Suppose that a monopolist firm’s demand curve is given by P = 1002y and its marg

Question

Suppose that a monopolist firm’s demand curve is given by P = 1002y and its marginal cost is given by MC(y) = y 2 . Answer the following questions: a. What is the profit-maximizing level of output and price for the monopolist? b. Now suppose that this firm is in a perfectly competitive industry, still facing the same marginal cost curve. (Assume that the average cost curve of this firm always lies below its marginal cost curve.) Calculate the price and quantity that we would observe in this industry. c. Calculate the consumer surplus under both the monopoly and the perfectly competitive industry structure. Compare your results.

Explanation / Answer

P = 100 - 2y

MC = y2

(a)

Total revenue, TR = P x y = 100y - 2y2

Marginal revenue, MR = dTR / dy = 100 - 4y

A monopolist maximizes profits by equating his MR with MC:

100 - 4y = y2

y2 + 4y - 100 = 0

This is a quadratic equation of the form (aX2 + bX + c = 0) where a = 1, b = 4, c = - 100.

The roots of the equation are:

[- b ±(b2 – 4ac)] / 2a = [- 4 ±(16 + 400)] / 2 = [4 ±(416)] / 2

= [- 4 ± 20.4] / 2

y = - 12.2 or y = 8.2

Since y >= 0, y = 8.2

P = 100 - 2y = 100 - (2 x 8.2) = 100 - 16.4 = 83.6

(b)

A perfectly competitive firm will equate P with MC:

y2 = 100 - 2y

y2 + 2y - 100 = 0

Solving for y as in part (a), we obtain y = 9 or y = - 11

Since y >=0, y = 9

P = 100 - (2 x 9) = 100 - 18 = 82

(c)

Consumer surplus (CS) is measured by the area between demand curve and price, and producer surplus (PS) is the area between supply curve & price.

From demand curve P = 100 - 2y, When y = 0, P = 100 [Vertical intercept]

From supply curve P (MC) = y2, when y = 0, P (MC) = 0 [Vertical intercept]

(1) Monopolist:

CS = (1/2) x (100 - 83.6) x 8.2 = 67.24

PS = (1/2) x (83.6 - 0) x 8.2 = 342.76

(2) Perfect competitor:

CS = (1/2) x (100 - 82) x 9 = 81

PS = (1/2) x (82 - 0) x 9 = 369

So, CS and PS are both higher in case of perfect competition.

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