Question

a. A cost minimizing firm’s production is given by Q=L1/2K1/2. Suppose the desired output is Q=10. Let w=12 and r=4. What is this firm’s cost minimizing combination of K & L? What it the total cost of producing this output?

b. Suppose the firm wishes to increase its output to Q=12. In the short run, the firm’s K is fixed at the amount found in (a), but L is variable. How much labor will the firm use? What will the total cost be?

c. With Q=12, what is the firm’s cost minimizing combination of K & L in the long-run? What it the total cost of producing this output?

Answer 1

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

Total cost (C) = wL + rK

C = 12L + 4K

(a) When Q = 10, we have

L^1/2K^1/2 = 10

LK = 100 (Squaring both sides)

Cost is minimized when MPL/MPK = w/r = 12/4 = 3

MPL = Q/L = (1/2) x (K/L)^1/2

MPK = Q/K = (1/2) x (L/K)^1/2

MPL/MPK = K/L = 3

K = 3L

Substituting in production function,

L x 3L = 100

L² = 100/3 = 33.33

L = 5.77

K = 3 x 5.77 = 17.31

C = 12 x 5.77 + 4 x 17.31 = 69.24 + 69.24 = 138.48

(b) When Q = 12,

LK = 144

Since K = 17.31,

L = 144/17.31 = 8.32

Total cost = 12 x 8.32 + 4 x 17.31 = 99.84 + 69.24 = 169.08

(c) When Q = 12,

LK = 144

Substituting K = 3L,

L x 3L = 144

L² = 144/3 = 48

L = 6.93

K = 3 x 6.93 = 20.79

Cost = 12 x 6.93 + 4 x 20.79 = 83.16 + 83.16 = 166.32