Determine his optimal choice of consumption and leisure and

Question

1. An expansion of production function is to separate workers based on their quality as below.

Y = z(K + 15000L_u)^0.6L^0.4,

where Y is output, z is TFP, K capital, L_u unskilled labor and L_s skilled labor.

The expression for the marginal product of skilled and unskilled labor are given below:

MP L_u = 0.6·z(K + 15000L_u)^-0.4L^0.4 ·15000

MP L_s = 0.4·z(K + 15000L_u)^0.6L^-0.6

In this production function capital and skilled labor complement each other, while capital and unskilled labor are substitutes. The picture you can have is that a trained operator using a bulldozer replacing a group of brawny man with shovels.

(a) In year 1990, z = 500, K = 30000, L_s = 1, L_u = 1. Calculate the marginal product of each labor type.

(b) In year 2010, z = 700, K = 65000, L_s = 1, L_u = 1. Calculate the marginal product of each labor type.

(c) Calculate the accumulated percentage increase of marginal product for each type of labor during this 20 year period.

2. A firm has production function Y = 150N^0.6. Suppose that w = 30.

(a) How much labor will this firm hire and how much can it produce?

(b) Suppose that the government subsidizes employment. That is, the government pays the firm 1 units of consumption goods for each unit of labor that the firm hires. (Direct subsidy to firms is rare. There are many federal and state job creation related grants and tax breaks that indirectly lower the cost of hiring.) Determine the effect of the subsidy on the firms hiring decision. (Adapted from chapter 4 page 135, Q12.)

(c) In the course of producing its output, this firm causes pollution. The government passes a law that requires the firm to stop polluting, and the firm discovers that it can prevent the pollution by hiring 0.2 workers for every worker that is producing output. Let N denote the amount of labor in production. The total employment at the firm is 1.2N . Cost of hiring a production worker includes both his wage and the wage of the worker who cleans after him, hence is 1.2w. Determine the effect of the pollution regulation on the firms profit-maximizing choice of labor input. (Adapted from chapter 4 page 136 Q15.)

3. Nemo has a preference where consumption and leisure time are perfect substitutes. His utility function is given by u(c, l) = c + 0.6l. Suppose that w = 0.7 and Π = 0.85. Total time endowment is normalized to be 1. Determine his optimal choice of consumption and leisure.