Question #201671

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 0.75L1/2 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?


1
Expert's answer
2021-06-02T12:08:45-0400

Solution:

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

MRTS = MPLMPK=wr\frac{MP_{L} }{MP_{K}} = \frac{w }{r}


Q = 0.75L1/2 K 1/2

MPL = QL=0.375L1/2K1/2\frac{\partial Q} {\partial L} = 0.375L^{-1/2} K^{1/2}


MPK = QL=0.375L1/2K1/2\frac{\partial Q} {\partial L} = 0.375L^{1/2} K^{-1/2}


MRTS = 0.375L1/2K1/20.375L1/2K1/2=KL\frac{0.375L^{-1/2} K^{1/2} }{0.375L^{1/2} K^{-1/2}} = \frac{K}{L}


MRTS = KL\frac{K}{L}


KL=wr\frac{K}{L} = \frac{w}{r}


KL=616\frac{K}{L} = \frac{6}{16}


KL=0.375\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

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