lI'Zonsider the decision problem of a representative household optimally choosing a sequence of consumption {er} and asset holdings {arid} over an infinite horizon, taking as given the real interest rate r [assumed to be constant}, the initial wealth on, and the sequence of labor income {cut}. The household's preferences are represented by ED Eofut'ct) where at E 1311 + p), p 3: I]. Take the utility function to be amen] 2 ac; - (1 3' 22m? for l] E c; E a {we take a: big enough so consumption in this economy never attains that level]. The budget constraint for each period I is c; + em = (1 + at + rot. For all questions below assume r = p. I. - Take the rst order conditions of the problem and obtain the liuler equation. Combine the budget constraints into an intertemporal, espeeted present value budget constraint; use the liuler equation together with the present value budget constraint to solve for the optimal decision rule that will determine the level otconsmnption ct (the "permanent income" rule}. Let's consider now some alternative processes thr the lahor income sequence {rot}. 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 at: = at her all r (with a posnive initial wealth on}. Saving soul is dened as total income minus consumption, where total income includes lab-or income or; and nancial income rot : i.e. sou, = to! + rot ct. What is the optimal level of consumption at? - Use the budget constraints ofthe rst two or three periods to deduce what the amount ofsaving and asset holdings or would be in each of the rst few periods Show graphically (putting time t on the horizontal axis} what the time path of labor income, consumption, saving and asset holdings look like {qualitatively}. 3. Suppose there is no uncertainty, hut labor income is subject to regular, perfectly anticipated uctuations: on = or + d in even periods{r=[}, 2, cl, ...} and on = to (1 + t). in odd periods tr=l, 3, 5, }. There is a positive initial wealth on. Repeat the analysis of the previous question: - What is the optimal path of consumption cy'? - Use the budget constraints of the lirst two or three periods to deduce what the amount of saving and asset holdings or would be in each of the first few periods - Show graphically {qualitatively} the time path of labor income, consumption, saving, asset holdings over time, and compare briefly with the previous case