ECOS3021 Business Cycles and Asset Markets University of Sydney
2021 Semester 1
Tutorial #3
1. Consider a simple household consumption and savings problem:
max logC1 +βlogC2 C1 ,C2 ,S
s.t C1+S=Y1 C2 = Y2 + SR
a) Derive the first order condition(s) for the household’s problem, and solve for an expression that describes the household’s savings behaviour.
b) Rearrange the expression for savings to derive the savings supply curve. This curve expresses the interest rate R as a function of savings S and incomes Y1,Y2. Draw the savings supply curve on a graph, with S on the x-axis and R on the y-axis. Now assume that the savings demand curve is constant/inelastic with respect to the interest rate. Add this demand curve to the figure. Label the equilibrium interest rate in the market for savings.
c) On two separate figures, show the effect of (I) a decrease in period 1 income Y1, (II) a decrease in period 2 income Y2. Provide economic intuition for equilibrium outcomes following the the changes in incomes.
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2. Consider the following three-period household consumption and savings problem. In this problem, a household lives for three periods. In the final period of life the household retires, earns no income, and must consume any available retirement savings, S3. Retirement savings must be put aside in the first period, but nothing can be saved from period 2. In period 1, the household can also put aside savings (or borrow), for period 2, S2. In all, the household makes two savings decisions in period 1 – S2,S3 – and one consumption decision in each of the three periods – C1, C2, C3. The household problem can be described as follows:
max logC1 +βlogC2 +β2 logC3 C1 ,C2 ,C3 ,S2 ,S3
s.t C1+S2+S3=Y1 C2 = Y2 + S2R2
C3 = S3R3
where R2 is the interest rate earned on savings in period 2, and R3 is the interest rate earning on
retirement savings available in period 3.
a) Substitute each of the budget constraints into the utility function in order to rewrite the household
problem as an unconstrained optimization problem in the two choice variables S2 and S3.
b) Take the first order conditions for the two savings decisions (S1, S2). Note: Because there are two
choice variables, there are two separate first order conditions in this case.
c) Use the first order conditions to solve for the two savings decisions as functions of incomes and interest rates. Note: It may be easiest to solve for S2 explicitly, but leave the expression for S3 as a function of S2.
d) How does the household use the available assets to finance its retirement? How does your answer change when second period income, Y2, is very large or very small?
e) How are retirement savings affected by the second period interest rate R2? How are retirement savings affected by the third period interest rate, R3?
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3. Consider the following two-period household consumption and savings problem. In this problem we will use a power utility function for the household’s objective function.1
C 1−σ C 1−σ max 1 +β2
C1,C2,S 1−σ 1−σ s.t C1+S=Y1
C2 = Y2 + SR
where R is the gross interest rate.
a) Write down the inter-temporal budget constraint. With reference to the inter-temporal budget constraint,
describe how the interest rate is related to the price of consumption in period 2.
b) Solve for the first order conditions of the problem. Use the first order conditions and the budget constraint(s) to find consumption demands as functions of the interest rate and income.
c) Show that when σ = 1, the consumption functions you derived above are identical to those under the log-utility specification discussed in Lecture 3.
d) Continue to assume that σ = 1. What is the effect of an increase in the interest rate R on C1 and C2? Use math to justify your answer.
e) Suppose that income is constant across periods, so Y1 = Y2 = Y , and that R > 1. Now assume that β = 0. Under this parameterization, is C1 or C2 more sensitive to changes in the interest rate?
f) Suppose that income is constant across periods, so Y1 = Y2 = Y , and that R > 1. Now assume that β = 1. Under this parameterization, is C1 or C2 more sensitive to changes in the interest rate?
1This utility function is also often to as the Constant Relative Risk Aversion utility function. 3