Question

There are two firms in a market, producing the same good. The firms simultaneously choose their output levels, q1 for firm 1 and q2 for firm 2. The price adjusts according to the inverse demand function p = 65−(q1 + q2). Each firm has a per-unit (average) cost of 5. Each firm’s payoff is its profit.

a. (5 pts) Find firm 1’s profit as a function of q1 and q2 (profit equals revenue minus total cost).

b. (10 pts) Find firm 1’s best response to a given output q2 of firm 2.

c. (7 pts) Find the Nash equilibrium of this game.

d. (8 pts) Find the cartel quantities (i.e. the output that maximizes the sum of the two firms’ profits, assuming that the firms split the total output evenly).

Answer 1

(a)

For firm 1,

TR1 = p x q1 = 65q1 - q12 - q1q2

TC1 = AC x q1 = 5q1

Profit (Z1) = TR1 - TC1 = 65q1 - q12 - q1q2 - 5q1 = 60q1 - q12 - q1q2

(b)

Setting Z1/q1 = 0,

60 - 2q1 - q2 = 0

2q1 + q2 = 60......(1) (Best response, firm 1)

(c)

For firm 2,

TR2 = p x q2 = 65q2 - q1q2 - q22

TC2 = AC x q2 = 5q2

Profit (Z2) = TR2 - TC2 = 65q2 - q1q2 - q22 - 5q1 = 60q2 - q1q2 - q22

Setting Z2/q2 = 0,

60 - q1 - 2q2 = 0

q1 + 2q2 = 60......(2) (Best response, firm 2)

Nash equilibrium is obtained by solving (1) and (2).

Multiplying (2) by 2,

2q1 + 4q2 = 120........(3), and

2q1 + q2 = 60..........(1)

(3) - (1) yields: 3q2 = 60

q2 = 20

q1 = 60 - 2q2 [from (2)] = 60 - (2 x 20) = 60 - 40 = 20

Q = q1 + q2 = 20 + 20 = 40

P = 65 - 40 = 25

(d)

For a Cartel, MR = MC.

P = 65 - Q

TR = PQ = 65Q - Q2

MR = dTR/dQ = 65 - 2Q

65 - 2Q = 5

2Q = 60

Q = 30

P = 65 - 30 = 35