Consider a numerical example for the firm's profit maximization problem. Suppose that the firm's production function is y = lnK + lnN. The firm's initial capital is K = 3. The current period wage rate is w = 0.1, and the future period wage rate is w' = 0.1. The depreciation rate is d = 0.2, and the market real interest rate is r = 0.05. The firm can sell leftover undepreciated capital in the future period at a price 1 (as in class). (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' and obtain values for firm's optimal choices of N, N', K’, and investment I. Compute profit today and tomorrow, n and t', and the value of the firm, V. (b) Now suppose that the real interest rate decreases from r = 0.05 to rnew = 0.01. Compute the new optimal levels of investment Inew and capital Khow and use these values to compute the new value of the firm Vnew. Compare these values to the values obtained in part (a) and discuss any differences.

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