3 Optimal Consumption with Financial Wealth In lecture 2, we saw an example where the household only lives for two periods. Now suppose the household lives for N periods, from Period 1 to N. Household receives an exogenous income stream {y1,y2, ...yN}, and accumulate their financial wealth at in each period. The return on real financial wealth is r. Let household's problem be t= 1 max Egt-lin(ct) (1) Ct subject to at+1 = (1 +r)at + yt - ct (2) Ct 2 0 (3) aN+120 (4) where at is the financial wealth at the beginning of Period t and al = 0. 1. Interpret (1) (2) (3) and (4). 2. Write down the inter-temporal budget constraint and derive the Euler equation(s). 3. Assume B = 1, r = 0. Solve for the optimal consumption for each period, {ct }, t = 1, 2, ..., N. Explain how your results are related to permanent income theory.4. Assume ,3 = 1, and 'r > U in period 1,2,..N. Compute optimal {CL t = 1, 2, ...N} and compare it with that in 3. 5. Assume )3 = 1, r = 0. What is the marginal propensity to consume in Period 1? Explain the intuition. Likewise, what is the marginal propensity to consume in period 2? 4 Borrowing Constraint Consider a representative consumer who lives for three periods, he or she has an exogenous endow- ment stream given by '91, y2,y3 and can borrow and lend at a given interest rate it, which is the interest rate between period t 1 and period t. Assume that he / she starts out with no wealth (that is, fa = 0). The discount factor is ,8 = 1. The consumer's problem is: maxC1552,CB \"(01) + \"(02) + \"(03) subject to I91+~'51=y1 02+52=yz+(1+i1)'51 03=ya+(1+i2)-52 1. For simplicity, assume that it = 0. Further assume the following form for the utility function: 15(0) = En(c). Solve for the optinaal consumption plan for the three periods. 2. Suppose y1 = 93