Solve the constrained optimization problem: max py rk wl subject to k ^a l^ b y, where 0 < a < 1, 0 < b < 1, and 0 < a + b < 1, and take p, r and w as given.

a. Set up the Lagrangian for this problem, letting denote the multiplier on the constraint. b. Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values y , k , and l that solve the firms problem, together with the associated value for the multiplier. c. Assume that the constraint binds at the optimum (can you tell under what conditions this will be true?), and use your results from above to solve for y , k , l , and in terms of the models parameters: a, b, p, r, and w. d. If we interpret y as the product produced by a firm, k as the capital and l as labor used by the firm. Then p can be viewed as the unit price of product sold by the firm, r can be viewed as the rents of each unit of capital, and w can be viewed as the wage for each unit of labor. Then use your solutions from above to answer the following questions: i. What happens to the optimal y , k , and l when the product price p rises, holding all other parameters fixed? In each case, does the optimal choice rise, fall, or stay the same? ii. What happens to the optimal y , k , and l when the rental rate for capital r rises, holding all other parameters fixed? iii. What happens to the optimal y , k , and l when the wage rate w rises, holding all other parameters fixed? iv. What happens to the optimal y , k , and l when p, r, and w all double at the same time?