Question

Suppose a firm's production function is given by LaTeX: Q\left(L,K\right)=4L^{0.65}K^{0.35}Q ( L , K ) = 4 L^0.65 K^0.35. The wage is w= 25/ hour and the rent for capital is r= 25/hour. To produce 350 units per hour, what is the minimum hourly cost of production? Enter to the nearest $0.1. [number only, no $ sign]

Answer 1

We have Q ( L , K ) = 4 L^0.65 K^0.35

w= 25/ hour

r= 25/hour.

Q=350

So, 4 L^0.65 K^0.35=350

L^0.65 K^0.35=87.5

L=(87.5/K^0.35)^(1/0.65)=972.086019023/K^0.53846153846

We have to minimize 25 L + 25 K = 25*972.086019023/K^0.53846153846+25K

Deriving wrt K, we have 25-13085.773333/k^1.53846153846=0

k=(1/0.00191047172)^(1/1.53846153846) = 58.5138

L=972.086019023/58.5138401982^0.53846153846 = 108.66855

So, the minimum cost an hour will be 25*(K+L) = 25*(58.5138+108.66855) = $4179.56 (Ans)