Question

Answer the following questions on the basis of the table shown here. Q_B is type B labor, and VMP_Bx and VMP_By are the industry values of the marginal products of this labor in producing x and y, the only two goods in the economy.

a. Explain why the VMPs in the table decline as more units of labor are employed.

b. If the supply price or opportunity cost of labor P_L is $9, how many units of type B labor need to be used in producing x and y to achieve an efficient allocation of labor? What will be the combined total value of the two outputs?

c. Suppose P_L is $15 and that presently five units of labor are being allocated to producing x while two units are being allocated to y. Is this an efficient allocation of labor? Why or why not? If not, what is the efficient allocation of type B labor?

d. Suppose P_L is $25 and three units of labor are being allocated to producing x, while six units are being allocated to producing y. Explain why this is not an efficient allocation of labor. What is the efficient allocation of this type of labor? What gain in the total value of leisure, alternative outputs, or home production results from this reallocation of labor?

e. Suppose product x is sold in a perfectly competitive product market. Also ignore the VMP_By column and assume that the VMP_Bx schedule is representative of each firm hiring workers in a perfectly competitive labor market. If the market wage rate is $12, what will be each firm’s MWC? What will be their MRPs at their profit-maximizing level of employment? Explain why an efficient allocation of labor will occur in this industry.

Answer 1

There are two goods industry where the Q_B labor is employed x and y. The value of marginal product of this labor is given below in Table.

QB	VMPBX	VMPBY
1	$18	$23
2	$15	$19
3	12	15
4	9	11
5	6	9
6	3	5

a.

At the initial level of output produced, the first unit of labor hired will be more productive. Hence, as the level of output produced increases, lower productive labor is employed. Thus, the value of marginal productivity declines as more labor is employed.

b.

The wage of labor is $9. The labor is employed till the point in the two goods industry where the wage rate equates the value of marginal product in the two industries. The labor employed in the Good x industry is 4 units where the value of marginal product in the good x industry equates the wage rate.

The labor employed in the Good y industry is 5 units where the value of marginal product in the good y industry equates the wage rate.

The total product in Good x industry when 4 units of labor is employed is the summation of marginal product at all levels of labor. Thus, the total product of Good x is calculated as given below.

The total product in Good y industry when 5 units of labor is employed is the summation of marginal product at all levels of labor. Thus, the total product of Good y is calculated as given below.

The total output in the Good x and y industry is 54+77= $131.

c.

When 5 units of labor is employed in Good x industry and 2 units of labor is employed in Good y industry, the value of marginal product in the Good x industry is 6 units of output and the value of marginal product in the Good y industry is 19 units of output. Thus this is not an efficient allocation because the value of marginal product in the two industries is unequal.

The unit of labor employed in Good y industry should be increased as the marginal product is higher and the unit of labor employed in Good x industry should be decreased as the marginal product is lower. Thus, the output in the two industries can be maximized when the labor is shifted from the production of Good y to the production of Good x.

The efficient allocation of labor is when the value of the marginal product in the two industries equates the wage rate. Thus, the labor employed in the good x industry is 2 units and the labor employed in the good y industry is 3 units.

d.

When 3 units of labor is employed in Good x industry and 6 units of labor is employed in Good y industry, the value of marginal product in the Good x industry is 12 units of output and the value of marginal product in the Good y industry is 5 units of output. The value of marginal product in the two industries at all level of unit of labor employed is greater than the wage rate that is equal to $25.

Thus, no labor is employed in the production of Good x and Good y when the wage rate is $25.

The gain in the value of leisure, alternative output or home production is total output produced by the workers employed deducted from the total wages given. The output produced by 3 units of labor is employed in Good x industry and 6 units of labor is employed in Good y industry is calculated as given below.

The total wages given is

The gain in the value of leisure, alternative output or home production is

e.

The marginal wage cost is set equal to the wage rate. Thus, the marginal wage cost in the two firms is equal to $12.

The marginal revenue product in the two markets is set equal the wage rate to maximize the profits of the form. Thus the marginal revenue product in the two industries is equal to $12.