Question

 Suppose you have used the following Production Function to estimate the Industry’s average and marginal products for its inputs:

Q = 150 L^1/4K^1/2M^1/5.

      Where Q stands for output; L is labor; K is capital (machine hors) and M is management;

Say, you currently have 120 full-time employees, and 20 Managers and use 200 units of machine hours per day.

1. What are the average product of your labor and managers at current level of inputs?

2. What are the marginal products of labor and managers?

3. Does this industry currently experience decreasing returns to scale or increasing returns to scale? Why?

4. If the firm decides to open another factory in Mexico anticipating a huge increase in the demand for its products which of the factors (all or any) would be most hired and why?

Answer 1

When K = 200,

Q = 150 L^1/4(200)^1/2 M^1/5 = 150 x 14.14 x L^1/4M^1/5 = 2,121 x L^1/4M^1/5

(a)

APL = Q / L = (2,121 x M^1/5) / (L^3/4) = [2,121 x (20)^1/5] / [(120)^3/4] = (2,121 x 1.82) / 36.26 = 106.47

APM = Q / M = (2,121 x L^1/4) / (M^4/5) = [2,121 x (120)^1/4] / [(20)^4/5] = (2,121 x 3.31) / 10.99 = 639.06

(b)

MPL = Q / L = [(1/4) x (2,121 x M^1/5)] / (L^3/4) = [530.25 x (20)^1/5] / [(120)^3/4] = (530.25 x 1.82) / 36.26 = 26.61

MPM = Q / M = [(1/5) x (2,121 x L^1/4)] / (M^4/5) = [424.2 x (120)^1/4] / [(20)^4/5] = (424.2 x 3.31) / 10.99 = 127.76

(c)

When all inputs are doubled, new production function is

Q* = 150 x (2L)^1/4(2K)^1/2(2M)^1/5 = 150 x (2)^1/4 x (2)^1/2 x (2)^1/5 x L^1/4K^1/2M^1/5 = (2)^0.95 x [150 x L^1/4K^1/2M^1/5 ]

= 1.93Q

Q*/Q = 1.93 < 2

Since doubling all inputs less than doubles output, there are decreasing returns to scale.

(d)

MPK = Q/K = [150 x (1/2) x L^1/4M^1/5] / (K^1/2) = [75 x (120)^1/4 x (20)^1/5] / [(200)^1/2] = (75 x 3.31 x 1.82) / 14.14

= 31.95

Since Management has highest marginal product (MPM being highest), Management should be used in the expansion.