Answers to End-of-Chapter 6 Exercises

7. MP_K = 200

10. a. K = 4 → q = 660.22

K = 5 → q = 789.25 → MP_K = 129.03

K = 6 → q = 913.19 → MP_K = 123.94

K = 7 → q = 1033.04 → MP_K = 119.85

L = 49 → q = 660.22

L = 50 → q = 662.89 → MP_L = 2.67

L = 51 → q = 665.52 → MP_L = 2.63

L = 52 → q = 668.11 → MP_L = 2.59

b. Constant returns to scale

Answers to End-of-Chapter 7 Exercises

9. a. FC = 200

b. AVC = $55,000

c. MC = $55,000

d. AFC = $2000

e. C = 250 + 45q + 3i

Answers to End-of-Chapter 8 Exercises

3. a. The firm will produce 8 or more units depending on the market price and will not produce in the 0 – 7 units because in this range MC is less than AVC.

b. When P = 38, each firm will produce 8 units. Therefore when P =38, Q = 800 units. When P = 45, Q = 900. When P =55, Q = 1000. When P =65 and Q = 1100.

4. a. q = 25

b. Profit = $1050

c. MC = 4q. AVC = 2q. MC is greater than AVC for any quantity greater than 0. The firm produces in the SR as long as P is positive.

7. a. VC = 4q²

FC = 16

AC = 4q + 16/q

AVC = 4q

AFC = 16/q

b. Graph

c. Minimum average cost occurs where MC = AC

q = 2

d. MC is above AC at all output levels, so the firm will supply positive output at any positive price.

e. When P less than AC, firm earns negative profit. Minimum AC occurs where q = 2. Plug this in average cost function to find AC = 16. The firm will earn negative profit if P is below 16.

f. When P is above 16.

9. a. C(q) = 4000q²/81 + 1000

b. P = 987654q or q = .010125P

c. q = .010125(1000) = 10.125

Profit = $4062.50

10. a. P = 5, Q = 6000

q = 500 Profit = 528

b. Entry. Because firms are making economic profit. Entry will equilibrium price to fall. Firm’s output will fall. Profits will fall to zero, no further entry.

c. P = $3.80

13. a. 100

b. No, P will be $1.50

c. 26

d. P = 2

e. $2

Answers to End-of-Chapter 9 Exercises

3. Japanese farmers would be indifferent between the subsidy and the tariff because the increase in producer surplus is the same under both policies. The government may prefer the tariff because it does not require any government spending.

5. a. If both demand and supply are very elastic then the program could cost more than $50 million per year. If both are relatively price inelastic then it could cost less than $50 million per year.

b. If demand is perfectly inelastic then the loss in consumer surplus will be exactly $50 million. Otherwise, it would cost less than $50 million per year.

6. a. Slope is rise over run 10-16/15-12 = -2

Constant, use point (15,10)

10 = constant - 2(15), or constant = 40

So Q_D = 40 -2P

Similarly, for supply function, slope is rise over run (4-2)/(6-3) = 2/3

4 = constant + (2/3)(6), or constant = 0

So Q_S = (2/3)P

b. P_w = 9

Q_S = 6 million pounds

Q_D = 22 million pounds

Imports = 22 – 6 = 16 million pounds

c. P_US = 9 + 3 = 12

Q_S = 8 million pounds

Q_D = 16 million pounds

Imports = 16 – 8 = 8 million pounds

Government collects 3(8) = $24 million

Deadweight loss = 0.5(12-9)(8-6) + 0.5(12-9)(22-16) = $12 million

d. With import quota of 8 million pounds, P_US = 12

Cost to consumers is reduction in consumer surplus = (12-9)(16) + (0.5)(12-9)(22-16) = $ 57 million

Gain to producers is increase in producer surplus = (12-9)(6) + (0.5)(8-6)(12-9) = $ 21 million

7. a. No tariff Price is 8 + 2 = 10. Q_D = 150

b. With tariff Price is 10 + 2 = 12. Q_D = 130

c. (12-10)(130) + 0.5(12-10)(150-130) = $280 million

d. $2 (130) = $260 million

e. Net loss because the gain of $260 million is less than the loss of $280 million.

9. a. Fraction of tax borne by consumers is E_S/(E_S – E_D) = 4/(4-(-0.2)) = 4/4.2 = 0.95

b. Demand for beer will increase. If supply of beer is infinitely elastic, equilibrium price of beer will not change, and the quantity of beer consumed will increase.

Answers to End-of-Chapter 10 Exercises

8. a. AR is the demand curve, P = 700 – 5Q

MR = 700 – 10Q

MC₁ = dC₁/dQ₁ = 20 Q₁

MC₂ = dC₂/dQ₂ = 40 Q₂

Q = Q₁ + Q₂ = MC₁/20 + MC₂/40 = 3MC_T/40 or MC_T = 40Q/3

Profit maximization occurs when MC_T = MR

b. 40Q/3 = 700 – 10Q, or Q = 30

MR = 700 – (10)(30) = 400.

MC₁ = 400 = 20Q₁, or Q₁ = 20

MC₂ = 400 = 40Q₂, or Q₂ = 10

P = 700 – 5(30) = $550.

c. MC₁ shifts left, causing MC_T to shift left, which will intersect MR curve at a lower total quantity and higher MR. At a higher level of MR, Q₂ is greater. Since Q_T falls and Q₂ rises, Q₁ must fall. Since Q_T falls, P must rise.

9. Only MC₂ is relevant because the other MC lies above the demand curve. So demand curve is just P = 20 – 3Q₂

MR then is equal to 20 – 6Q₂

MR = MC₂ means 20 – 6Q₂ = 10 + 5Q₂ or Q₂ = 0.91.

Q₁ = 0 Therefore P = 20 – 3(0.91) = $17.27.

12. a. SRMC = 5

P = 100Q^-1/2

TR = PQ = 100Q^1/2

MR = dTR/dQ = 50Q^-1/2

Set SRMC = MR, or 5 = 50Q^-1/2, gives Q = 100, P = 100(100^-1/2) = $10.

Profit = TR – SRTC = -$1500.

b. LRMC = 6

Following the same procedure as above, we get Q = 69.44 to find P = 12

Profit = TR - LRTC = $833.33 – 416.66 = $416.67. The firm should remain in business in the long run.

Answers to End-of-Chapter 11 Exercises

5. (i) MR₁ = MC = 3, Q₁ = 6

MR₂ = MC = 3, Q₂ = 5.5

P₁ = 15 – 6 = $9

P₂ = 25 – 2(5.5) = $14

Profit = 9(6) + 14(5.5) – [5 + 3 (11.5)] = $91.50

DWL₁ = (0.5)(12-6)(9-3) = $18

DWL₂ = (0.5)(11-5.5)(14-3) = $30.25

Total DWL = $48.25

(ii) P = 25 – 2Q if Q ≤ 5

18.33 – 0.67Q if Q > 5

MR = 25 – 4Q if Q ≤ 5

18.33 – 1.33Q if Q > 5

With MC = 3, 18.33 – 1.33Q is relevant.

Equating MR and MC:

18.33 – 1.33Q = 3, or Q = 11.5

P = 18.33 – (0.67)(11.5) = $10.67

With this price, Q₁ = 4.33 and Q₂ = 7.17

Profit is 10.67(11.5) – [5 + 3(11.5)] = $83.21

DWL₁ = (0.5)(12 – 4.33)(10.67 – 3) = $29.41

DWL₂ = (0.5)(11 – 7.17)(10.67 – 3) = $14.69

Total DWL = $44.10

Answers to End-of-Chapter 12 Exercises

4. a. Firm 2’s reaction curve is:

Q₂ = 24 – (Q₁/2)

Firm 1 does not have a reaction function because it makes its output decision before Firm 2, so there is nothing to react to.

b. Firm 1 chooses output Q₁ to maximize its profits subject to Firm 2’s reaction function:

Q₁ = 24

Substituting Q₁ into Firm 2’s reaction function gives Q₂

Q₂ = 24 – (24/2) = 12

Substituting Q₁ and Q₂ into the demand equation to find price

P = 53 – 24 – 12 = $17

Profits for each firm equal TR minus TC or:

Π₁ = (17)(24) – (5)(24) = $288

Π₂ = (17)(12) – (5)(12) = $144

Total industry profit = $288 + $144 = $432

Compared to Cournot equilibrium, total output has increased from 32 to 36, price has fallen from $21 to $17, and total profits have fallen from $512 to $432. Profits for Firm 1 have risen from $256 to $288, while Firm 2’s profits have declined sharply from $256 to $144.

6. a. Q₁ = Q₂ = 80

P = $140

Π₁ = $6400 = Π₂

b. Q = 120, P = $180, each will produce 60 units and profit for each firm is $7200.

c. If Firm 1 were the only firm, it would produce the entire 120 units and earn a profit of $14,400.

d. Firm 2 cheats by substituting Q₁ = 60 into its reaction function:

Q₂ = 120 – (60/2) = 90

Total industry output is equal to Q₁ + Q₂ = 150

P = 300 – 150 = $150

Π₁ = (150)(60) – (60)(60) = $5400

Π₂ = (150)(90) – (60)(90) = $8100

8. a. Texas Air’s reaction function:

Q₁ = 30 – (Q₂/2)

American’s reaction function:

Q₂ = 30 – (Q₁/2)

Substituting:

Q₁ = Q₂ = 20

Q = Q₁ + Q₂ = 40

P = $60

Π₁ = Π₂ = $400

b. Texas Air’s reaction function:

Q₁ = 37.5 – (Q₂/2)

American’s reaction function is same as before.

Q₁ = 30, Q₂ = 15

Q = Q₁ + Q₂ = 45

Compared to (a) equilibrium quantity has risen slightly.

c. P = 100- 30- 15 = $55

Texas Air’s profits would be:

(55)(30) – (25)(30) = $900

Difference is $500. So Texas Air should be willing to invest up to $500 to lower costs from 40 to 25 per unit.

Without investment, American’s profits would be:

(55)(15) – (40)(15) = $225

With investment:

Q₁ = 25 = Q₂

P = 100 – 50 = $50

American’s profits are:

(50)(25) – (25)(25) = $625

Difference in profits for American is $400 so American would be willing to invest up to $400 to reduce its MC to 25 if Texas Air also has MC of 25

10. a. q₁ = q₂ = 22.5

Q = q₁ + q₂ = 45

P = 300 – 3(45) = $165

Profit for both firms will be equal:

Π = $2278.13

b. Each firm should produce half the quantity that maximizes industry profits (i.e. half the monopoly output). If they have different cost functions, then it would not be optimal for them to split the monopoly output evenly.

Joint profits will be maximized at Q = 36. Each firm will produce 18 and the optimal price to charge is P = 300 – 3(36) = $192

Profits for each firm will be $2430.

c. We already know the profits if both choose the Cournot output or both choose the cartel output. If WW produces the Cournot output (22.5) and BBBS produces the collusive level (18), then Q = 22.5 + 18 = 40.5, P = 300 – 3(40.5) = $178.50

Profit for WW = $2581.88 and profit for BBBS = $2187

If WW choose collusive output and BBBS chooses Cournot output, profits will be reversed.

d. WW will use Stackelberg strategy. WW’s profits will be: $2316.86 and BBBS’s profits will be $2067.24

Answers to End-of-Chapter 15 Exercises

1. FV = $110 one year from now

FV = $121 two years from now

FV = $161.05 after five years

PDV = $90.91 one year from now

PDV = $82.64 two years from now

PDV = $62.09 five years from now

4. Find i such that

966 = (100)(1+i)^-1 + (1100)(1+i)^-2

Using the quadratic formula to solve for i

i = 0.12 or -2.017

Since negative interest rate does not make economic sense, the effective yield is 12 percent.

6. a. PDV of $500 today is $500. The present value of $540 next year is $514.29. Should take $540 next year

b. If take $500 loan, can invest for 4 years and pay back $500. Future value of $500 is 500(1.05)⁴ = $607.75

After paying back the loan you will have $107.75 to keep. The future value of $100 gift is 100(1.05)⁴ = $121.55. Take $100 gift.

c. Interest rate is 0 percent, which is 5 percent less than market rate. You save $400 = (0.05)($8000) one year from now. PDV of this $400 is

$400/1.05 = $380.95, which is greater than $350. Take the financing.

d. PDV = 50,000 + 50,000/1.05 + 50,000/(1.05)² + …+ 50,000/(1.05)¹⁹ = $654,266.04

e. PDV of $60,000 perpetuity is $60,000/0.05 = $1,200,000. Take the $60,000 per year payment.

f. Any gift of $N from parent to child could be made without taxation by lending the child $N(1+r)/r. To avoid taxes on $50,000 gift, parent would lend child $550,000, assuming a 10

percent interest rate. With that money child can earn $55,000 in interest after one year and still have $500,000 to pay back to parent. PV of $55,000 one year from now is $50,000.

People of more moderate incomes find these rules unfair because they might be able to give child $50,000 directly, but it would not be tax free.

7. After sixth year, Ralph’s income will be same with or without graduate education, so we can ignore all income after first six years. With graduate school, PV of income for next six years

- $15,000/(1.1)¹ - $15,000/(1.1)² + $60,000/(1.1)³ + $60,000/(1.1)⁴ +$60,000/(1.1)⁵ +$60,000/(1.1)⁶ = $131,150.35

Without graduate school, PV for next six years is

$30,000/(1.1)¹ + $30,000/(1.1)² + $30,000/(1.1)³ + $45,000/(1.1)⁴ + $45,000/(1.1)⁵ + $45,000/(1.1)⁶ = $158,683.95

Payoff from graduate school is not large enough to justify foregone income and tuition expense while Ralph is in school.

11. a. NPV of buying car is -20,000 + (12,000/(1.04)⁶) = -10,516.23

NPV of leasing is -3,600 – (3,600/(1.04)) - (3,600/(1.04)²) = -10,389.94

Better off leasing car

b. NPV of buying car is -13,920.43

NPV of leasing is -9,684.18

Still better off leasing car.

c. Indifferent if NPV’s are equal or

-20,000 + (12,000/(1+r)⁶) = -3,600 – (3,600/(1+r)) - (3,600/(1+r)²)

Solve for r. Easiest way is use spreadsheet and calculate NPV’s for different values of r. Interest rate will be in neighborhood of 3.8%.

Answers to End-of-Chapter 14 Exercises

10. a. L = 64

b. q = 64

c. $4800

d. L = 64, q = 64, profits = $3840

e. L = 64, q = 64, profits = $3840