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)
Solution:
a.). The optimal amount of labor and capital is where:
Q = 10L0.5K0.5
MPL = 5L-0.5K0.5
MPK = 5L0.5K-0.5
w = 1
r = 4
L = 4K
20 = 10L0.5K0.5 = 10(4K0.5) K0.5
K = 1
L = 4K = 4 1 = 4
Labor = 4 units
Capital = 1 unit
b.). Minimum cost is where: C = wL + rK
C = (1 4) + (4 1) = 4 + 4 = 8
The minimum cost = 8
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