Question

A firm’s production technology is given by the production function q = 0.25 LK where L represents labor hours, K machine hours and q the amount of output. The market wage and rental rates are, w= $16 and r = $256. The firm is operating in the long run where it can adjust both inputs.

(a) Suppose that the firm currently is using ten labor hours for each machine hour. Is it minimizing its long run total cost? If so why so and if not why not? Explain. If it is not minimizing its long run cost, how should it adjust its input usage? Explain. Provide appropriate calculations.

(b)Suppose that the firm wants to produce 100 units of output. Determine the cost minimizing combination of L and K. Calculate the resulting long run total cost. Show and explain all calculations.

Answer 1

q = 0.25LK

(a) Long run cost is minimized when MPL/MPK = w/r, or MPL/w = MPK/r

MPL = $\dpi{80} \partial$ q/$\dpi{80} \partial$L = 0.25K

MPK = $\dpi{80} \partial$ q/$\dpi{80} \partial$K = 0.25L

When cost is minimized,

MPL/MPK = K/L = 16/256 = 1/16, therefore

MPK/MPL = L/K = 16

However, current value of (L/K) = 10 (given) < 16, so firm is not minimizing cost.

To use optimal input ratio, firm has to use more labor and less capital until (MPL/MPK) = (w/r).

(b) When q = 100, we have

100 = 0.25LK

LK = 400.......(1)

Cost-minimizing condition: (MPL/MPK) = K/L = w/r = 16/256 = 1/16

L = 16K

Substituting in (1),

16K x K = 400

K² = 400/16 = 25

K = 5

L = 16 x 5 = 80

Total cost ($) = wL + rK = 16 x 80 + 256 x 5 = 1280 + 1280 = 2560