Question

The Ray Hill T-Shirt Company makes t-shirts to sell at the Nashville Sounds baseball games. They produce these shirt using Labor (L) and capital (K) according to the following production function:

Q = 300K^1/3L^1/2

Assume that the price of labor (w) is $30 and the price of capital (r) is $80.

a. The company gets an order for 19,200 t-shirts. What is the cost-minimizing combination of labor and capital that the company would use in order to produce 19,200 units of output?

b. What is the cost of producing these 19,200 shirts?

Answer 1

Q = 300K1/3L1/2

then Q=19200

19200=300*(K)^(1/3)*(L)^(1/2)

64=K^(1/3)*L^(1/2)

64^6=K^2*L^3...Headline Equation

Profit Function =@(L,K)=P*-(30*L+80*K)

d@/dL=d(P*300K1/3L1/2-30L-80K)/dL

300P(1/2)*K(1/3)*L^(-1/2)-30=0

P*K^(1/3)*L^(-1/2)=2/10=0.2...eq 1

d@/dK=d(P*300K1/3L1/2-30L-80K)/dK

=P*100K(-2/3)*L(1/2)-80=0

P(L^(0.5)*K^(-2/3))=0.8....eq 2

K/L=1/4=0.25

K=0.25L...equantion 3

From Headline Equation we have hence using equation 3

64^6=K^2*L^3=(0.25L)^2*L^3=0.625*L^5

(64^6/0.625)^(1/5)=L=161.53

K=0.25*161.53=40.38

Therefore Cost of producing 19200 shirts is

wL+rK=30*161.53+80*40.38=$8076.3=$8076