3. Consider the two-period model for a consumer with the following preferences over {C1, C2}: 1- 1 u(c1, c2) = 40 +850 1- 1 where the price of consumption in the two periods are P1 and P2 respectively. The consumer is endowed with an initial wealth position of Ao = 0 and wealth pays a positive nominal interest rate i. The consumer earns labor income in both periods (Y, Y2}. We will impose that A2 = 0. Lastly let interest income be subject to a proportional tax Ta > 0 if A0 and T = 0 if A 0. (a) Write down the two period-by-period budget constraints (Hint: be careful with how you handle the tax policy here!) (b) Derive the lifetime budget constraint. What happens in the lifetime budget constraint when first period consumption, c, is equal to 1? P1 (c) Write the Lagrangian optimization problem and derive the FOCs (d) Derive the optimality conditions for both cases (when A > 0 and when A 0). 1 (e) Solve for c and c2 as functions of exogenous variables. (f) Now assume that = 1 (log utility), under what conditions does the worker end up paying the tax on interest income in the second period?