61. When two firms selling identical products compete on price, the price continues to fall until P = MC. Thus, if marginal cost is a constant $10, the equilibrium price will be $10. To find the Bertrand equilibrium quantity, we can solve for inverse demand:

Q = 400 – 4P

P = 100 – 0.25Q

Next, set P = MC:

P = MC

100 – 0.25Q = 10

Q = 360 (split between the two Bertrand competitors)

62. a. For Firm 1, P = 60 – 2(q1 +q2) = 60 – 2q1 – 2q2

MR = 60 – 4q1 – 2q2

MR = MC

60 – 4q1 – 2q2 = $12 (Solve for q1 to get Firm 1’s reaction function.)

q1 = 12 – 0.5q2.

Because the two firms have the same marginal cost, Firm 2’s reaction function is the mirror image of Firm 1’s:

q2 = 12 – 0.5q1.

b. Use the reaction functions to solve for output. Substitute Firm 2’s reaction function into Firm 1’s reaction function:

q1 = 12 – 0.5(12 – 0.5q1) (solve for q1)

q1 = 8

Firm 2 produces:

q2 = 12 – 0.5(8) = 8.

c. Total market output is 16 (8 + 8). Plug 16 into inverse market demand: P = 60 – 2(16) = $28.

d. Profit can be calculated by (price – marginal cost) output. (If there were fixed costs, we would replace marginal cost with average total cost.) Each firm earns profit of (28 – 12)8 = $128.

63. a. A two-firm cartel produces where MR = MC. On the graph, this point occurs at Q = 4 and P = $24.

b. For Bertrand competition, P = MC, so P = $8 and Q = 8.

c. For Cournot competition, first derive each firm’s reaction function. Based on the graph, the market inverse demand curve is P = 40 – 4Q, where Q is the sum of each firm’s output, q1 + q2.

P = 40 – 4(q1 + q2) = 40 – 4q1 – 4q2, so Firm 1’s MR = 40 – 8q1 – 4q2.

Set MR = MC and solve for q1 to get Firm 1’s reaction function:

q1 = 4 – 0.5q2.

In a similar manner, Firm 2’s reaction function is:

q2 = 4 – 0.5q1.

Substitute Firm 2’s reaction function into Firm 1’s reaction function:

q1 = 4 – 0.5(4 – 0.5q1)

q1 = 2.67

q2 = 4 – 0.5(2.67) = 2.67

Market output: Q = 2.67 + 2.67 = 5.34

P = 40 – 4(5.34) = $18.64.

64. a. Firm 1 faces the inverse demand curve P = 260 – q1 – q2, giving rise to MR = 260 – 2q1 –q2.

Set MR = MC:

260 – 2q1 –– q2 = 20 (solve for q1)

q1 = 120 – 0.5q2

Firm 2’s reaction function is the mirror image of Firm 1’s:

q2 = 120 – 0.5q1.

Plug Firm 2’s reaction function into Firm 1’s reaction function:

q1 = 120 – 0.5(120 – 0.5q1)

q1 = 80

q2 = 120 – 0.5(80) = 80

P = 260 – (80 + 80) = 260 – 160 = $100

Each firm’s profit is calculated as (P – AC) × output:

(100 – 20)80 = $6,400.

b. Firm 1’s reaction function is unchanged: q1 = 120 – 0.5q2.

Firm 2’s reaction function is calculated by setting MR = MC:

260 –q1 – 2q2 = 80

q2 = 90 – 0.5q1

Plug Firm 2’s reaction function into Firm 1’s reaction function:

q1 = 120 – 0.5(90 – 0.5q1)

q1 = 100

q2 = 90 – 0.5(100) = 45

P = 260 – (100 + 45) = 260 – 145 = $115

Firm 1’s profit: (115 – 20)100 = $9,500

Firm 2’s profit: (115 – 80)45 = $1,575

66. To find the Nash equilibrium, first find the reaction function for each firm:

P = 100 – 10q1 – 10q2

MR = 100 – 20q1 – 10q2

MR = MC

100 – 20q1 – 10q2 = 25

q1 = 3.75 – 0.4q2

q2 = 3.75 – 0.4q1

q1 = 3.75 – 0.4(3.75 – 0.4q1)

q1 = 3.75 – 1.5 + 0.16q1

0.84 q1 = 2.25

q1 = q2 = 2.68

67. a. P = 18 – qK – qF, which leads to MR = 18 – qK – 2qF. Set MR = MC and solve for Flesh Not’s output, qF:

18 – qK – 2qF = 2

qF = 8 – 0.5qK.

b. P = 18 – qK – qF. Because Kibble sets output first, it will consider Flesh Not’s output response by incorporating Flesh Not’s reaction function in the market inverse demand curve.

P = 18 – qK – (8 – 0.5qK)

P = 10 – 0.5qK

MR = 10 – qK

c. Set Kibble’s MR = MC:

10 – qK = 2

qK = 8

d. Evaluate Flesh Not’s reaction function at qK = 8:

qF = 8 – 0.5(8) = 4

e. P = 18 – qK – qF = 18 – 8 – 4 = $6

68. a. MR = 4,000 – 4Q. Set MR = MC:

4,000 – 4Q = 1,000

Q = 750

P = 4,000 – 2(750) = $2,500

b. P = MC = $1,000

To find the market output, plug $1,000 into the inverse demand curve:

1,000 = 4,000 – 2Q

Q = 1,500

c. P = 4,000 – 2q1 – 2q2. Firm 1’s MR = 4,000 – 4q1 – 2q2.

Set MR = MC: 4,000 – 4q1 – 2q2 = 1,000

Solve for q1, Firm 1’s reaction function:

q1 = 750 – 0.5q2

Firm 2’s MR = 4,000 – 2q1 – 4q2.

Set MR = MC:

4,000 – 2q1 – 4q2 = 1,000

Solve for q2, Firm 2’s reaction function: q2 = 750 – 0.5q1.

Use the two-reaction function to solve for each firm’s output. Plug Firm 2’s reaction function into Firm 1’s reaction function:

q1 = 750 – 0.5(750 – 0.5q1)

q1 = 500

q2 = 500

Q = 500 + 500 = 1,000

P = 4,000 – 2(1,000) = $2,000

d. Assume that Firm 1 is the Stackelberg leader (i.e., the firm that sets the output level first), although the answers to this question would not change if Firm 2 were the Stackelberg leader. Plug Firm 2’s reaction function into the market inverse demand curve:

P = 4,000 – 2q1 – 2(750 – 0.5q1)

P = 2,500 – q1

MR = 2,500 – 2q1

MR = MC

2,500 – 2q1 = 1,000

q1 = 750 (plug 750 into Firm 2’s reaction function):

q2 = 750 – 0.5(750) = 375

P = 4,000 – 2(750 + 375) = $1,750

69. Solve for Firm 2’s reaction function by first finding its MR function:

MR = a – bq1 – 2bq2

Set MR = MC:

a – bq1 – 2bq2 = c

Solve for q2 to get Firm 2’s reaction function:

Substitute Firm 2’s reaction function into the market inverse demand curve:

MR =

Set MR = MC:

Solve for q1:

70. First, determine Polar’s reaction function:

P = 200 – 4(q1 + q2)

MR = 200 – 4q1 – 8q2

Set MR = MC

200 – 4q1–8q2 = 10

q2 = 23.75 – 0.5q1

Next, substitute for q2 in the inverse demand, determine MR, and then set equal to MC:

P = 200 – 4(q1 + q2)

P = 200 – 4q1– 4q2

P = 200 – 4q1 – 4(23.75 – 0.5q1)

P = 200 – 4q1– 95 + 2q1 = 105 – 2q1

MR = 105 – 4q1

MR = MC

105– 4q1 = 10

q1 = 23.75

q2 = 23.75 – 0.5(23.75) = 11.88

71. a. Aero’s total revenue: TR = PA(4,000 – 2PA + PP)

MR = 4,000 – 4PA + PP

b. Solve Aero’s MR for PA:

PA = 1,000 + 0.25PP

c. Pareto’s total revenue: TR = PP(4,000 – 2PP + PA)

MR = 4,000 – 4PP + PA

d. Solve Pareto’s MR for PP:

PP = 1,000 + 0.25PA

e. Substitute Pareto’s reaction function into Aero’s reaction function:

PA = 1,000 + 0.25(1,000 + 0.25PA)

PA = 1,250 + 0.0625PA

PA = $1,333.33 (plug into Pareto’s reaction function)

PP = 1,000 + 0.25(1,333.33)

PP = $1,333.33

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