Question

A firm produces output using the technology:

Q = (0.001)KL0.5

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 the United States WL = $10, and the hourly rental rate of capital is WK = 20 and the Price is $1.

(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). (6 Points)

(b) Compute the marginal products of labor and capital. Show that there are diminishing, but positive, returns to the labor input. (6 Points)

(c) Suppose that the firm has signed a contract to rent K = 1,000 machine hours over the course of the year. Determine the equilibrium level of Labor. (6 Points)

(d) If the wage level in China is $2, (given the same MPL in the US), what is the equilibrium level of Labor demanded in China. If wages in China fall to $1, what is the new level of demand for labor in China? (2 Points) How can the workers in the US combate this potential transfer of jobs to China (there are two possible solutions). (2 Points)

Answer 1

Q = (0.001)KL^0.5

W_L =10; W_K = 20

P = 1

1. If an inputs (capital & labor) is augmented A>0 times

Then Q₁ = 0.001 * AK * (AL)^0.5 = A^{0.5 + 1}(0.001)KL^0.5 = A^1.5 Q

Now , since A^1.5 > A ; hence Q(AK, AL) > A * Q(K,L)

Hence , the increase in factors leads to more than proportional increase in output . Hence there is increasing returns to scale.

2. Marginal productivity of labor = MPL = dQ/dL = 0.0005KL^-0.5

And MPK = dQ/dK = 0.001L^0.5

Now,

dMPL/dL = -0.00025KL ^-1.5 < 0

Hence increase in labor decreases the MPL thus diminishing marginal product of labor

3. K = 1000

When at equilibrium; wage = value of MPL = P *MPL

= > 10 = 1 * 0.0005KL^-0.5

= > 10 = 0.0005 *1000 *L^-0.5

= > 10= 0.5 * L^-0.5

= > L^0.5 = 0.05

= > L = 0.05²

4. Given the same levl of MPL

And wage = 2

= > 2 = 1 * 0.0005KL^-0.5

= > 2 = 0.0005*1000*L^-0.5

= > 2 =0.5* L^-0.5

= > L = 0.25²

Wage = 1

= > 1= 1 * 0.0005KL^-0.5

= > 1 = 0.0005*1000*L^-0.5

= > 1 =0.5* L^-0.5

= > L = 0.5²

US can stop the transfer of workers potential by lowering its wages or by increasing the MPL on the country by developing technology to make labor and capital more efficient.