.Consider the following production function,

Y={(1-u)L^b_h +uL^b_c}^1/b

where is a constant parameter with a value between 0 and 1, 0L_h stands for labor hours with high school degree, and L_c denotes labor hours with college degree. There are two types of technological change: is an aggregate neutral technological change; and captures technological change which complements college labor but not high school labor. In other words, captures skill-biased technological change.

Denote by P the price of one unit of output. Also, denote by W_h and W_cwage rates payed to high school labor and college labor, respectively.

a. Formulate the profit maximization problem for a perfectly competitive firm that hires two types of labor inputs, L_c and L_h, at respective wages W_c and W_h , to produce output Y using the production function described above (equation (1)).(8 points)

b. Derive the first order optimality conditions (explicitly derive expressions for marginal products of factor inputs). (16 points)

c. Using the optimality conditions obtained in part b, derive the expression for the college wage premium . What does the college wage premium depend on and how (positively or negatively)? (8 points)

d. In Homework 2 (see the pdf on iCollege), Question 6, Figure 1 shows the U-shaped evolution of the college wage premium in the United States from 1915-2005. Figure 2 shows that the relative supply of college workers was characterized by an upward trend during the same period. Can you use the expression for the college wage premium you derived in part c above to explain the observed U-shape? Explain the intuition (hint: think about skill-biased technological change (variable in the production function) and how it may affect the relative demand of high skill workers). (8 points).