Question

A high-tech firm produces a data processing service according to the production function Q = K + 2L.

a.) If the cost per hour of labor (L) and of machine time (K) are the same, what is the least-cost way to complete 100 data processing jobs? Will the firm ever use both inputs to complete processing jobs?

b.) What if the cost per hour of labor is twice as much as the cost of machine time, then what is the least-cost way to complete 100 data processing jobs?

Answer 1

Q = K + 2L

Total cost (C) = wL + rK

(a) w = r

C = wL + wK = rL + rK

For a linear production function, cost is minimized at one of the corner points of the linear isoquant. Either L or K will be used.

When Q = 100, we have

100 = K + 2L

When K = 0, L = 100/2 = 50 and C = w x 50 + w x 0 = 50w

When L = 0, K = 100 and C = w x 0 + w x 100 = 100w

Since cost is lower when L = 50 and K = 0, this is optimal input mix.

(b) w = 2r

C = 2rL + rK

When Q = 100, we have

100 = K + 2L

When K = 0, L = 100/2 = 50 and C = 2r x 50 + r x 0 = 100r

When L = 0, K = 100 and C = 2r x 0 + r x 100 = 100r

Since cost is equal for both input combinations, the firm will be indifferent between using 50 units of labor and no capital, or 100 units of capital and no labor.