Question

Consider the Leontief production function F(KL) = min {K,L], where capital K and labor L have respective positive input prices r and w. (a) Why is it that the cost-minimizing firm sets K 5. L? (b) What is the cost function? (c) How would your answer to part (b) change, if at all, if rw 0? Explain.

Answer 1

Q = F(K, L) = min{K, L}

(a)

This is a Leontief production function where K and L are used in fixed proportions. For 1 unit of K, 1 unit of L has to be employed in order to produce any output greater than zero. The expansion path will be a straight ray from origin that cuts across the inflection points of the isoquants (points where K = L). That is why cost-minimizing input mix is obtained by setting K = L.

(b)

When K = L,

Q = min{L, L} = L

K = L = Q

Total cost (C) = wL + rK

C = wQ + rQ

C = Q x (w + r) [Cost function]

(c)

If w = 0 and r > 0, it means r is positive.

C = Q x (0 + r)

C = Q x r [Cost function]