Consider the decision problem of a representative household optimally choosing a sequence of consumption {Ct} and asset holdings {at} over an innite horizon, taking as given the real interest rate r (assumed to be constant), the initial wealth (to, and the sequence of labor income {wt}. The household's preferences are represented by ED 22:033qu where [1' E 1/(1 + p), p > 0. Take the utility function to be u(ct) = act (1 / 2)th for O S C, S a (we take a: big enough so consumption in this economy never attains that level). The budget constraint for each period t is Ct + (EH1 = (1 + at + cot. For all questions below assume 7' = p. 1. - Take the rst order conditions of the problem and obtain the Euler equation. - Combine the budget constraints into an intertemporal, expected present value budget constraint; use the Euler equation together with the present value budget constraint to solve for the optimal decision rule that will determine the level of consumption or (the \"permanent income\" rule). Let's consider now some alternative processes for the labor income sequence {not}. For each case, you can use the above solution, but with the appropriate calculation of the expected present value of incomes. 2. Suppose there is no uncertainty and labor income is a constant ow not = w for all t (with a positive initial wealth (10). Saving savt is dened as total income minus consumption, where total income includes labor income to, and nancial income rat : i.e. savt = to, + rat ct. - What is the actual level of consumption ct? - Show graphically (putting time t on the horizontal axis) what would the time path of labor income, consumption, saving and asset holdings look like (qualitatively). [Note' you can use the budget constraints of the first two or three periods to deduce what is the amount of saving in each period and the evolution of asset holdings at] 3. Suppose there is no uncertainty, but labor income is subject to regular, perfectly anticipated fluctuations: not = to + A in evenperiods (t=0, 2, 4,...) and (or = to (1 + r)A in odd periods (t=1, 3, 5, .. .). There is a positive initial wealth (to. - What is the optimal path of consumption 6:? - What is the path of labor income, consumption, saving and asset holdings over time? Show graphically (qualitatively) and compare briey with the previous case