Question

1. Consider a market in which the demand function is P=50-2Q, where Q is total demand and P is the price. In the market, there are two firms whose cost function is TCi=10qi+qi^2+25, where qi1 is the quantity produced by firm i and Q=q1+q2
   1. Compute the marginal cost and the average cost.
   2. Compute the equilibrium (quantities, price, and profits) assuming that the firms choose simultaneously the output.

Answer 1

(a) TC = 10q + q² + 25

MC = dTC/dq = 10 + 2q, so

MC1 = 10 + 2q1

MC2 = 10 + 2q2

AC = TC/q = 10 + q + (25/q), so

AC1 = 10 + q1 + (25/q1)

AC2 = 10 + q2 + (25/q2)

(b) P = 50 - 2Q = 50 - 2q1 - 2q2 [since Q = q1 + q2]

For firm 1,

TR1 = P x q1 = 50q1 - 2q1² - 2q1q2

MR1 = TR1/q1 = 50 - 4q1 - 2q2

Setting MR1 = MC1,

50 - 4q1 - 2q2 = 10 + 2q1

6q1 + 2q2 = 40

3q1 + q2 = 20...........(1) (best response, firm 1)

For firm 2,

TR2 = P x q2 = 50q2 - 2q1q2 - 2q2²

MR2 = TR2/q2 = 50 - 2q1 - 4q2

Setting MR1 = MC1,

50 - 2q1 - 4q2 = 10 + 2q2

2q1 + 6q2 = 40

q1 + 3q2 = 20........(2) (best response, firm 2)

Multiplying (2) by 3:

3q1 + 9q2 = 60...........(3)

3q1 + q2 = 20...........(1)

(3) - (1): 8q2 = 40

q2 = 5

q1 = 20 - 3q2 [from (2)] = 20 - 3 x 5 = 20 - 15 = 5

Q = 5 + 5 = 10

P = 50 - 2 x 10 = 50 - 20 = 30

TR1 = 30 x 5 = 150

TC1 = 10 x 5 + 5 x 5 + 25 = 50 + 25 + 50 = 100

Profit, firm 1 = TR1 - TC1 = 150 - 100 = 50

TR2 = 30 x 5 = 150

TC2 = 10 x 5 + 5 x 5 + 25 = 50 + 25 + 50 = 100

Profit, firm 2 = TR2 - TC2 = 150 - 100 = 50