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Answers to Questions and Problems in the Text

Chapter 2 1.The statement “Talk is cheap because supply exceeds demand” makes sense if we interpret it to mean

that the quantity supplied of talk exceeds the quantity demanded at a price of zero. Imagine a downward-sloping demand curve that hits the horizontal, quantity, axis to the left of where the upward-sloping supply curve hits the axis. (The correct aphorism is “Talk is cheap until you hire a lawyer.”)

2. The demand curve shifts to the left from D1 to D2 by 30%, which is the distance between Q0 and Q4. For supply curve S1, the price drops from p0 to p1, a change less than 30%. For a steeper supply curve S2, the price decreases to p2, a larger decrease, yet still smaller than 30%. Accordingly, the equilibrium quantity changes less than 30% as well.

Figure 2.8

3. Both the demand and supply of guns have increased; i.e., demand shifted up to the right and supply shifted down to the right. However, the results suggest that the increase in demand was greater than the increase in supply and this led to an increase in both equilibrium price and quantity.

4. Suppose the original equilibrium price and quantity is P1 and Q1, respectively. A drop in production shifts the supply curve up to the left. Suppose the new equilibrium price and quantity is P2 and Q2. It is obvious that we will have Q2 Q1 and P2 P1. A price control at P1 after the supply curve has shifted creates excess demand, Q3 – Q1, where Q3 and Q1 are the quantities supplied (on the new supply curve) and demanded at price P1. A price ceiling may lead to the creation of a black market in which the prices as high as P2 could be charged. (See Figure 2.3.)

Figure 2.9

Chapter 2 Supply and Demand 5


5. Shifts of both the U.S. supply and the U.S. demand curves affected the U.S. equilibrium. U.S. beef consumers’ fear of mad cow disease caused their demand curve in the figure to shift slightly to the left from D1 to D2. In the short run, total U.S. production was essentially unchanged. Because of the ban on exports, beef that would have been sold in Japan and elsewhere was sold in the United States, causing the U.S. supply curve to shift to the right from S1 to S2. As a result, the U.S. equilibrium changed from e1 (where S1 intersects D1) to e2 (where S2 intersects D2). The U.S. price fell 15% from p1 to p2 0.85p1, while the quantity rose 43% from Q1 to Q2 1.43Q1. Comment: Depending on exactly how the U.S. supply and demand curves had shifted, it would have been possible for the U.S. price and quantity to have both fallen. For example, if D2 had shifted far enough left, it could have intersected S2 to the left of Q1, and the equilibrium quantity would have fallen.

Figure 2.10

6. The increase in demand for ethanol has increased the demand for corn, an input in the production of ethanol. The market demand for corn shifts to the right, increasing both the equilibrium price and quantity. The increase in price reduces the quantity of corn that is consumed as food. The increase in price induces more farmers to produce corn and increases the demand for agricultural land. See the New York Times article: “Ethanol is Feeding Hot Market for Farmland” by Monica Davey, August 8, 2007.

7. Audio-PowerPoint answer by James Dearden is also available (2B Brazilian Soybeans).

a. World demand decreased (China’s decrease in imports) and world supply increased (the increase in supply from the United States’ bumper crop was greater than the decrease in supply from Asian soy rust). The rightward shift of world supply and the leftward shift of world demand both work to lower the world price of soybeans.

b. In this case, since supply increased and demand decreased (the shifts were in opposite directions), the price is unambiguously lower. In general, however, we have to know the magnitude of the shifts in addition to their direction to predict accurately the effect on price and quantity.



8. After the quota was reimposed, the equilibrium price would increase and quantity decrease as shown in the figure below.

Figure 2.11

9. In the absence of price controls, the leftward shift of the supply curve as a result of Hurricane Katrina would push market prices up from p0 to p1 and reduce quantity from q0 to q1. At a government imposed maximum price of p2, consumers would want to purchase qd units but producers would only be willing to sell qs units. The resulting shortage would impose search costs on consumers, making them worse off. The reduced quantity and price also reduced firms’ profits.

Figure 2.12

10. Although the increase in supply (from the export ban) and the decrease in demand (the prohibition of government ministries from purchasing beef) worked to lower the equilibrium price, the increase in demand from higher incomes worked against these actions, keeping the price of beef high.



11. In the figure below, the no-quota total supply curve, S in panel c, is the horizontal sum of the U.S. domestic supply curve, Sd, and the no-quota foreign supply curve, Sf. At prices less than ,p foreign suppliers want to supply quantities less than the quota, .Q As a result, the foreign supply curve under the quota, ,fS is the same as the no-quota foreign supply curve, Sf, for prices less than .p At prices above ,p foreign suppliers want to supply more but are limited to .Q Thus the foreign supply curve with a quota, ,fS is vertical at Q for prices above .p The total supply curve with the quota, ,S is the horizontal sum of Sd and .fS At any price above ,p the total supply equals the quota plus the domestic supply. For example, at p*, the domestic supply is *

dQ and the foreign supply is ,fQ so the total supply is * .d fQ Q Above ,p S is the domestic supply curve shifted Q units to the right. As a result, the portion of S above p has the same slope as Sd. At prices less than or equal to ,p the same quantity is supplied with and without the quota, so S is the same as S. At prices above ,p less is supplied with the quota than without one, so S is steeper than S, indicating that a given increase in price raises the quantity supplied by less with a quota than without one.

Figure 2.13

12. Figure 2.14 reproduces the no-quota total American supply curve of steel, S, and the total supply curve under the quota, ,S which we derived in the answer to the previous question. At a price below

,p the two supply curves are identical because the quota is not binding: It is greater than the quantity foreign firms want to supply. Above ,p S lies to the left of S. Suppose that the American demand is relatively low at any given price so that the demand curve, Dl, intersects both the supply curves at a price below .p The equilibria both before and after the quota is imposed are at e1, where the equilibrium price, p1, is less than .p Thus if the demand curve lies near enough to the origin that the quota is not binding, the quota has no effect on the equilibrium. With a relatively high demand curve, Dh, the quota affects the equilibrium. The no-quota equilibrium is e2, where Dh intersects the no-quota total supply curve, S. After the quota is imposed, the equilibrium is e3, where Dh intersects the total supply curve with the quota, .S The quota raises the price of steel in the United States from p2 to p3 and reduces the quantity from Q2 to Q3.

Since the price of high-quality tequila has remained constant, in the face of increase in demand we should conclude that the supply is very elastic (almost horizontal).

Since the drop in the price of low-quality tequila is relatively large, the demand should be inelastic.



Figure 2.14

13. The problem states that it takes 7 years for an agave plant to be ready for harvesting, and obviously the amount of tequila distilled depends on the harvest. Therefore, changes in the price of tequila cannot immediately affect the supply of tequila. Thus in the short run, the supply is almost vertical (price elasticity is almost equal to zero) and price changes do not immediately result in changes in the quantity supplied. However, in response to lower prices, less agave plants will be planted and in the long run the harvest will go down. This means the long-run elasticity of supply is larger.

Since the price of high-quality tequila has remained constant, in the face of increase in demand we should conclude that the supply is very elastic (almost horizontal).

Since the drop in the price of low-quality tequila is relatively large, the demand should be inelastic.

14. As the rental price increases due to increase in demand, more apartments (in other uses) and hotel rooms become available. People will rent houses that otherwise would not be available for renting. However, there is a point beyond which capacity is reached and the hotel rooms and apartments cannot be increased in the short run. At this point the supply becomes vertical. The situation is depicted in Figure 2.15.

Figure 2.15



15. When Viagra was introduced, the demand curve for seal genitalia moved to the left from D1 to D2, resulting in a lower quantity and price. If the original supply curve is S2, then under demand curve D2, zero will be sold on the market even if there is a positive demand at various prices.

Figure 2.16

16. The numbers suggest that labor demand is inelastic. In Figure 2.17, the supply curve shifts to the right by 11%, yet the decrease in equilibrium wage is only 3.2%.

Figure 2.17

17. Audio-PowerPoint answer by James Dearden is also available (3A Economic PhDs).

a. In the very short run the supply is fixed. This means that the supply curve is vertical. In this case shift in demand only changes the price, and quantity exchanged remains the same. This means that QS% 0, and thus the short-run elasticity of supply is S %QS/%PS 0. Figure 2.18 illustrates this.

Figure 2.18



b. As salary remains high, more graduate students will be attracted to doctoral programs in economics. In the long run, the supply curve will become more elastic (upward sloping). This means that the long run elasticity of supply becomes positive, i.e., L %QL/%PL 0. Figure 2.19 illustrates this.

Figure 2.19

In general L S. This is true even if in the short run the supply curve is not perfectly inelastic. Figure 2.20 illustrates this point.

Figure 2.20

18. Use Equation 2.28 from the text to solve for the change in price.

.dP

d

a. If demand is perfectly inelastic, the demand curve is vertical. The supply curve shifts up by $1, and all of the incidence falls on consumers. Price increases by $1, and there is no change in quantity. Since 0, dP/d 1.

Figure 2.21



b. The demand curve is horizontal when perfectly elastic. The supply curve shifts up by $1. Price paid by consumers remains at p (the pre-tax level). Price received by sellers is p – (the price less the tax). Quantity falls to the intersection of the new supply curve and the original demand curve. Suppliers absorb the entire burden of the tax because consumers have no willingness to pay higher prices. Since , dP/d 0.

Figure 2.22

c. When supply is perfectly inelastic, the supply curve is vertical. Thus, shifting the supply curve upward would have no effect on the equilibrium quantity or price paid by consumers. Sellers would bear the entire burden of the tax. Since 0, dP/d 0.

Figure 2.23

d. When supply is perfectly elastic, the supply curve is horizontal. The supply curve shifts up by $1 increasing price by $1. The quantity falls and the incidence falls entirely on the consumer. Since , dP/d 1.



Figure 2.24 e. If the demand curve is perfectly elastic (horizontal), and the supply curve is perfectly inelastic

(vertical), the effect of a tax would be no change in equilibrium quantity and no change in price paid by consumers, and sellers would bear the entire burden of the tax. Since and 0, dP/d 0.

Figure 2.25

19. When the supply curve is upward sloping and the demand curve is vertical (perfectly inelastic demand), or when the demand is downward sloping and the supply curve is horizontal (perfectly elastic supply) the firms can pass all the tax burden completely to the consumers.

Figure 2.26

20. When the supply curve is upward sloping and the demand curve is horizontal (perfectly elastic demand), or when the demand curve is downward sloping and the supply curve is vertical (perfectly inelastic supply) there will be 0% pass through of the taxes.

Figure 2.27



21. We showed that, in a competitive market, the effect of a specific tax is the same whether it is placed on suppliers or demanders. Thus if the market for milk is competitive, consumers will pay the same price in equilibrium regardless of whether the government taxes consumers or stores.

22. The law would create a price ceiling (at 110% of the pre-emergency price). Because the supply curve shifts substantially to the left during the emergency, the price control will create a shortage: A smaller quantity will be supplied at the ceiling price than will be demanded.

23. The demand curve for pork is Q 171 20p 20pb 3pc 2Y. As a result, Q/Y 2. A $100 increase in income causes the quantity demanded to increase by 0.2 million kg per year.

24. To solve this problem, we first rewrite the inverse demand functions as demand functions and then add them together. The total demand function is:

1 2 (120 ) (60 1/2 )Q Q Q p p

180 1.5 .p

Figure 2.28

25. In Figure 2.29 below, line ad is the demand function for college students, and line ed is the demand function for other town residents. Line abc is the total demand function. Mathematically, the demand function can be expressed as:

Q 120 p when p 60

Q 240 3p when p 60.

Figure 2.29



26. The total demand is Q Qs Ql 15.6p0.563 16p0.296. At a price of 40 cents, small firms will demand 1.96 million Kbps and large firms will demand 5.37 million Kbps for a total of 7.33 million Kbps.

27. Price elasticity of demand for small firms is 0.563 and for large firms is 0.296. A demand function of the general form Q Ap has a constant elasticity of .

28. Equating the right sides of the supply and demand functions and using algebra, we find:

0.75 ln(p) 2.4 0.15 ln(pt)

ln(p) 3.2 0.2 ln(pt).

We then set pt 110, solve for ln(p)

ln(p) 3.2 0.2(4.7),

and exponentiate ln(p) to obtain the equilibrium price:

p $62.80/ton.

Substituting p into the supply curve and exponentiating, we determine the equilibrium quantity:

Q 11.91 million short tons/year.

29. The demand elasticity is 2.65/21 0.13 for prenatal smokers.

30. The elasticity of demand is (dQ/dp)(p/Q) (9.5 thousand metric tons per year per cent) (45 cents/1275 thousand metric tons per year) 0.34. That is, for every 1% fall in the price, a third of a percent more coconut oil is demanded. The cross-price elasticity of demand for coconut oil with respect to the price of palm oil is (dQ/dPp)(Pp/Q) 16.2 (31/1275) 0.39.

31. The expected percentage price change will be equal to the percentage change in quantity divided by the price elasticity. 10.4/(1.6) 6.5. The increase in supply would cause the price of beef to fall by 6.5%. Any other price change is likely a result of a change in demand.

32. Price elasticity is the percentage change in quantity divided by the percentage change in price. 8.3/21 0.395. An elasticity coefficient less than one indicates an inelastic demand.

33. Audio-PowerPoint answer by James Dearden is also available (3B Price Elasticity Health Ins.).

a. Suppose Q is the number of insured and Q is the change in the number of insured. An increase in the number of uninsured is the same as a decrease in the number of insured. Therefore an increase in the number of uninsured by 300,000 can be denoted by Q 300,000; i.e., number of insured has decreased by 300,000. Now we know:

Q%/P% (Q/Q)/(P/P)

and that

P/P 0.01.

When

Q 200,000,000,

we have:

(Q/Q)/(P/P)

(300,000/200,000,000)/0.01

0.15.



b. When Q 220,000,000, we have:

(Q/Q)/(P/P)

(300,000/220,000,000)/0.01

0.1364.

This shows that the price elasticity depends on the number of people who are insured.

34. Start by using Equation 2.2 with the usual values for income and chicken prices for demand 206 20 20 bQ p p , and Equation 2.7 88 40Q p for supply. As in solved Problem 2.1,

the effect of a change in the price of beef on the price of pork can be shown to be

/ 200.33.

/ / 40 ( 20)b

b

D pp

p S p D p

Thus a $0.60 increase in the price of beef will result in a $0.20 increase in the price of pork.

35. The demand curve is Q 114.8 0.656p. The supply curve without the ANWR production is Q 57.4 0.492p. p 50. The shock will change the supply function to Q 54.4 0.492p. Hence p 52.61, increasing by 5.2%. With the ANWR production, the supply function after the shock will be Q 55.2 0.492p. Hence p 51.92, indicating a 3.8% price increase.

36. A subsidy is essentially a negative tax. The supply curve shifts down by the amount of the tax, and the new equilibrium is determined by 286 20p = 40(p $1.05) The new equilibrium occurs at a price of $2.60 and a quantity of 234. Consumers receive $0.70 of the $1.05 subsidy.



37. The numbers suggest that cigarette demand is inelastic. Therefore, government can increase tax revenue by raising cigarette tax. The average federal and state tax is currently 84.5 c . A 10 c increase in federal tax coupled with a 2.8 c increase in state taxes will reduce the cigarette demand by 0.3 12.8/84.5 4.5%. Without an increase in state taxes, the increase in federal tax will reduce the cigarette demand by 0.3 10/84.5 3.6%.

38. Tax incidence for almonds: 12/(12 0.47) 0.96; for cotton, 0.73/(0.73 0.68) 0.52; for processing tomatoes, 0.64/(0.64 0.26) 0.71.

39. Differentiating quantity, Q(p()), with respect to , we learn that the change in quantity as the tax changes is (dQ/dp)(dp/d). Multiplying and dividing this expression by p/Q, we find that the change in quantity as the tax changes is (Q/p)(dp/d). Thus the closer is to zero, the less the quantity falls, all else the same.

Because R p()Q(p()), an increase in the tax rate changes revenues by

,dR dp dQ dp

Q pd d dp d

using the chain rule. Using algebra, we can rewrite this expression as

1 1 .dR dp dQ dp dQ p dp

Q p Q Qd d dp d dp Q d

Thus the effect of a change in on R depends on the elasticity of demand, . Revenue rises with the tax, given an inelastic demand (0 > > 1), and falls with an elastic demand, < 1.

40. We can determine how the total wage payment, W wL(w), varies with respect to w by differentiating. We then use algebra to express this result in terms of an elasticity:

1 (1 ),dW dL dL w

L w L Ldw dw dw L

where is the elasticity of demand of labor. The sign of dW/dw is the same as that of 1 . Thus total labor payment decreases as the minimum wage forces up the wage if labor demand is elastic, < 1, and increases if labor demand is inelastic, > 1.

41. a. The demand function is: 0.422000 Q P , and the inverse supply function is: 1.5 wP P ,

therefore the equilibrium price and quantity are:

*

* 0.42 0.42

1.5

2000 (1.5 ) 1686.83

w

w w

P P

Q P P

Therefore, *

1.42 1.421686.83 ( 0.42) 708.47 .w ww

dQP P

dP



b. As shown in Figure 2.41, The total $3.00 specific tax shifts up the supply curve by $3.00. Therefore the tax increases equilibrium retail price by $3.00 and decreases equilibrium retail quantity from Q1 to Q2, where:

0.421

0.422 1

2000 (1.5 )

2000 (1.5 3.00) .

w

w

Q P

Q P Q

Figure 2.41

Math: Suppose the total specific tax is 3.00 .

Since equilibrium retail price 1.5 wP P , then 1 0

P

, which implies that the specific

tax increases equilibrium retail price.

Since equilibrium retail quantity 0.422000 (1.5 ) wQ P , then:

1.422000 ( 0.42) (1.5 ) 0,

w

QP

which implies that the specific tax decreases equilibrium retail quantity. c. With the specific tax in place, the equilibrium price and quantity of cigarettes are:

*

* 0.42

1.5

2000 (1.5 ) .

w

w

P P

Q P

Then we have:

*1.42 1.422000 ( 0.42) 1.5 (1.5 ) 1260 (1.5 ) .

w w

w

QP P

P

Therefore: *

1.42

2.42

2.42

( ) ( 1260 (1.5 ) )

1260 ( 1.42) (1.5 )

1789.2 (1.5 ) .

w

QP w

w

w

P

P

P



Since the total specific tax is 3.00 , then: *

2.42( )

1789.2 (1.5 3) 0,

w

QP

wP

which implies that the change in the wholesale price will have a less negative effect on the equilibrium quantity if the tax rate increases.

42. (a) The inverse demand function is: 100020

Y

P Q , then the demand function is:

100020

Y

Q P

The inverse supply function is: 2 40

Q Y

P , then the supply function is:

220

Y

Q P

Setting the quantity demanded equal the quantity supplied, we have:

* * *1000 20001000 2 2

20 20 3 30 20 3 60

Y Y Y Y YP P P Q P

Therefore the equilibrium price and quantity in terms of Y are:

*

*

1000

3 302000

3 60

YP

YQ

(b) * 1

030

dP

dY, therefore a decrease in median income leads to a decrease in the equilibrium

rental price.

Chapter 3

1. Answers to parts a through d.

a. Imperfect substitutes b. Complements c. Independent (neither complements nor substitutes) d. Perfect substitutes for many consumers

2. In this case, charity is considered a good. Increases in charity increase his utility, but with diminishing returns, similar to most goods. Thus, indifference curves would have the typical convex shape, with quantity of charity giving on one axis and all other goods on the other.

3. If the neutral product is on the vertical axis, the indifference curves are parallel vertical lines.



4. U a(A B); where a is a positive constant, A denotes number of tickets to the opera, and B denotes number of tickets to the baseball games. Note that the diagram in the 4th edition is incorrect.

Figure 3.8

5. Sofia’s indifference curves are right angles (as in panel b of Figure 3.5). Her utility function is U min(H, W), where min means the minimum of the two arguments, H is the number of units of hot dogs, and W is the number of units of whipped cream.

6. Indifference curves are convex because the rate at which consumers are willing to trade one good for another varies with their endowments of the goods. Along a single indifference curve, utility is constant. In order to keep the consumer’s utility at the same level, some of one good must be forgone in order to consume more of the other without altering total utility. This trade-off creates the negative slope. The convexity comes from the changes in relative endowments as one moves along the curve. As one moves down an indifference curve, the marginal utility of the good on the X-axis falls while the marginal utility of the good on the Y-axis rises. Thus, as the endowment of Y falls, the consumer is willing to trade less and less of Y to get an additional unit of X, creating the nonlinear relationship. The slope of the indifference curve, the marginal rate of substitution, is the negative ratio of the marginal utilities. Diminishing marginal utilities makes the second derivative of the indifference curve positive.

7. In this case, the commodities are perfect substitutes above the threshold level, resulting in linear rather than convex indifference curves. The “more is better” assumption is violated below the threshold level. For example, an increase in X from 1 to 2 provides no additional utility unless Y is greater than 3.

Figure 3.9 8. Describe the indifference curves of the consumer for Canadian and American gasoline: Presumably the consumer views American gasoline and Canadian gasoline as perfect



substitutes, so the indifference curves are straight lines with a slope of 1. Draw the initial budget line where prices are relatively low in the United States, and show the optimum: For specificity, we assume that initially a gasoline budget of Y bought 15 gallons of gasoline in the United States but only 10 gallons in Canada: budget line L1. The consumer picks the highest indifference curve, I2, that touches this budget constraint. The optimum, e1, is a corner solution: The consumer buys gasoline only in the United States. More generally, as long as the budget line is flatter than the indifference curve, the consumer buys only American gasoline. Letting pa be the American price and pc be the Canadian price, the slope of the budget line, pa/pc, is greater (closer to zero) than the slope of the indifference curves, 1, because pa pc. Draw the new budget line where prices are relatively low in Canada and show the new optimum: After the price change, pa/pc 1, so the consumer buys only Canadian gasoline. In the figure, after the price change, Y buys 15 gallons in Canada and only 10 gallons in the United States: budget line L1. As a result, the consumer only buys in Canada at e2, where I2 hits L2. Thus, the consumer buys in whichever country the gasoline is cheapest: a corner solution.



9. In Figure 3.10, consumer can afford to buy up to 12 thousand gallons of water a week if not constrained. The opportunity set, area A and B, is bounded by the axes and the budget line. A vertical line at 10 thousand on the water axis indicates the quota. The new opportunity set, area A, is bounded by the axes, the budget line, and the quota line. Because of the rationing, the consumer loses part of the original opportunity set: the triangle B to the right of the 10-thousand-gallons quota line. The consumer has fewer opportunities because of rationing.

Figure 3.10

10. When all prices and income double, the budget constraint remains unchanged. The consumer is indifferent between income of $100 and prices of $10 and $20 for X and Z, respectively, and income of $50 and prices of $5 and $10 for X and Z.

11. Consumers in both cities will equate their respective marginal rates of substitution with the relative price in their local market. In Boston / 2/1A TMU MU and in San Diego / 1/2A TMU MU . Note that this does not require that consumers in Boston have the same tastes as the consumers in San Diego, only that both have well behaved preferences.

12. If the price of pizza decreases, then the budget line will become flatter. Spenser’s new optimal bundle, as illustrated below, is an interior bundle, with positive quantities of both burritos and pizza.

Figure 3.11



13. Different slopes of their budget lines are determined by the relative prices of the two kinds of cars. We cannot make any unambiguous statements about how much each would buy. However, if we know that Bob’s budget line goes through Nigel’s optimal bundle, then we know that they share the same budget. Now, we know that Nigel will buy more sedans because they are less expensive compared to SUVs for him.

14. There are 5 different scenarios possible with such a kinked utility function. In the diagram below, let MRS1 1( / )A BMU MU be the slope of the indifference curve above and to the left of the kink and

MRS2 2( / )A BMU MU be the slope of the indifference curve below and to the right of the kink.

a. If the absolute value of the relative price is greater than the absolute value of MRS1 we get a corner solution at B*.

b. If the absolute value of the relative price is less than the absolute value of MRS2 we get a corner solution at A*.

c. When the absolute value of the relative price is equal to the absolute value of MRS1 the optimal bundle lies anywhere between B* and C*.

d. When the absolute value of the relative price is equal to the absolute value of MRS1 the optimal bundle lies anywhere between A* and C*.

e. Finally, when the absolute value of the relative prices lies between the absolute values of MRS1 and MRS2, the optimal bundle will be C*.

Since the utility function is not differentiable at C*, the optimality conditions can not all be used.

Figure 3.12



15. Suppose that Dale purchases two goods at prices p1 and p2. If her original income is Y, the intercept of the budget line on the Good 1 axis (where the consumer buys only Good 1) is Y/p1. Similarly, the intercept is Y/p2 on the Good 2 axis. A 50% income tax lowers income to half its original level, Y/2. As a result, the budget line shifts inward toward the origin. The intercepts on the Good 1 and Good 2 axes are Y/(2p1) and Y/(2p2), respectively. The opportunity set shrinks by the area between the original budget line and the new line.

16. Both internet sales and total sales will fall if the internet tax is imposed. Both sales will fall more as the tax rates increase.

Figure 3.13

17. The change in the tax on ethanol-blended fuel will change the slope of the budget constraint.

a. When the two goods are perfect substitutes and prices are identical, the individual will choose any point on the line AB in the lefthand panel below. When the relative price of the ethanol blend falls, consumers move to the corner solution at C.

b. When the two goods are imperfect substitutes, the drop in the relative price of the ethanol-blend will induce consumers to purchase more. The sales of gasoline may rise, fall, or stay the same, depending on the size of the income effect.

Figure 3.14



18. In what follows it is assumed we have the number of $12 shoes on the vertical axis and the number of $3 shoes on the horizontal axis. After the tariff, the $3 shoes cost 3 1.67 5.01 and the $12 shoes cost 12 1.37 16.44. The slope of the budget constraint before the tariff is –3/12 0.25 and after tariff it becomes –5.01/16.44 0.305. Note that before tariff, Laura had to give up 0.25 units of expensive shoes to get a cheap shoe, but after the tariff, this rate increases to 0.3. This means that after the tariff cheap shoes have become relatively more expensive. Assuming the nominal income of Laura is fixed, as a result of the tariff the budget constraint shifts inward and becomes steeper. Since as a result of tariffs real income (the purchasing power) has decreased, Laura will buy less of both types of shoes, but since $3 shoes are now relatively more expensive, there will be some substitution away from $3 shoes to $12 shoes, so Laura will end up buying relatively more expensive shoes.

19. See Figure 3.15. Suppose you are at point O. Assume you are willing to accept AD units of Y in order to give up OA units of X. On other hand, to buy OB units of X(OA OB) you are willing to pay BC units of Y. Therefore you value the same unit of X differently, depending on whether you want to give it up or you want to buy it. The indifference curve would be DOC and has a kink at the endowment point. The graph shows different budget constraints at three different relative prices for X. As you can see, the optimum does not change for small changes of relative price. Therefore for a range of prices the demand curve for X will be vertical.

Figure 3.15

20. She will prefer the bundle with two anchovies and two boxes of biscuits because if MRS –1 at this point, and the indifference curve is convex, she will be unwilling to trade down to any lower quantity of biscuits at a one-for-one ratio. At the point where she has three anchovies and one box of biscuits, her indifference curve will have a slope of less than –1. Thus, if forced to trade one for one, her utility level would fall.

21. Andy’s marginal utility of apples divided by the price of apples is 3/2 1.5. The marginal utility

for kumquats is 5/4 1.2. That is, a dollar spent on apples gives him more extra utils than a dollar spent on kumquats. Thus he maximizes his utility by spending all his money on apples and buying 40/2 20 pounds of apples.

22. Given a quasilinear marginal utility function, U(q1, q2) u(q1) q2, the marginal utility of the first good is U1 U(q1, q2)/q1 du(q1)/dq1 > 0, which is independent of q2 because u(q1) is not a function of q2. The marginal utility of the second good is U2 U(q1, q2)/q2 1, which is independent of q1 and q2. Using Equation 3.3, we find that the marginal rate of substitution is MRS U1/U2 [du(q1)/q1]/1 [du(q1)/dq1] < 0, so the indifference curves are downward sloping. The MRS is independent of q2 because du(q1)/dq1 is independent of q2. Thus, at any given q1, the MRS, which is the slope of the indifference curve, must be the same for all the indifference curves. Using the same reasoning as in the text, these indifference curves must be parallel.



23. David’s marginal utility of B is U/B (B 2Z )/B 1 and his marginal utility of Z is 2. If we plot B on the vertical axis and Z on the horizontal axis, the slope of David’s indifference curve is UZ/UB 2. The marginal utility from one extra unit of Z is twice that from one extra unit of B. Thus if the price of Z is less than twice that of B, David buys only Z (the optimal bundle is on the Z axis at Y/pZ, where Y is his income and pZ is the price of Z). If the price of Z is more than twice that of B, David buys only B. If the price of Z is exactly twice as much as that of B, he is indifferent between buying any bundle along his budget line.

24. In order for him to be maximizing his utility, he must set his consumption such that the marginal utility per dollar (MU/p) of the last unit consumed is equal across commodities. In this case:

10/10 5/2.

Specifically, the marginal utility per dollar is greater for cookies. Therefore he should decrease his consumption of books and increase his consumption of cookies.

25. We can solve this problem by noting that Nadia determines her optimal bundle by equating the ratios of each good’s marginal utility to its price.

a. At the original prices, this condition is UR/10 2RC 2R2 UC/5. Thus by dividing both sides of the middle equality by 2R, we know that her optimal bundle has the property that R C. Her budget constraint is 90 10R 5C. Substituting C for R, we find that 15C 90, or C 6 R.

b. At the new price, the optimum condition requires that UR/10 2RC R2 UC/10, or 2C R. By substituting this condition into her budget constraint, 90 10R 10C, and solving, we learn that C 3 and R 6. Thus as the price of chickens doubles, she cuts her consumption of chicken in half but does not change how many slabs of ribs she eats.

26. The marginal rate of substitution is B/C.

The marginal condition is B/C 1/2.

B (1/2)C

Substituting into the budget constraint yields:

2B C 120

2C 120

C* 60, B* 30

When the price changes due to the tax, the new marginal condition is

B (1/3)C

Substituting into the new budget constraint yields:

3B C 120

2C 120

C* 60, B* 20;



Figure 3.16

27. For Linda, MUS 2T and MUT 2S.

a. 50S 50T 500

b. MRS –(U/S)/(U/T) T/S

c. MRT ps/pt

T/S 50/50

T S

Substitute into budget constraint

50T 50T 500

T 5, S 5

Figure 3.17

28. Diogo’s 2

1

3 .q

qMRS Equating this with the relative price (1/2) and substituting into the budget

constraint 2 2 2 2 2100 3 8 .P q P q q His optimal bundle is *1 75,q *

2 12.5.q



29. There are two steps to solve these problems. First, set MRS MRT, and then satisfy the budget constraint.

MRS MUX/MUZ

MRS (20XZ/10X2)

MRT (10/5)

(20XZ/10X2) (10/5)

Z X

10X 5Z 150

substituting Z for X, 15Z 150, Z* 10 and X* 10.

Figure 3.18

30. MUZ ABZ1 1

MUB 1 AB1Z

MRS MUB/MUZ (1 AB1Z)/(ABZ1 1).

31.

22 1 2 1 2 2

2 21 1 2 1 2

q q q q q qU

q q q q q

21 1 2 1 2 1

2 22 1 2 1 2

.q q q q q qU

q q q q q

Equating MRS with relative price:

22 1

21 2

q P

q P

and substituting into the budget constraint:

1

21

1 1 2 12

.P

Y Pq P qP

Solving for q1:

1 1

21 1 2

Yq

P P P



and by symmetry:

2 1

22 1 2

.Y

qP P P

32. Given the original utility function, U, the consumer’s marginal rate of substitution is U1/U2. If V(q1, q2) F(U(q1, q2)), the new marginal rate of substitution is V1/V2 [(dF/dU)U1]/[(dF/dU)U2] U1/U2, which is the same as originally.

33. Given the original utility function, U: the consumer’s marginal rate of substitution is U1/U2. If V(q1, q2) = F(U(q1, q2)), the new marginal rate of substitution is V1/V2 [(dF/dU)U1]/[(dF/dU)U2] U1/U2, which is the same as originally.

34. From the budget constraint: 1 2 2 1/q Y P q P (put parentheses around “Y P2q2”) and substituting into the utility function:

2 2 12 2

1

( ) .Y P q

U q qP

To maximize utility with respect to q2, set the first derivative to zero.

1

2 2 2 2 212 2

2 1 1 1

(1 ) 0.Y P q P Y P qU

q qq P P P

Collecting terms and solving for

2 1 22

(1 )( , , ) .

Yq P P Y

P

In turn,

1 1 21

( , , )Y

q P P YP

.

35. From equation 3.26 and 3.27 in the text:

1 1 21

( , , )Y

q P P YP

and 2 1 22

( , , ) (1 ) .Y

q P P YP

If we multiply all arguments by some positive value , then:

1 1 21 1

( , , )Y Y

q P P YP P

and 2 1 22 2

( , , ) (1 ) (1 ) .Y Y

q P P YP P

This is the same result that would occur if all prices and income in the budget constraint were multiplied by .

36. Consumers spent $932 million in total: $202 million (22%) on radios and $730 million (78%) on home theaters. The exponents of a Cobb-Douglas utility function determine the portion of income spent on each of the goods. Therefore we can estimate the utility function as: .22 .78

1 2 1 2( , ) .U q q Aq q

37. If we apply the transformation function F(x) x to the original utility function, we obtain the new utility function 1/

1 2 1 2 1 2 1 2( , ) ( ( , )) [( ) ] ,V q q F U q q q q q q which has the same preference properties as does the original function.



38. If we apply the transformation function F(x) x to the original utility function, we obtain the new utility function 1/

1 2 1 2 1 2 1 2( , ) ( ( , )) [( ) ] ,V q q F U q q q q q q which has the same preference properties as does the original function.

39. Using Equation 3.3, we find that the marginal rate of substitution is 11 2 1/U U q

1 12 1 2/( ) ( / ) .q q q

By symmetry:

2 1/1

22 1

1

.Y

qP

P PP

40. Solution provided in Jim Dearden’s audio presentation.

Jen’s Legrangian for cost minimization is

0.5 0.51 1 2 2 1 2( ( )).L p q q q U q q

Taking the derivative of this with respect to q1, q2, and , the first order conditions are

0.51 1

1

0.52 2

2

0.5 0.51 2

0.5 ( ) 0,

0.5 ( ) 0,

( ) 0.

dLp q

dq

dLp q and

dq

dLU q q

d

Dividing the first condition by the second, we get

0.51 1

0.52 2

0.5

1 2

2 1

2

12 1

2

0.5 ( )

0.5 ( )

.

p q

p q

p q

p q

pq q

p

(1)

Substituting this into the third condition and solving for q1,

0.52

10.51 1

2

0q

U q qp

10.5 0.51 1

2

pU q q

p

10.51

2

1p

U qp

1

2

0.51

(1 )pp

Uq



2

2 20.5 21 1

2 1 1 2

.( )

p U pq q U

p p p p

Substituting this for q1 in equation (1) above and simplifying,

2

1 22

1 2

pq U

p p

.

Substituting the optimal values for q1 and q2 into the expenditure function (E),

2 2

2 12 21 2

1 2 1 2

1 2 1 2 22

1 2

1 2 2

1 2

( )

( )

.( )

p pE p U p U

p p p p

p p p pE U

p p

p pE U

p p

Since 0,dE

dU an increase in money will make Jen better off.


Jim’s Legrangian is

0.5 0.5 ( ).C ML C M Y p C p M

Taking the derivative of the Legrangian with respect to C, M, and , the first order conditions are

0.5 0.5

0.5 0.5

0.5 0,

0.5 0, and

0.

C

M

C M

dLC M p

dCdL

C M pdMdL

Y p C p Md

Dividing the first condition by the second,

0.5 0.5

0.5 0.5

0.5

0.5

.

C

M

C

M

C

M

pC M

C M p

pM

C p

pM C

p

(1)

Substituting this into the third condition and solving for C,



0

0

2

.2

CC M

M

C C

C

C

pY p C p C

p

Y p C p C

Y p C

YC

p

Substituting this value for C into the equation (1) above and simplifying,

.2 M

YM

p

Since pC $1.20, pM $2.00, and Y $12.00, the optimal value for coffee is C* 5 and the optimal value of muffins is M* 3.

Figure 3.19

According to Figure 3.19, Jim will consume more coffee. For example, if he were to continue to purchase 5 cups of coffee and 3 muffins, then he would be given an additional (6) cup of coffee for free under the promotion.

Chapter 4

1. Perfect substitutes are goods that a consumer is completely indifferent as to which to consume. The indifference curves for such goods are straight, parallel lines with a slope of 1 everywhere along the curve. That is, marginal utility from each good is identical, so the marginal rate of substitution is equal to 1. However, the slope of indifference curves of perfect substitutes need not always be 1; it can be any constant rate. Therefore, the utility function over perfect substitutes is

1 2 ,U iq jq

where i and j are parameters. The corresponding indifference curves are straight lines with a slope or marginal rate of substitution of i

j . Let p1 be the price of manufactured diamonds, p2



be the price of natural diamonds, and Y be income. If the price of manufactured diamonds is greater than the price of natural diamonds, then consumers will demand 0 manufactured diamonds. If the price of manufactured diamonds is equal to the price of natural diamonds, then given the simplifying assumption, consumers will demand

2

Yp manufactured diamonds. Finally, if the price

of manufactured diamonds is less than the price of natural diamonds, then consumers will demand

1

Yp manufactured diamonds.

Figure 4.9

2. An individual is initially maximizing utility at consumption bundle e1 on budget line L1, as illustrated in the figure. When the price of cell phones decreases, the budget constraint will pivot outward along the horizontal axis because more cell phones are attainable at a lower price. This is illustrated in the figure by budget line L2. The new utility-maximizing consumption bundle (e2) will be where a higher indifference curve (I2) is just tangent to the new budget line. This new utility-maximizing consumption bundle will contain less tobacco, resulting in a downward-sloping price-consumption curve.

Figure 4.10



3. The figure will be similar to Panel (a) and Panel (c) of Figure 4.2 in the chapter, but relabel the vertical “wine” axis as “other goods” (or “fun”) in Panel (a) figure and the horizontal “beer” axis as “bequests” in both Panel (a) and (c) figures. The Engle curve will be steeper.

4. The income effect reinforces the substitution effect for normal goods. It partially offsets the substitution effect for inferior goods. When it more than offsets the substitution effect, it is known as a Giffen good.

5. a. The substitution effect causes her to buy more clothing. The convexity of indifference curves assures that the substitution effect will always be positive for a price decrease.

b. The income effect could be either positive or negative depending on whether clothing is a normal or inferior good for Michelle.

6. An opera performance must be a normal good for Don because he views the only other good he buys as an inferior good. To show this result in a graph, draw a figure similar to Figure 4.4, but relabel the vertical “Housing” axis as “Opera performances.” Don’s equilibrium will be in the upper-left quadrant at a point like a in Figure 4.4.

7. The CPI accurately reflects the true cost of living because Alix does not substitute between the goods as the relative prices change.

8. The low quality oranges are relatively less expensive in California due to the shipping costs. Likewise, the high quality oranges are relatively less expensive in New York. Therefore we expect to see relatively more high quality oranges sold in New York and relatively more low quality oranges sold in California.

9. On the graph, Lf is the budget line at the factory store and Lo is the constraint at the outlet store. At the factory store, the consumer maximum occurs at ef on indifference curve I f. Suppose that we increase the income of a consumer who shops at the outlet store to Y* so that the resulting budget line L* is tangent to the indifference curve I f. The consumer would buy Bundle e*. That is, the pure substitution effect (the movement from ef to e*) causes the consumer to buy relatively more firsts. The total effect (the movement from ef to eo) reflects both the substitution effect (firsts are now relatively less expensive) and the income effect (the consumer is worse off after paying for shipping). Presumably the income effect is small because the budget share of plates is small. An ad valorem tax has qualitatively the same effect as a specific tax because both taxes raise the relative price of firsts to seconds.

10. He will not be worse off. If he were to continue to buy all new books, the increase in income would just cover the price increase, leaving him at the corner solution in the diagram below. However, if he buys one used book and 7 new ones, he will have $27 left over to buy pizza and beer. Thus any combination of books that includes one or more used books leaves him with leftover income and therefore better off. Figure 4.9 shows a case where Ximing is better off. It is possible that the change in relative price is not sufficient to move him off the corner solution.

Figure 4.11



11. See Figure 4.12 Jean is equally well off at e2 compared to e1. Even if coffee becomes relatively cheaper, Jean won’t raise her utility by consuming more coffee because cream and coffee are prefect complements.

Figure 4.12

12. See Figure 4.13. Ann will buy more books and less ice cream this year and her utility will be better off this year on I2 than on I1. This is because the price of book rose by less than the price of ice cream. So Ann can gain higher utility by consuming more books and less ice cream.

Figure 4.13

13. The attractive feature of the Big Mac as an indicator of price index is its uniform composition. The component ingredients of the Big Mac are the same across a long period of time and among most countries. Therefore Big Mac is a standardized bundle of goods. For cross-country comparisons, we should assume other factors such as tax, tariff, and service costs are the same for all countries.

14. See Figure 4.14. The Paasche index effectively asks how much must my income decline in order for me to be able to buy my current bundle of goods at the old prices. In the figure, the price of Y increases in period 2. The new budget line is BC . As a result, the consumer moves from e0 to e1. However, for the consumer to achieve the same utility level at constant prices, as in the Paasche index, the consumer would move instead to point e2, which represents a lower level of income than at e1. Because the Paasche index is the ratio of the cost of the current bundle over the cost of the base year, the Paasche index underestimates the true CPI.



Figure 4.14

15. See Figure 4.15. In this case, the price of clothing increases, rotating the budget line in. The intercept on the food axis remains unchanged. Spencer maximizes utility on I1, rather than the original indifference curve I2.

Figure 4.15

16. Sven is indifferent between using cable or phone service, which means two services are perfect substitutes. Therefore his indifference curves are straight parallel lines with a marginal rate of substitution (slope) of 1. Earthlink sets the prices of cable and dial-up so that the relative price (slope of budget constraint) is 1. Sven chooses anywhere on the budget constraint that lies atop I2.



Given the taxes, the price for the phone service is higher than cable service. The slope of the budget constraint increases and Sven chooses the corner solution, consuming all cable internet with no change in utility. See Figure 4.16.

Figure 4.16

17. The original budget constraint is Y pz Qz pcQc; normally Ralph buys 1 pizza and 2 colas, which means Qz 1 and Qc 2. Therefore Y pz 2pc. New budget constraint: Y pz 0.5 pz(Qz 1) pcQc 0.5pz 0.5 pzQz pcQc. What Ralph will choose when faced with the new constraint depends on his indifference curve. The relative price change will induce him to consume more pizza. See Figure 4.17.

Figure 4.17



18. See Figure 4.18. Assume the budget is $100. Compare e1 and e2. We know that you would go to the pool fewer times if you did not purchase the membership.

Figure 4.18

19. See Figure 4.19. The company is still paying Alexx to afford the same bundle “A”; however, the original bundle is not optimal as the relative prices differ in the two cities. LA and LF have different slopes, and the new optimum cannot be at “A” regardless of whether LF is flatter or steeper than LA.

Figure 4.19



20. Eggs and toast are perfect complements. U min (3Qt, 2Qe). If the price of eggs increases but we compensate Guerdon to make him just as “happy” as he was before (which means his utility is the same as before), his consumption of eggs will still be the same as shown in Figure 4.18. There is only income effect but no substitution effect, because eggs and toast are perfect complements, Guerdon will not substitute toast with eggs even thought the price of eggs increases.

Figure 4.20

21. In the previous problem, consumers would simply purchase all manufactured diamonds or all real diamonds, depending on which had the lowest price. Since the marginal utility of both types of diamonds is decreasing, but they still enter the utility function additively and symmetrically, consumers will still purchase all of the cheapest diamond form, but will simply receive less utility than they would have in the previous problem.


The partial derivative of the utility function with respect to D is

0.5 0.50.5(50 ) ,dU

D AdD

which is negative, indicating D is an economic bad.

The budget constraint for this model is

60 ( 2.50) ,GY p D A

which is an upward-sloping line because the price of good D (pG 2.50) is negative.

Lan’s Legrangian is

0.5 0.5(50 ) ( 60 ( 2.50) ).GL D A Y p D A

The first order conditions are

0.5 0.5

0.5 0.5

0.5(50 ) ( 2.50)

0.5(50 ) , and

60 ( 2.50) 0.

G

G

dLD A p

dDdL

D AdAdL

Y p D Ad



Dividing the first two conditions by each other and solving for A,

0.5 0.5

0.5 0.5

( 2.50)0.5(50 )

0.5(50 )GpD A

D A

A (2.50 pG)(50 D). (1)

Substituting this into the third condition and solving for D, the optimal value for good D is

Y 60 (pG 2.50)D (2.50 pG)(50 D)

185 50

2 5G

G

Y pD

p

.

Substituting this value for good D into equation (1) above,

A 32.5 0.5Y 2.5pG.

The partial derivative of the optimal value of D with respect to the price of gasoline is

2

120 2,

(2 5)G G

dD Y

dp p

which is negative because 120 – 2Y is less than zero.

The partial derivative of the optimal value of D with respect to income is

1,

2 5G

dD

dY p

which is negative because 2pG 5 is negative.


First, the marginal utility of taste is

0.5dUN

dT

and the marginal utility of leisure is

0.50.5 .dU

TNdN

The marginal rate of substitution (MRS) is the maximum amount of one good a consumer will sacrifice to obtain one more unit of another good. Paul’s marginal rate of substitution, MRS, equals

0.5

0.5.

0.5 0.5T

L

MU N NMRS

MU TN T

The marginal rate of transformation (MRT) is the trade-off the market imposes on the consumer in terms of the amount of one good the consumer must give up to obtain more of the other good. That is, the marginal rate of transformation equals the slope of the budget line. Since the budget constraint is

24,TN p T



the slope of the budget line, which equals the marginal rate of transformation, MRT, is

.1T

T

pMRT p

Paul optimizes utility by selecting the consumption bundle that generates the highest level of utility subject to his budget constraint. That is, Paul selects the consumption bundle where an indifference curve is just tangent to his budget constraint. Therefore, Paul optimizes utility such that

.0.5 1

TpNMRS MRT

T

or

.2Tp T

N (1)

Substituting this into the budget constraint and solving for T,

* 16.

T

Tp

Substituting this value for T into (1) above, N* 8.

The effect of a change in the price of taste, pT, on the amount of taste consumed is

2

16.

T T

dT

dp p

Therefore, a decrease in the price of taste, pT, increases the amount of taste consumed by 216

Tp. This

illustrates that a decrease in the price of taste promotes weight gain: As it takes Paul less time to produce taste, he will consume tastier potatoes and gain weight.

24. Using Equation 4.1 from the textbook, where h is the income elasticity of housing, 0 is the income elasticity of all other goods, and is the budget share of housing and rearranging:

0 01 (1 ) 1 0.63.

0.37h

Since housing and all other goods are normal, 0 0 1 and 1 h 2.7. Again, using Equation 4.1 with h, 0 as previously defined, b and h and b the budget shares of housing and books respectively, then

0 01 (1 ) 1 0.016 0.614

0.37b b b h b

hh

and with 0 b, 0 1, 1 h 2.7.



25. The figure shows that the price-consumption curve is horizontal. The demand for CDs depends only on income and the own price, q1 0.6Y/p1.

Figure 4.21

26. The income elasticity of demand (or income elasticity) is the percentage change in the quantity demanded in response to a given percentage change in income, Y:

1

1

.qq

dq Y

dY q

First, consider the Cobb-Douglas utility function

1 2 1 2( , )U q q q q

where and are parameters. The demand for q1 is

11

.( )

Yq

p



The partial derivative of the demand for q1 with respect to Y is

1

1

.( )

dq

dY p

Therefore, the income elasticity of demand is

1

1 1

.( )q

Y

p q

Second, consider the utility function

1 2 1 2( , ) ,U q q q q

for perfect substitutes. Assume the price of good q1 is less than the price of good q2. If so, then the consumer will maximize utility at a corner solution, consuming only good q1, and the demand for q1 is

11

.Y

qp


1

1

1.

dq

dY p


11 1

1.q

Y

p q

Last, consider the utility function

U(q1, q2) min{q1, q2}

for perfect complements. Individuals will consume perfect complements in fixed proportions. The demand for q1 is

11 2

.Y

qp p


1

1 2

1.

dq

dY p p


11 2 1

1.

( )q

Y

p p q



27. Guerdon’s utility function is 11 2 1 22( , ) min( , ).U q q q q To maximize his utility, he always picks a

bundle at the corner of his right-angle indifference curves. That is, he chooses only combinations of the two goods such that 1

1 22 .q q Using that expression to substitute for q2 in his budget constraint,

we find that

Y p1q1 p2q2 p1q1 p2q1/2 (p1 0.5p2)q1.

Thus, his demand curve for bananas is q1 Y/(p1 0.5p2). The graph of this demand curve is downward sloping and convex to the origin (similar to the Cobb-Douglas demand curve in panel a of Figure 4.1).

28. Pie and ice cream are complements. Therefore we can assume Olivia always eats pie and ice cream in the same proportion, say Qp aQc Q, where a is a constant. Assume the prices of pie and ice cream are pp and pc, respectively. For a budget Y1, the relation of demand and price can be expressed in the equation Qpp Qpc/a Y1. Olivia’s weekly budget will have to rise by (pp apc) for her to buy one more piece of pie per week.

Figure 4.22

29. Barbara’s demand for CDs is 4.3, q1 0.6Y/p1. Consequently, her Engel curve is a straight line with a slope of dq1/dY 0.6/p1.

30. The tangency condition is

11 2

12 1

p q

p q

which implies

1/1

12 2 1

2

.p

p q qp

Substituting into the budget constraint:

1/1 1/1

1 11 1 1 1 1

2 2

.p p

Y p q q q pp p



31. From Question 29:

1/1

11 1

2

,p

Y q pp

so

11/1

11 1

2

pq Y p

p

and by symmetry:

11/1

22 2

1

.p

q Y pp

32. From Philip’s budget constraint, Y p1q1 q2, we know that q2 Y p1q1. Substituting that expression into his utility function, we have 1 1 1.U q Y p q Philip chooses q1 so as to maximize this unconstrained objective. His first-order condition is 1 11/(2 ) 0,q p so his demand function for the first good is q1 1/(4[p1]

2). Substituting this demand function into our earlier expression for q2 from the budget constraint, we learn that his demand function for the second good is q2 Y 1/(4p1). Because his demand function for q1 is independent of Y, a change in p1 has no income effect, so the total effect equals the substitution effect. This last result holds for any quasilinear utility function.


David’s Lagrangian for utility maximization is

L 10G0.25B0.75 + (Y pGG pBB).


0.75 0.75

0.25 0.25

2.5 0

7.5 0

0.

G

B

G B

dLG B p

dGdL

G B pdBdL

Y p G p Bd

Dividing the first condition by the second,

2.5

.7.5

G

B

pB

G p

Re-writing,

3 .G

B

pB G

p

(1)



Substituting this into the third condition and solving for G, David’s optimal value for gasoline is

3 0

3 0

4

.4

GG B

B

G G

G

G

pY p G p G

p

Y p G Gp

Y p G

YG

p

Substituting this into equation (1) on the previous page and solving for B, David’s optimal value of bread is

34

3.

4

G

G B

B

pYB

p p

YB

p

The partial derivative of David’s optimal value of gasoline is

2,

4( )G G

dG Y

dp p

which is negative, indicating that David will demand less gasoline when the price of gasoline increases.

The partial derivative of the effect of a change in the price of gasoline on David’s optimal quantity of gasoline is

2

2

1,

4( )G G

d G

dp dY p

which is negative, indicating that as income increases, David's response to a change in the price of gasoline on his demand for gasoline becomes more negative.



Chapter 5

1. According to the textbook footnote, consumer surplus lost (CS) when price increase by 5% is

CS Rx(1 0.5x),

where R is initial revenue, before the price increase, or p1 multiplied by Q1, x is the percentage change in price, or p/p (p2 p1)/p1, and is the price elasticity of demand.

Rx is equal to A 2B in the graph below (or A + B + C) and 0.5 R(x2) is equal to triangle B.

Assume the area of Rx can be expressed as A + 2B in the figure. Then, in the textbook footnote equation, subtract triangle B from Rx to get the lost consumer surplus (A 2B B A B). Triangle B is subtracted from Rx in the formula because the price elasticity of demand is negative.

(The price elasticity of demand and revenue are not provided in this question, but in Question 31 the price elasticity of demand is reported to be 0.6.)



Hong and Wolak estimate that the price elasticity of demand is 1.6. R is therefore

333 (1 0.5 )

333 (0.05)(1 0.5)( 1.6)(0.05)

333 (0.05)(1 0.04)

333 (0.05)(0.96)

$6937.5.

RX X

R

R

R

R

If R $6937.5 million, x 0.05, and 1.6, then the area of triangle B is

B 0.5 Rx2 (0.5)(0.0025)(6937.5)(1.6) 13.875 million

and the area of rectangle A is

A Rx 2B 6937.5*0.05 2B 346.875 27.75 319.125 million.

Adding the area of B to A, lost consumer surplus is 319.125 13.875 333.

Figure 5.3

2. Suppose computers and mobile phones have made the price elasticity of demand for the postal service more elastic. If so, then consumer surplus lost from postal price increases is smaller.

Figure 5.4



3. As shown in Figure 5.5, what is being measured is consumer surplus, which is the area A. An alternative question is to ask people how much they would be willing to pay to watch an extra hour of TV or how much they’d have to be paid to watch one hour less of TV.

Figure 5.5

4. If the government taxes gasoline at $1 per gallon, the budget line rotates in, with the intercept on the “all other goods” axis remaining unchanged. If the tax only applies to purchases over 10 gallons per week, then the budget line is nonlinear, increasing in slope beyond the 10 gallons per week level, as shown in Figure 5.6.

Figure 5.6

5. See Figure 5.7. e1 will be the same as e2 if the tangent point of the (dashed) indifference curve and the budget line is on the solid region of the budget line. If the tangent point is on the dashed region, e2 will be different to e1, and e2 will be the corner solution.

Figure 5.7



6. With the subsidy, the individual can buy up to Y dollars worth of all other goods, or up to Y 85 dollars worth of food (85 100 stamps $15 paid for the stamps). The first $100 of food can be purchased for $0.15 per dollar. All remaining expenditures on food would be at full price. See Figure 5.8.

Figure 5.8

7. The food stamps provide greater utility than the clothing stamps because expenditures on food are likely to exceed $100 per month. Most individuals do not need to spend $100 per month on clothing.

8. If the income elasticity of demand for food falls as income rises, a wealthy person is more likely to have a tangency at a point like f in Figure 5.7 in the textbook than a poor person. Point f indicates an income elasticity of zero for food.

9. No. Because rich people have a bigger budget set than poor people, the difference between the opportunity set with food stamps and the opportunity set with the cash transfer program is smaller for rich people than for poor people.

10. In normal cases, people will increase their expenditure on nonhousing expenditures with housing subsidy. However, if people’s tastes (or preference for housing) change after receiving the housing subsidy, it is possible that their expenditures on nonhousing will not increase. For example, one person paid $300 per month for housing before, but after receiving the $200 housing subsidy he might decide to rent an apartment for $500 per month. Then his nonhousing expenditures will not increase. See Figure 5.9. Instead of having an indifference curve I2, he now has the indifference curve I2.

Figure 5.9



11. See Figure 5.10

Figure 5.10

12. See Figure 5.11. The budget line under the voucher is the outer solid line. The budget with cash extends to the upper dotted line. Hence Doreen will consume bundle e1 under the voucher and e2 under cash. The value of the voucher is $V, as indicated in the figure.

Figure 5.11



13. See Figure 5.12. In the figure, it is assumed that the stamps can be sold for $0.50 per $1 worth of stamps. Thus, the budget line does not extend out to $1100 on the money income axis. Instead, it changes slope above $1000 and intersects the vertical axis at $1050.

Figure 5.12

14. See Figure 5.13. If the food stamps are free, the budget constraint is shifted up by the value (in food) of the stamps. It does not affect the budget for other goods, as the food stamps can only be used for food.

Figure 5.13

15. Parents who do not receive subsidies prefer that poor parents receive lump-sum payments rather than a subsidized hourly rate for child care. If the supply curve for day care services is upward sloping, by shifting the demand curve farther to the right, the price subsidy raises the price of day care for these other parents.

16. The government could give a smaller lump-sum subsidy that shifts the LLS curve down so that it is parallel to the original curve but tangent to indifference curve I2. This tangency point is to the left of e2, so the parents would use fewer hours of child care than with the original lump-sum payment.

17. Limiting the amount of the subsidy decreases utility from the child care subsidy programs, as long as the limitation is binding (restricts to less than would have been chosen otherwise). Parents are still better off with a lump-sum payment.



18. See Figure 5.14. Individuals with a high preference for leisure (with indifference curve IB) will accept the welfare payment. Those with a greater preference for income than leisure (with indifference curve IA) are likely to turn down the payment.

Figure 5.14

19. Leisure is not a Giffen good. When the wage increases, it increases the opportunity set of the individual. If leisure is a normal good, the individual will purchase more of it as wages rise, just as he or she may purchase more of other commodities. When the income effect dominates, it generates a backward-bending labor supply curve at high wages. If leisure is inferior, the income and substitution effects reinforce one another and leisure falls as the wage increases.

20. See Figure 5.15. Bessie is unambiguously worse off. Because her original optimal bundle lies in the dashed section in the figure, now she has to choose the corner solution as her optimal bundle.

Figure 5.15



21. See Figure 5.16. Whether he decides to work additional hours given the overtime premium depends on his taste for income versus leisure. Panel (a) shows no additional hours worked; panel (b) shows additional hours worked.

(a) (b)

Figure 5.16

22. Jerome’s budget line is kinked at 8 hours of work. At that point, the slope of the budget line increases from w to w*. See Figure 5.17.

Figure 5.17

23. If the higher-paying job had no hours restriction, as long as the lower-paying job provided no utility other than income, he would not work there, and his budget line would be linear, with slope w.



24. See Figure 5.18. If we assume that leisure is a normal good, both the nobleman and the peasant had to work more hours with poll tax than when there was no poll tax. Whether a nobleman or a peasant worked more hours depends on the shape of the indifference curves. For example, in Figure 5.16, the nobleman worked fewer hours than the peasant. Indifference curves with greater convexity could illustrate the opposite case.

Figure 5.18

25. Under progressive income taxes, the marginal tax is higher than the average tax.

26. The proposed tax system exempt an individual’s first $10,000 of income. Suppose that a flat 10% rate is charged on the remaining income. Someone who earns $20,000 has an average tax rate of 5%, whereas someone who earns $40,000 has an average tax rate of 7.5%, so this tax system is progressive.

27. a. See Figure 5.19. Assume all goods have unit price. George will work more hours under a lump-sum tax than under a per-hour tax. A per-hour tax reduces the effective wage but lump-sum tax does not; therefore when a lump-sum tax is used, the price for leisure is higher. Hence George will consume less leisure but works more hours under a lump-sum tax.

b. The income tax is likely to reduce George’s hours of work more. Income tax reduces the effective wage. However, inheritance tax is similar to lump-sum tax, which does not reduce the effective wage. Therefore, the price of leisure is lower when income tax is used, and George will consume more leisure but work less.

Figure 5.19



28. As the marginal tax rate on income increases, people substitute away from work due to the pure substitution effect. However, the income effect can be either positive or negative, so the net effect of a tax increase is ambiguous. Also, because wage rates differ across countries, the initial level of income differs, again adding to the theoretical ambiguity. If we know that people work less as the marginal tax rate increases, we can infer that the substitution effect and the income effect go in the same direction or that the substitution effect is larger. However, Prescott’s (2004) evidence alone about hours worked and marginal tax rates does not allow us to draw such an inference because U.S. and European workers may have different tastes and face different wages.

29. Figure 5.20 shows Julia’s original consumer equilibrium: Originally, Julia’s budget constraint was a straight line, L1 with a slope of w, which was tangent to her indifference curve I1 at e1, so she worked 12 hours a day and consumed Y1 12w goods. The maximum-hours restriction creates a kink in Julia’s new budget constraint, L2. This constraint is the same as L1 up to 8 hours of work, and is horizontal at Y 8w for more hours of work. The highest indifference curve that touches this constraint is I

2. Because of the restriction on the hours she can work, Julia chooses to work 8 hours a day and to consume Y2 8w goods, at e2. (She will not choose to work fewer than 8 hours. For her to do so, her indifference curve I2 would have to be tangent to the downward-sloping section of the new budget constraint. However, such an indifference curve would have to cross the original indifference curve, I1, which is impossible—see Chapter 3.) Thus, forcing Julia to restrict her hours lowers her utility: I2 must be below I1. Comment: When I was in college, I was offered a summer job in California. My employer said, “You’re lucky you’re a male.” He claimed that, in order to protect women (and children) from overwork, an archaic law required him to pay women, but not men, double overtime after 8 hours of work. As a result, he offered overtime work only to his male employees. Such clearly discriminatory rules and behavior are now prohibited. Today, however, both females and males must be paid higher overtime wages—typically 1.5 times as much as the usual wage. As a consequence, many employers do not let employees work overtime.

Figure 5.20



30. Hong and Wolak (2008) estimate that Area A is $215 million and area B is $118 ( 333 215) million (as you should have shown in your figure in answer to Question 1).

a. Given that the demand function is Q Xp1.6, the revenue function is R(p) pQ Xp0.6. Thus, the change in revenue, $215 million, equals R(39) R(37) X(39) 0.6 X(37)0.6 0.00356X. Solving 0.00356X 215, we find that X 60,353.

b. We follow the process in Solved Problem 5.1 39

391.6 0.6

13737

0.6 0.6

60,35360,353

0.6

100,588(39 37 )

100,588 ( 0.00356) 358.

CS p dp p

This total consumer surplus loss is larger than the one estimated by Hong and Wolak (2008) because they used a different demand function. Given this total consumer surplus loss, area B is $146 ( 358 215) million.

31. This is an application of the Slutsky equation. The Slutsky equation is

.Uility Constant

dL dL dL dY

dw dU dY dw

The change in income from a change in the wage is dYdw H (which is equal to T L). For example,

if the wage rises by $1, income increases by the number of hours worked. Thus, the Slutsky equation can be expressed as

.Uility Constant

dL dL dLH

dw dU dY

32. See Figure 5.21. Without the tax Cynthia chooses point e1 on budget constraint B1. An ad valorem tax changes the budget line to B2 with an optimal bundle e2. A lump sum tax with equal revenue generation changes the budget line to B3 with optimal bundle e3. The lump sum tax leaves her on a higher indifference curve.

Figure 5.21



33. Joe’s Lagrangian for utility maximization is

L U(L, X) (w(H)(T L) X).

Let L be leisure, X be other consumption, MRS be Joe’s marginal rate of substitution, and MU be marginal utility. Also, let the price of other consumption be normalized to be $1.00.

Recall that w(H) H (T L). Substituting this into the Lagrangian,

L U(L, X) + ((T L)2 X).


2

2( ) 0

0, and

( ) 0.

L

X

dLMU T L

dLdL

MUdXdL

T L Xd

Dividing the first two conditions by each other,

2 ( )L

X

MU T L

MU

or

2 ( ).MRS T L

This indicates that Joe’s budget constraint is convex (because 2(T L) is his marginal rate of transformation). His budget constraint is illustrated in the figure below. According to the graph, if Joe does not work, then he has no income with which to purchase other consumption. Thus, the budget line meets the horizontal axis at the time constraint. Joe’s budget constraint has a negative slope, indicating that as Joe takes less leisure, his income and consequently other consumption increases. Since Joe's budget constraint is convex, its slope and consequently the marginal rate of transformation depend on the amount of leisure consumed. That is, the number of hours he chooses to work depends on his tastes.

Time constraint

Figure 5.22



34. Initially, John purchases 1,000 units of each good. Subsequent to the move he purchases 667 units of each good. The compensating variation is $1000 and the equivalent variation is $667.

35. Jane will initially choose any combination of q1 and q2 that lie on her budget constraint since MRS is equal to the relative price everywhere on the budget constraint. Subsequent to the price rise, she will choose to consume all q1. Since there is no change in utility, both CV and EV are equal to zero.

36. See Figure 5.23. Starting with the original budget B1 the consumer chooses bundle e1. An ad valorem tax changes the budget line to B2 with an optimal bundle e2. A lump sum tax with equal revenue generation changes the budget line to B3. The difference between budget lines B1 and B4 is the equivalent variation, and the difference between B2 and B4 is the compensating variation.

Figure 5.23

37. Rx is equal to A + 2B in the graph below (or A B + C). This is because R in the textbook footnote equation is initial revenue, before the price increase, or p1 multiplied by Q1. Then, in the textbook footnote equation, you subtract triangle B from Rx to get the lost consumer surplus (A 2B B A B). Triangle B is subtracted from Rx in the formula because the price elasticity of demand is negative.

Figure 5.24




The price of the 20,000th bid is 1,000 0.4

1,000 0.4 (20,000)

$200.

p Q

p Z

p

Consumer surplus is the monetary difference between what a consumer is willing to pay for the quantity of the good purchased and what the good actually costs. That is, market consumer surplus is that area under the market inverse demand curve above the market price up to the quantity consumers buy. Market demand is

25,000 25 .Q p

Therefore, consumer surplus is the area under the demand curve between prices of $1000 and $200:

1000

200

2 1000200

(25,000 25 )

(25,000 12.5 )

$8,000,000.

CS p dp

CS p p

CS

At a $100 price, consumers will receive surplus of $8,000,000 as calculated above, plus the 20,000 consumers who acquire a ticket will have surplus equal to the difference in the $200 and $100 prices:

8,000,000 20,000(200 100)

8,000,000 2,000,000

$10,000,000.

CS

CS

CS

Consumer surplus with the $100 price is larger, so that will be how Springsteen chooses to price tickets for his concert.


An increase in income will shift Joe’s budget line upward such that the new budget constraint (L2) will be parallel to the original budget constraint (L1), as illustrated in Figure 5.25.

Figure 5.25



Let I be unearned income, w be the wage rate, p be the price of goods, and T be time. Joe’s optimal value for leisure is

2(120 ).

pN

w

Joe’s optimal value for consumption is

2( ) (120 ).

I wT w pY

wp

Joe’s demand for leisure is not a function of unearned income. Therefore,

0,dL dH

dI dI

indicating Joe does not change the number of hours he works in response to winning the lottery. The partial derivative of the demand for goods with respect to unearned income is

1,

dY

dI p

so Joe consumes more goods in response to winning the lottery.



Chapter 6

1. One worker produces one unit of output, two workers produce two units of output, and n workers produce n units of output. Thus the total product of labor equals the number of workers: q L. The total product of labor curve is a straight line with a slope of 1. Because we are told that each extra worker produces one more unit of output, we know that the marginal product of labor, dq/dL, is 1. By dividing both sides of the production function, q L, by L, we find that the average product of labor, q/L, is 1.

2. See Figure 6.3. After the 6th unit, marginal product of labor falls to zero. Total product remains at 6 units, and average product of labor falls after the 6th unit.

Figure 6.3

3. An indifference curve shows all combinations of goods that result in the same level of utility; an isoquant shows all the combinations of inputs that result in a given level of output.

4. If an isoquant were thick, it would imply that the addition of both capital and labor from a point on the inside edge of an isoquant to the outer edge of the same isoquant would not increase output.

5. a. See Figure 6.4(a).

b. See Figure 6.4(b) and 6.4(c).



Figure 6.4

6. The isoquant looks like the “right angle” ones in panel b of Figure 6.3 because the firm cannot substitute between discs and machines but must use them in equal proportions: one disc and one hour of machine services.

7. Michelle’s production process illustrates diminishing marginal returns to labor. This diminishing return to extra labor may be due to too many workers sharing too few machines, or to crowding.



8. a. See Figure 6.5(a)

Figure 6.5(a)

b. See Figure 6.5(b). Assume the number of copy machines K is fixed at 1. Then production function is Q 1000 * min(L, 3). For L 3, Q 1000L; for L 3, Q 3000.

Figure 6.5(b)

9. The law of diminishing marginal products indicates that if a firm keeps increasing one input while holding all other inputs and technology constant, the corresponding increases in output will become smaller eventually. For example, when the number of machines is fixed, with too many workers sharing too few machines, each extra worker produces less extra output.



10. The isoquant for q 10 is a straight line that hits the B axis at 10 and the G axis at 20. The marginal product of B is 1 everywhere along the isoquant. The marginal rate of technical substitution is 2 if B is on the horizontal axis.

11. The input are helicopters and pilots, the output is delivery of relief. Since the output will not double while one input is doubled, the production process exhibits diminishing marginal product of capital. Since helicopters do not fly themselves, the admiral was likely suggesting that there were enough pilots available to fly them and in that case, the production process had nearly constant returns to scale.

12. This question will be confusing if students assume all the printers are identical. Each printer is embodied with a different number of units of capital. For example, an all-in-one inkjet copier is very different from a high-speed laser copier with collating and stapling features. If each technology uses one printer and one worker, the production functions will be of the form Q min(L, K) and the isoquants will all have the typical L-shape. The corners will be along different rays out of the origin, as illustrated in Figure 6.6.

Figure 6.6

13. The amount of capital needed is proportional to the output. If the amount of labor to operate the catapult did not vary substantially with the size of the projectile, the marginal productivity of capital and scale economies will decrease.

14. See Figure 6.7. The technological progress is not neutral. Because of the technology progress, less labor was required to spin the same amount of cotton in order to produce the same amount of product.

Figure 6.7



15. See Figure 6.8.

Figure 6.8

16. In the short run, the marginal product of the first few workers will be greater, as they are now able to produce more than they could with the old machine on a per-person basis. However, given that the number of workers needed to correctly operate the machine has been reduced, marginal labor productivity will fall sooner than with the old machine. In order to determine the precise changes in the shapes of the curves, more information is needed about how much labor is saved, and how much more each worker can produce. This is partly dependent on the technology (i.e., how the workers interact with the machine). Returns to scale will also depend on the technology.

17. See Figure 6.7 in the text. Diminishing marginal returns is a short-run phenomenon, caused by the invariability of the fixed input, and they occur regardless of returns to scale. Returns to scale is a long-run phenomenon, which must be evaluated with all inputs variable. In Figure 6.5(a), returns to scale are constant, yet the isoquants are convex, due to the diminishing marginal returns of the inputs.

18. The marginal product of labor will increase if the firm experiences falling output and reduces its work force. If the work force remained constant and less output were produced, marginal product would fall.

19. Not enough information is given to answer this question. If we assume that Japanese and American firms have identical production functions and produce using the same ratio of factors during good times, Japanese firms will have a lower average product of labor during recessions because they are less likely to lay off workers. However, it is not clear how Japanese and American firms expand output during good times (do they hire the same number of extra workers?). As a result, we cannot predict which country has the higher average product of labor.

20. While this would be true for the productivity of labor at the amount of output observed, it may not be true for all output levels. Firm 1 may be capital intensive, giving it a higher average product of labor, while Firm 2 is labor intensive, giving it a higher average product of capital. If different production technologies are used, at another output level, Firm 2 may be more productive.

21. Audio-PowerPoint answer by James Dearden is also available (6A Studying).

a. WRMP (2.5)(0.64)A0.36R–0.36 1.6(A/R)0.36

DRMP (2.5)(0.75)A0.25R–0.25 1.875(A/R)0.25

b. MRTSW MPA/MPR (9/16)(R/A), assuming A on X axis and R on Y axis.

MRTSD MPA/MPR (1/3)(R/A), assuming A on X axis and R on Y axis. c. Yes. Consider the production functions:

GW 3A0.25R0.75 and GD 2.5A0.25R0.75.



These functions have different marginal products but the same MRTS.

22. The production function is 0.75 0.25.q L K

a. As a result, the average product of labor, holding capital fixed at ,K

is 0.25 0.25 0.25/ ( / ) .LAP q L L K K L

b. The marginal product of labor is 0.25

34d / / .LMP q dL K L

c. If we double both inputs, output doubles to 0.75 0.25 0.75 0.25(2 ) (2 ) 2 2 ,L K L K q where q is the original output level. Thus this production function has constant returns to scale.

23. Q L K.

24. Using Equation 6.8, we know that the marginal rate of technical substitution is 23/ .L KMRTS MP MP

25. a. No diminishing marginal returns to labor. With K fixed at any level, marginal product of labor is constant at 10.

2

20

Q

L

b. Diminishing marginal returns to labor. With capital fixed at 2 units, the production function becomes Q 1.414L0.5. Marginal product of labor, calculated using the derivative formula, is MPL 0.707L 0.5, which decreases as L is increased.

322

20.3535 0

QL

L

26. a. This production always displays constant return to scale.

b. The Cobb-Douglas production function has decreasing, constant, or increasing returns to scale as is less than, equal to, or greater than 1.

c. This production function has decreasing, constant, or increasing returns to scale as is less than, equal to, or greater than 1.

d. The CES production function has decreasing, constant, or increasing returns to scale as d is less than, equal to, or greater than 1.

27. The marginal product of labor of Firm 1 is only 90% of the marginal product of labor of Firm 2 for a particular level of inputs. Using calculus, we find that the MPL of Firm 1 is q1/L 0.9f(L, K)/L 0.9q2/L.

28. Audio-PowerPoint answer by James Dearden is also available (6C RTS and MP).

For Cobb-Douglas production function

Q ALK, if 1,

the function is CRS, if 1, the function is IRS, and if 1, it is DRS. We also know that

MPL AL 1K and MPK ALK 1.

For tobacco, the production function is DRS, and since

MPL 0.18AL 0.82K0.33 and MPK 0.33AL0.18K 0.67,

the law of diminishing returns holds for both capital and labor.



For electronics and equipment, the production function is IRS (almost CRS), and since



For primary metal, the production function is IRS, and since



29. This production function is a Cobb-Douglas. Even though it has three inputs instead of two, the same logic applies. Thus we can calculate the returns to scale as the sum of the exponents: 0.27 0.16 0.61 1.04. Thus it has (nearly) constant returns to scale. The marginal product of material is q/M 0.61L0.27K0.16M–0.39 0.61q/M.

30. If ( , ) ( , )f xL xK x f L K then differentiating with respect to L yields 1 1( , ) ( , )xf xL xK x f L K and 1

1 1( , ) ( , )f xL xK x f L K and differentiating with respect to K yields 12 2( , ) ( , ).f xL xK x f L K

31. From 5.30 above with 1, 1 1( , ) ( , )f xL xK f L K and 2 2( , ) ( , )f xL xK f L K which implies

1 1

2 2

( , ) ( , )

( , ) ( , )

f xL xK f L K

f xL xK f L K

and the MRTS is independent of x.

32. If ( , ) ( , )f xL xK x f L K then differentiating with respect to x yields 1 2( , ) ( , )Lf xL xK Kf xL xK 1 ( , )x f L K Set x 1 and 1 2( , ) ( , ) ( , ).Lf L K Kf L K f L K

33. If one divides the production function by 1/( )a b and also multiplies by this term, one derives the second expression, where c is the “share” (lies between 0 and 1).


The marginal product of an input such as labor, L, (MPL) is the change in total output resulting from using an extra unit of the input, holding other factors (such as other inputs) constant.

The marginal product of practice yards, MPy, is the derivative of the swim time production function with respect to y:

0.000050.00035 .ydte

dy

This first derivative is negative, indicating practice yards decrease swim time. The second derivative of the swim time production function with respect to practice yards, y, will show whether practice yards decrease swim time at an increasing or decreasing rate:

2

0.000050.0000000175 .yd te

dydy



This second derivative is positive, indicating practice yards decrease swim time at a diminishing rate. If Ben increases practice yards from 20,000 yards to 40,000 yards, his swim meet placement improves from tenth to third, and if he increases practice yards from 40,000 yards to 60,000 yards, his swim meet placement improves from third to first. Thus, the marginal productivity of increasing practice yards from 20,000 yards to 40,000 yards is 7 swim meet places (from tenth minus third), and the marginal productivity of increasing practice yards from 40,000 yards to 60,000 yards is 2 swim meet places (from third minus first). This also shows that his marginal productivity in terms of swim meet places is diminishing.


Given this production function, as approaches negative infinity, the players become perfect complements, where a player is unable to score without his teammate on the floor. As approaches 0, the production function takes the Cobb-Douglas functional form, where the marginal rate of technical substitution is diminishing. When equals 1, the players are perfect substitutes. In basketball, it makes sense for to lie between negative infinity and 0, and in baseball, is likely equal to one.

Chapter 7

1. If the plane cannot be resold, its purchase price is a sunk cost, which is unaffected by the number of times the plane is flown. Consequently, the average cost per flight falls with the number of flights, but the total cost of owning and operating the plane rises because of extra consumption of gasoline and maintenance. Thus the more frequently someone has reason to fly, the more likely that flying one’s own plane costs less per flight than a ticket on a commercial airline. However, by making extra (“unnecessary”) trips, Mr. Agassi raises his total cost of owning and operating the airplane.

2. Let q equal the number of offices cleaned per hour. Then C 2q (15 minutes of labor is required per office). Variable cost is also 2q because there are no fixed costs. Average variable cost and marginal cost are $2. See Figure 7.4.

Figure 7.4



3. The total cost of building a 1-cubic-foot crate is $6. It costs four times as much to build an 8-cubic-foot crate, $24. In general, as the height of a cube increases, the total cost of building it rises with the square of the height, but the volume increases with the cube of the height. Thus the cost per unit of volume falls.

4. You produce your output, exam points, using as inputs the time spent on Question 1, t1, and the time spent on Question 2, t2. If you have diminishing marginal returns to extra time on each problem, your isoquants have the usual shapes: They curve away from the origin. You face a constraint that you may spend no more than 60 minutes on the two questions: 60 t1 t2. The slope of the 60-minute isocost curve is 1: For every extra minute you spend on Question 1, you have one less minute to spend on Question 2. To maximize your test score, given that you can spend no more than 60 minutes on the exam, you want to pick the highest isoquant that is tangent to your 60-minute isocost curve. At the tangency, the slope of your isocost curve, 1, equals the slope of your isoquant, MP1/MP2. That is, your score on the exam is maximized when MP1 MP2, where the last minute spent on Question 1 would increase your score by as much as spending it on Question 2 would. Therefore, you’ve allocated your time on the exam wisely if you are indifferent as to which question to work on during the last minute of the exam.

5. The expansion path is B/C 1, or B C.

6. If there are constant returns to scale, the long-run expansion path will be a straight line, indicating a constant capital–labor ratio. As the firm expands under constant returns, proportional increases in both inputs yield the same proportional increase in output. For example, the doubling of both inputs doubles output.

7. When the wage increases, the firm will use a more capital-intensive input mix. The expansion path becomes steeper. See Figure 7.5.

Figure 7.5

8. When the long-run curve is sloping downward, the short-run curve touches the long-run curve to the left of its minimum. When the long-run curve is upward sloping, the short-run curve touches the long-run curve to the right of its minimum. At the minimum of the long-run curve, the short-run curve touches the long-run curve at its minimum.

9. The subsidy effectively reduces the wage rate, flattening the isocost line. With the new, lower effective wage, firms will use more labor and less capital to produce any given level of output.



10. Because the franchise tax is a lump-sum payment that does not vary with output, the more the firm produces, the less tax it pays per unit. The tax per unit is / .d+ (The lump-sum is a fixed cost, so the tax per unit is calculated the same way as we do to obtain the average fixed cost.) If the firm sells only 1 unit, its cost is ,+ however, if it sells 100 units, its tax payment per unit is only /100.+

The firm’s after-tax average cost, ACa, is the sum of its before-tax average cost, ACb, and its average tax payment per unit, / .d+ Because the average tax payment per unit falls with output, the gap between the after-tax average cost curve and the before-tax average cost curve also falls with output, as shown on the graph.

Because the franchise tax does not vary with out-put, it does not affect the marginal cost curve. The marginal cost curve crosses both average cost curves from below at their minimum points. Because the after-tax average cost curve lies above the before-tax average cost curve, the quantity, qa, at which the after-tax average cost curve reaches its minimum, is larger than the quantity qb at which the before-tax average cost curve achieves a minimum.

Figure 7.6

11. Because the tax is unrelated to how much output is produced, marginal cost and average variable cost are unaffected. The annual fee has the effect of increasing fixed costs, which shifts the long- and short-run total cost curves upward.

12. C VC F; AC AVC AF VC/q F/q. By going to supermarket, consumer can lower the fixed cost F, thus also lower AC.

13. From the information given and assuming that there are no economies of scale in shipping baseballs, it appears that balls are produced using a constant returns to scale, fixed-proportion production function. The corresponding cost function is C(q) (w s m)q, where w is the wage for the time period it takes to stitch one ball, s is the cost of shipping one ball, and m is the price of all material to produce a ball. Because the cost of all inputs other than labor and transportation are the same everywhere, the cost difference between Georgia and Costa Rica depends on w s in both locations. As firms choose to produce in Costa Rica, the extra shipping cost must be less than the labor savings in Costa Rica.



14. See Figure 7.7. Paddles and nets are perfect complements that the company always sells as a set one net with two paddles. The expansion path is a straight line with slope 2. The change in relative prices of paddles and nets does not change the expansion path.

Figure 7.7

15. According to Equation 7.11, if the firm were minimizing its cost, the extra output it gets from the last dollar spent on labor, / 50/200 0.25,LMP w should equal the extra output it derives from the last dollar spent on capital, / 200/1,000 0.2.KMP r Thus the firm is not minimizing its costs. It would do better if it used relatively less capital and more labor, from which it gets more extra output from the last dollar spent.

16. a. See Figure 7.8(a).

b. See Figure 7.8(a). The firm chooses labor-machine technology. c. See Figure 7.8(b).

Figure 7.8

17. Audio-PowerPoint answer by James Dearden is also available (7A Skilled and Unskilled Labor).

a. The equation of isoquant is (6/8)S (4/8)U 4 3S 2U 16. This equation implies that S and U are perfect substitutes. The isoquant is a negatively sloped straight line that hits the vertical axis (U) at 8 and the horizontal axis (S) at 16/3. The MRTS 1.5.

b. The isocost is 26S 16U 104. The slope is 26/16 1.625.

c. It is clear that | MRTS | 1.625. This means that at the optimum the firm will only use unskilled labor. It will hire only eight unskilled workers.



18. Audio-PowerPoint answer by James Dearden is also available (7B Corporate Jets).

a. The incremental cost is the marginal cost to a corporation of an additional flight. Some of these marginal costs are the costs of fuel, flight attendants (if they are paid for each flight), landing and taking off fees, etc.

b. The marginal opportunity cost of an executive’s flight is the price it could have earned from leasing the jet to someone else.

19. Audio-PowerPoint answer by James Dearden is also available (7C Music Publishing).

TC 900 5q,

AFC 900/q,

AVC 5,

ATC 900/q 5,

MC 5.

a. The graphs are produced in the audio-PowerPoint presentation. b. Note that ATC becomes smaller as the production increases, so there is an advantage to having

only one big music publisher. This is because the fixed cost is spread over a larger production, although the MC is constant and equal to 5.

c. Revenue publishing cost 300 15 (900 5 300) 4500 2400 2100. The publisher is willing to pay up to 2100 pfennigs to the composer.

20. If w/r is the same as the slope of the line segment connecting the wafer-handling stepper and the stepper technologies, then the isocost will lie on that line segment and the firm will be indifferent between using either of the two technologies (or any combination of the two). In all the isocost lines in the figure, the cost of capital is the same, and the wage varies. The wage such that the firm is indifferent lies between the relatively high wage on the C2 isocost line and the lower wage on the C3 isocost line.

21. If her production possibilities frontier is PPF1, she can pick 6 pints of mushrooms and 4 pints of strawberries in one day. If there were no scope economies, she could only pick 2.67 pints of mushrooms per day if she picks 4 pints of strawberries. Assuming she works 8-hour days, scope economies produce the equivalent of an additional 3.33 hours of work, valued at $16.65.

22. a. AFC 10/q.MC 10. AVC 10.AC 10/q 10.See Figure 7.9.



Figure 7.9

b. AFC 10/q. MC 2q. AVC q. AC 10/q q. See Figure 7.10.

Figure 7.10

c. AFC 10/q. MC 10 8q 3q2. AVC 10 4q q2. AC 10/q 10 4q q2. See Figure 7.11.

Figure 7.11

23. 0.67 2( ) 0.55 800AC q q q

0.33 3( )0.3685 1600 0

AC qq q

q

2.67 4341.93q

therefore

23q

with the tax:

0.67 2 1( ) 0.55 800 400AC q q q q

and

0.33 3 2( )0.3685 1600 400 0

AC qq q q

q

68.q



24. Let w be the cost of a unit of L and r be the cost of a unit of K. Because the two inputs are perfect substitutes in the production process, the firm uses only the less expensive of the two inputs. Therefore, the long-run cost function is C(q) wq if w r; otherwise, it is C(q) rq.

25. C q, MC AVC AC 1 for q less than or equal to 80 per day. C 80 1.5 (q 80) 1.5q 40, MC 1.5, AVC AC 1.5 40/q for all q greater than 80 per day. See Figure 7.12(a) and 7.12(b).

Figure 7.12

26. a. Variable cost is

VC(q) 10q bq2 q3.

Average variable cost is

AVC(q) 10 bq q2.

For this to be positive,

2

2

10 0

10

10/ .

bq q

q bq

b q q

b. The average cost curve is U-shaped. AC is minimized at dAC/dq Fq2 b 2q 0.

c. MC crosses AC when the functions are equal. MC AC where 10 2bq 3q2 F/q 10 bq q2. MC AVC where 10 2bq 3q2 10 bq q2.

d. AVC is minimized where dAVC/dq 0.

/ 2 0

or 2

dAVC dq b q

b q

MC AVC

where

10 2bq 3q2 10 bq q2.

Substituting 2q for b on both sides yields

10 q2 10 q2.



27. a. You must set dAC/dq 0 for each firm. The minimum point of AC1 is at q 2. At plant 2, the minimum point is at q 1.

b. The firm should produce three units in plant 1, and one unit in plant 2.

28. If the wage increases, the firm should use more capital and less labor at all output levels; thus the expansion path increases in slope.

29. In this case the production exhibits decreasing returns to scale, resulting in an increasing cost, or upward-sloping long-run cost function.

30. The average cost of producing one unit is (regardless of the value of ). If 0, the average cost does not change with volume. If learning by doing increases with volume, < 0, so the average cost falls with volume. Here the average cost falls exponentially (a smooth curve that asymptotically approaches the quantity axis).

31. In the short run, suppose capital is fixed at .K F r K 20 K ; VC wL 10L.

Total cost C F VC 20 K 10L.

q 10L0.32 K 0.56 L (0.1q K 0.56)1/0.32 (0.1q K 0.56)3.125

AVC VC/q 10L/q

Suppose K 1. L (0.1q)3.125; VC 10L 0.0075 q3.125.

AVC VC/q 0.0075q2.125.

MC dVC/dq d(0.0075 q3.125)/dq 0.023q2.125.

32. a. See Figure 7.13.

b. MPL/MPK (1/2q/L)/(1/2q/K) K/L w/r 1/4 > L 4K;

c. c wL rK L 4K 8K;

q 10(LK)1/2 20K K q/20; c 8K 2q/5

Figure 7.13

33. Using the same technology, the MRTS will be the same at home and abroad. Since the wage–rental ratio is the same, the capital labor ratio will also be the same at home and abroad. Cost at home will be twice as high as costs abroad.



34. MPL/MPK (1/2K1/2/L1/2)/(1/2L1/2/K1/2) K/L w/r

In U.S., w r 10 L K;

In Mexico, w* 5; r* 10 L 2K

Cus wL rK 20K; Cmexico 20K

qus (LK)1/2 K ; qmexico 21/2K

Cus 20qus; Cmexico 14.1qmexico

When q 100, for U.S., C 2000, L K 100;

for Mexico, C 1410, K 70.7, L 141.

35. The firm chooses its optimal labor–capital ratio using Equation 7.11: MPL/w MPK/r. That is,

1/2 /q wL 1/2 / ,q rK or L/K r/w. In the United States where w r 10, the optimal L/K 1,

or L K. The firm produces where q 100 L0.50K0.50 K0.50K0.50 K. Thus q K L 100. The cost is C wL rK 10 100 10 100 2,000. At its Asian plant, the optimal input ratio is L*/K* 1.1r/(w/1.1) 11/(10/1.1) 1.21. That is, L* 1.21K*. Thus q (1.21K*)0.50(K*)0.50 1.1K*. So K* 100/1.1 and L* 110. The cost is C* [(10/1.1) 110] [11 (100/1.1)] 2000. That is, the firm will use a different factor ratio in Asia, but the cost will be the same. If the firm could not substitute toward the less-expensive input, its cost in Asia would be C** [(10/1.1) 100] [11 100] 2009.09.

36. MPL/MPK (0.7q/L)/ (0.3q/K) 7K/3L w/r.

In the United States, w/r 7/3 L K; q L0.7K0.3 K. when q 100, K L 100, c 1000.

In Asia, w/r 7/9 L 3K; q L0.7K0.3 2.16K. When q 100, K 46.3, L 138.9, c 694.5. If it had to use the same factor quantities as in the United States (K L 100), c 350 450 800.

37. ( , , , ) ( , , , ) ( , , , ) ( , , , )C C w r e q wL w r e q rK w r e q eM w r e q where L(.), K(.) and M(.) are the factor demands. From the first derivatives of the Lagrangian:

,f

wL

f

rK

, and .f

eM

So

( , , , ) ( , , , ) ( , , , ) ( , , , ).f f f

C w r e q L w r e q K w r e q M w r e qL K M

Rearranging:

( , , , ) ( , , , ) ( , , , ) ( , , , ) .f f f

C w r e q L w r e q K w r e q M w r e qL K M

From Euler’s Theorem where f(L, K, M) is homogeneous of degree : ( , , , )C w r e q q where is the marginal cost of production.

38. From 7.22 1 1

1(1 )0

a a

a aC aw rw Aa a

q

for all a 0.

Shepard’s lemma shows that dCdw L . That is, the partial derivative of the cost function with respect

to the wage rate equals the optimal amount of labor.



39. L: [ ]wL rK q L K

FOCs

1

1

0

0

0

w L

r K

q L K

From the FOCs

1 1

1 1 1( )L qw w r

and 1 1

1 1 1( )K qr w r

.

In turn, the cost function, C, is

1

1 1( , , ) ( ) .C w r q q w r


The total cost of production is Rob’s wage multiplied by the number of hours he works: 150h. In terms of output, the cost function is

C(q) 150q2

because

h q2.

A firm’s marginal cost is the amount by which a firm’s cost changes if the firm produces one more unit of output.

( )( ) .

dC qM q

dq

Therefore, marginal cost is

( )300 .

dC qq

dq

With the new methodology, the cost function is

C(q) 150q1.333

because

h q1.333.

Marginal cost with the new methodology is

.333( )200 .

dC qq

dq

The marginal cost of production with the new teaching methodology is less than the original marginal cost. For example, if output is equal to one (q 1), then the original marginal cost is $300 and the marginal cost with the new methodology is $200.




An input’s marginal product is the change in total output resulting from using an extra unit of that input holding other factors (such as other inputs) constant. The marginal product of labor (MPL) is

1.522 3MPL

40

SL L

and the marginal product of capital (MPK) is

1.522 3MPK 0.025 .

40

SL S

An expansion path is the cost-minimizing combination of labor and capital for each output level. Use the tangency condition between the isocost and isoquant that determines the factor ratio when the firm is minimizing cost to derive the expansion path. That is where

0.33340.

rS L

w

Substituting $20 for w and $0.50 for r,

S L.

Thus the expansion path is a line with a slope of one that begins at the graph’s origin.

Substitute the cost-minimizing values for L and S into the expenditure function,

C wL rS,

to find the long-run cost function. The long-run cost function is

C (w0.667 (40) 0.333 r 0.667)0.5.

If the cost of capital decreases to $0.25, then the expansion path becomes steeper (indicating substitution toward capital, which is now relatively cheaper). In particular, the equation for the new expansion path is

S = 1.260L.



Chapter 8

1. The shutdown rule states that a firm should shut down when it can avoid additional losses by doing so. This occurs when losses would exceed fixed costs. If the firm can cover any portion of fixed costs by continuing production, it should do so.

2. How much the firm produces and whether it shuts down in the short run depend only on the firm’s variable costs. (The firm picks its output level so that its marginal cost—which depends only on variable costs—equals the market price, and it shuts down only if market price is less than its minimum average variable cost.) Learning that the amount spent on the plant was greater than previously believed should not change the output level that the manager chooses. The change in the bookkeeper’s valuation of the historical amount spent on the plant may affect the firm’s short-run business profit but does not affect the firm’s true economic profit. The economic profit is based on opportunity costs—the amount for which the firm could rent the plant to someone else—and not on historical payments.

3. Suppose that a U-shaped marginal cost curve cuts a competitive firm’s demand curve (price line) from above at q1 and from below at q2. By increasing output to q2 1, the firm earns extra profit because the last unit sells for price p, which is greater than the marginal cost of that last unit. Indeed, the price exceeds the marginal cost of all units between q1 and q2, so it is more profitable to produce q2 than q1. We can derive this result using calculus. The second-order condition for a competitive firm requires that marginal cost cut the demand line from below at q*, the profit-maximizing quantity:

dMC(q*)/dq > 0.

4. As shown in Figure 8.2, before low-carb food became popular, the demand curve was D1; it was not profitable to enter the market since the fixed cost was so high such that short-run average cost was about the market price. If the demand curve shifts rightward to D2 due to the popularity of low-carb food, the firms make a positive profit in the short run. However, once the low-carb food fell out of the fashion, the demand curve shifted leftward and the firms exited the market.

Figure 8.2



5. As shown in Figure 8.3, when demand increases from D1 to D2, the price increases as well. If the increase in demand is expected to be temporary, firms will produce Q1. On the other hand, if the higher demand is expected to be permanent, firms will produce Q2. In the long run, the market supply curve will be a horizontal line.

Figure 8.3

6. a. Assuming that the average cost and marginal cost all have a U-shape, a firm’s marginal and average costs are likely to rise with extra business.

b. As shown in Figure 8.4, since the increased demand is only seasonal, it will not affect firms’ long-run supply curve and the number of firms in the market if there is considerable entry cost. During peak demand period, firms will operate to the right of the minimum of the short run average cost curve.

Figure 8.4



7. With a competitive market, firms with higher average cost will be driven out of the market by lower-cost firm firms. The equilibrium profit will be zero. See Figure 8.5. The higher-cost firm with average cost curve AC 1 will be driven out of the market if there exists also lower-cost firm with average cost curve AC 2.

Figure 8.5

8. See Figure 8.6. With a relatively high price for Grade A milk, the firms can produce a large amount of Grade A milk with a lower average cost if they adopt the new technology. Hence the high price of Grade A milk gives the incentive to adopt the technology for its production.

Figure 8.6

9. Some farmers did not pick apples so as to avoid incurring the variable cost of harvesting apples. These farmers left open the question of whether they would harvest in the future if the price rose above the shutdown level. Other more pessimistic farmers did not expect price to rise anytime soon, so they bulldozed their trees, leaving the market for good. (Most farmers planted alternative apples such as Granny Smith and Gala, which are more popular with the public and sell at a price above the minimum average variable cost.)



10. See Figure 8.7. The lower of average minimum variable cost will extend firm’s supply curve downward. As a result, the market supply curve shifts rightward.

Figure 8.7

11. a. Since Lesotho is a price taker in the world market, the demand curve it faces is a horizontal line.

b. The increase in Chinese exports shifts its demand curve downward. c. The change in exchange rate makes its exports more expensive relatively, and further shifts down

its demand curve. d. See Figure 8.8. When the price falls below p1, the firms will experience an economic loss. If the

price falls below p2, the firms will shut down in the short run. If the losses continue in the long run, firms will exit the industry.

Figure 8.8



12. Figure 8.9 shows the effects of China’s entry into the generic art market. Without Chinese exports, the world supply curve starts with country A, whose cost is only higher than that of China. With large amount of Chinese exports, the world supply curve is dominated substantially by Chinese exports, as indicated by the first section of the supply curve.

Figure 8.9

13. As shown in Figure 8.10, due to the lower average cost, the long-run average cost curve shifts downward and firms’ supply curves shift rightward. As a result, the horizontal market supply curve shifts downward.

Figure 8.10

14. Audio-PowerPoint answer by James Dearden is also available (8A Kodak Exit Decision).

The firm exits if its profit is negative, i.e.,

p(q*) P*q* C(q*) 0,

where q* is the optimum output. Divide both sides by q*, and we get:

P* AC(q*) 0.

Therefore the firm exits if the price is less than its average cost.



15. Audio-PowerPoint answer by James Dearden is also available (8B Gasoline Prices).

a. The marginal cost at the retail level shifts up at each point vertically by the price increase in the wholesale price. We know the supply curve is the portion of MC above the minimum average variable cost. Therefore the firm supply curve shifts up vertically by the price increase in the wholesale price.

b. The market supply curve is the horizontal summation of each individual curve. Therefore the market supply curve also shifts up.

c. The supply curve shifts up by $x. Assuming the demand curve is downward sloping, the increase in price will be less than $x.

d. The average cost shifts up by $x, but the price rises by less than $x. Therefore the profit margin becomes smaller.

e. Suppose that there are two firms—one with high cost and the other with low cost. After the wholesale price increase by $x, the average and marginal cost curves for both firms go up by $x but the price will increase by less than $x. It is possible that at the optimum level of production (P MC) the high-cost firm loses money (its economic profit is negative), and for the low-cost firm the profit is positive. Therefore the high-cost firm should exit the market.

16. Audio-PowerPoint answer by James Dearden is also available (8C Natural Gas).

a. We assume there are three Fields: 1, 2, and 3. We also assume that Field 3 is the deepest and Field 2 is less deep, and Field 1 is the shallowest. This implies that the fixed cost and marginal cost of the first field will be lower than that of the other two fields. Let’s suppose that MC of each field is constant, therefore the cost function will be a positively-sloped straight line. If we put the cost function of each field on top of the other we get a step function. The cost function of the first field will be at the bottom, and for the Field 3 will be on the top. At each step the cost function rises by the fixed cost.

b. See Figure 8.11. The marginal cost curve will be a step function. If P* MC1, the firm will produce q1. If P* MC2, it will produce q2, and so on.

Figure 8.11

17. One reason that firms increase the price of coffee only 14% may be that the demand of coffee is elastic. On the other hand, since the roast coffee producers do not have good substitute for raw coffee beans, they have to bear the burden of a large portion of the increase in raw coffee beans.



18. a. See Figure 8.12, the firm’s supply curve shifts leftward due the storm. As a result, the market supply curve shifts leftward as well.

b. Whether Tomato Fest suffered an economic loss in this case depends on the demand elasticity, which determines how the price responds to a lower supply. In this situation, an “economic loss” is defined as negative profit.

Figure 8.12

19. As shown in Figure 8.13, the profit first increases with the days of vacations and then decreases with it. The profit is maximized if D* days of vacations are taken.

Figure 8.13

20. If the world demand curve crosses the supply curve in the flat section of the Brazil supply, there will be a unique equilibrium price and likely multiple equilibrium quantities along that section. Similarly, if the world demand curve crosses the supply curve in the following vertical section, there will be a unique equilibrium and likely multiple equilibrium prices. In either case, farmers in the United States will not produce cotton.

21. The logic behind the first claim is that the firm chooses not to charge the full price of one of the inputs, the rent. The logic for the second claim is that since it is charged a lower price for one of the inputs, the output price is also lower.



22. In either case, marginal cost is unaffected. The only difference is the magnitude of the upward shift in average cost. If the tax is collected every year instead of only once but the total amounts are equal, the average cost curve is shifted up by more due to the increase in the tax. Long-run supply is also shifted upward due to the increase in the minimum point of the average cost.

23. In the short run, since there is no change in marginal cost, there will be no change in the equilibrium output level or price. Profits will be reduced by the amount of the tax. The only exception is if the firms were close enough to the shutdown point such that payment of the tax increases losses beyond fixed costs (assuming payment of the tax is not required if the firm shuts down). In this case, with identical firms, equilibrium quantity drops to zero.

24. Such a tax will not affect the marginal cost. However, the tax incidence will be shared by the grocers and the consumers. The amount of tax passed on to consumers will be determined by the demand elasticity of grocery bags.

25. The shutdown notice reduces the firm’s flexibility, which matters in an uncertain market. If conditions suddenly change, the firm may have to operate at a loss for six months before it can shut down. This potential extra expense of shutting down may discourage some firms from entering the market initially.

26. See Figure 8.13. In this case the firm is producing more than the long-run profit-maximizing output level of 110. Profits are currently equal to area abcd but would be increased to area ebfg with the optimal plant size.

Figure 8.14

27. No. Although horizontal factor supply curves would generate a horizontal firm supply curve, it could also be that one input increases in cost at the same rate and that another decreases in cost. In either case, as firm and industry output expands, average cost will remain unchanged and the industry long-run supply curve will be horizontal.

28. If Arizona starts collecting a specific tax (a) on its firms, then they may be driven from the market. Assuming that firms in California and Arizona have initial costs to start, a specific tax on Arizona oranges could have two possible effects. If there is unlimited entry by California firms, the long-run supply curve will be unaffected and all Arizona firms will be driven from the market due to their cost disadvantage. If entry in California is limited, then the long-run supply curve becomes a step function, as shown in Figure 8.15. All output up to QC would be supplied by California firms, after which costs (and so the supply curve) shift upward by the amount of the tax for output supplied by Arizona firms.



Figure 8.15

29. See Figure 8.16. In panel (b), a decrease in the demand for real trees caused by the increase in artificial tree sales shifts the demand curve to the left. The supply curve shifts to the left due to the decreased number of tree growers. The result is an increase in price to $26.50, and a decrease in quantity to 33 million. In panel (a), each retail tree seller purchases trees at an average cost of $20, and sells for the market price of $26.50.

Figure 8.16

30. When the bribes lose their tax-deductible status, the cost of making a bribe increases. The supply curve shifts to the left, just as it would if a tax were enacted (similar to Figure 8.10). The increase in the price of bribes reduces the equilibrium quantity. In the product market, if we view the bribes as part of the cost of doing business, an increase in the cost of bribing foreign officials would shift the marginal cost curve upward in the product markets, reducing the supply.



31. If importers were allowed to bring gas to California at a 15-cent surcharge, it would not alter the normal equilibrium, as shown as p*, G* in Figure 8.17. However, a large leftward shift of the supply curve would prevent prices from rising above p* 0.15. In the graph, the horizontal portion of the supply curve represents imported gas at the surcharge price. As a result of these imports, prices rise only to p* 0.15 and quantity is G1, instead of a price increase to p and a drop to the “no import” level GNI.

Figure 8.17

32. Marginal cost is computed by taking the derivative dC/dq. Profits are maximized by setting MC MR p. For the function given, MC 10 2q q2. Thus profits are maximized when p 10 2q q2. The supply curve is p 10 2q q2 for P 9.25.

33. The competitive firm’s marginal cost function is found by differentiating its cost function with respect to quantity: MC(q) dC(q)/dq b 2cq 3dq2. The firm’s necessary profit-maximizing condition is p MC b 2cq 3dq2. The firm solves this equation for q for a specific price to determine its profit-maximizing output.

34. In the long run, price equals marginal cost, and profits are zero. Thus given that industry output Q nq, the following will be true in long-run equilibrium, p 24 nq. Therefore,

24 nq 2q

(24 nq)q 16 q2.

Solving these equations for q, n, Q, and p yields

q 4

n 4

Q 16

p 8.

35. Because the clinics are operating at minimum average cost, a lump-sum tax that causes the minimum average cost to rise by 10% would cause the market price of abortions to rise by 10%. Based on the estimated price elasticity of between 0.70 and 0.99, the number of abortions would fall to between 7% and 10%. A lump-sum tax shifts upward the average cost curve but does not affect the marginal cost curve. Consequently, the market supply curve, which is horizontal and the minimum of the average cost curve, shifts up in parallel.



36. To derive the expression for the elasticity of the residual or excess supply curve in Equation 8.17, we differentiate the residual supply curve, Equation 8.16, Sr(p) S(p) Do(p), with respect to p to obtain

.r odS dS dD

dp dp dp

Let Qr Sr(p), Q S(p), and Qo D(p). We multiply both sides of the differentiated expression by p/Qr, and for convenience, we also multiply the second term by Q/Q 1 and the last term by Qo/Qo 1:

.r o

o

r r r o

QdS p dS p Q dD p

dp Q dp Q Q dp Q Q

We can rewrite this expression as Equation 8.17 by noting that r (dSt/dp)(p/Qr) is the residual supply elasticity, (dS/dp)(p/Q) is the market supply elasticity, o (dDo/dp)(p/Qo) is the demand elasticity of the other countries, and Qr/Q is the residual country’s share of the world’s output (hence 1 Qo/Q is the share of the rest of the world). If there are n countries with equal outputs, then 1/ n, so this equation can be rewritten as r n (n 1) o.

37. See the text for details:

a. The incidence of the federal specific tax is shared equally between consumers and firms, whereas firms bear virtually none of the incidence of the state tax (they pass the tax on to consumers).

b. From Chapter 2, we know that the incidence of a tax that falls on consumers in a competitive market is approximately /( ). Although the national elasticity of supply may be a relatively small number, the residual supply elasticity facing a particular state is very large. Using the analysis about residual supply curves, we can infer that the supply curve to a particular state is likely to be nearly horizontal—nearly perfectly elastic. For example, if the price rises even slightly in Maine relative to Vermont, suppliers in Vermont will be willing to shift up to their entire supply to Maine. Thus we expect the incidence on consumers to be nearly one from a state tax but less from a federal tax, consistent with the empirical evidence.

c. If all 50 states were identical, we could write the residual elasticity of supply, Equation 8.17, as r 50 49o. Given this equation, the residual supply elasticity to one state is at least 50 times larger than the national elasticity of supply, r 50, because o < 0, so the 49o term is positive and increases the residual supply elasticity.

38. Each competitive firm wants to choose its output q to maximize its after-tax profit: pq C(q) – + . Its necessary condition to maximize profit is that price equals marginal cost: p dC(q)/dq 0. Industry supply is determined by entry, which occurs until profits are driven to zero (we ignore the problem of fractional firms and treat the number of firms, n, as a continuous variable): pq [C(q) + ] 0. In equilibrium, each firm produces the same output, q, so market output is Q nq, and the market inverse demand function is p p(Q) p(nq). By substituting the market inverse demand function into the necessary and sufficient condition, we determine the market equilibrium (n*, q*) by the two conditions:

* * *

* * * *

( ) ( )/ 0,

( ) [ ( ) ] 0.

p n q dC q dq

p n q q C q

+



For notational simplicity, we henceforth leave off the asterisks. To determine how the equilibrium is affected by an increase in the lump-sum tax, we evaluate the comparative statics at + 0. We totally differentiate our two equilibrium equations with respect to the two endogenous variables, n and q, and the exogenous variable, + :

dq(n[dp(nq)/dQ] d2C(q)/dq2) dn(q[dp(nq)/dQ]) d + (0) 0,

dq(n[qdp(nq)/dQ] p(nq) dC/dq) dn(q2[dp(nq)/dQ]) d + 0.

We can write these equations in matrix form (noting that p dC/dq 0 from the necessary condition) as

2

2

2

0.

1

dp d C dpn q

dqdQ dq dQd

dndp dpnq q

dQ dQ

+

There are several ways to solve these equations. One is to use Cramer’s rule. Define

2

2

2

22

2

22

20,

dp d C dpn q

dQ dq dQD

dp dpnq q

dQ dQ

dp d C dp dp dpn q q nq

dQ dq dQ dQ dQ

d C dpq

dq dQ

where the inequality follows from each firm’s sufficient condition. Using Cramer’s rule:

2

2

2

2

2

0

1

0, and

0

1

0.

dpq

dQ

dp dpq qdQdq dQ

d D D

dp d Cn

dQ dq

dpnq

dQdn

d Ddp d C

ndQ dq

D

+

+



The change in price is

2

2

2

2

0.

dp nq dp dn dqq n

d dQ d d

dp d C dpn q nqdQ dqdp dQ

dQ D D

d Cq

dp dq

dQ D

+ + +

39. After-tax profit is (1 ) ( )pq C q and the profit-maximizing output after the tax is imposed is:

( ) ( )(1 ) 0

q C qp

q q

or 1( )(1 ) 0p MC q

the first order condition. Differentiating this with respect to the ad valorem tax rate yields:

1 2(1 ) ( )(1 ) 0MC q

MC qq

and rearranging:

( )0

(1 ).

q MC qMC

q


a. The total cost function is:

3 37 37 169 37( ) 6860 6860 ,

12 27,000,000 12 27,000,000

TC q p t q q q q

then the marginal cost function is:

2( ) 169 37( )

12 9,000,000

C q

MC q qq

b. The average variable cost function is:

2 2( ) 7 37 169 37( ) .

12 27,000,000 12 27,000,000

T

VC qAVC q p t q q

q

Therefore, the shutdown price 2169 37 169min ( ) min .

12 27,000,000 12

q qAVC q q



c. Since the short-run supply function is the segment of the marginal cost function above the average variable cost function, and since

2 2169 37 169 37( ) ( ),

12 9,000,000 12 27,000,000 MC q q q AVC q

the marginal cost curve is always above the average variable cost curve. Thus the short-run supply function is just the solution q* to: p MC(q). Solving this equation, we can obtain the short-run supply function:

1 1

2 2169 7( )

12 12( ) 300 30037 37

Tp p p tS p q

d.

1

27( )( , ) 150 12 0

37 37

Tp p tS p t

t

1

2

11.5, 2

11.5, 2

1 1

2 2

7( )( , ) 150 12

37 37

169 169150 15012 1237 37 37 37

300 30

444 6253

T

T

T

p t

p t

p p tS p t

t

p p

p


a. 8-year-old Joe’s total cost function is: ( ) 180 0.15 . C q q

His marginal cost function is: ( )

( ) 0.15. dC q

MC qdq

His average variable cost function is: ( )

( ) 0.15. VC q

AVC qq

Therefore his shutdown price 0.15. b. 17-year-old Joe’s total cost function is:

2

2

180 0.15 6 180 1.35 , if 6 5 305

( )( 30) 30

180 36 0.15 6 216 1.05 0.04 , if 30.5 5

qq q q

C qq q

q q q q



Then his marginal cost function is:

1.35, if 30( )( )

1.05 0.08 , if 30.

qdC qMC q

q qdq

Since his average variable cost function is:

2

1.35, if 30( )

( ) ( 30) 5( 30)0.15 6 1.05, if 30,

5 25

qVC q

AVC q qq qq qq q

Therefore the shutdown price 1.35. c. The shutdown prices of 8-year-old Joe and 17-year-old Joe are different. Since 17-year old Joe’s

costs are higher, his shutdown price is also higher.

Chapter 9

1. See Figure 9.5. As demand becomes more elastic, the welfare effect of specific tax becomes larger because there is a larger change in the equilibrium quantity. In the graph, D0 is more elastic than D1 at point a. When a specific tax shifts the supply curve upwards, the welfare loss with the more elastic demand curve is abc. With D1, the welfare loss is only adf.

Figure 9.5



2. See Figure 9.6. The price ceiling (pc) on gasoline reduces welfare by the areas abd bcd. The new consumer surplus is area gacf, and producer surplus falls to area fch.

Figure 9.6

3. See Figure 9.7. An individual receiving a gift valued at G increases his or her utility from U0 to U1 (at point b). The same individual receiving cash in an amount equal to the cost of G increases his or her utility to U2 (at point c). The reason for the lower utility in the case of the gift is that it limits the individual’s ability to substitute between the gift good and other commodities (i.e., in the figure, the individual cannot reach point c, where the marginal rate of substitution equals the price ratio).

Figure 9.7

4. If the tax is based on economic profit, the tax has no long-run effect because the firms make zero economic profit. If the tax is based on business profit and business profit is greater than economic profit, the profit tax raises firms’ after-tax costs and results in fewer firms in the market. The exact effect of the tax depends on why business profit is greater than economic profit. For example, if the government ignores opportunity labor cost but includes all capital cost in computing profit, firms will substitute toward labor and away from capital.



5. See Figure 9.8. The welfare loss from the tax is equal to area abc. If the ad valorem tax is levied on firms, then the supply curve will shift leftward. Also note that it is not a parallel shift—as the price goes up, the original supply curve and the new supply curve will get farther apart, creating a cone shape (narrower near the origin of the graph and wider as you move farther away from the origin).

Figure 9.8

6. See Figure 9.9. The welfare loss due to the minimum wage is area abc. When the minimum is instituted, employment falls to L1 from L0. Because labor supply at the new higher wage increases to L2, unemployment is increased by the institution of the minimum. Because of the excess supply, less experienced workers are likely to be losers with the new policy. In addition, if discrimination exists on the basis of age, gender, or race, those workers in the less desired group are also likely to be hurt by the minimum wage.

Figure 9.9

7. Solved Problem 8.5 shows the long-run effect of a lump-sum tax in a competitive market. Consumer surplus falls by more than tax revenue increases, and producer surplus remains zero, so welfare falls.

8. Owners of existing billboards would not oppose the ban because it creates an entry barrier and so also market power for existing sellers. Producers would then be able to set prices above marginal cost, and transfer some of the consumer surplus to producers. Exclusive of external effects, welfare will fall, as a deadweight loss is created. The consumers are the purchasers of billboard space. The producers are billboard owners. Welfare is improved if the increase in billboards was creating a negative externality by making the scenery less enjoyable.



9. See Figure 9.10. With the price increase, producers gain A but lose D. The payment x must be enough to compensate producers for their net loss from the price increase (x D A). With this payment, producer welfare is unchanged but consumer welfare falls by A B. With a price support program, consumer surplus would be the same as with the lump-sum payment program, but producer surplus would be A B C D E F. A quota set at Q1 producer surplus is A B C D E. With the quota set at Q2, producer surplus is A C.

Figure 9.10

10. See Figure 9.11. At the original price, P0 , consumer surplus is A B C, and producer surplus is E F. If officials want to reduce the number of visits to Q*, they can either increase the price to P1, which reduces consumer surplus by B C, or they can institute a quota and leave the price unchanged. With the quota, consumers retain area A B, and area C is deadweight loss. Area F is eliminated under either scenario. Assuming the park is charging marginal cost for admission, which is constant, there are no welfare effects from the change in area F.

Figure 9.11



11. See Figures 9.12(a) and (b). In the sugar market, the quota increases price, reduces output, and causes a deadweight loss of A. In the corn sweetener market, demand is increased due to the increased cost of the substitute, raising prices from p1 to p2.

Figure 9.12

12. See Figure 9.13. The government prefers the tariff. In either case, consumer surplus is reduced to area A. With the quota, the government collects no revenue. With the tariff, the government collects B C D E as revenue.

Figure 9.13

13. See Figure 9.14. The subsidy increases consumer welfare by B C, but costs the government B C D. The net change in welfare is D.



Figure 9.14

14. If we assume that the Canadian supply curve is perfectly elastic (horizontal) at the open-market price, there would be no welfare effects in Canada. Figure 9.15 shows the effects on the U.S. markets. If the open market supply curve (U.S. plus Canadian) is S0, and the U.S. only supply curve is SUS, then the effect of the import restriction is to increase the price to U.S. water users from p to p. Consumer surplus falls from A B C D to D only. U.S. producers gain A, but B C represent deadweight loss.

Figure 9.15

15. With the subsidy, consumers benefit, but taxpayers (including those same consumers) bear the burden of the subsidy, and a deadweight loss results, as the number of apartments rented exceeds the unregulated quantity. The price ceiling is analogous to the figure shown in Solved Problem 9.3 (in the text). In this case consumers who are still able to find an apartment gain, but fewer apartments are available. With the price ceiling, the good is underprovided, which again leads to a deadweight loss, although in this case there is no tax burden as there is no government expenditure. The supply and demand elasticities are important determinants of the size of the deadweight loss, as in both cases, the loss is created by the wedge between supply and demand at the new quantity. The relative slopes of the curves (which are directly related to the elasticities) determine the size of the wedge.

16. Audio-PowerPoint answer by James Dearden is also available (9A New York Wine).

a. The new equilibrium price will be lower and will be set equal to the U.S. price. b. The consumer surplus will increase. c. As demand becomes more elastic (at the equilibrium point), the consumer surplus increases.

17. Audio-PowerPoint answer by James Dearden is also available (9B Ethanol).

a. Denote the price of ethanol and gasoline with Pe and Pg, respectively. Demand for ethanol is:

Q 0, if Pe Pg

anything between 0 up to market demand for gasoline at Pg, if Pe Pg

the whole market demand for gasoline. b. Demand for ethanol in case of legislation is:

Q 5% of gasoline market, if Pe Pg

anything between 0 up to market demand for gasoline at Pg, if Pe Pg

the whole market demand for gasoline. c. The inelastic part of market demand in part (b) is further to the right compared to part (a).



d. Since the MC of ethanol is higher than the MC of gasoline at all output levels, and the MC is the firm’s supply curve, there is a price below which only gasoline will be produced, unless there is a federal mandate.

18. Audio-PowerPoint answer by James Dearden is also available (9C Water Price Elasticity).

a. According to the problem, when price doubles, the agricultural water use decreases by 30%. Therefore the price elasticity of demand is –0.3/1 –0.3.

b. The more elastic the demand, the more responsive will be the quantity consumed to price changes. An increase in the price will lead to a larger decrease in the quantity demanded.

19. Audio-PowerPoint answer by James Dearden is also available (9D Click Fraud).

a. Saving nxzpc. b. The maximum willingness to pay for a click-fraud detective is equal to the savings. For the first 200 advertisers the savings is 600 0.3 0.8 9 1296. For the next 500

advertisers the savings is 700 0.2 0.8 5 560. For the next 300 advertisers the savings is 100 0.1 0.7 12 $84.

c. Consumer surplus 200(1296 – 500) 500(560 – 500) 159,200 30,000 189,200.

20. See Figure 9.16. Panel (a) shows the impacts of medallions or annual licenses. The long run supply curve changes from S1 to S2. The area C is the deadweight loss. With medallions, the producer surplus B goes to the medallion owner; while with annual licenses, the surplus goes to government. Panel (b) shows the case of daily tax. The area C is the deadweight loss and the area B is the tax revenue.

Figure 9.16

21. As shown in Figure 9.17, the ban of self-serve shifts the supply curve from S1 to S2. As a result, consumer surplus decreases by area (A B), the social deadweight loss is (B C), and gas stations get extra producer surplus equal to area (A C).



Figure 9.17

22. As shown in Figure 9.18, the USDA recommendation shifts the demand curve from D1 to D2. As the result, the consumer surplus increases from area (A B) to area (B C).

Figure 9.18

23. As shown in Figure 9.19, the government subsidy shifts the supply curve from S1 to S2. As a result, consumer surplus changes from (A B) to (A B C E), producer surplus changes from (C G) to (C G B D), and government expense changes from zero to (B C D E F).

Figure 9.19

24. As shown in Figure 9.20, under the subsidy, consumer surplus increases from A to (A B C), and produce surplus decreases from B D to D. If the government imposes a large tariff such that the domestic price is raised to the level without the subsidy, then the consumer and produce surpluses will return to the levels without the subsidy.



Figure 9.20

25. As shown in Figure 9.21, the government subsidy to the cotton producers shifts the supply curve from S1 to S2, and the subsidy to the buyers further lowers the price from p2 to p3. As a result, consumer (buyer) surplus increases by (C D G H I) and producer surplus increases by (L H C). The government expense increases from zero to (C D E F G H I J).

Figure 9.21

26. As shown in Figure 9.22, the tax on wireless service creates deadweight loss equal to area B. On the other hand, demand for landline service is almost perfectly inelastic; therefore this is virtually no deadweight loss.



Figure 9.22

27. See Figure 9.23. Consider an imported good first. With the embargo, the consumer surplus decreases by (B C), and producer surplus increases by B. On the other hand, for an exported good, the consumer surplus increases by (B C) and producer surplus decreases by B.

Figure 9.23

28. As shown in Figure 9.24, the tariff will decrease consumer surplus by the area (A B), while the tariff revenue is equal to area (A C). If C B, the welfare in the importing country improves.

Figure 9.24



29. The specific subsidy shifts the supply curve, S in the figure on the next page, down by s 11¢, to the curve labeled S 11¢. Consequently, the equilibrium shifts from e1 to e2, so the quantity sold increases (from 1.25 to 1.34 billion rose stems per year), the price that consumers pay falls (from 30¢ to 28¢ per stem), and the amount that suppliers receive, including the subsidy, rises (from 30¢ to 39¢), so that the differential between what the consumers pay and what the producers receive is 11¢. Consumers and producers of roses are delighted to be subsidized by other members of society. Because the price to customers drops, consumer surplus rises from A B to A B D E. Because firms receive more per stem after the subsidy, producer surplus rises from D G to B C D G (the area under the price they receive and above the original supply curve). Because the government pays a subsidy of 11¢ per stem for each stem sold, the government’s expenditures go from zero to the rectangle B C D E F. Thus the new welfare is the sum of the new consumer surplus and producer surplus minus the government’s expenses. Welfare falls from A B D G to A B D G F. The deadweight loss, this drop in welfare W F, results from producing too much: The marginal cost to producers of the last stem, 39¢, exceeds the marginal benefit to consumers, 28¢.

Figure 9.25

30. At the price of 30, the quantity demanded is 30, so the consumer surplus is 12 (30 30) 450,

because the demand curve is linear.

31. CS 0.5(a/2)(a/2b) a2/8b.

32. * *( 1)

*( 1) 1/

0( / ) .

1

Ap ApPS Ap Q A dQ

33. The initial equilibrium is Q* 30, p* 30. The tax reduces output to 29. Consumers pay $31 and producers receive $29. Tax revenue is $58, and deadweight loss is $1.

34. a. The initial equilibrium is determined by equating the quantity demanded to the quantity supplied: 100 10p 10p. That is, the equilibrium is p 5 and Q 50. At the support price, the quantity supplied is Qs 60. The market clearing price is p 4. The deficiency payment was

( ) (6 4)60 120.sD p p Q

b. Consumer surplus rises from 11 2 (10 5)50 125CS to 1

2 2 (10 4)60 180.CS Producer surplus rises from 1

1 2 (5 0)50 125PS to 12 2PS (6 0)60 180. Welfare falls from

CS1 PS1 125 125 250 to CS2 PS2 D 180 180 120 240. Thus the deadweight loss is 10.



35. a. The equilibrium price and quantity without the tax is p 5 and Q 50. With the tax, the demand function will be Q 100 10(p 1) and the supply function remains the same. Hence the equilibrium will be p 5.5 and q 45.

b. Consumer surplus decreases by (50 45) 0.5/2 23.75, producer surplus decreases by the same amount. Government tax revenue is 45. Hence the deadweight loss is 23.75*2 45 2.5.

36. a. With the price ceiling, the equilibrium will be p 3 and Q 30.

b. The consumer surplus increases by 2*30 2*(50 30)/2 40, producer surplus decreases by 2*(50 30)/2 80. So the social deadweight loss is 40.

37. Without the tariff, the U.S. supply curve of oil is horizontal at a price of $14.70 (S1 in Figure 9.9), and the equilibrium is determined by the intersection of this horizontal supply curve with the demand curve. With a new, small tariff of , the U.S. supply curve is horizontal at $14.70 , and the new equilibrium quantity is determined by substituting p 14.70 into the demand function: Q 35.41(14.70 )p0.37. Evaluated at 0, the equilibrium quantity remains at 13.1. The deadweight loss is the area to the right of the domestic supply curve and to the left of the demand curve between $14.70 and $14.70 (area C D E in Figure 9.9) minus the tariff revenues (area D):

14.70

14.70

14.700.67 0.33

14.70

[ ( ) ( )] [ ( ) ( )]

[3.54 3.35 ]

DWL D P S p dp D p S p

p p dp

0.67 0.33[3.54( ) 3.35( ) ].p p

To see how a change in affects welfare, we differentiate DWL with respect to :

14.70

14.70

[ ( ) ( )] [ (14.70 ) (14.70 )]

[ (14.70 ) (14.70 )] [ (14.70 )

(14.70 ) (14.70 )(14.70 )]

(14.70 ) (14.70 ).

dDWL dD p S p dp D S

d d

D S D

dD dSS

d d

dD dS

d d

If we evaluate this expression at 0, we find that dDWL/d 0. In short, applying a small tariff to the free-trade equilibrium has a negligible effect on quantity and deadweight loss. Only if the tariff is larger—as in Figure 9.9—do we see a measurable effect.


a. The equilibrium price and quantity are:



25 3

14 28125 125

7 14

3.

28

N

N

N

p p

Q p

dp

dp

b. If $30Np , the equilibrium price and quantity are:

5

250.

p

Q

The consumer surplus is:

30

510 (30 ) 3125. CS p dp

The producer surplus is:

5

2

( 2)375.

0.012

pPS dp


The equilibrium price and quantity are:

6.44

6743.

p

Q

The consumer surplus is:

1.076

6.4450,000 571,059.14.CS p dp

The producer surplus is:

6.44

7.208

00.01 5301.04. PS p dp

Chapter 10

1. A subsidy is a negative tax. Thus we can use the same analysis that we used in Solved Problem 10.1 to answer this question by reversing the signs of the effects.

2. With a local wage tax, the equilibrium wage rate in Philadelphia is higher to offset its residents’ loss in net of tax wage. The employment in Philadelphia will decrease, while employment in the surrounding area will increase. The impact on the total employment depends on the magnitudes of those two offsetting impacts.

3. The price ceiling will create excess demand in the city, which will spill out into the suburbs. As a result, rental prices will increase in the suburbs, the number of rentals will increase in the suburbs but



decrease in the city, and the total number of units available in the metro area will remain the same (assuming no change in the total population).

4. The labor demand in the “covered” sector decreases due to the tax. As a result, the total labor demand decreases. Workers shift between sectors until the new wage is equal in both sectors. The new wage and total employment decrease.

5. Because the subsidy is not tied to new job creation or per-hour wages, the new law does not directly create employment in either sector. Firms in the subsidized sector can simply pocket the subsidy as cash. The subsidy also causes secondary effects. Firms in the uncovered sector may switch to producing products in the covered sector in order to be eligible for the subsidy. In this case, supply decreases in the uncovered sector and increases in the covered sector. The changes in output cause employment in the covered sector to increase and employment in the uncovered sector to decrease.

6. The tariff will decrease the total quantity of the good sold. The quantity sold in Europe will be lower and the price will be higher, while the quantity and price in the United States will not be affected.

7. In the consumer market, the price is lower and quantity sold is lower. If the markets are sealed, there will be no direct effect in the canners’ market. If unsatisfied consumer demanders of fresh peaches substitute for canned peaches, then the canners’ demand will increase.

8. The law simply reduces the number of buyers in the market. Therefore, the quantity sold in New York will be lower and so will the price. The price elsewhere will be lower and so will the quantity.

9. The price in the foreign market will be higher, while the quantity sold will be lower. On the other hand, the price in the home country will be lower and quantity sold will be higher, assuming total production remains the same.



10. See Figure 10.4.

Figure 10.4

11. As Chapter 4 shows, the slope of the budget constraint facing an individual equals the negative of that person’s wage. Panel (a) of the figure illustrates that Pat’s budget constraint is steeper than Chris’s because Pat’s wage is larger than Chris’s. Panel (b) shows their combined budget constraint after they marry. Before they marry, each spends some time in the marketplace earning money and other time at home cooking, cleaning, and consuming leisure. After they marry, one of them can specialize in earning money and the other at working at home. If they are both equally skilled at household work (or if Chris is better), then Pat has a comparative advantage (see Figure 10.5) in working in the marketplace, and Chris has a comparative advantage in working at home. Of course, if both enjoy consuming leisure, they may not fully specialize. As an example, suppose that before they got married, Chris and Pat each spent 10 hours a day in sleep and leisure activities, 5 hours working in the marketplace, and 9 hours working at home. Because Chris earns $10 an hour and Pat earns $20, they collectively earned $150 a day and worked 18 hours a day at home. After they marry, they can benefit from specialization. If Chris works entirely at home and Pat works 10 hours in the marketplace and the rest at home, they collectively earn $200 a day (a one-third increase) and still have 18 hours of work at home. If they do not need to spend as much time working at home because of economies of scale, one or both could work more hours in the marketplace, and they will have even greater disposable income.

Figure 10.5



12. Yes, they may want to trade. If two individuals are consuming different bundles and have identical preferences, their marginal rates of substitution may be unequal and they may gain by trading. For example, if each had the utility function U XY, an initial allocation of 4X, 2Y for one person and 2X, 4Y for the other, each would have a utility level of 8, and their marginal rates of substitution would differ. By trading 1 unit of X for 1 unit of Y, each can achieve a utility level of 9.

13. See Figure 10.6. At point A, the individual’s indifference curves are tangent. At point B, however, Joe is indifferent to point A, but Mary is happier, making the allocation implied by B Pareto-superior to that at A.

Figure 10.6

14. No. Trade is only valuable if one party has at least a comparative advantage over the other. With identically sloped production possibility frontiers, there is no comparative advantage.

15. Yes. In this case, Britain is relatively more efficient at producing cloth than food (MRT 2), and Greece is relatively more efficient at producing food than cloth (MRT –0.5). Suppose that Britain has a high preference for food relative to cloth and Greece has a high preference for cloth relative to food. Thus if Greece trades 1 unit of food to Britain for 2 units of cloth, Greece would end up with 1 unit of food and 3 units of cloth per day, which is beyond its current production possibility frontier. Britain gets to consume 9 units of cloth and 6 units of food per day, which it cannot achieve without trade.

16. Assume a four-person economy with two goods. The welfare function W (U1)(U2)(U3)(U4), where for each individual Ui XiYi would be maximized when each person receives an equal allocation of each good.

17. If you draw the convex production possibility frontier on Figure 10.5, you will see that it lies strictly inside the concave production possibility frontier. Thus more output can be obtained if Jane and Denise use the concave frontier. That is, each should specialize in producing the good for which she has a comparative advantage.



18. In an economy with many individuals but the goal of maximizing the utility of only one, the welfare function is W max{U1, U2,…, Un}. An allocation with all goods for the best-off individual and no goods for any other person would maximize this function. The individual chosen would be the person who would receive the highest utility level from consuming all goods.

19. a. In the absence of trade, the United States can produce 30 units of food and 15 units of toys. Mexico can produce 10 units of food and 2 units of toys.

b. The United States has comparative advantages in producing toys and Mexico has comparative advantages in producing food.

c. Panels (a) and (b) in Figure 10.7 show the production frontier for the United States and Mexico. d. Panel (c) in Figure 10.7 shows the joint production frontier. e. As shown in panel (c), if the United States produces 5 units of food and

12.5 units of toys and Mexico produces 10 units of food, the total amount of goods is larger than when there is no trade. Hence both countries can benefit from trade.

Figure 10.7

20. a. The marginal rate of substitution is 1 for either person.

b. The contract curve is on the 45-degree line, which is the shaded section within the lens as shown in Figure 10.8.

Figure 10.8



21. a. w (a c e)/(b d f ). L1 a b(a c e)/(b d f ). L2 c d(a c e)/(b d f ).

b. L1 a bw. To solve for w2, set L2 L – L1 e fw2 (a bw), so w2 (a c e bw)/(d f ). L2 c d(a c e bw)/(d f ).

c. w w. L1 a bw. L2 c dw. L a c (b d)w.

22. Because there are five units of each good, we can set Q1 Q2 and solve for the price ratio, which turns out to be p1 p2. Substituting back into the original equations yields p1 p2 5.

23. Using the partial equilibrium condition, QSi QDi in each market will yield functions that can be solved for either price (e.g., p1). These equations can be used to solve for the other price (p2). The prices can then be used to solve for the quantities in each market.

D1(p1, p2) S1(p1)

p1 3.25 1/4p2

D2(p1, p2) S2(p2)

p2 5/3 1/3p1

p1 $4.00, p2 $3.00

Q1 6, Q2 4

24. Following Equation 10.13, the contract curve is where each individual’s MRS are equal to each other. Note that MRSt (Ht/Gt) and MRSm (Hm/2Gm). Also note that Gt Gm 100, and Ht Hm 50. As in Solved Problem 10.3, equating the MRSs and using the information about endowments given in the problem we get the following contract curve: 100Gm 100Hm HmGm 0.

25. Using Equations 3.26 and 3.27, the demands for each good are found to be Gt Yt /2P; Ht Yt /2; Gm Ym/3P; and Hm 2/3Ym. To find the price of G with the price of H normalized to 1, note that the sum of the individual demands for G equals the supply of G, as in Solved Problem 10.4, or Gt + Gm 100. Substituting demands into this equation and rearranging yields the equilibrium price:

( /200) ( /300)

.(1 ( /200) ( /300))

t m

t m

H HP

G G

Further simplifying,

3 2

.600 3 2

t m

t m

H Hp

G G

Substituting 100 – Gt for Gm and 50 – Ht for Hm and simplifying,

100.

400

t

t

Hp

G

For a step-by-step solution, see the guided solution for problem 10.25 in MyEconLab.




a. Demand functions:

25 19820,000

CF F

nQ p

(1)

100 150

3T

T

pQ

(2)

Supply functions:

50 10F F TQ p p (3)

50 10T T FQ p p (4)

The solution to the four equations above is:

1497600 87

523 1046002988000 3

523 2092043650 7

523 41840068490 1

.523 2092000

F C

T C

F C

T C

Q n

Q n

p n

p n

If 40,000,Cn then the equilibrium prices and quantities are:

2897

5707

84.13

130.97.

F

T

F

T

Q

Q

p

p

The effects of Cn on the equilibrium prices are:

7

418400

1.

2092000

F

C

T

C

dp

dn

dp

dn

b. Now 10Fp in the demand equations, and 35Fp in the supply equations.

Demand side:

25 198 1020,000

CF

nQ

(1)

100(150 )

3T

T

pQ

(2)



Supply side:

50 35 10F TQ p (3)

50 10 35T TQ p (4)

The solution to equations (2), (3) and (4) is:

454

6128

129.57.

F

T

T

Q

Q

p

c. Assume all the revenues go to the salary. Then in the competitive medical checkup market, the equilibrium salary of dentists per day is: 130.97 5707 747445.79,T Tp Q while in the insurance-company-dictated medical-doctor payments, the equilibrium salary of dentists per day is: 129.57 6129 794134.53 747445.79;T Tp Q therefore the equilibrium salary of dentists increases after the shift.


A utility possibilities frontier is shown in Figure 10.9.

Figure 10.9

The social welfare choice problem is:

1 2

1 1 2 2,

2 1. . ( ).

U UMax U U

s t U F U

The Lagrangian is:

1 2 1 1 2 2 2 1( , , ) ( ) .L U U U U U F U



The F.O.C.s are:

1 11

22

2 1

( ) 0 (1)

0 (2)

( ) 0 (3)

LF U

U

L

U

LU F U

From (1) and (2) we have: 1

21( )F U . Therefore the social welfare is maximized at the point on

the utility possibilities frontier with a slope of 1

2.

As 1

2

increases, 1( )F U decreases, implying a steeper tangent line, and hence, as shown in Figure

10.9, Person 1 benefits and Person 2 is harmed, since U1 becomes greater and U2 becomes smaller.

Chapter 11

1. See Figure 11.6. Because there is no supply curve, if the demand curve shifts from D0 to D1, output increases from Q0 to Q1, but price remains unchanged.

Figure 11.6

2. The effect of a franchise tax or lump sum tax on a monopoly is to reduce profits by the amount of the tax. Because there is no change in marginal cost, the profit-maximizing/loss-minimizing output and price remain unchanged, with one exception. If the tax is large enough, losses may exceed variable costs. If that is the case, the firm will shut down (produce no output) in order to minimize losses.

3. When the average total cost curve lies above the demand curve at all output levels, the monopolist cannot earn positive profits.

4. See Figure 11.7. The values of price and quantity depend on the demand curve drawn by the student. In the given graph, profit-maximizing quantity is 4 and profit-maximizing price is 8. Profits are area abcd and the deadweight loss is area bef.



Figure 11.7

5. Yes. As the “Electric Power Utilities” application illustrates, the demand curve could cut the average cost curve only in its downward-sloping section. Consequently, the average cost is strictly downward sloping in the relevant region.

6. No. In order for a firm to be a natural monopoly, its production must exhibit economies of scale; that is, firm’s average cost curve must be downward sloping. If the firm operates in the upward sloping region of its average cost curve, it is possible that two or more firms could produce in the same industry more efficiently than one firm.

7. Utilities are often government-created monopolies. In addition, the government essentially creates monopolies through patents and copyrights.

8. See Figure 11.8. A competitive firm maximizes long-run profits by setting LMC p, as long as price exceeds average cost (it should shut down if p LAC). Because marginal cost is above average cost only where AC slopes up, the firm will never operate in an area where average cost is falling. A monopolist maximizes profit by setting MR MC, which can occur on either the upward- or downward-sloping portion of the LAC curve. In the graph, DM represents the demand curve faced by the monopolist, and dC represents the demand curve faced by a competitive firm.

Figure 11.8

9. A monopolist may set price equal to marginal cost if other firms can enter costlessly. If there is free entry, any price above marginal cost will attract other firms. Thus even though the firm has no current competitors, it sets price equal to marginal cost to deter entry of potential competitors. Also, if not all customers are charged the same price (price discrimination), the firm may want to sell the last unit where price equals marginal cost (Chapter 12).

10. See Figure 11.9. If the government sets a price cap between the monopoly price and the socially optimal price, output increases from QM to QR, and deadweight loss is reduced from area abc to cdf.



Figure 11.9

11. See Figure 11.10. To be on the contract curve (Pareto efficiency) requires that all goods be traded at competitive prices. Because Jane is a monopolist, she sets the price of wood above the competitive price. At this higher price, she receives more candy bars per unit of wood than with competitive prices, and consumers end up with less wood than with competitive prices. The monopoly price line in the graph depicts this higher price ratio. Instead of reaching the Pareto-optimal solution at a, Jane is able to use her market power to force the solution at b, which is off the contract curve and Pareto inferior.

Figure 11.10

12. See Figure 11.11. In this case, MR curve coincides with the demand curve. Equilibrium quantity is 100 and equilibrium price is $100. Consumer surplus is zero and producer surplus is ($100 $10) 100 $9000. If a price ceiling of $30 is imposed, consumer surplus increases and producer surplus decreases by area abcd. There is no deadweight loss.



Figure 11.11

13. The wholesale price of milk represents the marginal cost for retailers, who sell to consumers. In Figure 11.12, suppose prices at the wholesale level fall 30.3% from $2.00 to $1.394 per gallon, or $0.606 per gallon. That is, for retailers, marginal cost shifts from MC to MC. If the original retail price were $3.00 per gallon, a 30.3% price decrease would mean the new price of $2.09, a drop of $0.91 per gallon. However, the negative slope of the demand curve results in a price decrease that is less than $0.91. The new retail price, for the given demand curve, is $2.50. In other words, price at the retail level may not fall by the same percentage as the wholesale level due to the slope of the demand curve.

Figure 11.12

14. See Figure 11.13. Summing the demand curves horizontally yields the market demand curve. Once the price is set, quantity in each market is determined from the individual product demand curves. If the firm charged different prices in each market, physicians could simply prescribe the lower-priced version of the product.



Figure 11.13

15. Once a book is online, it is available to consumers for free. As a result, the publisher may lose some consumers after the copyright expires. Limiting the length of a copyright (in the United States or elsewhere) would encourage the publisher to charge a higher price for the novel, as the publisher attempts to recoup the cost of publishing the novel and earn profit in a shorter period before the copyright expires. After the copyright runs out, the publisher may lower the price of the novel to stay competitive with the online version.

16. This test is relevant because if the club were maximizing revenue, it would be operating at the level MR 0, where the elasticity is 1.

17. Consider a small hotel where the jammer would cost $25,000. If the expected profit from room phone service is $A per day per room and the number of rooms in the hotel is B. Then if $365AB $25,000, it is profitable to install a jammer.

18. See Figure 11.14. Assume that Bleyer Industries Inc. has a production process described by the marginal cost curve 1MC . As a monopoly, Bleyer Industries Inc. can charge price 1P , while selling

1Q plastic Easter eggs. Since 1P is above Bleyer’s average total cost AC at the level of 1Q , Bleyer is profitable. The Chinese firms have lower production cost; we can denote their marginal cost as 2MC . Once the Chinese firms start competing with Bleyer, they can charge a price as low as 2P . Such a low price would force Bleyer to exit the market because 2P is below AC and Bleyer cannot make positive profit.



Figure 11.14

19. a. Assume demand is 1D . Then, the consumer surplus is ,a d h the producer surplus is b i,

and social deadweight loss is c e. If the monopoly behaves like a price taker, the quantity will be Qc, as shown in the figure.

b. When the new demand curve, 2 ,D is tangent to the original one at (Q*, P*), as shown in Figure 11.15, the price and quantity will not change. However, the quantity if the monopoly behaves like a price taker will change to Q2. Consumer surplus will decrease and will be the area a h under the concave demand curve; producer surplus will remain as b i; deadweight loss will decrease to .c

c. Under the convex demand curve, 3 ,D the price and quantity will be the same as in the linear case. But under the price-taker assumption, the quantity supplied will be Q3. Consumer surplus will increase and will be the area a d h f; producer surplus will remain as b i; deadweight loss will increase to c e g.

Figure 11.15

20. See Figure 11.16. After the remove of the tariff, the former monopoly will sell its products at the world price pw. The new quantity Qw is determined by the intersection of its MC and world price. The consumer benefits from removal of the tariff, and the former monopoly suffers losses.



Figure 11.16

21. When the hoi polloi buy the chocolate, the snobs won’t buy. So the monopoly is facing a relatively flat demand curve, which suggests a low price, high quantity outcome. On the contrary, if the hoi polloi do not buy the chocolate, the snobs will buy. In this situation, the monopoly will face a steep demand curve, resulting in a high price, low quantity outcome. If the demand curve for the snobs is substantially steeper than that for the hoi polloi such that the profit from the former is larger than that from the latter, the monopoly will choose to cater to the snobs.

22. For a general linear inverse demand function, p(Q) a bQ, dQ/dp 1/b, so the elasticity is p/(bQ). The demand curve hits the horizontal (quantity) axis at a/b. At half that quantity (the midpoint of the demand curve), the quantity is a/(2b), and the price is a/2. Thus the elasticity of demand is p/(bQ) (a/2)/[ab/(2b)] 1 at the midpoint of any linear demand curve. As the chapter shows, a monopoly will not operate in the inelastic section of its demand curve, so a monopoly will not operate in the right half of its linear demand curve.

23. Set MC MR and solve:

*

*

100 2

5

5 100 2

47.5

52.5

MR Q

MC

Q

Q

p

If ( ) 100 5 , C Q Q the answer does not change because marginal cost is still 5,MC and therefore the profit-maximizing condition is still the same.

24. Set MC MR and solve:

1/ 2 1/ 2 1/ 2

*

*

(10 ) (10 ) 5

5

1

10

dR d dMR Q Q Q Q

dQ dQ dQ

MC

Q

p

25. The total profit is total revenue minus total cost and tax.

( ) ( ) ( ( ) ) TR TC tQ P Q Q C Q tQ

The first-order condition is



( ( )) ( ) 0.

d dTR TC tQ MR MC t

dQ dQ

Imposing tax is equivalent to increasing the marginal cost, i.e. . MC MC t Hence after tax is imposed, output will drop, price will increase, and profit will be smaller.

26. When the demand curve is linear, the marginal revenue curve will always be linear with twice the slope of the demand curve. This is true because when multiplying the demand curve by Q to obtain total revenue, we always obtain a squared term in the revenue function, which then doubles the slope when we take the derivative to obtain marginal revenue. For example, if the demand curve is

P a bQ

TR aQ bQ2

2 .dTR

MR a bQdQ

Because the slope of MR is double of the slope of the linear demand curve, MR curve always crosses the MC segment in the middle between the y-axis and point ce (see figure in the Botox application). Hence triangles A and C have one equal side and three equal angles, which implies that these triangles are equal.

27. The inverse demand function is p 775 375Q. Imposing a specific tax of $75 will be equivalent to shifting the demand curve to p 700 375Q. With MC 25, profit-maximizing quantity and price are Q 0.9 and p 437.5. The new deadweight loss will be (437.5 25)(2 0.9)$226.875 million.

28. With a price ceiling of $200, the monopoly will produce Q (775 200)/375 1.53 million vials and the deadweight loss will be (200 25) (2 1.53)/2 $41.1 million.

29. Audio-PowerPoint answer by James Dearden is also available (11A State Wine Stores).

a. Setting MR MC we get 5 2 0.001Q 2. Solving the equation we get Q* 1500. Plugging the quantity back in the inverse demand function we get P* 3.50. The profit is (3.5 2) 1500 $2250.

b. Setting P MC we get 5 0.001Q 2. Solving the equation we get Q* 3000. The competitive price will be P* 2.

c. The supply function in New Jersey is perfectly elastic, thus any tax will be paid by suppliers. A specific tax of $1.50 will raise the supply curve up to $3.50. The price buyers pay will be $3.50, the price received by sellers will be $2, and the quantity exchanged will be 1500. The tax revenue is 1500 1.5 2250. The State of New Jersey’s tax revenue is equal to Pennsylvania’s monopoly profit.

30. Audio-PowerPoint answer by James Dearden is also available (11B iPod).

a. Setting MR MC we get 1.98 2 0.00198Q 0. Solving the equation we get Q* 500. Plugging the quantity back in the inverse demand function we get P* 0.99.

b. Setting MR MC we get 2.58 2 0.0129Q 0. Solving the equation we get Q* 100. Plugging the quantity back in the inverse demand function we get P* 1.29.

31. The price/marginal cost ratio is 99/45.37 2.18. Lerner Index is / 99 45.37/99 0.542. P MC P Using the formula for the Lerner Index, the elasticity is .85.



32. See MyEconLab Chapter 11, “Humana Hospitals,” for more examples. For saline solution, p/MC 55.4 and the Lerner Index is (p MC)/p 0.98. From Equation 11.9, we know that (p MC)/p 0.98 1/, so 1.02.

33. The Lerner Index is (1840.8 100)/1840.8 0.95. Hence Tenet believes that the elasticity it faces is 1.06.

34. The price/marginal cost ratio is 5000/2000 2.5. The Lerner Index is (5000 2000)/5000 0.6. Hence Segway believes it faces a demand elasticity of 1.67.

35. The price/marginal cost ratio is 499/258 1.93. The Lerner Index is (499 258)/499 0.48, and the elasticity Apple believes it faces is 2.08.

36. The Lerner Index is (84.95 37)/84.95 0.56. Hence Stamps.com believes that it faces a demand elasticity of 1.79.

37. A profit tax (of less than 100%) has no effect on a firm’s profit-maximizing behavior. Suppose the government’s share of the profit is . Then the firm wants to maximize its after-tax profit, which is (1 ). However, whatever choice of Q (or p) maximizes will also maximize (1 ). Figure 19.3 gives a graphical example where 1/3. Consequently, the tribe’s behavior is unaffected by a change in the share that the government receives. We can also answer this problem using calculus. The before-tax profit is B R(Q) C(Q), and the after-tax profit is A (1 )[R(Q) C(Q)]. For both, the first-order condition is marginal revenue equals marginal cost: dR(Q)/dQ dC(Q)/dQ.

38. In the competitive case, equilibrium is found by equating supply and demand curves, D S:

1.787 0.0004641 0.496 0.00020165 .Q Q

Then

*

*

3429.2151 lb

0.19550 $/lb.C

C

Q

P

Once the specific tax $0.01 is imposed, the supply curve, which is also MC, shifts up (see Figure 11.17a). To find the after-tax equilibrium, we solve

1.787 0.0004641 0.496 0.00020165 0.01

3414.1945 lb

0.20247 $/lb.C

C

D S S

Q Q

Q

P

This suggests that tax incidence in the case of competition is

' * 0.20247 0.195500.7 70%.

0.01C CP Pp

In the monopoly case, pretax profit-maximizing output and price levels are found by equating MC and MR (see Figure 11.17b). Marginal revenue is

( ( ) ) ((1.787 0.0004641 ) )

1.787 0.0009282 .

dR d dMR P Q Q Q Q

dQ dQ dQ

Q



Marginal cost is the same as in the competitive case. Then, MR MC means that

*

*

1.787 0.009282 0.496 0.00020165

2020.71

1.787 0.0004641 2020.71 0.8492$ / .M

M

Q Q

Q lb

P lb

After specific tax is imposed, MC shifts to MC and monopoly chooses output that satisfies the following condition

.MR MC MC

Then

1.787 0.0009282 0.496 0.00020165 0.01

2011.7714 lb

1.787 0.0004641 2011.7714 0.8533 $/lb.M

M

Q Q

Q

Q

Tax incidence on consumers in the case of monopoly is

* 0.8533 0.84920.41 41%.

0.01M MP Pp

Figure 11.17



39. Consider constant marginal cost and suppose the monopoly is facing a linear demand function with inverse demand function p a bQ. The monopoly will produce Q (a MC)/2b, where MC is the lower of two marginal costs at the factory with lower MC, and zero units at the factory with higher MC. Suppose both factories have increasing marginal cost, the monopoly will produce at two factories Q1 and Q2 such that MC(Q1) MC(Q2) MR(Q).

40. a. If the consumer cannot steal music, the total demand function function will be p 120 Q/2. The monopoly will set MR 120 Q 20, such that Q 100 and p 70. Consumer surplus will be 2500, profit and producer surplus will be 5000, and deadweight loss will be 2500.

b. If the dishonest customer can steal music, then the total demand function will be p 120 Q. The monopoly will set MR 120 2Q 20, such that Q 50 and p 70.

c. When dishonest customers can pirate the music, consumer surplus will consist of consumer surplus of honest and dishonest customers. Consumer surplus of dishonest customers will be 7200 and consumer surplus of honest customers will be 1250; therefore, total consumer surplus will be 7200 1250 8450. Producer will receive profit and surplus only by selling to the honest customers. Profit and producer surplus will be 2500. Deadweight loss will be 1250.

41. Given that the demand curve is p 10 Q, its marginal revenue curve is MR 10 2Q. Thus the output that maximizes the monopoly’s profit is determined by MR 10 2Q 2 MC, or Q* 4. At that output level, its price is p* 6 and its profit is * 16. If the monopoly chooses to sell 8 units in the first period (it has no incentive to sell more), its price is 2 and it makes no profit. Given that the firm sells 8 units in the first period, its demand curve in the second period is p 10 Q/, so its marginal revenue function is MR 10 2Q/. The output that leads to its maximum profit is determined by MR 10 2Q/ 2 MC, or its output is 4. Thus its price is 6 and its profit is 16. It pays for the firm to set a low price in the first period if the lost profit, 16, is less than the extra profit in the second period, which is 16( 1). Thus it pays to set a low price in the first period if 16 16( 1), or 2 .

42. a. To solve for the expansion path, set the ratio of the marginal products equal to the ratio of the input prices (see Appendix 7C).

L

K

MP

MP r

This gives the equation for the expansion path 0.25 rK L L . See Figure 11.18.

b. Substituting the expansion path into the production function yields L 2Q, and K 0.5Q. Thus C(Q) wL rK w2Q r 0.5Q 1(2)Q 4(0.5)Q 4Q.

c. Long-run profit-maximizing output is where LMC MR.

2

4

( ( ) ) (100 ) 100 2

dCLMC

dQ

dR d dMR P Q Q Q Q Q

dQ dQ dQ

Q* 48

p* 52



d. To solve, plug the capital labor ratio K 0.25L and the output level (48) into the production function.

1/ 2 1/ 2 1/ 2 1/ 248 (0.25 ) 0.5 L K L L L

L* 96

K* 0.25 96 24

See Figure 11.18.

Figure 11.18

43. Solution also provided in Jim Dearden’s audio presentation.

a. Total revenue is: 2

20 20( ) (9 ) 9 ,Q QTR Q p Q Q Q therefore marginal revenue is:

( )

( ) 9 .10

dTR Q QMR Q

dQ

Marginal cost function is: ( ) 2( ) 10 8 2 .dC QdQm Q Q Q

Marginal revenue curve (MR) and marginal cost curve (m(Q)) are shown in Figure 11.19.

1 2

79 5441 79 5441( ) ( ) , .

40 40MR Q m Q Q Q

b. Profit function is: 2 22 3

20 3( ) ( ) ( ) 9 (10 10 4 );QQ TR Q C Q Q Q Q Q the F.O.C. is:

12 2

2

79 54410

( ) 79 409 10 8 2 1 2 010 10 79 5441

0.40

Qd Q Q

Q Q Q QdQ

Q

The S.O.C. is:

21

12

( ) 79 54414 0,

10 10

d QQ

dQ

22

22

( ) 79 54414 0.

10 10

d QQ

dQ

Therefore the S.O.C. is satisfied at 79 54412 40 ,Q i.e. the profit-maximizing output is

2

79 5441.

40Q



Figure 11.19


a. The revenue function is: 1000( ) (10 ) ;QR Q p Q Q then the F.O.C. is:

( )10 0 5000 10 5.

500 1000

dR Q Q QQ p

dQ

The revenue-maximizing price is $5.

b. The sum of the cable car revenues and the economic impact is:

( ) ( ) ( ) 10 4 141000 1000

Q QS Q R Q EI Q Q Q Q.

The F.O.C. is:

( )14 0 7000 10 3.

500 1000

dS Q Q QQ p

dQ

The optimal price is $3.

Chapter 12

1. In order to price discriminate, Alexx must have market power—the ability to set prices. Consumers must have varying price sensitivities, and Alexx must be able to identify individual consumers or groups of individuals based on willingness to pay. Alexx must also be able to prevent reselling after the initial sale.

2. This policy allows the firm to maximize its profit by price discriminating if people who put a lower value on their time (so are willing to drive to the store and move their purchases themselves) have a higher elasticity of demand than people who want to order over the phone and have the goods delivered.

3. The colleges may be providing scholarships as a form of charity, or they may be price discriminating by lowering the final price for less wealthy families (who presumably have higher elasticities of demand).



4. The pharmaceutical firms offer the discount to low-income seniors because the seniors would probably not purchase the medicines if they were not discounted. By segregating this portion of the market, they are able to price discriminate profitably because the marginal cost of producing the extra medications is very low. Thus if they can still charge higher prices to the rest of the market, they will profit from the discounted prescriptions as well. (In addition to the short-run profits, the pharmaceutical firms get good publicity, which could forestall more costly regulation in the future.)

5. Since adults cannot use children’s tickets to enter Disneyland, there won’t be a resale problem. On the other hand, for nonlocal visitors, a small difference in ticket price does not mean much compared to their travel cost. In other words, their willingness to pay the price is higher than locals.

6. When there is a big price different across the border and shipping the car from Canada to the United States is relatively cheap, consumers in Canada are able to make a profit by reselling their cars in the United States. To prevent this kind of resale that would decrease Ford’s profit from price discrimination, Ford required Canadian dealers to sign an agreement that prohibited moving vehicles to the United States. Since the prevention might not be effective, in the following year Ford cut the supply in Canada to only 2000 cars to practically eliminate the resale problem.

7. Google is essentially practicing price discrimination perfectly by taking advantage of advertisers’ desire to reach small, difficult-to-find segments of the population and varying the price of ads according to advertisers’ willingness to pay. The amount that firms will pay for advertisements depends on the difficulty of making a match. Firms will pay more to advertise when there are fewer self-identified potential customers—fewer people searching for a phrase. That is, firms bid more when there are fewer customers and the need to target the advertisement is greater. Therefore, a firm that provides local services should be willing to pay more for an ad in a small town because there are fewer self-identified potential customers.

8. If the difference in the cost of a car renting service is equal to the difference in the rental price between the two cities, then there is no price discrimination. Otherwise, there may be price discrimination, especially as resales of the service is practically impossible.

9. Lower profit margin indicates the PC market is becoming more competitive. In other words, the market power of PC producers is decreasing, therefore lowering their ability to price discriminate.

10. a. This is third-degree price discrimination. Suppose 1000 tickets were available; then price is determined by the bidder with the 1000th highest willingness to pay. All bidders with higher willingness to pay get the same price.

b. This is first-degree price discrimination, as each bidder is charged at a price equal to his or her willingness to pay.

11. The difference in profit between a single price monopoly and perfect price discrimination is the area A and C in the application, which is $375 million.

12. This is a called a two-part tariff. The union can set the price at the level w* where demand and supply curves cross and request a lump-sum contribution whose total is equal to the area below the demand curve and above the price level w*. If the workers are not identical, say, if their ages are different, then the value of that uniform lump sum contribution may be different for different workers.



13. See Figure 12.3. The monopolist produces where price equals marginal cost. Total revenue is the area under the demand curve from the origin to Q*, or OafQ*. Total cost is 0dbQ*. Profits are adg gbf.

Figure 12.3

14. Yes. Even if a consumer purchased 40 units per day, the average price would just equal the monopoly price, but the consumer surplus would fall from $450 to $400. In addition, the consumer in panel (b) pays $60 for 30 units, but the consumer in panel (a) must purchase 40 units to achieve the same average price. If they only bought 30 units, their average price would be $63.33.

15. See Figure 12.4. Output expands, as do profit and consumer surplus. When the markets are combined, the monopolist sells *

2Q for $5, all to customers in Market 2. When the markets can be separated,

price and quantity remain unchanged in Market 2, but the monopolist also sells *1Q for p1.

Figure 12.4

16. Before the item is discounted, the department store attempts to sell the item to customers with relatively high reservation prices. If after a while the item is not sold out, the department store may try to sell the item to customers with lower reservation prices by putting the item on sale.

17. Yes. The monopoly’s ability to price discriminate depends on the marginal cost. Suppose there are two groups of potential consumers. If the marginal cost is so high that one group will not buy the good at a price acceptable to the monopoly, then the monopoly is not able to price discriminate in this case.



18. No. If no consumers want the second product, profits are reduced by bundling it with the first product. The reason is that because there is no demand for the second product, consumer demand for the bundled products will be equal to the demand for the first (desired) product alone. If there is no secondary market for the undesired product and the consumer must pay to dispose of it, then profits will be reduced rather than increased, as consumer demand falls to reflect the disposal cost.

19. If m 7, the marginal revenue curve crosses the marginal cost curve above the kink point. The monopoly will charge the monopoly price as if there is only one country, the country with higher demand. If m 4, the marginal cost curve crosses the marginal revenue curve three times, including the vertical portion of the marginal revenue cost. At the first cross point to the left of the kink point, we have 9 2Q 4. Hence Q 2.5 and p 6.5, the profit is 2.5 (6.5 4) 5.25. For the cross point on the vertical section, Q 3 and p 6, the profit is 3*(6 4) 6. For the cross point to the right of the kink point, 4 7 (2/3)Q. Hence Q 4.5 and p 5.5. The profit is 4.5 (5.5 4) 6.75. Hence the monopoly will set the price at 5.5.

20. Abbott raised the price of Norvir such that it can make more profit from it if the demand elasticity is low. At the same time, not raising the price of Kaletra will give its own product a price advantage on the market.

21. Audio-PowerPoint answer by James Dearden is also available (12A GM Pricing).

a. Multi market price discrimination. b. In the market the demand is relatively more elastic; the price will be relatively lower. c. Auto in future is a substitute of auto today. So as if there is an expectation that the future price of

auto is going to rise, the demand for auto today increases. The national discounting plan that is targeted to reduce slumping sales is NOT a form of price discrimination. That is a response to current lower demand.

22. Audio-PowerPoint answer by James Dearden is also available (12B Mets Pricing).

a. The demand for more popular games is relatively less elastic, therefore a higher price can be charged. That is price discrimination based on different characteristics of demand in two markets.

b. A lower price for cheap seats for unpopular games decreases the demand for tickets for Yankee games. Assume for simplicity that the total cost is zero. Then the goal is to maximize total revenue TRY TRM PYQY PMQM. Taking derivative with respect to PM and putting it equal to zero, we get:

( )0

Y M Y M

Y M MM M M

TR TR Q QP Q P

P P P .

Y MY M M

M M

Q QP Q P

P P

Note that

0

Y

M

Q

P and 0.

M

M

Q

P

Therefore it sets PM such that the above equality holds.

23. Audio-PowerPoint answer by James Dearden is also available (12C Grocery Store Pricing).

a. The supermarket cannot approach perfect price discrimination. Therefore it is multi market price discrimination.

b. The price markup (P MC) is inversely related to elasticity. The demand, which is less elastic, can be charged a higher price.



24. Audio-PowerPoint answer by James Dearden is also available (12D Publisher Pricing).

a. The university will only buy Journals A and B.

b. The net gain for the university if it buys the journals individually is (2000 1600) (1100 800) 400 300 700. The net gain of buying a bundle cannot be less than $700, therefore the maximum it is willing to pay is (2000 1100 1400) 700 $3800.

c. The individual prices that maximize revenue for the publisher are $1800 for Journal A, $1100 for Journal B, and $1400 for Journal C. In this case the total revenue will be $7500. The maximum revenue for mixed bundles is to sell Bundles A, B, and C for $4300 to University 1 and the same bundle for $4000 to University 2. In this case its revenue will be $8300.

25. a. The marginal cost is zero, so the MC curve is the X axis. The amount of tickets sold will be T*, where the MR curve intersects X axis.

b. The concerts’ failure may indicate monopoly price, but not that the monopoly set too high a price. c. If the monopoly can perfectly price discriminate, it can obtain all the producer surplus below

the demand curve.

26. No, it is not reasonable to conclude that U.S. drivers subsidize European gasoline prices. If oil companies have market power in the United States and Europe and can price discriminate, they will treat each market as a separate market setting prices and quantities at profit-maximizing levels independent of the other market.

27. The larger a magazine’s or newspaper’s circulation, the more advertisers will pay per advertisement. Therefore, many magazines and newspapers will drop their subscription price to boost circulation and in turn increase advertising revenue.

Monopolies maximize profit by producing such that marginal revenue equals marginal cost and then charging the price indicated by the demand curve. Advertising revenue is much like a specific (per-unit) subsidy or negative tax. Advertising revenue shifts the demand curve for subscriptions up, as a subsidy would. This shifts the marginal revenue curve up as well. Since the marginal revenue curve with the advertising subsidy is above the marginal revenue curve without advertising, the profit-maximizing quantity is higher and the corresponding price indicated by the original demand curve is lower. Thus advertising increases output and decreases subscription prices. As advertising dollars shrink, the demand curve (and the marginal revenue curve) with advertising will shift down toward the demand curve with no advertising, and the profit-maximizing quantity will decrease and the corresponding price will increase.

28. Equating the right-hand sides of the demand and supply functions, 100 w w 20, and solving, we find that w 60. Substituting that into either the demand or supply function, we find that H* 100 60 60 20 40. To find w*, we need to equate areas A and C in the figure in Solved Problem 12.1. We could integrate, but with a linear demand function it is easier to calculate the area of triangles. The area of A is 1

2 (100 w*)2, while the area of B is 12 (w* 60)2. Equating these areas and solving,

we find that w* 80. Substituting that into the demand function, we obtain H 20.

29. Setting marginal revenue equal to marginal cost yields Q* 30, p* 60. Profit is $900, consumer surplus is $450, welfare is $1350 (PS CS), and deadweight loss is $450.

30. The profit function of the monopoly described in Figure 12.3 is

1 1 2 2 1 3 3 2 3.p Q p Q Q p Q Q mQ



The monopoly faces a demand curve 90 p Q . Then, 90 Q p and we can write

1 1

2 2

3 3

90

90

90 .

Q p

Q p

Q p

Using these expressions, the profit function can be rewritten as

1 1 2 2 1 3 3 2 3

1 1 2 1 2 3 2 3 3

2 2 21 1 1 2 2 2 3 3 3

(90 ) (90 90 ) (90 90 ) (90 )

(90 ) ( ) ( ) (90 )

90 90 .

p p p p p p p p m p

p p p p p p p p m p

p p p p p p p p m mp

FOC:

1 21

1 2 32

2 33

90 2 0

2 0

2 0

p pp

p p pp

p p mp

Here 30.m Solving simultaneously, we find that profit-maximizing prices are 1 75,p 2 60,p

and 3 45.p

31. a. The monopoly can set the price to be 60 and the minimum amount to be 60 to achieve the same outcome as in the perfect price discrimination case.

b. For an initial price of 90, the total demand will be zero.

32. See MyEconLab Chapter 12, “Aibo,” for more details. The two marginal revenue curves are MRJ 3,500 QJ and MRA 4,500 2QA. Equating the marginal revenues with the marginal cost of $500, we find that QJ 3,000 and QA 2,000. Substituting these quantities into the inverse demand curves, we learn that pJ $2,000 and pA $2,500. Rearranging Equation 11.9, we know that the elasticities of demand are J p/(MC p) 2,000/(500 2,000) 4

3 and A 2,500/(500 2,500) 5

4 . Thus using Equation 12.3, we find that

54

43

1 1/( ) 1 1/2,0000.8 .

2,500 1 1/( ) 1 1/

J A

A J

p

p

The profit in Japan is (pJ m)QJ ($2,000 $500) 3,000 $4.5 million, and the U.S. profit is $4 million. The deadweight loss is greater in Japan, $2.25 million ( 1

2 $1,500 3,000), than in the United States, $2 million ( 1

2 $2,000 2,000).



33. By differentiating, we find that the American marginal revenue function is MRA 100 2QA, and the Japanese one is MRJ 80 4QJ. To determine how many units to sell in the United States, the monopoly sets its American marginal revenue equal to its marginal cost, MRA 100 2QA 20, and solves for the optimal quantity, QA 40 units. Similarly, because MRJ 80 4QJ 20, the optimal quantity is QJ 15 units in Japan. Substituting QA 40 into the American demand function, we find that pA 100 40 $60. Similarly, substituting QJ 15 units into the Japanese demand function, we learn that pJ 80 (2 15) $50. Thus the price-discriminating monopoly charges 20% more in the United States than in Japan. We can also show this result using elasticities. From Equation 2.22, we know that the elasticity of demand is A pA/QA in the United States and

1/2 / J J Jp Q in Japan. In the equilibrium, A 60/40 3/2 and J 50/(2 15) 5/3. As Equation 12.3 shows, the ratio of the prices depends on the relative elasticities of demand: pA/pJ 60/50 (1 1/J)/(1 1/A) (1 3/5)/(1 2/3) 6/5.

34. With multimarket price discrimination, a monopoly will equate the marginal revenue for each group to its common marginal cost, MC m, such that the marginal revenues for the two countries are equal:

MRC m MRJ.

In this example, with multimarket price discrimination, a monopoly will set price in Canada such that

C pC/(m pC),

where C is the price elasticity of demand in Canada, pC is the price in Canada, and m is the marginal cost of production. Substituting $21.40 for pC and $1.00 for m,

C = 21.40/(1.00-21.40)

C 1.049.

Similarly, with multimarket price discrimination, a monopoly will set price in Japan such that

J pJ/(m pJ),

where J is the price elasticity of demand in Japan, pJ is the price in Japan, and m is the marginal cost of production. Substituting $32.00 for pJ and $1.00 for m,

J 32/(1.00 32)

J 1.032.

35. From the problem, we know that the profit-maximizing Chinese price is p 3 and that the quantity is Q 0.1 (million). The marginal cost is m 1. Using Equation 11.11, (pC m)/pC (3 1)/3 1/C, so C 3/2. If the Chinese inverse demand curve is p a bQ, then the corresponding marginal revenue curve is MR a 2bQ. Warner maximizes its profit where MR a 2bQ m 1, so its optimal Q (a 1)/(2b). Substituting this expression into the inverse demand curve, we find that its optimal p (a 1)/2 3, or a 5. Substituting that result into the output equation, we have Q (5 1)/(2b) 0.1 (million). Thus b 20, the inverse demand function is p 5 20Q, and the marginal revenue function is MR 5 40Q. Using this information, you can draw a figure similar to Figure 12.4.

36. Using Eq. 12.2,

pUS (1 1/2) 10 pJ (1 1/5)

pUS $20, pJ $12.50.



37. Set marginal revenue in each market equal to marginal cost to determine the quantities. Plug the quantities into the demand functions to determine prices.

MR1 100 – 2Q1 30 MC

MR2 120 – 4Q2 30 MC

Q1 35; p1 65

Q2 22.5; p2 75

38. Marginal revenue depends on price and elasticity of demand:

11

1

22

2

11

11

MR p

MR p

From Solved Problem 12.3 we know that 1 2 60 p p . Since multimarket price discrimination leads

to prices that equate marginal revenues, that is, 1 2MR MR , we may write

1 21 2

1 2

1 11 1

1 160 1 60 1

p p

1 2 .

39. Suppose a two-part tariff includes a fixed entry fee F, plus a per-unit cost m. In this case, the average price per unit is F/q m, which exceeds the marginal price m. Thus consumers who purchase more units pay a lower average or per-unit price.

40. See figure in Solved Problem 12.4. Without advertising, the optimal number of subscriptions is determined by the intercept of 1MR and MC . This gives optimal output and price levels 1Q and 1.p Presence of advertising has the same effect on the demand curve as a subsidy, that is, demand shifts outward, from 1D to 2.D The new optimal point is set by the intercept of 2MR and .MC With advertising, the new equilibrium is a pair 2Q and 2.p Increasing a increases subscriptions.

41. Giving a lump-sum subsidy to Canadian publishers lowers their marginal cost regardless of how many subscriptions they sell. If MC drops, the publisher can charge a lower price for a subscription, which will increase subscription sales.

42. Assume one unit of advertising costs $1. Then monopoly solves

,

max ( , ) ( ) .p A

R p A C p A

First order conditions are

0

1 0.

R dC

p p dp

R

A A

A pair *p and *A that solves the first-order conditions maximizes the monopoly’s profit.



43. Assume one unit of advertising costs $1. The profit function is

p (100 – Q A1/2 )Q – 10Q – A.

The resulting first-order conditions are

p/A (Q/2)A–1/2 – 1 0

p/Q 100 – 2Q A1/2 – 10 0

A* 900

Q* 60

p* 70.

44. Given the demand and cost information, the profit function is

p (a – bQ cAa )Q – mQ – A.

The resulting first-order conditions are

p/A acQAa–1 – 1 0

p/Q a – 2bQ cAa – m 0.

Solving for A* and Q* yields

A* (acQ*)–(1/a–1)

with Q* is solution to (a – m) – 2bQ c(acQ*)–(a/a–1) 0.

45. If pharmaceutical firms are rational, then they advertise because advertising increases their profit. Profit is revenue less total cost. It is reasonable then to infer that as sales increase due to advertising, production and distribution costs of pharmaceutical companies increase more slowly than revenue.


The optimal lump-sum fee 1 2seniors 2 (4 )L CS p , where p is the optimal price.

The profit function is:

21 2400 400 800 400 (5 ) 400 (4 ) 400(4 ) .pq pq L p p p p p

The F.O.C. is:

2

2000 800 1600 800 3200 800 400 800 0

0.5

1(4 ) 6.125,

2

dp p p p

dp

p

L p

i.e. the optimal lump-sum fee is $6.125L and the optimal price is $0.5p .




a. Without price discrimination, the profit function is:

[(10000 100 ) (9000 100 )] ( 5).p p p

The F.O.C. is:

19000 200 200( 5) 0 50.d

p p pdp

The profit-maximizing quantity in the gate market is 10000 100 50 5000,GQ and the

profit-maximizing quantity in the municipal offices is 9000 100 50 4000.MQ

The maximum possible profit is

( ) [(10000 100 50) (9000 100 50)] (50 5) 405000.p

b. With price discrimination, the profit function in the gate market is:

(10000 100 ) ( 5).G G Gp p

The profit function in the municipal offices is:

(9000 100 ) ( 5).M G Gp p

The F.O.C. for the gate market is:

10000 100 100( 5) 0

52.5

10000 100 4750.

GG G

G

G

G G

dp p

dp

p

Q p

The F.O.C. for the municipal offices market is:

9000 100 100( 5) 0

47.5

9000 100 4250.

MM M

M

M

M M

dp p

dp

p

Q p

The maximum possible profit in both markets is:

( 5) ( 5) 406250.G M G G M MQ p Q p

Chapter 13

1. In each panel, the dominant strategy for each firm is to advertise. This implies that pairs advertise-advertise are Nash equilibria in both panels of Table 14.4.

2. The payoff matrix in this prisoners’ dilemma game is

Duncan



Squeal Stay Silent

–2 –5 Squeal

–2 0

0 –1 Larry

Stay Silent –5 –1

If Duncan stays silent, Larry gets 0 if he squeals and 1 (a year in jail) if he stays silent. If Duncan confesses, Larry gets 2 if he squeals and 5 if he does not. Thus Larry is better off squealing in either case, so squealing is his dominant strategy. By the same reasoning, squealing is also Duncan’s dominant strategy. As a result, the Nash equilibrium is for both to confess.

3. A Nash equilibrium is a set of strategies such that, holding the strategies of all other firms constant, no firm can obtain a higher profit by choosing a different strategy. The Nash equilibria are for Firm 1 to pick low and Firm 2 to pick medium, and for Firm 1 to pick medium and Firm 2 to pick low. At those outcomes, neither firm can increase profit by changing their behavior given what the other firm selects.

Conversely, it is not a Nash equilibrium for Firm 1 to pick low and Firm 2 to pick low. Firm 1 could increase profit from $1 to $21 by instead picking medium. Firm 2 could increase profit from $11 to $16 by instead picking medium.

It is not a Nash equilibrium for Firm 1 to pick low and Firm 2 to pick high. Firm 1 could increase profit from $24 to $26 by instead picking medium. Firm 2 could increase profit from $5 to $16 by instead picking medium.

It is not a Nash equilibrium for Firm 1 to pick medium and Firm 2 to pick medium. Firm 1 could increase profit from $2 to $25 by instead picking low. Firm 2 could increase profit from $3 to $19 by instead picking low.

It is not a Nash equilibrium for Firm 1 to pick medium and Firm 2 to pick high. Firm 2 could increase profit from $14 to $19 by instead picking low.

It is not a Nash equilibrium for Firm 1 to pick high and Firm 2 to pick low. Firm 1 could increase profit from $12 to $21 by instead picking medium. Firm 2 could increase profit from $6 to $23 by instead picking medium.

It is not a Nash equilibrium for Firm 1 to pick high and Firm 2 to pick medium. Firm 1 could increase profit from $4 to $25 by instead picking low.

It is not a Nash equilibrium for Firm 1 to pick high and Firm 2 to pick high. Firm 1 could increase profit from $11 to $26 by instead picking medium. Firm 2 could increase profit from $17 to $23 by instead picking medium.



4. Assume you’re Lori. If Max works, your best strategy is to give no bonus (your payoff is 3), rather than give a bonus (your payoff is 1). If Max loafs, again, your best strategy is to give no bonus (your payoff is 0), rather than give a bonus (your payoff is 1). Hence “No Bonus” is Lori’s dominant strategy. Now assume you’re Max. If Lori offers you a bonus, your best strategy is to loaf because the payoff is 3, rather than 2 when you work. If Lori gives no bonus, again, your best strategy is to loaf (payoff of 0 vs. payoff of 1 if you work). This means that “Loaf” is Max’s dominant strategy. Combining two dominant strategies together suggests that the pair No Bonus – Loaf is the Nash equilibrium outcome of this static game.

5. a. There are two Nash equilibria (the off diagonals). If either firm produces 20 while the other produces 10, neither player has an incentive to change strategies given the strategy of the other player.

b. If Firm 1 can choose first, it will commit to selling 20 units, and Firm 2 sells 10 units. If Firm 1 were to choose 10 units, Firm 2 would choose to produce 20 units, reducing Firm 1’s payoff by $10.

c. If Firm 2 can choose first, it will sell 10 units, and Firm 1 will sell 20. If Firm 2 were to produce 20 units, Firm 1 would produce only 10, reducing Firm 2’s payoff by $5.

6. a. If both must move simultaneously, neither has a dominant strategy because neither can credibly commit to producing the television.

b. The two Nash equilibria are on the off diagonals where one firm enters and the other does not. c. With the subsidy, Zenith can credibly commit to entering the market, because the worst it can do

is to gain 10 if Panasonic also enters. In this case, Zenith will enter and Panasonic will not. d. The equilibrium with the head start is the same as that with the subsidy. Once Zenith commits to

producing the new product, there is no benefit to Panasonic if Panasonic produces it also.

7. There are no pure-strategy Nash equilibria in this game. In each cell, one of the players always would prefer to switch, given the move of the other.

8. The incumbent must compare the two-period profits under two scenarios. In the first scenario, the incumbent overproduces in the first period in order to reduce marginal cost in the second period, knowing that it will be a monopolist in the second period. In the second scenario, the incumbent produces the profit-maximizing output level in the first period, resulting in duopoly profits (with higher marginal cost) in the second period. In the first game tree below, the monopolist overproduces in the first period, resulting in total profits of $1000, which exceeds the profits with profit-maximizing production in the first period. In the second game tree, profits are greater if the incumbent does not overproduce in the first period.



Figure 14.1

9. Given the information, an optimal strategy for the rulers may be: (i) spend little resource in catapult research and development; (ii) buy latest technology from other countries; (iii) publicly announce the deployment of catapult. Suppose catapult research and development required a substantial fixed cost, then this strategy is particularly suitable for small countries. Since new technology was not well protected, public announcement of research and development would encourage free riding of other countries. On the other hand, credible announcement of catapult deployment would help to discourage possible attacks.

10. Audio-PowerPoint answer by James Dearden is also available (14A Thugs).

a. The safe owner (S) moves first and decides whether to “open” the safe or “don’t open” the safe. Then the thug decides to “kill” or “don’t kill.”

Figure 14.2

b. The subgame perfect Nash equilibrium is “don’t open” for the safe owner and “don’t kill” for the thug irrespective of whether the owner opens the safe or not. So the safer owner will not open the safe and the thug will not kill.

c. The thug’s threat is not credible. Therefore the safe owner should not believe it. d. He will not open the safe.

11. Audio-PowerPoint answer by James Dearden is also available (14B Film Release).

a. The pure strategy Nash equilibrium in this game is for both Warner Bros. and the T-3 producer to release their movies on July 4. Given that the T-3 producer releases its movie on July 4, the best choice for Warner Bros. is to release its movie on July 4. On the other hand, given that Warner



Bros. releases its movie on July 4, the best choice for the T-3 producer is to release its movie on July 4.

b. The release on July 18 by the T-3 producer and on July 4 by Warner Bros. maximizes joint profit. Note that 90 30 120 is the greatest sum of profits.

c. The maximum Warner Bros. is willing to pay is the difference between its profit when both movies are released on July 4th, and its payoff when it has bought the release of T-3 (released on July 18) and Matrix (released on July 4), i.e., 90 50 40. The profit the T-3 producer earns if it does not sell its right of release is 50. Therefore the minimum price the T-3 producer accepts for the sale of its right of release is 50. Since the maximum that Warner Bros. is willing to pay (40) is less than the minimum that the T-3 producer is willing to accept (50), there is no mutually beneficial price at which trade can take place.

d. Warner Bros. will release its movie on July 4th and T-3 on July 18.

12. The advertising might be strategic to increase Microsoft’s own market share by attracting customers from competitors. On the other hand, it does not necessarily reflect its “fear” of competitors. The advertising might target new customers. Overall, the optimal advertising is determined jointly by the marginal cost and marginal benefit.

13. Figure 14.3(a) offers an extensive form representation of the sequential game when Mimi moves first. Using backward induction we conclude that Jeff will not choose actions that are double-crossed because alternatives offer Jeff better payoffs (4 vs. 2 and 1 vs. 0). Given Jeff’s preferred actions, Mimi receives the same payoff of 1 regardless of what she does. Since supporting Jeff will not improve Mimi’s payoff, she may choose not to support him.

If Jeff moves first, the game looks as in Figure 14.3(b). When Mimi makes a decision, she chooses actions that give her the highest payoffs; we crossed actions that Mimi will not choose. Then Jeff realizes that if he looks for a job, his payoff will be 2, and if he does not, his payoff will be 0. Maximizing his payoff, Jeff chooses to look for a job. This means that the subgame perfect Nash equilibrium is the strategy look-support with the outcome of (4, 2).

Figure 14.3



14. Suppose the payoff is 1 if you win, 0 if you tie, and 1 if you lose. The payoff matrix will be the following:

Sotheby’s

Rock Paper Scissors

Christie’s Rock 0, 0 1, 1 1, 1 Paper 1, 1 0, 0 1, 1 Scissors 1, 1 1, 1 0, 0

Assuming the 11-year-old girls provide the correct insight and the rival would be very likely to take their advice (of selecting scissors), then a pure strategy of rock should be recommended (assuming the rival didn’t know that we knew their consultation with the girls).

15. It is not clear what the answer to the second part of this question is because we don’t know the payoffs.

16. The payoff matrix is:

B

Swerve Don’t

Swerve Swerve 1, 1 0, 2 A

Don’t swerve 2, 0 10, 10

The two Nash equilibria are the cells with outcomes (2, 0) and (0, 2).

17. A Nash equilibrium is a set of strategies such that, holding the strategies of all other firms constant, no firm can obtain a higher profit by choosing a different strategy. If driver 1 swerves and driver 2 does not swerve, then neither driver can increase his payoff. For example, if driver 1 instead does not swerve, then his payoff decreases from 0 to 2. If driver 2 instead swerves, then his payoff decreases from 2 to 1. Therefore, neither driver has an incentive to change his behavior. The game is symmetric, so if it is a Nash equilibrium for driver 1 to swerve and driver 2 to not swerve, then it is also a Nash equilibrium for driver 1 to not swerve and driver 2 to swerve.

18. See Figure 14.4. Taliban decides if kidnapping an Italian journalist will pay off. If Taliban kidnaps the journalist, Italy should pay because if it pays, Italy gets the journalist back (payoff of 1), but if Italy does not pay, the journalist is not going to be freed (payoff of 1 for Italy). If Italy pays, Taliban’s payoff is 5 (5 Taliban prisoners are released) and if Italy does not pay, the payoff for Taliban is 5 (5 Taliban prisoners are not released). If Taliban does not kidnap the journalist, the payoff to Italy is always 0, whether or not Italy chooses to release Taliban prisoners; hence Italy is not motivated to pay. For Taliban, the payoffs are 5 if Italy pays (i.e., releases Taliban prisoners) and 5 if Italy does not pay (i.e., does not release prisoners). Backward induction suggests that the subgame perfect Nash equilibrium is the strategy kidnap-pay with outcome (5, 1). This is exactly what happened. The other governments were upset because the situation demonstrated to Taliban that a government may act as a rational agent. Hence Taliban may decide that it is beneficial to “play” this “game” in the future again.



Figure 14.4

19. We start by checking for dominant strategies. (See Figure 14.5) Given the payoff matrix, Toyota always does at least as well by entering the market. If GM enters, Toyota earns 10 by entering and 0 by staying out of the market. If GM does not enter, Toyota earns 250 if it enters and 0 otherwise. Thus entering is Toyota’s dominant strategy. GM does not have a dominant strategy. It wants to enter if Toyota does not enter (earning 200 rather than 0), and it wants to stay out if Toyota enters (earning 0 rather than 40). Because GM knows that Toyota will enter (entering is Toyota’s dominant strategy), GM stays out of the market. Toyota’s entering and GM’s not entering is a Nash equilibrium. Given the other firm’s strategy, neither firm wants to change its strategy. Next we examine how the subsidy affects the payoff matrix and dominant strategies. The subsidy does not affect Toyota’s payoff, so Toyota still has a dominant strategy: It enters the market. With the subsidy, GM’s payoff if it enters increases by 50: GM earns 10 if both enter and 250 if it enters and Toyota does not. With the subsidy, entering is a dominant strategy for GM. Thus both firms’ entering is a Nash equilibrium.

20. If GM gets no subsidy but can move first, it faces a situation as described in Figure 14.5. Backward induction suggests that the Nash equilibrium is a pair of strategies “don’t enter” for GM and “enter” for Toyota, with the outcome of 250 for Toyota and 0 for GM.

Figure 14.5

21. If firms only care about current profits, then the firms will not use any intertemporal strategies. They will solve a simple single-period prisoners’ dilemma game in each period.

22. The game tree in Figure 14.6 illustrates why the incumbent may install the robotic arms to discourage entry even though its total cost rises. If the incumbent fears that a rival is poised to enter, it invests to discourage entry. The incumbent can invest in equipment that lowers its marginal cost. With the lowered marginal cost, it is credible that the incumbent will produce larger quantities of output, which discourages entry. The incumbent’s monopoly (no-entry) profit drops from $900 to $500 if it makes the investment because the investment raises its total cost. If the incumbent doesn’t buy the robotic arms, the rival enters because it makes $300 by entering and nothing if it stays out of the market. With entry, the



incumbent’s profit is $400. With the investment, the rival loses $36 if it enters, so it stays out of the market, losing nothing. Because of the investment, the incumbent earns $500. Nonetheless, earning $500 is better than earning only $400, so the incumbent invests.

Figure 14.6

23. The incumbent firm has a first-mover advantage, as the game tree on the facing page illustrates. Moving first allows the incumbent or leader firm to commit to producing a relatively large quantity. If the incumbent does not make a commitment before its rival enters, entry occurs and the incumbent earns a relatively low profit. By committing to produce such a large output level that the potential entrant decides not to enter because it cannot make a positive profit, the incumbent’s commitment discourages entry. Moving backward in time (moving to the left in the diagram), we examine the incumbent’s choice. If the incumbent commits to the small quantity, its rival enters and the incumbent earns $450. If the incumbent commits to the larger quantity, its rival does not enter and the incumbent earns $800. Clearly, the incumbent should commit to the larger quantity because it earns a larger profit and the potential entrant chooses to stay out of the market. Their chosen paths are identified by the darker blue in the figure.

Figure 14.7

24. It is worth more to the monopoly to keep the potential entrant out than it is worth to the potential entrant to enter, as the figure shows. Before the pollution-control device requirement, the entrant would pay up to $3 to enter, whereas the incumbent would pay up to i d $7 to exclude the potential entrant. The incumbent’s profit is $6 if entry does not occur, and it loses $1 if entry occurs. Because the new firm would lose $1 if it enters, it does not enter. Thus the incumbent has an incentive to raise costs by $4 to both firms. The incumbent’s profit is $6 if it raises costs rather than $3 if it does not.



Figure 14.8

25. The payoff matrix is shown below. Given these payoffs, mixed strategies are required. There is no Nash equilibrium because in any given cell the outcome (winner) changes if one player changes strategies. Thus in any given cell the loser would always prefer to switch given the opponent’s strategy.

26. Let the probability that a firm sets a low price be 1 for Firm 1 and 2 for Firm 2. If the firms choose their prices independently, then 12 is the probability that both set a low price, (1 1)(1 2) is the probability that both set a high price, 1(1 2) is the probability that Firm 1 prices low and Firm 2 prices high, and (1 1)2 is the probability that Firm 1 prices high and Firm 2 prices low. Firm 2’s expected payoff is E(2) 212 (0)1(1 2) (1 1)2 6(1 1)(1 2) (6 61) (5 71)2. Similarly, Firm 1’s expected payoff is E(1) (0)12 71(1 2) 2(1 1)2 6(1 1)(1 2) (6 42) (1 32)1. Each firm forms a belief about its rival’s behavior. For example, suppose that Firm 1 believes that Firm 2 will choose a low price with a probability 2

ˆ . If 2̂ is less than 13

(Firm 2 is relatively unlikely to choose a low price), it pays for Firm 1 to choose the low price because the second term in E (1), 2 1

ˆ(1 ) , is positive, so as 1 increases, E(1) increases. Because the highest possible 1 is 1, Firm 1 chooses the low price with certainty. Similarly, if Firm 1 believes

2̂ is greater than 13 , it sets a high price with certainty (1 0).

If Firm 2 believes that Firm 1 thinks 2̂ is slightly below 13 , Firm 2 believes that Firm 1 will

choose a low price with certainty, and hence Firm 2 will also choose a low price. That outcome, 2 1, however, is not consistent with Firm 1’s expectation that 2̂ is a fraction. Indeed, it is only rational for Firm 2 to believe that Firm 1 believes Firm 2 will use a mixed strategy if Firm 1’s belief about Firm 2 makes Firm 1 unpredictable. That is, Firm 1 uses a mixed strategy only if it is indifferent between setting a high or a low price. It is indifferent only if it believes 2̂ is exactly 1

3 . By similar reasoning, Firm 2 will use a mixed strategy only if its belief is that Firm 1 chooses a low price with probability 5

1 7ˆ . Thus the only possible Nash equilibrium is 5

1 7 and 12 3 .

27. In the battle of sexes game the payoff matrix is:

Wife Mountain Ocean

Mountain 2, 1 0, 0 Husband

Ocean 0, 0 1, 2



This game has two Nash equilibria (mountain, mountain) and (ocean, ocean), which means that the game theory cannot offer a unique solution and cannot predict what the couple will end up doing. If, however, this is a repeated game (the couple makes the same decision before every vacation), the husband and wife may agree that they alternate vacation spots. Then once the initial vacation is chosen, all the following vacation locations can be predicted.

28. Audio-PowerPoint answer by James Dearden is also available (13B Chicken Pie).

a.

Emil’s Diner

0 1 2 3 4 5 6

0 (120, 120) (240, 0) (240, 0) (240, 0) (240, 0) (240, 0) (240, 0)

1 (0, 240) (50, 50) (100, 0) (100, 0) (100, 0) (100, 0) (100, 0)

2 (0, 240) (0, 100) (0, 0) (0, 0) (0, 0) (0, 0) (0, 0)

3 (0, 240) (0, 100) (0, 0) (30, 30) (60, 0) (60, 0) (60, 0)

4 (0, 240) (0, 100) (0, 0) (0, 60) (40, 40) (80, 0) (80, 0)

5 (0, 240) (0, 100) (0, 0) (0, 60) (0, 80) (30, 30) (60, 0)

Bobby’s Diner

6 (0, 240) (0, 100) (0, 0) (0, 60) (0, 80) (0, 60) (0, 0)

b. The Nash equilibriua are as follows:

Bobby Emil

2 2 3 3

c. The inverse demand curve is

P 6 0.05Q,

thus

MR 6 0.1Q.

The profit is maximized where

MR MC 6 0.1Q 2 Q 40, P 4, profit 160 80 80.

d. At the (3, 3) Nash equilibrium, each diner’s profit is 30. However, if they collude, they can set the price at 4, the monopoly price, get the monopoly profit and then divide it between themselves and in this way increase their profit to 40 for each.



29. Audio-PowerPoint answer by James Dearden is also available (13C Warranties).

a. i Ri Ci 27000wi /(wA wB) 2000wi. b. (numbers in $ thousands).

Acura

1 2 3 4 5

1 (11.5, 11.5) (7, 14) (4.75, 14.25) (3.4, 13.6) (2.5, 12.2)2 (14, 7) (9.5, 9.5) (6.8, 10.2) (5, 10) (3.7, 9.3)

3 (14.25, 4.75) (10.2, 6.8) (7.5, 7.5) (5.6, 7.4) (4.2, 6.9)4 (13.6, 3.4) (10, 5) (7.4, 5.6) (5.5, 5.5) (4, 5)

Volvo

5 (12.5, 2.5) (9.3, 3.7) (6.9, 4.2) (5, 4) (3.5, 3.5)

c. The Nash equilibrium is for both to offer three years warranty. Offering three years warranty is the dominant strategy for each company.

d. They both offer the same warranty, because this is best response to each other’s strategy. e. If they collude, they will still provide three-year warranties. Following other strategies makes one

better off and the other worse off. f.

Acura

1 2 3 4 5

1 (12.5, 11.5) (8, 14) (5.75, 14.25) (4.4, 13.6) (3.5, 12.5)

2 (16, 7) (11.5, 9.5) (8.8, 10.2) (7, 10) (5.7, 9.3)

3 (17.25, 4.75) (13.2, 6.8) (10.5, 7.5) (8.6, 7.4) (7.2, 6.9)

4 (17.6, 3.4) (14, 5) (11.4, 5.6) (9.5, 5.5) (8, 5)

Volvo

5 (17.5, 2.5) (14.3, 3.7) (11.9, 4.2) (10, 4) (8.5, 3.5)

The Nash equilibrium is now for Volvo to offer five years warranty and for Acura to offer three years. Now the Volvo profit is larger compared with when its costs were higher.

30. Solution is also provided in Jim Dearden’s audio presentation.

a. The expected profit function for university i is:

( , , or , ).ii i i

i j

mv m i A j B i B j A

m m

The F.O.C. for university A is:

2

1 0.( )

A A A

A A B

v m

m m m (1)

The F.O.C. for university B is:

2

1 0.( )

B B B

B A B

v m

m m m (2)

The solution to equations (1) and (2) is:



2

2

2

2

( )

,( )

A BA

A B

B AB

A B

v vm

v v

v vm

v v

i.e., in the Nash equilibrium, university A wins

2

2( )A B

A B

v vA v v

m , and university B wins

2

2.

( )B A

BA B

v vm

v v

If vA vB v in the Nash equilibrium, each school wins 4 .vA Bm m Since

142

4 4

v

v v, so

each school wins one-half of its game.

b. Since 4A B

vm m , then

1

04

A Bm m

v v, therefore if vA vB increases, each school spends

more on its teams. Moreover, since

142

4 4

v

v v still holds after the increase, each school

continues to win one-half of its games. c. This is not a zero-sum game.

31. a. It seems in this model the demand does not depend on the price that companies charge but on how much they spend on advertisement.

The market share of Firm A is

qA/(qA qB) [a b(AA AB)0.5]/{2[a b(AA AB)0.5]} 0.5.

Therefore, irrespective of how much each firm spends on advertisement, each firm’s market share will not change.

b. Each firm maximizes its profit, given the other firm’s level of expenditures on advertisement and the price it charges:

di /dAi 0.5bpi(Ai Aj)–0.5 1 0,

i A, B.

Solving the two equations we get that

PA PB P

and

AA AB 0.125b2P2.

As b increases, the expenditure on advertisement will also increase.



32. The table below illustrates this game. “S” denotes taking steroids and “NS” denotes not taking steroids. If they both take steroids they tie and the payoff for each is 10 6 4. If one takes steroids and one does not, the person who has taken steroids wins and gets 20 6 14, and the other person loses and gets 0. If they both don’t take steroids, they tie again and each gets 10.

200-Meter Star

S NS

S (4, 4) (14, 0)100-Meter Star

NS (0, 14) (10, 10)

a. The Nash equilibrium is for both to take steroids, even though they will both be better off if they don’t take steroids. This is a prisoner’s dilemma.

b. Suppose the utility of the 100-meter star receives from taking the steroids is 12, while it is 6 for the 200 meter star. The payoff table for this game is illustrated below.

200-Meter Star

S NS

S (2, 4) (8, 0)100-Meter Star

NS (0, 14) (10, 10)

Now the Nash equilibrium is “don’t take steroids” for the 100-meter star and “take steroids” for the 200-meter star. Both players have dominant strategies, but cooperation by not taking steroids does not improve both. This is not a prisoner’s dilemma game.

33. This game has two possible outcomes, one in which Westley drinks from the poisoned glass and Vizzini drinks from the glass that is not poisoned, and in the other outcome the opposite happens. Each of the outcomes can happen with the probability of 50%.

34. Xavier makes the first move and Ying, observing Xavier’s decision, makes his choice. The setup is like the Stackelberg model of noncooperative behavior. Xavier knows that Ying maximizes his utility and considers his optimal choice when making his own decision. Ying chooses how much he needs to work to maximize his utility:

dUY /dhY 0 9(hX hY)0.5 1 0

hX hY 81

hY 81 hX.

Knowing Ying’s choice, Xavier maximizes his utility:

UX 18(hX hY)0.5 hX UX 18(hX 81 hX)0.5 hX

UX 162 hX.

The optimum for Xavier is to not work at all; i.e., * 0,Xh and thus the optimum for Ying will be to

work * 81.Yh Therefore Ying’s threat that he will not work is not credible.

35. In the auction, Anna’s, Bill’s, and Cameron’s optimal bids will be $20,000, $18,500, and $16,800, respectively. Anna will win the auction and the price she pays will be $18,500, and her surplus will be $20,000 $18,500 $1500.



36. a.

Colts

R P

DR (1, 1) (10, 10)Patriots

DP (6, 6) (2, 2)

Where “R” denotes run, “DR” denotes defense against run, “P” denotes pass, and “DP” denotes defense against pass. There is no pure Nash equilibrium. We can find the mixed-strategy Nash equilibrium. Suppose Colts follow “R” by probability and “P” by probability 1 . Then the expected payoff of following “DR” and “DP” by Patriots is:

E(DR) 10(1 ) 11 10

and

E(DP) 6 2(1 ) 8 2.

At the equilibrium we have:

E(DR) E(DP) 11 10 8 2

12/19.

Suppose Patriots follow “DR” by probability and “DP” by probability 1 , then the expected pay off of following “R” and “P” by Colts is:

E(R) 6 (1 )

7 6

and

E(P) 10 2(1 )

12 2.


E(R) E(P) 7 6 12 2

8/19.

Therefore the mixed strategy Nash equilibrium is {[12/19, 7/19], [8/19, 11/19]}. b. The payoff matrix is now:

Colts

R P

DR (1, 1) (10, 10)Patriots

DP (6, 6) (, )

There are no pure Nash equilibria. The maixed-strategy Nash equilibrium is as follows:

Suppose Colts follow “R” by probability and “P” by probability 1 . Then the expected payoff of following “DR” and “DP” by Patriots is:



E(DR) 10(1 ) 11 10

and

E(DP) 6 2(1 ) 8 2.


E(DR) E(DP) 11 10 8 2

12/19.

Suppose Patriots follow “DR” by probability and “DP” by probability 1 , then the expected pay off of following “R” and “P” by Colts is:

E(R) 6 (1 )

7 6

and

E(P) 10 2(1 )

12 2.


E(R) E(P) 7 6 12 2

8/19.

Therefore the mixed strategy Nash equilibrium is {[12/19, 7/17], [8/19, 11/19]}.

Chapter 14

1. See Figure 14.1 in the text. Each cartel member has incentive to cheat, reasoning that one country increasing the output will not change the price much. At the same time, since the marginal revenue is above marginal cost, by producing more oil than the agreed-upon amount, it can make additional profit.

2. The monopoly will make more profit than the duopoly will, so the monopoly is willing to pay the college more rent. Although granting monopoly rights may be attractive to the college in terms of higher rent, students will suffer (lose consumer surplus) because of the higher textbook prices.

3. Assume that markets with multiple bail-bond firms (n firms) are oligopolistic, produce a homogenous good, face a linear inverse market demand of p a bQ, and have constant marginal costs of m per unit. Also assume that the market outcome is a Cournot-Nash equilibrium.

Firm 1’s profit function is

1 1 1 2 1( ( ... )) . nq a b q q q mq

Firm 1 maximizes profit by first taking the derivative of their profit function with respect their output (q1):

11 2

1

(2 ... ) .

n

da b q q q m

dq



If all n firms are identical, then in equilibrium q1 q2 … qn q. Therefore, setting the first order condition equal to zero and solving for q,

.( 1)

a m

qn b

Total market supply is

( ).

( 1)

n a m

Qn b

Substituting this back into the inverse demand function and solve for p to find the market price the price consumers pay before the tax is

.1

a mnp

n

If markets with one or two firms charge prices (fees) that are essentially equal to the maximum, then the state maximum must be no more than 2

3( ),a m which is the optimal price with n equal to 2, but if markets with more than two firms charge prices (fees) that are less than the maximum, then the state maximum must be greater than 3

4( ),a m which is the optimal price with n equal to 3.

4. With only one firm, the deadweight loss is equal to the deadweight loss of a monopoly; that is, (243 147) (192 96)/2 (243 147) (243 147)/2 4608 (see Figure 14.2(a) in the text). With three firms, the deadweight loss is (195 147) (195 147)/2 1152, decreasing by 75%.

5. See Figure 14.2. The graph shows response curves for the two airlines. The airline with the lower marginal cost produces more: Southwest transports 1Q passengers and US Airways transports 2Q passengers. This result is shown algebraically in Solved Problem 14.1.

Figure 14.2

6. The increase in price after the exit of one firm is consistent with a Cournot equilibrium, where the equilibrium price is decreasing in the number of firms. We have to assume that the painkillers are homogeneous for this model to work.

7. The best response function of Firm 2 is

2 1

2 .2

a m bqq

b



Figure 14.3 shows best-response functions for Firm 1 and Firm 2. If marginal costs of the two firms are equal, then Nash-Cournot outputs of the firms are also equal (equilibrium 1e ). Increasing marginal cost of the second firm shifts its best-response function inward, which leads to lower output of Firm 2 (equilibrium 2e ).

a m

b

x1e

2q

1q2

a m

b

2

2

a m

b

2a m

b

2e

2m m

2m m x

Figure 14.3

8. Governmental subsidy that reduces fixed cost of each firm in an industry would shift each firm’s best-response curve rightward, resulting in higher output and lower price in a Cournot monopolistic equilibrium.

9. If there are no fixed costs, MC AC. The two conditions that must hold for a monopolistically competitive equilibrium MC MR, and p AC, cannot hold simultaneously. When MC MR, firms earn positive profits, which violates p AC. The solution is indeterminate.

10. Because firms must bear part of the burden of the tax, profits are reduced. With each firm earning smaller profits, beginning from a point where all firm profits are zero, some firms must exit the industry to restore equilibrium.

11. No. As with the monopolist, there is no unique relationship between price and output. A change in demand may produce a change in price but no change in output, a change in output but not price, or a change in both.

12. In Figure 14.11 in the text, a specific tax increases marginal cost by the amount of the tax. Coke’s best-response function shifts upward, and Pepsi’s best-response function shifts to the right. The result is that both firms charge higher prices.

13. If the firms are price setters, then we can assume that they are currently in a Bertrand equilibrium. We can evaluate the increase in cost using a figure similar to Figure 14.11 in the text. The increase in marginal cost of between $70 and $270 per vehicle will shift the best-response functions outward on their respective axes. The new equilibrium will result in higher prices, but those price increases will not be on the order of $7000, and will not likely be much different than the increase in marginal cost.



14. If output is homogeneous, the market inverse demand function is p(Q), where Q, the total market output, is the sum of the output of each of the n firms Q = q1 q2 … qn. Each of the n identical firms has the same cost function, C(qi). Firm 1’s profit function is

1 1 1 2 1( ... ) ( ). nq p q q q C q


1 11

1 1 1

( )( )( ) .

d dC qdp Q dQp Q q

dq dQ dq dq

If all n firms are identical, then in equilibrium q1 q2 … qn. Therefore, setting the first order condition equal to zero and solving for q,

1 1

1 1

( )[1 ] .

d dC qq dpp

dq p dQ dq

Multiplying and dividing the last term in parentheses by n, noting that Q nq (given that all firms are identical), and substituting in the market elasticity of demand, , we can rewrite the first-order condition as

1 ( )1

dC qp

n dq

or

1,

1

n

MCp

where ( )

.dC q

MCdq

Therefore, the optimal price depends on marginal costs but not fixed costs. Furthermore, the effect of an increase in marginal cost (MC) is

1

1.

1

n

dp

dMC

If consumers are price sensitive (if consumer demand is not perfectly inelastic), then does not

equal 0 and dpdMC does not equal 1. That is, if does not equal zero, then prices will not increase

proportionally with marginal costs.

15. In the Bertrand equilibrium, the price is equal to the competitive price for homogeneous good when there are at least two firms. Hence increasing the number of firms beyond two will not affect the market price.

16. Given that the duopolies produce identical goods, the equilibrium price is lower if the duopolies set price rather than quantity. If the goods are heterogeneous, we cannot answer this question definitively.



17. By differentiating its product, a firm makes the residual demand curve it faces less elastic everywhere. For example, no consumer will buy from that firm if its rival charges less and the goods are homogeneous. In contrast, some consumers who prefer this firm’s product to that of its rival will still buy from this firm even if its rival charges less. As the chapter shows, a firm sets a higher price the lower the elasticity of demand at the equilibrium.

18. The best-response curve of the praised firm shifts out, while the best-response curve of the other firm does not shift. This is the result of an outward shift of the demand curve for the praised firm but not the other. The prices of both firms increase, but the price of the praised firm increases by more. See Figure 14.4 below.

Figure 14.4

19. Beginning with Equation 14.4 in the text and substituting the new marginal cost levels, the best-response functions and output levels become

qU 119.5 1/2 qA qA 69.5 – 1/2 qU

qU 113 qA 13.

20. Because the profit-maximizing output level is determined by equating residual marginal revenue to marginal cost, the addition of a fixed cost for both firms does not shift the response functions. However, each response function is truncated at the point where output is too low for the firm to cover variable cost (the shutdown point). Thus as long as profits are greater than –FC, the output decisions are unchanged.

21. a. If there is a collusion, Firm 1 should produce all output due to its lower marginal cost. The monopoly price and output levels are determined by

MR 120 2Q MC 20 120 2Q 20 Q* 50 p* 70.



b. To calculate the Cournot equilibrium, derive the response function and solve each by setting it equal to the appropriate marginal cost for each firm. Then, solve the response functions simultaneously to determine output.

p 120 q1 q2

1rMR 120 2q1 q2

2rMR 120 q1 2q2

120 2q1 q2 20

q1 50 – 1/2 q2

120 – q1 2q2 40

q2 40 1/2 q1

q1* 40

q2* 20

Q* 60

p* 60

22. The inverse demand curve is p 1 0.001Q. The first firm’s profit is 1 [1 0.001(q1 q2)]q1 0.28q1. Its first-order condition is d1/dq1 1 0.001(2q1 q2) 0.28 0. If we rearrange the terms, the first firm’s best-response function is 1

1 22360 . q q Similarly, the second firm’s best-response function is 1

2 12360 . q q By substituting one of these best-response functions into the other, we learn that the Nash-Cournot equilibrium occurs at q1 q2 240, so the equilibrium price is 52¢.

23. The response functions and output levels with the subsidy are

qU 120 1/2qA

qA 120 – 1/2qU

qU qA 80.

24. Given that the firm’s after-tax marginal cost is m, the Nash-Cournot equilibrium price is p = (a + n[m + ])/(n + 1), using Equation 14.17. Thus, the consumer incidence of the tax is dp/d = n/(n + 1) < 1 ( 100%).

25. See Solved Problem 14.2. The equilibrium quantities are q1 (a 2m1 m2)/3b (90 30)/6 20 and q2 (90 60)/6 5. As a result, the equilibrium price is p 90 20 5 65.

26. Consider a Cournot equilibrium where each of n firms faces a constant marginal cost of m and the market demand curve is

p a bQ.

Firm 1’s profit function is

1 1 1 2 1( ( ... )) . nq a b q q q mq


11 2

1

(2 ... ) .

n

da b q q q m

dq



If all n firms are identical, then in equilibrium q1 q2 … qn q. Therefore, setting the first order condition equal to zero and solving for q,

.( 1)

a m

qn b

Total market supply is

( ).

( 1)

n a m

Qn b

Substituting this back into the inverse demand function and solve for p to find the market price., the price consumers pay before the tax is

.1

a mnp

n

The effect of a shift in demand on price is

1,

1

dp

da n which is positive.

The effect of an increase in marginal cost on price is

,1

dp n

dm n which is positive.

The effect of an increase in the number of firms on price is

,1

dp m a

dn n which is negative.

In sum, a leftward shift in demand cannot explain a price increase, but market price would increase if demand shifts to the right, marginal costs increase, or the number of firms decreases.

27. When there are multiple followers instead of one, the response function of the followers becomes

qi (a m)/nb q1/n.

The leader maximizes profits, taking the best-response functions as given. The output of the leader is the same as the one follower example.

q1 (a m)/2b

Industry output is Q [(a – m)/2b][(2n 1)/n].

28. Firm 1 wants to maximize its profit:

1 (p1 10)q1 (p1 10)(100 2p1 p2).

Its first-order condition is d1/dp1 100 4p1 p2 20 0, so its best-response function is 1

1 2430 . p p Similarly, Firm 2’s best-response function is 12 1430 . p p Solving, the

Nash-Bertrand equilibrium prices are p1 p2 40. Each firm produces 60 units.



29. If marginal cost is equal to zero, we can use the same methodology described above to solve for the new prices. In this case, the response functions are

p1 25 0.25p2

p2 25 0.25p1

and the new equilibrium prices are p1 33.33, p2 33.33.

30. This problem is solved using the same methodology as in the previous two problems, except that now, the response functions are asymmetric due to the difference in marginal cost. The best-response functions are

p1 40 0.25p2

p2 30 0.25p1

and the new equilibrium prices are p1 50.67, p2 42.67.

31. One approach is to show that a rise in marginal cost or a fall in the number of firms tends to cause the price to rise. Solved Problem 14.4 shows the effect of a decrease in marginal cost due to a subsidy (the opposite effect). The section titled “The Cournot Equilibrium with Two or More Firms” shows that as the number of firms falls, market power increases and the markup of price over marginal cost increases. The two effects reinforce each other. Suppose that the market demand curve has a constant elasticity of . We can rewrite Equation 14.10 as p m/[1 1/(n)] m, where 1/[1 1/(n)] is the markup factor. Suppose that marginal cost increases to (1 )m and that the drop in the number of firms causes the markup factor to rise to (1 ); then the change in price is [(1 )m (1 )] m ( )m. That is, price increases by the fractional increase in the marginal cost, , plus the fractional increase in the markup factor, , plus the interaction of the two, .

32. Audio-PowerPoint answer by James Dearden is also available (14A Computer Chips).

a. Suppose that there is a positive fixed and sunk cost F. At the Bertrand equilibrium

* * 0 A BP P and * * . A Bp p F

b. No. With product differentiation, the firms can raise their price above marginal cost, but still their profit can be “razor thin.”

c. In what follows we assume F 0. We have:

pV ( PV PA)(PV m)

and

pA ( PA PV)(PA m).

Taking the derivative of the profit function with respect to PV and PA and putting it equal to zero, we get:

(PV m) PV PA 0

and

(PA – m) PA + PV 0.

Solving the equations we get:

PV PA (m )/(2 – )



and

V A [( m m)]2/(2 )2.

We see if:

m( )

then

PV PA m and V A 0.

Thus even though the products are differentiated, we see that under certain conditions the profits can be zero.


a. Wawa’s profit function is: ( 2)(680 500 400 )W W W Sp p p .

Sunoco’s profit function is: ( 2)(680 500 400 ).S S S Wp p p

The F.O.C. for Wawa is:

680 500 400 500( 2) 400 1000 1680 0. (1)W

W S W S WW

p p p p pp

The F.O.C. for Sunoco is:

680 500 400 500( 2) 400 1000 1680 0. (2)S

S W S W SS

p p p p pp

Solving equations (1) and (2) simultaneously, we can obtain the Nash equilibrium prices:

2.8

2.8.W

S

p

p

b. With the salty snacks, Wawa’s profit function is:

( 0.25 2)(680 500 400 ).W W W Sp p p

Sunoco’s profit function is still: ( 2)(680 500 400 ).S S S Wp p p

The F.O.C. for Wawa is now:

680 500 400 500( 1.75) 400 1000 1555 0. (3)W

W S W S WW

p p p p pp

The F.O.C. for Sunoco is still equation (2). Solving equations (2) and (3) simultaneously, we can obtain the new Nash equilibrium prices:

2.65

2.74.W

S

p

p




Farmer A’s market profit function is: [10 0.1( )] .Am Am Bm Cm Amq q q q

Farmer A’s home profit function is: (5 0.1 ) .Ah Ah Ahq q

Farmer A’s total profit function is:

[10 0.1( )] (5 0.1 )

[10 0.1( )] [5 0.1(50 )](50 ).A Am Ah Am Bm Cm Am Ah Ah

Am Bm Cm Am Am Am

q q q q q q

q q q q q q

The F.O.C. for farmer A is:

0.1 10 0.1( ) 0.1(50 ) [5 0.1(50 )]

15 0.4 0.1( ) 0. (1)

AAm Am Bm Cm Am Am

Am

Am Bm Cm

q q q q q qq

q q q

Since farmers A, B, and C are symmetric, then: .Am Bm Cmq q q Substitute this relationship into

equation (1): 15 0.4 0.1( ) 0 25.Am Am Am Amq q q q

Therefore the Nash-Cournot equilibrium quantities are: 25.Am Bm Cmq q q

The market price is 10 0.1( ) 2.5,m Am Bm Cmp q q q and the roadside stand prices for farmer A,

B, and C are all: 5 0.1 2.5.Ah Bh Ch Amp p p q

35. Audio-PowerPoint answer by James Dearden is also available (14C Warranties).

a. i Ri Ci 32,000wi /(wA wB) 2000wi. b. The Nash equilibrium is for both to offer four years warranty. c. If they collude, they will provide one-year warranties.


a. The sum of the profits of auction houses Sotheby’s (S) and Christie’s (C) are:

( ) ( ) 2 ( ) ( ) ( ) 2 ( ) .S C S C S Crp D r D r F v D r D r rpD r F vD r

where: D(r) is the market demand ( ) ( ).S CD r D r

b. The F.O.C. for maximizing the sum of profits is:

( ) ( ) ( ) ( )

( ) ( ) ( ) 0.S C D r D r D rpD r rp v pD r rp v

r r r r

In terms of the monopoly’s Lerner Index and price elasticity of market demand:

1 1 1( ) .

r

D rp v Drp v pD

D D rr rp rr r D

where: r is the price elasticity of market demand.



c. If they jointly set the commission rate, the F.O.C. for the profit-maximizing problem is:

1

.r

rp v

rp

If Christie’s cheats on their agreement, the F.O.C. for Christie’s profit-maximizing problem is:

( , ) 1

( , ) ( ) 0 .C

C

r

C C CC C C

C C

D r r r p vpD r r r p v

r r r p

where: Cr

is the price elasticity of demand for Christie’s.

Since the market demand is less price elastic than the demand for Christie’s, then:

1 1

;C

C

r r Cr r

r r

i.e., Christie’s has incentives to cheat on their agreement by lowering its own commission, and gets greater profit. People will then substitute away from Sotheby’s toward Christie’s; hence Sotheby’s will be worse off if it continues to charge r.

37. a. The profits of Highland Park Hospital and Evanston Northwestern Hospital are:

2

( 2000)(50 0.01 0.005 )

70 0.01 0.005 10 100,000.

H H H N

H H H N N

p P P P

P P P P P

2

( 2000)(500 0.01 0.005 )

520 0.01 0.005 10 1000,000.

N N N H

N N H N H

p P P P

P P P P P

Taking the derivative of PH with respect to PH and PN with respect to PN and putting equal to zero we get:

4PH PN 14,000 and 4PN PH 104,000.

Solving the two equations we get the Bertrand equilibrium prices:

* 10666.67HP and * 28666.67.NP

b. After the merger, the new entity maximizes the following profit function:

2 260 0.01 0.01 510 0.01 1,100,000.H N

H H H N N N

p p p

p p p p p p

Taking the derivative with respect to PH and PN and putting equal to zero we get the following two equations:

2PH PN 6000 and 2PN PH 51000.

Solving the two equations we get the prices after the merger:

* 21000HP and * 36000.NP

The effect of change in PN on PH is 0.005PH 10 0.005 21000 10 95.

The effect of change in PH on PN is 0.005PN 10 0.005 36000 10 170. c. The prices charged after the merger are higher than before the merger.



38. You can solve this problem using calculus or the formulas for the linear demand and constant marginal cost Cournot model from the chapter. a. For the duopoly, q1 (15 2 2)/3 5, q2 (15 4 1)/3 4, pd 6, 1 (6 1)5 25, 2

(6 2)4 16. Total output is Qd 5 4 9. Total profit is d 25 16 41. Consumer surplus is 1/2(15 6)9 81/2 40.5. dCS At the efficient price (equal to marginal cost of 1), the output is 14. The deadweight loss is 1/2(6 1)(14 9) 25/2 12.5. dDWL

b. A monopoly equates its marginal revenue and marginal cost: MR 15 2Qm 1 MC. Thus Qm 7, pm 8, m (8 1)7 49. Consumer surplus is 1/2 15 8 7 49/2 24.5. mCS The deadweight loss is 1/2(8 1)(14 7) 49/2 24.5. mDWL

c. The average cost of production for the duopoly is [(5 1) (4 2)]/(5 4) 1.44, whereas the average cost of production for the monopoly is 1. The increase in market power effect swamps the efficiency gain, so consumer surplus falls while deadweight loss nearly doubles.

39. The answers are:

a. In the Cournot equilibrium, qi (a m)/(3b) (150 60)/3 30, Q 60, p 90.

b. In the Stackelberg equilibrium in which Firm 1 moves first, q1 (a m)/(2b) 50 60)/2 45, q2 (a m)/(4b) (150 60)/4 22.5, Q 67.5, and p 82.5.

40. The answers are:

a. The Cournot equilibrium in the absence of government intervention is q1 30, q2 40, p 50, 1 900, and 2 1,600.

b. The Cournot equilibrium is now q1 33.3, q2 33.3, p 53.3, 1 1108.9, and 2 1108.9. c. Because Firm 2’s profit was 1,600 in part (a), a fixed cost slightly greater than 1600 will prevent

entry.

41. Firm 1 earns revenue 21 1 1 2 1 1 1 1 2(1 ) R pq q q q q q q q , which corresponds to the marginal

revenue of

11 1 2

1

1 2 .R

MR q qq

Firm 1 maximizes its profit by satisfying the first order condition (FOC) 1 1MR MC . Due to the

government subsidy, the firm faces marginal cost 1 MC m s . Hence, the FOC is

1 21 2 q q m s . The corresponding response curve is

21

1 1.

2 2

q mq s

(1)

Similarly, Firm 2 maximizes its profit by equating its marginal revenue to its marginal cost. Since Firm 2 receives no subsidy, 2 .MC m Then for Firm 2 we have:

22 2 1 2 2 2 2 1 2

2 2 1

2 2

2 1

(1 )

1 2

1 2 .

R pq q q q q q q q

MR q q

MR MC

q q m



The reaction function of Firm 2 is

12

1.

2

q mq

(2)

Solving Equations (1) and (2) simultaneously, we obtain equilibrium values:

*1

*2

1 2

3 31 1

.3 3

mq s

mq s

(3)

The net national income of Government 1 is:

1 1

1 1 1 1

1 1

1 2 1 1

( )

(1 ) .

NNI sq

pq mq sq sq

pq mq

q q q mq

(4)

After substituting Equations (3) into (4) and completing some algebra, we find that

1 2 1

.3 3

m s s mNNI

Government 1 maximizes its NNI by solving the FOC:

1 2 1

3 3

1 40.

9

m s s mNNI

s s

s m

which suggests that the optimal subsidy is

* 1.

4

ms

(5)

Substituting Equation (5) into Equation (3) gives

*1

*2

1

21

.4

mq

mq

which are the outputs of the Stackelberg leader and follower respectively.

42. In the Cournot model of Solved Problem 14.4, a subsidy causes the best-response function of Firm 1 to shift outward.



Each firm i’s profit function is:

2 .2i i j i i i

j i

a b q q q q q

The F.O.C. is:

2 (2 ) 0.ii j i i j

j i j ii

a bq b q q a b q b qq

By the symmetry among all the firms, we have: , .j iq q j i Substitute this relationship into the

F.O.C. above, we can solve for each firm’s Nash equilibrium output:

(2 ) ( 1) 0 .( 1)i i i

aa b q b n q q i

b n

Each firm’s equilibrium profit is:

2

22

( ) (2 )( ) .

2 2 ( 1)i i j i i i

j i

a ba b q q q q q

b n

The equilibrium price is:

( ).

( 1)i jj i

a b a np a b q q

b n

Since 2

2

( ) (2 )

2[ ( 1) ]lim lim 0,a b

i b nn n

then each firm’s equilibrium profit approaches zero as n

approaches infinity. The Cournot oligopoly market approaches a competitive market as the number of firms goes to infinity if each firm produces identical products.

Chapter 15

1. The competitive firm’s demand curve for labor is given by the equation w MRPL p MPL. Because price is constant, when marginal product is rising, the demand curve for labor slopes upward (the firm will continue to hire labor throughout this stage). When marginal product is negative, demand for labor is negative (the firm will reduce labor).

2. Before the tax, the competitive firm’s labor demand was p MPL. After the tax, the firm’s effective price is (1 )p, so its labor demand becomes (1 )p MPL.

3. There are two opposing effects. With a change in relative factor prices, a firm that can easily substitute capital for labor will do so, which has a negative effect on the demand for labor. However, when the cost of capital falls, the firm’s demand for labor typically increases in the long run because with less expensive capital, the firm can profitably expand output (i.e., if inputs are complementary). In either case, the long-run demand curve for labor is more elastic than the short-run curve.

4. With the entry of a second firm and a resulting Cournot equilibrium, the demand curve of the dominant firm shifts to the left. Accordingly, its labor demand curve also shifts to the left.



5. If the labor supply curve is horizontal, the effect is the same in either case. If it is positively sloped, the effect is larger if the output market is monopolistic. Because price is larger than marginal revenue for a monopolist, marginal revenue product curves lie to the left and are steeper (less elastic) than value of marginal product curves. When the labor supply curve shifts, the quantity of labor changes by less, but the wage changes more than if the market were competitive.

6. In the short run, capital is fixed. The monopolist sets w MRPL. When L K, MPL 0. Thus no workers are hired beyond this point. When L K, MPL is positive, the firm hires labor until w MRPL.

7. The monopsonist will advertise if the increase in pre-advertising expense profits exceed $1000. This occurs if the advertisement causes the wage bill to fall by more than $1000. The ad causes the wage and marginal expense for labor to fall. Marginal cost for the firm is reduced, and employment of that input increases, as does output. The marginal revenue product of the additional output must exceed the marginal expense of the input, plus the cost of the advertisement.

8. Refer to Figure 15.4. The portion of the minimum wage line that lies to the left of the intersection with the supply curve becomes the new marginal expense curve. If the minimum wage is not set at the competitive wage level, deadweight loss is created. If it is above the competitive level, the firm hires up to where the minimum wage line crosses the demand curve. If the minimum is below the competitive level but above the monopsony wage w1, the firm hires up to where the minimum wage line crosses the supply curve, L*. Note that in both cases, L L2, where L2 is the competitive equilibrium.

Figure 15.4



9. No. There can be no monopsony power if the input supply curve is horizontal. The marginal expense will be the same as the average expense per unit.

A monopsony chooses a price-quantity combination from the industry supply curve that maximizes its profit. The monopsony’s marginal expenditure (for example, the additional cost of hiring one more worker) depends on the shape of the supply curve. If the supply curve is upward sloping, then the marginal expenditure is greater than the average expenditure, and the monopsony will maximize profit by buying a quantity that is less than that which would be purchased by competitive buyers at a price below the price that competitive buyers would pay. However, if the supply curve is horizontal at the market price, then a monopsony’s marginal expenditure to purchase one more unit of a good would be the good’s price. In this case, the monopsony would pay the same price that a competitive firm would pay.

10. A price support above the price set by a monopsony will increase price and the amount of input employed. In particular, if the price support is set at the price where the supply curve intersects the demand curve, the equilibrium will be identical competitive equilibrium.

11. See Figure 15.5 in text. When a major share of a small country’s workforce is incapacitated, the supply of labor decreases and the labor supply curve shifts to the left. The marginal expenditure curve also shifts to the left. Consequently, wages increase.

12. Audio-PowerPoint answer by James Dearden is also available (15A Loggers and Truckers).

Truckers (T) and loggers (L) are perfect complements and the product function is Q Min(L, 6T). The marginal revenue product of truckers is 80.000 MR if T L /6 and zero if T L /6. For a perfectly competitive market MR P.

Suppose there are 18 loggers. Then the marginal revenue product of truckers is 80.000 MR if T 3 and zero if T 3. Suppose W < 80.000 MR. Therefore 3 truckers will be hired. Now if the number of loggers is reduced to 12 then the marginal revenue product of truckers is 80.000 MR if T 2 and zero if T 2. Now only 2 truckers will be hired. Hence as the number of loggers decreases, the number of truckers hired will also decrease.

13. Audio-PowerPoint answer by James Dearden is also available (15B Oil Drilling).

As technology improves we can assume that MC decreases. Therefore the MC curve shifts down. We know MRP MR MP. Because of an increase in oil prices, MR increases. On the other hand, depending on the type of technological improvement, MP (the increase in oil yield by drilling another well) might increase. Therefore there will be an additional shift up in the MRP curve. As a result of MC shifting down and MRP shifting up, the amount of well drilling increases.

14. Audio-PowerPoint answer by James Dearden is also available (15C House of Garlic).

a. We know g is the number of garlic cloves in a dish. If Jacqueline works H hours and there are 250 customers (250 dishes), then g 120H/250. The marginal product of Jacqueline is 120/250 per hour per dish and

MR 250 (price/z) (z/g)

250 0.4 0.5 0.5 g0.5

25g0.5.

Thus

MRP 25(120H/250)0.5(120/250)



12(120H/250)0.5.

b. The number of hours that Jacqueline works is where

MRP w 12(120H/250)0.5 10

(120H/250)0.5 10/12

(120H/250)0.5 0.833

((120H/250)0.5)2 (0.833)2

(120H/250)1 0.694

250/120H 0.694

0.694H 250/120

H ( 250/120)*(1/0.694)

H 3.

c. 120 3/250 360/250 1.44.

15. An individual with a zero discount rate views current and future consumption as equally attractive. An individual with an infinite discount rate cares only about current consumption and puts no value on future consumption.

16. The lower the interest rate, the lower the cost of borrowing for tuition, and the lower the discount on future earnings. At an interest rate of zero, there is no discounting, so if the area labeled benefits exceeds the area labeled costs, the individual should attend college.

17. An individual would have to use a discount rate of zero in order for this rule to apply.

18. Assume the 3% interest rate is an annual rate. Then the present values is

$4$48,666.67.

(0.03 / 365)

fPV

i

19. The demanded quantity of labor and capital decrease with their factor prices ( w and r respectively) and increase with the output price. This is suggested by the signs of the corresponding derivates of the factor demand functions:

(1 )1

(1 )1

1( )

1( ) .

b bdd d

d

a add d

d

dL a bAp A

dp w r d

dK a bAp A

dp w r d

20. The marginal product of labor is 0.2L0.8K0.3. Suppose the marginal revenue is MR, then the marginal revenue product MRPL MR 0.2L0.8K0.3.

21. The competitive firm’s marginal revenue of labor is MRPL p(1 2K).

22. Yes. In the production function given, labor and capital are perfect substitutes. The firm will produce using whichever input is cheaper. Long-run average cost will equal marginal cost (min{w, r}). Thus firm size is indeterminate.



23. An example of a constant elasticity demand curve is p AQb. Marginal revenue product of labor is defined as . L LMRP MR MP Then

/

( 1)

( / )[( 1) ].

L

b

bL

MP aQ L

MR b Aq

MRP aQ L b AQ

24. To get the marginal expenditure curve, multiply the supply curve by Q, then take the derivative with respect to Q. ME 10 2Q.

25. The monopsony determines the amount of the factor to purchase by setting marginal expenditure equal to demand:

10 2 50

3 40

40/3.

LME D

Q Q

Q

Q

Substituting this quantity into the supply equation,

10

10 40/3

70/3.

P Q

P

P

The competitive equilibrium is Q 20, p 30, obtained by setting supply equal to demand (10 Q 50 – Q).

26. If a firm has a monopoly in the output market and is a monopsony in the labor market, its profit is

( ( )) ( ) ( ) , p Q L Q L w L L

where Q(L) is the production function, p(Q)Q is its revenue, and wL—the wage times the number of workers—is its cost of production. The firm maximizes its profit by setting the derivative of profit with respect to labor equal to zero (if the second-order condition holds):

( ) ( ) 0.

dp dQ dwp Q L w L L

dQ dL dL

Rearranging terms in the first-order condition, we find that the maximization condition is that the marginal revenue product of labor,

1( ) ( ) 1

L L

dp dQ dQMRP MR MP p Q L p

dQ dL dL

equals the marginal expenditure,

( ) ( ) 1

1( ) 1

dw L dwME w L L w L

dL w dL

w L

where is the elasticity of demand in the output market and is the supply elasticity of labor.



27. The present value is $285.93 100 100/1.05 100/1.052.

28. A principal of $2000 would earn $200 in interest per year at 10% compounded annually.

29. The present value is PV [100/(1 i)] (100/(1 i)2.

30. Whether you should buy or rent depends on how long you believe the phone will last, and the interest rate. Even if the interest (discount) rate is zero, the phone would have to last 10 years for the payments to be equal. At an interest rate of 10%, the cost is equal only if the phone lasts forever. Thus the individual is better off renting.

31. The cost to buy the washer now is $800. If we don’t buy the washer now, the present discounted value of 5 years’ higher operating cost by using the older washer (assuming they are realized at year-end) and buying the washer 5 years later (assuming the price is the same) is:

PV 80[1/(1.05)1 1/(1.05)2 · · · 1/(1.05)5] 800/(1.05)5

$346.66 626.82 973.48 800.

So the washer should be purchased.

32. The resale value of the refrigerator today is worth $90.70 100/(1 i)2. Subtracting this from the $200 purchase price yields a net price of $109.30.

33. The question is whether saving $10 per year for 10 years is worth $100 now (the difference in the prices of the machines). Given that the total savings over the 10-year period is just $100, any interest rate above zero would make the more expensive machine a bad choice.

34. Solving for irr, we find that irr equals 1 or 9. This approach fails to give us a unique solution, so we should use the NPV approach instead. The NPV 1 12/1.07 20/1.072 7.254, which is positive, so that the firm should invest.

35. The price of oil would have to be more than p(1 i).

36. In the first period,

1/

1

2 1

[1 (1 2)]

(1 ).

Ap

Q

p p i

37. Using Equation 15.21 gives a close approximation of the precise answer, which would be derived from Equation 15.20. (If you want to try this, I suggest either a finance calculator, or Excel.) The breakeven discount rate is approximately 16%. Because most of the enlisted men took the cash, they are implicitly discounting at a rate greater than that. Officers are about evenly divided.

38. Thirty-seven days per year is about 1/10 of a year. Thus they earn about one-tenth of the APR or 4% per year on the value of input purchases. For every million dollars spent on inputs, they save $4000.

39. Using Equation 15.18, the value of the stock at 5% per year is $21,609.71 in 30 years. At 4.75% it is $20,118.28. Thus Alexx will accumulate $1491.43 more than Spencer.



40. Currently, you are buying 600 gallons of gas at a cost of $1,200 per year. With a more gas-efficient car, you would spend only $600 per year, saving $600 per year in gas payments. If we assume that these payments are made at the end of each year, the present value of these savings for five years is $2,580 at a 5% annual interest rate and $2,280 at 10%. The present value of the amount you must spend to buy the car in five years is $6,240 at 5% and $4,960 at 10%. Thus the present value of the additional cost of buying now rather than later is $1,760 ( $8,000 $6,240) at 5% and $3,040 at 10%. The benefit from buying now is the present value of the reduced gas payments. The cost is the present value of the additional cost of buying the car sooner rather than later. At 5%, the benefit is $2,580 and the cost is $1,760, so you should buy now. However, at 10%, the benefit, $2,280, is less than the cost, $3,040, so you should buy later.

41. As was shown in Solved Problem 15.4, the present value of the expected returns is $196.7 million. If the purchase price is $205 million, it is not worth to buy it.

42. The internal rate of return is 20/400 0.05.

43. Because the first contract is paid immediately, its present value equals the contract payment of $1 million. Our pro can use Equation 15.19 and a calculator to determine the present value of the second contract (or hire you to do the job for him). The present value of a $2 million payment 10 years from now is $2,000,000/(1.05)10 $1,227,827 at 5% and $2,000,000/(1.2)10 $323,011 at 20%. Consequently, the present values are as shown in the table.

Payment Present Value at 5% Present Value at 20%

$500,000 today $ 500,000 $500,000

$2 million in 10 years $1,227,827 $323,011

Total $1,727,827 $823,011

Thus at 5%, he should accept Contract B, with a present value of $1,727,827, which is much greater than the present value of Contract A, $1 million. At 20%, he should sign Contract A.

44. Audio-PowerPoint answer by James Dearden is also available (16A Insurance and Hiring).

a. When somebody is hired, there is a series of obligations, such as health insurance, created that firm should pay over time. As long as the employee is hired, the firm has the obligation to pay these future costs.

b. The PV of these future health care costs should be used in making hiring decisions, exactly because the firm can make investments equal to the PV in risk-free assets and earn the amount of future costs.

45. Audio-PowerPoint answer by James Dearden is also available (16B Auto Financing).

a. PV of Cash back 20,000 500 19,500.

PV of loan 5000 (5000/1.04) (5000/1.042) (5000/1.043)

18,875.45.

You should choose the zero interest loan.

b. 5,000 (5000/1 i) (5000/(1 i)2) (5000/(1 i)3) 19,500 i 0.0171.



46. Audio-PowerPoint answer by James Dearden is also available (16C New Car or Public Transportation).

PV of purchasing car 20,000 0

3,000

1.04

X

nn

PV of public transportation 0

5,000.

1.04

X

nn

For X 11, PV of purchasing car 49,281 and PV of public transportation 48,802.

For X 12, PV of purchasing car 51,155 and PV of public transportation 51,922.

Therefore, as long as X 12, public transportation should be chosen.

47. See Figure 15.5. Suppose the value of the good goes up first, then decreases, as shown in the figure. The two upward sloping curves represent present value of the good if it was sold this year with different interest rate. The optimal harvest time is the time when the gap between the value of the good and the present value of the good if it was sold this year. As one can tell from the figure, the higher is the interest rate, the earlier is the optimal harvest time (T1 as shown in the figure). If the interest rate is zero, then the optimal harvest time is when the value curve reaches its peak.

Figure 15.5

48. a. 0.4

1.185L

PMRP

L

b. 4ME L

c. 1/1.40.4194L P

Chapter 16

1. Zero pollution is probably impossible to attain even if it were desirable. Any type of production results in some level of pollution (even spring water has to be delivered, which requires trucks). Thus no pollution means no production, and the elimination of the consumer and producer surplus that it creates. The marginal cost of pollution elimination would be extremely high for the last units of pollution, and the marginal benefits would be very low. Figure 16.3 in the text illustrates this trade-off.



2. An externality occurs when a person’s well-being or a firm’s production capability is directly affected by the actions of other consumers or firms rather than indirect through changes in prices. In this example, individuals sick with swine flu do not account for the effects of their actions on co-workers (and other students). In particular, if individuals sick with swine flu report for work (or attend school), then they risk infecting co-workers. Prior to the law, the marginal private benefit of staying home if sick was less than the marginal social benefit. By essentially subsidizing staying at home if sick, the law raises the marginal private benefit of staying home if sick such that it is closer to (or perhaps equal) to the marginal social benefit.

3. As Figure 16.3 shows, a specific tax of $84 per ton of output or per unit of emissions (gunk) leads to the social optimum.

4. When the marginal private cost of output falls, output increases (as does the quantity of gunk produced). If the tax remains unchanged at $84, as shown by the dashed line in Figure 16.4, it is not high enough to decrease the firm’s output to the socially optimal level. The new output level is Q, which is greater than the optimal output Q*. Thus, although consumer surplus increases because the price falls, there is a deadweight loss associated with the excess production beyond the socially optimal level.

Figure 16.4

5. See Figure 16.5. When the tax is imposed, price increases from pM to pS. Although prices are increased, a deadweight loss of area E results. Area A B C D is the reduction in social cost due to the tax, resulting in a net increase in social welfare.

Figure 16.5



6. Granting the chemical company the right to dump 1 ton per day results in that firm’s dumping 1 ton and the boat rental company’s maintaining one boat, which maximizes joint profit at $20.

7. Pure public goods have two important characteristics. They are nonrival in consumption, as are broadcast and cable television, and they are nonexcludible. While broadcast television is nonexcludible, consumers can be excluded from cable television. Therefore, cable television is a public good and broadcast television is a pure public good. Broadcast television is privately provided (in the United States) to solve the free rider problem. Commercials allow for payment for services given that the services are provided without exclusion.

8. As noted in the text, textbooks are typically owned by individuals, making them private goods, but the information in the books is public. Although copyright laws create a mechanism for exclusion, they do not cover all forms of communication (e.g., a lecture on the material in the book). Thus the books create a positive externality, and are underprovided, if priced as pure private goods.

9. If there were no market for any of the city’s garbage, it would all represent a negative externality at a cost of $125 per ton. Because farmers are willing to take some of the garbage at a reduced rate, the cost of the externality is reduced. The garbage taken by the farmers represents a positive externality, as it helps others in the agriculture market. The fact that they consume the garbage means that it is beneficial for them to do so at current prices (which are negative). If the market was not used, the farmers would have to retrieve the garbage after it had been disposed of at full price—a less efficient solution.

10. If a government has sufficient knowledge about pollution damage, the demand curve, costs, and the production technology, it can force a competitive market to produce the social optimum. The government might control pollution directly by restricting the amount of pollution that firms may produce (with an emissions standard) or by taxing them for pollution they create (with an emissions tax). With the emissions fee, the government can achieve the social optimum by taxing output or the amount of pollution produced. With an emissions standard, the government can achieve the social optimum by forcing the optimal quantity of pollution or output.

Unfortunately, the government usually does not know enough to regulate optimally. To set quantity restrictions on output optimally, the government must know how the marginal social cost curve, the demand curve, and pollution vary with output. Whether it is optimal to use fees or standards depends on the government’s degree of uncertainty and the shape of the marginal benefit and marginal costs curves for pollution abatement.



11. The sink and the dishes are both a form of public good for those who live in the house. If there were no one else there who might clean up, each individual would benefit from cleaning up the mess. However, the possibility that the other person might do the dishes makes it seem that an optimal strategy is to simply leave dishes and wait for the other person to do them. Unfortunately, they have the same dominant strategy, and the dishes accumulate. As with Table 16.4 and the example of the security guard, each benefits the most by letting the other bear the cost.

12. Because of the negative externality generated by an SUV, the government can impose an SUV tax such that people have to pay more to drive an SUV.

13. Having Michael Jordan in the NBA increased the total revenue of the league and therefore each team benefited. In terms of social welfare, the huge amount of money spent in advertising may be a social waste.

14. Those games and movies are not designed to impact people’s productivity in any way. However, the firms do not control their employees’ after-work activity, although they clearly affect people’s work performance. If we consider these examples a negative externality, we might as well label all holidays the same way.

15. Since the marginal benefit is significantly higher than marginal cost, the industry advertising is actually not optimal.

16. Using the information in Figure 16.5 in the text. At the market price of guard service of $10 per hour, the TV store will hire four guards and the ice-cream store will hire none. Now with a $2 per hour subsidy from the mall owner, the private price of the TV store will be $8 per hour. Subsequently, the TV store will hire 5 guards, which makes the social marginal benefit equal to social marginal cost and therefore achieves social optimal.

17. Audio-PowerPoint answer by James Dearden is also available (16A First Class Externalities).

a. The minimum price that the traveler can offer the family not to travel in first class is $500 $300 $200.

b. Any price between $200 and $600 is mutually agreeable. c. At price $200, both are indifferent.


a. Yes, there is a negative externality. b. For simplicity, assume that each motorcyclist rides without a helmet. As shown in Figure 16.6,

the social marginal cost curve is the sum of the private marginal cost and the marginal cost of the negative externality. If the government sets a no-helmet specific t tax |

S

externalityQ QMC as

shown in the figure, then the socially desirable level of motorcycle sales will be achieved at QS.



Figure 16.6

19. Audio-PowerPoint answer by James Dearden is also available (16C Line Cutting).

a. For now we can ignore Clara (she has the lowest values) and consider Alan and Ben without changing the nature of the game. The following table illustrates the payoff in the game.

Ben

1 2 3

1 4.5, 0.5 6, 1 6, 0.52 3, 0 4.5, 0.5 3, 0.5Alan

3 3, 0 3, 1 3, 0.33

As an example of how payoffs are calculated, suppose both Alan and Ben announce first position. In this case they both can be equally assigned first or second position. Alan’s payoff is then 0.5(12 6) 0.5(5 2) 4.5 and Ben’s payoff is 0.5(6 6) 0.5(3 2) 0.5. The other cells are calculated in the same manner. In this game the Nash equilibrium is for Alan to announce first and Ben second. Given this, then Clara will be better off or indifferent announcing third. Therefore Alan will go first, Ben second, and Clara third.

b. This is an example of crowding externality. Without Alan, Ben will go first and Clara second, and their total value is 8. However, with Alan, they go second and third, and the total value they get drops to 3. Therefore Alan causes $5 of negative externality for Ben and Clara.

c. Yes. The price of $6 is a tax and a little bit higher than the negative externality of $5. d. The total value of line order ABC (Alan first, Ben second, Clara third) is 15. The total value of

any other line order will be less.

20. We care only about the marginal harm of gunk at the social optimum, which we know is MCg $84 (because it is the same at every level of output). Thus the social optimum is the same as in our graphical example (and no algebra is necessary). We can also solve the problem using algebra. Using the equations from the chapter, we set the inverse demand function, p 450 2Q, equal to the new social marginal cost, MCs MCp 84 30 2Q 84 114 2Q, and we find that the socially optimal quantity is Qs (450 114)/(2 2) 84.

21. a. The unregulated equilibrium (MCP p) is Q 60, p 140.

b. The socially optimal equilibrium (MCS MCP MCg p) is Q 40, p 160. A specific tax of $40 per unit results in this outcome. With the tax, MCP 120 Q.



c. The unregulated monopoly output is the same as the socially optimal output. Q 40, p 160. d. The monopolist is already producing the socially optimal output level, and thus does not require

regulation.

22. The parameter A must be positive in order for gunk reduction to be beneficial, and must be between zero and 1 in order for marginal benefits of gunk reduction to decline. If costs are increasing at an increasing rate, must be greater than one.

23. To determine this level, set marginal benefit equal to marginal cost.

1 1

( )

1/( )

/( )

[ /( )

AH H

H A

H A

24. To find the social demand for the service, add the demand curves vertically (you must solve for p first). Individual inverted demands are p (–a1/b1) (1/b1)q, and p (–a2/b2) (1/b2)q. When added, the resulting curve will be nonlinear unless the demand curves have the same horizontal axis intercept. Note that b1, b2 0. For the solution below, assume that a1/b1 a2/b2.

p (–a1/b1 – a2/b2) (1/b1 1/b2) q if q a1

p (–a2/b2) (1/b2)q if q a1

25. We differentiate the utility function to find that

1

1

11

1 1

1

1

.

a

a

Ua A B

A

UA

B

Then

11

11

1

1

1 1 1 11

aU

AaU

B

a A B a BMRS

AA

Due to the symmetry of the problem,

2 2

2 2

2

12 2 2 2

2

U aA

U aB

a A B a BMRS

A A

The optimal allocation toward a public good is achieved when Equation 16.5 is satisfied; that is, 1 2 1.MRS MRS Then

1 1 2 2

1 1 2 2

1

.

a B a B

A AA a B a B

26. The value of crossing the bridge depends on the number of cars on the bridge (in the following, n is the number of cars on the bridge):



VA 4, if n 60; 3, if 60 n 120; 2, if 120 n 180; 1, if n 180.

VB 3, if n 60; 2, if 60 n 120; 1, if 120 n 180; 0, if n 180.

VC 2, if n 60; 1, if 60 n 120; 0, if 120 n 180; 1, if n 180.

VD 1, if n 60; 0, if 60 n 120; 1, if 120 n 180; 2, if n 180.

a. If the cost of crossing is zero, all groups will cross the bridge. This is because each group does not take into account the negative externality it creates for others.

b. If only cars in Group A pass the bridge, the total utility is 240. If Group B also passes the bridge, the value will be 60 3 60 2 300. If Group C also passes the bridge, the total value will be 60 2 60 1 60 0 180. With the addition of Group D, the total value is 60 1 60 0 60 1 60 2 120. The total utility is maximized when 120 (Groups A and B) pass the bridge.

27. In Nash equilibrium, each person maximizes its utility taking the number of hours the other works as given. Taking the partial derivative of UA with respect to tA and putting it equal to zero we get 24tA tB 552. Taking the partial derivative of UB with respect to tB and putting it equal to zero we get 24tB tA 552. Solving these two equations we get tA tB 22.08.

To find the number of hours that maximizes the sum of utilities, we take the partial derivative of the sum, once with respect to tA and once with respect to tB, and put them equal to zero. We get the two equations, 47tA tB 1,104 and tA 47tB 1,104. Solving these two equations we get tA tB 23. Therefore Anna and Bess, while maximizing their utilities, would ride free.

Chapter 17

1. Figure 17.2 plots ( ) .U W W Since it is a concave function, Jen is a risk-averse individual.

5 10 15 20 25W

1

2

3

4

5

6U

Figure 17.2

2. Figure 17.3 plots ( ) ln .U W W Since it is a concave function, Irma is a risk-averse individual.



5 10 15 20 25W

1

2

3

4

5

6U

Figure 17.3

3. As textbook Figure 17.2 shows, Irma’s expected utility of 133 at point f (where her expected wealth is $64) is the same as her utility from a certain wealth of Y.

4. See Figure 17.4. When x increases from x0 to x1, the chord showing expected utility shifts downward. Initially the risk premium is $10. After the increase in x, the risk premium increases to $40.

Figure 17.4

5. The expected punishment for violating traffic laws is V, where is the probability of being caught and fined and V is the fine. If people care only about the expected punishment (that is, there’s no additional psychological pain from the experience), increasing the expected punishment by increasing or V works equally well in discouraging bad behavior. The government prefers to increase the fine, V, which is costless, rather than to raise , which is costly due to the extra police, district attorneys, and courts required.

6. The argument is rather logical. Given a fixed probability of being caught and convicted, regardless of how low it might be, the expected punishment is much more severe if death penalty is imposed on those convicted.

7. In the decision tree below, the individual decides to have the transplant because the expected utility from the transplant is greater than remaining on dialysis.



Figure 17.5

8. The plaintiff must believe that X is at least $83,333 ($50,000/0.6) in order for the expected value of not settling to exceed the certain settlement. If the plaintiff is risk averse, he would accept a smaller offer of settlement to avoid taking the risk.

Figure 17.6

9. No. Risk-neutral individuals are indifferent between certainty and a fair bet. Thus they may or may not purchase fair insurance, but would not be willing to purchase unfair insurance. Because the risk premium for a risk-neutral person is zero, he or she is unwilling to pay anything to avoid taking a risk.

10. See Figure 17.7. If individuals know that the government will provide subsidies to homeowners with losses, they have an incentive to purchase less insurance. In the figure, initially, the individual has an expected utility of 2( ),U w because he or she receives w0 if there is a flood, and w3 if there is no flood. This results in a risk premium of 2 2( ).w w When the government offers the subsidy, the potential loss decreases, even though the probability of a loss does not change. The expected utility chord swings upward, resulting in a higher utility level if there is a flood [U(w1)]. The risk premium of the individual decreases to 2 2 ,w w decreasing the person’s willingness to pay for insurance.

Figure 17.7



11. See Figure 17.8, which depicts the market for loans among high-risk individuals. In the graph, the equilibrium interest rate (i*) is above the maximum limit set by the usury law (iu). The price ceiling creates a shortage in the market for high-risk loans of Q1 – Qu.

Figure 17.8

12. See Figure 17.9. If the individual has wealth w2 and faces the possibility of a loss to w0, he or she is risk averse, as the expected utility EUL is less than the utility of w1 with certainty. The same individual is risk preferring with respect to a possible gain to w4, as the expected utility EUG is greater than the utility of w3 with certainty.

Figure 17.9

13. For individuals, gambling at a casino and buying a stock might not be too different. Both serve to redistribute wealth within a society. For the society as whole, stock market places an important role in the financial market, which influences the general economy directly. On the other hand, gambling is generally considered an undesirable good, or a “bad,” as it does not contribute to the productivity of the society.

14. According to prospect theory, people are concerned about gains and losses—the changes in wealth—rather than the level of wealth, as in expected utility theory. People start with a reference point and consider lower outcomes as losses and higher ones as gains, using their initial endowment as a reference point. To determine the value of a gamble, individuals use decision weights, where the weight function assigns different weights than the original probabilities of the gamble. For example, people might assign disproportionately high weights to rare events. The value function in prospect



theory passes through the reference point because gains and losses are determined relative to the initial situation. The value curve is concave to the horizontal, outcome axis. Finally, the value function’s curve is asymmetric with respect to gains and losses, with people treating gains and losses differently, in contrast to the predictions of expected utility theory. For example, the value function’s curve might show a bigger impact to a loss than to a comparably-sized gain.

15. Kahneman and Tversky’s (1979) prospect theory is an alternative theory of decision-making under uncertainty that can explain some of the choices people make that are inconsistent with expected utility theory.

In expected utility theory, if an individual does not take a gamble, then his utility is U(W), where W is initial wealth. Expected utility (EU) with the gamble is

EU U(W A) + (1 ) U (W B).

The individual will take the gamble if

EU > U(W).

In prospect utility theory, people are concerned about gains and losses—changes in wealth—rather than the level of wealth, as in expected utility theory. People start with a reference point and consider lower outcomes as losses and higher ones as gains. To determine the value from taking a gamble, individuals do not calculate the expectation using the probabilities and (1 ), as they would with expected utility theory. Rather, an individual would use decision weights w( ) and w(1 ), where the w function assigns different weights than the original probabilities. Furthermore, in prospect theory, the value function’s curve is asymmetric, potentially with a bigger impact to a loss than to a comparably-sized gain. An individual gambles if the value from not gambling, V(0), is less than her evaluation of the gamble, which is the weighted average of her values in the two outcomes:

V(0) < [w()V(A) (1 ) V(B)].

Because prospect theory differs from expected utility theory in both the valuation of outcomes and how they are weighted, it is not possible to state conditions for which someone who acts as described in prospect theory is always more or less likely to take a gamble than someone who acts as described in expected utility theory. For example, even if w(θ) were less than θ, it may be the case that someone who acts as described in prospect theory is less likely to take a gamble than someone who acts as described in expected utility theory if the value function reflects loss aversion, where people hate making losses more than they like making gains.

16. The following shows how to find the risk premium if ( ) ln .U W W

2 1144 225 171

3 32 1

ln144 ln225 5.123 3

EW

EU

Then

5.12

( )

(171 ) 5.12

171

3.66.

U EW P EU

U P

P e

P



17. Since

2

2

100 2

2,

dUW

dWd U

dW

the Arrow-Pratt measure is

2

2 2( ) .

100 2

d U

dWWdU WdW

The measure is shown graphically in Figure 17.10. As it is apparent from the graph, the measure is positive and increasing in wealth when 50.W

Figure 17.10

18. If they were married, Andy would receive half the potential earnings whether they stayed married or not. As a result, Andy will receive $12,000 in present-value terms from Kim’s additional earnings. Because the returns to the investment exceed the cost, Andy will make this investment (unless a better investment is available). However, if they stay unmarried and split, Andy’s expected return on the investment is the probability of their staying together, 1/2, times Kim’s half of the returns if they stay together, $12,000. Thus Andy’s expected return on the investment, $6,000, is less than the cost of the education, so Andy is unwilling to make that investment (regardless of other investment opportunities).

19. Audio-PowerPoint answer by James Dearden is also available (17A Speeding Tickets).

a. E(FNJ) 0.25 300 0.75 0 75 E(YNJ) 0.25 0 0.75 300 225 E(UNJ) 0.25 (0)0.5 0.75 (300)0.5 12.99.

b. E(FPN) 0.5 200 0.5 0 100 E(YPN) 0.5 100 0.5 300 200 E(UPN) 0.5 (100)0.5 0.5 (300)0.5 13.66.

c. E(UPN) E(UNJ) but E(YPN) E(YNJ). Sylvia will choose Pennsylvania because she is risk averse.



20. Audio-PowerPoint answer by James Dearden is also available (17B Insurance, Risk Aversion and Wealth).

a. Risk Neutral: Y is the wealth

UINS Y 150.

The expected utility of no insurance (1/36)(Y 400) (35/36)Y Y 11.11. A risk-neutral person doesn’t buy insurance.

Risk Averse and poor: Y is the wealth and is equal to 4000.

UINS (4,000 150)0.5 62.04

The expected utility of no insurance (1/36)(0)0.5 (35/36)(4,000)0.5 61.49. A risk-averse and poor person purchases insurance.

Risk Averse and wealthy: Y is the wealth and equal to 500,000.

UINS (500,000 150)0.5 707

The expected utility of no insurance (1/36)(500,000 4000)0.5 (35/36)(500,000)0.5 726.67. A risk-averse and wealthy person does not purchase insurance.

b. The loss for a wealthy person is inconsequential and he or she behaves like a risk-neutral person.

21. Audio-PowerPoint answer by James Dearden is also available (17C DVDs).

a. Let’s denote E[R(i)] as the expected profit of the retailer if it orders i copies, and E[R(i)] as the expected profit of the studio if the retailer orders i copies. Now we have:

E[R(1)] 7

E[R(2)] 0.2 7 0.8 14 0.2 2 12.2

E[R(3)] 0.2 7 0.3 14 0.5 12 0.2 4 0.3 2 14.7

E[R(4)] 0.2 7 0.3 14 0.3 21 0.2 28 0.2 6 0.3 4 0.3 2 14.5. Therefore the optimum number of orders that maximizes expected profit for the retailer is 3.

E[N(1)] 7

E[N(2)] 0.2 7 0.8 14 0.2 1 12.4

E[N(3)] 15.4

E[N(4)] 16.0. Therefore the optimum number of orders that maximizes expected profit of the studio is 4.

b. E[R(1)] 7

E[R(2)] 12.26

E[R(3)] 16.1

E[R(4)] 17.5. Now the optimum number of orders is 4.

c. As parts (a) and (b) show, the optimum does depend on who pays the shipping cost.



22. a. The budget constraint is 8G 32L 8650.

b. The indifference curve for U 0.9 (Figure 17.11).

Figure 17.11

c. (0.64/0.36)(G/L) PL/PG (G/L) 2.25. d. L* (0.64 8650)/32 173 and G* (0.36 8650)/8 389.25. e. U (1/400)(389.25)0.36(173)0.64 0.58. f. If Hall has less than 8650 minutes per week, the probability of being accepted by Duke decreases.

This is because how much he spends on G and L depends on how much time overall he has to study, and the less he spends studying overall, the less he will spend on G and L at the optimum.

23. If the objective is to maximize net revenue, then the lottery is designed optimally. If the demand elasticity is smaller than 1 (or larger than 1 in absolute value), they can decrease the price to increase the sales, and therefore revenue. On the other hand, if the elasticity is larger than 1 (or smaller than 1 in absolute value), they can raise the price to increase the revenue. Operating at the point where the elasticity is 1 is optimal.

24. Assuming that the painting is not insured against fire, its expected value is

$550 (0.2 $1,000) (0.1 $0) (0.7 $500).

25. The probability of being caught must be at least 0.625, because $800 0.625 $500.

26. If she sends them together, the expected utility is qU(0) (1 – q)U($2000). If they are sent separately, and losses are independent events, she might lose both, one, or neither. The expected utility is (1 – q2)U($2000) 2q(1 – q)U(1000) (q2)U($0).

27. Yes, Mary is risk averse because she has a declining marginal utility of wealth (MUW 1/3 W2/3).

28. The risk premium is the amount that a risk-averse person would pay to avoid taking a risk. The expected value (EV) of the gamble is

EV 0.50 (29,791) 0.50 (24,389) $27,091.

Mary’s expected utility (EU) from the gamble is

EU 0.50 (29,7911/3) 0.50 (24,3891/3) 30.



The amount of money with certainty that would give Mary utility of 30 is

W1/3 30

W 27,000.

So, the gamble’s risk premium is $27,091 $27,000 $91. That is, she would pay $91 to avoid assuming the risk associated with the gamble.

29. It is not consistent because the two experiments have identical payoffs. The second choice probably is more popular in Scenario B for most individuals because they are starting from a higher level. They reason that only one flip of a coin is required from them to be able to keep all of their winnings, and if they lose, they still have $10,000. In the first scenario, beginning from a lower level, the $2500 sure thing is too tempting for many to resist.

30. a. EV 0.6(100,000) 0.4(–20,000) $52,000. Var [0.6($48,0002) 0.4(–72,0002)] $3456 million.

b. Yes, she would accept the offer because the expected value of the harvest ($52,000) is below that of the sure thing offer of $70,000.

c. One reason is that Ethan may not understand the laws of probability, and is unable to calculate the expected value. He may also not know the probabilities involved. If he believes that the probability of good weather is greater than it actually is, he may believe that he is making a good investment even if he is risk averse. Finally, he may be risk preferring. If his expected utility from the harvest is greater than the utility of $70,000 with certainty, he will make the purchase. See Figure 17.12.

Figure 17.12


a. Expected utilities for street parking are:

0.4 0.4Carolyn

0.4 0.4Sanjay

EU 0.5(80,000 10,000) 0.5(80,000) 89.08

EU 0.5(20,000 10,000) 0.5(20,000) 46.17

Let pC be the largest amount that Carolyn is willing to pay for a garage, and let pS be the maximum amount that Sanjay is willing to pay; then:

0.4Carolyn

0.4Sanjay

(80,000 ) EU 89.08 5100.11

(20,000 ) EU 46.17 5515.13

C C

S S

p p

p p



b. 5100.11 5515.13C Sp p because Sanjay is more risk averse than Carolyn.

The Arrow-Pratt measure of risk aversion at any wealth level W is:

1.6

0.6

( ) 0.24 0.6( ) ,

( ) 0.4

U W Wr W

U W W W

which is a decreasing function of the wealth level W. Thus Sanjay is more risk averse than Carolyn because Carolyn is wealthier than Sanjay.


a. The retailer’s expected utility of stocking a copy is:

0.5 0.51EU ( ) 0.5(10 15 ) 0.5(10 2) .p p

The retailer’s utility of purchasing no copies is: 0.51 10U .

Suppose the greatest amount the retailer is willing to pay to stock a copy is p1, then:

0.5 0.5 0.51 1 1 1

1

EU ( ) 0.5(10 15 ) 0.5(10 2) 10

12.78.

p p U

p

b. Now the retailer’s (expected) utility of purchasing two copies is:

0.52 ( ) (10 15 ) .U p p

The retailer’s utility of purchasing no copies is still: 0.51 10 .U

Suppose the greatest amount the retailer is willing to pay per film to purchase the two copies is p2; then:

0.5 0.52 2 2 1

2 1

( ) (10 15 ) 10

13 12.78 .

U p p U

p p

The diversification increases the retailer’s willingness to pay because it eliminates the risk.

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