Download Chapter 13 Imperfect Competition SOLUTIONS TO EXERCISES

Firms, Prices & Markets
Timothy Van Zandt
© August 2012
Chapter 13
Imperfect Competition
SOLUTIONS TO EXERCISES
Exercise 13.1. Suppose the two goods are complements rather than substitutes.
a. Write down an example of a linear demand function for good 1 such that good 2 is a complementary good.
Write down and then graph the residual demand curve for two different prices of good 2.
Solution:
Demand function: Q1 = 88 − 2P1 − P2 .
Residual demand when P2 = 16:
Q1 = 72 − 2P1
(solid line in figure).
Residual demand when P2 = 28:
Q1 = 60 − 2P1
(dashed line in figure).
Figure S1
P1
40
30
20
10
20
40
60
80 Q
1
b. Analyze the volume and price-sensitivity effects of an increase in the price of good 2.
Solution:
Both effects lead to a decrease of P1 in response.
Volume effect. An increase in P2 reduces demand for good 1: the new residual demand
curve has lower volume. (This is true by the definition of a complementary good and so
would happen even if the demand function were not linear.) The volume effect is thus
toward a lower price by firm 1, if firm 1 has increasing marginal cost.
Price-sensitivity effect. An increase in P2 lowers the choke price of the demand for
good 1. The new residual demand curve is thus more elastic. (This conclusion is usually true
for complementary goods even if the demand function is not linear.) The price-sensitivity
effect is thus also in the direction of lower price.
c. In strategic competition with complementary goods, would you therefore conclude that the prices are strategic
complements or strategic substitutes?
Firms, Prices & Markets • Solutions for Chapter 13
Solution:
(Imperfect Competition)
The pricing decisions are thus strategic substitutes.
Exercise 13.2. Consider a price competition model with two firms, 1 and 2, whose demand functions are as
follows:
Q1 = 90 − 3P1 + 2P2 ;
Q2 = 90 − 3P2 + 2P1 .
Each firm’s marginal cost is 6. This game is symmetric, and you can use this symmetry to simplify some calculations.
a. Write down firm 1’s profit function Π1 (P1 , P2 ).
Solution:
Per-unit profit is P1 − 6 and sales equals 90 − 3P1 + 2P2 . Hence,
Π1 (P1 , P2 ) = (P1 − 6) (90 − 3P1 + 2P2 ) .
b. For any P2 , write firm 1’s residual demand curve in the linear form Q1 = A − BP1 . Specifically, what are the
values of A and B (which may include P2 as parameters) and what is the choke price?
Solution:
Firm 1’s residual demand curve is
d 1r (P1 ) = (90 + 2P2 ) −3P1 .
↑
A
B
This has the form Q1 = A − BP1 if A = (90 + 2P2 ) and B = 3. The choke price is
P̄1 =
A
= 30 + 23 P2 .
B
c. Using the midpoint pricing rule, what is firm 1’s optimal price given P2 ? That is, what is firm 1’s reaction
curve in this game?
Solution:
When firm 2’s price is P2 , firm 1’s best response is
6 + (30 + 23 P2 )
MC1 + P̄1
=
= 18 + 13 P2 .
2
2
Thus firm 1’s reaction curve is b 1 (P2 ) = 18 + 13 P2 .
d. Because the game is symmetric, you can obtain firm 2’s reaction curve by changing the indices in firm 1’s
reaction curve. Graph the two curves on the same axes, with firm 1’s price on the horizontal axis and firm 2’s price
on the vertical axis. Estimate from the graph the Nash equilibrium prices.
Solution:
2
Firms, Prices & Markets • Solutions for Chapter 13
(Imperfect Competition)
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Figure S2
firm 1’s
reaction curve
P2
55
P1 = b 1 (P2 )
50
firm 2’s
reaction curve
45
40
Equilibrium:
35
P2∗
P2 = b 2 (P1 )
P1∗ = 27
P2∗ = 27
30
25
20
15
10
5
P1
5
10 15 20 25 30 35 40 45 50 55
P1∗
e. Because the game is symmetric, there is a symmetric equilibrium. Let P ∗ be the common price. Write down
and solve the single equation that defines a symmetric equilibrium.
Solution:
The symmetric equilibrium is thus the solution to P ∗ = 18+ 13 P ∗ , or P ∗ = 27.
f. Calculate the profit of each firm in the Nash equilibrium.
Solution:
The profit is
Π1 (27, 27) = (27 − 6) (90 − (3 × 27) + (2 × 27)) = 21 × 63 = 1323 .
Exercise 13.3. Two identical firms produce 128-Mb memory chips in plants that are capacity constrained.
The cost per unit of capacity (including short-run marginal cost) is 30, and the market’s inverse demand curve is
P = 150 − Q .
You are to analyze the Cournot model of quantity competition.
a. Write down firm 1’s revenue and profit as a function of the quantity levels for the two firms.
Solution:
As long as the price is not driven to zero:
Π1 (Q1 , Q2 ) = (150 − (Q1 + Q2 ) − 30) Q1 = (120 − (Q1 + Q2 )) Q1 .
b. Find each firm’s reaction curve by solving MR = MC.
Firms, Prices & Markets • Solutions for Chapter 13
(Imperfect Competition)
4
Solution: When firm 1’s quantity is Q1 and firm 2’s quantity is Q2 , the total quantity is
Q1 + Q2 and the price is p (Q1 + Q2 ) = 150 − (Q1 + Q2 ). From firm 1’s point of view,
if it expects firm 2’s quantity to be Q2 then it faces the following residual inverse demand
curve:
p r1 (Q1 ) = (150 − Q2 ) − Q1 .
The revenue and marginal revenue curves are then
r1 (Q1 ) = (150 − Q2 )Q1 − Q21 ,
mr1 (Q1 ) = (150 − Q2 ) − 2Q2 .
Firm 1’s marginal cost is 30. Hence, the MR = MC condition is
150 − Q2 − 2Q1 = 30
120 − Q2 = 2Q1
Q1 = 60 − 12 Q2 .
Thus, firm 1’s reaction curve is b 1 (Q2 ) = 60 − 12 Q2 . Because the two firm’s have the same
marginal cost, firm 2’s reaction curve has the same form, but with the roles of Q1 and Q2
switched: b 2 (Q1 ) = 60 − 12 Q1 .
c. Graph the reaction curves, with firm 1’s quantity on the horizontal axis and firm 2’s quantity on the vertical
axis. Estimate the Nash equilibrium as the intersection of these two curves.
Solution:
Figure S3
120
110
Q2
firm 1’s
reaction curve
Q1 = b 1 (Q2 )
100
90
80
70
60
Equilibrium:
Q∗1 = 40
Q∗2 = 40
50
Q∗2 40
30
firm 2’s
reaction curve
20
Q2 = b 2 (Q1 )
10
Q1
10
20
30
40
Q∗1
50
60
70
80
90
100 110 120
Firms, Prices & Markets • Solutions for Chapter 13
(Imperfect Competition)
d. Calculate the Cournot equilibrium capacities.
Solution: Q∗ is an equilibrium quantity for the two firms if Q∗ = b 1 (Q∗ ), i.e., if Q∗ =
60 − 12 Q∗ or Q∗ = 40. Hence, in the Cournot equilibrium both firms have capacities of 40.
e. Calculate the Cournot equilibrium market price and the profit for each firm.
Solution: Total quantity is 80. Hence the price is 150 − 80 = 70. Each firm’s profit is
(70 − 30) 40 = 1600.
f. Suppose the firms were to behave as in the model of perfect competition. What is the equilibrium price and
total output? Compare these values with the Cournot equilibrium price and total output.
Solution: The firm’s long-run cost curve has constant average cost of 30. Hence, the
competitive equilibrium price is 30. Total output is 150 − 30 = 120. The Cournot output is
lower and the Cournot price is higher than in the competitive equilibrium.
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