Question

5. Suppose a firm’s production function is Q = K.25L.25. The MPK = .25K-.25L.25 and MPL = .25K.25L-.25. The price of K is 1 and the price of L is 2.

• Derive the conditional demand functions for K and L

• Derive the long-run cost function

Answer 1

5.

Q = K0.25L0.25

MPK= 0.25K-0.25L0.25

MPL= 0.25K0.25L-0.25

• For Demand functions:

MPL/MPK = Cost of labor/cost of capital

0.25K0.25L-0.25 / 0.25K-0.25L0.25 = 2/1

K0.5/L0.5 = 2

Squaring both sides

K/L= 4

K= 4L

Put value of K in production function:

Q = (4L)0.25L0.25= 40.25L0.5

L0.5= Q/40.25

Squaring both sides

L= Q2/40.5= Q2/2 Demand function of L

Use the value of L in K=4L

K= 4 x Q2/2= 2Q2 Demand function of K

• Derive the long-run cost function:

TC= Total cost= Lw+Kr, Here L is the quantity of labor, w is the price of L, K is the quantity of capital and r is the price of K.

Put value of Demand functions of L and K into Total cost:

TC= Q2/2 x w + 2Q2 x r

Put value of w and r

TC= Q2/2 x 2 +2Q2 x 1

TC= Q2 +2Q2

TC= 3Q2 (Long run cost function)