Please show work for each step leading up to the solutions

3 Capital Investment and the Savings Rate 111 the Solow growth model, we have assumed that the savings rate was exogenous, i.e., we supposed that the economy saves a fraction 3 of GDP K. In this question, we want to think about how the incentives to save may depend on the existing capital stock. Consider an economy with two periods and no population growth, i.e., L1 : L2 = 1. The initial capital stock is K1 > 0. Firms produce according to the production function F (K, L) : AK'ILl'Q, where or E (0, 1). Note that GDP at time 1 is given by Y1 : AK? Time 2 GDP depends on K2 which is consequence of total savings in the economy. Suppose that capital fully depreciates, 6 : 1, so that K2=I, where I denotes investment today. Suppose furthermore that the interest rate is given by r > D. 1. 61 Suppose that, at time 2, rms pay if to rent capital and 102 to hire workers. Write the standard prot maximization problem of the rm and show that K _ ctr1 r2 ocAK2 . An investor is deciding how much to invest and whether to apply that income to capital or to put that money in a savings account, which obtains the return '1". Argue that the net present value of investing I today is given by 715:1 1+1\" NPV : I+ Argue that r5: = 1 + 1\". Use that relationship to how that (1A ]11\" K2: [1+r' What happens to K2 (and investment I) if A increases? What if 2" increases? Provide intuition. Show that the savings rate 1 i_ Y; Q m i 70, [1+rl A \"Kl 3 What happens to the savings rate if rr increases? \\Vhat if A increases? Relate these answers to your intuition in 3. Suppose that a earthquake destroys 10% of the capital stock in this economy, i.e., after the earthquake K; = 0.9K}. Relate the savings rate after the earthquake, s', to the savings rate before the earthquake, s. (Hint: Divide s' 2 [air] 17? A (K970; by the savings rate .9 and notice that afew terms dr0p Out.) \"thich savings rate is higher? Provide intuition for the investment consequences of this disaster in terms of total investment I and the savings rate .3