Question

A firm produces output using the technology:

Q = 5K^0.33L^0.5

where capital, K, is measured in machine-hours, labor, L, is measured in person-hours, and Q denotes the yearly output. The hourly wage rate in China W_L = $10, and the hourly rental rate of capital (IN EFFECT CAPITAL COSTS $2 PER MACHINE HOUR) is W_K = 2 and the Price is $10.

a.) Does this production function display increasing returns to scale, constant returns to scale or decreasing returns to scale(hint—use labor).Please show how you arrived at this conclusion (this requires a mathematical answer that you must show).

b.) Compute the marginal products of labor and capital.Show that there are diminishing, but positive, returns to the labor input.

(c) Suppose that the firm has signed a contract to rent K = 800 machine hours over the course of the year. Determine the equilibrium level of Labor.

(d) If the wage level in England is $4, (given the same MP_L in China), what is the equilibrium level of Labor demanded in England. If wages in England fall to $2, what is the new level of demand for labor in England? How can the workers in the US combate this potential transfer of jobs to England (there are two possible solutions).  

Answer 1

A firm produces output using the technology Q = 5K^0.33L^0.5. The hourly wage rate in China W_L = $10, and the hourly rental rate of capital in per machine hour is W_K= 2 and the Price is $10.

a.) This production function display decreasing returns to scale

Increase inputs by multiplying them with 2

New Q = 5(2K)^0.33 (2L)^0.5

= (2^0.33)(2^0.5) x 5K^0.33L^0.5

= 1.77 x Old Q

Since output is increased by 1.77 while inputs were increased by 2, it implies that output is increased with a lower proportion than the proportion of increase in inputs. There are decreasing returns to scale.

b.) MPL = dQ/dL = 5*0.5K^0.33L^-0.5. Similarly, MPK = dQ/dK = 5*0.33K^-0.67L^0.5. Marginal product of labor is a decreasing function of L but increasing function of K, and marginal product of capital is a decreasing function of K but increasing function of L. This is because there partial derivates MPL’(L) and MPK’(K) are negative but the partial derivates MPL’(K) and MPK’(L) are positive.

(c) Now K = 800 machine hours. With K being fixed, the equilibrium level of Labor is found by following the rule VMPL = wage

2.5(800^0.33)L^-0.5 x 10 = 10

L^0.5 = 2.5(800^0.33)

L* = 515 (approx.)

(d) If the wage level in England is $4, (given the same MP_L in China), the equilibrium level of Labor demanded in England is

2.5(800^0.33)L^-0.5 x 10 = 4

L^0.5 = 6.25(800^0.33)

LE = 3220 (approx.)

If wages in England fall to $2, the new level of demand for labor in England is

2.5(800^0.33)L^-0.5 x 10 = 2

L^0.5 = 12.5(800^0.33)

LE = 12878 (approx.)

The workers in the US combat this potential transfer of jobs to England by increasing there marginal product of labor so that VMPL rises.