7. Consider the neoclassical model of investment. There is a production function for output Y=F(K) that depends on the stock of capital K. The capital stock is K : (I * CDKO + I Where I is investment spending, K0 is the past capital stock, and 5 is the rate of depreciation of capital. The optimal level of the capital stock is determined by the equation: F '(K) : r + 5 where F '(K) is the marginal product of capital and r is the real interest rate. Assume that the production function isY = ARJZ , where A is the level of (exogenous) total factor productivity, and L is the size of the labour force (also exogenous). (a) i. ii. Derive an expression for the marginal product of capital for the production function given above and show that it is diminishing as the capital stock increases. Show how the optimal capital stock K can be determined using a diagram, and explain how this determines the level of investment spending in the short run (when the past capital stock K0 is xed) and in the long run. (b) Explain how the level of investment in the short run and the long run is affected by the following: i. ii. iii. iv. v. An increase in desired saving that reduces the real interest rate r. Aging of the population that reduces the labour force L. Adverse climate change that increases the cost of maintaining buildings (interpreted as an increase in the rate of depreciation 3). Discovery of new technologies raising total factor productivity A. ii. A war that destroys a portion of the existing capital stock K0. (0) Suppose output Y is used either for consumption C or investment I, so C + I = Y. Explain how to nd the capital stock that maximises consumption in the long run (when K = K0). Find the real interest rate that would provide incentives for rms to undertake the level of investment required to reach this capital stock