Consider the following real business cycle model. A representative household worker seeks to maximize is present discounted utility given by Ett (Inc -n) where c is consumption and n is the fraction of time spent working. The household produces output, Y, for a representative firm using the following constant returns to scale technology: Yt = Ka Na . The household faces a per-period budget constraint of wtnt + rt (1 T) kt + (18) kt + Rt Bt = ct + Bt+1 +kt+1 where 8 (0, 1) is the depreciation rate of capital and Tt is the rental income tax rate. In some instances, the government will increase the rental income tax rate following an AR(1) process: Tt = PTt-1 + Et where p < 1 and Et is a random draw from a normal distribution with a mean of zero and variance of o Assume that all agents form rational expectations and make optimal decisions. a) Solve for the first order conditions of the household and firm. b) What are the equilibrium conditions for the economy? c) How do the equilibrium conditions depend on the rental income tax, Tt? Explain intuitively how this income tax affects the optimal decisions of the household or firm, and any market prices. d) Suppose the government decided to suddenly increase the rental income tax rate, ie. To > 0 for period 0. The increase in the tax was a purely one-time occurrence with no persistence. Describe the transition of the shock throughout the economy on impact of the shock, and in the periods that follow. Draw the transition paths of key macroeconomic variables to make it easier to understand your intuition. Make sure to explain in detail why the economy experiences this transition. Make sure to discuss the tradeoffs the household faces in response to this shock. Also explain how the firm responds to this unexpected tax shock immediately and in the later periods.