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E-32 Answers to Selected Problems I know the answer! The answer lies within the heart of all mankind! The answer is twelve? I think I’m in the wrong building. —Charles Schultz Chapter 2 1.1 The demand curve for pork is Q = 171 - 20p + 20p b + 3p c + 2Y. As a result, 0 Q/ 0 Y = 2. A $100 increase in income causes the quantity demanded to increase by 0.2 million kg per year. 1.2 To solve this problem, we first rewrite the inverse demand functions as demand functions and then add them together. The total demand function is Q = Q 1 + Q 2 = (120 - p) + 160 - 1 2 p 2 = 180 - 1.5p. 2.3 In the figure, the no-quota total supply curve, S in panel c, is the horizontal sum of the U.S. domestic supply curve, S d , and the no-quota foreign supply curve, S f . 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, S f , is the same as the no-quota foreign supply curve, S f , 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, S f is vertical at Q for prices above p . The total supply curve with the quota, S , is the horizontal sum of S d and S f At any price above p , the total supply equals the quota plus the domestic supply. For example at p*, the domestic supply is Q * d and the foreign sup- ply is Q f , so the total supply is Q * d + Q f . 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 S d . 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. For Chapter 2, Exercise 2.3 p, Price per ton p, Price per ton p, Price per ton S d Q, Tons per year (a) U.S. Domestic Supply (b) Foreign Supply (c) Total Supply p * p * p * p p p S S Q d Q f Q d , Tons per year Q f , Tons per year Q d * Q f * S f S f Q d * + Q f * Q d * + Q f Q d + Q f

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Page 1: Answers to Selected Problems - Semantic Scholar · 2017-08-27 · Answers to Selected Problems E-33 both before and after the quota is imposed are at e 1, where the equilibrium price,

E-32

Answers to Selected Problems

I know the answer! The answer lies within the heart of all mankind! The answer is twelve? I think I’m in the wrong building. —Charles Schultz

Chapter 2

1.1 The demand curve for pork is Q = 171 - 20p +

20pb + 3pc + 2Y. As a result, 0Q/0Y = 2. A $100 increase in income causes the quantity demanded to increase by 0.2 million kg per year.

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

Q1 + Q2 = (120 - p) + 160 -12 p2 = 180 - 1.5p.

2.3 In the figure, 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, Sf, 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, Sf is vertical at Q for prices above p. The total supply curve with the quota, S, is the horizontal sum of Sd and Sf At any price above p, the total supply equals the quota plus the domestic supply. For example at p*, the domestic supply is Q*

d and the foreign sup-ply is Qf , so the total supply is Q*

d + Qf . 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.

For Chapter 2, Exercise 2.3

p, P

rice

per

ton

p, P

rice

per

ton

p, P

rice

per

ton

S d

Q, Tons per year

(a) U.S. Domestic Supply (b) Foreign Supply (c) Total Supply

p* p* p*

p– p– p–

S–

S

Qd–

Qf–

Qd, Tons per year Qf , Tons per year

Qd* Qf

*

S f–

S f

Qd* + Qf

*–Qd

* + Qf––

Qd + Qf

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E-33Answers to Selected Problems

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.

5.8 The elasticity of demand is (dQ/dp)(p/Q) = (-9.5 thousand metric tons per year per cent) * (45./1,275 thousand metric tons per year) L -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 * (31/1,275) L 0.39.

6.4 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.

6.8 Differentiating quantity, Q(p(τ)), with respect to τ, we learn that the change in quantity as the tax changes is (dQ/dp)(dp/dτ). Multiplying and divid-ing this expression by p/Q, we find that the change in quantity as the tax changes is ε(Q/p)(dp/dτ). Thus, the closer ε is to zero, the less the quantity falls, all else the same.

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

dRdτ

=dp

dτ Q + p

dQ

dp dp

dτ,

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

dRdτ

=dp

dτ ¢Q + p

dQ

dp≤ =

dp

dτ Q¢1 +

dQ

dp p

Q≤ =

dp

dτ Q(1 + ε).

Thus, the effect of a change in τ on R depends on the elasticity of demand, ε. Revenue rises with the tax if demand is inelastic (-1 6 ε 6 0) and falls if demand is elastic (ε 6 -1).

7.3 A usury law is a price ceiling, which causes the quantity that firms want to supply to fall.

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

dWdw

= L + w dLdw

= L¢1 +dLdw

wL≤ = L(1 + ε),

3.1 The statement “Talk is cheap because supply exceeds demand” makes sense if we interpret it to mean that the quantity of talk supplied exceeds the quantity demanded at a price of zero. Imagine a downward-sloping demand curve that hits the horizontal, quan-tity 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.”)

3.3 Equating the right-hand sides of the tomato supply and demand functions and using algebra, we find that ln p = 3.2 + 0.2 ln pt. We then set pt = 110, solve for ln p, and exponentiate ln p to obtain the equilibrium price, p L $62.80 per ton. Substitut-ing p into the supply curve and exponentiating, we determine the equilibrium quantity, Q L 11.91 mil-lion short tons per year.

4.3 To determine the equilibrium price, we equate the right-hand sides of the supply function, Q = 20 + 3p - 20r, and the demand function, Q =

220 - 2p, to obtain 20 + 3p - 20r = 220 - 2p. Using algebra, we can rewrite the equilibrium price equation as p = 40 + 4r. Substituting this expres-sion into the demand function, we learn that the equilibrium quantity is Q = 220 - 2(40 + 4r), or Q = 140 - 8r. By differentiating our two equilib-rium conditions with respect to r, we obtain our com-parative statics results: dp/dr = 4 and dQ/dr = -8.

4.7 The graph 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 Exercise 2.3. 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

For Chapter 2, Exercise 4.7

p, P

rice

of s

teel

per

ton

Q2Q3

Dh (high)

Q1

S (no quota)

Q,Tons of steel per year

p2

p3 e2

e3

e1p1

S (quota)–

p–

Dl (low)

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E-34 Answers to Selected Problems

q2ρ)1/ρ]ρ = q1

ρ + q2ρ, which has the same preference

properties as does the original function.

2.5 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 sub-stitution is -V1/V2 = - [(dF/dU)U1]/[(dF/dU)U2] =

-U1/U2, which is the same as originally.

2.6 By differentiating we know that U1 = a(aqρ

1 + [1 - a]qρ2)

(1 - ρ)/ρqρ - 11 and

U2 = [1 - a](aqρ1 + [1 - a]qρ

2)(1- ρ)/ρqρ -1

2 .

Thus, MRS = -U1/U2 = - [(1 - a)/a](q1/q2)ρ -1.

3.1 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 ori-gin. The intercepts on the Good 1 and Good 2 axes are Y/(2p1) and Y/(2p2), respectively. The oppor-tunity set shrinks by the area between the original budget line and the new line.

3.3 In the figure, the consumer can afford to buy up to 12 thousand gallons of water a week if not con-strained. 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 con-sumer has fewer opportunities because of rationing.

where ε is the elasticity of demand of labor. The sign of dW/dw is the same as that of 1 + ε. Thus, total labor payment decreases as the minimum wage forces up the wage if labor demand is elastic, ε 6 -1, and increases if labor demand is inelastic, ε 7 -1.

9.2 Shifts of both the U.S. supply and 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 essen-tially 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. sup-ply 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.

For Chapter 2, Exercise 9.2

p, P

rice

per

poun

d

Q, Tons of beef per year

p1

p2 = 0.85p1

S1

D1

D2

S 2

e1

e2

Q1 Q2 = 1.43Q1

Chapter 3

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

2.2 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.

2.4 If we apply the transformation function F(x) = xρ to the original utility function, we obtain the new utility function V(q1, q2) = F(U(q1, q2)) = [(q1

ρ +

For Chapter 3, Exercise 3.3

Oth

er g

oods

per

wee

k

100 12

Water, thousand gallons per month

Budget line

Quota

A B

4.3 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

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E-35Answers to Selected Problems

2.4 Barbara’s demand for CDs is q1 = 0.6Y/p1. Con-sequently, her Engel curve is a straight line with a slope of dq1/dY = 0.6/p1.

3.2 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.

3.5 On a graph show Lf, the budget line at the factory store, and Lo, the budget constraint at the outlet store. At the factory store, the consumer maximum occurs at ef on indifference curve If. Suppose that we increase the income of a consumer who shops at the outlet store to Y* so that the resulting budget

on apples gives him more extra utils than a dollar spent on kumquats. Thus, Andy maximizes his util-ity by spending all his money on apples and buying 40/2 = 20 pounds of apples.

4.14 David’s marginal utility of q1 is 1 and his marginal util-ity of q2 is 2. The slope of David’s indifference curve is -U1/U2 = -

12. Because the marginal utility from one

extra unit of q2 = 2 is twice that from one extra unit of q1, if the price of q2 is less than twice that of q1, David buys only q2 = Y/p2, where Y is his income and p2 is the price. If the price of q2 is more than twice that of q1, David buys only q1. If the price of q2 is exactly twice as much as that of q1, he is indifferent between buying any bundle along his budget line.

4.15 Vasco determines his optimal bundle by equating the ratios of each good’s marginal utility to its price.a. At the original prices, this condition is

U1/10 = 2q1q2 = 2q12 = U2/5. Thus, by divid-

ing both sides of the middle equality by 2q1, we know that his optimal bundle has the prop-erty that q1 = q2. His budget constraint is 90 = 10q1 + 5q2. Substituting q2 for q1, we find that 15q2 = 90, or q2 = 6 = q1.

b. At the new price, the optimum condition requires that U1/10 = 2q1q2 = 2q1

2 = U2/10, or 2q2 = q1. By substituting this condition into his budget constraint, 90 = 10q1 + 10q2, and solv-ing, we learn that q2 = 3 and q1 = 6. Thus, as the price of chickens doubles, he cuts his con-sumption of chicken in half but does not change how many slabs of ribs he eats.

6.2 Change the labels on the figure in the Challenge Solution to illustrate the answer to this question: When the price in Canada is relative low, the motor-ist buys gasoline in Canada, and vice versa.

Chapter 4

1.7 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.

2.2 Guerdon’s utility function is U(q1, q2) = min(0.5q1, q2). 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 0.5q1 = q2. Using that expres-sion 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).

For Chapter 4, Exercise 1.7

6

45

15

6

0

L1

p1,

$ p

er u

nits

L2 L3

40 12 30

e3e2e1

E3

E2

E1

I 1I 2

I 3

q1, Music CDs,Units per year

q1, Music CDs,Units per year

CD demand curve

Price-consumption curveq 2,

Mov

ie D

VD

s, U

nits

per

yea

r

(a) Indifference Curves and Budget Constraints

4 12 30

(b) CD Demand Curve

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E-36 Answers to Selected Problems

million (as you should have shown in your figure in the answer to Exercise 1.3).a. Given that the demand function is Q = Xp-1.6,

the revenue function is R(p) = pQ = Xp-0.6. Thus, the change in revenue, -$215 million, equals R(39) - R(37) = X(39)-0.6 - X(37)-0.6 L

-0.00356X. Solving -0.00356X = -215, we find that X L 60,353.

b. We follow the process in Solved Problem 5.1

∆CS = - L39

3760,353p-1.6dp1 =

60,3530.6

p-0.6 2 39

37 L 100,588(39-0.6 - 37-0.6) L 100,588 * (-0.00356) L -358.

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.

2.2 Because the good is inferior, the compensated demand curves cut the uncompensated demand curve, D, from below as the figure shows. Consequently, � CV � = A, � ∆CS � = A + B, � EV � = A + B + C. � CV � 6 � ∆CS � 6 � EV � .

line L* is tangent to the indifference curve If. 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 con-sumer is worse off after paying for shipping). The income effect is small if (as seems reasonable) 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.

3.7 We can determine the optimal bundle, e1, at the original prices p1 = p2 = 1 by using the demand equation from Table 4.1: q1 = 4(p2/p1)

2 = 4 and q2 = Y/p2 - 4(p2/p1) = 10 - 4 = 6. This opti-mal bundle is on an indifference curve where U = 4(4)0.5 + 6 = 14.

At the new bundle, e2, where p1 = 2 and p2 = 1, q1 = 4(1/2)2 = 1, and q2 = 10 - 4(1) = 8. This optimal bundle is on an indifference curve where U = 4(1)0.5 + 8 = 12.

To determine e*, we want to stay on the origi-nal indifference curve. We know that the tangency condition will give the same q1 as at e2 because q1 depends on only the relative prices, so q1 = 1. The question is what Y will compensate Phillip for the higher price so that he can stay on the original indif-ference curve. Because q2 = Y - 4(1/2) = Y - 4, the utility is U = 1 + (Y - 4) = Y - 3. So the Y that results in U = 14 is Y = 17. Thus, the substi-tution effect is -3 (based on the movement from e1 to e*) and the income effect is 0 (the movement from e* to e2), so the total effect is -3 (movement from e1 to e2).

3.9 At Sylvia’s optimal bundle, q1 = jq2 (see Chapter 3). Otherwise, she could reduce her expenditure on one of the goods and attain the same level of utility. Because at the optimal bundle U = min(q1, jq2), the Hicksian demands are q1 = H1(p1, p2, U) = U and q2 = H2(p1, p2, U) = U/j. The expenditure function is E = p1q1 + p2q2 = p1U + p2U/j = (p1 + p2/j)U.

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

Chapter 5

1.1 At a price of 30, the quantity demanded is 30, so the consumer surplus is 12 (30 * 30) = 450, because the demand curve is linear.

1.4 Hong and Wolak (2008) estimate that Area A is $215 million and area B is $118 (= 333 - 215)

For Chapter 5, Exercise 2.2

p, $

per

uni

t

e1

e2

HEV

H CV

D

q1, Units per quarter

A C

B

p1

p2

3.4 The two demand curves cross at e1 in the diagram. The price elasticity of demand, ε = (dQ/dp)(p/Q), equals 1 over the slope of the demand curve, dp/dQ, times the ratio of the price to the quantity. Thus, at e1 where both demand curves have the same price, p1, and the same quantity, Q1, the steeper the demand curve, the lower the elasticity of demand. If the price rises from p1 to p2, the consumer surplus falls from A + C to A with the relatively elastic demand curve (a loss of C) and from A + B + C + D to A + B (a loss of C + D) with the relatively inelastic demand curve.

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E-37Answers to Selected Problems

Julia chooses to work eight hours a day and to con-sume Y2 = 8w goods, at e2. (She will not choose to work fewer than eight hours. For her to do so, her indifference curve I2 would have to be tangent to the downward-sloping section of the new budget con-straint. 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, to protect women (and children) from overwork, an archaic law required him to pay women, but not men, double overtime after eight 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. Consequently, many employers do not let employees work overtime.

5.8 The proposed tax system exempts 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.

5.10 As the marginal tax rate on income increases, people substitute away from work due to the pure substi-tution 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 dif-fer across countries, the initial level of income dif-fers, again adding to the theoretical ambiguity. If we know that people work less as the marginal tax rate increases, we can infer that the substitu-tion 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.

5.11 The figure shows Julia’s original consumer equi-librium: Originally, Julia’s budget constraint was a straight line, L1 with a slope of -w, which was tan-gent 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 eight hours of work, and is horizon-tal at Y = 8w for more hours of work. The highest indifference curve that touches this constraint is I2. Because of the restriction on the hours she can work,

For Chapter 5, Exercise 3.4p,

$ p

er u

nit

Q, Units per weekQ1Q3 Q2

p1

p2

e1

e2e3 D

C

BA

Relatively elastic demand (at e1)

Relatively inelastic demand (at e1)

For Chapter 5, Problem 5.11Y

, Goo

ds p

er d

ay Timeconstraint

H1 = 12 H2 = 824 H, Work hours

I2I1L2

L1

e2

Y1 12w

Y2 = 8w

e1

per day

=

6.2 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 child-care services is upward slop-ing, by shifting the demand curve farther to the right, the price subsidy raises the price of child-care for these other parents.

6.3 The government could give a smaller lump-sum sub-sidy that shifts the LLS curve down so that it is par-allel 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.

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E-38 Answers to Selected Problems

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.

Chapter 7

1.3 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 some-one has a 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.

2.5 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.

2.12 Because the franchise tax is a lump-sum tax that does not vary with output, the more the firm produces, the less tax it pays per unit, l/q. 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, l/q. Because the franchise tax does not vary with output, it does not affect the marginal cost curve. The mar-ginal cost curve crosses both average cost curves from below at their minimum points. 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.

Chapter 6

3.1 One worker produces one unit of output, two work-ers produce two units of output, and n workers pro-duce 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.

3.4 (a) Given that the production function is q = L0.75K0.25, the average product of labor, holding capital fixed at K, is APL = q/L = L-0.25K0.25 =

(K/L)0.25. (b) The marginal product of labor is MPL = dq/dL =

34 (K/L)0.25. (c) At K = 16,

APL = 2L0.25 and MPL = 1.5L0.25.

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

4.8 Using Equation 6.8, we know that the mar-ginal rate of technical substitution is MRTS =

-MPL/MPK = -23.

4.9 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 MPB = 0q/ 0B = 1 everywhere along the isoquant. Similarly, MPG = 0.5. Given that B is on the horizontal axis, MRTS = -MPB /MPG

= -1/0.5 = -2.

5.4 This production function is a Cobb-Douglas pro-duction function. 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 expo-nents: γ = 0.27 + 0.16 + 0.61 = 1.04. That is, it has (nearly) constant returns to scale. The marginal product of material is

0q/0M = 0.61L0.27K0.16M-0.39 = 0.61q/M.

6.4 The marginal product of labor of Firm 1 is only 90% of the marginal product of labor of Firm 2 for a par-ticular level of inputs. Using calculus, we find that the MPL of Firm 1 is 0q1/0L = 0.9 0 f(L, K)/0L

= 0.9 0q2/0L.

7.2 We do not have enough information to answer this question. If we assume that Japanese and American firms have identical production functions and pro-duce 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

For Chapter 7, Exercise 2.12

Cos

ts p

er u

nit,

$

/q

q, Units per day

AC b

qa

MC

qb

AC a = AC b + /q

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E-39Answers to Selected Problems

3.1 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.

3.2 According to Equation 7.11, if the firm were mini-mizing its cost, the extra output it gets from the last dollar spent on labor, MPL/w = 50/200 = 0.25, should equal the extra output it derives from the last dollar spent on capital, MPK/r = 200/1,000 = 0.2. Thus, the firm is not minimizing its costs. It would save money if it used relatively less capital and more labor, from which it gets more extra output from the last dollar spent.

3.4 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 mar-ginal 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 ques-tions: 60 = t1 + t2. The slope of the 60-minute iso-cost 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.

3.6 From the information given and assuming that there are no economies of scale in shipping base-balls, it appears that balls are produced using a constant returns to scale, fixed-proportion produc-tion 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 one ball. Because the cost of all inputs other than labor and transportation are the same every-where, 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.

4.2 The average cost of producing one unit is α (regard-less of the value of β). If β = 0, the average cost does not change with volume. If learning by doing

increases with volume, β 6 0, so the average cost falls with volume. Here, the average cost falls exponentially (a smooth curve that asymptotically approaches the quantity axis).

6.1 If -w/r is the same as the slope of the line segment connecting the wafer-handling stepper and the step-per 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 com-bination 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.

6.3 The firm chooses its optimal labor-capital ratio using Equation 7.11: MPL/w = MPK /r. That is, 1

2q/(wL) =12q/(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 = L0.5K0.5 =

K0.5K0.5 = 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*)0.5(K*)0.5 =

1.1K*. So K* = 100/1.1 and L* = 110. The cost is C* = [(10/1.1) * 110] + [11 * (100/1.1)] = 2,000. 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] = 2,009.09.

Chapter 8

2.3 How much the firm produces and whether it shuts down in the short run depend only on the firm’s vari-able 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 aver-age variable cost.) Learning that the amount spent on the plant was greater than previously believed should not change the output level that the man-ager chooses. The change in the bookkeeper’s valu-ation 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.

2.5 The first-order condition to maximize profit is the derivative of the profit function with respect to q set equal to zero: 120 - 40 - 20q = 0. Thus,

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E-40 Answers to Selected Problems

the second term by Q/Q = 1 and the last term by Qo/Qo = 1:

dSr

dp

p

Qr

=dSdp

p

Qr

Q

Q-

dDo

dp

p

Qr

Qo

Qo

.

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

4.6 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 η/(η - ε). 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 in Maine rises even slightly relative to the price in Vermont, suppliers in Ver-mont will be willing to shift their entire supply to Maine. Thus, we expect the nearly full incidence to fall on consumers 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 ηr = 50η - 49εo. Given this equation, the residual supply elasticity to one state is at least 50 times larger than the national elasticity of supply, ηr Ú 50η, because εo 6 0, so the -49εo term is positive and increases the residual supply elasticity.

5.5 Because the clinics are operating at minimum aver-age 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 esti-mated 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.

5.6 Each competitive firm wants to choose its output q to maximize its after-tax profit: π = pq - C(q) - l.

profit is maximized where q = 4, so that R(4) =

120 * 4 = 480, VC(4) = (40 * 4) + (10 * 16) =

320, π(4) = R(4) - VC(4) - F = 480 - 320 - 200 =

-40. The firm should operate in the short run because its revenue exceeds its variable cost: 480 7 320.

3.9 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 farm-ers did not expect the 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 mini-mum average variable cost.)

3.11 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. We can use the qua-dratic formula to solve this equation for q for a spe-cific price to determine its profit-maximizing output.

3.13 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 out-put to q1 + 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. Thus, the firm should either produce q2 or shut down (if it is making a loss at q2). 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 7 0.

4.2 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 dis-courage some firms from entering the market initially.

4.5 To derive the expression for the elasticity of the resid-ual or excess supply curve in Equation 8.17, we dif-ferentiate the residual supply curve, Equation 8.16, Sr(p) = S(p) - Do(p), with respect to p to obtain

dSr

dp=

dSdp

-dDo

dp.

Let Qr = Sr(p), Q = S(p), and Qo = D(p). We multiply both sides of the differentiated expres-sion by p/Qr, and for convenience, we also multiply

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E-41Answers to Selected Problems

dndl

=

4 n dp

dQ-

d2C

dq2 0

nq dp

dQ1

4D

=

n dp

dQ-

d2C

dq2

D6 0.

The change in price is

dp(nq)

dl=

dp

dQ Jq

dndl

+ n dq

dlR

=dp

dQ D ¢n

dp

dQ-

d2C

dq2 ≤q

D-

nq dp

dQ

DT

=dp

dQ § -

d2C

dq2 q

D¥ 7 0.

Chapter 9

5.5 The specific subsidy shifts the supply curve, S in the figure, 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 differ-ential between what the consumers pay and what the producers receive is 11¢. Consumers and pro-ducers of roses are delighted to be subsidized by other members of society. Because the price to cus-tomers 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 sup-ply curve). Because the government pays a subsidy of 11¢ per stem for each stem sold, the govern-ment’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 sur-plus 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¢.

5.7 If the tax is based on economic profit, the tax has no long-run effect because the firms make zero eco-nomic profit. If the tax is based on business profit

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) + l] = 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 substi-tuting the market inverse demand function into the necessary and sufficient condition, we determine the market equilibrium (n*, q*) by the two conditions:

p(n*q*) - dC(q*)/dq = 0,

p(n*q*)q* - [C(q*) + l] = 0.

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

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

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

We can write these equations in matrix form (noting that p - dC/dq = 0 from the necessary condition) as4 n

dp

dQ-

d2C

dq2 q dp

dQ

nq dp

dQq2

dp

dQ

4 Jdqdn

R = J01Rdl.

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

D = 4 n dp

dQ-

d2C

dq2 q dp

dQ

nq dp

dQq2

dp

dQ

4 = ¢n

dp

dQ-

d2C

dq2 ≤q2 dp

dQ- q

dp

dQ ¢nq

dp

dQ≤

= -d2C

dq2 q2 dp

dQ7 0,

where the inequality follows from each firm’s suf-ficient condition. Using Cramer’s rule:

dq

dl=

4 0 q dp

dQ

1 q2 dp

dQ

4D

=

-q dp

dQ

D7 0,

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E-42 Answers to Selected Problems

intersection of this horizontal supply curve with the demand curve. With a new, small tariff of τ, the U.S. supply curve is horizontal at $14.70 + τ, and the new equilibrium quantity is determined by substi-tuting p = 14.70 + τ into the demand function: Q = 35.41(14.70 + τ)p-0.37. Evaluated at τ = 0, the equilibrium quantity remains at 13.1. The dead-weight 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 + τ (area C + D + E in Figure 9.9) minus the tariff revenues (area D):

DWL = L 14.70+ τ

14.70

[D(p) - S(p)]dp - τ[D(p + τ) - S(p + τ)]

= L 14.70+ τ

14.70

33.54p-0.67 - 3.35p0.334dp

-τ33.54(p + τ)-0.67 - 3.35(p + τ)0.334. To see how a change in τ affects welfare, we differ-

entiate DWL with respect to τ:

dDWLdτ

=ddτ

b L 14.70+ τ

14.70

[D(p) - S(p)]dp

- τ[D(14.70 + τ) - S(14.70 + τ)] r= [D(14.70 + τ) - S(14.70 + τ)] - [D(14.70 + τ)

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 less than economic profit. For example, if the government ignores opportunity labor cost but includes all capi-tal cost in computing profit, firms will substitute toward labor and away from capital.

5.8 The Challenge Solution in Chapter 8 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.

5.10 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 clear-ing price was p = 4. The deficiency payment was D = (p - p)Qs = (6 - 4)60 = 120.

b. Consumer surplus rises from CS1 =12 (10 - 5)

50 = 125 to CS2 =12 (10 - 4)60 = 180. Pro-

ducer surplus rises from PS1 =12 (5 - 0)50 = 125

to PS2 =12 * (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.

6.5 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

For Chapter 9, Problem 5.5

p, ¢

per

ste

m

Q, Billions of rose stems per year

39¢

30¢28¢

s = 11¢

1.25 1.34

e1

e2D

G

S

Demand

C

E

B

A

F

S − 11¢

s = 11¢

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E-43Answers to Selected Problems

(or if Chris is better), then Pat has a comparative advantage (see Figure 10.5) in working in the mar-ketplace, and Chris has a comparative advantage in working at home. Of course, if both enjoy con-suming 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 market-place, and 9 hours working at home. Because Chris earns $10 an hour and Pat earns $20 an hour, 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.

Chapter 11

1.4 For a general linear inverse demand function, p(Q) = a - bQ, dQ/dp = -1/b, so the elasticity is ε = -p/(bQ). The demand curve hits the horizontal (quantity) axis at a/b. At half that quantity (the mid-point of the demand curve), the quantity is a/(2b), and the price is a/2. Thus, the elasticity of demand is ε = -p/(bQ) = - (a/2)/[ab/(2b)] = -1 at the mid-point 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.

2.2 Amazon’s Lerner Index was (p - MC)/p =

(359 - 159)/359 L 0.557. Using Equation 11.11, we know that (p - MC)/p L 0.557 = -1/ε, so ε L -1.795.

-S(14.70 + τ)] - τ JdD(14.70 + τ)

dτ-

dS(14.70 + τ)

dτR

= -τ JdD(14.70 + τ)

dτ-

dS(14.70 + τ)

dτR .

If we evaluate this expression at τ = 0, we find that dDWL/dτ = 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.

Chapter 10

1.7 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.

4.1 If you draw the convex production possibility fron-tier 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.

4.2 As Chapter 4 shows, the slope of the budget con-straint 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

For Chapter 10, Exercise 4.2

24 0

Time constraint

Y, G

oods

per

day

H, Work hours per day

(a) Unmarried

LP

LC

2448 0

Time constraint

Y, G

oods

per

day

H, Work hours per day

(b) Married

LCombined

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E-44 Answers to Selected Problems

16β. 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(β - 1). Thus, it pays to set a low price in the first period if 16 6 16(β - 1), or 2 6 β.

7.6 If a firm has a monopoly in the output market and is a monopsony in the labor market, its profit is π = p(Q(L))Q(L) - w(L)L,where Q(L) is the pro-duction function, p(Q)Q is its revenue, and w(L)L—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):¢p + Q(L)

dp

dQ≤

dQ

dL- w(L) -

dwdL

L = 0.

Rearranging terms in the first-order condition, we find that the maximization condition is that the marginal revenue product of labor,

MRPL = MR * MPL = ¢p + Q(L) dp

dQ≤

dQ

dL

= p¢1 +1ε≤

dQ

dL,

equals the marginal expenditure,

ME = w(L) +dwdL

L = w(L)¢1 +Lw

dwdL

≤ = w(L)¢1 +

1η≤,

where ε is the elasticity of demand in the output market and η is the supply elasticity of labor.

Chapter 12

1.3 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 transport their purchases themselves) have a higher elasticity of demand than people who want to order by phone and have the goods delivered.

1.4 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).

3.5 See MyEconLab, Chapter Resources, 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,

2.4 Given that Apple’s marginal cost was constant, its average variable cost equaled its marginal cost, $200. Its average fixed cost was its fixed cost divided by the quantity produced, 736/Q. Thus, its average cost was AC = 200 + 736/Q. Because the inverse demand function was p = 600 - 25Q, Apple’s revenue function was R = 600Q - 25Q2, so MR = dR/dQ = 600 - 50Q. Apple maximized its profit where MR = 600 - 50Q = 200 = MC. Solving this equation for the profit-maximizing output, we find that Q = 8 million units. By sub-stituting this quantity into the inverse demand equation, we determine that the profit-maximizing price was p = $400 per unit, as the figure shows. The firm’s profit was π = (p - AC)Q = [400 -

(200 + 736/8)]8 = $864 million. Apple’s Lerner Index was (p - MC)/p = [400 - 200]/400 =

12.

According to Equation 11.11, a profit-maximizing monopoly operates where (p - MC)/p = -1/ε. Combining that equation with the Lerner Index from the previous step, we learn that 1

2 = -1/ε, or ε = -2.

3.4 A tax on economic profit (of less than 100%) has no effect on a firm’s profit-maximizing behavior. Suppose the government’s share of the profit is β. Then the firm wants to maximize its after-tax profit, which is (1 - γ)π. However, whatever choice of Q (or p) maximizes π will also maximize (1 - γ)π. Figure 19.3 gives a graphical example where γ =

13.

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 πB = R(Q) - C(Q), and the after-tax profit is πA = (1 - γ)[R(Q) - C(Q)]. For both, the first-order condition is marginal reve-nue equals marginal cost: dR(Q)/dQ = dC(Q)/dQ.

4.1 Yes. The demand curve could cut the average cost curve only in its downward-sloping section. Conse-quently, the average cost is strictly downward slop-ing in the relevant region.

6.1 Given 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 π* = 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/β, so its marginal revenue function is MR = 10 - 2Q/β. The output that leads to its maximum profit is determined by MR = 10 - 2Q/β = 2 = MC, or its output is 4β. Thus, its price is $6 and its profit is

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E-45Answers to Selected Problems

3.11 If a monopoly manufacturer can price discriminate, its price is pi = m/(1 + 1/εi) in Country i, i = 1, 2. If the monopoly cannot price discriminate, it charges everyone the same price. Its total demand is Q =

Q1 + Q2 = n1 pε1 + n2 p

ε2. Differentiating with respect to p, we obtain dQ/dp = ε1Q1/p + ε2Q2/p. Multiplying through by p/Q, we learn that the weighted sum of the two groups’ elasticities is ε = s1ε1 + s2ε2, where si = Qi /Q. Thus, a profit-maximizing, single-price monopoly charges p =

m/(1 + 1/ε).

Chapter 13

1.1 The payoff matrix in this prisoners’ dilemma game is

If Duncan stays silent, Larry gets 0 if he squeals and -1 (a year in jail) if he stays silent. If Duncan con-fesses, 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 domi-nant strategy. As a result, the Nash equilibrium is for both to confess.

1.3 No strategies are dominant, so we use the best-response approach to determine the pure-strategy Nash equilibria. First, identify each firm’s best responses given each of the other firms’ strategies (as we did in Solved Problem 13.1). This game has two Nash equilibria: (a) Firm 1 medium and Firm 2 low, and (b) Firm 1 low and Firm 2 medium.

1.8 Let the probability that a firm sets a low price be θ1 for Firm 1 and θ2 for Firm 2. If the firms choose their prices independently, then θ1θ2 is the probabil-ity that both set a low price, (1 - θ1)(1 - θ2) is the probability that both set a high price, θ1(1 - θ2) is the probability that Firm 1 prices low and Firm 2 prices high, and (1 - θ1)θ2 is the probability that Firm 1 prices high and Firm 2 prices low. Firm 2’s expected payoff is E(π2) = 2θ1θ2 + (0)θ1(1 - θ2) + (1 - θ1)θ2+ 6(1 - θ1)(1 - θ2) = (6 - 6θ1) - (5 - 7θ1)θ2. Similarly, Firm 1’s expected payoff is E(π1) =

(0)θ1θ2 + 7θ1(1 - θ2) + 2(1 - θ1)θ2 + 6(1 - θ1)(1 - θ2) = (6 - 4θ2) - (1 - 3θ2)θ1. Each firm forms a

we learn that pJ = $2,000 and pA = $2,500. As the chapter shows, the elasticities of demand are εJ = p/(MC - p) = 2,000/(500 - 2,000) = -

43

and εA = 2,500/(500 - 2,500) = -54. Using Equa-

tion 12.9, we find that

pJ

pA

=2,0002,500

= 0.8 =1 + 1/1-

542

1 + 1/1-432 =

1 + 1/εA

1 + 1/εJ

.

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= 12 * $1,500 * 3,0002, than in the United States, $2 million 1= 12 * $2,000 * 2,0002.

3.6 By differentiating, we find that the American mar-ginal 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 opti-mal quantity is QJ = 15 units in Japan. Substitut-ing QA = 40 into the American demand function, we find that pA = 100 - 40 = $60. Similarly, sub-stituting 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. Because dQA/dpA = -1 the elasticity of demand is εA =

-pA/QA in the United States and εJ = -12 PJ /QJ in

Japan. In the equilibrium, εA = -60/40 = -3/2 and εJ = -50/(2 * 15) = -5/3. As Equation 12.9 shows, the ratio of the prices depends on the relative elasticities of demand: pA/pJ = 60/50 =

(1 + 1/εJ)/(1 + 1/εA) = (1 - 3/5)/(1 - 2/3) = 6/5.

3.8 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/εC, so εC = -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. Substitut-ing 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 12.3.

Duncan

Squeal

Squeal

–2 –5

Larry –2

–5

0

Silent

Silent

0 –1

–1

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E-46 Answers to Selected Problems

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 strat-egy for GM. Thus, both firms’ entering is a Nash equilibrium.

2.3 If the airline game is known to end in five periods, the equilibrium is the same as the one-period equi-librium. If the game is played indefinitely but one or both firms care only about current profit, then the equilibrium is the one-period one because future punishments and rewards are irrelevant to it.

2.9 The game tree 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 discour-age entry. The incumbent can invest in equipment that lowers its marginal cost. With the lowered mar-ginal cost, it is credible that the incumbent will pro-duce larger quantities of output, which discourages entry. The incumbent’s monopoly (no-entry) profit drops from $900 to $500 if it makes the invest-ment 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. (If the rival were to enter, the incumbent would earn $132.) Because of the investment, the incumbent earns $500. Nonetheless, earning $500 is better than earning $400, so the incumbent invests.

2.10 The incumbent firm has a first-mover advantage, as the game tree 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

belief about its rival’s behavior. For example, sup-pose that Firm 1 believes that Firm 2 will choose a low price with a probability θn2. If θn2 is less than 1

3 (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(π1), (1 - 3θn2)θ1, is positive, so as θ1 increases, E(π1) increases. Because the highest possible θ1 is 1, Firm 1 chooses the low price with certainty. Similarly, if Firm 1 believes θn2 is greater than 1

3, it sets a high price with certainty (θ1 = 0). If Firm 2 believes that Firm 1 thinks θn2 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, θ2 = 1, however, is not consistent with Firm 1’s expectation that θn2 is a frac-tion. 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 unpre-dictable. 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 θn2 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 θn1 =

57. Thus, the only possible

Nash equilibrium is θ2 =57 and θ2 =

13

1.9 We start by checking for dominant strategies. 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 (enter-ing is Toyota’s dominant strategy), GM stays out. 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 exam-ine how the subsidy affects the payoff matrix and

For Chapter 13, Exercise 2.9

Incumbent

Enter

Do not enter($900, $0)

($400, $300)

Do not invest

Enter

Do not enter($500, $0)

($132, –$36)

Invest

Entrant

Entrant

Profits (πi, πe)Second stageFirst stage

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E-47Answers to Selected Problems

Chapter 14

3.1 The inverse demand curve is p = 1 - 0.001Q. The first firm’s profit is π1 = [1 - 0.001(q1 + q2)]q1 -

0.28q1. Its first-order condition is dπ1/dq1 = 1 -

0.001(2q1 + q2) - 0.28 = 0. If we rearrange the terms, the first firm’s best-response function is q1 =360 -

12 q2. Similarly, the second firm’s best-

response function is q2 = 360 -12 q1 By substitut-

ing 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¢.

3.5 Given that the firm’s after-tax marginal cost is m + τ, the Nash-Cournot equilibrium price is

p = (a + n [m + τ])/(n + 1),

using Equation 14.17. Thus, the consumer incidence of the tax is dp/dτ = n/(n + 1) 6 1 (= 100%).

3.6 The monopoly will make more profit than the duopoly will, so the monopoly is willing to pay the college more rent. Although granting monop-oly rights may be attractive to the college in terms of higher rent, students will suffer (lose consumer surplus) because of the higher textbook prices.

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 incum-bent’s commitment discourages entry. Moving back-ward 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 com-mits to the larger quantity, its rival does not enter and the incumbent earns $800. Clearly, the incum-bent 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.

2.11 It is worth more to the monopoly to keep the potential entrant out than it is worth to the poten-tial 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 πi - πd = $7 to exclude the poten-tial entrant. The incumbent’s profit is $6 if entry does not occur, and its loss is $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.

For Chapter 13, Exercise 2.10

Accommodate (qi small)

Second stageFirst stage

Incumbent

Enter

Do not enter($900, $0)

($450, $125)

Enter

Do not enter($800, $0)

($400, $0)

Deter (qi large)

Entrant

Entrant

Profits (πi, πe)

For Chapter 13, Exercise 2.11

Incumbent

Enter

Do not enter($10, $0)

($3, $3)

Do not raise costs

Enter

Do not enter($6, $0)

(–$1, –$1)

Raise costs $4

Entrant

Entrant

Profits (πi, πe)Second stageFirst stage

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E-48 Answers to Selected Problems

b. The Nash-Cournot equilibrium is now q1 = 33.3, q2 = 33.3, p = 53.3, π1 = 1,108.9, and π2 = 1,108.9.

c. Because Firm 2’s profit was 1,600 in part a, a fixed cost slightly greater than 1,600 will prevent entry.

4.1 a. Using Equation 14.16, the Nash-Cournot equi-librium quantity is qi = (a - m)/(nb) = (150 -

60)/3 = 30, so Q = 60, and p = 90.b. In the Stackelberg equilibrium (Equations 14.31

and 14.32) if Firm 1 moves first, then q1 = (a -

m)/(2b) = (150 - 60)/2 = 45, q2 = (a - m)/(4b) = (150 - 60)/4 = 22.5, Q = 67.5, and p = 82.5.

5.2 Given that the duopolies produce identical goods, the equilibrium price is lower if the duopolies set price rather than quantity. If the goods are heteroge-neous, we cannot answer this question definitively.

5.3 Firm 1 wants to maximize its profit: π1 = (p1 - 10)q1 = (p1 - 10)(100 - 2p1 + p2). Its first-order con-dition is dπ1/dp1 = 100 - 4p1 + p2 + 20 = 0, so its best-response function is p1 = 30 +

14 p2. Similarly,

Firm 2’s best-response function is p2 = 30 +14 p1.

Solving, the Nash-Bertrand equilibrium prices are p1 = p2 = 40. Each firm produces 60 units.

6.5 In the long-run equilibrium, a monopolistically competitive firm operates where its downward sloping demand curve is tangent to its average cost curve as Figure 14.9 illustrates. Because its demand curve is downward sloping, its average cost curve must also be downward sloping in the equilibrium. Thus, the firm chooses to operate at less than full capacity in equilibrium.

Chapter 15

1.2 Before the tax, the competitive firm’s labor demand was p * MPL. After the tax, the firm’s effective price is (1 - α)p, so its labor demand becomes (1 - α)p * MPL.

1.8 The competitive firm’s marginal revenue of labor is MRPL = pMPL = p(Lρ + Kρ)1/ρ -1Lρ -1.

2.1 An individual with a zero discount rate views cur-rent 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.

2.7 Because the first contract is paid immediately, its present value equals the contract payment of $1 mil-lion. Our pro can use Equation 15.15 and a calcu-lator 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 L $1,227,827 at 5%

3.11 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. The Challenge Solution shows the effect of a decrease in marginal cost due to a subsidy (the oppo-site effect). The section titled “The Cournot Equilib-rium with Many Firms” shows that a decrease in the number of firms causes market power (the markup of price over marginal cost) to increase. The two effects reinforce each other. Suppose that the market demand curve has a constant elasticity of ε. We can rewrite Equation 14.10 as p = m/[1 + 1/(nε)] = mµ, where µ = 1/[1 + 1/(nε)] is the markup factor. Suppose that marginal cost increases to (1 + a)m and that the drop in the number of firms causes the markup factor to rise to (1 + b)µ; then the change in price is [(1 + a)m * (1 + b)µ] - mµ = (a + b + ab)mµ. That is, price increases by the fractional increase in the marginal cost, a, plus the fractional increase in the markup factor, b, plus the interaction of the two, ab.

3.12 By differentiating its product, a firm makes the residual demand curve it faces less elastic every-where. 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.

3.17 You can solve this problem using calculus or the for-mulas in Solved Problem 14.1.a. Using Equations 14.21 and 14.22 for the duopoly,

q1 = (15 - 1 + 1)/3 = 5, q2 = (15 - 1 - 2)/3 = 4, pd = 6, π1 = (6 - 1)5 = 25, π2 = (6 - 2)4 = 16. Total output is Qd = 5 + 4 = 9. Total profit is πd = 25 + 16 = 41. Consumer surplus is CSd =

12(15 - 6)9 = 81/2 = 40.5. At the effi-

cient price (equal to marginal cost of 1), the output is 14. The deadweight loss is DWLd =12(6 - 1)(14 - 9) = 25/2 = 12.5.

b. The monopoly equates its marginal revenue and (its lowest) marginal cost: MR = 15 -

2Qm = 1 = MC. Thus, Qm = 7, pm = 8, πm =

(8 - 1)7 = 49. Consumer surplus is CSm =12(15 - 8)7 = 49/2 = 24.5. The deadweight loss is DWLm =

12(8 - 1)(14 - 7) = 49/2 = 24.5.

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.

3.19 a. The Nash-Cournot equilibrium in the absence of government intervention is q1 = 30, q2 = 40, p = 50, π1 = 900, and π2 = 1,600.

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E-49Answers to Selected Problems

expected punishment (that is, there’s no additional psychological pain from the experience), increas-ing the expected punishment by increasing θ or V works equally well in discouraging bad behavior. The government prefers to increase the fine, V, which is costless, rather than to raise θ, which is costly because doing so requires extra police, dis-trict attorneys, and courts.

2.3 The expected value for Stock A, (0.5 * 100) +

(0.5 * 200) = 150, is the same as for Stock B, (0.5 * 50) + (0.5 * 250) = 150. However, the variance of Stock A, (0.5 * [100 - 150]2) +

(0.5 * [200 - 150]2) = 2,500, is less than that of Stock B, (0.5 * [50 - 150]2) + (0.5 * [250 - 150]2) = 10,000. Consequently, Jen’s expected utility from Stock A, (0.5 * 1000.5) + (0.5 *

2000.5) L 12.07, is greater than from Stock B, (0.5 * 500.5) + (0.5 * 2500.5) L 11.44, so she pre-fers Stock A.

2.5 As Figure 16.2 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.

2.7 Hugo’s expected wealth is EW = 123 * 1442 +

113 * 2252 = 96 + 75 = 171. His expected utility is

EU = 323 * U(144)4 + 31

3 * U(225)4= 32

3 * 21444 + 313 * 22254

= 323 * 124 + 31

3 * 154 = 13.

He would pay up to an amount P to avoid bearing the risk, where U(EW - P) equals his expected util-ity from the risky stock, EU. That is, U(EW - P) =

U(171 - P) = 2171 - P = 13 = EU. Squaring both sides, we find that that 171 - P = 169, or P = 2. That is, Hugo would accept an offer for his stock today of $169 (or more), which reflects a risk pre-mium of $2.

4.1 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 earn-ings. 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 stay-ing 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).

and $2,000,000/(1.2)10 L $323,011 at 20%. Con-sequently, the present values are as shown in the table.

Payment Present Value at 5%

Present Value at 20%

$500,000 today $50,000 $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.

2.12 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 L 7.254, which is positive, so that the firm should invest.

2.16 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 pres-ent 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 ben-efit 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.

Chapter 16

1.2 Assuming that the painting is not insured against fire, its expected value is

(0.2 * $1,000) + (0.1 * $0) + (0.7 * $500) = $550.

1.3 The expected value of the stock is (0.25 * 400) +

(0.75 * 200) = 250. The variance is (0.25 * [400 -

250]2) + (0.75 * [200 - 250]2) = 7,500.

1.6 The expected punishment for violating traffic laws is θV, where θ is the probability of being caught and fined and V is the fine. If people care only about the

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E-50 Answers to Selected Problems

amount of gunk actually produced depends only on the marginal cost of abatement and not on the marginal benefit. Because the fee and standard lead to the same level of abatement at e, they cause the same deadweight loss.

6.9 No. The marginal benefit of advertising exceeds the marginal cost.

7.1 There are several ways to demonstrate that welfare can go up despite the pollution. For example, one could redraw panel b with flatter supply curves so that area C became smaller than A (area A remains unchanged). Similarly, if the marginal pollution harm is very small, then we are very close to the no-distortion case, so that welfare will increase.

7.2 See Figure 9.7 (which corresponds to panel a). Going from no trade to free trade, consumers gain areas B and C, while domestic firms lose B. Thus, if consum-ers give firms an amount between B and B + C, both groups will be better off than with no trade.

Chapter 18

1.2 Because insurance costs do not vary with soil type, buying insurance is unattractive for houses on good soil and relatively attractive for houses on bad soil. These incentives create a moral hazard problem: Relatively more homeowners with houses on poor soil buy insurance, so the state insurance agency will face disproportionately many bad outcomes in the next earthquake.

1.3 Brand names allow consumers to identify a particu-lar company’s product in the future. If a mushroom company expects to remain in business over time, it would be foolish for it to brand its product if its mushrooms are of inferior quality. (Just ask Babar’s grandfather.) Thus, all else the same, we would expect branded mushrooms to be of higher quality than unbranded ones.

3.3 Because buyers are risk neutral, if they believe that the probability of getting a lemon is θ, the most they are willing to pay for a car of unknown quality is p = p1(1 - θ) + p2θ. If p is greater than both v1 and v2, all cars are sold. If v1 7 p 7 v2, only lem-ons are sold. If p is less than both v1 and v2, no cars are sold. However, we know that v2 6 p2 and that p2 6 p, so owners of lemons are certainly willing to sell them. (If sellers bear a transaction cost of c and p 6 v2 + c, no cars are sold.)

4.1 If almost all consumers know the true prices, and all but one firm charges the full-information com-petitive price, then it does not pay for a firm to set a high price. It gains a little from charging ignorant consumers the high price, but it sells to no informed

Chapter 17

3.4 As Figure 17.3 shows, a specific tax of $84 per ton of output or per unit of emissions (gunk) leads to the social optimum.

3.7 a. Setting the inverse demand function, p = 450 -

2Q, equal to the private marginal cost, MCp = 30 + 2Q, we find that the unregulated equilibrium quantity is Qp = (450 - 30) ,

(2 + 2) = 105. The equilibrium price is pp = 450 - (2 * 105) = 240.

b. Setting the inverse demand function, p = 450 -

2Q, equal to the new social marginal cost, MCs = 30 + 3Q, we find that the socially opti-mal quantity is Qs = (450 - 30)/(2 + 3) = 84. The socially optimal price is ps = 450 -

(2 * 84) = 282.c. Adding a specific tax τ, the private marginal cost

becomes MCp = 30 + 2Q, so the equilibrium quantity is Q = (450 - 30 - τ)/4. Setting that equal to Qs = 282 and solving, we find that τ = 84.

3.10 As the figure shows, the government uses its expected marginal benefit curve to set a standard at S or a fee at f. If the true marginal benefit curve is MB1, the optimal standard is S1 and the optimal fee is f1. The deadweight loss from setting either the fee or the standard too high is the same, DWL1. Similarly, if the true marginal benefit curve is MB2, both the fee and the standard are set too low, but both have the same deadweight loss, DWL2. Thus, the deadweight loss from a mistaken belief about the marginal benefit does not depend on whether the government uses a fee or a standard. When the government sets an emissions fee or standard, the

For Chapter 17, Exercise 3.10

Fee

, mar

gina

l ben

efit,

mar

gina

l cos

t, $

Units of gunk abated per day

S S2S1

MB1

MB2

DWL1

DWL2

f1

f

f2

e

MC of abatement

Expected MBof abatement

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E-51Answers to Selected Problems

the store is the partner’s best alternative use of time. A partner could earn money working for someone else or use the time to have fun. Because a partner bears the full marginal cost but gets only half the marginal benefit (the extra business profit) from an extra hour of work, each partner works only up to the point at which the marginal cost equals half the marginal benefit. Thus, each has an incentive to put in less effort than the level that maximizes their joint profit, where the marginal cost equals the mar-ginal benefit.

2.4 If Paula pays Arthur a fixed-fee salary of $168, Arthur has no incentive to buy any carvings for resale, given that the $12 per carving cost comes out of his pocket. Thus, Arthur sells no carvings if he receives a fixed salary and can sell as many or as few carvings as he wants. The contract is not incentive compatible. For Arthur to behave efficiently, this fixed-fee contract must be modified. For example, the contract could specify that Arthur gets a salary of $168 and that he must obtain and sell 12 carvings. Paula must monitor his behavior. (Paula’s residual profit is the joint profit minus $168, so she gets the marginal profit from each additional sale and wants to sell the joint-profit-maximizing number of carv-ings.) Arthur makes $24 = $168 - $144, so he is willing to participate. Joint profit is maximized at $72, and Paula gets the maximum possible residual profit of $48.

4.2 The minimum bond that deters stealing is $2,500.

customer. Thus, the full-information competitive price is charged in this market.

Chapter 19

1.2 By making this commitment, a company may be trying to assure customers who cannot judge how quickly a product will deteriorate that the product is durable enough to maintain at least a certain value in the future. The firm is trying to eliminate asymmetric information to increase the demand for its product.

1.3 Presumably, the promoter collects a percentage of the revenue of each restaurant. If customers can pay cash, the restaurants may not report the total amount of food they sell. The scrip makes such opportunistic behavior difficult.

2.1 This agreement led to very long conversations. Whichever of them was enjoying the call more apparently figured that he or she would get the full marginal benefit of one more minute of talking while having to pay only half the marginal cost. From this experience, I learned not to open our phone bill so as to avoid being shocked by the amount due (back in an era when long-distance phone calls were expensive).

2.2 A partner who works an extra hour bears the full opportunity cost of this extra hour but gets only half the marginal benefit from the extra business profit. The opportunity cost of extra time spent at

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