uonsmer the creelsion promem or a representative nousenom opumauy cnoosmg a sequence of consumption {ct} and asset holdings {a,+ 1} over a two-period horizon: t = 0 is the current period, t= 1 the future period. The household takes as given the real interest rate r (assumed to be constant), the initial wealth no (that here for simplicity we take to be zero) and the sequence of (possibly uncertain) labor income {yr} The household's preferences are represented by the discounted ow of expected utility E02 'u(ct) = a(c0 )+ E0u(cl) where I3 E 1/(1+P) , P>0, and E0 indicates expectation at time 0. The budget constraints for period 0 and period 1 respectively are 00+ 3.15 yo and C1 5 yl + (1+1') all where a1 is the amount of assets that the household decides to carry over to period 1; a] can be negative, meaning debt, since we assume that households can borrow as well as lend at the interest rate r [to save time we do not include a2 in the second constraint since we know that, being the last period, it will be chosen to be zero]. Decisions for C0 and a] are taken in the current period t= 0 after observing the realization of current income yo, so on and a] are deterministic; but y] and hence (:1 are random variables, so we have to take expectations (conditional on information available at time 0) of any term depending on those variables (utility terms but also the multiplier of the constraint for the future period t= 1). 1. - Write the Lagrangian for the expected utility maximization problem, including both budget constraints separately, each with its own multiplier (i.e. without combining them yet into one intertemporal constraint). [For algebraic convenience, apply the subjective discount factor I3 both to next period utility and to the multiplier of next period constraint it's just an innocuous normalization] - Obtain the rst order conditions of the problem. Then combine them to get the Euler equation (the intertemporal relation between co and c1). 2. Suppose there is no uncertainty (i.e. both current and future income yo and y1 are known with certainty) and r = p. Combine the budget constraints into an intertemporal, present value budget constraint; then use the Euler equation and that budget constraint to solve for the optimal c0 and c1 as a function of r, ya and y1. What would the level of consumption in each period be if labor income is a constant ow y] = yo = y? What would be the amount of asset holdings a1? 3. Consider a temporary increase in current income in the amount Ayn. What would be the change Ace in current consumption? If the interest rate is r = 0.04 (same as p), how much umnld that hp? TInui much urmrlrl SIQQPf hnldinao 9. P1191109? 4. Consider a permanent increase in income, that is both yo and 311 increase by Ay. What would be the change Ac in current consumption? How much would asset holdings a] change? 5. Consider an increase in future income only in the amount Ayl (no change in current income yo). What would be the change A00 in current consumption? If the interest rate is r = 0.04 (same as P), how much would that be? How much would asset holdings a] change? 6. For what follows take the utility function to be \"(9) = 5'0, '(1/ 2)C,2 for 0 5 cr