Question

Suppose a certain contractor wants to maximize profit from building one bridge. The

contractor uses both labor and capital, and efficient combinations of Labor and capital

that are sufficient to make a bridge is by the function 0.75L^1/2 K ^1/2. If the prices of labor

(w) and capital (r) are $ 6 and $ 16 respectively

A) Find the least cost combination of L and K?

B) Find the minimum cost?

Answer 1

Solution:

A.). The least cost combination of L and K is where:

MRTS = $\frac{MP_{L} }{MP_{K}} = \frac{w }{r}$

Q = 0.75L^1/2 K ^1/2

MP_L = $\frac{\partial Q} {\partial L} = 0.375L^{-1/2} K^{1/2}$

MP_K = $\frac{\partial Q} {\partial L} = 0.375L^{1/2} K^{-1/2}$

MRTS = $\frac{0.375L^{-1/2} K^{1/2} }{0.375L^{1/2} K^{-1/2}} = \frac{K}{L}$

MRTS = $\frac{K}{L}$

$\frac{K}{L} = \frac{w}{r}$

$\frac{K}{L} = \frac{6}{16}$

$\frac{K}{L} = 0.375$

K = 0.375L

The least cost combination of L and K is where: K = 0.375L where the contractor uses 0.375 as much capital as labor.

B.). The minimum cost:

C = wL + rK

C = 6L + 16K

K = 0.375L

C = 6L + 16(0.375L)

C = 6L + 6L

C = 12L

The minimum cost: C = 12L