PLEASE SOLVE ALL OF THIS, MY HOMEWORK IS DUE TOMORROW.

The output of the representative firm in period t is given by Yt = At Kt where At is productivity and Kt is capital in period t. Denote the marginal product of capital in period t by MPKt and note that MPKt = At 2 Kt . There are two periods with productivity A1 and A2 for the first and second period, respectively. K1 is the capital stock of period 1 and is the outcome of past decisions. K2 is the capital stock of period 2 and it is determined by K2 = (1 d)K1 + I where d is the depreciation rate and I is the firms period 1 investment. At the end of period 2, the undepreciated capital (1 d)K2 is sold at a price of 1 per unit of capital. In period 1 the firm decides how much to invest, taking as given the interest rate R and its expectations about future productivity A2. The firms goal is to maximize the intertemporal value of the firm which is equal to the discounted stream of profits: V = 1 + 2 1+R where 1 = Y1 I and 2 = Y2 + (1 d) K2. The firms period 1 capital stock is K1 = 25, the depreciation rate is d = 0.1, the interest rate is R = 0.25 and productivity is given by A1 = 4 and A2 = 4.

1. Calculate the level of output in period 1.

2. Suppose the firm increases its investment by one unit. Describe in words and equations (i.e. without calculating an actual number) the change in first period output, first period profits, second period output and second period profits.

3. Based on your previous answer write down the first period marginal cost and marginal benefit of an additional unit of investment, MC(I) and MB(I).

4. Equate the marginal cost and marginal benefit of additional investment to find the optimal level of capital for the second period as a function of productivity, the depreciation rate and the interest rate. Use the numbers for R, d and productivity to calculate the second period level of output and profits.

5. Calculate first period investment and first period profits. Calculate the firms value. Suppose that the interest rate drops to R0 = 0.2. 6. Explain in words how this change will affect the firms investment decision.

7. Calculate how much the firm will invest, I 0 , after the change in the interest rate. Calculate the second period level of output, Y2.

8. Suppose that A1 drops but A2 remains the same. Describe in words how first period output, first period investment and first period profits are affected by this change.

9. Suppose that A1 remains the same but A2 increases. Describe in words how Y1, I, 1, Y2 and 2 are affected.

10. A journalist from the business press observes the drop in first period profits and reports that the decline in profits is due to the firms ambitious investment programme. Suppose that you cannot directly observe A1 or the firms investment. What other information might be useful for assessing whether the journalists opinion is correct?