Question

Exercise 3. Consider a firm with the Cobb-Douglas production function Q = 4L^1/3*K^1/2. Assume that the firm faces input prices of w = $7 per unit of labor, and r = $10 per unit of capital.

a) Solve the firm’s cost minimization problem, to obtain the combination of inputs (labor and capital) that minimizes the firm’s cost of production a given amount of output, Q.

b) Use your results form part (a) to find the firm’s cost function. This is the long-run total cost, as all inputs can be altered.

c) Find the firm’s marginal cost function, and its average cost function. Interpret.

d) Assume that the amount of capital is held fixed at K = 3 units. Solve the firm’s cost minimization problem again to find the amount of labor that minimizes the firm’s cost.

e) Use your result from part (d) to find the firm’s short-run cost function (in the short-run, the firm can alter the amount of labor, but without changing the units of capital).

Answer 1

Q = 4L^1/3K^1/2

(a)

Cost is minimized when MPL/MPK = w/r = 7/10

MPL = Q/L = [4 x (1/3) x K^1/2] / (L^2/3)

MPK = Q/K = [4 x (1/2) x L^1/3] / (K^1/2)

MPL/MPK = (2/3) x (K/L) = 7/10

2K / 3L = 7/10

K = 21L / 20

Plugging in production function,

4L^1/3(21L / 20)^1/2 = Q

4L^1/3(1.05L)^1/2 = Q

4L^1/3 x 1.02 x L^1/2 = Q

4.08 x L^5/6 = Q

L^5/6 = 0.245Q

L = (0.245)^6/5 = 0.185 x Q

K = (21 x 0.185 x Q) / 20 = 0.194 x Q

(b)

Total cost (TC) = wL + rK

TC = 7 x 0.185 x Q + 10 x 0.194 x Q

TC = 1.295 x Q + 1.94 x Q

TC = 3.235 x Q

(c)

MC = dC/dQ = 3.235

AC = C/Q = 3.235

Since cost function is linear with zero fixed costs, AC and MC are equal and fixed at a level of 3.235 per unit.

(d)

When K = 3,

4L^1/3(3)^1/2 = Q

4 x 1.73 x L^1/3 = Q

6.92 x L^1/3 = Q

L^1/3 = 0.145 x Q

Cubing,

L = 0.003 x Q³

With fixed capital, cost is minimized when MPL = w = 7

MPL = dQ/dL = 0.009 x Q² = 7

Q² = 765.37

Q = 27.67

L = 0.003 x (27.67)³ = 63.52

(e)

Short run TC = 7 x 0.003 x Q³ + 10 x 3

Short run TC = 0.021 x Q³ + 30