Question

A monopolist firm has the following cost function:

C(q)=0.5q^2 +10q+2

The (inverse) demand function for the monopolist’s output is as follows:

P(q) = 100 - q

a. Assume that the monopolist must charge the same price for all units of output (i.e. the monopolist cannot price discriminate). How many units of output will the monopolist produce to maximize profits? At what price will this output be sold? Illustrate this outcome in a graph.

b. How much profit does this monopolist earn?

c. Compute the own price elasticity of demand at the monopoly price and quantity. What is the value of the Lerner Index at the monopoly outcome?

d. Does this monopoly outcome maximize welfare? Why or why not?

If this outcome is inefficient, calculate the value of the deadweight loss the monopoly imposes on society. Which region on your graph represents this deadweight loss?

Answer 1

a)

P = 100 - q

Total Revenue = P*q = (100-q)*q

Marginal Revenue = dTR/dq = 100 - 2q

C = 0.5q^2 +10q+2

Marginal Cost = dC/dq = q + 10

The monopolist optimises where MR = MC

100 - 2q = q + 10

90 = 3q

q = 30

P = 100 - q = 70

MC = q+10 = 30+10 = 40

b)

Profit = Total Revenue - Total Cost = P*Q - MC*Q = 70*30 - 40*30 = 900

c)

Price elasticity of demand = dQ/dP*P/Q

dQ/dP = -1*70/30 = -2.33

Lerner Index = P-MC/P = 70-40/70 = 30/70 = 0.43

d)

Welfare is maximised when the firm produces at level where price equals the marginal cost.

When P = MC

100 - q = q + 10

q = 45

P = 55.

Thus we can see that the monopolist outcome is not efficient.

Deadweight loss is the orangle triangle between the demand and MC curve. Area of triangle = 1/2*(70-40)*(45-30) = 225.