Question

(Deriving SR Cost Curves) In this problem, we’ll work through deriving short-run total cost and marginal cost functions from a production function. Such cost functions show how costs vary when quantity changes (you’re typical cost curves from intro to micro!). A firm has a production function ?? = 0.25????^0.5, the rental rate of capital is $100, and the wage rate is $25. In the short-run, capital is fixed at 100 units.

a. What is the firm’s short-run production function, q(L)? In one sentence, explain what this tells you.

b. What is the short-run demand for labor as a function of quantity, q(L) ? In one sentence, explain what this tells you.

c. Write the firm’s cost as a function of labor, ??(L) ? In one sentence, explain what this tells you.

d. Use what you found in (b) and (c) to derive the firm’s short-run cost function, ??(q) . In one sentence, explain what this tells you.

e. What is the firm’s short-run marginal cost function, ????(q) . In one sentence, explain what this tells you.

f. If the firm produces 125 units, what will be its total cost and marginal cost?

Answer 1

q = 0.25KL^0.5

(a) When K = 100,

q = 0.25 x 100 x L^0.5

q = 25 x L^0.5

This relationship depicts the quantity of output that is possible to produce using L units of labor and 100 units of capital.

(b) Since q = 25 x L^0.5,

L^0.5 = q/25

Squaring both sides,

L = q²/625

This relationship depicts the amount of labor required to produce q units of output, while using 100 units of capital.

(c)

Total cost (C) = wL + rK

C ($) = 25L + (100 x 100)

C ($) = 25L + 10,000

This signifies that total cost of production using L units of labor is the sum of fixed cost (= 10,000) and total variable cost (= 25 x L).

(d) C = 25L + 10,000

Since L = q²/625,

C = 25 x (q²/625) + 10,000

C = (q²/25) + 10,000

This signifies that Total cost of producing q units of output using 100 units of capital is the sum of fixed cost (= 10,000) and total variable cost (= q²/25).

(e)

Marginal cost (MC) = dC/dq = 2q/25

It signifies that additional cost of producing one additional unit of output equals (2q/25).

(f) When q = 125,

Total cost ($) = (125 x 125 / 25) + 10,000 = 625 + 10,000 = 10,625

Marginal cost ($) = (2 x 125 / 25) = 10