Consider a numerical example for the firm's profit maximization problem. Suppose that the firm's production function is y=ln(K) +ln(N) . The firm's initial capital is K= 1.5. The current period wage rate is w= 0.1, the future period wage rate is w′= 0.1. The depreciation rate is d = 0.3 and the market real interest rate is r = 0.2. The price at which the firm can sell leftover undepreciated capital in the future period is 1.
A. Write down the firm's optimization problem. Take the first order conditions with respect to the first period labor N , the future period labor N' and the future period capital K' to get the firm's optimal decisions of N,N ' and K'.
B. Use these answers to compute profit today and tomorrow π ,π′ and the value of the firm, V.
C. Now suppose the current period wage rate increases from w= 0.1 to w2= 0.2. If the firm does not re-optimize, compute its profit in the current period given the new wage, and compare it with the profit from part b. Now compute the new optimal level of labor N2 . What is the firm's profit at the new wage and new level of labor?
D. Now suppose that the wage is unchanged, but instead the real interest rate increases from r= 0.2 to r2= 0.3. Compute the value of the firm at the new interest rate if it does not re-optimize and compare it to the value obtained in point b. Now compute the new optima level of investment I2 and capital new K′2 and use these values to compute the new value of the firm V2 . Compare this value to the value obtain in part b and discuss any differences