When disucssing the business cycles, and introducing the IS curve, we stated that investment demand is the most volatile part of expenditure. In this exercise, you are going to work through an example that helps explaining why investment might be so volatile, and sheds some light on how the IS curve is based on the actual optimizing decisions made by firms. Consider a simple model of a representative firm, similiar to the one we discussed in Chapter 4. The firm currently has a stock of capital K and has to decide about its stock of capital in the next period (say, year- let's call it period 2), K'. The firm determines the desired level of K' based on two parameters: expected future productivity z, and the real interest rate R it faces. Once the firm decides how much capital next period it wants (what is the desired level K'), the firm undertakes investment I to achieve this level of capital. K' is determined through a standard law of motion for capital, like the one we used in the Solow model: where is the depreciation rate. K' = (1-8)K+I Next period, the firm uses the capital stock K' it achieved to produce output Y using a Cobb-Douglas production function: Y = (K') - we assume that the labor input N is constant over time, so we don't have to worry about it. From Chapter 4, we know that the marginal product of capital (MPK) for this production function is given by: MPK = az (K')a-1. It can be shown that the the optimal amount of capital is given by the standard condition: MPK = R a. Use the optimality condition (MPK = R) to derive the optimal level of future capital K' for this firm as a function of parameters and prices (K, a, z, R, and 8). This should take the form of an equation where you have K on the right-hand side, and all the parameters on the left-hand side. Does the optimal amount of capital in period 2 (K'), depend on the initial value of capital (K)? b. Use the result from part a) and the law of motion for capital to solve for optimal investment (I) the firm should do between periods 1 and 2. c. (For the last two parts, the numbers are not going to be very round) In the remaining part of this problem, assume the following values of parameters: a = 0.3, and 8 = 0.1. Suppose that the firm starts with K = 25 units of capital. Suppose that in the second period, the value of z the firm expect to happen is equal to z = 1. If the real interest rate equals R = 0.03, how much capital will the firm want to have in Period 2? Given the initial value of capital stock, and the value of depreciation parameter 5, what will be the investment demand of this firm? d. Now, suppose that the firm faces a slightly smaller interest rate of R = 0.025. Other parameters remain the same as in the previous part of the exercise. What will be the new optimal level of capital the firm is going to choose in period 2? To get to that level, how much will the firm have to invest in Period 1? Compare the percentage changes in the desired levels of capital and in investment demand between points c) and d). What can you tell about changes in the size of investment relative to the underlying change in the real interest rate? e. Suppose that the firm faces the same interest rate as in part c) (R = 0.03), but instead becomes more optimistic about its productivity in Period 2, expecting z = 1.2. How will it change the optimal level of capital, and investment relative to parameter values in part c) of the problem? Compare the percentage changes in the desired levels of capital and in investment demand between points c) and e). What can you tell about changes in the size of investment relative to the underlying change in the expected productivity?