Question

Suppose a firm's short run-run production function is q = 2 * L^0.5 , the firm faces a fixed price of L at PL = 4, and the firm’s fixed cost is 50.

a. Derive the short-run total cost and total variable cost functions, and then solve for the marginal cost, average variable cost, and average total cost functions. Assume the firm is a marginal price taker at P = 60, solve for the firm’s profit-maximizing amount of L to employ (assume P_L = 4)

b. Now assume the firm is a monopolist in its output market facing the (inverse) demand function P = 90 - q. Solve for the profit maximizing output and profit. How much will the firm employ?

Answer 1

(a)

Short Run Total Cost(SRTC) = fixed cost + P_L*L

Here q = 2L^0.5 => L = q²/4

Hence, Short Run Total Cost(SRTC) = fixed cost + P_L*L = 50 + 4*q²/4

=> SRTC = 50 + q²

Variable cost = P_L*L = 4*q²/4 = q²

Average variable cost = Variable cost/q = q

Average Total cost = SRTC/q = 50/q + q.

Marginal cost = d(SRTC)/dq = 2q.

In order to maximize profit a price taking produces that quantity at which bP = MC.

Here P= 60 and MC = 2q.

Thus P = MC => 60 = 2q => q = 30.

Thus L = q²/4 = 30²/4 = 225

Hence, the firm’s profit-maximizing amount of L to employ = 225.

(b)

In order to maximize profit a profit maximizing firm produces that quantity at which MR = MC(Marginal cost).

Here MR = d(TR)/dq = d(P*q)/dq = 90 - 2q, where TR = Total Revenue = P*q.

Thus MR = MC => 90 - 2q = 2q => q =  22.5

P = 90 - q = 90 - 22.5 = 67.5

Profit = TR - SRTC = P*Q - SRTC = 67.5*22.5 - (50 + 22.5²) = 962.5

Amount of a labor firm will employ = q²/4 = 22.5²/4 = 126.56 units