ECOS3021 Business Cycles and Asset Markets University of Sydney
2021 Semester 1
Tutorial #5
1. Consider the following household problem:
max log C1 + βE [log C2]
C1 ,C2 ,S
s.t. C1+S=Y ̄
C 2 = Y ̃ 2 + S
where savings earn no interest (i.e. r = 0), and the realisation of income in period 2 is uncertain:
of an increase in the probability α? Provide an intuitive explanation for why this is the case.
b) What is the effect on expected income of an increase in the random payoff x? What is the effect of an
increase in x when: i) α = 0.5, ii) α < 0.5. Provide intuitive explanations for your answers.
c) Substitute the budget constraints into the utility function. Now substitute in the uncertain income Y2, and expand the expectations term to show how utility depends on the two possible outcomes for Y2. What is the effect on expected utility of an increase in α? What is the effect on expected utility of an increase in x (does this depend on the value of α)? Provide intuition for your answers.
d) Derive the Consumption Euler Equation for the household’s problem. Write the Euler Equation as a function of the two possible outcomes for consumption in period 2: C2+ = Y ̄ +x+S and C2− = Y ̄ −x+S. Recall from class that one side of the Euler Equation represents the Marginal Utility of Consumption in period 1, while the other side of the equation represents the (Expected) Marginal Utility of Consumption in period 2. What is the effect on the expected marginal utility of consumption in period 2 of an increase in α? What do you expect this increase in α to have on consumption in period 1? What about savings? Note: You do not need to explicitly solve for consumption functions to answer this question. Instead, use the Euler equation to justify your answer.
f) Again referring to the Consumption Euler Equation. What is the effect on the expected marginal utility of consumption in period 2 of an increase in x? What do you expect this increase in x to have on consumption in period 1? What about savings? Note: You do not need to explicitly solve for consumption functions to answer this question. Instead, use the Euler equation to justify your answer. You may also assume that α is close to zero.
Y ̄ + x with probability α
Y ̄ − x with probability 1 − α
Y2 =
a) Write an expression for the expected value of income in period 2. What is the effect on expected income
1
2. Consider the following household problem:
max log C1 + βE [log C2]
C1 ,C2 ,B
s.t. C1+B=Y1
C 2 = Y 2 + R ̃ B where the bond B has an uncertain payoff R ̃:
R with probability α
0 with probability 1 − α
In the good state of the world, the bond pays off with interest R. In the bad state of the world, the bond pays out nothing: the household loses both the interest payment and the initial investment in the bond. You can think of this as a situation where bonds may suffer from default risk, where borrowers are unable to repay the bond in bad states of the world.
a) Substitute the budget constraints into the utility function. Now substitute in the uncertain bond payoff R ̃, and expand the expectations term to show how utility depends on the two possible outcomes for the interest rate R ̃. What is the effect on expected utility of an increase in α?
b) Solve for the household’s bond equation. What is the effect on household bond holdings of an increase in α? What is the effect on bond holdings of an increase in the interest rate in the good state, R? Provide economic intuition for your answers.
c) Consider households that have more or less income in period 1 (i.e. high Y1 vs. low Y1). Which of these households are more sensitive to changes in the probability of a good payoff (i.e. an increase in α)? Provide some economic intuition for your answer.
d) Consider an extreme case where the interest rate on the bond goes to infinity (R → ∞) but the probability of the good state goes to zero (α → 0). What is the household’s desired bond holdings in this case? Provide economic intuition for your answer.
R ̃ =
2