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Homework answers / question archive / Suppose the size of the inifial inflation shock is given by 01
Suppose the size of the inifial inflation shock is given by 01. For simplicity, assume that the economy begins with the inflation rate at its steady state (Tt-1 = 7). Recall that the AS curve is given by T4 = Tt-1+?Ýi + ot, and the AD curve is Yt = ? – bm(Tt – ?). For the rest of the problem, we will assume that there are no demand shocks to the economy, so you can set ?= 0 for all t and simply ignore this term. a. Use the expressions for AS and AD curves to solve for the values of 71 and Ý , i.e. the inflation rate and output in the first period right after the shock occurred. Solve for these values in terms of paramters of the model (i.e. without plugging in any numbers). The problem is a system of two equations in two unknowns. To solve the problem means to express the values of T1 and Ý? as functions of parameters, without any terms including 71 and Ý. b. Suppose the inflation shock is only temporary and it dies out after one period (so 77 = 0 for all t > 1). Also, we continue with the assumption that we have no demand shocks (? = 0). Use the expressions for AS and AD curves to solve for the future path of inflation and output, assuming no further inflation shocks. The most convenient way to do this will be to plug in the equation for the AD curve into the equation for the AS curve. That way, you should obtain a first-order difference equation in it only, i.e. an equation where it is only a function of 7t-1, and parameters of the model (5, m, ?, and 7). Once you have the expression for it, the output (?) is easy to calculate using the AD curve. c. Calculate the paths of inflation and output for five periods following the shock, assuming the following pa- rameter values: 0 3%, ? = 0, h = 0.4, ñ = 0.5, ? = 0.5, and 7 = 2%. Comment briefly on your results. d. Suppose that following the inflation shock, the central bank announces that it will do all that is necessary not to allow the inflation to remain elevated and bring it down to the inflation target. Moreover, suppose this is a credible central bank, so that people believe it, and thus set their expectations accordingly at Ti = 7. What will happen to inflation and output in period 2 and in later periods? Explain. When disuessing 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 those we discussed in Chapter 4. The firm lives for two periods. In the first period, the firm is born with an exogenous capital stock of Ko. After that, it observes the (real) expected interest rate, R, and its expected productivity in the second period, given by z. The firm's decision problem is to choose how much capital to install before production (what level of K to choose) in order to maximize profits in the second period. This level of capital K will be determined by investment I on top of the stock of capital the firm already has, Ko, minus depreciation, through a standard law of motion for capital: K = (1-5)K, +I. To close the model, suppose that the production function of the firm in the second firm is a Cobb-Douglas production function, with Y = 2K Nl-. From Chapter 4. we know that the marginal product of capital (MPK) for this production function is equal to MPK = oz Ka-1N1-4. To make life easy, we will assume that the labor supply in the economy is completely inelastic, so we can set N = 1 throughout the whole exercise, so you can start ignoring it from now on. Formally, the firm's problem is given by: max{zKºni-- R-1)} ? subject to: K= (1-5)Ko +I It is easy to show that the the optimal amount of capital is given by the standard condition: MPK = R a. Derive the optimal level of capital in Period 2 for this firm as a function of parameters and prices(K), Q, 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. You can drop any terms related to N, since we assume it to be a fixed and equal to 1. In this formulation of the problem, will the optimal amount of capital in period 2 (K), depend on the value of Ko? 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 Ko = 12 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=0.05, 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 8, what will be the investment demand of this firm? d. Now, suppose that the firm faces a slightly smaller interest rate of R=0.045. 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=0.05), but instead becomes more optimistic about its productivity in Period 2, expecting z = 1.1. 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?
