You have been hired to replace the manager of a firm that used only two inputs,

Question

You have been hired to replace the manager of a firm that used only two inputs, capital (K) and labor (L), to produce output. The production function is given by Q = L1/3K2/3. The firm can hire as much labor as it wants at a wage of SI per hour and can rent as much capital as it wants at a price of $2 per hour. Solve for the optimal input combination to produce 10 units of output. For arbitrary values of w, r, and Q, solve for the (long run) input demand curves. Suppose that the firm's capital is fixed in the short run at K. Find the firm's short run demand for labor.

Explanation / Answer

a) Min L+2K subject to 10 = L1/3K2/3

1000/K2 +2K

Differentiate this w.r.t K

-2000/K3 + 2 = 0

K = 10

L= 10

B) Minimize wL+rK s.t Y=L1/3K2/3

Differentiate w(Y)3/K2 + rK wrt K

-2wY3/K3 + r = 0

2wY3 /r = K

L = (Y3)r2/ 4w2Y6

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