CHAPTER 11UNCERTAINTY AND GAME THORY

I. CHAPTER OVERVIEW

In our discussions up to this point, firms and households have approached their decisions armed with completeand perfect information; actors in the economic play, in short, have been blessed with a clear picture of theireconomic circumstance and an equally clear portrait of the ramifications of their decisions. It is, however, neverthat easy. In real life, imprecise and fuzzy perceptions of economic circumstance complicate every decision, andeconomists have begun to consider seriously the effects of these complications. Chapter 11 introduces you totwo major avenues of analysis.

Section A looks into decisions made under circumstances of uncertainty from two sources. The first isderived from the fact that the future is unknown. The best information available is frequently no better than acollection of possible futures with no prospect of resolution before a decision must be made. The second isderived from the potential that sudden and unforeseen events might dramatically alter even the basiccomponents of an actor’s economic condition. Hedging, speculation, and insurance all come into play in thisarena.

Section B takes explicit note of the interdependence of decisions taken by more than one actor. The rangeof outcomes that one person must consider as he or she makes a specific decision can easily depend not onlyupon prior decisions by other people but also upon their response to the very decision being considered.Economists have long applied game-theoretic constructs to the analysis of strategic interaction amongdecisions, and those applications are now being assimilated into mainstream economic theory even as theygrow significantly in number.

II. LEARNING OBJECTIVES

After you have read Chapter 11 in your text and completed the exercises in this Study Guide chapter, youshould be able to:

1. Define a role in the economy for speculators, arbitrage, and hedging.2. Illustrate the relationship between risk aversion and diminishing marginal utility of income, andapply these concepts to the notion of risk spreading in insurance and capital markets.3. Understand the importance of moral hazard and adverse selection in establishing insurance marketsthat function properly.4. Define and interpret the basic terms used in game theory: players, strategies, payoffs, and payofftables.5. Use payoff tables to describe optimal strategies for players in a game. Find a dominant strategy for afirm, if one exists.6. Use payoff tables to describe the concept of Nash equilibrium as a noncooperative solution to agame. Apply these concepts to “real world” scenarios in which unregulated markets may or may not leadto efficient equilibria.

III. REVIEW OF KEY CONCEPTS

Match the following terms from column A with their definitions in column B.A B

__ Speculation 1. Implies that the displeasure from losing a given amount of income is greaterthan the pleasure from gaining the same amount of income.

__ Arbitrage 2. The profits (or losses) earned when a particular set of strategies are pursued.__ Hedging 3. Arises when the people with the highest risk are the most likely ones to buy

insurance.__ Risk aversion 4. Occurs when each player in a game chooses his or her own strategy without

collusion or cooperation.__ Diminishing 5. Actions or sets of actions that a player follows.

marginal utilityof income

__ Risk spreading 6. Each additional dollar that an individual earns brings less additional utility.

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__ Moral hazard 7. Involves making profits from the fluctuations in prices.__ Adverse selection 8. Occurs when no player can improve his or her payoff given the other player’s

strategy.__ Players 9. Economic agents, or actors, who are engaged in strategic behavior.__ Strategies l0. Implies that individuals who own insurance engage in riskier behavior than they

would engage in without insurance.__ Payoffs l1. The purchase of a good or asset in one market for immediate resale in another

market.__ Nash equilibrium 12. Takes risks that would be large for one person and spreads them around so that

they are small risks for a large number of people.__ Dominant 13. The activity of avoiding a risk by making a counteracting sale or investment.

strategy__ Noncooperative 14. Arises when one player has a best strategy no matter what strategy the other

equilibrium player follows.__ Cooperative 15. Situations in which the payoffs are determined primarily by

equilibrium relative merit.__ Winner-take-all 16. Exists when players are expected to keep promises and carry out

games threats.__ Credibility 17. Comes when players act in unison to find strategies that will maximize their

joint payoffs.

IV. SUMMARY AND CHAPTER OUTLINE

This section summarizes the key concepts from the chapter.

A. Economics of Risk and Uncertainty1. In the “real world” life is filled with uncertainty and risks. Some of these risks are more important thanothers, to be sure, but almost every decision that we make as economists involves some degree of uncertaintyabout the course of future events. Uncertainty means that we do not know for sure what the future holds, whilerisk implies that there is some probability that an unfavorable event will occur in the future.2. Due to the existence of uncertainty, speculators look to buy low and sell high across geographicallydefined markets, across time, and across states of nature. Speculators often exchange goods without takingpossession of them; the point is not to buy what they need to use but rather to buy what can fetch a higherprice in the future.3. Speculators have three important functions. First, they buy goods in regions in which they are abundantand sell them in regions in which they are relatively scarce. For example, a speculator might buy apples fromsmall farmers in Washington State and sell them to grocers in Arizona. This redistributes goods from areas inwhich oversupply would drive prices down to areas in which undersupply would drive prices up, and thus helpsto equalize prices across markets.

Second, speculators buy goods when they are in abundance and commanding relatively low prices and sellthem later when they are more scarce and commanding relatively high prices. This lowers supply in times ofabundance, raises the supply in times of scarcity, and smoothes prices over time, providing greater stability inmarkets. For example, a speculator might buy the same apples from small farmers in Washington State duringthe month of September when apples are plentiful; after storing them for six months, the speculator might thenresell the apples to an applesauce company, when apples are much less plentiful. This helps to smooth pricesover the product’s life cycle. This behavior is referred to as arbitrage.

Third, speculators engage in hedging. This involves the speculator in absorbing risks that others do notwant to bear. For example, if the small orchard owners are afraid that the price of apples will fall precipitouslybefore they get their crops to market, they might decide to sell the future crop to the speculator in June for afixed price. The speculator actually purchases the crop in September for the agreed-upon price, hoping toimmediately resell for a profit. Thus, the farmer is able to limit his or her uncertainty, and the speculator is leftwith the potential profits or losses on the sale of the apples.4. The self motivated actions of speculators and arbitragers tend to moderate price and consumptioninstability in a wide variety of markets. Acting in their own individual best interest, these people generate animprovement in social welfare above and beyond the level of their individual gains. The source of this gain canbe found in the principle of diminishing marginal utility, because it is diminishing marginal utility that leads

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to the conclusion that stable consumption is economically beneficial. The underlying point is simply put:Starting at some average level of consumption, diminishing marginal utility means that an increase in utilitygenerated by the next unit of consumption is smaller than the increase in utility generated by the previous unitof consumption. When a product is abundant, marginal utility and prices are low; when a product is scarce,marginal utility and prices are high. By smoothing consumption over time and space, speculators are able todampen these price swings, leading to greater economic efficiency.5. A person is risk-averse when the displeasure from losing a given amount of income is greater than thepleasure from gaining the same amount of income. Simpler put, a risk-averse individual experiencesdiminishing marginal utility of income due to the fact that an additional dollar will bring less utility than theprevious dollar. The individual will lose more utility with the loss of a dollar than he or she would gain fromacquiring a dollar. Therefore, a person who is risk-averse prefers a sure thing, or a “dollar in hand,” to uncertainlevels of consumption, other things equal.6. Insurance companies are involved in risk spreading. This means that they take what would be a large riskfor any single person and “pool” it, or spread it across a large number of people so that each individual’s shareis relatively small. For example, most people who own cars buy auto insurance. In fact, some states requireowners to purchase insurance. There is some probability that any single driver will have an accident on anygiven day; that probability is higher for some people and in some areas. Insurance companies spread this riskover a larger number of people and across a wider area. Risk-averse individuals are willing to pay premiums, atleast to some degree, to avoid the risk of an uncertain future. Private insurance markets can exist when there aremany independent events and when there is little chance of moral hazard and/or adverse selection. Socialinsurance is sometimes provided in situations which cannot support private insurance markets.

In capital markets, this same principle holds. Oftentimes, a new venture is too risky for a single investor,but a group of investors working together may each be willing to accept a smaller share of the total risk.7. Sometimes markets fail because people do not have perfect information about the uncertainty and risks thatthey face. Moral hazard occurs when ownership of insurance reduces an individual’s incentive to avoid riskybehavior. For example, knowing he has insurance, a particular motorist might drive faster than he otherwisewould. Adverse selection occurs because the people who buy insurance are often those most likely to need it.The example in your text on health insurance is a good one. In the United States, most people who own healthinsurance are relatively wealthy and/or sick. Hence, the pool of insured people is not representative of the entirepopulation; those who consider themselves relatively healthy may use their money on other things and forgoinsurance. This drives rates up for those in the pool, and limits the gains to insurance.8. In summary, efficient insurance markets can operate (a) when there exist many independent events or statesof nature and (b) when there is little chance of moral hazard and/or adverse selection. Governments sometimesintervene by providing social insurance when private markets fail to operate.

B. Game Theoryl. Economic life is full of circumstances in which competition among a few individuals, firms, or evennations degenerates into a process of jockeying for positions of strategic dominance. Game theory has beenapplied by economists to provide the structure for systematic analysis of these sorts of situations. The basicstructure involves (a) identifying the players, (b) specifying the various actions or strategies available to eachplayer, and (c) completing a payoff table which records the outcomes of each combination of players’ strategies.2. Underlying this structure is the notion that all players (a) identify their individual goals, (b) recognize thegoals of the other player(s), and (c) think through all the possible outcomes for each and every possiblestrategy. Players then pick the strategy with the highest payoff relative to their goals, assuming that the otherplayers do likewise. (Note that in some situations, there may only be two players in the game.)3. There sometimes exists a dominant strategy for a given player—a strategy which provides the bestoutcome regardless of what the other players do. If all players have a dominant strategy, then the game issolved by a dominant equilibrium.4. Equilibria are more frequently Nash in character—that is, no player can make himself or herself better offby switching strategies, given the actions of the other players. Cooperative and noncooperative equilibria canthen exist; with cooperative equilibrium, firms collude to determine the best strategies to pursue, while withnoncooperative equilibrium, each firm pursues its optimal strategy without accounting for the welfare of rivals.5. Perfect competition in the absence of externalities supports a noncooperative Nash equilibrium which isefficient. In many other situations, however, a cooperative equilibrium, which must be supported by some sortof intervention, can be socially superior to the noncooperative Nash equilibrium.

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V. HELPFUL HINTS

l. Some people claim that speculators and arbitragers perform no productive function in an economy because,rather than producing any good or service, they simply buy and sell existing products. However, it isimportant not to understate the importance of the service that they provide to a market economy. Large supplyfluctuations over time would lead to price volatility and to much greater uncertainty in markets. This in turncould lead to higher prices for consumers and to fewer consumers in general. The stability provided by theseeconomic agents provides benefits to all agents in the market.2. The term expected value is used throughout this chapter. You will learn more about it in a later chapter.For now, note that the expected value of an event is the weighted average of the possible outcomes where theprobabilities of occurrence serve as weights. For example, when you toss a coin, there are two possibleoutcomes: heads or tails. Each of these outcomes has a 50 percent chance of occurring. With the $1000 betdescribed in your textbook, you will win $1000 if the coin turns up heads and lose $1000 if the coin turns uptails. The expected value of the event is:

.5 ($1000) + .5 ( - $1000) = 0

Expected utility can be calculated in the same way. The challenge, as always with utility, is to find someacceptable measure of utility to use in the calculation.3. Game theory is an exciting and relatively new area of economic analysis. It allows us to consider the kindof mutual interdependence that is pervasive in economic life. Most often, individual decisions about resourceallocation are dependent upon the decisions made by others; most often, firm decisions concerning price andoutput are affected by the behavior of rivals or potential rivals. Hence game-theoretic concepts have beenapplied in a wide range of areas, from union-managemcnt bargaining, to household and family behavior, tointernational trade wars, to public policy and optimal taxation.4. Students sometimes have trouble differentiating between a Nash equilibrium and a dominant strategy forplayers. The key here is that a Nash equilibrium occurs when no player can improve his or her payoff given theother player’s strategy. A dominant strategy occurs when one player has a best strategy no matter whatstrategy the other player follows. With Nash, we are saying that a firm is doing the best it can based uponwhat rivals are doing. With a dominant strategy, we are saying that a firm is doing the best it can.

VI. MULTIPLE CHOICE QUESTIONS

These questions are organized by topic from the chapter outline. Choose the best answer from the optionsavailable.

A. Economics Risk and of Uncertaintyl. Speculation involves:

a. buying inputs that your firm will use to produce final goods and services.b. knowledge of future events with certainty.c. riskless activities.d. making profits from the fluctuations in prices.

2. Speculative activity tends to:a. spread consumption more evenly over time.b. concentrate consumption in particular regions in which production is most abundant.c. spread consumption more evenly across regions or areas.d. increase volatility of prices.e. both a and c.

3. The welfare gain associated with smoothing consumption is derived from:a. diminishing income effects.b. increasing returns to scale.c. the law of comparative advantage.d. diminishing marginal utility.e. increasing marginal productivity.

Panels (a) and (b) of Figure 11-1 show the demand for some good Y in two separate regions of the country:regions A and B. There are 4 units of Y supplied initially to region A and 5 units supplied to region B. Use

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Figure 11-1 to answer questions 4 through 6.

Figure 11-1

4. The welfare-maximizing transfer of Y would, in this case, without any transportation cost, involve moving:a. 2 units of Y from A to B.b. 2 units of Y from B to A.c. 1 unit of Y from B to A.d. 1 unit of Y from A to B.e. all the Y units to region A.

5. The increase in total welfare generated by the maximally efficient transfer of Y identified in question 4 isequal to:

a. $1.b. $2.c. $3.d. $4.e. $5.

6. The transfer of no units of Y would be welfare-maximizing if the cost of transporting each unit from oneregion to the other were greater than:

a. $0.67.b. $1.00.c. $1.33.d. $1.67.e. $2.00.

7. In general, total welfare is maximized across space, time, or states of nature in the absence of transfer costswhen the:

a. marginal utilities of consumption are equal everywhere.b. level of consumption is equal everywhere.c. total utility derived from consumption is equal everywhere.d. total value added in production is equal everywhere.e. slopes of the demand curves are equal everywhere.

Suppose that Table 11-1 describes Janet’s utility function for income. Use it to answer questions 8 and 9.

TABLE 11-1Total

Income Utility$ 5,000 10010,000 25015,000 45020,000 60025,000 70030,000 750

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8. The marginal utility of the 20,000th dollar is:a. .03 utils.b. 33.33 utils.c. 150 utils.d. 5000 utils.e. none of the above.

9. Janet is:a. indifferent to risk.b. risk seeking.c. risk averse.d. uncertain about risk.e. satiated with money.

An individual has an income of $1000 and possesses the utility schedule shown in Figure 11-2. Use thisschedule to answer questions 10 through 12.10. Suppose this person faces a 50 percent chance of losing $100. Her expected utility would equal:

a. 50 utils.b. 45 utils.c. 40 utils.d. 35 utils.e. 30 utils.

Figure 11-2

11. Which answer to question 10 correctly identifies her level of utility if she could arrange to receive herexpected level of income with certainty?

a. 50 utils.b. 45 utils.c. 40 utils.d. 35 utils.e. 30 utils.

12. How much would the individual described in question 10 be willing to pay as an insurance premium toguarantee an invariant income (prior to paying the premium) of $950—her expected income?

a. $100.b. $75.c. $50.d. $25.e. It is impossible to tell from the information provided.

13. In 1994, as part of their transition to a market economy, the Russians decided to outlaw bankruptcy. Thismeant that regardless of the productivity or behavior of the owners and managers, firms were not allowed to go

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out of business. This would be most likely to create a(n):a. adverse selection problem.b. moral hazard problem.c. Nash equilibrium.d. hedging opportunity.e. dominant strategy.

14. Adverse selection occurs when:a. insurance reduces a person’s incentive to avoid or prevent the risky event.b. both players in a payoff matrix have a dominant strategy.c. speculators engage in hedging.d. the people with the highest risk are the most likely ones to buy insurance.e. the people that are most risk-averse and most careful are the most likely to buy insurance.

15. Any person who places larger value on gaining $2,000 than on losing $2,000 is:a. risk averse.b. risk loving.c. risk neutral.d. not enough information to answer the question.

B. Game Theory16. Game theory is most useful for analyzing:

a. the results of the Super Bowl.b. the ways that two or more players choose strategies that jointly affect each participant.c. uncertainty in economic analysis.d. equilibrium in a perfectly competitive market.e. adverse selection in markets for insurance.

17. In a game-theoretic model, a dominant strategy exists when:a. a player has a best strategy no matter what strategy the other player follows.b. no better strategy exists given the behavior of the other player.c. players cooperate and coordinate their behavior in order to maximize joint profits.d. players engage in a rivalry game involving price.e. the payoffs to the players are equal.

Figure 11-3 shows a payoff table for two firms, A and B. Each firm must choose to play one of two strategies.The payoffs to the players depend not only on the strategies they choose individually but also on the strategythat the rival plays. The numbers inside the cells show the payoffs to the two firms depending upon theirchoice of strategy. For example, if Firm A chooses strategy I and Firm B chooses strategy III, Firm A gets$20 and Firm B gets $50. Use Figure 11-3 to answer questions 17 through 20.

Figure 11-3

18. Which strategy is a dominant strategy for firm A?a. Strategy I.b. Strategy II.c. Strategy III.

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d. Strategy IV.e. None of the strategies available to firm A is a dominant strategy.

19. Which strategy is a dominant strategy for firm B?a. Strategy I.b. Strategy II.c. Strategy III.d. Strategy IV.e. None of the strategies available to firm B is a dominant strategy.

20. Which cell is a dominant equilibrium?a. Cell a.b. Cell b.c. Cell c.d. Cell d.e. None of the cells is a dominant equilibrium.

21. Which cell is a Nash equilibrium?a. Cell a.b. Cell b.c. Cell c.d. Cell d.e. None of the cells is a Nash equilibrium.

22. A cooperative equilibrium:a. occurs when the parties act in unison to find strategies that will benefit their joint payoffs.b. occurs when firms form a cartel.c. is unstable because firms have an incentive to cheat on agreements.d. all the above.e. none of the above.

23. Which of the following is an example of a “winner-take-all” game?a. UPS workers end a strike by securing a 5% wage increase for all workers.b. Russia and the United States end the cold war.c. A merit pay scheme awards a $50,000 bonus to the top salesperson of the year.d. Schools award tenure to all professors who are recommended by a faculty committee.e. All of the above are examples of winner-take-all games.

VII. PROBLEM SOLVING

The following problems are designed to help you apply the concepts that you learned in this chapter.

A. Economics of Risk and Uncertainty1. Assume that the demand curve for some good Q drawn in Figure 11-4 applies equally well to two differentregions of the country, region A and region B. Suppose, additionally, that 10 units of Q are available initiallyin region A, while only 6 units are available in region B.

a. Given this demand curve, the marginal utility of the last unit supplied in A would equal $___, whilethe marginal utility of the next unit which could be supplied in B would equal $___. Without anytransportation cost, the transfer of 1 unit from A to B would cause utility in A to (rise / fall) by $___ andutility in B to (rise / fall) by $___ for a net gain in welfare of $___. The transfer of a second unit of Qfrom A to B would similarly (increase / decrease) total welfare, this time by $___. Total welfare acrossthe two regions would, in fact, be maximized when the supply available in A equaled ___ units and thesupply, available in B equaled ___ units. In that circumstance, the marginal utility of the last unitsupplied to A would be $___ and the last unit supplied to B would be $___. The total transfer of ___units from A to B would, in fact, maximally (increase / reduce) total welfare by $___. In general, totalwelfare is maximized where (the marginal utilities of the last unit supplied are equal across regions /the total utilities of consumption are equal across regions / the quantities supplied across regionsare equal).

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Figure 11-4

b. Now suppose that it costs 75 cents per unit to transport Q from one region to the other. The first unittransferred (from A to B / from B to A) would again improve total welfare, this time by $___ . The nextunit transferred would see (another increase / a reduction) in total welfare of $___. The advent oftransportation costs would (increase / reduce) the gain in total welfare by ___ percent, from $___ to $___.There would exist no beneficial transfers of supply if transportation costs were greater than $___ per unit.

2. Consider Figure 11-5. It shows the demand for some good X in every period of a three-period supplycycle. Assume that 30 units of X are supplied in the first period of the cycle and then nothing is supplied ineither period 2 or period 3.

Figure 11-5

a. If every unit of X were consumed in the first period, then consumption in periods 2 and 3 would be___ units, and total welfare (the sum of consumer and producer surplus) over the entire cycle would equal___ the (area under the demand curve to the left of 30 units on the horizontal axis / value assumedby the demand curve above 30 units on the horizontal axis / total area under the demand curve).b. Now assume that consumption is smoothed over the three periods so that consumption is 10 units ineach. Without any carrying costs, the price in each period would equal $___. Total welfare in each periodwould equal $___, for a total over the three periods of $___. Total welfare (would / would not) bemaximized in this case because (the marginal utility of consumption in each period would be

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equalized / the total utility of consumption in each period would be equalized). This is true despitethe fact that the price of X would be ___ percent higher in period 1 than it was before.c. (advanced concept) Assume, as an additional complication, that it costs $5 in interest and storage feesto hold 1 unit of X for one period and $10 to hold 1 unit for two periods. As a result, the price of Xwould be $___ (higher / lower) in period 2 than in period 1 and $___ (higher / lower) in period 3 than inperiod 1. The only distribution of consumption over the three periods which would (1) support this pricepattern and (2) exhaust total supply over the three period cycle would have consumption equal to ___units, ___ units, and ___ units in periods 1, 2, and 3, respectively. (Hint: The holding fees suggest thatP2 = Pl + $5 and P3 = Pl + $10. The slope of the demand curve is -1 so X2 = Xl -$5 and X3 = Xl -$10.The solution must then satisfy the condition that (Xl + (Xl -$5) + (Xl - $10)) = 30.) The price of X wouldtherefore be $___ in period 1, $___ in period 2, and $___ in period 3. Welfare in period 1 would thenamount to $___; in period 2, $___; and in period 3, $___. Total welfare over the cycle would then be$___ (higher / lower) than the lumpy consumption pattern of part a and (higher / lower) than thecostless-storage situation described in part b. In comparing the costless-storage case with the costlystorage case described in part c, however, welfare falls by $___, an amount which is (greater than / lessthan) the total storage fees of $___ .d. Draw the price pattern for several cycles in Figure 11-6 and confirm that it corresponds to the patternsuggested in Figure 11-1 in the text.

Figure 11-6

Figure 11-7

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3. Consult Figure 11-7. A cardinal utility schedule dependent upon the level of income is drawn there; itdisplays a (diminishing / constant / increasing) marginal utility of income.

a. Suppose that income is $50,000 per year; in terms of utility measured on the vertical axis in utils, this$50,000 would produce utility equal to ___ utils.b. Now suppose that we consider a “lottery” that has an expected payoff of $0 but which will return again of $10,000 with a probability of 50 percent or a loss of $10,000 with a probability of 50 percent.Would playing this lottery improve welfare? ____ To see why or why not, notice that $60,000 wouldproduce ___ utils—a gain of ___ utils over $50,000. An income of $40,000 would, meanwhile, produce___ utils—a reduction of ___ utils from $50,000. Losing $10,000 therefore produces a (larger / equal /smaller) change in utility than gaining an extra $10,000 because marginal utility is (increasing / constant/ decreasing).c. To put this another way, the average level of utility that could be expected if the lottery were playedwould be ____. This level could be achieved with a fixed income of $___. The risk involved in thelottery therefore costs, in terms of constant-income alternatives, $___, the difference between $50,000without the lottery and the $___ value of taking the lottery.e. (advanced concept) Finally, assume that the lottery is not optional. Assume, instead, that itnecessarily faces an individual with the utility schedule graphed in Figure 11-7. How much would thisperson pay to get out of the uncertainty of receiving $40,000 with a 50 percent probability or $60,000 witha 50 percent probability? The lottery yields the same average utility as a constant income of $___. Theperson would, therefore, pay $___ for an insurance policy that would remove the risk of thelottery—thereby guaranteeing an income of $___ even though the lottery produces an average income of$___.

Figure 11-8

4. Now suppose that the utility schedule is given by the straight line drawn in Figure 11-8. To avoid anylottery (e.g., the case in question 3 with $10,000 on either side of $50,000) with an expected return of $0, anindividual with this schedule would be willing to pay $___. Why?

5. a. Efficient insurance markets can exist if there are (many / only a few) independent events and if there islittle chance of ___ or ___.b. Indicate whether each of the following circumstances illustrates a problem of (a) moral hazard, (b)adverse selection, or (c) too few independent events.

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___ (1) The existence of some cataclysmic event can cause enormous damage in a small but populatedarea.___ (2) The purchase of insurance against automobile theft makes owners less cautious about where theypark their cars.___ (3) High premium charges drive careful car owners to forgo theft insurance in lieu of “self-insurance.”___ (4) The existence of the FDIC makes bankers more risk-seeking than they would otherwise be.c. Failure of private insurance markets frequently leads to government financed programs of ___. List afew examples:(1)(2) (3)

B. Game Theory

Figure 11-9

6. Consult Figure 11-9. It is constructed just like the payoff matrices discussed in the text, with two firms(A and B) and four strategies (I and II for firm A and III and IV for firm B). The numbers in the boxes indicateoutcomes: the number in the upper right corner applies to firm B, while the number in the lower left cornerapplies to A. Cell c, for example, corresponds with firm A playing strategy II and firm B playing strategy III;note, in this case, that firm A would lose $200 while firm B would lose only $20.

The payoff matrix in Figure 11-9 shows that strategy I (is / is not) a dominant strategy for firm A. If Awere to choose strategy I over II when B chooses III, then A would (earn $50 / lose $200 / lose $20 / lose$100) instead of (earning $50 / losing $200 / losing $20 / losing $100); on the other hand, if A were tochoose strategy I over II when B chooses IV, then A would (earn $50 / lose $200 / lose $20 / lose $100)instead of (earning $50 / losing $200 / losing $20 / losing $100).

Strategy III (is / is not) a dominant strategy for firm B, as well, so cell (a / b / c / d) is a dominantequilibrium. It (is / is not) a Nash equilibrium because:

7. Figure 11-10 displays a different payoff matrix for two firms, A and B. (Firm A has / Firm B has /Both firms have / Neither firm has) a dominant strategy; it is strategy ___ for firm A and strategy ___ for B.The Nash equilibrium is indicated by cell (a / b / c / d) because firm B will (always / sometimes / never)choose ___ strategy and firm A will certainly choose strategy ___ as soon as it realizes that B will play strategy___.

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Figure 11-10

8. Figures 11-11 through 11-14 display four more payoff matrices for firms A and B. Complete Table 11-2by identifying A’s dominant strategy, B’s dominant strategy, the cell indicating a dominant equilibrium, thecell indicating a Nash equilibrium, and the cell indicating a cooperative equilibrium. If any of these categoriesis not exhibited by any figure, write “n/a” in the corresponding blank.9. Pollution abatement can cost any firm money, raising its products’ prices and damaging its market share (ifit acts alone). The key to effective abatement regulation is therefore frequently seen to be one of getting all thepolluting firms to act together to reduce pollution from all sources simultaneously. Figure 11-15 displays apayoff matrix which illustrates this point.

Figure 11-11

Figure 11-12

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Figure 11-13

Figure 11-14

TABLE 11-2Fig. Fig. Fig. Fig.11-11 11-12 11-13 11-14

A’s dominant strategy ____ ____ ____ ____B’s dominant strategy ____ ____ ____ ____Dominant equilibrium ____ ____ ____ ____Nash equilibrium ____ ____ ____ ____Cooperative equilibrium ____ ____ ____ ____

The two firms can choose to control their emissions or not. If they both choose to control their emissions,they each make $___ in profits; this is the situation of cell (a / b / c / d). This (is / is not), however, a Nashequilibrium. Each firm has an incentive (not to control emissions / to control emissions) if it knows that theother will be acting to control emissions; it would then earn $___ at the expense of the other firm, whoseprofits fall to $___. The Nash equilibrium, therefore, is indicated by cell (a / b / c / d) where each firm earns$___. Only a (noncooperative / cooperative) equilibrium supports the socially desirable control-controloutcome of cell a. (This is socially optimal as long as we assume that the society desires low levels ofpollution.)

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Figure 11-15

VIII. DISCUSSION QUESTIONS

Answer the following questions, making sure that you can explain the work you did to arrive at the answers.

1. Discuss the important functions of speculators. Why might it be efficient to support the efforts of thesepeople, even though they produce no output in the economy?2. Define hedging, and describe a situation in which this service might be useful for some risk-averse people.3. Explain why adverse selection might cause problems in the markets for auto insurance.4. Define the term moral hazard. Why might the savings and loan crisis of the 1990s have been initiated bymoral hazard problems?5. What is the difference between a game that has an equilibrium in dominant strategies and a game that has aNash equilibrium?6. Explain why the perfectly competitive equilibrium is a Nash equilibrium.7. Insurance is a useful device for reducing risk; however, sometimes insurance is not available. Explainunder what conditions insurance will be available to consumers who want to reduce their risk.

IX. ANSWERS TO STUDY GUIDE QUESTIONS

III. Review of Key Concepts 7 Speculation l1 Arbitrage 13 Hedging 1 Risk aversion

6 Diminishing marginal utility of income12 Risk spreading10 Moral hazard3 Adverse selection9 Players5 Strategies2 Payoffs8 Nash equilibrium

14 Dominant strategy4 Noncooperative equilibrium

17 Cooperative equilibrium15 Winner-take-all games16 Credibility

VI. Multiple Choice Questions1. D 2. E 3. D 4. B 5. B 6. B7. A 8. A 9. C 10. B 11. B 12. C13. B 14. D 15. A 16. B 17. A 18. B

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19. E 20. E 21. D 22. D 23. C

VII. Problem Solvingl. a. $2.50, $4.50, fall, $2.50, rise, $4.50, $2.00, increase, $1.00, 8 units, 8 units, $3.50, $3.50, 2,

increase, $3.00, the marginal utilities of the last unit supplied were equal across regions.b. from A to B, $1.25, another increase, 25 cents, reduce, 50%, $3.00, $1.50, $1.50

2. a. 0, $1050, area under the demand curve to the left of 30 units on the horizontal axis.b. $40, $450, $1350, would, the marginal utility of consumption in each period would be equalized, 100percent.c. $5, higher, $10, higher, 15, 10, 5, $35, $40, $45, $637.50, $450.00, $237.50, $1325, higher, lower,$25, $100

3. diminishinga. 800b. no, 900, 100, 600, 200, larger, decreasingc. 750, $46,500 (approximately, the following answers are based on this estimate), $3500, $46,500d. $46,500, $3500, $46,500, $50,000

4. $0. Since total utility is linear, the individual is risk neutral, or indifferent to risk. Therefore he or she isnot willing to pay anything to avoid risk.

5. a. many, adverse selection, moral hazardb. (1) too few independent events

(2) moral hazard(3) adverse selection(4) moral hazard

c. social insurance(1) social security(2) Medicaid, health care for the poor(3) systems of national health care, such as the Canadian system

6. is, earn $50, losing $200, lose $20, losing $100, is, a, is, any dominant equilibrium is a Nash equilibriumbecause neither player would have an incentive to switch strategies, given what the other player is doing.7. Firm B has, neither I nor II, IV, d, always choose strategy IV, II, IV.8. See Table 11-2.9. $100, a, is not, not to control emissions, $170, $20, d, $150, cooperative.

TABLE 11-2Fig. Fig. Fig. Fig.11-11 11-12 11-13 11-14

A’s dominant strategy I n/a n/a IIB’s dominant strategy n/a III n/a IVDominant equilibrium n/a n/a n/a cell dNash equilibrium cell a cell c cell d cell dCooperative equilibrium cell a cell c cell a cell d

VIII. Discussion Questions1. Speculators help to smooth consumption across time, space, or uncertain states of nature.2. Hedging is the activity of avoiding a risk by making a counteracting sale or investment. This servicemight be useful for an individual who will have goods to sell in the future, but who is worried about uncertainmarket events that might erode prices between now and the point of sale.3. Adverse selection might cause problems in the markets for auto insurance if those who are most likely toget in an auto accident are most likely to get insurance. This would mean that rates are higher for everyone inthe insured pool because there will be more accidents occurring among people with insurance.4. Moral hazard occurs when insurance reduces an individual’s incentive to avoid or prevent risky events. Thesavings and loan crisis might have been initiated by moral hazard problems if bank directors were followinginvestment strategies that were more risky than they would have been without government deposit insuranceprovided by the FDIC.

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5. A dominant strategy occurs for a player when there is a best strategy for him or her no matter what strategythe other player follows. A dominant equilibrium occurs when both players have and follow their dominantstrategies. A Nash equilibrium occurs when both players follow their best strategies given the strategy of theiropponent.6. The perfectly competitive equilibrium is a Nash equilibrium because no buyer or seller has a better strategyto follow given the behavior of all other market participants. Firms set marginal cost equal to marginalrevenue, given prices dictated by the market. Consumers set the ratios of marginal utilities to price equal forall goods in their market baskets. In a noncooperative setting, there is no strategy for any firm that willincrease profits, and no strategy for consumers that will increase utility.7. First, there must be a large number of insurable events. Additionally, there must be sufficient experienceregarding such events so that insurance companies can reliably estimate the losses. Finally, the insurance mustbe relatively free of moral hazard.

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