Infinitely repeated games

Slides:



Advertisements
Similar presentations
Copyright © 2008 by the McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Managerial Economics, 9e Managerial Economics Thomas Maurice.
Advertisements

19 CHAPTER Uncertainty and Information.
Choices Involving Risk
Price Discrimination RESERVATION PRICE: A customer's reservation price is the most he is willing to pay for a unit of purchase. If I will pay up to 12.
6 - 1 Copyright © 2002 by Harcourt, Inc All rights reserved. CHAPTER 6 Risk and Return: The Basics Basic return concepts Basic risk concepts Stand-alone.
Economics of Information (ECON3016)
Optimal Contracts under Adverse Selection
Information Economics Consider the following variants on the game of poker: The Certainty Game – 5 cards dealt face up so that all players can see them.
From risk to opportunity Lecture 11 John Hey and Carmen Pasca.
Slides 8a: Introduction
AAEC 3315 Agricultural Price Theory
Chapter Outline 7.1 Risk Aversion and Demand for Insurance by Individuals The Effects of Insurance on Wealth Risk Aversion Other Factors Affecting an Individual’s.
“A little knowledge is a dangerous thing. So is a lot.”
ECON 202: Principles of Microeconomics
Hal Varian Intermediate Microeconomics Chapter Thirty-Six
Chapter 12 Capturing Surplus.
Risk and Return Learning Module.
Games with continuous payoffs. The Cournot game In all the games discussed so far, firms had a discrete set of choices (high – medium – low, enter – not.
Economics of incomplete information
Copyright©2004 South-Western 15 Monopoly. Copyright © 2004 South-Western While a competitive firm is a price taker, a monopoly firm is a price maker.
FIRM BEHAVIOR AND THE ORGANIZATION OF INDUSTRY
State Preference Theory
Utility Theory.
Fall 2008 Version Professor Dan C. Jones FINA 4355 Class Problem.
Michael R. Baye, Managerial Economics and Business Strategy, 3e. ©The McGraw-Hill Companies, Inc., 1999 Managerial Economics & Business Strategy Chapter.
The Economics of Information
Choices Involving Risk
…. The lemons model, continued from the last class:
Adverse Selection Asymmetric information is feature of many markets
Incomplete asymmetric information “Asymmetric” in this case points at the fact that one party is less informed than the other. The following analogy may.
The Economics of Information. Risk a situation in which there is a probability that an event will occur. People tend to prefer greater certainty and less.
Lecture Presentation Software to accompany Investment Analysis and Portfolio Management Seventh Edition by Frank K. Reilly & Keith C. Brown Chapter.
Chapter 2: Basic Microeconomic Tools 1 Basic Microeconomic Tools.
Uncertainty and Consumer Behavior
Chapter 9 THE ECONOMICS OF INFORMATION Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved. MICROECONOMIC THEORY BASIC.
Extensions to Consumer theory Inter-temporal choice Uncertainty Revealed preferences.
13. The Economics of Information and Uncertainty Risk aversion Asymmetric information (pages ) Risk aversion Asymmetric information (pages )
Managerial Economics & Business Strategy
Basic Tools of Finance Finance is the field that studies how people make decisions regarding the allocation of resources over time and the handling of.
The Pricing of Risky Financial Assets Risk and Return.
Industrial Economics Fall INFORMATION Basic economic theories: Full (perfect) information In reality, information is limited. Consumers do not know.
Economics of Information Economics 230 J.F. O’Connor.
Asymmetric Information
Some Background Assumptions Markowitz Portfolio Theory
Investment and portfolio management MGT 531.  Lecture #31.
CHAPTER 21 PURE COMPETITION COMPETITION.
Competition and Market Power
Risks and Rates of Return
Asymmetric Information
Copyright © 2004 South-Western 27 The Basic Tools of Finance.
McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved. The Competitive Firm Chapter 7.
TOPIC THREE Chapter 4: Understanding Risk and Return By Diana Beal and Michelle Goyen.
Chapter 3 Arbitrage and Financial Decision Making
Chapter 5 Uncertainty and Consumer Behavior. ©2005 Pearson Education, Inc.Chapter 52 Q: Value of Stock Investment in offshore drilling exploration: Two.
Investment Analysis and Portfolio Management First Canadian Edition By Reilly, Brown, Hedges, Chang 6.
Chapter 5 Choice Under Uncertainty. Chapter 5Slide 2 Topics to be Discussed Describing Risk Preferences Toward Risk Reducing Risk The Demand for Risky.
PERFECT COMPETITION 11 CHAPTER. Objectives After studying this chapter, you will able to  Define perfect competition  Explain how price and output are.
Chapter 7: Pure Competition. McGraw-Hill/Irwin Copyright  2007 by The McGraw-Hill Companies, Inc. All rights reserved. What is a Pure Competition? Pure.
Chapter 7: Pure Competition Copyright © 2007 by the McGraw-Hill Companies, Inc. All rights reserved.
1 Risk and Information Chapter Chapter Fifteen Overview 1.Introduction: Amazon.com 2.Describing Risky Outcome – Basic Tools Lotteries and Probabilities.
Chapter 19: Consumer Choice Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin 13e.
Decision theory under uncertainty
1 THE FUTURE: RISK AND RETURN. 2 RISK AND RETURN If the future is known with certainty, all investors will hold assets offering the highest rate of return.
Auctions serve the dual purpose of eliciting preferences and allocating resources between competing uses. A less fundamental but more practical reason.
1 Theory of the firm: Profit maximization Theory of the firm: Profit maximization.
Econ 2610: Principles of Microeconomics Yogesh Uppal
19-1 Consumer Choice  Prices are important in determining consumer behavior.  New products have to be priced correctly. The price could be set too high.
Money and Banking Lecture 11. Review of the Previous Lecture Application of Present Value Concept Internal Rate of Return Bond Pricing Real Vs Nominal.
Decisions Under Risk and Uncertainty
Lecture 8 Asymmetric Information: Adverse Selection
Presentation transcript:

Infinitely repeated games The concept of present value (see pp.14-18): Profit today is more valuable than profit one year from today. The present value of the future profit is Profit PV = ---------- , (1+i) where i is the discounting factor, usually set equal to the interest rate.

A firm that is believed to exist and earn profits infinitely into the future has the present value of (Such an expression is called an infinite series.)

A couple of useful facts about infinite series: If the profits, π, are the same in each period and we start counting with the present period, then If the first period in the series is “a year from now”, then

Airline pricing game, revisited Firm 2 Low price High price Firm 1 3, 3 8, 1 1, 8 6, 6 What if this game is played repeatedly?

Firms use “trigger strategies” – strategies contingent on the past play of a game (a certain action “triggers” a certain response). Suppose both firms are currently keeping their prices high. An example of a trigger strategy: “I will continue to play “HIGH” as long as you are playing “HIGH”. Once you cheat by playing “LOW”, I will play “LOW” in every period thereafter.” If firm 1 continues to cooperate, its present value is

Firms use “trigger strategies” – strategies contingent on the past play of a game (a certain action “triggers” a certain response). Suppose both firms are currently keeping their prices high. An example of a trigger strategy: “I will continue to play “HIGH” as long as you are playing “HIGH”. Once you cheat by playing “LOW”, I will play “LOW” in every period thereafter.” If firm 1 continues to cooperate, its present value is

If firm 1 cheats, its present value is   Firm 1 will prefer to cooperate if or 6 i + 6 > 8 i + 3   If the rate of discounting is less than 66.7%, then both firms prefer to cooperate  3 > 2 i   i < 66.7%  

Another example of a trigger strategy: Quality choice by a firm. The good can be purchased repeatedly. Firm Low quality High quality Consumer Don’t buy 0; 0 0; - 10 Buy - 10; 10 1; 1

A trigger strategy by consumers that can support the mutually beneficial outcome: “I will buy your product as long as you produce high quality. Once you produce low quality, I will never buy your product again”

Economics of incomplete information Traditional microeconomic analysis deals with economic agents making decisions under complete information.   Examples of such assumptions: Consumers know the utility they get from a good; Firms know demand schedules; Firms know each others’ prices; and so on. Real life is more complicated and less certain.

Comparing projects with uncertain outcomes Every project has two characteristics, the expected value and the degree of risk. A certain outcome = no risk. The expected value, or the mean: Computed as the weighted sum of all possible payoffs (“weighted” = multiplied by the probabilities of each respective outcome): E[x] = q1x1 + q2x2 + … + qnxn, where xi is payoff i, qi is the probability that payoff i occurs, and q1 + q2 + … + qn= 1.

Variance (a measure of risk): The sum of (the probabilities of each outcome multiplied by the squared differences between the value of the random variable and its mean: Var = q1(x1 – E[x])2 + q2(x2 – E[x])2 + … + qn(xn – E[x])2 The standard deviation, σ, is the square root of the variance. The larger the variance (or the standard deviation), the riskier the project. (For events occurring with certainty, Var = 0)

I personally would pay… $30 How much would you be willing to pay for a lottery ticket that pays $100 with a 50% probability and nothing with a 50% probability? I personally would pay… $30 What is the expected value of this lottery? E[x] = q1x1 + q2x2 = 0.5 ∙ 100 + 0.5 ∙ 0 = $50 Why the difference?

The attitude to risk may vary. Individuals who prefer less risk to more risk, all other things being equal, are called “risk averse”. It is believed that most people belong to this group. Those who don’t care about the degree of risk and care only about the expected value are called “risk neutral”. Individuals who prefer more risk to less risk, all other things being equal, are called “risk preferring”, or “risk loving”. (Sort of an anomaly.) In the example above, the person in question is . . . risk averse.

In the example above, my risk premium is $20. “Risk premium” – the minimum reward that would induce a risk averse person to accept risk while preserving the same expected value. Alternatively, risk premium is the maximum amount of money an individual will be willing to pay to replace an uncertain situation with a certainty situation that has the same expected value. Risk premium depends on the characteristics of the lottery as well as on individual preferences. In the example above, my risk premium is $20. I will pay only $30 for a lottery; in other words, will trade certain $50 for this lottery only for a premium of 50 – 30 = $20

Project selection Suppose we have two projects: A: expected value = $1000, Var = 5000 B: expected value = $1000, Var = 1500 Which one will each type choose? Risk averse – Risk neutral – Risk loving – will choose B indifferent; either A or B will choose A

What if the expected values differ as well? A: expected value = $1200, Var = 5000 B: expected value = $1000, Var = 1500 Everyone prefers higher expected value to lower expected value, all other things being equal. Risk preferences of each type are the same as on the previous slide. Type of individual Preferences Exp. value Risk Overall Risk averse A B It depends Risk neutral either Risk loving

What can a risk averse individual do to reduce risk? 1. Be informed. Useful information has value!!! 2. Diversify.

Diversification, or “spreading the risk”. You are considering investing $100 into one or two assets (stocks, for concreteness). Each stock is worth $50 now, and you believe that within the next three months its value can with equal probability either increase to $80 or drop to $40. For now, let us assume that what happens to one stock is not correlated to what happens to the other one.

Expected value of the investment: If you invest in the shares of only one of the companies: E[x] = 2 (0.5·80 + 0.5·40) = $120 If you buy one share of each of the two companies: E[x] = (0.5·80 + 0.5·40) + (0.5·80 + 0.5·40) = $120

The variance: If you invest in one company: Var = 0.5 (160 – 120)2 + 0.5 (80 – 120)2 = = 0.5·1600 + 0.5·1600 = 1600 If you invest in both… Possible outcomes: W/prob ¼ both stocks go up, x = $160 W/prob ¼ stock A goes up, stock B falls, x = $120 W/prob ¼ stock B goes up, stock A falls, x = $120 W/prob ¼ both stocks fall, x = $80 Var = 0.25 (160 – 120)2 + 0.5 (120 – 120)2 + 0.25 (80 – 120)2 = = 0.25·1600 + 0.25·1600 = 800

If the payoffs from two assets are negatively correlated, then diversification becomes even more attractive. Example: Two companies compete for a large government contract. After the winner is announced, the winner’s stock goes up and the loser’s stock falls. When firms undertake many projects at the same time, it is best for them to be risk neutral. Moreover, shareholders WANT managers to act in a risk-neutral manner (to care only about expected values).

Summary: Everyone prefers a higher expected value to a lower expected value. Most individuals are risk averse. This means they prefer less risk to more risk, provided the expected value stays the same. If one project offers a higher expected value and a higher risk at the same time, then we need more information to tell which of the two will a risk averse person choose. Firms can be assumed to be risk neutral. They evaluate projects based solely on their expected values.

Pricing and output decisions under uncertainty Consider a modification of problem 4 on p.469. You are the manager of a firm that sells soybeans in a perfectly competitive market. Your cost function is C(Q) = 2Q +2Q2. Due to production lags, you must make your output decision prior to knowing what the market price is going to be. You believe that there is a 25% chance the market price will be $120 and a 75% chance it will be $160. What is the optimal quantity of output ? The good news: All the rules we have learned before (MR=MC, etc.) still apply but “expected” appear in them as needed. Let’s do it step by step.

Normally, the rule we’d apply would be P=MC. Here, we replace P with its expected value, E(P). Calculate the expected market price. E(P) = 0.25·120 + 0.75·160 = 30 + 120 = 150 b. What output should you produce to maximize expected profits? E(P) = MC TC = 2Q +2Q2 therefore MC = 2 + 4Q 150 = 2 + 4Q 148 = 4Q Q = 37

c. What are your profits under each outcome and the expected profits? You produce Q=37 which determines your cost, TC = 2·37 + 2·372 = 74 + 2738 = $ 2,812 If P = 120, your profit is = 120·37 – 2812 = $ 1,628 (happens w/prob ¼ ) If P = 160, your profit is = 160·37 – 2812 = $ 3,108 (happens w/prob ¾ ) Expected profit = ¼ · 1628 + ¾ · 3108 = $ 2,738

Looks like in one case we are underproducing and in the other case – overproducing. Wouldn’t it be better to bet on the most likely outcome? P = 160 MC = 2 + 4 Q 4 Q = 158 Q = 39.5 and TC = 2·39.5 + 2·39.52 = $ 3,199.50 If P = 160, our profit = 160·39.5 – 3199.50 = $ 3,120.50 If P = 120, our profit = 120·39.5 – 3199.50 = $ 1,540.50 Expected profit = ¼ · 1540.5 + ¾ · 3120.5 = $ 2,725.50

The same approach can be extended to the imperfectly competitive market case. Consider the following problem: A firm with market power produces at constant marginal (and average) cost of $1. There is a 50% chance of a recession and a 50% chance of an economic boom. During a boom, the inverse demand for firm’s product will be P = 10 – 0.5 Q If there is a recession, the inverse demand will be P = 6 – 0.5 Q The firm is risk neutral and must set output before demand is known. How much output should it produce to maximize expected profit?

Normally, we would look for the point where MR = MC. This time, we will do E(MR) = MC. There are two equally good ways to find expected marginal revenue: Find expected demand, then expected marginal revenue: Expected inverse demand: E(P) = 0.5 (10 – 0.5 Q) + 0.5 (6 – 0.5 Q) = … = 8 – 0.5 Q E(MR) = 8 – Q 2. Find the marginal revenue under each scenario, then find the expected MR: Boom: P = 10 – 0.5 Q MR = 10 – Q Recession: P = 6 – 0.5 Q MR = 6 – Q E(MR) = 0.5 (10 – Q) + 0.5 (6 – Q) = 8 – Q

The rest is trivial: E(MR) = 8 – Q MC = 1 E(MR) = MC 8 – Q = 1 Q = 7 If boom, then P = 10 – 0.5 Q = $6.50 Profit = (P – AC) Q = (6.50 – 1) · 7 = $38.50 If recession, then P = 6 – 0.5 Q = $2.50 Profit = (P – AC) Q = (2.50 – 1) · 7 = $10.50 Expected profit = 0.5 · 38.50 + 0.5 · 10.50 = $24.50 Or directly: “Expected price” given Q = 7 is E(P) = 8 – 0.5 Q = $4.50 Exp.profit = (E(P) – AC) Q = (4.50 – 1) · 7 = $24.50

Consumer search for the best price and implications for the firm’s behavior General idea: A consumer samples several stores and obtains a price quote from each. The cost of obtaining each quote is the same. The total number of stores is large, so “drawing” one of them doesn’t affect the odds. After several quotes, you can always return to the store with the best price. It makes sense to continue searching as long as the (expected) benefit exceeds the cost of search.

Expected benefit: Joe wants to buy a DVD player. He thinks one-third of the stores charge $130 for a DVD player, one-third charge $100, and one-third charge $85. He sampled one store and the price was $100. What is the expected benefit from sampling another store? w/prob 1/3 next P = $85, a $15 benefit w/prob 1/3 next P = $100, no benefit w/prob 1/3 next P = $130, since he can return to the first store, no benefit Exp.benefit = (1/3)·15 = $5

How will the answer change if the best price found so far is $130? w/prob 1/3 next P = $85, a $45 benefit w/prob 1/3 next P = $100, a $30 benefit w/prob 1/3 next P = $130, no benefit Exp.benefit = (1/3)·45 + (1/3)·30 = $25

In general, if you sample a store and the price is high, the expected benefit from further search is greater (it makes more sense to keep searching). The lower the observed price, the more sense it makes to stop searching and buy. This principle holds even if the distribution of prices is not known.

If observed P is at or below this level, we stop searching and buy Cost/benefit of continuing to search Exp.benefit of another search Cost of another search Price observed If observed P is at or below this level, we stop searching and buy

What happens if the cost of search increases? Cost/benefit of continuing to search Exp.benefit of another search Cost of another search Price observed Consumers are more likely to settle for higher prices.

What has happened with the advent of the Internet? Search cost decreased; As a result, consumers buy same goods at lower prices Are all industries affected equally? (Durable) consumer goods – very much so, Groceries and expendable household items –

What has happened with the advent of the Internet? Search cost decreased; As a result, consumers buy same goods at lower prices Are all industries affected equally? (Durable) consumer goods – very much so, Groceries and expendable household items – less; Travel fares – affected a lot; Industrial shipping rates – less. Why? Insurance rates, phone rates – also affected a lot

Socially optimal risk sharing   If individuals are risk averse while firms are risk neutral, what is the optimal risk sharing between consumers and firms? Getting out of risk has more value for consumers than for firms. Therefore there is room for mutually beneficial exchange, where consumers reward the firm for accepting some of their risk for them. Example: insurance industry.

- have better information about the odds than their clients; Insurance companies are able to make money selling insurance because they may   - have better information about the odds than their clients; - differ from clients in their attitude to risk; - diversify. The larger the group of the insured, the smaller the variance, hence the lower the risk.

Incomplete asymmetric information   “Asymmetric” in this case points at the fact that one party is less informed than the other. The following analogy may be helpful: A card game may be played under different rules: All cards are dealt face up – complete information Some (or all) cards are dealt face down – incomplete symmetric information. Some (or all) cards are dealt face down but player can look at some of his own cards – incomplete asymmetric information.

An example of asymmetric information: Sellers know product quality, buyers do not. The only way for buyers to find out the true value is to try the product. (“Experience goods”) Under certainty, the rule for rational behavior is:   Buy if Value > P, where “value” (a.k.a. “utility”) stands for the subjective value the buyer gets from the product. Under uncertainty, it becomes Buy if Exp. Value > P (for a risk neutral consumer) or Buy if Exp. Value – risk premium > P (for a risk averse consumer)

Say the product has a value of $100 for a consumer if it works as expected/promised. The consumer believes that there is a 90% chance the product will deliver services; a 10% chance it will break down immediately (value = 0). Up to what price will a risk neutral consumer pay for the product?   Max P = Exp Value = 0.9·100 + 0.1· 0 = $90   What happens if the consumer is risk averse?   Max P = $90 – risk premium < $90 If buyers’ subjective valuations of the good are below the producer’s cost of making it, the market breaks down – nothing will be sold.  Both buyers and sellers are hurt by that.

Example: The market for “lemons” (Akerlof, 1973) analyzes the market for “lemons”, or cars with hidden defects. Asymmetric information is reflected in the fact that the quality of cars in the market is known to sellers but not to buyers. There are two types of used cars offered for sale in the market, 1,000 good cars and 1,000 “lemons”.

The number of potential buyers exceeds the number of cars available (a case of “sellers’ market”). All buyers are identical – each of them will pay up to $1,000 for a lemon and $2,000 for a good car. (Those numbers are also called “reservation prices”.) The sellers’ “reservation price” (the lowest price they would agree to sell for) is $800 for a lemon and $1600 for a good car.

We have two separate markets: Case 1. Symmetric complete information – the true quality of each car is known to both parties. We have two separate markets: P P 2000 2000 P=$2000 1600 1600 P=$1000 1000 1000 800 800 Q Q 1000 1000 Good cars Lemons All cars are sold.

Case 2. Symmetric incomplete information – the true quality of a particular car is not known to anybody. Each car is either a good one (with a 50% probability) or a lemon (with a 50% probability). Neither buyers nor sellers can tell one from another. For simplicity, we are going to assume both sides are risk neutral. Therefore they base their reservation prices on expected values. For sellers, EV = 0.5 ∙ 1600 + 0.5 ∙ 800 = $1,200 For buyers, EV = 0.5 ∙ 2000 + 0.5 ∙ 1000 = $1,500

Equilibrium quantity = 2000 P 2000 1500 1200 1000 800 Q 1000 2000 Equilibrium price = $1500 Equilibrium quantity = 2000

Case 3. Asymmetric incomplete information – sellers know the quality, buyers don’t. For buyers, the situation is the same as in the previous case – they will pay $1500 for any car. Sellers, however, can tell good cars from lemons, and their reservation price is different for each category. P 2000 1500 1200 1000 800 Q 1000 2000

As a result, only lemons are sold. This is an example of adverse selection, or a situation when poor quality products drive high quality products out of the market. Adverse selection prevents markets from operating efficiently and is detrimental for both buyers and sellers. After buyers realize that no good cars are being traded, their EV drops to $1000. What happens to the market price? It also decreases to $1000.

Asymmetric information does not necessarily result in adverse selection. For instance, if sellers’ reservation price for a good car is $1200, then efficiency is restored. See below. P 2000 1500 1200 1000 800 Q 1000 2000

A similar example: Adverse selection in the health insurance market. An individual knows his probability of accident, illness, etc. better than the insurance company. Insurance companies know only the composition of the population. If they offer a uniform insurance contract and price it based on the average degree of risk, then it is attractive only for the high-risk individuals. Low-risk individuals don’t buy insurance, and the average probability of accident/illness exceeds the initial estimate.

Ways to overcome the undesirable consequences of information asymmetry involved making the uninformed party better informed or reducing the amount at stake for them: laws protecting consumers; consumer reports; “screening”; “signaling”. The last two deserve some discussion. (To be continued….)