中级微观经济学习题及答案

ANSWERS

1 The Market

1. Suppose that there were 25 people who had a reservation price of $500, and the 26th person had a reservation price of $200. What would the demand curve look like?

1.1. It would be constant at $500 for 25 apartments and then drop to $200.

2. In the above example, what would the equilibrium price be if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?

1.2. In the ?rst case, $500, and in the second case, $200. In the third case, the equilibrium price would be any price between $200 and $500.

3. If people have di?erent reservation prices, why does the market demand curve slope down?

1.3. Because if we want to rent one more apartment, we have to o?er a lower price. The number of people who have reservation prices greater than p must always increase as p decreases.

4. In the text we assumed that the condominium purchasers came from the inner-ring people—people who were already renting apartments.

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What would happen to the price of inner-ring apartments if all of the condominium purchasers were outer-ring people—the people who were not currently renting apartments in the inner ring?

1.4. The price of apartments in the inner ring would go up since demand for apartments would not change but supply would decrease.

5. Suppose now that the condominium purchasers were all inner-ring people, but that each condominium was constructed from two apartments. What would happen to the price of apartments? 1.5. The price of apartments in the inner ring would rise.

6. What do you suppose the e?ect of a tax would be on the number of apartments that would be built in the long run?

1.6. A tax would undoubtedly reduce the number of apartments supplied in the long run.

7. Suppose the demand curve is D(p) = 100?2p. What price would the monopolist set if he had 60 apartments? How many would he rent? What price would he set if he had 40 apartments? How many would he rent?

1.7. He would set a price of 25 and rent 50 apartments. In the second case he would rent all 40 apartments at the maximum price the market

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would bear. This would be given by the solution to D(p) = 100?2p = 40, which is p? = 30.

8. If our model of rent control allowed for unrestricted subletting, who would end up getting apartments in the inner circle? Would the outcome be Pareto e?cient?

1.8. Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the ?nal outcome would be Pareto e?cient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of ine?ciency.)

2 Budget Constraint

1. Originally the consumer faces the budget line p1x1 + p2x2 = m. Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income. 2.1. The new budget line is given by 2p1x1 +8p2x2 =4m.

2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant?

2.2. The vertical intercept (???? axis) decreases and the horizontal intercept (???? axis) stays the same. Thus the budget line becomes ?atter.

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3. If the price of good 1 doubles and the price of good 2 triples, does the budget line become ?atter or steeper? 2.3. Flatter. The slope is ?2????/3????.

4. What is the de?nition of a numeraire good?

2.4. A good whose price has been set to 1; all other goods’ prices are measured relative to the numeraire good’s price.

5. Suppose that the government puts a tax of 15 cents a gallon on gasoline and then later decides to put a subsidy on gasoline at a rate of 7 cents a gallon. What net tax is this combination equivalent to? 2.5. A tax of 8 cents a gallon.

6. Suppose that a budget equation is given by

???????? +???????? = m. The government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, and a quantity subsidy on good 2 of s. What is the formula for the new budget line? 2.6. (????+ t) ???? +(?????s) ???? = m?u.

7. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least

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as well-o??

2.7. Yes, since all of the bundles the consumer could a?ord before are a?ordable at the new prices and income.

3 Preferences

1. If we observe a consumer choosing (????, ????) when (????, ????) is available one time, are we justi?ed in concluding that (????, ????) >(????, ????)? 3.1. No. It might be that the consumer was indi?erent between the two bundles. All we are justi?ed in concluding is that (????, ????)> (????, ????).

2. Consider a group of people A, B, C and the relation “at least as tall as,” as in “A is at least as tall as B.” Is this relation transitive? Is it complete? 3.2. Yes to both.

3. Take the same group of people and consider the relation “strictly taller than.” Is this relation transitive? Is it re?exive? Is it complete? 3.3. It is transitive, but it is not complete—two people might be the same height. It is not re?exive since it is false that a person is strictly taller than himself.

4. A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Is this preference relation transitive? Is it complete?

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3.4. It is transitive, but not complete. What if A were bigger but slower than B? Which one would he prefer?

5. Can an indi?erence curve cross itself? For example, could Figure 3.2 depict a single indi?erence curve?

3.5. Yes. An indi?erence curve can cross itself, it just can’t cross another distinct indi?erence curve.

6. Could Figure 3.2 be a single indi?erence curve if preferences are monotonic?

3.6. No, because there are bundles on the indi?erence curve that have strictly more of both goods than other bundles on the (alleged) indi?erence curve.

7. If both pepperoni and anchovies are bads, will the indi?erence curve have a positive or a negative slope?

3.7. A negative slope. If you give the consumer more anchovies, you’ve made him worse o?, so you have to take away some pepperoni to get him back on his indi?erence curve. In this case the direction of increasing utility is toward the origin.

8. Explain why convex preferences means that “averages are preferred

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to extremes.”

3.8. Because the consumer weakly prefers the weighted average of two bundles to either bundle.

9. What is your marginal rate of substitution of $1 bills for $5 bills? 3.9. If you give up one $5 bill, how many $1 bills do you need to compensate you? Five $1 bills will do nicely. Hence the answer is ?5 or?1/5, depending on which good you put on the horizontal axis.

10. If good 1 is a “neutral,” what is its marginal rate of substitution for good 2?

3.10. Zero—if you take away some of good 1, the consumer needs zero units of good 2 to compensate him for his loss. ANSWERS A13

11. Think of some other goods for which your preferences might be concave.

3.11. Anchovies and peanut butter, scotch and Kool Aid, and other similar repulsive combinations.

4 Utility

1. The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even

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power? Is this a monotonic transformation? (Hint: consider the case f(u)=u^2.)

4.1. The function f(u)=u^2 is a monotonic transformation for positive u, but not for negative u.

2. Which of the following are monotonic transformations?

(1) u =2 v?13; (2) u = ?1/v^2; (3)u =1/v^2; (4)u = ln v; (5)u = ?e^?v; (6)u = v^2; (7) u = v^2 for v>0; (8) u = v^2 for v<0.

4.2. (1) Yes. (2) No (works for v positive). (3) No (works for v negative). (4) Yes (only de?ned for v positive). (5) Yes. (6) No. (7) Yes. (8) No.

3. We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indi?erence curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indi?erence curve twice?)

4.3. Suppose that the diagonal intersected a given indi?erence curve at two points, say (x,x) and (y,y). Then either x>y or y>x, which means that one of the bundles has more of both goods. But if preferences are monotonic, then one of the bundles would have to be preferred to the other.

4. What kind of preferences are represented by a utility function of the

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form u(x1,x2)= ???? + ????? What about the utility function v(x1,x2)= 13x1 + 13x2?

4.4. Both represent perfect substitutes.

5. What kind of preferences are represented by a utility function of the form u(x1,x2)=x1 + ????? Is the utility function v(x1,x2)=x2 1 +2x1 ???? +x2 a monotonic transformation of u(x1,x2)? 4.5. Quasilinear preferences. Yes.

6. Consider the utility function u(x1,x2)= ???? ???? . What kind of pref- erences does it represent? Is the function v(????, ????) = ????????????a monotonic transformation of u(????, ????)? Is the function w(????, ????) = ???????????? a monotonic transformation of u(????, ????)?

4.6. The utility function represents Cobb-Douglas preferences. No. Yes.

7. Can you explain why taking a monotonic transformation of a utility function doesn’t change the marginal rate of substitution?

4.7. Because the MRS is measured along an indi?erence curve, and utility remains constant along an indi?erence curve.

5 Choice

1. If two goods are perfect substitutes, what is the demand function for good 2?

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5.1. ????=0 when p2>p??, ???? = m/p?? when p??

2. Suppose that indi?erence curves are described by straight lines with a slope of ?b. Given arbitrary prices and money income p1, p2, and m, what will the consumer’s optimal choices look like?

5.2. The optimal choices will be x1 = m/p1 and x2 = 0 ifp1/p2 b, and any amount on the budget line if p1/p2 = b.

3. Suppose that a consumer always consumes 2 spoons of sugar with each cup of co?ee. If the price of sugar is p1 per spoonful and the price of co?ee is p2 per cup and the consumer has m dollars to spend on co?ee and sugar, how much will he or she want to purchase?

5.3. Let z be the number of cups of co?ee the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint 2p??z + p??z = m. Solving for z we have z =.

4. Suppose that you have highly nonconvex preferences for ice cream

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m

2p??+ p??

and olives, like those given in the text, and that you face prices p1, p2 and have m dollars to spend. List the choices for the optimal consumption bundles.

5.4. We know that you’ll either consume all ice cream or all olives. Thus the two choices for the optimal consumption bundles will be x1 = m/p??, x2 = 0, or x1 = 0, x2 = m/p??.

5. If a consumer has a utility function u(x1,x2)=x1x4 2, what fraction of her income will she spend on good 2?

5.5. This is a Cobb-Douglas utility function, so she will spend 4/(1 + 4) = 4/5 of her income on good 2.

6. For what kind of preferences will the consumer be just as well-o? facing a quantity tax as an income tax?

5.6. For kinked preferences, such as perfect complements, where the change in price doesn’t induce any change in demand.

6 Demand

1. If the consumer is consuming exactly two goods, and she is always spending all of her money, can both of them be inferior goods? 6.1. No. If her income increases, and she spends it all, she must be purchasing more of at least one good.

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2. Show that perfect substitutes are an example of homothetic preferences.

6.2. The utility function for perfect substitutes is u( x??, x??)= x?? + x??. Thus if u( x??, x??) >u ( y??, y??), we have x?? + x??> y?? + y??. It follows that t x?? + t x?? > t y?? + ty??, so that u(t x??,t x??) >u (t y??, ty??).

3. Show that Cobb-Douglas preferences are homothetic preferences. 6.3. The Cobb-Douglas utility function has the property that u(t x??,t x??)=( t x??)a( t x??)1?a = tat1?a x??a x??1?a 2 = t x??a x??1?a2 = t*u(x1, x??). Thus if u( x??, x??) >u ( y??, y??), we know that u(t x??,t x??) >u (t y??,t y??), so that Cobb-Douglas preferences are indeed homothetic.

4. The income o?er curve is to the Engel curve as the price o?er curve is to ...?

6.4. The demand curve.

5. If the preferences are concave will the consumer ever consume both of the goods together?

6.5. No. Concave preferences can only give rise to optimal consumption bundles that involve zero consumption of one of the goods.

6. Are hamburgers and buns complements or substitutes?

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6.6. Normally they would be complements, at least for non-vegetarians.

7. What is the form of the inverse demand function for good 1 in the case of perfect complements?

6.7. We know that x1 = m/(p1 + p2). Solving for p1 as a function of the other variables, we have p1 = m x1 ?p2.

8. True or false? If the demand function is x1 = ?p1, then the inverse demand function is x = ?1/p1. 6.8. False.

7 Revealed Preference

1. When prices are (p1,p2) = (1 ,2) a consumer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (2 ,1) the consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the model of maximizing behavior?

7.1. No. This consumer violates the Weak Axiom of Revealed Preference since when he bought (x1,x2) he could have bought (y1,y2) and vice versa. In symbols:

p1x1 + p2x2 =1×1+2×2=5> 4=1×2+2×1=p1y1 + p2y2 and

q1y1 + q2y2 =2×2+1×1=5> 4=2×1+1×2=q1x1 + q2x2.

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2. When prices are (p1,p2) = (2 ,1) a consumer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (1 ,2) the consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the model of maximizing behavior?

7.2. Yes. No violations of WARP are present, since the y-bundle is not a?ordable when the x-bundle was purchased and vice versa.

3. In the preceding exercise, which bundle is preferred by the consumer, the x-bundle or the y-bundle?

7.3. Since the y-bundle was more expensive than the x-bundle when the x-bundle was purchased and vice versa, there is no way to tell which bundle is preferred.

4. We saw that the Social Security adjustment for changing prices would typically make recipients at least as well-o? as they were at the base year. What kind of price changes would leave them just as well-o?, no matter what kind of preferences they had?

7.4. If both prices changed by the same amount. Then the base-year bundle would still be optimal.

5. In the same framework as the above question, what kind of preferences would leave the consumer just as well-o? as he was in the

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base year, for all price changes? 7.5. Perfect complements.

8 Slutsky Equation

1. Suppose a consumer has preferences between two goods that are perfect substitutes. Can you change prices in such a way that the entire demand response is due to the income e?ect?

8.1. Yes. To see this, use our favorite example of red pencils and blue pencils. Suppose red pencils cost 10 cents a piece, and blue pencils cost 5 cents a piece, and the consumer spends $1 on pencils. She would then consume 20 blue pencils. If the price of blue pencils falls to 4 cents a piece, she would consume 25 blue pencils, a change which is entirely due to the income e?ect.

2. Suppose that preferences are concave. Is it still the case that the substitution e?ect is negative? 8.2. Yes.

3. In the case of the gasoline tax, what would happen if the rebate to the consumers were based on their original consumption of gasoline, x, rather than on their ?nal consumption of gasoline, x’?

8.3. Then the income e?ect would cancel out. All that would be left would be the pure substitution e?ect, which would automatically be

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negative.

4. In the case described in the preceding question, would the government be paying out more or less than it received in tax revenues? 8.4. They are receiving tx’ in revenues and paying out tx, so they are losing money.

5. In this case would the consumers be better o? or worse o? if the tax with rebate based on original consumption were in e?ect?

8.5. Since their old consumption is a?ordable, the consumers would have to be at least as well-o?. This happens because the government is giving them back more money than they are losing due to the higher price of gasoline.

9 Buying and Selling

1. If a consumer’s net demands are (5,?3) and her endowment is (4,4), what are her gross demands? 9.1. Her gross demands are (9,1).

2. The prices are (p1,p2) = (2 ,3), and the consumer is currently consuming (x1,x2) = (4 ,4). There is a perfect market for the two goods in which they can be bought and sold costlessly. Will the consumer necessarily prefer consuming the bundle (y1,y2) = (3 ,5)? Will she

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necessarily prefer having the bundle (y1,y2)?

9.2. The bundle (y1,y2) = (3 ,5) costs more than the bundle (4,4) at the current prices. The consumer will not necessarily prefer consuming this bundle, but would certainly prefer to own it, since she could sell it and purchase a bundle that she would prefer.

3. The prices are (p1,p2) = (2 ,3), and the consumer is currently consuming (x1,x2) = (4 ,4). Now the prices change to (q1,q2) = (2 ,4). Could the consumer be better o? under these new prices?

9.3. Sure. It depends on whether she was a net buyer or a net seller of the good that became more expensive.

4. The U.S. currently imports about half of the petroleum that it uses. The rest of its needs are met by domestic production. Could the price of oil rise so much that the U.S. would be made better o?? 9.4. Yes, but only if the U.S. switched to being a net exporter of oil.

5. Suppose that by some miracle the number of hours in the day increased from 24 to 30 hours (with luck this would happen shortly before exam week). How would this a?ect the budget constraint? 9.5. The new budget line would shift outward and remain parallel to the old one, since the increase in the number of hours in the day is a pure

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endowment e?ect.

6. If leisure is an inferior good, what can you say about the slope of the labor supply curve?

9.6. The slope will be positive.

10 Intertemporal Choice

1. How much is $1 million to be delivered 20 years in the future worth today if the interest rate is 20 percent?

10.1. According to Table 10.1, $1 20 years from now is worth 3 cents today at a 20 percent interest rate. Thus $1 million is worth .03×1,000,000 = $30,000 today.

2. As the interest rate rises, does the intertemporal budget constraint be- come steeper or ?atter?

10.2. The slope of the intertemporal budget constraint is equal to ?(1+r). Thus as r increases the slope becomes more negative (steeper).

3. Would the assumption that goods are perfect substitutes be valid in a study of intertemporal food purchases?

10.3. If goods are perfect substitutes, then consumers will only purchase the cheaper good. In the case of intertemporal food purchases, this implies that consumers only buy food in one period, which may not be

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very realistic.

4. A consumer, who is initially a lender, remains a lender even after a decline in interest rates. Is this consumer better o? or worse o? after the change in interest rates? If the consumer becomes a borrower after the change is he better o? or worse o??

10.4. In order to remain a lender after the change in interest rates, the consumer must be choosing a point that he could have chosen under the old interest rates, but decided not to. Thus the consumer must be worse o?. If the consumer becomes a borrower after the change, then he is choosing a previously unavailable point that cannot be compared to the initial point (since the initial point is no longer available under the new budget constraint), and therefore the change in the consumer’s welfare is unknown.

5. What is the present value of $100 one year from now if the interest rate is 10%? What is the present value if the interest rate is 5%? 10.5. At an interest rate of 10%, the present value of $100 is $90.91. At a rate of 5% the present value is $95.24.

11 Asset Markets

1. Suppose asset A can be sold for $11 next period. If assets similar to A are paying a rate of return of 10%, what must be asset A’s current

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price?

11.1. Asset A must be selling for 11/(1 + 0.10) = $10.

2. A house, which you could rent for $10,000 a year and sell for $110,000 a year from now, can be purchased for $100,000. What is the rate of return on this house?

11.2. The rate of return is equal to (10,000 + 10,000)/100,000 = 20%.

3. The payments of certain types of bonds (e.g., municipal bonds) are not taxable. If similar taxable bonds are paying 10% and everyone faces a marginal tax rate of 40%, what rate of return must the nontaxable bonds pay?

11.3. We know that the rate of return on the nontaxable bonds, r, must be such that (1?t)rt = r, therefore (1?0.40)*0.10 =0 .06 = r.

4. Suppose that a scarce resource, facing a constant demand, will be exhausted in 10 years. If an alternative resource will be available at a price of $40 and if the interest rate is 10%, what must the price of the scarce resource be today?

11.4. The price today must be 40/(1 +0 .10)^10 = $15.42.

12 Uncertainty

1. How can one reach the consumption points to the left of the

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endowment in Figure 12.1?

12.1. We need a way to reduce consumption in the bad state and increase consumption in the good state. To do this you would have to sell insurance against the loss rather than buy it.

2. Which of the following utility functions have the expected utility property? (a) u(c1,c2,π1,π2)=a(π1c1 + π2c2), (b) u(c1,c2,π1,π2)=π1c1 + π2c2 2, (c)u(c1,c2,π1,π2)=π1 lnc1 + π2 lnc2 + 17.

12.2. Functions (a) and (c) have the expected utility property (they are a?ne transformations of the functions discussed in the chapter), while (b) does not.

3. A risk-averse individual is o?ered a choice between a gamble that pays $1000 with a probability of 25% and $100 with a probability of 75%, or a payment of $325. Which would he choose?

12.3. Since he is risk-averse, he prefers the expected value of the gamble, $325, to the gamble itself, and therefore he would take the payment.

4. What if the payment was $320?

12.4. If the payment is $320 the decision will depend on the form of the utility function; we can’t say anything in general.

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5. Draw a utility function that exhibits risk-loving behavior for small gambles and risk-averse behavior for larger gambles.

12.5. Your picture should show a function that is initially convex, but then becomes concave.

6. Why might a neighborhood group have a harder time self insuring for ?ood damage versus ?re damage?

12.6. In order to self-insure, the risks must be independent. However, this does not hold in the case of ?ood damage. If one house in the neighborhood is damaged by a ?ood it is likely that all of the houses will be damaged.

13 Risky Assets

1. If the risk-free rate of return is 6%, and if a risky asset is available with a return of 9% and a standard deviation of 3%, what is the maximum rate of return you can achieve if you are willing to accept a standard deviation of 2%? What percentage of your wealth would have to be invested in the risky asset?

13.1. To achieve a standard deviation of 2% you will need to invest x = σx/σm =2 /3 of your wealth in the risky asset. This will result in a rate of return equal to (2/3)0.09 + (1?2/3)0.06 = 8%.

2. What is the price of risk in the above exercise?

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13.2. The price of risk is equal to (rm ? rf)/σm = (9? 6)/3 = 1. That is, for every additional percent of standard deviation you can gain 1% of return.

3. If a stock has a β of 1.5, the return on the market is 10%, and the risk- free rate of return is 5%, what expected rate of return should this stock o?er according to the Capital Asset Pricing Model? If the expected value of the stock is $100, what price should the stock be selling for today?

13.3. According to the CAPM pricing equation, the stock should o?er an expected rate of return of rf + β(rm ?rf)=0.05 + 1.5(0.10?0.05) =0 .125 or 12.5%. The stock should be selling for its expected present value, which is equal to 100/1.125 = $88.89.

14 Consumer’s Surplus

1. A good can be produced in a competitive industry at a cost of $10 per unit. There are 100 consumers are each willing to pay $12 each to consume a single unit of the good (additional units have no value to them.) What is the equilibrium price and quantity sold? The government imposes a tax of $1 on the good. What is the deadweight loss of this tax?

14.1. The equilibrium price is $10 and the quantity sold is 100 units. If the tax is imposed, the price rises to $11, but 100 units of the good will still be sold, so there is no deadweight loss.

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2. Suppose that the demand curve is given by D(p) = 10?p. What is the gross bene?t from consuming 6 units of the good?

14.2. We want to compute the area under the demand curve to the left of the quantity 6. Break this up into the area of a triangle with a base of 6 and a height of 6 and a rectangle with base 6 and height 4. Applying the formulas from high school geometry, the triangle has area 18 and the rectangle has area 24. Thus gross bene?t is 42.

3. In the above example, if the price changes from 4 to 6, what is the change in consumer’s surplus?

14.3. When the price is 4, the consumer’s surplus is given by the area of a triangle with a base of 6 and a height of 6; i.e., the consumer’s surplus is 18. When the price is 6, the triangle has a base of 4 and a height of 4, giving an area of 8. Thus the price change has reduced consumer’s surplus by $10.

4. Suppose that a consumer is consuming 10 units of a discrete good and the price increases from $5 per unit to $6. However, after the price change the consumer continues to consume 10 units of the discrete good. What is the loss in the consumer’s surplus from this price change?

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14.4. Ten dollars. Since the demand for the discrete good hasn’t changed, all that has happened is that the consumer has had to reduce his expenditure on other goods by ten dollars.

15 Market Demand

1. If the market demand curve is D(p) = 100 ? .5p, what is the inverse demand curve?

15.1. The inverse demand curve is P(q) = 200?2q.

2. An addict’s demand function for a drug may be very inelastic, but the market demand function might be quite elastic. How can this be? 15.2. The decision about whether to consume the drug at all could well be price sensitive, so the adjustment of market demand on the extensive margin would contribute to the elasticity of the market demand.

3. If D(p) = 12?2p, what price will maximize revenue? 15.3. Revenue is R(p) = 12 p?2p2, which is maximized at p = 3.

4. Suppose that the demand curve for a good is given by D(p) = 100/p. What price will maximize revenue?

15.4. Revenue is pD(p) = 100, regardless of the price, so all prices maximize revenue.

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5. True or false? In a two good model if one good is an inferior good the other good must be a luxury good.

15.5. True. The weighted average of the income elasticities must be 1, so if one good has a negative income elasticity, the other good must have an elasticity greater than 1 to get the average to be 1.

16 Equilibrium

1. What is the e?ect of a subsidy in a market with a horizontal supply curve? With a vertical supply curve?

16.1. The entire subsidy gets passed along to the consumers if the supply curve is ?at, but the subsidy is totally received by the producers when the supply curve is vertical.

2. Suppose that the demand curve is vertical while the supply curve slopes upward. If a tax is imposed in this market who ends up paying it?

16.2. The consumer.

3. Suppose that all consumers view red pencils and blue pencils as perfect substitutes. Suppose that the supply curve for red pencils is upward sloping. Let the price of red pencils and blue pencils be ???? and ????. What would happen if the government put a tax only on red pencils? 16.3. In this case the demand curve for red pencils is horizontal at the

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price ????, since that is the most that they would be willing to pay for a red pencil. Thus, if a tax is imposed on red pencils, consumers will end up paying ???? for them, so the entire amount of the tax will end up being borne by the producers (if any red pencils are sold at all—it could be that the tax would induce the producer to get out of the red pencil business).

4. The United States imports about half of its petroleum needs. Suppose that the rest of the oil producers are willing to supply as much oil as the United States wants at a constant price of $25 a barrel. What would happen to the price of domestic oil if a tax of $5 a barrel were placed on foreign oil?

16.4. Here the supply curve of foreign oil is ?at at $25. Thus the price to the consumers must rise by the $5 amount of the tax, so that the net price to the consumers becomes $30. Since foreign oil and domestic oil are perfect substitutes as far as the consumers are concerned, the domestic producers will sell their oil for $30 as well and get a windfall gain of $5 per barrel.

5. Suppose that the supply curve is vertical. What is the deadweight loss of a tax in this market?

16.5. Zero. The deadweight loss measures the value of lost output. Since the same amount is supplied before and after the tax, there is no

27

deadweight loss. Put another way: the suppliers are paying the entire amount of the tax, and everything they pay goes to the government. The amount that the suppliers would pay to avoid the tax is simply the tax revenue the government receives, so there is no excess burden of the tax.

6. Consider the tax treatment of borrowing and lending described in the text. How much revenue does this tax system raise if borrowers and lenders are in the same tax bracket? 16.6. Zero revenue.

7. Does such a tax system raise a positive or negative amount of revenue when ????

16.7. It raises negative revenue, since in this case we have a net subsidy of borrowing.

17 Auctions

1. Consider an auction of antique quilts to collectors. Is this a private-value or a common-value auction?

17.1. Since the collectors likely have their own values for the quilts, and don’t particularly care about the other bidders’ values, it is a private-value auction.

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2. Suppose that there are only two bidders with values of $8 and $10 for an item with a bid increment of $1. What should the reservation price be in a pro?t-maximizing English auction?

17.2. Following the analysis in the text, there are four equally likely con?gurations of bidders: (8,8), (8,10), (10,8), and (10,10). With zero reservation price, the optimal bids will be (8,9,9,10), resulting in expected pro?t of $9. The only candidate for a reservation price is $10, which yields expected pro?t of 30/4 = $7 .50. Hence zero is a pro?t-maximizing reservation price in this auction.

3. Suppose that we have two copies of Intermediate Microeconomics to sell to three (enthusiastic) students. How can we use a sealed-bid auction that will guarantee that the bidders with the two highest values get the books?

17.3. Have each person write down a value, then award the two books to the students with the two highest values, but just charge them the bid of the third highest student.

4. Consider the Ucom example in the text. Was the auction design e?cient? Did it maximize pro?ts?

17.4. It was e?cient in the sense that it awarded the license to the ?rm that valued it most highly. But it took a year for this to happen, which is

29

ine?cient. A Vickrey auction or an English auction would have achieved the same result more quickly.

5. A game theorist ?lls a jar with pennies and auctions it o? on the ?rst day of class using an English auction. Is this a private-value or a common-value auction? Do you think the winning bidder usually makes a pro?t?

17.5. This is a common-value auction since the value of the prize is the same to all bidders. Normally, the winning bidder overestimates the number of pennies in the jar, illustrating the winner’s curse.

18 Technology

22

1. Consider the production function f(x1, ????)= x1 x2. Does this exhibit

constant, increasing, or decreasing returns to scale? 18.1. Increasing returns to scale.

2. Consider the production function f(x1,x2)=4 ???? ????. Does this exhibit constant, increasing, or decreasing returns to scale? 18.2. Decreasing returns to scale.

????3. The Cobb-Douglas production function is given by f(x1,x2)=A???? ????.

??/??

??/??

It turns out that the type of returns to scale of this function will depend on the magnitude of a + b. Which values of a + b will be associated with

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the di?erent kinds of returns to scale?

18.3. If a + b = 1, we have constant returns to scale, a + b<1 gives decreasing returns to scale, and a + b>1 gives increasing returns to scale.

4. The technical rate of substitution between factors x2 and x1 is ?4. If you desire to produce the same amount of output but cut your use of x1 by 3 units, how many more units of x2 will you need? 18.4. 4×3 = 12 units.

5. True or false? If the law of diminishing marginal product did not hold, the world’s food supply could be grown in a ?owerpot. 18.5. True.

6. In a production process is it possible to have decreasing marginal product in an input and yet increasing returns to scale? 18.6. Yes.

19 Pro?t Maximization

1. In the short run, if the price of the ?xed factor is increased, what will happen to pro?ts? 19.1. Pro?ts will decrease.

2. If a ?rm had everywhere increasing returns to scale, what would

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happen to its pro?ts if prices remained ?xed and if it doubled its scale of operation?

19.2. Pro?t would increase, since output would go up more than the cost of the inputs.

3. If a ?rm had decreasing returns to scale at all levels of output and it divided up into two equal-size smaller ?rms, what would happen to its overall pro?ts?

19.3. If the ?rm really had decreasing returns to scale, dividing the scale of all inputs by 2 would produce more than half as much output. Thus the subdivided ?rm would make more pro?ts than the big ?rm. This is one argument why having everywhere decreasing returns to scale is implausible.

4. A gardener exclaims: “For only $1 in seeds I’ve grown over $20 in pro- duce!” Besides the fact that most of the produce is in the form of zucchini, what other observations would a cynical economist make about this situation?

19.4. The gardener has ignored opportunity costs. In order to accurately account for the true costs, the gardener must include the cost of her own time used in the production of the crop, even if no explicit wage was paid.

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5. Is maximizing a ?rm’s pro?ts always identical to maximizing the ?rm’s stock market value?

19.5. Not in general. For example, consider the case of uncertainty.

6. If pMP1 >w 1, then should the ?rm increase or decrease the amount of factor 1 in order to increase pro?ts? 19.6. Increase.

7. Suppose a ?rm is maximizing pro?ts in the short run with variable factor x1 and ?xed factor x2. If the price of x2 goes down, what happens to the ?rm’s use of x1? What happens to the ?rm’s level of pro?ts?

19.7. The use of x1 does not change, and pro?ts will increase.

8. A pro?t-maximizing competitive ?rm that is making positive pro?ts in long-run equilibrium (may/may not) have a technology with constant returns to scale. 19.8. May not.

20 Cost Minimization

1. Prove that a pro?t-maximizing ?rm will always minimize costs. 20.1. Since pro?t is equal to total revenue minus total costs, if a ?rm is

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not minimizing costs then there exists a way for the ?rm to increase pro?ts; however, this contradicts the fact that the ?rm is a pro?t maximizer.

2. If a ?rm is producing where MP1/w1 > MP 2/w2, what can it do to reduce costs but maintain the same output?

20.2. Increase the use of factor 1 and decrease the use of factor 2.

3. Suppose that a cost-minimizing ?rm uses two inputs that are perfect substitutes. If the two inputs are priced the same, what do the conditional factor demands look like for the inputs?

20.3. Since the inputs are identically priced perfect substitutes, the ?rm will be indi?erent between which of the inputs it uses. Thus the ?rm will use any amounts of the two inputs such that x1 + x2 = y.

4. The price of paper used by a cost-minimizing ?rm increases. The ?rm responds to this price change by changing its demand for certain inputs, but it keeps its output constant. What happens to the ?rm’s use of paper?

20.4. The demand for paper either goes down or stays constant.

5. If a ?rm uses n inputs (n>2), what inequality does the theory of

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revealed cost minimization imply about changes in factor prices (Δwi) and the changes in factor demands (Δxi) for a given level of output? 20.5. It implies that

35

tsts

nΔwiΔxi ≤ 0, where Δwi = w ?wand Δxi = x?xi=1 iiii.

21 Cost Curves

1. Which of the following are true?

(1) Average ?xed costs never increase with output;

(2) average total costs are always greater than or equal to average variable costs;

(3) average cost can never rise while marginal costs are declining. 21.1. True, true, false.

2. A ?rm produces identical outputs at two di?erent plants. If the marginal cost at the ?rst plant exceeds the marginal cost at the second plant, how can the ?rm reduce costs and maintain the same level of output?

21.2. By simultaneously producing more output at the second plant and reducing production at the ?rst plant, the ?rm can reduce costs.

3. True or false? In the long run a ?rm always operates at the mini- mum level of average costs for the optimally sized plant to produce a given amount of output. 21.3. False.

22 Firm Supply

1. A ?rm has a cost function given by c(y) = 10 ????+ 1000. What is its

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supply curve?

22.1. The inverse supply curve is p = 20 y, so the supply curve is y = p/20.

2. A ?rm has a cost function given by c(y) = 10 ???? +1000. At what output is average cost minimized?

22.2. Set AC = MC to ?nd 10y + 1000/y = 20 y. Solve to get y? = 10.

3. If the supply curve is given by S(p) = 100+20p, what is the formula for the inverse supply curve?

22.3. Solve for p to get Ps(y)=( y?100)/20.

4. A ?rm has a supply function given by S(p)=4 p. Its ?xed costs are 100. If the price changes from 10 to 20, what is the change in its pro?ts? 22.4. At 10 the supply is 40 and at 20 the supply is 80. The producer’s surplus is composed of a rectangle of area 10×40 plus a triangle of area 1 2 ×10×40, which gives a total change in producer’s surplus of 600. This is the same as the change in pro?ts, since the ?xed costs don’t change.

5. If the long-run cost function is c(y)= ????+1, what is the long-run supply curve of the ?rm?

22.5. The supply curve is given by y = p/2 for all p ≥ 2, and y = 0 for all p ≤ 2. At p = 2 the ?rm is indi?erent between supplying 1 unit of

37

output or not supplying it.

6. Classify each of the following as either technological or market constraints: the price of inputs, the number of other ?rms in the market, the quantity of output produced, and the ability to produce more given the current input levels.

22.6. Mostly technical (in more advanced models this could be market), market, could be either market or technical, technical.

7. What is the major assumption that characterizes a purely competitive market?

22.7. That all ?rms in the industry take the market price as given.

8. In a purely competitive market a ?rm’s marginal revenue is always equal to what? A pro?t-maximizing ?rm in such a market will operate at what level of output?

22.8. The market price. A pro?t-maximizing ?rm will set its output such that the marginal cost of producing the last unit of output is equal to its marginal revenue, which in the case of pure competition is equal to the market price.

9. If average variable costs exceed the market price, what level of

38

output should the ?rm produce? What if there are no ?xed costs? 22.9. The ?rm should produce zero output (with or without ?xed costs).

10. Is it ever better for a perfectly competitive ?rm to produce output even though it is losing money? If so, when?

22.10. In the short run, if the market price is greater than the average variable cost, a ?rm should produce some output even though it is losing money. This is true because the ?rm would have lost more had it not produced since it must still pay ?xed costs. However, in the long run there are no ?xed costs, and therefore any ?rm that is losing money can produce zero output and lose a maximum of zero dollars.

11. In a perfectly competitive market what is the relationship between the market price and the cost of production for all ?rms in the industry?

22.11. The market price must be equal to the marginal cost of production for all ?rms in the industry.

23 Industry Supply

1. If S1(p)=p?10 and S2(p)=p?15, then at what price does the industry supply curve have a kink in it?

23.1. The inverse supply curves are P1(y1) = 10+y1 and P2(y2) = 15+y2. When the price is below 10 neither ?rm supplies output. When the price

39

is 15 ?rm 2 will enter the market, and at any price above 15, both ?rms are in the market. Thus the kink occurs at a price of 15.

2. In the short run the demand for cigarettes is totally inelastic. In the long run, suppose that it is perfectly elastic. What is the impact of a cigarette tax on the price that consumers pay in the short run and in the long run?

23.2. In the short run, the consumers pay the entire amount of the tax. In the long run it is paid by the producers.

3. True or false? Convenience stores near the campus have high prices because they have to pay high rents.

23.3. False. A better statement would be: convenience stores can charge high prices because they are near the campus. Because of the high prices the stores are able to charge, the landowners can in turn charge high rents for the use of the convenient location.

4. True or false? In long-run industry equilibrium no ?rm will be losing money. 23.4. True.

5. According to the model presented in this chapter, what determines

40

the amount of entry or exit a given industry experiences?

23.5. The pro?ts or losses of the ?rms that are currently operating in the industry.

6. The model of entry presented in this chapter implies that the more ?rms in a given industry, the (steeper, ?atter) is the long-run industry supply curve. 23.6. Flatter.

7. A New York City cab operator appears to be making positive pro?ts in the long run after carefully accounting for the operating and labor costs. Does this violate the competitive model? Why or why not? 23.7. No, it does not violate the model. In accounting for the costs we failed to value the rent on the license.

24 Monopoly

1. The market demand curve for heroin is said to be highly inelastic. Heroin supply is also said to be monopolized by the Ma?a, which we assume to be interested in maximizing pro?ts. Are these two statements consistent?

24.1. No. A pro?t-maximizing monopolist would never operate where the demand for its product was inelastic.

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2. The monopolist faces a demand curve given by D(p) = 100?2p. Its cost function is c(y)=2 y. What is its optimal level of output and price? 24.2. First solve for the inverse demand curve to get p(y) = 50? y/2. Thus the marginal revenue is given by MR(y) = 50? y. Set this equal to marginal cost of 2, and solve to get y = 48. To determine the price, substitute into the inverse demand function, p(48) = 50?48/2 = 26.

3. The monopolist faces a demand curve given by D(p) = 10 p?3. Its cost function is c(y)=2 y. What is its optimal level of output and price? 24.3. The demand curve has a constant elasticity of ?3. Using the formula p[1 + 1/ +=MC, we substitute to get p*1 ? 1/3] = 2. Solving, we get p = 3. Substitute back into the demand function to get the quantity produced: D(3) = 10×3?3.

4. If D(p) = 100/p and c(y)=????, what is the optimal level of output of the monopolist? (Be careful.)

24.4. The demand curve has a constant elasticity of ?1. Thus marginal revenue is zero for all levels of output. Hence it can never be equal to marginal cost.

5. A monopolist is operating at an output level where |ε| = 3. The government imposes a quantity tax of $6 per unit of output. If the

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demand curve facing the monopolist is linear, how much does the price rise?

24.5. For a linear demand curve the price rises by half the change in cost. In this case, the answer is $3.

6. What is the answer to the above question if the demand curve facing the monopolist has constant elasticity?

24.6. In this case p = k MC, where k =1 /(1?1/3) = 3/2. Thus the pricerises by $9.

7. If the demand curve facing the monopolist has a constant elasticity of 2, then what will be the monopolist’s markup on marginal cost? 24.7. Price will be two times marginal cost.

8. The government is considering subsidizing the marginal costs of the monopolist described in the question above. What level of subsidy should the government choose if it wants the monopolist to produce the socially optimal amount of output?

24.8. A subsidy of 50 percent, so the marginal costs facing the monopo- list are half the actual marginal costs. This will ensure that price equals marginal cost at the monopolist’s choice of output.

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9. Show mathematically that a monopolist always sets its price above marginal cost.

24.9. A monopolist operates where p(y)+yΔp/Δy = MC(y). Rearranging, we have p(y)=MC(y)?yΔp/Δy. Since demand curves have a negative slope, we know that Δp/Δy<0, which proves that p(y) >MC (y).

10. True or false? Imposing a quantity tax on a monopolist will always cause the market price to increase by the amount of the tax.

24.10. False. Imposing a tax on a monopolist may cause the market price to rise more than, the same as, or less than the amount of the tax.

11. What problems face a regulatory agency attempting to force a monopolist to charge the perfectly competitive price?

24.11. A number of problems arise, including: determining the true marginal costs for the ?rm, making sure that all customers will be served, and ensuring that the monopolist will not make a loss at the new price and output level.

12. What kinds of economic and technological conditions are conducive to the formation of monopolies?

24.12. Some appropriate conditions are: large ?xed costs and small marginal costs, large minimum e?cient scale relative to the market, ease

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of collusion, etc.

25 Monopoly Behavior

1. Will a monopoly ever provide a Pareto e?cient level of output on its own?

25.1. Yes, if it can perfectly price discriminate.

2. Suppose that a monopolist sells to two groups that have constant elasticity demand curves, with elasticity ε1 and ε2. The marginal cost of production is constant at c. What price is charged to each group? 25.2. pi =εic/(1 +εi) for i =1 ,2.

3. Suppose that the amusement park owner can practice perfect ?rst-degree price discrimination by charging a di?erent price for each ride. Assume that all rides have zero marginal cost and all consumers have the same tastes. Will the monopolist do better charging for rides and setting a zero price for admission, or better by charging for admission and setting a zero price for rides?

25.3. If he can perfectly price discriminate, he can extract the entire consumers’ surplus; if he can charge for admission, he can do the same. Hence, the monopolist does equally well under either pricing policy. (In practice, it is much easier to charge for admission than to charge a di?erent price for every ride.)

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4. Disneyland also o?ers a discount on admissions to residents of Southern California. (You show them your zip code at the gate.) What kind of price discrimination is this? What does this imply about the elasticity of demand for Disney attractions by Southern Californians? 25.4. This is third-degree price discrimination. Apparently the Disneyland administrators believe that residents of Southern California have more elastic demands than other visitors to their park.

26 Factor Markets

1. We saw that a monopolist never produced where the demand for output was inelastic. Will a monopsonist produce where a factor is inelastically supplied?

26.1. Sure. A monopsonist can produce at any level of supply elasticity.

2. In our example of the minimum wage, what would happen if the labor market was dominated by a monopsonist and the government set a wage that was above the competitive wage?

26.2. Since the demand for labor would exceed the supply at such a wage, we would presumably see unemployment.

3. In our examination of the upstream and downstream monopolists we de- rived expressions for the total output produced. What are the

46

appropriate expressions for the equilibrium prices, p and k?

26.3. We ?nd the equilibrium prices by substituting into the demand functions. Since p = a?by, we can use the solution for y to ?nd p =

3a + c4

Since k = a?2bx, we can use the solution for x to ?nd k =

a + c2

.

27 Oligopoly

1. Suppose that we have two ?rms that face a linear demand curve p(Y )= a ? bY and have constant marginal costs, c, for each ?rm. Solve for the Cournot equilibrium output.

27.1. In equilibrium each ?rm will produce (a?c)/3b, so the total industry output is 2(a?c)/3b.

2. Consider a cartel in which each ?rm has identical and constant marginal costs. If the cartel maximizes total industry pro?ts, what does this imply about the division of output between the ?rms?

27.2. Nothing. Since all ?rms have the same marginal cost, it doesn’t matter which of them produces the output.

3. Can the leader ever get a lower pro?t in a Stackelberg equilibrium than he would get in the Cournot equilibrium?

27.3. No, because one of the choices open to the Stackelberg leader is to

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choose the level of output it would have in the Cournot equilibrium. So it always has to be able to do at least this well.

4. Suppose there are n identical ?rms in a Cournot equilibrium. Show that the absolute value of the elasticity of the market demand curve must be greater than 1/n. (Hint: in the case of a monopolist, n = 1, and this simply says that a monopolist operates at an elastic part of the demand curve. Apply the logic that we used to establish that fact to this problem.)

27.4. We know from the text that we must have p*1?1/n|ε|]=MC. Since MC > 0, and p>0, we must have 1 ? 1/n|ε| > 0. Rearranging this inequality gives the result.

5. Draw a set of reaction curves that result in an unstable equilibrium 27.5. Make f2( y1) steeper than f1( y2).

6. Do oligopolies produce an e?cient level of output?

27.6. In general, no. Only in the case of the Bertrand solution does price equal the marginal cost.

28 Game Theory

1. Consider the tit-for-tat strategy in the repeated prisoner’s dilemma. Suppose that one player makes a mistake and defects when he meant

48

to cooperate. If both players continue to play tit for tat after that, what happens?

28.1. The second player will defect in response to the ?rst player’s (mistaken) defection. But then the ?rst player will defect in response to that, and each player will continue to defect in response to the other’s defection! This example shows that tit-for-tat may not be a very good strategy when players can make mistakes in either their actions or their perceptions of the other player’s actions.

2. Are dominant strategy equilibria always Nash equilibria? Are Nash equilibria always dominant strategy equilibria?

28.2. Yes and no. A player prefers to play a dominant strategy regardless of the strategy of the opponent (even if the opponent plays her own dominant strategy). Thus, if all of the players are using dominant strategies then it is the case that they are all playing a strategy that is optimal given the strategy of their opponents, and therefore a Nash equilibrium exists. How- ever, not all Nash equilibria are dominant strategy equilibria; for example, see Table 28.2.

3. Suppose your opponent is not playing her Nash equilibrium strategy. Should you play your Nash equilibrium strategy?

28.3. Not necessarily. We know that your Nash equilibrium strategy is

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the best thing for you to do as long as your opponent is playing her Nash equilibrium strategy, but if she is not then perhaps there is a better strategy for you to pursue.

4. We know that the single-shot prisoner’s dilemma game results in a dominant Nash equilibrium strategy that is Pareto ine?cient. Suppose we allow the two prisoners to retaliate after their respective prison terms. Formally, what aspect of the game would this a?ect? Could a Pareto e?cient outcome result?

28.4. Formally, if the prisoners are allowed to retaliate the payo?s in the game may change. This could result in a Pareto e?cient outcome for the game (for example, think of the case where the prisoners both agree that they will kill anyone who confesses, and assume death has a very low utility).

5. What is the dominant Nash equilibrium strategy for the repeated prisoner’s dilemma game when both players know that the game will end after one million repetitions? If you were going to run an experiment with human players for such a scenario, would you predict that players would use this strategy?

28.5. The dominant Nash equilibrium strategy is to defect in every round. This strategy is derived via the same backward induction process that

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