Question

Suppose, the production function for X company is given by ? = 5(??) 0.5 where, Q is the amount of output produced, K is the amount of capital employed in production and L is the amount of labor employed in production. The prices of capital and labor are given by ?? = $48 and ?? = $75.

a)Express the total cost in terms of K and Q.

b)Derive the expression of marginal cost of capital.

c)Derive the long-run cost function in terms of Q.

Answer 1

a)

We are given

Squaring both sides

Total Cost, TC is given as

TC=P_k*K+P_L*L

b)

In this case, price of capital is constant. So,

Marginal Cost of capital=PK=$48

Marginal cost of output keeping K constant is given as

MC of product=dTC/dQ=6Q

c)

In long run

K/L=25/16

K=(25/16)*L

Set K=(25/16)*L in production function

or L=0.8Q

K=(25/16)*L=(25/16)*0.8*Q=1.25Q

Long run cost is given as

LRTC=KP_K+LP_L

LRTC=(1.25Q)*48+(0.8Q)*75

LRTC=120Q