Question

A firm with market power faces the demand function q=4000-40P and a total cost function TC(q)=10q+0.001q²+1000.

a. If the firm acts as a simple monopoly, calculate the firm’s optimal price, quantity and the profits at that point.

b. What is the efficient level of output in this market (i.e., the one that maximizes total surplus, not just producer surplus)?

c. Calculate the deadweight loss from part a).

Answer 1

q = 4,000 - 40P

40P = 4,000 - q

P = 100 - 0.025q

TC = 10q + 0.001q² + 1,000

Marginal cost (MC) = dTC/dq = 10 + 0.002q

(a) A monopolist will maximize profit by equating Marginal revenue (MR) with MC.

Total revenue (TR) = P x q = 100q - 0.025q²

MR = dTR/dq = 100 - 0.05q

Equating with MC,

100 - 0.05q = 10 + 0.002q

90 = 0.052q

q = 1,731 (taking integer value for quantity)

P = 100 - (0.025 x 1,731) = 100 - 43.28 = 56.72

TR = 56.72 x 1,731 = 98,182.32

TC = (10 x 1,731) + (0.001 x 1,731 x 1,731) + 1,000 = 17,310 + 2,996.36 + 1,000 = 21,306.36

Profit = TR - TC = 98,182.32 - 21,306.36 = 76,875.96

(b) Output is efficient when total surplus is maximized by equating Price with MC.

100 - 0.025q = 10 + 0.002q

0.027q = 90

q = 3,333 (Taking interger value for quantity)

(c) When q = 3,333, P = 100 - (0.025 x 3,333) = 100 - 83.33 = 16.67

Deadweight loss = (1/2) x Difference in price x Difference in quantity = (1/2) x (56.72 - 16.67) x (3,333 - 1,731)

= (1/2) x 40.05 x 1,602 = 32,080.05