Question

A hat manufacturing firm has the following production function with capital and labor being the inputs:...

A hat manufacturing firm has the following production function with capital and labor being the inputs: Q = min(5L,3K) (it has a fixed-proportions production function). If w is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm’s optimal inputs, long-run total cost curve, average cost curve, and marginal cost curve in terms of the input prices and Q.

b) A firm has the linear production function Q = 2L + 7K. Derive the expression for the optimal inputs and the 1ong-run total cost that the firm incurs, as a function of Q and the factor prices, w and r

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Answer #1

a. Q = min(5L,3K)
This is a fixed proportions production function. So, Q = 5L = 3K
So, optimal inputs are: L = Q/5 and K = Q/3
Long run total cost (TC) = wL + rK = (wQ/5) + (rQ/3)
Average cost (AC) = TC/Q =

Marginal cost (MC) =

b. Q = 2L + 7K
MRTS = MPL/MPK = 2/7

When MRTS > w/r, that is, w/r < 2/7 then only L will be used. So, K = 0
So, Q = 2L + 7(0) = 2L
So, L = Q/2
Optimal combination is L = Q/2 and K = 0
Total cost (TC) = wL + rk = w(Q/2) + r(0) = wQ/2

When MRTS < w/r, that is, w/r > 2/7 then only K will be used. So, L = 0
So, Q = 2(0) + 7K = 7K
So, K = Q/7
Optimal combination is L = 0 and K = Q/7
Total cost (TC) = wL + rk = w(0) + r(K/7) = rK/7

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