Question

Suppose the quantity demanded for a product is given by  ?=20(?+1)−?/4q=20(A+1)−p/4, where ?p is the price of the good. The good is sold by a monopoly firm with constant marginal cost equal to 20 and fixed cost ?=400(?+1)2f=400(A+1)2. Let ?A be 4 and answer the following:

a) Suppose the firm must charge the same price to all consumers. Derive the profit maximizing price and quantity if the firm were to serve the consumers interested in this good. .

b) Suppose the firm were to charge every consumer the same price. Will the firm find it profitable to operate?

c) Suppose the firm were able to practice perfect price discrimination. What would be the firm's profit in this case? Would the outcome be efficient?

d) What is the efficiency loss of the outcome in part (b) ?

Answer 1

a).

Consider the given problem here the demand function is, “q = 20*(A+1) – P/4”, where “A=4”.

=> Q = 20*5 – P/4, => Q = 100 – P/4, => P = 400 – 4*Q.

The fixed cost is “F = 400*(A+1)^2 = 400*25 = 10,000” and the marginal cost is 20. So, the total cost function is, TC = F+MC*Q = 10,000 + 20*Q.

The marginal revenue function is, MR = 400 – 8*Q, and the at the equilibrium the MR must be equal to MC.

=> MR = MC, => 400 – 8*Q = 20, => Q = 380/8 = 47.5, => Q = 47.5.

The market price is, => P=400-4*Q = 400 - 4*47.5 = 210, => P = 210.

So, the profit maximizing price and the output are “P=210” and “Q=47.5” respectively.

b).

So, here the monopolist is charging “210” and produce “47.5 units of output”. So, the profit of the monopolist is given below.

=> A = P*Q – TC = 210*47.5 – 10,000 – 20*47.5 = (210-20)*47.5 – 10,000 = 9,025 – 10,000 = (-975).

=> A = (-975) < 0.

The monopolist is incurring losses. So, it is not profitable to operate.

c).

Let’s assume the monopolist is able to practice perfect price discrimination, => the monopolist can charge different buyers their willingness to pay. Here the monopolist will charge each buyer their maximum willingness to pay implied P=MR. So, the monopolist will determine the optimum production by “P=MC”.

=> P=MC, => 400 – 4*Q = 20, => Q=380/4 = 95, => Q=95. Here at “Q=95” the price is equal to MC implied the level of output is exactly equal to the competitive level of output. So, it is efficient.

Here the TR is the area under the demand curve up to “Q=95”. So, the profit of the monopolist is given below.

=> A = TR – TC = [MC*Q + 0.5*(400-MC)*Q] – F – MC*Q = 0.5*(400-20)*95 – F.

=> A = 18,050 – 10,000 = 8,050 > 0.

d).

Here the efficiency loss of output in part (b) is the dead weight loss associated with the lower production. So, the DWL is “0.5*(Qc-Qm)*(Pm-MC) = 0.5*(95-47.5)*(210-20) = $4,512.5.