Question

The can industry is composed of two firms. Suppose that the demand curve for cans is P=100-Q where P is the price (in cents) of a can and Q is the quantity demanded (in millions per month) of cans. Suppose the total cost function of each firm is TC=2+15Q where TC is total cost (in tens of thousands of dollars) per month and Q is the quantity produced (in millions) per month by the firm. a) what are the price and output if managers set price equal to marginal cost? b) What are the profit - maximizing price and output if the managers - collide and act like a monopolist? c) Do the managers make a higher combined profit if they collide than if they set price equal to marginal cost? If so, how much higher is their combined profit?

Answer 1

Demand curve : P = 100 - Q

Total Cost: TC = 2+15Q

a)

For marginal cost differentiate TC with respect with Q we get,

MC =

When P = MC = 15.

Put P = 15 in the demand function we have,

15 = 100 - Q

Q = 85

Thus each firm produces 85/2 = 42.5 units of output.

b)

A monopolist equates MR to MC

Total Revenue(TR) = P*Q = (100-Q)*Q = 100Q - Q²

For marginal revenue(MR) differentiate TR with respect with Q we get,

Equating MR and MC we get,

100 - 2Q = 15

Q = 42.5

Put Q = 42.5 in the demand function we get,

P = 100 - 42.5

P = 57.5

c)

Profit when managers collied = TR - TC = P*Q - MC*Q = 57.5 *42.5 - 15*42.5 = 1806.25

When they act independently Profit = TR - TC = P*Q - MC*Q = 15*85 - 15*85 = 0

Thus they make higher profit when they collide. Profit higher by 1806.25 - 0 = 1806.25