1. 3-Period Consumption with Certainty
Assume that a household seeks to optimize their sum of discounted utility over consumption Ct 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
rL, with probability 0.5
r = rH, with probability 0.5
for some rL < rH . Let r¯ = 0.5∗ (rL +rH ) and note that Er = 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 = rL and r = rH. 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) = C1-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.