Consider a representative household who lives for three periods. The period-by-period budget constraints of the household are:

1+ 0(1 + ) = 1+ 12+ 1(1 + ) = 2+ 23+ 2(1 + ) = 3+ 3

whereis interest rate,is price level,is consumption in periodt,is income in periodt, andis bonds in periodt.

(A)Derive the household'spresent value lifetime budget constraint. The goal of the household is to

max {(1) + (2) + 2(3) }1, 2, 3

subject to the present value lifetime budget constraint obtained in (A).(. )denotes the utilityfunction, is discount factor, and 0<<1. The household takes the interest rate and the price level as given, and income in each period is exogenous. Suppose()=ln ,0=0, and3=0.

1. (B)Write the Lagrangian expression for the household's constrained maximization problem, andderive the first-order conditions.
2. (C)Derive the Euler equation between2and1.
3. (D)Derive the Euler equation between3and2.
4. (E)Solve for thehousehold'soptimal choices of1,2, and3.