Question 1. (Consumption under Uncertainty) Consider the representative household who is uncertain about its future income stream, {Yt}T=1t=0. At t = 0, the representative household maximizes the following expected lifetime utility
U = Eo∑t=0∞βtu(Ct)
subject to subject to given Ao ≥ 0, and
At+1 = At(1+ r) + Yt - Ct, 0 ≤ t ≤ T -1. (*)
(a) Construct the Lagrangian for this problem, and derive the first-order Kuhn-Tucker conditions for the interior solution. Combining them, obtain the Euler equation and interpret it.
(b) Hereafter, assume (i) u(C) = C - 1/2C2, (ii) Ρ (the rate of time preference) equals r. Show that consumption obeys a martingale process. Then, show that consumption has the following representation.
Ct+1 = Ct + εt+1
where εt+1 is the forecasting error.
Question 2. (Lucas Model of CAPM)
Consider the Lucas tree model where there is one asset (tree) that pays a stochastic dividend dt at time t. The representative consumer maximizes the following lifetime expected utility
U = E0∑t=0∞βtu(Ct)
subject to (Pt + dt)st>ct + ptst+1+1 qtbt+1
(a) Construct the Lagrangian for this problem. Assuming the interior solution, obtain the first order conditions for ct, st+1 and bt+1. Then, derive the Euler equations for the bond holdings (bt+1) and asset holdings (st+1).
(b) Let pt and xt genetically denote the price of an asset and the payoff, respectively. Then, the pricing equations from can be rep resented by Pt = Etmt.t+1xt,t+1. CarefUlly discuss how Etmt.t+1, Etxt.t+1 and Covt(mt,t+1,xt,t+1) will affect pt and provide intuitive explanations.