3. RBC Model With Only Capital: The economy consists of a unit mass of households and a unit mass of firms and lasts two periods. There is no uncertainty. Households have no initial wealth and receive income Y1 in period 1 and Y2 in period 2, where Y1 and Y2 are output from firms that they own. They also own shares of the firms and receive the dividends. They take the interest rate rt as given, and the price of the consumption good is 1. They maximize U (C1) U (C2) where U > 0 and U < 0. Firms produce the consumption good with a technology Yt = tK t They enter with capital K1 and invest I1 in period 1, which comes on line in period 2, at a cost of 1 per unit of capital. Capital that is used depreciates at rate . There are no adjustment costs, and the firm maximizes the PDV of dividends discounted at the interest rate r. At the end of period 2, the firms sell the capital they own at a price of 1 per unit of capital after depreciation. Goods markets clear, so C1 + I1 = Y1 C2 = Y2 + (1 ) K2 (a) Write down the households' optimization problem and optimality conditions. If you use results from the consumption section of the course, you may skip straight from the optimization problem to the optimality condition. (b) Write down the firms' optimization problem and optimality conditions. If you use results from the investment section of the course, you may skip straight from the optimization problem to the optimality condition. 2 (c) Write down the equilibrium conditions. i. What pins down the net real interest rate r ? ii. Combine the equilibrium conditions into one equation in K2 and plot the two sides of the equation vs. K2 to show equilibrium diagrammatically. (d) What happens if the economy gets news at time t = 1 that productivity will rise at time t = 2? (e) Now assume there is a government that finances itself by lump-sum taxes. Assume government spending is valued at U (Ct) + V (Gt) and government spending is included in goods market clearing so that C1 + I1 + G1 = Y1 C2 + G2 = Y2 + (1 ) K2 Write down the government's present-value budget constraint and the equilibrium conditions. How does the one equation in K2 change? (f) Use your diagram from part c to show what happens to K2 and r when government spending G1 increases. How does the change in government spending affect I1? Explain intuitively whether government spending crowds out or crowds in private savings and why.