Consider the decision problem of a representative household optimally choosing a sequence of consumption {ct} and asset holdings {at+1} over an infinite horizon, taking as given the real interest rate r (assumed to be constant), the initial wealth a0, and the sequence of labor income {t}. The household's preferences are represented by E0t=0tu(ct) where 1/(1+),>0. Take the utility function to be u(ct)=ct(1/2)ct2 for 0ct (we take big enough so consumption in this economy never attains that level). The budget constraint for each period t is ct+at+1=(1+r)at+t. For all questions below assume r=. 1. - Take the first 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 ct (the "permanent income" rule). Let's consider now some alternative processes for the labor income sequence {t}. For each case, you can use the above solution, but with the appropriate calculation of the expected present value of incomes. Consider the decision problem of a representative household optimally choosing a sequence of consumption {ct} and asset holdings {at+1} over an infinite horizon, taking as given the real interest rate r (assumed to be constant), the initial wealth a0, and the sequence of labor income {t}. The household's preferences are represented by E0t=0tu(ct) where 1/(1+),>0. Take the utility function to be u(ct)=ct(1/2)ct2 for 0ct (we take big enough so consumption in this economy never attains that level). The budget constraint for each period t is ct+at+1=(1+r)at+t. For all questions below assume r=. 1. - Take the first 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 ct (the "permanent income" rule). Let's consider now some alternative processes for the labor income sequence {t}. For each case, you can use the above solution, but with the appropriate calculation of the expected present value of incomes