A firm has the following production function f(K, L)= K 1/3 L 2/3 . Where K is c

Question

A firm has the following production function f(K, L)= K1/3L2/3. Where K is capital and L is labor. It will produce 80 units of output and faces prices for labor and capital as follows:wk= 15, wL=10. Find the cost miniminizing bundle of labor and capital.

A firm has the following production function f(K,L)-K½?2/3, where K is capital and L is labor. It will produce 80 units of output and faces prices for labor and capital as follows: WK -15, WL 10. Find the cost minimizing bundle of labor and capital 1.

Explanation / Answer

From the production function, MRTS = MPL/MPK = (2/3)(K/L)^(1/3) divided by (1/3)(L/K)^(2/3) which gives 2K/L. At the optimum input mix, MRTS = wL/wK so we have 2K/L = 10/15 or L = 3K. For 80 units, we have 80 = K^1/3 L ^2/3

80 = K^1/3 (3K)^2/3

80 = 3^2/3 K

K = 38.46

L = 115.38

This is the cost mininizing bundle.

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