Question

A monopolist faces the inverse demand for its output: p = 30 − Q. The monopolist also has a constant marginal and average cost of $4/unit.

(a) (10 points) What is the monopolist’s profit-maximizing level of output? What is the monopolist’s profit at this level of output?

(b) (10 points) Calculate the consumer surplus and show it in a graph along with the monopolist’s equilibrium; label the monopolist’s equilibrium e M.

(c) (2.5 points) What would the competitive equilibrium be if the demand curve were the same as the one faced by the monopolist and the market supply curve were equal to the monopolist’s marginal cost, i.e. P = MC? Show this equilibrium in the same graph as above and label it e C.

(d) (2.5 points) What is the gain in consumer surplus if this market were perfectly competitive instead of a monopoly?

Answer 1

p = 30 - Q

MC = AC = $4

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

Total revenue (TR) = p x Q = 30Q - Q²

MR = dTR/dQ = 30 - 2Q

Equating MR & MC,

30 - 2Q = 4

2Q = 26

Q = 13

P = 30 - 13 = $17

Profit = Q x (P - MC) = 13 x $(17 - 4) = 13 x $13 = $169

(b)

From demand function, When Q = 0, P = 30 (Vertical intercept) & when P = 0, Q = 30 (Horizontal intercept)

From MR function, When Q = 0, P = 30 (Vertical intercept) & when P = 0, Q = 30/2 = 15 (Horizontal intercept)

In following graph, D, MR & MC are demand, MR and MC functions with above intercepts. Equilibrium is at point e_M where MR intersects MC with price P_M (= $17) and quantity Q_M (= 13). Consumer surplus (CS) is area Ae_MP_M.

CS = Area between demand curve & market price = (1/2) x $(30 - 17) x 13 = (1/2) x $13 x 13 = $84.5

(c) A perfect competitor will equate P (Demand) with MC:

30 - Q = 4

Q = 26

P = MC = $4

In above graph, perfectly competitive equilibrium is at point e_C where MR intersects MC with price P_C (= MC = $4) and quantity Q_C (= 26). CS equals area Ae_CP_C.

(d) In perfect competition, CS = (1/2) x $(30 - 4) x 26 = 13 x $26 = $338

Gain in CS ($) = 338 - 84.5 = 253.5