Question

The city of Springfield has granted the Springfield Nuclear Power Plant the right to be the sole provider of power to the city! As a monopolist, the Power Plant faces a demand for power given by p(Q) = 40 - 2Q and has a cost function, C(Q)=2Q. (8 points total) (a) In order to maximize profit, how much power should the Springfield Nuclear Power Plant sell? What will be the resulting price? How much will the Power Plant earn in profits? (2 points) (b) Calculate the Lerner index at the profit-maximizing quantity for the Power Plant and provide an interpretation of this value. (2 points) (c) Calculate the consumer surplus, producer surplus, and deadweight loss in this market. (2 points) (d) Suppose that Mayor Quimby and other city officials in Springfield desperately want to improve social welfare and have been able to obtain perfect information about demand and the Power Plant's marginal cost. What price ceiling should city officials set in order to optimally regulate the Power Plant? How would consumer surplus, producer surplus, and deadweight loss change under optimal price regulation? (2 points)

Answer 1

P = 40 - 2Q

TR = P* Q = (40 -2Q)Q = 40Q - 2Q²

C = 2Q²

a) Monopolist profit maximizing quantity is at MR = MC

MR = dTR/ dQ

MR = 40 - 4Q

MC = dC/dQ

MC = 4Q

Equating MR = MC

40 -4Q = 4Q

Q = 5

P = 40 - 2Q

P = 30

The profit maximizing quantity will be 5 and the price will be 30

Profit = TR -TC

Profit = 40Q - 2Q² - 2Q²  = 40q

Hence, profit = 40 * 5 = 200

b) Lerner Index = (P - MC)/ P

Lerner Index = (30 - 4Q)/ 30

Lerner Index = (30 - 20) / 30 = 10/30 = 1/3

Lerner Index is 1/3. This means that the company Springfield nuclear power plant has 1/3rd market power which is its power to set prices above marginal cost.

45 40 MC 35 E 30 25 F 20 D G 15 10 5 MR 0 0 2 4 6 00 10 12 Quantity

c) Consumer Surplus = Area between price line and the demand curve = 1/2 * 5 * 10 = 25

Producer Surplus = Area between price line and the MC curve= (10 * 5) + (1/2 *20* 5)= 100

Point F is calculated by the intersection of demand curve and MC curve. F = 40/6

Dead Weight loss = Area of triangle EFG = 1/2 * 10 * 1.67 = 8.33

d) Price ceiling should be put at the point where P = MC

40 - 2Q = 4Q

Gives Q = 20/3 = 6.67

and P = 80/3 = 26.67, which should be the price ceiling.

With this optimal pricing

consumer surplus = 1/2 * 6.67 * 13.34 = 44.44

Producer Surplus = 1/2 * 6.67 * 26.67 = 88.89

Deadweight loss = 0

Note that the total of producers surplus, consumers surplus and deadweight loss in part c and d are equal. This always will be equal.

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