Question

A company has the following production function:

Q=20L0.5K0.5, Q=quantity of output produced per hour, L = amount of labor, and K= capital expended. The company’s total cost function is TC= 10L + 40K. To meet customer demand, the firm needs to produce 100 units of the product per day at the minimum cost. How many units of labor and capital are needed to meet these conditions?

Answer 1

Given

Q=20L^0.5K^0.5

Marginal Product of labor=MPL=dQ/dL=0.5*20L^-0.5K^0.5=10L^-0.5K^0.5

Marginal Product of capital=MPK=dQ/dK=0.5*20L^0.5K^-0.5=10L^0.5K^-0.5

Total Cost is given by

TC=10*L+40*K

It gives that

w=Coefficient of L=$10

r=Coefficient of K=$40

Cost minimization requires that

MPL/MPK=w/r

(10L^-0.5K^0.5)/(10L^0.5K^-0.5)=10/40

K/L=1/4

L=4K

Set L=4K and Q=100 in production function

Q=20L^0.5K^0.5

100=20(4K)^0.5K^0.5

100=40K

K=2.5 (Minimum amount of capital)

L=4K=4*2.5=10 (Minimum amount of labor)