Construct the lagrangian for this problem and derive the

Question

Question 1. (Consumption under Uncertainty) Consider the representative household who is uncertain about its future income stream, {Y_t}^T=1_t=0. At t = 0, the representative household maximizes the following expected lifetime utility

U = E_o∑_t=0^∞β^tu(C_t)

subject to subject to given A_o ≥ 0, and

A_t+1 = A_t(1+ r) + Y_t - C_t, 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.

C_t+1 = C_t + ε_t+1

where ε_t+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 d_t at time t. The representative consumer maximizes the following lifetime expected utility

U = E₀∑_t=0^∞β_tu(C_t)

subject to (P_t + d_t)s_t>c_t + p_ts_t+1+1 q_tb_t+1

(a) Construct the Lagrangian for this problem. Assuming the interior solution, obtain the first order conditions for c_t, s_t+1 and b_t+1. Then, derive the Euler equations for the bond holdings (b_t+1) and asset holdings (s_t+1).

(b) Let p_t and x_t genetically denote the price of an asset and the payoff, respectively. Then, the pricing equations from can be rep resented by P_t = E_tm_t.t+1x_t,t+1. CarefUlly discuss how E_tm_t.t+1, E_tx_t.t+1 and Cov_t(m_t,_t+1,x_t,_t+1) will affect p_t and provide intuitive explanations.