3. (4 p.) Consider the neoclassical model of investment. Assume that the production function is Y = AV/KL , where A is the level of (exogenous) total factor productivity, and L is the size of the labour force (also exogenous) and K = (1-8)K, + I where ) is investment spending, K. is the past capital stock, and o is the rate of depreciation of capital. The optimal level of the capital stock is determined by the equation: Fi =r+8. (a) Find algebraically the optimal level of investment spending in the short run (when the past capital stock Ke is fixed) and in the long run (when K = K,)- (b) Assume that initially /" = 6. Explain and find algebraically how the level of investment in the short run and the long run is affected by the following: i. Discovery of new technologies raising total factor productivity A by 10%. ii. A war that destroys 10% of the existing capital stock Ko. ili. Adverse climate change that increases the cost of maintaining buildings (interpreted as 10% increase in the rate of depreciation 5). (c) Suppose output Y is used either for consumption C or investment I, so C + 1 = Y. Find the capital stock that maximizes consumption in the long run (when K = Ko). Find the real interest rate that would provide incentives for firms to undertake the level of investment required to reach this capital stock