Question

Suppose a firm produces an output level according to the simple production function: Q = 5 L K, which implies M P L = 5 K and M P K = 5 L.

Further suppose a firm must pay labor (L) a wage rate (w) of $5 per unit, and the rental rate (r) on capital (K) is $25 per unit.

A. Find the marginal rate of technical substitution.

B. Write the equation for the isocost line. What is the slope?

C. Given the above information, find the cost-minimizing combination of labor and capital for an output level of Q = 2 , 500. What is the total cost of this combination of inputs?

D. Draw a graph of an isocost line and isoquant representing your answer to part (C).

Answer 1

(A)

MRTS = MPL/MPK = 5K / 5L = K/L

(B)

Isocost equation: C = wL + rK

C = 5L + 25K

Slope of isocost = - w/r = - 5/25 = - 0.2

(C)

Cost is minimized when MRTS = w/r = 5/25 = 1/5

K/L = 1/5

L = 5K

Substituting in production function,

5LK = 2,500

LK = 500

5K x K = 500

K² = 100

K = 10

L = 5 x 10 = 50

Total cost = 5 x 50 + 25 x 10 = 250 + 250 = 500

(D)

Isocost equation for case (C): 500 = 5L + 25K

100 = L + 5K

When L = 0, K = 100/5 = 20 (vertical intercept) and when K = 0, L = 100 (horizontal intercept).

In following graph, AB is the isocost line which is tangent to isoquant Q0 (= 2,500) at point E with optimal input mix (L0, K0) = (50, 10).

Ke 10 100 Lo