Question

A monopolist has access to an industry with market demand

P = 10 − y

where y is the firm’s quantity. Its cost function is C(y) = 2y

a. Determine the firm’s profit maximizing quantity. Show your outcome on a graph. What is the firm’s profit? Compute the point-elasticity of demand at the profit-maximizing output.

b. Now suppose the firm’s cost function is C(y) = 4y

Again determine the profit-maximizing quantity, profit and the elasticity at the profit-maximizing quantity. (No graph is required in this case.)  

c. Essentially, we have two types of monopolist. Which monopolist type operates at the higher level of elasticity? Why?

d. Prove that for any linear demand, p = a − by, and constant marginal cost, c, that a monopolist would never ever operate at a point elasticity less than 1.

Answer 1

A monopolist has access to an industry with market demand P = 10 ? y or y = 10 - P, where y is the firm’s quantity. Its cost function is C(y) = 2y. This implies that MR = 10 - 2y and MC = 2.

a. The firm’s profit maximizing quantity is the one where MR = MC. This is shown by point A in the graph. This is determined at 10 - 2y = 2 or y = 4 units. The price is 10 - 4 = $6. At this level firm’s profit is (TR - TC) = (6*4 - 4*2) = $16. The point-elasticity of demand at the profit-maximizing output is given by ed = price coefficient x P*/Q* = -1 x 6/4 = -1.5

b. Now suppose the firm’s cost function is C(y) = 4y. MC is now 4. The firm’s profit maximizing quantity is the one where MR = MC. This is shown by point B in the graph. This is determined at 10 - 2y = 4 or y = 3 units. The price is 10 - 3 = $7. At this level firm’s profit is (TR - TC) = (3*7 - 3*2) = $15. The point-elasticity of demand at the profit-maximizing output is given by ed = price coefficient x P*/Q* = -1 x 7/3 = -2.33

c. Essentially, we have two types of monopolist. The second monopolist type which has a cost of C(y) = 4y, operates at the higher level of elasticity. This is because the elasticity at that point is 2.33 which is greater than that of 1.5.

d. For a linear demand, p = a ? by, , MR = a - 2by. With a constant marginal cost, c, MR = MC or a - 2by = c or y* = 0.5(a - c)/b. This is profit maximizing quantity and so the price is p = a - b*(a - c)/2b or p = 0.5(a + c).

Now elasticity = -(1/b)*(P/Q) = -(1/b)*(0.5(a + c)/0.5(a - c)/b) = -(a + c)/(a - c).

See that c can be zero. But a is greater than zero. Hence a > c and so absolute elasticity is always greater than or equal to 1 for these demand function.