Question #283513

Suppose you have the following production function: Q =f(L,K)=10L^1/2K^1/2 in addition, the price of labor is $1 and the price of capital is $4

a) what is the optimal amount of labor and capital if you want to produce 20 units?

b) what is the level of minimum cost ? (Ans L=4 and K=1, Min C=$8)


1
Expert's answer
2021-12-29T12:30:29-0500

Solution:

a.). The optimal amount of labor and capital is where: MPLMPK=wr\frac{MP_{L} }{MP_{K} } = \frac{w}{r}

Q = 10L0.5K0.5

MPL = 5L-0.5K0.5

MPK = 5L0.5K-0.5

w = 1

r = 4

5L0.5K0.55L0.5K0.5=14\frac{5L^{-0.5}K^{0.5} }{5L^{0.5}K^{-0.5}} = \frac{1}{4}


L = 4K

20 = 10L0.5K0.5 = 10(4K0.5) K0.5

K = 1

L = 4K = 4 ×\times 1 = 4

Labor = 4 units

Capital = 1 unit

 

b.). Minimum cost is where: C = wL + rK

C = (1 ×\times 4) + (4 ×\times 1) = 4 + 4 = 8

The minimum cost = 8


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