Microeconomics 2nd Edition By Goolsbee – Test Bank

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