The output of the representative firm in period t is given by Yt = At ...

   

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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 firm's 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 firm's goal is to maximize the intertemporal value of the firm which is equal to the discounted stream of profits: V = 1 + 12 1+R where П1 = Y1 - 1 and 2 = Y2 + (1 - d) * K2. The firm's period 1 capital stock is K1 = 10, the depreciation rate is d = 0.15, the interest rate is R = 0.3 and productivity is given by A1 = 3 and A2 = 3. 1. Calculate the level of output for period 1. Calculate the optimal level of investment in two steps: First calculate the optimal level of second period capital by equating the marginal cost and marginal benefit of additional capital; Second, calculate how much investment is necessary to reach that level of second period capital? 2. Calculate first period profits. Calculate second period output and second period profits. Calculate the value of the firm V? 3. The firm makes an important technological discovery which will increase its future productivity but does not affect its current productivity, i.e. A2 increases and A1 remains constant. Describe how investment will be affected by this change. What about first period output and first period profits? How about second period profits and the firm's value? 4. A financial journalist comments that if a firm's profits drop, then its value must also fall. Does this statement hold under all circumstances? At time t = 1 the capital stock of the economy is K1 = 500 and that the optimal level of capital for the next period is the same: K2 = 500 (i.e. the economy is at steady state). The depreciation rate is d = 0.15.? 5. Calculate the level of investment in period 1, 11.? (NOTE: you only need to use the transition equation for capital). 6. Suppose that the optimal level of capital for all subsequent periods (t = 3, t = 4...) is Kt = 500. Calculate how much will be invested in periods 2 and 3 (12 and 13) and the proportional change of investment between periods 1 and 2 and between periods 2 and 3? 7. Draw a graph with time on the horizontal axis and investment on the vertical axis and show the evolution of investment over time? 8. Suppose that at time t = 2 expectations about future productivity improve and the optimal level of capital for all subsequent periods (t = 3, t = 4...) is KO t = 550. Calculate the level of investment for periods 2, 3 and 4 (102, 103, 104) and the growth of investment (i.e. the proportional change in investment) between periods 1 and 2, periods 2 and 3 and period 3 and 4? 9. Draw a graph with time on the horizontal axis and investment on the vertical axis and show the evolution of investment over time under this scenario? 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 firm's 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 firm's goal is to maximize the intertemporal value of the firm which is equal to the discounted stream of profits: V = 1 + 12 1+R where П1 = Y1 - 1 and 2 = Y2 + (1 - d) * K2. The firm's period 1 capital stock is K1 = 10, the depreciation rate is d = 0.15, the interest rate is R = 0.3 and productivity is given by A1 = 3 and A2 = 3. 1. Calculate the level of output for period 1. Calculate the optimal level of investment in two steps: First calculate the optimal level of second period capital by equating the marginal cost and marginal benefit of additional capital; Second, calculate how much investment is necessary to reach that level of second period capital? 2. Calculate first period profits. Calculate second period output and second period profits. Calculate the value of the firm V? 3. The firm makes an important technological discovery which will increase its future productivity but does not affect its current productivity, i.e. A2 increases and A1 remains constant. Describe how investment will be affected by this change. What about first period output and first period profits? How about second period profits and the firm's value? 4. A financial journalist comments that if a firm's profits drop, then its value must also fall. Does this statement hold under all circumstances? At time t = 1 the capital stock of the economy is K1 = 500 and that the optimal level of capital for the next period is the same: K2 = 500 (i.e. the economy is at steady state). The depreciation rate is d = 0.15.? 5. Calculate the level of investment in period 1, 11.? (NOTE: you only need to use the transition equation for capital). 6. Suppose that the optimal level of capital for all subsequent periods (t = 3, t = 4...) is Kt = 500. Calculate how much will be invested in periods 2 and 3 (12 and 13) and the proportional change of investment between periods 1 and 2 and between periods 2 and 3? 7. Draw a graph with time on the horizontal axis and investment on the vertical axis and show the evolution of investment over time? 8. Suppose that at time t = 2 expectations about future productivity improve and the optimal level of capital for all subsequent periods (t = 3, t = 4...) is KO t = 550. Calculate the level of investment for periods 2, 3 and 4 (102, 103, 104) and the growth of investment (i.e. the proportional change in investment) between periods 1 and 2, periods 2 and 3 and period 3 and 4? 9. Draw a graph with time on the horizontal axis and investment on the vertical axis and show the evolution of investment over time under this scenario?

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ANSWER 1 The level of output for period 1 is given by Y1 A1 K1 3 10 949 To find the optimal level of investment we need to maximize the intertemporal value of the firm Lets start by finding the optima... View the full answer