserve price: * Optimal reserve price = (value of the object up for auction, if it does not sell) + (half the managerial estimate of the highest reservation price) * Table 13.1: Auction * Table 13.2: Posted Price 7. Value of Information • •

Managers can use auctions to gather more information about demand before announcing prices. Example: Repurchase tender offers (RTOs) ÿ Repurchase tender offers: Offers used by managers to buy back stock shares from current shareholders * To induce selling, RTOs generally offer a premium above the current market price, given that shareholders are not required to sell. * Distinguish between: Fixed-price RTO: One price is announced and modified Dutch auction RTO: generally acquire shares at lower price. * Prior to 1981, RTOs were fixed price and supply was estimated. ÿ Since 1981, the use of a modified Dutch auction format has been more commonly used. ÿ Modified Dutch auction * The firm announces a range of prices at which it is willing to repurchase tendered shares, with the minimum set at a small premium over current market valuation. * Shareholders willing to tender send a price schedule that indicates the number of shares they are willing to sell at each price. * The firm can then construct a price schedule and, based on the number of shares it wishes to repurchase, determine the optimal pricing strategy. ÿ In the fixed-price offer, managers set a price before knowing the supply schedule. ÿ In the modified Dutch auction, managers set the price after seeing the supply schedule. ÿ Example: Citizens First Financial Corp. (CFFC)

212 | Chapter 13 * * * * * * * * * * *

*

CFFC believes that the share price of its stock will rise to $20 in the near future. Table 13.3: Shareholder Supply Schedule—Three Scenarios A fixed price of $15 maximizes expected profit ($1,685,000). If managers choose a fixed-price RTO, they must set a price before knowing the supply curve. Managers might choose a price based on expected value (EV). EV($15) = $2,000,000(0.40) + $1,550,000(0.30) + $1,400,000(0.30) = $1,685,000 EV($16) = $1,660,000(0.40) + $1,600,000(0.30) + $1,260,000(0.30) = $1,522,000 EV($17) = $1,800,000(0.40) + $1,245,000(0.30) + $1,200,000(0.30) = $1,453,500 If CFFC managers have used such a strategy, they might set a tender price of $15, which is the highest expected value. Question: How might managers have increased firm value by structuring the RTO as a modified Dutch auction? If the willingness to tender was strong, they would announce a price of $15, if it was medium, they would announce a price of $16 (given that maximum profit is $1,600,000), and if it was weak, a  price of $15 would be set (given that the maximum profit is $1,400,000). Using a modified Dutch auction, managers have perfect information about the shareholder supply schedule and can set the price accordingly. This results in a higher expected profit of EV(auction) = 0.40 ($2,000,000) + 0.30($1,600,000) + 0.30($1,400,000) = $1,700,000 $15 Payoff: $2,000,000

Strong 0.40

0.30

Medium

$16 Payoff: $1,600,000

0.30 Weak $15 Payoff: $1,400,000

Auctions | 213 •

•

Managers are generally better off using a modified Dutch auction RTO. They get shareholders to reveal valuations and hence can buy back shares at a lower price than if they used a fixed price. The expected number of shareholders is 0.4(400,000) + 0.30(400,000) + 0.30(280,000) = 364,000.

STRATEGY SESSION: The Use of Sophisticated Pricing within an Auction Format Discussion Question 1. Priceline.com does not offer its service for tickets to sporting events. Why do you think that it does not take advantage of this market? 8. Risk Aversion • •

• •

Most individuals are risk-averse. Question: What are the effects of risk aversion on bidding behavior? * In second-price auction (SPA), risk preferences do not influence bidding strategy. * Risk preferences affect bidding behavior in a first-price auction (FPA). * Recall: FPAs have no dominant strategies. Managers must anticipate the bids of others and hence can only partially control this uncertainty. * A high bid increases the probability of winning. In second-price auctions, risk aversion has no effect on the optimal bidding strategy. First-price auctions ÿ A risk-averse bidder will pay to avoid loss, so he or she raises the bid by some amount of “bidding insurance.” Thus: ÿ A risk-averse bidder will bid higher than a risk-neutral bidder (although not higher than their reservation price), with the difference equal to a risk premium representing “bidding insurance” that makes it more likely that the bidder will win. ÿ Sellers prefer a first-price auction because it causes bidders to reveal their risk preferences and premiums and because it reduces the variability in expected return compared to second-price auctions. ÿ There is more risk in SPAs, though expected revenues are the same from the four auction formats.

9. Number of Bidders •

Auction markets have many buyers (bidders) and only one, or a few, sellers.

214 | Chapter 13 • • •

The more bidders there are in an auction market, the higher is the expected price. In all auction formats, the expected is given by the reservation price of the second-highest bidder. Example: ÿ Assume that bidder reservation prices are uniformly distributed between zero and $100. ÿ Seller’s expected revenue = [(N − 1)/(N + 1)](Reservation price of bidders) ÿ As additional bidders enter an auction, the expected revenue of the seller approaches the reservation price. ÿ Figure 13.2: Expected Revenue versus Number of Bidders

10. Winner’s Curse Winner’s Curse: When the bidder with the highest estimated value bids the most and wins the auction, but the bid amount may exceed the true value of the object ÿ The idea: Managers who overestimate the true value have a higher probability of winning the auction, but they pay more than the item’s worth. ÿ The value of the thing being auctioned off is not known with certainty, although it has the same value to all bidders. ÿ In a sealed-bid, first-price auction, the bidder with the highest estimated value (and bid) wins the auction. • Figure 13.3: The Winner’s Curse • Example: The winner’s curse in bidding for oil rights • Three issues to be considered by managers when contemplating their bids: 1. A bidder with relatively less information than others should bid lower. 2. A bidder with relatively little confidence in his or her estimate of the value of the thing at auction should bid lower. 3. The more bidders there are in the auction, the lower the bid should be.

•

STRATEGY SESSION: The Winner’s Curve in Bidding for Oil Rights Discussion Question 1. When you buy a house or a car, you negotiate with a seller in what is actually an auction. As a bidder, you don’t know the seller’s reservation price or the number and size of any bids already placed. Which of the four types of auctions is this? Answer: First-price, sealed bid

Auctions | 215 11. Concerns in Auction Design • Two issues to consider: 1. The ability of bidders to collude is a concern to sellers. ÿ Bidders form a “ring” in which no one bids against the designated bidder. The things that are purchased are then distributed among members of the ring after the auction. ÿ Colluding bidders do not bid up prices. 2. The ability of the seller to attract bidders is a concern to sellers. ÿ If it is clear to all bidders that one par ticular bidder will win, they are less likely to participate in the auction. ÿ If the seller sets a reserve price that is too high, then bidders are less likely to participate in the auction. ÿ If the seller sets a reserve price that is too low, it encourages collusion.

Chapter 13: Problem Solutions 1. Consultant.com is a company that employs business professors as virtual consultants who supply answers to other companies’ problems. Consultant. com wants to raise funds with a private equity issue. Unfortunately, because of fluctuations in the stock market, it is uncertain about the demand for its offering. It hopes to issue the stock at either $45 or $50. The demand is categorized into four possible scenarios. The following table shows demand for each scenario–price combination along with the beliefs regarding the probability of each possible state. Consultant.com must pay 10% of the generated funds to the investment bank that helped it identify potential investors. The company wants to maximize the funds raised. Price/Share

State 1

State 2

State 3

State 4

$45 $50 Probability

1,750 1,200 0.35

1,975 1,415 0.20

2,220 2,001 0.30

2,445 2,305 0.15

What is the expected value of the stock offering if Consultant.com sets its price without knowing the future demand state? If Consultant.com can determine the future demand state by using a modified Dutch auction, what is its expected profit? If someone approached Consultant.com and told managers she could predict the future demand state, how much would that information be worth to them?

216 | Chapter 13 Solution: Notice that if the stock price is $45, then the company must pay $4.50 to the investment bank that helped it identify potential investors and therefore retains $40.50. Similarly, if the price is $0, then the company retains $45. EV($45) = ($40.50)[(1,750)(0.35) + (1,975) (0.20) + (2,220) (0.30) + (2,445) (0.15)] EV($45) = $82,630.10 EV($50) = ($45)[(1,200)(.35) + (1,415)(0.20) + (2,001)(0.30) + (2,305) (0.15) EV($50) = $74,207.30 EV(auction) = $2,4806.25 + 15,997.50 + 27,013.50 + 15,558.75 = $83,376.00 Value of information = $83,376.00 − $82,630.10, or $745.88 2. Your company is planning to auction off a manufacturing plant in Asia. You are asked to determine the auction design that will generate the highest revenue for the company. You believe that bidders will value the plant independently. Which design would you choose, and why? Solution: All auction designs yield the same revenues for the company under these conditions. 3. There are 100 bidders in an English auction. A random sample of 40 bidders shows the following reservation prices: Number of Bidders

Reservation Price

1 3 6 5 8 6 7 3 1

$10 $20 $30 $40 $50 $60 $70 $80 $100

Assume the bidding distribution is normal. a. What is the mean value of bids across the 100 bidders? b. What is the probability of a bid being less than $80?

Auctions | 217 Solution: a. In an English auction, bidders have an incentive to bid up to their reservation price. The mean bid will be $68. b. Let us assume the distribution of bids is standard normal; that is, the bids have a normal distribution with mean 0 and variance 1. The probability of a bid being less than $80 is about 0.19 or 19%. 4. Your company is bidding for a service contract in a first-price, sealed-bid auction. You value the contract at $12 million. You believe the distribution of bids will be uniform, with a high value of $16 million and a low value of $3 million. What is your optimal bidding strategy with a. 5 bidders? b. 10 bidders? c. 20 bidders? Solution: As the number of bidders increases, the optimal bid becomes closer to the reservation price. We have v = $12 million L = $3 million a. For n = 5, we obtain b = 12 − [(12 − 3)/5] = $10.2 million b. For n = 10, we obtain b = 12 − [(12 − 3)/10] = $11.1 million c. For n = 20, we have b = 12 − [(12 − 3)/20] = $11.55 million 5. The Philadelphia Eagles of the National Football League build a stadium. One revenue source during the construction of the stadium is a personal seat license (PSL), a one-time, up-front payment charged to season ticket holders before the stadium is built. It gives the buyer the right to purchase tickets for a par ticular seat. The Eagles are uncertain about demand for seats in the new stadium. They have selected three price points for PSLs ($6,000, $7,000, and $8,000). Management also has estimated that demand for PSLs could be low, medium, or high. Their beliefs are reflected in this table: Price

Low Demand Probability = 0.4

Medium Demand Probability = 0.35

High Demand Probability = 0.25

$6,000 $7,000 $8,000

24,500 PSLs sold 21,500 PSLs sold 17,500 PSLs sold

28,000 PSLs sold 24,000 PSLs sold 22,000 PSLs sold

40,000 PSLs sold 32,000 PSLs sold 25,000 PSLs sold

218 | Chapter 13 Some of the stadium funding is provided by the city of Philadelphia. Because Eagles fans view PSLs as an attempt to take away consumer surplus, they resent them and have put pressure on the city government to limit their use. Therefore, the city has set a target of 25,000 seats assigned to PSLs. If the Eagles sell fewer than 25,000 PSLs, the city will grant the team a tax benefit of $10/seat for each seat under 25,000. If the Eagles sell 25,000 or more PSLs, no tax break will be given. A consulting group has told the Eagles that the team is better off using a modified Dutch auction to sell the PSLs. The group has estimated the cost of running the auction at an additional $5.1 million. Eagles management has come to you for help. The managers want to know whether they should use an auction and what the expected benefits will be. What will you tell them? Solution: If they choose to price their PSLs at $6,000, their expected revenue is $58,802,000 + $58,800,000 + $60,000 = $177,602,000. If they choose to price their PSLs at $7,000, their expected revenue is $60,214,000 + $58,803,500 + $56,000,000 = $175,017,500. If they choose to price their PSLs at $8,000, their expected revenue is $56,030,000 + $61,610,500 + $50,000,000 = $167,640,500. So, if they set the price and don’t use an auction, they should charge $6,000/ PSL. If the Eagles use a modified Dutch auction, their expected revenue is $60,214,000 + $61,610,500 + $60,000 = $181,824,500. Relative to setting a price of $6,000/PSL, the auction would increase expected revenue by $4,222,500. But since auction costs are $5,100,000, the Eagles are better off pricing the PSLs at $6,000. 6. Your company is bidding for a broadband spectrum license. You have been asked to submit an optimal bidding strategy. You expect that bidders will have independent private values for the licenses because each bidder presently has a different structure in place. You believe the valuations for these licenses will be between $200 million and $700 million. Your own valuation is $650 million. There is some uncertainty about the auction design that will be used, so you must suggest an optimal bidding strategy for the following auction designs: a. Second-price, sealed-bid auction. b. English auction. c. Dutch auction. Please describe your strategy. Is it a dominant strategy?

Auctions | 219 Solution: a. Bid the reservation price, $650 million. b. Be willing to bid up to the reservation price, $650 million. c. Bid = $650 million − ($650 million − $200 million)/n, where n is the number of bidders. By design, second-price auctions have a dominant strategy. However, we know that for the Dutch, or first-price, sealed-bid auctions, managers must consider the bids of the other players in the game. Unlike the ascending format, neither the descending nor the first-price, sealed-bid auction has a dominant strategy.

CHAPTER 14

Risk Analysis

Lecture Notes 1. Introduction •

•

220

Objectives ÿ To explain how managers should make strategic decisions when faced with incomplete or imperfect information or complete, accurate information ÿ To present a variety of tools to help managers improve their decision making ÿ To discuss the concepts of risk and probability ÿ To compare expected profit ÿ To discuss expected utility theory ÿ To examine the concept of expected value of perfect information ÿ To study attitudes toward risk ÿ To develop some measures of risk ÿ To discuss the concept of certainty equivalence and its application to the insurance market Some Useful Managerial Tools ÿ Expected value ÿ Decision trees ÿ Techniques to reduce uncertainty ÿ Expected utility theory. * Expected value: Summarizes a set of possible outcomes into a single representative value * Decision trees: Give a visual and intuitive representation through the web of possible consequences and allow one to structure a decision in a simple, sequential way

Risk Analysis | 221 2. Risk and Probability • •

•

•

In making decisions, managers must form beliefs about the future. Risk: Hazard or Chance of Loss ÿ The bigger the chance of loss or the greater the size of a potential loss, the more risky a par ticular course of action is. Probability: The likelihood or chance that something will happen ÿ Example: * Throw a single fair die. * Outcomes of the experiment: 1, 2, 3, 4, 5, 6 * A probability is a number attached to each possible outcome. Frequency Definition of Probability: An event’s limit of frequency in a large number of trials ÿ Thus, if an experiment is repeated many times, R, and the outcome A occurs r times, the probability of A is given by Probability of event A = P(A) = r/R

•

* R = Large number of trials * r = Number of times event A occurs * Probability: The proportion of times the outcome occurs over the long run if the situation is repeated many, many times Subjective Definition of Probability: The degree of a manager’s confidence or belief that the event will occur.

PROBLEM SOLVED: Pfizer and Its New Cholesterol Drug Discussion Questions 1. The question of risk, benefit, and regulation in the pharmaceuticals industry is a difficult one. If you were afflicted with a deadly disease and a new (experimental) drug was being tested that might, just might, prolong your life, would you want regulators to keep you from using it for fear of side effects? 2. The drug approval process in the United States is the result of horrible events that occurred prior to the beginning of regulation. Many countries are currently virtually unregulated with regard to the development and distribution of prescription drugs. What are the pros and cons of letting consumers choose how to treat their maladies without government intervention? 3. Probability Distributions and Expected Values •

Rules of Probability ÿ Probabilities may not be less than zero or greater than one.

222 | Chapter 14

•

ÿ Given a list of mutually exclusive events, a collectively exhaustive list of the events that can occur in a given situation, the sum of the probabilities of the events must be equal to one. Probability Distribution (the discrete case): A table that lists all possible outcomes along with their corresponding probability of occurrence N

•

∑ P

Expected Profit = E ( ) =

i i

i =1

•

ÿ i = Profit associated with the outcome i ÿ Pi = Probability of outcome i Example: Suppose managers will earn a profit of $1 million if they develop a new robot with probability of 0.6 in one year and lose $600,000 if they do not. Then the probability distribution along with the expected profit is given as follows. Profit

Probability

(Probability) (Profit)

$1,000,000 −600,000

0.60 0.40

$600,000 −240000

Expected Profit = $360,000 4. Comparisons of Expected Profit • •

The idea: To decide which of a number of courses of action to take, managers can compare the expected profits. Example: Jones Corporation is considering a decision involving pricing and advertising. The payoffs and their corresponding probability are given in the following table. The expected value if they raise the price is calculated from the probability distribution as follows. Profit

Probability

(Probability) (Profit)

$800,000 −600,000

0.50 0.50

$400,000 −300,000

Expected Profit = $100,000 The expected payoff from not increasing the price is $200,000, compared with the expected payoff of $100,000 from increasing the price. Thus, if managers want to maximize expected profit, they should not increase the price. This is the optimal strategy for the managers. Question: Under what circumstance is it rational to maximize expected profit? 5. Road Map to Decision •

Managers frequently face choices over alternative strategies, and the payoff of the chosen strategy will depend on the actions of others.

Risk Analysis | 223 • •

Decision Tree: A diagram that helps managers visualize their strategic future Figure 14.1: Decision Tree, Jones Corporation

PROBLEM SOLVED: Should Managers at Genco Exploration Drill? Discussion Questions • Two Sectors: Natural gas and oil production from shale rock formations • Managers’ Decision: Drill or not drill • Geologists’ report led managers to believe that if a well were drilled, there was a 0.60 probability of finding no deposits, a 0.15 probability of finding the equivalent of 10,000 barrels, a 0.15 probability of finding 20,000 barrels, and a 0.10 probability of finding 30,000 barrels. • Managers estimate they will realize a $90,000 loss if they find no oil, a $100,000 profit if they find 10,000 barrels of oil, a $300,000 profit if they find 30,000 barrels, and a $500,000 profit if they find 30,000 barrels. Question: Based on these beliefs, should the managers drill the well? Answer: The answer to the question is best found by using a decision tree. The expected profit if the managers decided to drill is given by E(Profit) = 0.60($90,0 0 0) + 0.15($10 0,0 0 0) + 0.15($30 0,0 0 0) + 0.10($500,000) = $56,000. If the managers do not drill, then expected profit is $0. So if managers want to maximize expected profit, they should drill the well! 6. The Expected Value of Perfect Information • • • •

Sometimes managers can obtain information that will dispel uncertainty. Question: If managers can buy such information, how much should they be willing to pay for it? Value of Perfect Information: The increase in expected profit if the manager can obtain completely accurate information concerning future outcomes Expected Value of Perfect Information (EVPI) ÿ The increase in expected profit from completely accurate information concerning future outcomes ÿ Example: The Jones Case (Figure 14.1) * The idea: If managers can obtain perfect information, choosing the optimal strategy becomes easier. * Given perfect information, the company will increase price and earn a profit of $800,000 if the campaign will be successful and will not increase price if the campaign will not be successful and earn a profit of $200,000.

224 | Chapter 14 Forecasters: There is a 0.50 probability the campaign will be successful, and there is a 0.50 probability the campaign will fail. * To managers with perfect information, the expected profit is 0.50($8,000,000) + 0.5($200,000) = $500,000. Therefore, if managers obtain access to a perfect forecast (which is revealed after the payment is made) the expected profit is $500,000. Therefore, the expected value of perfect information is * EVPI = $500,000 − $200,000 = $300,000. * $300,000 is the manager’s reservation price for this information. Some sources of perfect information: * Testing services * Research organizations * News bureaus * Credit-rating agencies *

•

PROBLEM SOLVED: Evaluating an Investment in a New Chemical Plant Discussion Questions 1. Do you think that the managers should spend $10.1 million on information about by-product quality and level of impurities? Why or why not? How much should they have spent? Answer: If they are risk-averse, they should spend less than the EVPI. 2. If calculation of the EVPI does not tell a firm exactly how much to spend on information, is it still a useful calculation to managers? Answer: Yes, it helps to rank alternatives. STRATEGY SESSION: FedEx’s and Ata Holdings Corporation’s Disclosures about Risk Discussion Questions 1. FedEx’s projections were way off with regard to both interest rates (which fell very significantly going into 2009) and fuel costs (which more than doubled before falling again in 2009). Does this suggest that management was not effectively managing risk? Why or why not? 2. Do you think laws, like SOX, that require management to disclose risk to investors are effective in ensuring that investors make better informed decisions?

Risk Analysis | 225 7. Measuring Attitudes toward Risk: The Utility Approach • •

Goal: To examine how risk affects managerial behavior Illustration ÿ A small business is offered the following choice: 1. A certain profit of $2,000,000 2. A gamble with a 50–50 chance of $4,100,000 profit or a $60,000 loss. The expected value of the gamble is E(Profit of gamble) = 0.5($4,100,000) + 0.5(−$60,000) = $2,020,000

• • • •

ÿ Managers should choose the gamble over the certainty of $2,000,000 if they want to maximize expected profit. ÿ If the business is risk-averse, it is likely to take the certain profit. ÿ Fact: We need not assume that managers want to maximize expected profit. ÿ We can construct a utility function for managers that measures their attitude toward risk. Utility Function: Function used to identify the optimal strategy for managers conditional on their attitude toward risk Rational Manager: One who maximizes expected utility The manager’s utility function represents the level of satisfaction she attaches to each possible outcome. Constructing a Utility Function ÿ Expected utility: The sum of the utility of each outcome times the probability of the outcome’s occurrence Example: The Genco Exploration case ÿ Procedure: 1. Arbitrarily assign utility levels to two payoffs, with the higher payoff set to a higher level of utility certain profit of $2,000,000. For the Genco Exploration example, set U(−90) = 0 and U(500) = 50. 2. Next, ask the decision maker what value of P (probability) would make him or her indifferent between a certain amount (say 100) and the following gamble: A gain of $500,000 with probability P and a loss of $90,000 with probability (1 − P). 3. Suppose we want to find U(100). Suppose the decision maker sets P = 0.4, then U(100) = 0.40U(500)+0.60U(−90). That is, U(100) = (0.4)(50) + (0.6)(0) = 20. 4. Continuing with this method will give all the derivation of utility for any possible payoff. ÿ Figure 14.2: Utility Function

226 | Chapter 14 PROBLEM SOLVED: To Drill or Not Discussion Questions 1. Suppose that the probabilities associated with the different outcomes of drilling by Genco are defined as follows: Probability

Payoff

0.7 0.1 0.1 0.1

−90 100 300 500

What is Genco’s expected utility, and what is its optimal choice? Answer: E(U ) = 11, so the optimal choice is still to drill. 8. Attitudes toward Risk: Three Types •

•

•

• •

1. Risk averter 2. Risk lover 3. Risk-neutral Risk Averters: When managers prefer a choice with a more certain outcome to one with a less certain outcome, when confronted with gambles offering equal expected wealth ÿ The utility function has diminishing marginal utility. That is, the (Bernoulli) utility function is concave, where the utility function is defined over wealth. Risk Lovers: When managers prefer a gamble with a less certain outcome to one with a more certain outcome, when confronted with gambles offering equal expected wealth ÿ The utility function has increasing marginal utility. In other words, the (Bernoulli) utility function u is convex. Risk-Neutral: When a manager maximizes expected wealth, regardless of risk ÿ Utility function is linear and marginal utility is constant. Given a gamble ((p1, p), (p2, 1 − p)), then the individual is called ÿ risk averter if u[pp1 + (1 − p) p2] > p u(p1) + (1 − p)u(p2); ÿ risk lover if u[pp1 + (1 − p) p2] < p u(p1) + (1 − p)u(p2); ÿ risk-neutral if the (Bernoulli) utility function u is a linear function of wealth u(p) = a + bp,

•

where p is wealth.

Given the linear utility function, expected utility is given by E(u)  E(a  bp)  a  bE(p)

Risk Analysis | 227 But for the gamble, we have E(profit) = pp1 + (1 − p)p2

•

Now, maximizing E(u) = a + bE(p) is equivalent to maximizing E(p), given that the first expression is a linear combination of the latter. Therefore, for risk-neutral individuals, maximizing expected utility is equivalent to maximizing expected value of the gamble. Figure 14.3: Three Types of Utility Functions

9. The Standard Deviation and Coefficient of Variation: Measures of Risk • • •

Risk has many meanings. Consider N outcomes p1 , ... , pN with corresponding probability p1, ... , pN . Standard Deviation: The most frequently used metric for dispersion in a probability distribution. N

ÿ =

∑ P [ i

i

− E ( )]2

i =1

• • • • •

Coefficient of variation: V  s/E() Note: Although the standard deviation is often a useful measure of risk, it may not always be the best measure. A larger standard deviation implies greater risk. Figure 14.4: Probability Distribution of the Profit from an Investment in a New Plant Example: Pi

pi

(Pi) (pi)

[pi  E(p)]2

Pi [pi  E (p)]2

0.3 0.4 0.3

1.0 0.2 0.6

0.30 0.08 0.18

0.6400 0.0000 0.6400

0.1920 0.0000 0.1920

E(p)  0.2

s2  0.3840 s  0.6197 V  3.10

10. Adjusting the Valuation Model for Risk •

Certainty Equivalent Approach: When a manager is indifferent between a certain payoff and a gamble, the certainty equivalent (rather than the expected profit) can identify if the manager is a risk averter, risk lover, or risk-neutral. ÿ If the certainty equivalent is less than the expected value, then the decision maker is risk-averse.

228 | Chapter 14

•

•

ÿ If the certainty equivalent is equal to the expected value, then the decision maker is risk-neutral. ÿ If the certainty equivalent is greater than the expected value, then the decision maker is risk loving. The present value of future profits, which managers seek to maximize, can be adjusted for risk by using the certainty equivalent profit in place of the expected profit. Indifference Curves ÿ Figure 14.5: Manager’s Indifference Curve between Expected Profit and Risk ÿ With expected value on the horizontal axis, the horizontal intercept of an indifference curve is the certainty equivalent of the risky payoffs represented by the curve. ÿ If a decision maker is risk-neutral, indifference curves will be vertical.

11. Certainty Equivalence and the Market for Insurance •

•

Example ÿ Managers hold $900 million in debt. * There is a 25% chance that the value of the bonds will fall to $400 million. * There is a 75% chance that the value of the bonds will remain constant. * Expected value = $775 million ÿ Managers have the following utility function defined on wealth (W): U = W 0.5 * Expected utility = 27.5 2 2 * Certainty equivalent = W* = U = (27.5) = $756.25 million * The certainty equivalent should be the manager’s reservation price for selling the bonds at a discount. * The certainty equivalent is the monetary sum that would make the manager indifferent between having that monetary sum for certain and holding the bonds (which have an expected utility of 27.5). * Certainty equivalents are also used to create metrics in the insurance industry. Example ÿ LBI Insurance Company provides full coverage of loss and is risk-neutral. * LBI’s expected payout is $125 million, so that is the minimum price for the policy. * The most that the policy is worth to the buyer is the difference between $900 million and the certainty equivalent of $756.25 million, or $143.75 million.

Risk Analysis | 229 *

The risk premium is the difference between LBI’s reservation price for the policy and the maximum amount the buyer would pay: $143.75 − $125 = $18.75 million.

STRATEGY SESSION: Pepsico Risk Management Discussion Question 1. Hedging allows a firm to “lock in” prices for purchases that will take place at a future point in time. As such, it is a form of insurance against price fluctuations. Futures markets have been criticized by some as a speculative environment that tends to exaggerate price fluctuations. Do you agree? Why or why not? STRATEGY SESSION: The Use of Risk-Adjusted Discount Rates Discussion Questions 1. Compare the graph shown here with Figure 14.5. How do they differ? Answer: The axes are switched, and risk is measured by the standard deviation instead of the coefficient of variation. 2. What does the increasing slope of the indifference curve imply about the manager’s attitude toward risk and return? Answer: As risk increases, the manager requires greater and greater increases in return to maintain a constant level of utility.

Chapter 14: Problem Solutions 1. The president of the Martin Company is considering two alternative investments, X and Y. If each investment is carried out, there are four possible outcomes. The present value of net profit and probability of each outcome follow:

Outcome 1 2 3 4

Investment X Net Present Value Probability $20 million 0.2 8 million 0.3 10 million 0.4 3 million 0.1

Outcome A B C D

Investment Y Net Present Value Probability $12 million 0.1 9 million 0.3 6 million 0.1 11 million 0.5

230 | Chapter 14 a. What are the expected present value, standard deviation, and coefficient of variation of investment X? b. What are the expected present value, standard deviation, and coefficient of variation of investment Y? c. Which investment is riskier? d. The president of the Martin Company has the utility function U  10  4P  0.2P2 where U is utility and P is net present value. Which investment should she choose? Solution: A $20 million

0.2

X

0.3 0.4

0.1

B $8 million

C $10 million

D $3 million

A

$12 million

B

$9 million

C

$6 million

D

$11 million

0.1 0.3 Y

0.1

0.5

Risk Analysis | 231 a. E(Investment X)  (0.2)(20)  (0.3)(8)  (0.4)(10)  (0.1)(3)  $10.7 million s X  [(20  10.7)2(0.2)  (8  10.7)2(0.3)  (10  10.7)2(0.4)  (3  10.7)2(0.1)]1/2 sX  5.0606. V X  sX/E(X)  5.0606/10.7  0.4729 b. E(Investment Y)  (.1)(12)  (.3)(9)  (.1)(6)  .5(11)  $10 million sY  [(12  10)2(.1)  (9  10)2(.3)  (16  10)2(.1)  (11  10)2(.5)]1/2  1.673 V Y  s Y /E(pY)  1.673/10  0.1673 c. V X V Y → X is riskier. d. An easy way to compute the expected utility of X and Y is to use the linearity property of the expectation operator. Doing so, we have E(U)  10  5E(P)  0.2E(P2) For investment X, we have E(X)  $10.7 million, so that E(P)  $10.7 million. Therefore, we obtain E(P2) 400(0.2)  64(0.3)  100(0.4)  9(0.1)  140.1 and, therefore, expected utility of X is given by 10  4(10.7)  0.2(140.1)  24.78 Similarly, for investment Y, we obtain E(P2) 144(0.1)  81(0.3)  36(0.1)  121(0.5)  102.8. Therefore, E[U(X)]  10  40  0.2(102.8)  29.44. Since investment Y has a higher expected utility than investment X, then investment Y should be chosen assuming the investor is rational and therefore maximizes expected utility. 2. William J. Bryan is the general manager of an electrical equipment plant. He must decide whether to install a number of assembly robots in his plant. This investment would be risky because both management and the workforce have no real experience with the introduction or operation of such robots. His indifference curve between expected rate of return and risk is as shown in the figure.

232 | Chapter 14

a. If the riskiness (s) of this investment equals 3, what risk premium does he require? b. What is the riskless rate of return? c. What is the risk-adjusted discount rate? d. In calculating the present value of future profit from this investment, what interest rate should be used? Solution: If the risk is 0—that is, there is no risk—then expected return is 18 percent. a. 18 percent  6 percent  12 percent. b. 6 percent. c. 18 percent. d. 18 percent. 3. The Zodiac Company is considering the development of a new type of plastic. Whether the plastic will be successful depends on the outcome of a research project being carried out at a major university. Zodiac’s executives have no reliable means of estimating the university research team’s probability of success. Zodiac’s gains (or losses), depending on the outcome of the university research project, are as follows:

Risk Analysis | 233 Outcome of University Research Project Success Failure

Action Zodiac develops plastic Zodiac does not develop plastic

$50 million 0

$8 million 0

On the basis of the information given, can you calculate the expected value of perfect information? Why or why not? (You may assume that Zodiac is risk-neutral.) Solution: No. The expected value of perfect information can only be calculated if probabilities are known. 4. The Electro Corporation, which manufactures television sets, has a fixed cost of $1 million per year. The gross profit from each TV set sold—that is, the price less the average variable cost—is $20. The expected value of the number of sets the company sells per year is 100,000. The standard deviation of the number of sets sold per year is 10,000. a. What is the expected value of the firm’s annual profit? b. What is the standard deviation of the firm’s annual profit? c. What is the coefficient of variation of the firm’s annual profit? Solution: Let Q be the number of TV sets the company sells per year. We can derive a formula for the company’s annual profit. We have Profit : P  revenue  cost  20Q  100,000 Now, it is easy to compute the expected value of P, including the standard deviation and the coefficient of variation. We are given the following information: E(Q)  100,000 and standard deviation  10,000 a. Using the linearity property of the expectation operator, we see that E(P)  20E(Q)  1,000,000  20(100,000)  1,000,000  $1,000,000 b. For the standard deviation, given that P is a linear function of Q, then we have sP  20sQ  20 (10,000)  $200,000 c. V  $200,000/$1,000,000  0.2

234 | Chapter 14 5. Richard Miller, a Wall Street trader, says he is risk-neutral. Suppose we let 0 be the utility he attaches to $100,000 and 1 be the utility he attaches to $200,000. If what he says is true, what is the utility he attaches to (a) $400,000? (b) $40,000? (c) $20,000? Solution: If the individual is risk-neutral, then his or her utility function is linear in wealth. That is, U(W)  a  bW. Now, given that U(100)  0 and U(200)  1, we obtain the following equations: A  100b  0 and a  200b  1 Solving for a and b, by elimination, for example, gives b  0.01 and a  1. Therefore, the utility function is given by U(W)  1  0.01W, where W is in thousands. Hence, we have a. U(400) 1  0.01(400)  3. b. U(40) 1  0.01(40)  0.6. c. U(20)1  0.01(20)  1.2. 6. The chief executive officer of a publishing company says she is indifferent between the certainty of receiving $7,500 and a gamble where there is a 0.5 chance of receiving $5,000 and a 0.5 chance of receiving $10,000. Also, she says she is indifferent between the certainty of receiving $10,000 and a gamble where there is a 0.5 chance of receiving $7,500 and a 0.5 chance of receiving $12,500. a. Draw (on a piece of graph paper) four points on the utility function of this publishing executive. b. Does she seem to be a risk averter, a risk lover, or risk-neutral? Explain. Solution: a. We have U(7,500)  0.5U(5,000)  0.5U(10,000) and U(10,000)  0.5U(7,500)  0.5U(12,500).

Risk Analysis | 235

b. She is risk-neutral, since her utility function is linear. 7. The Oahu Trading Company is considering the purchase of a small firm that produces clocks. Oahu’s management feels there is a 50–50 chance, if Oahu buys the firm, that it can mold the firm into an effective producer of washing machine parts. If the firm can be transformed in this way, Oahu believes that it will make $500,000 if it buys the firm; if it cannot be transformed in this way, Oahu believes that it will lose $400,000. a. Construct a decision tree to represent Oahu’s problem. b. What are the decision forks? (Are there more than one?) c. What are the chance forks? (Are there more than one?) d. Use the decision tree to solve Oahu’s problem. In other words, assuming that the firm wants to maximize the expected profit, should Oahu buy the firm? e. Before Oahu makes a decision concerning the purchase of the firm, Oahu’s president learns that if the clock producer cannot be made into an effective producer of washing machine parts, there is a 0.2 probability that it can be resold to a Saudi Arabian syndicate at a profit of $100,000. (If the firm cannot be resold, Oahu will lose $400,000.) (1) Does this information alter the decision tree? (2) Can you think of three mutually exclusive outcomes if Oahu buys the firm? (3) What is the probability of each of these outcomes? (4) What is the monetary value to Oahu of each of these outcomes?

236 | Chapter 14 f. Use your results in part (e) to solve Oahu’s problem under this new set of conditions. In other words, on the basis of this new information, should Oahu buy the firm? g. Oahu’s executive vice president discovers an error in the estimate of how much Oahu will gain if it buys the clock manufacturer and turns it into an effective producer of washing machine parts. (1) Under the circumstances in part (d), how big would this error have to be to reverse the indicated decision? (2) Under the circumstances in part (e), how big would the error have to be to reverse the indicated decision? Solution: a.

b. Only one: To buy or not to buy the firm. c. Only one: The firm is effective or it is not. d. Yes, Oahu should buy the firm because the expected return of $50,000 is greater than the alternative of zero. e. (1) Yes, the “won’t work” branch has another chance fork attached on which 0.2 leads to $100,000 and 0.8 leads to $400,000. (2) Yes, (a) the project works, (b) the project is resold, and (c) the project loses money. (3) (a) 0.5, (b) 0.5(0.2)  0.1, (c) 0 0.5(0.8)  0.4 (4) (a) $500,000, (b) $100,000, (c) $400,000 f. E(p)  500,000(0.5)  100,000(0.1)  400,000(0.4)  $100,000. Buy the firm.

Risk Analysis | 237 g. (1) E(p)  (500,000  X)(0.5)  400,000(0.5)  0 → X  $100,000 (2) E(p)  (500,000  X)0(.5)  100,000(0.1)  400,000(0.4)  0 → X  $200,000 8. The National Aeronautics and Space Administration (NASA) estimated the probability of a crash of the space shuttle to be 1 in 100,000, whereas the probability was in fact closer to about 0.010.02. If a decision tree had been used to determine whether to attempt a launch of the shuttle, what difference, if any, would this have made? Solution: Given the low-risk factor NASA used, the attractiveness of a launch appeared greater than an unbiased appraisal would have suggested. 9. The East Chester Tribune must decide whether to publish a Sunday edition. The publisher thinks the probability is 0.6 that a Sunday edition would be a success and 0.4 that it would be a failure. If it is a success, she will gain $100,000. If it is a failure, she will lose $80,000. a. Construct a decision tree corresponding to the problem, and use backward induction to solve the problem. (Assume that the publisher is risk-neutral.) b. List all forks in the decision tree you constructed; then indicate whether each is a decision fork or a chance fork and state why. Solution: a.

238 | Chapter 14 b. The first fork is a decision fork to publish or not to publish. The second fork is a chance fork, where publishing either succeeds or fails. 10. Roy Lamb has an option on a par ticular piece of land, and he must decide whether to drill on the land before the expiration of the option or give up his rights. If he drills, he believes that the cost will be $200,000. If he finds oil, he expects to receive $1 million; if he does not find oil, he expects to receive nothing. a. Construct a decision tree to represent Lamb’s decision. b. Can you tell whether he should drill on the basis of the available information? Why or why not? Lamb believes that the probability of finding oil if he drills on this piece of land is 0.25, and the probability of not finding oil if he drills there is 0.75. c. Can you tell whether he should drill on the basis of the available information? Why or why not? d. Suppose Lamb can be demonstrated to be a risk lover. Should he drill? Why or why not? e. Suppose Lamb is risk-neutral. Should he drill? Solution: a.

b. No, without the probabilities, the problem can’t be solved. c. 1/4(800)  3/4(200)  50 0, so a person who is risk-neutral would drill. However, a person who is very risk-averse would not want to drill. d. Yes, since the project has both a positive expected value and contains risk, Mr. Lamb will be doubly pleased. e. Yes, Mr. Lamb cares only about expected value, which is positive for this project.

CHAPTER 15

Principal–Agent Issues and Managerial Compensation

Lecture Notes 1. Introduction •

Objectives ÿ To discuss issues related to principal–agent relationships ÿ To formalize the principal–agent problem ÿ To analyze the effect of risk, information, and compensation on principal–agent issues ÿ To examine a possible solution to the incentive conflict when output is risky and effort is not observable ÿ To examine some refinement to managerial compensation ÿ To discuss principal–agent issues in other contexts ÿ To show how incentive schemes such as * Bonuses * Equity * Options help converge preferences and resolve much of the principal–agent conflict ÿ To discuss the issue of product liability and the safety of consumer goods

2. Principal–Agent Issues • • •

Assume managers seek to maximize the value for shareholders. Managers face situations where their personal utility function conflicts with that of being an agent of the firm. Principal–Agent Issues: When managers (agents) make decisions that affect the wealth of shareholders (principals).

239

240 | Chapter 15

•

•

ÿ Conflict of interest: Personal utility of decision maker (agent) conflicts with objectives of the principal Factors ÿ Uncertainty is a factor when the effects of an agent’s decisions are not deterministic. Outcomes may be bad when decisions are good. ÿ Information asymmetry arises when: * agents and principals do not share common sets of information. * agents (managers) have more information about the action than does the principal. * the agent’s action is not directly observable by the principal, and the outcome of the action is not completely determined by the agent’s action. * agents and principals play a noncooperative game. * the principals determine compensation rules for the agents. * the consequences of decisions made by agents are not entirely predictable. ÿ The issue vanishes when principal and agent have identical objectives. * Example: Manager is the business owner. Managers can anticipate issues associated with the principal–agent problem and devise incentive compensation strategies that will help to minimize the problem. ÿ Product liability laws provide incentives for agents (product manufacturers) to produce safe products for the principals (consumers).

3. The Diverging Paths of Owners and Managers • • • •

Firm’s Owners: Shareholders who purchase the stock as an investment Shareholders are concerned about the value of their shares. Manager’s Goal: ÿ Maximize the value of the firm Alternative Management Goals ÿ Minimize effort. * Increasing profit often takes hard work, and there is always disutility to work. * Managerial behavior is largely driven by how owners structure compensation. ÿ Maximize job security. * Many decisions of managers involve risk. * Often, risky projects are characterized by high potential reward or large potential loss. * Managers may be disinclined to make risky choices that could jeopardize job employment. ÿ Avoid failure.

Principal–Agent Issues and Managerial Compensation | 241 Managers can be rewarded for good performance and penalized for bad results. * Managers might be disinclined to take risks. ÿ Enhance reputation and employment opportunities. ÿ Maximize perquisites. Examples: Luxury travel, expensive artwork in the office, corporate donations to favorite charities, employing favored people ÿ Maximize pay. * Managers work for pay. * Both the level and structure of the compensation package become an important part of the principal–agent problem. *

4. The Principal–Agent Situation Illustration: Figure 15.1 * The principal employs an agent who performs a task that results in a benefit (output) to the principal. * The principal must pay the agent. This compensation can be a fixed sum, or it can depend on the output. * The level of output depends on the quantity and the quality of effort provided by the agent. * Effort is not always observable or measurable. * Principal–agent problem: Effort cannot be perfectly monitored by the principal and therefore cannot be directly rewarded. * Efficient solution: Requires some alignment of interests of the two parties • Examples of Principal– Agent Issues: 1. Choice of distribution channel by managers at a life insurance company ÿ Existing channel: Low risk, low expected profit * Managers may prefer this choice because of low risk. They are playing it safe. ÿ New channel: High risk, high potential profit * Stockholders may prefer this choice because of the high potential profit and because they can reduce risk by holding a diversified portfolio. 2. Charitable giving by a firm ÿ Stockholders prefer charitable giving that will provide strategic benefit to the firm. ÿ Some level of charitable giving benefits shareholders. Example: * Stimulate demand for firm’s product. * Charitable giving can cast the firm in a favorable light with legislators and regulators. •

242 | Chapter 15 ÿ Managers may have personal agendas that lead to charitable giving that does not provide strategic benefit to the firm. 5. The Effect of Risk, Information, and Compensation on Principal–Agent Issues •

•

•

Managerial Behavior and Effort ÿ Disutility of effort: A measure of the cost to the manager of supplying effort ÿ Effort imposes personal costs on managers, who therefore prefer to exert less effort. ÿ Effort contributes to increasing the value of the firm, so stockholders prefer managers to exert more effort. Model ÿ p = p (e): A function of the manager’s effort * p is profit and e is manager’s effort. ÿ Assume profit is not risky: Once the manager chooses effort, we can forecast the profit with certainty. ÿ p = Total revenue − Total cost ÿ Divide total cost between the manager’s compensation, S, and all other costs, C. ÿ p (e) = R(e) − (S + C) * R(e) is revenue as a function of manager’s effort, S is manager’s compensation, and C is other costs. ÿ Figure 15.2: The Principal–Agent Problem with a Flat Salary * u(e) is the disutility of effort for the manager, or the cost of supplying effort, represented as linear with a positive slope. * Net benefit to the manager is B(e) = S − u(e). dB/de = − du/de < 0. Hence, net benefit, B(e), must be downward sloping. ÿ Figure 15.3: The Principal–Agent Problem with Flat Pay Resolving the Incentive Conflict If Effort Is Observable * Question: How can owners motivate managers to work harder? * One solution: Include incentive pay: The manager is now persuaded to increase effort, which in turn increases revenues and profit. Owners structure the compensation in two parts. ÿ Model: S(e) = K + U(e) * S(e) is compensation as a function of observable effort, K is the fixed component of compensation, and U(e) is the portion of compensation that depends on effort. * Profit: p (e) = R(e) − S(e) − C = R(e) − [K + U(e)] − C * Profit-maximizing compensation: Obtained by setting dp(e)/de = dR(e)/de − dU(e)/de = 0, that is dR(e)/de = dU(e)/de

Principal–Agent Issues and Managerial Compensation | 243

•

Interpretation: Marginal benefit from effort, in terms of increased revenues, R(e), must equal the marginal cost of compensating the managers for effort. * Manager net benefit: B(e) = S(e) − u(e) = K + U(e) − u(e) * If U(e) = u(e), then B(e) = K, and managers will be willing to supply any desired effort level (net benefit is always positive as long as K is positive). The stockholders just need to communicate the optimal level of effort to managers and they will provide it. ÿ Figure 15.4: Motivating Managers When Effort Is Observable. ÿ Figure 15.5: The Principal–Agent Problem with Pay as a Function of Effort. ÿ In practice, managers have better information than shareholders, and it is too costly for shareholders to perfectly monitor managerial effort. Resolving the Incentive Conflict If Effort Is Not Observable ÿ If effort is not observable, then it is difficult to reward or penalize effort levels. ÿ Model * Assume that the firm’s revenue R(e) is riskless and determined solely by e, the effort put forth by the manager. * The level of managerial effort can be inferred from R when revenue is riskless, but this does not ensure that managers select the optimal level of effort, e*. This problem is resolved by replacing K with a bonus: A share, a, of profits that is selected by owners. * Divide the manager’s compensation into two parts: Compensation = S(e) = salary + bonus = U(e) + ap(e) * Profit = p(e) = R(e) − U(e) − C * Net benefit to manager = B(e) = S(e) − u(e) = U(e) + ap(e) − u(e) = ap(e), assuming U(e) = u(e). Managers get a net benefit a (e), and the owners get the remainder (1 − a)p (e). Thus, both are interested in maximizing net profit, p (e). Hence, * Managers’ and owners’ incentives converge. Both seek to maximize net profit. * The idea: By giving managers a share of profits, owners align their preferences with those of managers. * Two stages of the process: 1. The manager chooses a level of effort to maximize profit, p (e). This is achieved where the marginal benefit of effort equals the marginal disutility of the cost of effort. 2. The firm’s owners choose the level of a so that the compensation package is competitive. * Incentive compatibility: When the agent and the owners share in the profit of the firm and the agent’s effort maximizes the principal’s profit

244 | Chapter 15 *

As before: ° The manager chooses a level of effort to maximize profit. ° The owners choose a such that the compensation package is competitive.

PROBLEM SOLVED: Setting Optimal Compensation When Output Is Not Risky and Effort Is Not Observable Discussion Questions Assume that Revenue: R(e) = 3,500 + 100e0.5 and u(e) = 853.55 + 7.07e. In addition to paying the manager a salary, the owners must pay a production cost of $1,000. To persuade managers to work, the manager receives a net benefit of $1,000. If she receives less, she will leave and take another job. 1. What is the optimal level of effort for the manager? 2. What is the weekly profit? 3. How much profit will be paid to the manager (what is alpha)? Solution: 1. Profit before payment is p(e) = R(e) − U(e) − C = R(e) − u(e) − C. She will choose to maximize her bonus, which is the same as maximizing profit because a bonus is a proportion of profit. First-order condition gives dR/de = du/de. That is, 100(0.5)e−0.5 = 7.07. That is, e−0.5 = 7.07/50 or e = 50. Hence, the manager works 50 hours per week. 2. Weekly profit: p(e) = R(e) − u(e) − C = [3,500 + 100(50)0.5] − [853.55 + (7.07)(50)] − 1,000 = 2,000 3. Notice that B(e) = U(e) + ap(e) − u(e) = ap(e) If B(e) = 1,000, then we obtain = 1,000. That is, ap(e) = 1,000 or a(2,000) = 1,000 or a = 50%.

•

Therefore, if the manager is to receive a net benefit of $1,000, then she must receive 50% of the profit. Managers are generally given considerable freedom in operating discretion. It is too costly for owners to perfectly monitor managerial effort.

6. Resolving the Incentive Conflict When Output Is Risky and Effort Is Not Observable • •

The point: Revenue is risky, and effort is not observable. With no risk, owners can infer the level of managerial effort from the firm’s profit.

Principal–Agent Issues and Managerial Compensation | 245 •

•

•

When output is risky, owners cannot infer the level of managerial effort from observed profits. Owners are rarely certain whether high profit is due to high effort or simply good luck (a strong economy) and whether low profit is due to low effort or bad luck. The lesson: Poor management can occasionally result in short-term high profit due to random events. Bonus Plans ÿ contribute to efficiency by aligning managers’ incentives with owners’ objectives. ÿ impose risk on managers in that performance measures that determine bonuses are not entirely under managers’ control. Risk Sharing ÿ Risk premium: The minimum difference a manager requires in compensation to be willing to take a risk ÿ Owners can reduce risk through portfolio diversification. Managers are averse to risk in their compensation plan. This suggests that owners should absorb all risk. ÿ Model * Revenue = R(e) = R1(e) + R2 * Revenue consists of two components: R1(e) is determined by the manager’s effort and is under managerial control, and R2 is outside of the manager’s control. * Compensation should consist of a fixed component, K, which is independent of effort, and a bonus, ap(e), that depends on profit. * S = K + ap(e) * Profit = p(e) = R1 (e) + R2 − K − C * The manager is exposed to risk if she receives a bonus. * Net benefit is the expected utility of compensation minus the disutility of effort B(e) = EU(S) − u(e) = EU[K + ap(e)] − u(e) ÿ Elements of the principal–agent conflict in this example * Principal sets incentive compensation to align manager and owner interests. * Agent selects a level of effort, given the compensation plan, to maximize expected utility. * Result: When profit is determined, the manager receives compensation, and the principal keeps the residual profit. * Figure 15.6: The Principal–Agent Problem When Effort Is Not Observable

246 | Chapter 15 PROBLEM SOLVED: Two Compensation Schemes Based Solely on Risk Sharing Discussion Questions 1. If managers are risk-averse and owners are risk-neutral, will owners benefit from paying managers using a bonus plan instead of a flat salary? Why or why not? Answer: If the efficiency effect of the bonus on profit, resulting from higher effort, exceeds the risk premium paid to the manager, then using a bonus compensation system will benefit owners. If the efficiency effect is ignored, and only risk sharing is considered, then owners should always pay a flat salary. PROBLEM SOLVED: Setting Compensation for Managers Discussion Questions The manager has received an offer of a flat salary of $1.44 million. 1. What is the manager’s utility with this salary? Answer: 1,200 2. What proportion of equity must be offered to the manager to provide an equal expected utility? Answer: 0.98826 3. What is the expected compensation of the manager under the bonus plan? Answer: $1,482,390 4. What is the expected equity to shareholders? Answer: $13,517,610 7. Some Refinements to Managerial Compensation •

•

Motivating the Manager with Profit Sharing ÿ Figure 15.7: The Effect of Compensation Schemes on Managerial Effort * Utility of flat salary B = U(B) with low effort * With high effort, EU = (0.4)U(A) + (0.6)U(C) = U(B) * Risk premium plus disutility of high effort = D − B Motivating the Manager with an Income Guarantee and Stock Options ÿ Figure 15.8: Reducing Managerial Risk with Stock Options * Manager is guaranteed a salary of E plus stock options that will pay F with probability 0.35 and nothing with probability 0.65.

Principal–Agent Issues and Managerial Compensation | 247 *

EU of compensation is the same as in Figure 15.7, but there is a guaranteed minimum income.

STRATEGY SESSION: Getting the Board to Focus on the Long Term Discussion Questions 1. Do you think the change in compensation will encourage board members to focus on long-term growth? Why or why not? 2. Many, if not all, incentive-based compensation plans provide opportunities for managers to “game the system” by altering the timing of revenues and costs in ways that do not increase value. Does this mean that such systems should be abandoned? Why or why not? • Motivating the Manager with an Income Guarantee and Stock Options Owners can use stock options. 1. The manager is paid a flat salary of E. This salary is riskless; it is paid regardless of performance. 2. The manager receives a call option on the firm’s stock, which is at risk. ÿ Call option: Option that gives managers the right to purchase a firm’s shares at some future date ÿ Strike price (or exercise price): The fixed price at which the stock can be purchased • Indexed Stock Options * Both profit-sharing and stock option plans motivate managers to exert high effort; but even if they exert this greater effort, they are not guaranteed high payments. * Factors beyond managerial control (such as interest rates) can affect firm performance. * One plan that has attracted attention is the index stock option. ÿ Indexed stock options have a striking price that is linked to an index of stock prices. They reward performance relative to the market that is indexed. STRATEGY SESSION: Call Options Discussion Question 1. How do the people who sell call options make money? What outcome are they betting on? 8. Principal–Agent Issues in Other Contexts (Market for Insurance) •

Two Parties: An insured party and an insurance company—Issues between the two parties are known as moral hazard.

248 | Chapter 15 • •

Moral Hazard: When a party insured against risks behaves differently from the way it would behave if it were uninsured against these risks Insurance moral hazard can be divided into two types: ex-ante moral hazard and ex-post moral hazard. ÿ Ex-ante moral hazard: Refers to the tendency of insured entities to take less care to prevent future losses when they have insurance ÿ Ex-post moral hazard: Refers to the reluctance of policyholders who have already suffered some misfortune to keep the cost of the event under control

STRATEGY SESSION: The Good and Bad of Incentive Pay Discussion Question 1. When people are paid to do something that can be measured, and their pay is based on the measurement, they are likely to make an effort to increase the measurement’s value. How are incentives likely to influence doctors, who are paid based on the number of patients they see, or teachers, who are paid based on their students’ test performance? • Asset Substitution ÿ Example * Shareholders have control over the decisions of managers through the compensation scheme, but creditors do not. * A drug company is exposed to risk. Expected future earnings could be 100 or 200 with equal probability. Expected value of the firm is 150. The firm has 100 in debt. Hence, after paying off the bonds, there is 50 in expected equity. * The drug company has an opportunity to invest in a new drug. Formula A is relatively risk-free, and formula B is potentially more profitable but more risky. Capital cost of the new drug is 200. * Certain value of formula A = 220 − 200 = 20 * Expected value of formula B = (0.5)(20) + (0.5)(310) − 200 = −35 * Table 15.2: Firm Value if Project A Is Chosen: Equity = 70 * Table 15.3: Firm Value if Project B Is Chosen: Equity = 80 * The firm is better off with Project B because, if the project fails, the firm will go bankrupt, bondholders won’t get paid, and stockholders walk away. If the project succeeds, then stockholders will reap the rewards. Bondholders, recognizing the optimal action of the firm, will refuse * to provide financing above the 135 amount, which is what bondholders would receive if Project B were selected and it was unsuccessful. ÿ Face value: The principal amount of the bond ÿ Residual claim: What is left of the dividend value ÿ Figure 15.9: Will Shareholders Pull the Bait and Switch?

Principal–Agent Issues and Managerial Compensation | 249 Game theory representation of the example as a sequential decision problem where bondholders make the first move ÿ Possible solutions to the asset substitution problem * Fund with equity. * Establish a reputation for protecting creditors. * Precommit to hedge or insure risk. *

STRATEGY SESSION: The Song of the Sirens Discussion Question 1. Are managers doing anything inappropriate when they adopt high-risk strategies in order to increase the expected value of their options? Answer: No. They’re doing what they’re supposed to be doing. Remember, stockholders can diversity their portfolios to reduce risk. They want managers to be risk takers. 9. Product Liability and the Safety of Consumer Goods • •

•

•

Product liability laws provide an incentive for managers to make safer products because the firm must compensate victims. How safe would products be without a product liability law? Consider the following example: ÿ Total cost of safety to firm = s ÿ Marginal cost of safety = 1 ÿ Expected cost of accidents = 4,000 − 20s0.5 ÿ Expected marginal benefit of safety = 10/s0.5 ÿ In the absence of product safety laws, s = 0 Safety under a Product Liability Law ÿ With an optimal product safety law, s = 100, and expected cost of accidents = 3,800. Optimal Safety under a Market Mechanism ÿ Assume Q = 1,000 and customers are willing to pay a price = 30 − (4 − 0.02s 0.5), which is the value of a known perfectly safe product minus the expected cost of accidents (a discount for lack of safety). Also assume that customers have perfect information. ÿ Total revenue = (1,000)[30 − (4 − 0.02s0.5)] ÿ Profit = (1,000)[30 − (4 − 0.02s0.5)] − s − 10,000 ÿ Profit is maximized where s = 100. ÿ This example demonstrates that market mechanisms can substitute for regulations.

250 | Chapter 15 STRATEGY SESSION: Moral Hazard in the Financial Market: The Rescue of the Investment Bank Bear Stearns Discussion Question 1. Did the Fed create expectations of future bailouts and thus increase the likelihood that such bailouts will be needed? Chapter 15: Problem Solutions 1. Your business generates the following profits (these are stated before compensation is paid to the manager):

Low effort High effort

Low Demand [0.3]

Medium Demand [0.4]

High Demand [0.3]

$5 million $7 million

$10 million $12 million

$15 million $17 million

You see that profit depends on both the level of effort chosen by the manager and the level of demand. The demand level is random, and the probabilities of each demand level are shown. So with low effort, expected profit is $10 million; with high effort, it is $12 million. The manager has a utility function that is either Utility = (Wealth)0.5 if effort is low or Utility = (Wealth)0.5 − 100 if effort is high Therefore −100 is the disutility of effort. You are interested in maximizing the expected profit after deduction of compensation. You consider three different compensation packages: • A flat salary of $575,000 • A payment of 6% of profit • A flat payment of $500,000 plus half of any profit in excess of $15 million Which compensation do you choose? Solution: Write the utility function as U(W) = W 0.5 if effort is low and U(W) = W 0.5 − 100 if effort is high where W denotes the level of wealth. Case 1: A flat salary A flat salary of $575,000 yields Utility: U(575,000) = (575,000)0.5 = $758.29 if effort is low and

Principal–Agent Issues and Managerial Compensation | 251 Utility: U(575,000) = −100 + (575,000)0.5 = $658.29 if effort is high. Therefore, it is clear that the manager will choose to provide a low effort assuming she is maximizing expected utility. Now, given this choice, the owner’s expected profit is given by * 0.3(5million) + 0.4(10) million) + 0.3 (15million = $10,000,000 −$575,000 = $9,425,000. Case 2: 6% of profit The manager decides whether to provide a low effort or a high effort. If the manager provides a low effort, her expected utility is given by EU(W) = 0.3U(0.6($5 million)) + 0.4U(0.6($10 million)) + 0.3U(0.6($15million)) = 0.3(0.6(5 million))0.5 + 0.4(0.6(10 million))0.5 + 0.3(0.6(15 million))0.5 = 758.76042 If the manager provides a high effort, her expected utility is given by EU(W) = −100 + 3U(0.6($7 million)) + 0.4U(0.6($12 million)) + 0.3U(0.6($17 million)) = 0.3(0.6(7million))0.5 + 0.4(0.6(12 million))0.5 + 0.3(0.6(17 million))0.5 = 736.8186 The manager will then choose a low effort, which gives her a higher expected profit, and the expected profit of the firm, given this choice by the manager, is given by $10 million − .06(10 million) = $9.4 million Case 3: $500,000 plus half of profits in excess of $15 million If the manager chooses a low effort, there is no profit level greater than $15 million. Therefore, the manager’s compensation is just $500,000. Her utility is given by, for low effort U(500,000) = (500,000)0.5 = 707.1067 If the manager chooses a high effort, then the manager’s compensation is $500,000 with a probability of 0.7 and 0.5(17million − $15 million) + $0.5 million = $2.5 million with probability 0.3. Her utility is given by EU(W) = −100 + 0.7($500,000)0.5 + 0.3($2.5 million)0.5 = 762.40 Therefore, since the manager is an expected utility maximizer, she will choose high effort. The owner’s net expected profit, given that the manager has chosen to provide high effort, is given by expected profit minus total compensation. That is, expected profit = $12 million − [$0.5m + 0.3(0.5)($2 million)] = $12 million − $800,000 = $11.2 million.

252 | Chapter 15 Given these three profit streams, it is clear that the owner will choose the third package plan given that it leads to the highest net profit. 2. Suppose the typical Florida resident has wealth of $500,000, of which his or her home is worth $100,000. Unfortunately, Florida is infamous for its hurricanes, and it is believed there is a 10% chance of a hurricane that could totally destroy a house (a loss of $100,000). However, it is possible to retrofit the house with various protective devices (shutters, roof bolts, and so on) for a cost of $2,000. This reduces the 10% chance of a loss of $100,000 to a 5% chance of a loss of $50,000. The homeowner must decide whether to retrofit and thereby reduce the expected loss. The problem for an insurance company is that it does not know whether the retrofit will be installed and therefore cannot quote a premium conditioned on the policyholder choosing this action. Nevertheless, the insurance company offers the following two policies from which the homeowner can choose: (1) The premium for insurance covering total loss is $12,000 or (2) the premium for insurance covering only 50% of loss is $1,500. The typical homeowner has a utility function equal to the square root of wealth. Will the homeowner retrofit the house, and which insurance policy will the homeowner buy? Will the insurance company make a profit (on average) given the homeowner’s choice? Solution: Write the utility function as U(W) = W0.5, where W denotes wealth. Suppose the homeowner buys the full coverage for a premium of $12,000 and does not retrofit. Wealth will be $500,000 minus a premium of $12,000. Thus, wealth will not change whether or not a loss occurs (since the loss will be fully paid by the insurers). The expected utility is EU = (500,000 − 12,000)0.5 = 698.57 If the policyholder chooses this option, the insurer’s expected profit is the premium minus expected claims payments: Ep = 12,000 − (0.1)(100,000) = $2,000 If the homeowner buys the full coverage for a premium of $12,000 but does retrofit, wealth will be $500,000 minus a premium of $12,000 minus the $2,000 cost of retrofit, giving a total of $486,000, whether or not a loss occurs. Thus, the expected utility and expected profit are EU = (500,000 − 12,000 − 2,000)0.5 = 697.14 Ep = 12,000 − (0.05)(100,000) = $7,000 If the homeowner buys the 50% coverage and does not retrofit, she faces a 0.1 chance of realizing the $100,000 loss, of which only half will be paid by the insurer.

Principal–Agent Issues and Managerial Compensation | 253 EU = 703.73 Ep = 1,500 − (0.05)(25,000) = $250 Thus, the homeowner will choose to retrofit and buy the 50% insurance coverage because this offers her the highest expected utility. Given this choice by the homeowner, the insurance company will make an expected profit of $250. 3. The expected profit of your firm is 1,000, plus 500 if the manager works hard. The manager receives a flat salary of 100 plus a portion x of any profit in excess of 1,300. The manager’s utility function is EU = [(compensation)0.5] if she does not work hard EU = [(compensation)0.5 − 1] if she works hard What portion x must be paid to the manager to ensure that she chooses to work hard? This new compensation package must be competitive with the 100 flat salary. Solution: If the manager does not work hard, she will receive a utility of [100]0.5 = 10. If the manager works hard, she will receive [100 + x(15,000 − 13,000)]0.5 − 1. To calculate the minimum level of x necessary to ensure that expected compensation will be higher with hard work, set the expected utility with hard work equal to that without hard work: [100 + x(15,000 − 13,000)]0.5 − 1 = 10 [100 + 200x]0.5 = 11 Square both sides to get 100 + 200x = 112 = 121. That is x = 21/200 = 0.105 Thus, if the manager gets a little over 0.105 times equity in excess of 13,000, she will work hard. 4. A firm used to have productive assets that generated an income stream with a present value (PV) of 3,000. However, a fire occurred, and most of those assets were destroyed. The remaining undamaged assets produce an income stream that has a present value of only 1,000. Therefore the fire has reduced the value of the firm from 3,000 to 1,000. The firm could reconstruct the damaged assets for a capital cost of 1,500, which would restore the income stream to its pre-loss level (PV = 3,000). The firm has existing debt of 2,000, which is a senior claim. Would the shareholders choose to reinvest by issuing new equity to pay for the loss, or are they better off walking away from the firm? Would the decision made by the shareholders be in the best interests of the bondholders? In answering this question, remember that the shareholders have limited liability, and therefore the share value cannot be negative.

254 | Chapter 15 Solution: The decisions and payouts can be represented as shown in the following table. Note that the total gain in value for the firm is $2,000. If the firm does not reinvest, then total value is $1,000 but $2,000 is owed in debt. The firm is bankrupt, so the shareholders will use limited liability to walk away and equity will be worth zero. In this case, the firm defaults on the debt and it will be worth only $1,000 (there is simply not enough value in the firm to pay the whole $2,000 owing to the creditors). If the shareholders decide to reinvest, they must raise $1,500. This seems like a good deal to the firm as a whole since the investment of $1,500 raises the total value by $2,000 (from $1,000 to $3,000). But where does this gain go? First, the creditors can now be paid in full (they are senior and have priority over shareholders). This leaves a gain of only $1,000 for the shareholders. From the shareholders’ perspective, they have paid $1,500 to raise the value of their equity by only $1,000 (from zero to $1,000), yielding a net loss of $500. The shareholders would not choose to reinvest. This leaves the creditors in a default situation, getting only 50 cents payback on each dollar of debt. Value of firm, debt, and equity with and without reinvestment Decision Reinvest Not reinvest Gain Cost of reinvestment Net gain to stakeholders

Firm

Debt

Equity

$3,000 $1,000 $2,000

$2,000 $1,000 $1,000

$1,000 $0 $1,000 $1,500 $−500

$1,000

5. CareLess Industries has two divisions. Division 1 makes cleaning products, and the net worth of this division (PV of cash flows) is 500. Division 2 makes a chemical product. The net worth of division 2 is 300, absent any potential liability. However, there is a chance that division 2 could have a 700 liability for pollution damage. The potential victims have no contractual relationship with the firm. The probability of such a loss is 0.2/(1 + s), where s is the amount the firm spends on safety. The firm must choose the level of s. If you could sell off division 2, would you do so? What is the gain from splitting the firm in this way? Assume a separated division 2 (as a standalone firm) is protected by limited liability. Note also that the derivative of a/ (1 + s) with respect to s is −a/(1 + s)2; that is, [a/(1 + s)]/  s = −a/(1 + s)2. Solution: The total value of the firm is given by V = 500 + 300 − 700[0.2/(1 + s)] − s = 800 − (140/(1 + s)) −s

Principal–Agent Issues and Managerial Compensation | 255 Now, the goal is to find the value of s that maximizes the value of the firm. This is a simple unconstrained maximization problem. The first-order condition indicates taking the first derivative of the objective function and setting it equal to zero and then solving for s. We have dV/ds = 140/(1 + s)2 −1 = 0 (using the quotient rule for derivative). That is, 140/(1 + s)2 = 1 or equivalently (1+s)2 = 140. Taking the square root of both sides and subtracting 1 gives s = 10.8321. The (optimum) value of the firm is 800 − (140/11.8321) − 10.8321 = 777.3 The stand-alone value of division 2 is V = 300 + 300 − 300[0.2/(1 + s)] − s = 300 − (60/(1 + s)) −s Using the same method as above gives s = 6.746, and the optimum value is 285.5. Also, division 1 is riskless and has a stand-alone value of 500. Therefore, the total value with split up is 500 + 285.5 = 785.5 and the corresponding gain from split up is 785.5 − 777.34 = 8.21 6. SubAquatics (SA) sells scuba diving equipment. Its clients typically read specialist journals and are well informed about the price, reliability, and safety of SA and competitors’ products. SA has estimated that of 100,000 units sold each year at a price of $100 each, there are 4/(1 − s) fatal accidents due to defective equipment. The value s is the amount spent by SA on safety in millions of dollars. a. Assuming that SA is fully liable for such accidents and that the average settlement of each fatal accident is $1 million, how much should SA spend on safety? Now assume that SA can escape this liability by selling its products at a lower price under a contract that allocates all responsibility for accidents to the purchaser (assume that courts enforce such contracts). If SA spends s (expressed in millions of dollars) on safety, the expected cost of accidents to any consumer is [4/(1 − s)]($1m/100,000) = $40/(1 + s). Note that consumers are willing to pay $100 when all liability is assumed by SA (assuming consumers are risk-neutral). b. How much would consumers be willing to pay when they bear the cost of accidents? c. How much would SA spend on safety? d. Assuming that customers cannot observe the level of safety and there is no liability law, how much would SA spend on safety and how much would customers pay for the product?

256 | Chapter 15 Solution: a. SA’s spending on safety is $1 million given that the company is fully liable and that the marginal cost of safety is 1. b. SA would not spend on safety since it cannot be rewarded with a higher price. Customers would rationally anticipate that SA would spend zero on safety and the number of accidents would increase to 4/(1 − 0) = 4. Customers would buy the product for $100 − $40/(1 − 0) = $60. c. The company is a profit maximizer. Without a product liability law, SA would not spend on safety since it cannot be rewarded with a higher price and therefore sets s = $0. d. If consumers cannot observe the level of safety, then there is a moral hazard problem. Still, without a product liability law, the company would not spend on safety. In such a case, given the moral hazard problem, customers will still pay $100 for the product. 7. A firm has existing operations that generate an earnings stream with a present value, PV, of 300 or 600, each with 0.5 probability. The firm has 250 in existing debt. The firm wishes to undertake one of the following mutually exclusive new investments: Capital Cost Project A Project B

400 400

PV of Earnings

NPV

420 0; probability 0.5 or 700; probability 0.5

−20 −50

The capital cost of each project (400) is financed with new junior debt (face value 400). Is there an asset substitution problem? (Will shareholders try to choose the lower NPV project?) Show whether any asset substitution problem would disappear if the new project were financed with an equity issue of 400 instead of new debt. Solution: Analysis with debt financing: There is an asset-substitution problem. To show this, we should value the firm as a whole, and each of the stakeholders’ claims, first assuming A is chosen, then assuming B is chosen. We can then see which project selection leads to the higher value of equity. This is the one shareholders would naturally favor. Value of the firm if project A is chosen: First note that the value of the firm will be either 720 (300 from existing operations and 420 from the new project) or 1,020 (600 from existing operations and 420 from the new project), depending on the success of existing operations. This value must be divided up by paying off first old debt, next new debt, and finally equity.

Principal–Agent Issues and Managerial Compensation | 257 Value of the firm = 0.5(720 + 1,020) = 870 Old debt = 0.5(250 + 250) = 250 New debt = 0.5(400 + 400) = 400 Equity = 0.5(70 + 370) = 220 Value of the firm if project B is chosen: The value of the firm will be 300, 600, 1,000, or 1,300. These figures come from the different combinations of the two possible values for the existing operations (300 and 600) and the two values for the new project (0 and 700). Value of the firm = 0.25(300 + 600 + 1,000 + 1,300) = 800 Old debt = 0.25(250 + 250 + 250 + 250) = 250 New debt = 0.25(50 + 350 + 400 + 400) = 300 Equity = 0.25(0 + 0 + 350 + 650) = 250 Shareholders would like to choose B if they had the creditors’ money. But since investors would pay only 300 for new debt, this cannot be funded. So neither project can be undertaken if debt financing is used. Analysis with equity financing: The analysis can be repeated using equity financing for the new project. The values of the firm will be the same as above, but these will be allocated first to the existing debt (for which 250 is owed) and any residual will accrue to equity. Value of the firm if project A is chosen: Value of the firm = 0.5(720 + 1,020) = 870 Old debt = 0.5(250 + 250) = 250 Equity = 0.5(470 + 770) = 620 Value of the firm if project B is chosen: Value of the firm = 0.25(300 + 600 + 1,000 + 1,300) = 800 Old debt = 0.25(250 + 250 + 250 + 250) = 250 Equity = 0.25(50 + 350 + 750 + 1,050) = 550 Now shareholders will naturally choose project A, with its higher net present value, since it has the higher equity value. The asset-substitution problem is solved.

CHAPTER 16

Adverse Selection

Lecture Notes 1. Introduction •

Objectives ÿ To explain how managers can use their informational advantage to increase performance ÿ To illustrate how managers at an informational disadvantage can mitigate the effect by using creative defenses ÿ To explore the market for “lemons” ÿ To examine adverse selection in the automobile insurance market ÿ To explore the market for annuities ÿ To show how to potentially resolve adverse selection through a selfselection menu ÿ To show how warranties and education can be used as signals in the product and job market, respectively

2. The Market for “Lemons” • •

•

258

Lemon: A used car that turns out to have many faults not apparent at the time of sale (hence the sour taste) Akerlof’s idea: A systematic process may ensure that a disproportionate number of lemons turn up in the used car market due to information differences between buyers and sellers. Akerlof’s model ÿ Used cars are either gems (which is good) or lemons (which is bad). ÿ Information asymmetry means that sellers have more information about the quality of the car they are selling than buyers do.

Adverse Selection | 259 ÿ Assumption: The buyer is willing to pay, at most, the value of a car of average quality. ÿ Owners of gems are less willing to sell at the average price because they know that gems are worth more than the average. ÿ Owners of lemons are eager to sell at the average price because they know that lemons are worth less than the average. ÿ As a result, most of the used cars on the market are lemons. Eventually, the average price of a used car will be equal to the value of a lemon because no one will sell a gem. ÿ This is a case of adverse selection in that the market dynamic leads to only lemons being offered for sale on the used car market. PROBLEM SOLVED: Equilibrium in the Used Car Market Discussion Questions 1. Market for Toyota Camrys for 2006 * Some used cars are lemons, some are gems. * Consumers’ reservation price for gems: $10,000 * Consumers’ reservation price for lemons: $5,000 The problem: Consumers do not know which car is a lemon and which one is a gem. * They are willing to pay the average price of $7,500. * Owners of gems would not sell for $7,500. * Owners of lemons would be happy to sell for $7,500. * Cheap talk is not convincing because it can be mimicked by the owners of lemons. The fact: * Buyers are not totally ignorant, and sellers do not know everything. * There are mechanisms involving sophisticated signaling to separate the gems from the lemons. 2. If you know that your car is a gem, and you want to get an appropriate price when you sell it, how can you convince the buyer that it is a gem? Answer: Used car dealers often sell cars that are “certified” and carry a limited warranty for some period of time. The warranty is a credible signal that a car is a gem. The same method could be used by an individual. 3. Adverse Selection in Automobile Insurance •

Insurance markets to be considered: ÿ Automobile insurance ÿ Annuities ÿ Life insurance

260 | Chapter 16 •

•

•

Model ÿ Drivers are either high risk or low risk. ÿ Both types of drivers start with a wealth level of 125, and a loss can reduce wealth to 25. ÿ High-risk drivers have a loss with probability 0.75, and their expected loss is therefore (0.75)(100) = 75. ÿ Low-risk drivers have a loss with probability 0.25, and their expected loss is therefore (0.25)(100) = 25. Perfect Information ÿ We claim that if the insurer could distinguish between the two groups, managers could charge competitive premiums, and each group buy insurance. So, ÿ Given perfect information, high-risk drivers will be charged 75, and low-risk drivers will be charged 25, and because both are risk-averse, both will buy insurance. ÿ Assuming that U = (wealth)0.5 for both types of drivers: 0.5 * High risk: Utility without insurance U = (0.25)(125) + 0.75) 0.5 (25) = 6.545 0.5 * High risk: Utility with insurance U = (125 − 75) = 7.071 0.5 * Low risk: Utility without insurance U = (0.75)(125) + (0.25) 0.5 (25) = 9.635 0.5 * Low risk: Utility with insurance U = (125 − 25) = 10 Implication: High-risk group: Buy insurance because utility with insurance is larger than utility without insurance. ÿ Illustration: Figure 16.1: Adverse Selection in Automobile Insurance Asymmetric Information ÿ If the insurer cannot distinguish between high- and low-risk drivers, and there are equal numbers of each, then the insurer can break even by charging the average premium of 0.5(25 + 75) = 50. Question: Will each group now continue to buy insurance? * High-risk group: Utility with insurance U = (125 − 50)0.5 = 8.660 Utility without insurance U = (0.25)(125)0.5 + (0.75)(25)0.5 = 6.545 High-risk drivers will buy insurance at this price. * Low-risk group: Utility with insurance U = (125 − 50)0.5 = 8.660 Utility without insurance U = (0.75)(125)0.5 + (0.25)(25)0.5 = 9.635 Low-risk drivers will not buy insurance. Since only high-risk drivers will buy insurance, the insurance premium must increase to 75. But good drivers will not buy the policy. Utility with insurance U = (125 − 75)0.5 = 7.071 Utility without insurance U = (0.75)(125)0.5 + (0.25)(25)0.5 = 9.63

Adverse Selection | 261 ÿ Insurers can compete by collecting better information so that lower premiums can be charged to low-risk drivers, but perfect information is not attainable. STRATEGY SESSION: Adverse Selection and the National Health Care Debate: The Federal Mandate on Compulsory Insurance • • • •

The case for mandating health insurance appeals to adverse selection. If insurance is not mandated, some will buy it and some will not. The lack of risk-related premiums implies that those in good health will subsidize those in poor health. The Congressional Budget Office (CBO): Even with a mandate, there would still be 20 million uninsured. Moral hazard problem: When people are insured, they tend to pay less attention to costs.

STRATEGY SESSION: Information Asymmetries and Brownfields Discussion Questions 1. There are two asymmetries at work in this situation. One is informational. The seller is likely to know more about how polluted a property is than the buyer. The second is wealth. The seller is likely to have less wealth than the buyer and therefore has more to lose if a cleanup is necessary. Which of these do you think is more important in causing a property to be abandoned? 2. The second way managers can help restore markets is through strategic design. Is there any way to induce policyholders to reveal information about themselves in a credible way? Illustration: Two choices: A, B 3. Can managers design these alternatives so that only a low-risk driver will choose A and only high-risk drivers will choose B? If managers can develop such a self-selecting menu, they create the socalled separating equilibrium. 4. The Market for Annuities •

An annuity is a financial instrument that is purchased for a fixed amount and pays a fixed amount per time period to a person over his or her entire life. ÿ Question: What are the incentives of the managers at the annuity firm?

262 | Chapter 16 •

The Cost of Information Asymmetries in Annuity Markets ÿ Illustration: * Managers at the annuity firm sell many annuities. * Consider 1,000 65-year-old women, all of whom buy annuities. * Two cases: 1. Perfect information: Health status of each individual is known both to that person and to managers at the annuity firm. 2. Asymmetric information: The person knows her health status, but the managers do not. ÿ Annuity markets with full information: Managers have perfect information. * Population health distribution is as follows: 25% in poor health with a life expectancy of 5 years 50% in average health with a life expectancy of 15 years 25% in excellent health with a life expectancy of 25 years * *

All members of the population have $300,000 in capital to buy an annuity. Population annuity payments are as follows: Poor health: $60,000 per year Average health: $20,000 per year Excellent health: $12,000 per year

ÿ Annuity markets with asymmetric information: Adverse selection * Suppose that everyone gets a payment of $20,000 per year until death but pays a premium of $300,000. Total income for the firm: (1,000)(300,000) = $300,000,000 Annuity payments: 250 die after 5 years and pay = (250)($20,000)(5) = $25,000,000 500 die after 15 years and pay = (500)($20,000)(25) = $150,000,000 250 die after 25 years and pay = (250)($20,000)(25) = $125,000,000 Total payments = $300,000,000 * *

Therefore, the firm breaks even. Alternatively, proceed as follows: ° Poor health: Pay $300,000 for the annuity and receive (5) ($20,000) = $100,000 The annuity firm makes a profit of $200,000 on each of these individuals. ° Average health: Pay $300,000 and receive (15)($20,000) = $300,000 The annuity firm breaks even on these individuals. ° Excellent health: Pay $300,000 and receive (25)($20,000) = $500,000

Adverse Selection | 263

*

* *

The annuity firm loses $200,000 on each of these individuals. Average cost: (0.25)(100) + (0.50)(300) + (0.25)(500) = $300,000 Question: Who buys the annuity? ° People in poor health are unlikely to buy the annuity paying $20,000. ° Average health: People may be inclined to buy because annuities remove some of the uncertainty. ° Excellent health: A great deal: They pay $300,000 and get back $500,000. Conclusion: Only those in average or better-than-average health will buy the annuity. The average cost goes up to (2/3)(300) + (1/3)(500) = $366,667

If the price is increased (or payment reduced), then only people in excellent health will buy the annuity, and it will cost $500,000 or will pay $12,000 per year. ÿ Evidence of adverse selection in the annuity and life insurance markets * Evidence indicates that people who buy annuities tend to live longer, on average, than the general population. This supports the premise that adverse selection makes a difference in purchasing annuities. ÿ The absence of adverse selection in life insurance * Annuities insure people against living “too long” and running out of money. * Life insurance protects the survivors of people who “die too soon.” *

ÿ Do managers at health insurance firms have less information than policyholders? * Those who seek life insurance are required to undergo a medical examination. * Those who have life insurance tend to live longer than the average population, suggesting an absence of adverse selection in life insurance. STRATEGY SESSION: Adverse Selection in the Federal Government Prescription Drug Plan Discussion Question 1. What effect would universal health insurance coverage have on adverse selection issues that exist in the Medicare prescription drug plan (PDP)? 5. Resolving Adverse Selection through Self-Selection • •

Akerlof described the adverse selection problem. Joseph Stiglitz and Michael Rothschild laid out an elegant solution to adverse selection.

264 | Chapter 16 •

The idea: Asking is not enough.

•

Question: How can an uninformed manager obtain credible information?

•

Example: An automobile insurer does know the following: ÿ Some drivers are high risk and some are low risk. ÿ The drivers themselves know whether they are good or bad drivers. * Manager’s task: To design policies that separate behavior between good and bad drivers Managers induce drivers to reveal their risk type by the policy they choose. ÿ A choice problem: Full insurance or a high deductible: * Full insurance: When every loss is paid in full * Deductible: When the insurer does not pay the full loss but pays the loss minus some fixed amount * Self-selection menu: When buyers act in their own self-interest and use their private information about their loss probabilities to select policies * Example 1 ° Policy A: High premium and full insurance (designed to appeal to high-risk drivers) ° Policy B: Low premium and high deductible (designed to appeal to low-risk drivers) ° Suppose the policy has a $2,000 deductible. Suppose there is a $20,000 loss. ° The insurer will pay $20,000 − $2,000 = $18,000. ° If the loss is less than $2,000, the insurer pays nothing. ° The idea: The high deductible can be a big deterrent for bad drivers. ° Good drivers select the cheaper deductible policy, and bad drivers select the more expensive, full coverage. * Example 2 ° Policy C: High premium that is constant from year to year (designed to appeal to high-risk drivers) ° Policy D: High premium at first that declines from year to year if there are no claims (designed to appeal to low-risk drivers) Simple Adverse Selection ÿ Policy 1: Premium of 75 and full insurance (designed to appeal to highrisk drivers); that is, pays for the loss of 100 if a crash occurs. Then

•

wealth: always 125 − 75 = 50 if buying insurance. ÿ Policy 2: Premium is 2.5 and pays a fixed sum of 10 (designed to appeal to low-risk drivers) if a loss occurs. Utility function: U(W) = W 0.5 *

High-risk drivers:

Adverse Selection | 265 Utility without insurance U = (0.25)(125)0.5 + 0.75) (125 − 100)0.5 = 6.545 Utility with policy 1 = (125 − 75)0.5 = 7.071 Utility with policy 2 = (0.25)(125 − 2.5)0.5 + 0.75(125 − 100 − 2.5 + 10)0.5 = 7.043 They will choose policy 1. * Low-risk drivers: Utility with policy 1 = (125 − 75)0.5 = 7.071 Utility with policy 2 = (0.25)(125 − 100 − 2.5 + 10)0.5 + 0.75(125 − 2.5)0.5 = 9.726 They will choose policy 2. The two types of drivers choose different policies. Call this a “self-selection menu.” * Only high-risk drivers buy policy 1. Expected claims = 0.75(100) = 75, which matches the premium of 75. * Only low-risk drivers buy policy 2. Expected claims = 0.25(10) = 2.5, which matches the premium of 2.5. This solution to the adverse selection problem is called a separating equilibrium: induces policyholders to select their respective risk types.

6. Using Education as a Signal: Adverse Selection in the Job Market •

• • •

Introduced by Michael Spence ÿ Question: How can managers predict which job applicants will have good work skills? Information asymmetry in job markets: Applicants know more about their job skills, abilities, and ambitions than a potential employer. Managers need a credible method for separating those who know they have good job skills from those who lack them. Premise: Highly skilled applicants can complete courses at a lower cost than applicants with low skills. Therefore, the employer can get applicants to self-select based on the number of courses they are required to complete to get a higher-paying job.

7. Using Warranties As Signals: Adverse Selection in the Product Market •

How Managers Can Construct Warranties to Mitigate Adverse Selection ÿ Experience goods: Goods that can only be evaluated with regard to their quality after they have been consumed.

266 | Chapter 16 ÿ Examples: Autos, appliances, consumer electronics ÿ There is an incentive for producers of high-quality goods to signal their  quality and increase the willingness of buyers to pay a higher price. ÿ Product warranties can accomplish this goal by acting as separating mechanisms. ÿ If constructed correctly, a warranty is a credible signal of product quality. ÿ Model—Two products: A high-quality and a low-quality product by two rival firms * PH = consumer reservation price for high-quality good * PL = consumer reservation price for low-quality good (PL , PH) * CH = cost of producing the high-quality good * CL = cost of producing the low-quality good (CL , CH). * Y = number of years of warranty * Warranty cost of a high-quality good is WH YH, and that of a lowquality good is WL YL , where WL . WH ÿ Scenario 1: Consumers perceive any good with a warranty (YH) to be a high-quality good. The longest warranty a high-quality producer can afford to give is one where profit from signaling high quality just equals the profit from not signaling at all, that is, low quality. * Profit from a high-quality good with a warranty is PH − CH − WHYH. * Profit from a high-quality good without a warranty is PL − CH. The high-quality producer will not issue a warranty if PL − CH , PH − CH − WHYH. * Profit from a low-quality good with a warranty is PH − CL − YLWL . * Profit from a low-quality good without a warranty is PL − CL. The lowquality producer will not issue a warranty if PH − CL − YLWL , PL − CL. * Credible warranty: (PH − PL)/WH . Y . (PH − PL)/WH ÿ Scenario 2: Consumers perceive the good with the longer warranty (Y ) as the high-quality good. * The longest warranty a low-quality producer can afford to offer is YL , where YL = (PH − PL)/WL , obtained by setting PH − CL − YLWL = PL − CL (1) *

The longest warranty a high-quality producer can afford to offer is YH, where YH = (PH − PL)/WH, obtained by setting PH − CH − WHYH = PL − CH (2)

* From equation (1), we have PH − PL = YLWL * From equation (2), we have PH − PL = WHYH *

This implies WHYH = YLWL . But WL . WH, by assumption. Hence, we have WHYH , WLYH , = . YLWL , WLYH , = . YH . WL .

Adverse Selection | 267 Therefore, the high-quality producer can afford to warranty the low-quality producer by setting YH = YL + a small amount. * In practice, the high-quality product will have a warranty YH = YL + 1, and the low-quality product will not have a warranty. PROBLEM SOLVED: Determining Warranty Length Discussion Questions Suppose that a third company, Whitman Homes, enters the market. They can build a home for $300,000 that will sell for $600,000 if it is perceived as high quality and $400,000 if it is perceived as low quality. The warranty cost on their homes is $12,500 per year. 1. What is the break-even warranty length for Whitman Homes? Answer: The break-even warranty length is 16 years. 2. Will Whitman Homes choose to implement a warranty, and if so, how long a warranty will be implemented? Answer: Since Tole Brothers’s break-even warranty is 10 years, Whitman will implement a warranty to 10+ years. The other housing manufacturers will then choose not to offer a warranty.

Chapter 16: Problem Solutions 1. Sellers of used cars know the cars’ quality, but buyers do not. Imagine that used Toyota Corollas are worth $10,000 if they are of high quality and $5,000 if they are of poor quality. Although buyers may not know the quality of a specific car, they do know that 25% will be of poor quality. In such a market, what cars will be sold on the secondhand market and at what price? Solution: Given the information buyers have about the used cars, namely, the proportion of bad and good cars, they will not be willing to pay $10,000 for a used car. One price they can consider paying is the average price given by 0.25($5,000) + 0.75($10,000) = $8,750. The problem is that, assuming sellers of good-quality cars know about the quality of their cars, they will unlikely be willing to sell at the average price of $8,750 given that they know that their car is really worth $10,000. Therefore, only sellers with lemons will be on the market and will be willing to settle for a price of $5,000. Given that buyers know that only lemons will be on the market, they will be willing to pay $5,000 for any used Toyota Corolla. Hence, it can be seen that only lemons will be sold in the secondhand car market and the price will be $5,000.

268 | Chapter 16 2. The market for digital cameras is relatively new. Ajax Inc. produces what it regards as a high-quality digital camera. Knockoff Inc. produces what it regards as a low-quality digital camera. However, because the market is so new, reputations for quality have not yet developed, and consumers cannot tell the quality difference between an Ajax digital and a Knockoff digital just by looking at them. If consumers knew the difference, they’d be willing to pay $200 for a highquality camera, and they’d be willing to pay $100 for a low-quality camera. It costs Ajax $85 to produce a high-quality camera, and it costs Knockoff $55 to produce a low-quality camera. A recent MBA hire at Ajax suggests that Ajax could differentiate its camera from Knockoff’s by offering a fullcoverage warranty (which would fully cover any defect in the camera at no cost to the customer). The MBA estimates that it would cost Ajax $20 per year to offer such a warranty. The MBA also estimates that it would cost Knockoff $40 per year should Knockoff attempt to copy Ajax’s warranty strategy. Consumers will feel that the camera with the longest warranty is high-quality and that with the shortest warranty is low-quality. The camera companies want to maximize the profit per camera. What is Ajax’s profit per camera in the digital camera market? Solution: With no warranties, the profit on a high-quality camera is $140 and the profit on a low-quality camera is $75. However, consumers cannot distinguish between high- and low-quality cameras, so they buy low-quality cameras. With the warranty, the profit on a high-quality camera falls to $120, but Ajax is able to attract customers away from Knockoff, whose profits without the warranty remain at $75. If Knockoff offers a warranty, its profits fall to $35 per camera. 3. No-State Insurance Company has made the following estimate of auto damage for several groups of potential customers who own cars worth $10,000. There are an equal number of customers in each group. No-State is risk-neutral. Group

Initial Value of Car

Probability of Accident that Devalues Car to $5,000

A B C

$10,000 $10,000 $10,000

0.2 0.3 0.4

State regulation mandates that every customer must pay the same premium regardless of his or her group, and this premium must be sufficient to cover all expected claims from those who purchase insurance from No-State. There are no additional costs to the company other than paying off claims.

Adverse Selection | 269 All consumers have the following utility function (U): U = W 0.5 W is the consumer’s wealth as represented by the value of the car. What premium should No-State offer for full-coverage insurance? Solution: Notice that given the concave shape of the utility function, consumers are risk-averse. Given that the insurer must charge every consumer the same price, they will charge $2,000 for the policy. This allows them to cover their losses on the type C drivers, but it prices the type A and B drivers out of the market. 4. Some people are good drivers, and others are bad drivers. The former have a 10% chance of crashing their cars, and the latter have a 30% chance. All have a total wealth of 400, but this will fall to 100 if they crash their cars. In other words, each will lose 300 of wealth if they crash. You are an insurance company manager who wishes to offer a pair of policies to all drivers. Each policy is designed to break even (zero profit) given the people who choose to buy that policy. The first policy has a premium of 90 and covers all losses (it will pay 300 in the event of a crash). The second policy has a premium of 5 and will pay 50 in the event of a crash. Who will buy which policy? Will the insurance company make a profit, break even, or lose money? Each person has a utility function as follows: Utility = (Wealth)0.5 Solution: * Good drivers: 10% chance of a loss and 90% chance of no loss * Bad drivers: 30% chance of a loss and 70% chance of no loss * Wealth: 400 * Potential loss: 300 * Policy 1: Full insurance: Premium 90 Deductible: 0 Premium 5 Deductible: 50 * Policy 2: Deductible: Utility function: U(W) = W 0.5 ÿ Good drivers: Utility with no insurance = 0.90(400)0.5 + 0.10(100)0.5 = 19 Policy 1 utility = 0.90(400 − 90)0.5 + 0.10(400 − 90)0.5 = 17.6068 Policy 2 utility = 0.90(400 − 5)0.5 + 0.10(400 − 300 − 5 + 50)0.5 = 19.0913

270 | Chapter 16 Therefore, good drivers, being expected utility maximizers, will choose policy 2. ÿ Bad drivers: Utility with no insurance = 0.70(400)0.5 + 0.30(100)0.5 = 17 Policy 1 utility = 0.70(400 − 90)0.5 + 0.30(400 − 90)0.5 = 17.6068 Policy 2 utility = 0.70(400 − 5)0.5 + 0.30(400 − 300 − 5 + 50)0.5 = 17.5247 Bad drivers will choose policy 1. Expected claims from good drivers are 0.10(50) = 5. Hence, the firm will break even. Expected claims from bad drivers are 0.30(300) = 90 and the firm breaks even. 5. Consider a market for annuities for 70-year-old men in which people differ in terms of both their expected remaining years of life and their risk preferences. Of the population of 200, half have a life expectancy of 9 years, and the remaining half have a life expectancy of 11 years. We can express risk preference in the following way. The risk people are worried about is that of running out of wealth before they die. The more risk-averse you are, the higher the up-front price you are willing to pay for the annuity. More risk-averse people are willing to pay 1.3 times x times A, where x is the expected years of life remaining and A is the dollar amount paid each year to the annuitant. Less risk-averse people are willing to pay only 1.1 times x times A. Assume that of the 100 people in each health group, half are more risk-averse and half less risk-averse. The annuity firm sells an annuity of $50,000 per year for as long as the buyer lives, and the price of the annuity is $550,000. Because the annuity firm cannot tell whether any applicant has a short or long life expectancy, it must accept any application for its product. What is the expected profit of the annuity firm? (You may ignore discounting in this example.) Solution: Given the information, we have the following. The product will be purchased by those in good health as well as those in poor health. Both groups are composed of risk-averse individuals. Therefore, Expected utility for more risk-averse in poor health = 50($550,000) − 50(9) (50,000) = $5,000,000 Expected utility for less risk-averse in good health = 50($550,000) − 50(11) (50,000) = $0 Expected utility for more risk-averse in poor health = 50($550,000) − 50(11) (50,000) = $0 Firm’s total profit is then given by $5,000,000.

CHAPTER 17

Government and Business

Lecture Notes 1. Introduction •

Objectives ÿ To ensure that managers understand the legal environment of business such as: * Antitrust laws * Fair trade laws * Employment laws * Safety laws * Environmental issues and laws * Securities laws * Patents and copyrights ÿ To ensure that managers understand the role of government in the business environment * Economic regulation (banks) * Noneconomic regulation (safety) * Tax and subsidy policies * Price controls * Spending * Infrastructure maintenance * Controls in cases of market failure (externalities like pollution) ÿ To discuss the role of government in providing public goods and in correcting market failures

271

272 | Chapter 17 STRATEGY SESSION: Government Actions and the Financial Crisis of 2008 Discussion Questions As this book is being written, financial markets continue to teeter on the brink of catastrophe, according to government officials and the business media. Unemployment has risen to 8.1% and government deficits are at record levels. 1. What is the current state of financial markets? Have they recovered, or do they continue in crisis? 2. High levels of deficit-supported government spending combined with loose monetary policy have led to accelerating inflation in the past. Has this occurred, or have consumer prices stabilized? 3. Prevailing opinion suggested that the U.S. economy was likely to experience a prolonged recession with unemployment rates rising above 10%. Has that occurred? Is the economy in recovery, or has it stagnated? 4. What is the role of government and its impact on managerial behavior? 5. What is the role of government and its impact as a result of the crisis? Notes: 1. Monetary policy action by the Federal Reserve Bank: To get credit flowing so banks can lend money for working capital 2. Fiscal policy action: Designed to invest in public infrastructure, create jobs, and generate a multiplier effect 3. A major push to re-regulate the segments of the economy that are blamed for precipitating the financial crisis to promote transparency and integrity. The Financial Crisis Inquiry Commission has put the blame for the crisis on widespread failure in government regulation, corporate management, and heedless risk-taking by Wall Street. 4. Some facts: * Banks were not required to keep enough reserves to cover for losses from the risk-taking practices of negligent mortgage banking. * The five largest investment banks held only $1 in capital to cover losses for every $40 of assets they held. 2. Competition versus Monopoly •

• •

According to most economists, and the U.S. Supreme Court, competition is generally preferable to monopoly because it results in a better allocation of resources. A monopolist tends to restrict output, driving up prices. In the United States, commissions such as the Federal Communications Commission (FCC) regulate the behavior of monopolists. Antitrust laws enacted by Congress are meant to promote competition and control monopoly.

Government and Business | 273 • • •

Global policies are often ambiguous with regard to promoting competition. The United States also promotes monopoly with patents, licenses, and copyrights. The patent system is beneficial even though it creates temporary monopolies.

3. Regulation of Monopoly • •

•

Monopolies that exist because of economies of scale (natural monopolies) are often regulated. Example: Acme Water Company ÿ Figure 17.1: Regulation of Acme Water Company: Maximum Price ÿ Without regulation, price = P0 and output = Q 0. ÿ With regulation that sets a maximum price of P1, output is equal to Q1. ÿ Figure 17.2: Regulation of Acme Water Company: Fair Rate of Return ÿ Price is set equal to P2, where average total cost is equal to demand. ÿ Valuation based on a fair rate of return involves controversies. Regulatory commissions often establish the price (or the maximum price) at the level at which it equals average total cost, including a fair rate of return on the company’s investment.

STRATEGY SESSION: The Social Cost of Monopoly See the figure on page 662 of the textbook. • Under a monopoly market, CS = A and PS = B + C and social welfare (SW) = A + B + C •

Under perfect competition, CS = A + B + D and PS = C + E and SW = CS + PS = A + B + C + D + E

Therefore, the social welfare under perfect competition is D + E greater than under a monopoly. This is often called the social welfare triangle or deadweight loss (welfare cost of monopoly). Part of the rationale of antitrust policies and regulations is to ensure that society captures part of this D + E triangle. Discussion Questions (An additional example) Consider that a monopolist faces a demand curve QD = 120 − 2P and charges a price P0 = 45. If the market was competitive, the market price would be P1 = 38. 1. What is consumer surplus under perfect competition? Answer: (60 − 38)(44/2) = 462

274 | Chapter 17 2. What is consumer surplus under monopoly? Answer: (60 − 45)(30/2) = 225 3. What is the deadweight loss? Answer: (45 − 38)(44 − 30)/2 = 49 4. The One Star Gas Company: A Case Study We look at an illustration of how regulatory commissions work. • The One Star Gas Company requested a price increase in 1978. • The regulatory commission determined that ÿ the value of capital based on historical cost and depreciation was $185 million. ÿ the weighted average cost of capital was 11.1%. ÿ profit should therefore be (0.111)(185) = $20.535 million. ÿ Because actual profit was $9.8 million, the rate increase was granted. PROBLEM SOLVED: The Trenton Gas Company The aim is to illustrate the workings of public utility regulation. Company’s asset: $300 million The state PUC decides a fair rate of return for the firm of 10% so that Trent Gas is allowed a profit of 0.10 ($300 million) = $30 million per year. Supposed the demand curve for gas provided by the firm is P = 30 − 0.1Q, where Q is the number of units of gas demanded (in millions). Also, the firm’s total cost is TC = 10 + 5Q + 0.9Q2

(17.2)

Discussion Questions 1. What price and output will the manager choose? Answer: The firm’s accounting profit is p = −Q2 + 25Q − 10

(17.3)

Because the commission has decided the firm’s accounting profit should be $30,000, we set −Q2 + 25Q − 10 = 30 or Q2 − 25Q + 40 = 0 (17.4) Solving this quadratic equation using the quadratic formula gives Q = 1.72 or Q = 23.28 Because commissions generally want public utilities to serve as many customers as possible, the quantity choice is Q = 23.28. The relevant price is given by P = 30 − 0.1(23.28) = 27.67.

Government and Business | 275 Therefore, the company’s price will be $27.67, and it will deliver 23.28 million units of gas. 2. Suppose that an increase in the weighted average cost of capital leads the commission to increase the level of profit to $40 million per year. What price should the commission set? Answer: Q = 22.81 and P = 27.72 To see that, proceed as before, setting −Q2 + 25Q − 10 = 40. 3. Suppose that the company’s cost changes to TC = 20 + 10Q + 0.9Q2, and the commission changes the level of profit to $40 million per year. What price should the commission set? Answer: Q = 18.94 and P = 28.11 5. Effects of Regulation on Efficiency • •

• • •

Regulators try to prevent a monopoly from earning excessive profits. Regulated monopolies are guaranteed a fair rate of return no matter how poorly they are managed. Managers have no incentive to increase efficiency. Managers need to anticipate a regulatory process that is characterized by long delays, called regulatory lag. Regulatory Lag: A delay between a proposed price change and its ultimate disposition Regulatory lags that are long penalize inefficiency and reward efficiency. An increase in efficiency enhances profits for a longer period with long regulatory lags. The opposite is true for a failure to increase efficiency.

6. The Concentration of Economic Power • • • • •

Government regulatory commissions are not the only device used by society to deal with the problem of monopoly. Another device: Antitrust laws Antitrust laws address the concentration of market power. These laws reflect a feeling that excessive power lies in the hands of relatively few firms. Antitrust laws are aimed at promoting competition and limiting monopoly. To measure how closed a par ticular industry is to being perfectly competitive (or monopolized), economists usually consider the concentration ratios. The two most commonly used of such ratios are: * The four-firm concentration ratio (C4) * The Herfindahl-Hirschman index (HHI): (H)

276 | Chapter 17 •

•

•

Four-firm concentration ratio: Percentage of total sales or production accounted for by an industry’s four largest firms * The higher the ratio, the more concentrated the industry. * This ratio varies widely from industry to industry. * The ratio takes no account of competition from foreign suppliers. Herfindahl-Hirschman index (HHI): An index that equals the sum of the squared market shares of all the firms in the market of manufacturing industries ÿ HHI can range from 10,000 (one firm) to 0 (atomistic competition). ÿ If the HHI will be 1,000 or less after a merger, the merger is unlikely to be challenged. ÿ If the HHI will be between 1,000 and 1,800 and the index changes by less than 100 points as a result of the merger, the merger is unlikely to be challenged. ÿ If the HHI will be greater than 1,800 and the index changes by less than 50 points as a result of the merger, the merger is unlikely to be challenged. ÿ The government has rules about disclosing information about firms. Table 17.1: Concentration Ratios and Herfindahl-Hirschman Indexes (HHI) by Economic Sectors for Largest (by Revenues or Value of Shipments) Subsector for Each Three-Digit NAICS Sector (Bold) and by Largest and Smallest Four-Firm Concentration Ratio or HHI for Each Sector: 2007

7. The Sherman Act • • • • • •

The first federal antitrust law Passed by Congress in 1890 Prohibits restraint of trade and monopolistic practices in trade among states or foreign nations Violations, as indicated in sections 1 and 2, were considered misdemeanors. Prohibits collusion to fix prices among firms engaged in interstate or international trade In 1974, the Sherman Act was amended, making violations felonies rather than misdemeanors.

8. The Clayton Act, the Robinson-Patman Act, and the Federal Trade Commission Act •

•

The ineffectiveness of the Sherman Act (during its first 20 years) led Congress in 1914 to pass the Clayton Act and the Federal Trade Commission Act. Clayton Act ÿ Passed in 1914 and amended in 1936 ÿ Outlawed unjustified price discrimination

Government and Business | 277

•

•

ÿ Outlawed tying contracts unless a firm needed to maintain control over complementary goods and services to ensure that its product worked properly ÿ Outlawed mergers that substantially lessened competition Celler-Kefauver Act ÿ Passed in 1950 ÿ Expanded the Clayton Act to include mergers accomplished through acquisition Federal Trade Commission Act ÿ Prevents undesirable and unfair, or predatory, competitive practices ÿ Outlaws untrue and deceptive advertising

9. Interpretation of the Antitrust Laws •

•

•

•

Charges are brought against a firm or group of firms by the Antitrust Division of the U.S. Department of Justice (DOJ); a trial is held, and a decision is reached. In the 1920s and 1930s, the courts, including the conservative Supreme Court, interpreted the antitrust laws in such a way that they had little impact, using the “Rule of Reason.” Rule of reason: Rule stating that only unreasonable combinations in restraint of trade, not all trusts, require conviction under the Sherman Act ÿ Put forth by the Supreme Court in 1911 in cases brought against the Standard Oil Company and the American Tobacco Company. The companies were required to divest much of their holdings. ÿ 1920: Interpreted in favor of U.S. Steel, saying monopoly itself is not a crime. ÿ 1945: Prosecution of Alcoa (Aluminum Company of America) resulted in a conviction of violation of antitrust laws. Antitrust laws are vague and ambiguous. ÿ Mergers that significantly lessen competition are frowned on, but it is not clear in many cases what “the market” actually is. ÿ Example: Staples and Office Depot * Court defined the market as “category killers” that provided a broad range of products and services. ÿ Antitrust policies, and their enforcement, vary between political administrations in the United States. ÿ It is difficult to define market boundaries.

278 | Chapter 17 STRATEGY SESSION: Antitrust Violations Discussion Questions 1. Given the frequency with which violators of antitrust and price-fixing laws are caught, and the fines that are levied, what can you conclude about the incentives to engage in these crimes? 2. In a small town with only two gas stations that is 30 miles away from the next closest town, the owners of the two stations have breakfast every morning at the town cafe. The local newspaper notes that the average price of regular unleaded is usually three cents higher than the price in other towns in the region. Discuss. STRATEGY SESSION: Antitrust on the Continent Discussion Questions 1. The antitrust divisions of governments are limited by the resources allocated to their activities. Given the likelihood of violations of antitrust laws, why do you suppose the resources actually available to these agencies tend to be very modest? Answer: Because of the costs and benefits to the government administration 2. In a small town with only two gas stations that is 30 miles away from the next closest town, the owners of the two stations have breakfast every morning at the town cafe. The local newspaper notes that the average price of regular unleaded is usually three cents higher than the price in other towns in the region. Do you think that the costs and benefits of bringing an antitrust case against the gas station owners would justify action? PROBLEM SOLVED: Government Purchase of Toxic Assets Discussion Question 1. One of the difficulties connected with the current financial crisis is that owners of securities do not know the value of the securities they hold. Are they toxic or are they merely a little stinky? How does this affect the choices that will be made by bondholders? Answer: Assume that bondholders believe that it is equally likely that they have toxic or illiquid bonds. Their expected utility is EU  (0.5)[(0.1)(00.5)  (0.9)(10000.5)]  (0.5)[(0.3)(00.5)  (0.7)(10000.5)] EU  (0.5)(1.6)(10000.5)  25.30

Government and Business | 279 The certainty equivalent is $640, which means that both would accept an offer of $700 by the government. The government’s expected revenue is $800 per bond. 10. The Patent System •

•

•

•

U.S. patent laws grant an inventor exclusive control over the use of an invention for 20 years (from initial filing) in exchange for making the invention public knowledge. Justification for patent laws: Three principal arguments: ÿ Incentive to induce inventors to produce inventions ÿ Incentive to induce managers to take the risk of implementing a new technology ÿ Incentive to disclose inventions to get patent protection Argument against patent laws ÿ Once created, the marginal cost of using new knowledge is virtually zero. Patent laws, by limiting the use of new knowledge, reduce economic efficiency. Patent protection is less effective than it appears because of the opportunity for imitation.

11. Trade and Trade Policy STRATEGY SESSION: Making Whistle-Blowing Pay Off Discussion Questions 1. How would you describe the change in antitrust activity in the United States and Europe over the period described? To what would you attribute the differences? 2. Why does trade occur among countries? Answer: Trade permits specialization, and specialization increases output. International differences in resource endowments and the relative quantity of various types of human and nonhuman resources are important bases for specialization. •

•

Foreign Trade ÿ It accounts for a very significant part of every modern country’s output and consumption. ÿ Trade allows specialization, and specialization increases output. ÿ Countries specialize in the production of goods and services in which they have a comparative advantage. Using Demand and Supply to Determine the Country of Import and the Country of Export

280 | Chapter 17 • •

Question: How can managers predict whether their country has a comparative advantage in the production of a par ticular good? Illustration: The one-good, two-country case ÿ Countries: the United States and the Netherlands ÿ Exchange rate: 2 dollars per pound ÿ Assume that the cost of transporting a product between countries is zero and that the price of the product after trade will be the same in both countries after accounting for the exchange rate. Also, assume that there is no government intervention in the market and that the market is competitive. ÿ Example U * In the United States, demand is QD = 8 − PU and supply is U QS = −2 + PU, so autarky equilibrium is PU = $5 and QU = 3. U * In the Netherlands, demand is QD = 6 − 2PN and supply is N QS = −2 + 2PN, so autarky equilibrium is PN = 2 euros and QN = 2.

•

Question: Will the product be exported and if so by which country?

•

Law of One Price: The prices in both counties are the same, based on the prevailing exchange rate. ÿ Exchange rate: 0.5PU = PN, so the autarky equilibrium price in the United States in euro terms is 2.5 euros and the autarky equilibrium price in the Netherlands in terms of dollars is $4. ÿ Note: Since the autarky price in the Netherlands is lower than the U.S. price, after accounting for the exchange rate, the Netherlands will export to the United States under free trade. ÿ Equilibrium under free trade: QDU + QDN = QSU + QSN 8 − PU + 6 − 2PN = −2 + PU − 2 + 2PN 8 − PU + 6 − PU = − 2 + PU − 2 + PU 14 − 2PU = − 4 + 2PU PU = $4.5 and PN = 2.25 euros

•

ÿ Monthly quantity demanded in the United States, QDU = 3.5 million units and QSU = 2.5 million units, so the United States is importing 1 million units per month. ÿ In the Netherlands, QDN = 1.5 million units and QSN = 2.5 million units, so the Netherlands is exporting 1 million units. Analyzing the Argument for the Government’s Advocacy of Free Trade Using Producer and Consumer Surplus ÿ Figure 17.3: Consumer and Producer Surplus in the United States Before and After Trade ÿ Figure 17.4: Consumer and Producer Surplus in the Netherlands Before and After Trade ÿ U.S. imports must match Dutch exports in a two-country, one-good trading world. The United States has a trade deficit, and the Netherlands

Government and Business | 281

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•

•

has a trade surplus, equal to the value of the traded goods. The United States must obtain foreign exchange sufficient to compensate the sellers. Use of Tariffs and Quotas to Mitigate the Gains from Trade ÿ Consumers gain and producers lose when a good is imported. * There are relatively few producers, so there is an opportunity to organize lobbying to seek protectionist legislation. * Quotas specify the maximum a country can export to the importing country. Tariffs are a tax on imports. Figure 17.5: Consumer and Producer Surplus in the United States Before and After Trade with an Import Quota of QDAQ − QSAQ Units ÿ An equivalent tariff would reduce the loss of consumer surplus to the United States. ÿ The lesson: If the government wants to restrict trade, a tariff is a more efficient way (for domestic social welfare) to do so. Trade Policy When the Market Is Not Perfectly Competitive ÿ Markets are not perfectly competitive. ÿ There are strategies for governments to use in situations where trade involves imperfect competitive situations. ÿ Traditionally, economists argue that “free trade” is the best policy to promote the interests of society as a whole. ÿ Strategic trade policies: Some economists argue that government should control the access of foreign firms to our domestic markets and promote the activities of our firms in foreign markets. * Government should use subsidies or tariffs to promote U.S. interests when appropriate, for example, economies of scale. * Strategic industries should be protected in this way. It is not clear how government can identify “strategic industries,” and industries have an incentive to argue that they are strategic whether they are or not. ÿ Using game theory to illustrate strategic trade policy: The case of Boeing (U.S.) and Airbus (European Union) ÿ Game-theoretic model of strategic trade policy: * There are two firms, Boeing and Airbus (the players of the game), capable of producing a new 250-seat aircraft. * Boeing moves first. * Figure 17.6: Payoff Matrix: Airbus and Boeing * There are two Nash equilibria in the game where only one firm will produce the aircraft. * So, if either firm is the sole producer, then production will be profitable. Otherwise, it will not. * But Boeing is a U.S. firm, whereas Airbus is 67% owned by EU firms as a joint venture of French, British, German, and Spanish aerospace firms. Now, suppose a new game where the EU decides

282 | Chapter 17

* *

*

to pay Airbus a subsidy of $10 billion if and only if it produces the airplane. Figure 17.7: New Payoff Matrix: Airbus and Boeing (assumes that Airbus will get a subsidy of $10 billion if it produces the plane). Note that as a result of the subsidy, Airbus managers have a dominant strategy whereby they will produce the new plane regardless of the strategy used by Boeing managers. The game has a unique NE: (Airbus produces the new plane, Boeing does not produce the new plane). Government intervention of this sort can pay off. But things are not so simple in practice: Retaliation by the U.S. government can follow.

12. Government Price Ceilings and Price Floors •

•

•

Price Floor: Where the government will not allow a price to fall to its market level because of a belief or political pressure that the marketdetermined price is too low ÿ Example of price floors: * Minimum wage laws * Agricultural price supports ÿ Figure 17.8: Impact of a Government Price Floor ÿ Tools used to evaluate social welfare impacts: Consumer surplus (CS) and producer surplus (PS) Price Ceiling: Where the government will not allow a price to rise to its market level because of a belief or political pressure that the marketdetermined price is too high ÿ Example: Rent control ÿ Figure 17.9: Impact of a Government Price Ceiling Deadweight Loss: Social welfare under perfect competition minus social welfare under alternative pricing

13. The Welfare Impacts of Taxes •

• • • •

Suppose the government imposes a per-unit tax of t on a good and requires sellers to pay this tax to the government. This tax drives a wedge of magnitude t between the price a seller receives and what a demander pays. With the tax the demander pays q price of PD and consumes Qt. On net, the supplier receives PS = PD − t. Consider Figure 17.10: The Incidence and Welfare Costs of a Per-Unit Tax Without the tax, the deadweight loss is 0, and social welfare is SW = A + B + C + E + F + G. With the tax, SW = A + G + (B + E) = A + B + E + G.

Government and Business | 283 • • • •

The deadweight loss caused by the tax is DWL = C + F. Question: Who bears the brunt of the tax: The supplier or the demander? Answer: It depends on the relative elasticities of demand and supply: The lesson: The supplier would bear more of the tax with a less elastic supply curve. More generally: ÿ The burden of the tax goes in the direction of the less elastic market participant.

14. Regulation of Environmental Pollution • •

•

•

Claim: In the absence of government action, the economy is likely to generate too much pollution. External Economies and Diseconomies ÿ External economy: Occurs when an action by a firm or individual gives uncompensated benefits to others * Example: An employee moved to another firm after being trained by a previous firm ÿ External diseconomy: Occurs when an action by a firm or an individual results in uncompensated costs or harm to others * Example: A firm may generate smoke that harms neighboring families and businesses. * Fact: In general, activities resulting in external diseconomies tend to be overperformed from society’s point of view. The Genesis of the Pollution Problem ÿ Managers should be sensitive to the effects of their actions on society as a whole, as well as their firm’s interests. ÿ The actual level of pollution exceeds the optimal level because of external diseconomies. The polluter does not pay the full social cost of pollution. The Optimal Level of Pollution Control ÿ Figure 17.11: Pollution Cost * The more untreated waste the industry discharges into the environment, the greater are the total costs. ÿ Figure 17.12: Pollution Control Cost * The more the industry reduces the amount of wastes it discharges, the higher are its cost of pollution control. ÿ Figure 17.13: Sum of the Pollution Cost and the Pollution Control Cost * From society’s viewpoint, the optimal level of pollution in the industry is at point B, where the sum of the two costs is at a minimum. ÿ Figure 17.14: Marginal Cost of Pollution and Margin Cost of Pollution Control * The socially optimal level of pollution for the industry is the point where the two curves intersect.

284 | Chapter 17 •

•

Forms of Government Regulation ÿ Direct regulation of pollution ÿ Effluent fee: The fee a polluter must pay to the government for discharging waste. ÿ Illustration: Figure 17.14: Marginal Cost of Pollution and Marginal Cost of Pollution Control ÿ Transferable emissions permits: Permits to generate a par ticular amount of pollution ÿ Coase theorem: A law that makes someone liable for the damage caused by pollution can lead to a negotiated agreement that is efficient. Effects of the Regulation-Induced Cost Increase on Price and Output ÿ Model: Assume the paper industry is perfectly competitive. * Marginal cost before regulation: MC = 20 + 40Q (17.11) * Profit-maximizing output: St P = MC or P = 20 + 40Q or Q = −0.5 + 0.025P * Supply curve for 1,000 producers: QS = −500 + 25P (17.12) * Market demand: QD = 3,500 − 15P (17.13) * Equilibrium: P = 100 and Q = 2,000 * Suppose a regulation raises the MC of producing paper by 25%. Hence, ° Marginal cost after regulation: MC = 1.25(20 + 40Q) = 25 + 50Q ° Profit-maximizing output: Set P = 25 + 50Q or Q = −0.5 + 0.02P ° Supply curve for 1,000 producers: QS = −500 + 20P ° Market demand: QD = 3,500 − 15P ° Equilibrium: P = 114.29 and Q = 1,785.71 * The new regulation increases price (from $100 to $114.29 per ton) and reduces output. * Incidence of cost increase depends on the price elasticities of supply and demand.

STRATEGY SESSION: Buying and Selling the Right to Emit Greenhouse Gases Discussion Question 1. How do tradable pollution permits increase social welfare? Answer: Polluters who can reduce pollution at low cost will do so and then sell their pollution permits to polluters who can only reduce pollution at high cost. This encourages efficiency in pollution control. 15. Public Goods •

The government also provides goods and services. ÿ Example: National defense

Government and Business | 285 • •

•

Question: Why does the government provide some goods and not others? Answer: Some goods—so-called public goods—are unlikely to be produced in sufficient amounts by the private sector of the economy. Therefore the government is given the task of providing these goods. Public Good: A good that can be consumed by one person without reducing the amount available to others. It cannot be provided by markets. ÿ National defense ÿ Flood control ÿ Environmental protection ÿ Figure 17.15: Classification of Goods ÿ Public goods are nonrival (one person’s consumption does not influence the availability of the good to others) and nonexcludable (if the good is available to anyone, it is available to everyone). ÿ Marketable public goods are nonrival and excludable. ÿ Common property goods are rival and nonexcludable. ÿ Private goods are both rival and excludable.

STRATEGY SESSION: Entrance Fees to National Parks Discussion Questions 1. Do you think that the entrance fees at national parks should be increased to the point where demand does not exceed capacity? Why or why not? 2. Do you think that entrance fees should remain low and that parks should simply bar entrance when some limit is reached? Why or why not?

Chapter 17: Problem Solutions 1. In 1985 United Airlines purchased Pan Am’s Pacific Division for $750 million. The Department of Justice opposed the purchase, but it was approved by the U.S. Department of Transportation. The percentages of total passengers carried across the Pacific by each airline in 1984 were as follows: Firm Northeast JAL Pan Am Korean Air

Percentage 27.5 21.9 18.5 9.3

Firm United China Airlines Singapore Airlines Other

Percentage 7.3 6.8 2.9 5.8

a. What was the concentration ratio before the purchase? Was it relatively high? b. What was the concentration ratio after the purchase?

286 | Chapter 17 Solution: a. Here, we are asked to find the four-firm concentration ratio, C4. It is given by the percentage of sales of the four largest fi rms in the industry. Here, the four largest firms are the first four in the table. Therefore, the concentration ratio is given by C4 = 27.5 + 21.9 + 18.5 + 9.3 = 77.2%. Yes, it is quite high. b . After the purchase, they become one single firm and make up 18.5 + 7.3 or 25.8% of sales in the industry and the industry would have seven firms. To find the concentration ratio after the purchase, we again consider the four largest firms in the industry. We have C4 = 27.5 + 21.9 + (18.5 + 7.3) + 9.3 = 84.5% 2. The Chicago Board of Trade voted to create a private market for rights to emit sulfur dioxide. The Clean Air Act of 1990 established a limit, beginning in 1995, on total emissions of sulfur dioxide from 110 power plants. Firms finding it relatively expensive to cut their sulfur dioxide emissions are likely to buy pollution permits because such permits cost less than cutting their emissions. Given that firms can exceed their legal limits and pay fines of $2,000 per ton, do you think that the price of a right to emit a ton of sulfur dioxide exceeds $2,000? Why or why not? Solution: No. If a firm can pay a fine of $2,000 in order to emit a ton of sulfur dioxide, the firm would have no incentive to purchase a right to emit that cost more than $2,000. 3. The Miller-Lyons Electric Company is engaged in a rate case with the local regulatory commission. The demand curve for the firm’s product is P = 1,000 − 2Q where P is price per unit of output (in dollars) and Q is the output (in thousands of units per year). The total cost (excluding the opportunity cost of the capital invested in the firm by its owners) is TC = 50 + 0.25Q where TC is expressed in millions of dollars and Q is the output (in units per year). a. The Miller-Lyons Electric Company has requested an annual rate (that is, price) of $480. If the firm has assets of $100 million, what would be its rate of return on its assets if this request is granted? b. How much greater would the firm’s accounting profit be if it were deregulated?

Government and Business | 287 Solution: a. P = 1,000 − 2Q, TC = 50 + 0.25Q Set P = 480 → Q = (1,000 − 480)/2 = 260 Profit = TR − TC = $124,800,000 − $115,000,000 = $9.8 million. The firm’s rate of return would be 0.098, or 9.8%. b. With no regulation, the firm becomes a typical profit maximizer and chooses the quantity where MR = MC. The firm’s profits in millions of dollars can be written as p = 0.001[1,000 − 2Q − (50 + 0.25Q)] p = Q − 0.002Q2 − 50 − 0.25Q = 0.75Q − 0.002Q2 − 50 dp/dQ = 0.75 − 0.004Q = 0 → Q = 187.5 and the maximum profit is p = 0.75(187.5) − 0.002(187.5)2 − 50 = $20.3125 million. Thus the firm’s profits would be $10.5125 million greater if it were deregulated. 4. The cost of pollution (in billions of dollars) originating in the paper industry is CP = 2P + P2 where P is the quantity of pollutants emitted (in thousands of tons). The cost of pollution control (in billions of dollars) for this industry is CC = 5 − 3P a. What is the optimal level of pollution? b. At this level of pollution, what is the marginal cost of pollution? c. At this level of pollution, what is the marginal cost of pollution control? Solution: a. The socially optimal level of pollution is at the point where the sum of the costs is minimized. Given the above information, the sum of the costs is given by S = 2P + P2 + 5 − 3P = P2 − P + 5 From the first-order condition for minimization, we set 2P − 1 = 0 or P = 1/2 or P = 0.5 Of course, the second derivative ensures that the sum is indeed being minimized given that the second derivative of S is 2, which is positive, an indication of an optimizer. b. Because the marginal cost of pollution is just the derivative, that is, 2 + 2P, we have MCP(P = 1/2) = 2 + 2(1/2) = 3 c. MCC = −3 because the MC of pollution cost is −3, independently of P.

288 | Chapter 17 5. Seven firms produce kitchen tables. Suppose their sales in the year 2011 are as follows: Firm

Sales (Millions of Dollars)

A B C D E F G

100 50 40 30 20 5 5

a. What is the concentration ratio in this industry? b. Would you regard this industry as oligopolistic? Why or why not? c. Suppose that firm A merges with firm G. What is the new concentration ratio in this industry? d. Suppose that after they merge, firms A and G go out of business. What is the subsequent concentration ratio in this industry? Solution: a. Look at the percentage of sales of the four largest firms in the industry. Industry sales = 100 + 50 + 40 + 30 + 20 + 5 + 5 = 250 For firm A, market share = 100/250 = 0.4 or 40% of industry sales For firm B (second largest), market share = 50/250 = 0.2 or 20% of industry sales For firm C (third largest), market share = 40/250 = 0.16 or 16% of industry sales For firm D (fourth largest), market share = 30/250 = 0.12 or 12% of industry sales Therefore, the four-firm concentration ratio is given by C4 = 0.4 + 0.2 + 0.16 + 0.12 = 0.88 or 88% of industry sales b. This industry is highly concentrated because the four largest firms in the industry make up 88% of industry sales, which is a usual characteristic of an oligopolistic industry, so without contradictory information we would expect this industry to be oligopolistic. c. C4 = 105/250 + 50/250 + 40/250 + 30/250 = 0.90 or 90% of industry sales d. C4 = 50/145 + 40/145 + 30/145 + 20/145 = 0.9655 or 97% of industry sales 6. The cost of pollution emanating from the chemical industry (in billions of dollars) is CP = 3P + 3P2

Government and Business | 289 where P is the quantity of pollutants emitted (in thousands of tons). The cost of pollution control (in billions of dollars) is CC = 7 − 5P a. What is the optimal effluent fee? b. If the cost of pollution control falls by $1 billion at each level of pollution, does this alter your answer to part (a)? Solution: a. The socially optimal level of pollution is obtained at the point where the sum of the costs is minimized. Hence, first-order condition indicates setting the sum of the marginal cost of pollution and the marginal cost of pollution control equal to zero. MCP + MCC = 3 + 6P − 5 = 0 → P = 1/3 and MCP(P = 1/3) = 3 + 6(1/3) = 5 and MCC = −5. The optimal effluent fee is $5 million per ton. b. The optimal effluent fee does not change. 7. In the cardboard box industry, the minimum average cost is reached when a firm produces 1,000 units of output per month. At this output rate, the average cost is $1 per unit of output. The demand curve for this product is as follows: Price (Dollars per Unit of Output) 3.00 2.00 1.00 0.50

Quantity (Units Demanded per Month) 1,000 8,000 12,000 20,000

a. Is this industry a natural monopoly? Why or why not? b. If the price is $2, how many firms, each of which is producing output such that average cost is at a minimum, can the market support? Solution: a. No, the market can support 12 firms operating where price equals minimum average total cost. b. The market can support 8 firms each producing 1,000 units at a price of $2. 8. Bethlehem and Youngstown, two major steel producers, accounted for about 21% of the national steel market in the late 1950s, when they proposed to merge. a. Should the two steel companies have been allowed to merge? Why or why not?

290 | Chapter 17 b. According to the companies, Bethlehem sold most of its output in the East, whereas Youngstown sold most of its output in the Midwest. Was this fact relevant? Why or why not? c. The district court did not allow Bethlehem and Youngstown to merge. Yet in 1985 (as we saw in problem 1), the Department of Transportation allowed United Airlines (with about 7% of the service between Japan and the U.S. mainland) to acquire Pan Am’s Pacific Division (with about 19%). How can you explain this? Solution: a. Economists are sharply divided over which size mergers should be allowed. The courts’ views have evolved from the point in the 1950s and 1960s where mergers involving even minuscule concentrations were blocked on the basis of the incipiency doctrine to where today mergers of even major firms are allowed on the basis of the notion of contestability. This merger, though blocked in the late 1950s, would therefore likely be allowed today. In part, this change would be especially likely in the steel industry where competition from foreign producers is strong. The definition of the market now includes international producers; as transportation and communications links have become much more sophisticated in recent years, we can no longer look only at competition among domestic producers in many industries. However, whether or not the merger would be blocked today depends on the amount by which their market share increases in the industry, information that is not provided here, in the setting of the problem. b. Assuming that transport costs are sufficiently low and that steel from different regions competed for the same customers on a national market, then where the firms currently sell their steel is irrelevant. c. As discussed in part (a), prevailing economic wisdom and court opinions have changed dramatically in the last 40 years. 9. The New York State Electric and Gas Corporation filed a request for a 10.7% increase in electric revenues. The reasons given to justify the increase were that the value of the firm’s plant and equipment had increased by $140 million, operating costs had increased, and investors required a higher rate of return. a. Why should an increase in the value of the firm’s plant and equipment result in an increase in the amount of revenue allowed by the Public Service Commission? b. Why should an increase in operating costs have the same effect? c. Why should the attitude of investors regarding what they require as a rate of return be relevant here?

Government and Business | 291 Solution: a. Commissions attempt to set rates so that utility companies’ shareholders earn returns that equal zero economic profits. The larger the absolute value of the capital investment, the larger must be the absolute value of the annual return, other things being equal. b. The higher the operating costs, the higher must be operating revenues in order to yield zero economic profits, other things being equal. c. The accounting rate of return consistent with zero economic profits increases with shareholders’ perceptions of risk, other things being equal. 10. Since early in this century, an enormous amount of attention was devoted to global warming. According to many scientists, increases in carbon dioxide and other greenhouse gases may produce significant climatic changes over the next century. To cope with this potential problem, it has been suggested that firms reduce energy consumption and switch to nonfossil fuels. William Nordhaus, a leading expert on this topic, estimated that the worldwide costs (in 1989 U.S. dollars) of various percentage reductions in the quantity of greenhouse gases emitted into the atmosphere would be as shown in the following figure.

a. Does this graph show the cost of pollution or the cost of pollution control? b. Can this graph alone indicate the socially optimal amount of greenhouse gases that should be emitted into the atmosphere? Why or why not?

292 | Chapter 17 c. If world output is about $20 trillion, by what percentage would that world output be reduced if the countries of the world agreed to cut greenhouse gas emission by 50%? d. The single most common policy proposed to decrease greenhouse gas emissions is a carbon tax—a tax on fossil fuels in proportion to the amount of carbon they emit when burned. Why would such a tax have the desired effect? Solution: a. The curve indicates that the higher the percentage of reduction, the higher the total cost of reduction. Therefore, the figure indicates the cost of pollution control. b. The graph does not show the cost of pollution (which is the benefit side of pollution control) and therefore cannot be used to determine the socially optimal amount of abatement. c. 1%. d. Pollution taxes, if set at a rate that reflects the social cost of the pollution externality, cause polluters to internalize all their costs and therefore lead to the socially efficient level of pollution.

CHAPTER 18

Optimization Techniques

Lecture Notes 1. Introduction •

Objectives ÿ To discuss functional relations ÿ To discuss marginal analysis ÿ To develop the concept of derivatives ÿ To develop some optimization techniques for one-variable functions ÿ To discuss optimization techniques for multivariable functions ÿ To introduce the Lagrange multiplier technique ÿ To compare incremental revenue with incremental cost

2. Functional Relationships •

A relationship between economic variables is frequently represented by either a table, a graph, or an equation. ÿ Example: The relationship between quantity demanded of a product and the product price can be expressed as Q = f(P) (18.1) * read “Q is a function of P”. ÿ While equation 18.1 is useful in specifying the relationship, it does not tell us how the number of units sold depends on P, the price of the product. In practice, the function f must be specified. A linear relationship between Q and P can be specified as follows: Q = a − bP, where a and b are given positive constants, called the parameters of the model. An example is Q = 200 − 5P (18.2) ÿ For example, if the price is $10, then Q = 200 − 5(10) = 150 units sold per period.

293

294 | Chapter 18 3. Marginal Analysis •

•

• •

Marginal Value: The change in the dependent variable associated with a one-unit change in a par ticular independent variable ÿ Illustration: Table 18.1: Relationship between Output and Profit: Roland Corporation Marginal Profit: The change in total profit associated with a one-unit change in output ÿ Formula: Marginal Profit = dp/dQ (if the relationship is given by an equation) = p/Q (if the relationship is given by a table or a graph) Average Profit = Total profit divided by quantity = p(Q)/Q Note: The dependent variable is maximized where its marginal value shifts from positive to negative.

4. Relationships among Total, Marginal, and Average Values • • •

Illustration: Figure 18.1: Total Profit, Average Profit, and Marginal Profit: Roland Corporation Figure 18.2: Marginal Profit Equals the Slope of the Tangent to the Total Profit Curve Proposition: The average profit curve must be rising if it is below the marginal profit curve, and it must be falling if it is above the marginal profit curve.

5. The Concept of a Derivative ÿ ÿ ÿ ÿ

Let Y be the dependent variable and X the independent variable. Relation: Y = f(X) Change in Y: Y and change in X: X The marginal value of X is estimated by Y/X = change in Y/change in X (18.4) ÿ If f is a linear function and therefore is represented graphically by a straight line, the above ratio is constant. To see this, suppose that Y = a + bX. For a given change X in the independent variable X, the corresponding change in the dependent variable, Y, is given by Y = a + b(X + X) − (a + bX) = a + bX + bX − a − bX = bX Therefore, we obtain Y/X = bX/X = b, which is a constant, since b is assumed to be a constant. Hence, b is the slope of the straight line representing the relationship.

Optimization Techniques | 295

•

* A steep curve: A small change in X results in a large change in Y. * A flat curve: A large change in X results in a small change in Y. Derivative * A derivative is basically a rate of change. It tells the (instantaneous) change in the dependent variable resulting from a small, infinitesimal change in the independent variable. * The derivative can be approximated by the (average) rate of change Y . The approximation becomes better and better by making the X change in X smaller and smaller. * Assuming Y = f(X), the derivative of Y (the dependent variable) dY (or with respect to X (the independent variable) is denoted by dX Y f’(X)) and is defined as the limit of the ratio as X approaches  X zero. We write this as dY Y = lim X → 0 dX X

(18.5)

Read equation 18.5 as: “The derivative of Y with respect to X equals the limit of the ratio Y as X approaches zero).” X Example: Consider the linear function Y = f(X) = X − 2. * Find the limit of the derivative of the function f using the definition. Solution: We have

Y = f(X + X) −f(X) = X + X − 2 −(X − 2) = X + X − 2 −X + 2 = X

Therefore, we have dY Y dY X = lim X → 0 = = lim X → 0 = lim (1) = 1. dX X dX X X → 0 More generally, for a linear function Y = a + bX, we have Y = f(X + X) − f(X) = a + b(X + X) − (a + bX) = bX Hence,

296 | Chapter 18 Y = b and since the limit of a constant is the constant itself, we X dY obtain = b. This result says that for a linear function as above, dX the coefficient of X represents the slope of the line at any point on the line. 6. How to Find a Derivative • •

• •

Managers want to know how to optimize performance. If the dependent variable Y is some measure of organizational performance and X, the dependent variable, is a variable under a particular manager’s control, he or she would like to know the value of X that maximizes Y. Fact: The tools of a derivative are very useful to solving optimization problems. The idea: The definition of derivative through the use of limit is not commonly useful in calculating derivatives. To come up with a tractable methodology in calculating the derivative of a given function, there are some useful mathematical properties. We present some of them below. ÿ Derivatives of a constant * Suppose Y = a, where a is a constant. Then the derivative of Y with dY = 0. (18.6) respect to X is 0. That is dX * This property says that the derivative of a constant alone is zero. * This is obvious from the fact that given that Y is constant, then Y = 0 and therefore the ratio Y = 0, no matter how small X is. X dY =0 * Example: Let Y = 6. Then dX ÿ Derivatives of power functions b * A power function can be represented as Y = aX , where a and b are given constants and are real numbers. dY dY Power rule: Then = a(bX b−1) or equivalently, = abX b−1. dX dX * Basically, the power rule for a derivative says that to find the derivative of a power function as expressed above, keep the multiplicative constant, then down the power and finally subtract 1 from the exponent. * Example 1: Suppose Y = 3X. Then clearly this is a power function where a = 3, and b = 1. Then * dY = 3(1X0) = 3(1)(1) = 3 dX

Optimization Techniques | 297 *

Example 2: Suppose Y = 2X2. Then clearly this is a power function where a = 2, and b = 2. Then, we have dY = 2(2X) = 4X dX

ÿ Derivatives of a sum and differences * Suppose that Y = U(X) + W(X). Then * Sum of functions rule: dY dU dW (18.8) = + dX dX dX * Similarly, suppose that Y = U(X) − W(X). Then * Difference of functions rule:

*

*

dY dU dW (18.9) = − dX dX dX The above properties say that the derivative of a sum (a difference) of functions is the sum (respectively, the difference) of the derivatives of the individual functions. Example: Suppose Y = 3X3 + 4X2. Then we have d d dY = (3X3) + (4X2) = 9X2 + 8X dX dX dX

(18.10)

ÿ Derivatives of products Suppose Y = U(X)W(X). That is, Y is a product of two functions U and W. Then * Product Rule: dY ⎛ dU ⎞ ⎛ dW ⎞ =⎜ ⎟⎠ W ( X ) + U ( X ) ⎜⎝ ⎟ (18.12) ⎝ dX dX dX ⎠ *

Example: Suppose that Y = 6X(3 − X2). Q: Find the derivative of Y. Solution: This is a product of two functions, where U(X) = 6X and W(X) = 3 − X2. Then dU dW = 6 and = −2X and we obtain dX dX

dY ⎛ dU ⎞ ⎛ dW ⎞ =⎜ W (X ) + U (X ) ⎜ ⎟ ⎝ dX ⎟⎠ dX ⎝ dX ⎠ = 6(3 − X2) + 6X(−2X) = 18 − 18X2.

298 | Chapter 18 ÿ Derivatives of quotients *

Quotient rule: Y = U(X)/W(X)

*

Example: Let Y = 5X3/(3 − 4X). Then

dY / dX =

W dU /dX − U dW /dX W2

dY = (45X2 − 40X3)/(3 − 4X)2 dX ÿ Derivatives of a function of a function (the chain rule) * Sometimes a variable depends on another variable, which in turn depends on another variable. * Suppose Y = f(W) and W = g(X). * This can be simply written as Y = f[W(X)], so that Y is a function of a function of X. Ultimately, of course, Y is a function of X. We often call f the outer function and W the inner function. We have * Chain rule: If Y = f[W(X)], then dY/dX = dW/dX df/dW. * This rule says that to find the derivative of a composite function, first differentiate the inner function and then find the derivative of the outer function evaluated at the inner function. dY 3 2 . * Example: Suppose that Y = 4W + W and W = 3X . Find dX Solution: dW Here, the inner function is W = 3X2. We have = 6X and dX dY = 4 + 3W2 = 4 + 3(3X2)2. Then, dW dY = (6X)(4 + 27X4) = 24X + 162X4. dX 7. Using Derivatives to Solve Maximization and Minimization Problems ÿ Goal: We try to find the value (or values) of the dependent variable that maximizes or minimizes Y = f(X). ÿ Illustration: Figure 18.9 ÿ The key point: To find the value(s) of X that maximizes or minimizes Y, it is necessary (but not sufficient) to find the value of X where the derivative equals 0. ÿ In other words, to optimize a function one variable subject to no constraint, set the derivative equal to zero and solve for X. The obtained value (or values) of X is called a critical point or optimizer. * To distinguish between a maximize or a minimizer, we use the so-called second derivative test. To do so, we find the second derivative of Y, which is the derivative of the first derivative, and then evaluate it at the critical values of X.

Optimization Techniques | 299 ÿ The second derivative test: * If the second derivative is positive at the critical point, then the critical point indicates that Y is minimized at this point. * If the second derivative is negative at the critical point, then the critical point indicates that Y is maximized at this point. ÿ Illustration: Figure 18.10: Using the Second Derivative to Distinguish Maxima from Minima ÿ Example: Suppose the relationship between profit and output at the Kantor Corporation is given by Y = −1 + 9X − 6X2 + X3, where Y equals annual profit (in millions of dollars) and X equals annual output (in millions of units). Capacity limitations prevent the firm from producing more than 3 million units per year. Find the values of output that maximize or minimize profit. ÿ Solution: dY = 9 − 12X + 3X2 = 0. * First-order condition: Set dX * Solving this quadratic formula for X gives X = 1 and X = 3. * Second-order condition: * First, find the second derivative. We have f”X) = −12 + 6X, where we write Y = f(X). * Next, we have f’’(3) = 6 0, and Y attains a minimum at X = 3. F”(1) = −6 0 and this indicates that profit is maximized when the output level is 3 million units and the maximum profit is 1 million. 8. Marginal Cost Equals Marginal Revenue and the Calculus of Optimization ÿ Profit: p = TR − TC, where * p is total profit * TR = total revenue and * TC = total cost ÿ Suppose that only one output is being produced at the level of Q. Marginal profit is d dTR dTC = − dQ dQ dQ ÿ For profit to be maximized, this derivative must be zero. This implies that dTR dTC = dQ dQ ÿ But

dTR dTC = MR: marginal revenue and = MC: marginal cost. dQ dQ

300 | Chapter 18 ÿ The first-order condition for profit maximization states that one choose the level of output at which MR = MC. Of course, one has to also use the second derivative test. ÿ Illustration:

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Optimization Techniques | 301 9. Partial Differentiation and the Maximization of Multivariable Functions •

In many situations, the dependent variable depends on several dependent variables, not just one. ÿ Illustration: Suppose the Merrimack Company produces two goods at the levels Q1 and Q2. Then its profit depends on the amount that is produced of each good. That is p = f(Q1,Q2)

(18.19)

Partial derivative of p with respect to Q1 is the derivative obtained by keeping Q2 constant. ÿ Notation: ∂p/∂Q1 Similarly, the partial derivative of p with respect to Q1 is the derivative obtained by keeping Q1 constant. ÿ Notation: ∂p/∂Q2 ÿ Example: Suppose the relationship between the Merrimack Company’s profit (in thousands of dollars) and its output level is given by *

p = −20 + 113.75Q1 + 80Q2 − 10Q12 − 10Q22 − 5Q1Q2.

(18.20)

Then ∂p/∂Q1 = 113.75 − 20Q1 − 5Q2 and ∂p/∂Q2 = 80 − 20Q2 − 5Q1 ÿ To find the values of the independent variables that maximize profit, set the partial derivatives equal to zero. 113.75 − 20Q1 − 5Q2 = 0 and 80 − 20Q2 − 5Q1 = 0 ÿ Equivalently, write 20Q1 + 5Q2 = 113.75 and 5Q1 + 20Q2 = 80 ÿ This is a system of two equations with two unknowns. Proceeding by elimination, we can eliminate Q2 by multiplying the first equation by −4 and add the resulting two equations. One gets Q1 = 5 and Q2 = 2.75, and the maximum profit is given by 374,375 units. 10. Constrained Optimization •

Managers of firms and other organizations generally face constraints that limit their operations. For example, a production manager may want to minimize the firm’s cost but may not be permitted to produce less than is required to meet the firm’s contracts with its customers. ÿ Example: The one-constraint case

302 | Chapter 18 Suppose the Kloster Company faces the following constraint minimization problem. * Minimize the total cost under the constraint that the sum of outputs from the products must be 30. The firm’s total cost function is given by *

TC = 4Q12 + 5Q12 − 5Q1Q2 * The constraint is written as Q1 + Q2 = 30. * One way to solve this constrained problem is to transform it into an unconstrained problem. One can solve the constraint for Q1 and plug in the result into the objective function. 2 * Doing so gives total cost as TC = 3600 − 270Q2 + 10Q . * The first-order condition gives Q1 = 13.5 and therefore from the constraint we have Q2 = 16.5. 11. Lagrangian Multipliers •

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The technique described above may not be feasible because there may be too many constraints and the relationship can be complex so that one variable cannot be solved for the other. A more general method, known as the Lagrange multiplier method or the Lagrangian method, is used. The Lagrangian method consists of introducing a new function, called the Lagrangian function, formed by adding a transformed version of the constraint(s) to the objective function and then solving the problem as an unconstrained problem based on the Lagrangian function. The Lagrangian function has the following properties: ÿ When this function is maximized (or minimized), the original function we want to maximize (minimize) is in fact maximized (minimized). ÿ All the constraints are satisfied. Example: Solve the previous problem using the Lagrange multiplier method. Solution: Rewrite the constraint as 30 − Q1 − Q2 = 0 ÿ Now, for the Lagrangian function as LTC = 4Q12 + 5Q12 − 5Q1Q2 + (30 − Q1 − Q2). ÿ At this stage, we can minimize L as an unconstrained function using the first-order conditions by setting the partial derivatives equal to zero. Here, L is a function of three variables. That gives a linear system of three equations with three unknowns. Thus, Q1 = 13.5

•

Q2 = 16.5 and = 118.5

Interpretation of the Lagrange Multiplier

Optimization Techniques | 303 ÿ The Lagrange multiplier measures the change in the objective function if the constraint is relaxed by one unit. * Illustration: If the company’s managers want to minimize TC subject to the constraint that the total output of both products is 32, rather than 30, the value of indicates by how much the minimum value of TC will increase. Hence, = 118.5 indicates that if the constraint is relaxed so that total output is 31 rather than 30, TC will go up by $118.50. * An additional example: Suppose the goal is to minimize the cost of producing 200 units of output according to the production function Q = KL when K costs $5 per unit and L costs $10 per unit. ° Employing the requirement that the ratio of marginal products must equal the ratio of factor prices at a cost-minimizing combination of K and L, we conclude that K* = 2L*. 2 ° Substituting this into the output constraint yields 200 = 2(L*) , or L* = 10 and K* = 20. Further, noting that MC = MPK/PK = MPL/ PL at the cost-minimizing combination of K and L, we can determine that the marginal cost of output at 200 units is 1/2. ° Using the Lagrangian function for this problem, we minimize Z where is known as the Lagrangian multiplier. The Lagrangian function Z is the original objective function, in this case total cost, plus the constraint multiplied by . Z = 5K + 10L + (200 − KL) °

° ° °

Setting ∂Z/∂K = 5 − L and ∂Z/∂L = 10 − K both equal to zero and solving simultaneously, we get K* = 2L*. Setting ∂Z/ = 200 − KL = 0 and solving with the first two first-order conditions, we get K* = 20 and L* = 10. Substituting either K* or L* into their first-order conditions yields * = 1/2. The Lagrangian multiplier at the solution is the rate at which the objective function increases as the constraint is relaxed. In this case if the output constraint were to produce 201 units instead of 200 units, then total cost would go up by approximately 1⁄2.

12. Comparing Incremental Costs with Incremental Revenues • •

Incremental Cost: The extra cost from an output increase that may be substantial Incremental Revenue: The extra revenue from an output increase that may be substantial ÿ Many business decisions often require comparing incremental costs with incremental revenues.

304 | Chapter 18 ÿ Note: Incremental cost is not the same as marginal cost, and incremental revenue is not the same as marginal revenue. Recall that MC is the extra cost from a small (1-unit) change in output and MR is the extra revenue from a small (1-unit) change in output. For incremental costs and incremental revenue, the increments in output may be substantial, not necessarily small. Chapter 18 Problem Solutions 1. One very important question facing hospitals is this: How big must a hospital be (in terms of patient-days of care) to minimize the cost per patient-day? According to one well-known study, the total cost (in dollars) of operating a hospital (of a par ticular type) can be approximated by: C = 4,700,000 + 0.00013X2 where X is the number of patient-days. a. Derive a formula for the relationship between cost per patient-day and the number of patient-days. b. On the basis of the results of this study, how big must a hospital be (in terms of patient-days) to minimize the cost per patient-day? c. Show that your result minimizes, rather than maximizes, the cost per patient-day. Solution: a. Since Y = C/X, the desired relationship is Y = 4,700,000/X + 0.00013X = 4, 700,000X−1 + 0.00013X. b. To find the value of X that minimizes Y, we set the derivative of Y with respect to X equal to zero: dY/dX = −4,700,000/X2 + 0.00013 = 0 Thus, X = (4,700,000/0.00013)1/2, or approximately 190,141.6476 patient-days. c. Since d2Y/dX2 = 2(4,700,000)/X3, d2Y/dX2 must be positive (because X is positive). Thus, Y must be a minimum, not a maximum, at the point where dY/dX = 0. 2. The Trumbull Company has developed a new product. Trumbull’s chairperson estimates that the new product will increase the firm’s revenues by $5 million per year and result in extra out-of-pocket costs of $4 million per year, the fully allocated costs (including a percentage of overhead, depreciation, and insurance) being $5.5 million. a. Trumbull’s chairperson feels that it would not be profitable to introduce this new product. Is the chairperson right? Why or why not? b. Trumbull’s vice president for research argues that since the development of this product has already cost about $10 million, the firm has little choice but to introduce it. Is the vice president right? Why or why not?

Optimization Techniques | 305 Solution: a. Trumbull’s chairperson is wrong. Introducing the product will increase the firm’s profits. The decision whether to introduce the product should only depend on the consequences of that decision. If revenues are going to increase by $5 million and costs are going to increase by $4 million, the project should proceed. The overhead costs that will be incurred regardless of the decision on this particular project should not be included in the decision calculus. b. As in part (a), the only costs and benefits that belong in the decision calculus are those that will occur as a result of the decision. In this case, the past research and development expenditures are immaterial; they are sunk costs and should be ignored. 3. For the Martin Corporation, the relationship between profit and output is the following: Output (number of units per day)

Profit (thousands of dollars per day)

0 1 2 3 4 5 6 7 8 9 10

−10 −8 −5 0 2 7 12 21 22 23 20

a. What is the marginal profit when output is between 5 and 6 units per day? When output is between 9 and 10 units per day? b. At what output is average profit a maximum? c. Should the Martin Corporation produce the output where average profit is at a maximum? Why or why not? Solution: a. Marginal profit = change in profit/change in output = (12 − 7)/(6 − 5) = $5,000 per day; and between 9 and 10, we have MP = (20 − 23)/1 = −$3,000 per day. b. 7 units per day at which average profit is $3,000 c. Although average profit is maximized at an output of 7 units, the total profit is only $21,000 when $23,000 per day can be earned if 9 units are produced instead.

306 | Chapter 18 4. Determine the first derivative of each of the following functions: a. Y = 3 + 10X + 5X2 b. Y = 2X(4 + X3) c. Y = 3X/(4 + X3) d. Y = 4X/(X − 3) Solution: a. dY/dX = 10 + 10X b. Write Y = 8X + 2X4. Hence, dY/dX = 8 + 8X3. c. Use the quotient rule to obtain dY/dX = (12 − 6X3)/(4 + X3)2. d. Using the product rule, we get dY/dX = −12/(X − 3)2. 5. The total cost function at the Duemer Company is TC = 100 + 4Q + 8Q2, where TC is total costs, and Q is the output. a. What is marginal cost when output is 10? b. What is marginal cost when output is 12? c. What is marginal cost when output is 20? Solution: MC = Derivative of total cost = 4 + 16Q a. Q = 10 → MC = 164 b. Q = 12 → MC = 196 c. Q = 20 → MC = 324 6. The Bartholomew Company’s profit is related in the following way to its output: p = −40 + 20Q − 3Q2, where p is total profit and Q is output. a. If the firm’s output equals 8, what is its marginal profit? b. Derive an equation relating the firm’s marginal profit to its output. c. What output maximizes the firm’s profit? Solution: a. dp/dQ = 20 − 6Q, Q = 8 → dp/dQ = −$28 b. dp/dQ = 20 − 6Q c. dp/dQ = 0 → Q* = 10/3 7. Determine the second derivative of the following functions: a. Y = 4 + 9X + 3X2 b. Y = 4X(3 + X2) c. Y = 4X(2 + X3) d. Y = (4/X) + 3 Solution: a. dY/dX = 9 + 6X; d2Y/dX2 = 6 b. dY/dX = 12 + 12X2; d2Y/dX2 = 24X

Optimization Techniques | 307 c. dY/dX = 8 + 16X3; d2Y/dX2 = 48X2 d. dY/dX = −4/X2 = −4X−2; d2Y/dX2 = 8/X3 8. The Mineola Corporation hires a consultant to estimate the relationship between its profit and its output. The consultant reports that the relationship is p = −10 − 6Q + 5.5Q2 − 2Q3 + 0.25Q4 a. The consultant says that the firm should set Q equal to 1 to maximize profit. Is it true that dp/dQ = 0 when Q = 1? Is p at a maximum when Q = 1? b. Mineola’s executive vice president says that the firm’s profit is a maximum when Q = 2. Is this true? c. If you were the chief executive officer of the Mineola Corporation, would you accept the consultant’s estimate of the relationship between profit and output as correct? Solution: dp//dQ = −6 + 11Q − 6Q2 + Q3; d2p/dQ2 = 11 − 12Q + 3Q2 a. At Q = 1, the first derivative is indeed zero, but the second derivative is positive; this suggests that Q = 1 is (at least locally) a minimum profit output, not a maximum profit output. b. At Q = 2, the first derivative is again zero, and the second derivative is negative; this suggests that Q = 2 is (at least locally) a maximum profit output. Still, Q = 2 is not the profit-maximizing output! c. At Q = 2, profits equal −12, but at Q = 10, profits equal 980! The maximum and minimum suggested above were only local, not global, extreme points. Profits rise without bound with output, and so I would doubt the consultant’s profit/output relationship. 9. Find the partial derivative of Y with respect to X in each of the following cases: a. Y = 10 + 3Z + 2X b. Y = 18Z2 + 4X3 c. Y = Z 0.2 X0.8 d. Y = 3Z/(4 + X) Solution: ∂Y =2 ∂X ∂Y b. = 12X2 ∂X ∂Y c. = 0.8X−0.2 Z 0.2 ∂X a.

308 | Chapter 18

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∂Y = −3Z/(4 + X)2 ∂X

10. The Stock Corporation makes two products, paper and cardboard. The relationship between p, the firm’s annual profit (in thousands of dollars), and its output of each good is p = −50 + 40Q1 + 30Q2 − 5Q12 − 4Q22 − 3Q1Q2 where Q1 is the firm’s annual output of paper (in tons) and Q2 is the firm’s annual output of cardboard (in tons). a. Find the output of each good that the Stock Corporation should produce if it wants to maximize profit. b. If the community in which the firm is located imposes a tax of $5,000 per year on the firm, will this alter the answer to part (a)? If so, how will the answer change? Solution: a. p = −50 + 40Q1 + 30Q2 − 5Q12 − 4Q22 − 3Q1Q2 Solve ∂p/∂Q1 = 40 − 10Q1 − 3Q2 = 0 and ∂p/∂Q2 = 30 − 8Q2 − 3Q1 = 0. Rewrite these equations as 10Q1 + 3Q2 = 40 and 3Q1 + 8Q2 = 30. Multiply the first equation by 8 and and the second by −3 and add the resulting two equations to obtain Q*1 = 230/71 and Q*2 = 180/71. b. The tax is a lump sum and therefore a fixed cost. In determining what level of output to produce, this tax will matter only if it is so large as to make it unprofitable to continue producing. Profits before the tax are about $50,000, and so the $5,000 tax does not affect Q*1 and Q*2. 11. The Miller Company uses skilled and unskilled labor to do a par ticular construction project. The cost of doing the project depends on the number of hours of skilled labor and the number of hours of unskilled labor used, the relationship being C = 4 − 3X1 − 4X2 + 2X 12 + 3X 22 + X1X2 where C is cost (in thousands of dollars), X1 is the number of hours (in thousands) of skilled labor, and X2 is the number of hours (in thousands) of unskilled labor. a. Find the number of hours of skilled labor and the number of hours of unskilled labor that will minimize the cost of doing the project. b. If the Miller Company has to purchase a license costing $2,000 to do this project (and if the cost of this license is not included in C), will this alter the answer to part (a)? If so, how will the answer change?

Optimization Techniques | 309 Solution: a. C = 4 − 3X1 − 4X2 + 2X21 + 3X22 + X1X2 Solve ∂C/∂X1 = −3 + 4X1 + X2 = 0 and ∂C/∂X2 = −4 + 6X2 + X1 = 0. Rewrite the equations as 4X1 + X2 = 3 and X1 + 6X2 = 4. Multiply the first equation by −6 and the resulting two equations and obtain X*1 = 14/23 and X*2 = 13/23. b. As in Problem 8, the tax is a lump sum, and therefore in determining what level of inputs to consume, this tax will matter only if it is so large as to make it unprofitable to continue producing. We are not given enough information to know whether it is still worth producing after the tax. 12. Ilona Stafford manages a small firm that produces wool rugs and cotton rugs. Her total cost per day (in dollars) equals C = 7X 12 + 9X 22 − 1.5X1X2 where X1 equals the number of cotton rugs produced per day and X2 equals the number of wool rugs produced per day. Because of commitments to retail stores that sell her rugs to consumers, she must produce ten rugs per day, but any mix of wool and cotton rugs is acceptable. a. If she wants to minimize her costs (without violating her commitment to the retail stores), how many cotton rugs and wool rugs should she produce per day? (Do not use the method of Lagrangian multipliers.) b. Does it seem reasonable that she would want to minimize cost in a situation of this sort? Why or why not? c. Can she produce fractional numbers of rugs per day? Solution: a. C = 7X12 + 9X 22 − 1.5X1X2 Since X1 = 10 − X2, C(X2) = 700 − 155X2 + 17.5X 22 dC/dX1 = −155 + 35X2 = 0 → X*1 = 195/35 and X*2 = 155/35 d2C(X2)/dX 22 = 35 0 → we found a minimum point. An alternative, intuitive approach would have been to argue that we would choose the outputs so that their marginal costs are equal. Setting MC1 = MC2 and substituting into the constraint that output equals 10 would yield the same results. b. Given the contract to supply exactly 10 rugs per day, the cost-minimization goal is equivalent to a profit-maximization goal. As long as the marginal revenue from an eleventh rug is less than marginal cost, then cost minimization, subject to output’s equaling 10, is equivalent to profit maximization.

310 | Chapter 18 c. By producing different proportions of whole rugs on different days, she may be able to produce precise fractional amounts on average over a long time period. 13. a. Use the method of Lagrangian multipliers to solve Problem 12. b. Do you get the same answer as you do without using this method? c. What does equal? What does this mean? Solution: The Lagrangian function is Z = 7X21 + 9X 22 − 1.5X1X2 + (10 − X1 − X2). a. The first-order conditions are: (1) ∂Z/∂X1 = 14X1 − 1.5X2 − = 0 (2) ∂Z/∂X2 = 18X2 − 1.5X1 − = 0 (3) ∂Z/∂ = 10 − X1 − X2 = 0 (1) and (2) yield X *1 = (195/155)X *2, which substituted into (3) yields X *1 = 195/35 and X *2 = 155/35. b. Yes. c. From equations (1) and (2) above, we get l = 14X1 − 1.5X2 and l = 18X2 − 1.5X1. Now, we have 14X1 − 1.5X2 = 18X2 − 1.5X1 or 15.5X1 − 19.5X2 = 0, or equivalently 31X1 − 39X2 = 0. Combining this new equation with equation (3) gives 31X1 −39X2 = 0 and X1 + X2 = 10. Multiply (2) by 39 and add the resulting equation to the previous one to get 70X1 = 390 or X1 = 39/7. Now, plugging this value into the constraint gives X2 = 10 − 39/7 = 31/7. Finally, we can solve for by substituting these values into (1) or (2) from part (c) to get l* = 71.36. is the marginal cost of rugs at the cost-minimizing combination.