Question

Exercise 2. A Los Angeles firm uses a single input (labor) to produce a recreational commodity according to a production function Q(L)4VL, where L is the amount of labor it uses. The commodity sells for $100 per unit. The input costs $50 per unit. Write down a function that states the firm's profit as a function of the amount of input. b. What is the profit-maximizing amount of input? What is the profit- maximizing amount of output? How much profit does it make when it maximizes profits? c. Suppose that the firm is taxed $20 per unit of its output and the price of its input is subsidized by $10 (per unit of input). What is the new profit- maximizing amount of input? What is the new profit-maximizing amount of output? How much profit does it make now? а.

Answer 1

From production function we get L = Q^2/16

Cost function is C = wL or C = 50L. This becomes C = 25Q^2/8

a) profit is given by π = revenue – cost

π = 100Q - 25Q^2/8

b) Profit is maximized when π’(0) = 0

100 – 50Q/8 = 0

Q = 100*8/50 = 16 units

L = 16^2/16 = 16 units

Profit π = 100*16 – 25*16^2/8 = 800

c) New input price is 40 and so new cost is C = 40L or C = 5Q^2/2 + 20Q

Profit is now π = 100Q - 5Q^2/2 - 20Q or π = 80Q – 5Q^2/2

Profit is maximized when π’(0) = 0

80 – 5Q = 0

Q = 16 units

L = 16 units

Profit = 80*16 – 5*16^2/2 = $640.