1. 3-Period Consumption with Certainty

Assume that a household seeks to optimize their sum of discounted utility over consumption C_t in periods t = 0, 1, 2. The household receives income in each period - known with cer- tainty - given by Y₀, Y₁, and Y₂. The household may save the amounts S₁ and S₂ in periods 0 and 1, respectively.

The fixed interest rate on savings is given by 1 + r, and the household discounts the future with rate β. Therefore, the household solves the following problem

max U (C₀) + βU(C₁) + β²U(C₂)
^{S1 ,S2}

s.t. C₀ + S₁ = Y₀

C₁ + S₂ = Y₁ + (1 + r)S₁

C₂ = Y₂ + (1 + r)S₂

(a) If S₁ is positive, is the household saving or borrowing in period 0?

(b) Using the budget constraints of the household, derive the unified lifetime present value budget constraint. Show your work. Note that your answer should involve the term Y₀ + Y₁/1+r + Y₂/(1+r)² , which is the present discounted value of income or human wealth.

For notational simplicity, refer to this term with the constant H.

(c) Write the Euler equation or first-order condition for optimal savings S₁. Explain the intuition behind this equation in words, e.g. "The LHS represents the ... The RHS rep- resents the .... Optimal consumption behavior involves setting..."

(d) Write the Euler equation or first-order condition for optimal savings S₂. Explain the intuition behind this equation in words.

(e) Assume that the period utility function U(C) = log C. Use the two Euler equations, together with the lifetime present value budget constraint, to derive formulas for C₀, C₁, and C₂ in terms of β, r, and H only.

(f) Assume that β < 1/1 + r. Is consumption C₂ in period 2 lower or higher than consumption C₀ in period 0?

2. 2-Period Consumption with Interest Rate Uncertainty

Assume that a household is optimally choosing consumption C₀ and C₁ in periods 0 and 1 to maximize expected discounted utility

U (C₀) + βEU (C₁).

The household's incomes Y₀ and Y₁ in both periods are known with certainty, and the house- hold can save in the amount S with budget constraints

C₀ + S = Y₀

C₁ = Y₁ + (1 + r)S.

However, the interest rate r is a random variable, with
       r_L, with probability 0.5
r =  r_H, with probability 0.5

for some r_L < r_H . Let r¯ = 0.5∗ (r_L +r_H ) and note that E_r = r¯. In other words, the household faces interest rate risk in period 1 implying uncertainty about the return to the household's savings.

(a) Consumption in period 1 is a random variable because of the uncertainty in r. Given some savings level S, write a formula describing the value of C₁ in the case of r = r_L and r = r_H. What is the expected value EC1?

(b) Write the Euler equation or first-order condition for optimal savings behavior S. How does this Euler equation differ from the example shown in class in the 2-period model with uncertainty over income?

(c) Assume that period utility takes a CRRA form with U (C) = C^1-γ/1-γ for γ > 0. Write the Euler equation in this particular case.

(d) Given CRRA utility, is consumption C₀ in period 0 larger than, smaller than, or equal to the expected value of consumption in period 1 E(C₁)? You may assume that the function f (x) = (1 + x)(Y₁ + (1 + x)S)^-γ is convex as a function of x, and you may also assume that β(1 + r¯) > 1.