1. Consider the two-period model we studied in class. A consumer’s income in the current period is Y1 = 110, and income in the future period is Y2 = 110. The interest rate R is
10%. Assume that the price level equals 1 in both periods.
(a) Write down the present-value budget constraint. What is the lifetime wealth of this con- sumer? Suppose the consumer has logarithm utility function. The total utility is log (C1) +
β log(C2). Suppose the discount factor β = 0.95. Write down the optimality condition of this consumer. Solve for the optimal consumption in both periods. Is this consumer a borrower or a saver
in the current period?
(b) Now suppose the current-period income for this individual increases from 110 to 115. Solve for the optimal consumption in both periods for this individual. What is the marginal
propensity to consume out of the additional income?
2. Consider the two period model of consumption that we described in class. Consider the following interpretation of the model. The first period corresponds to the working period of life, and
the second period corresponds to the retirement period of life. Consistent with this, assume that income in the first period is Y1, and that income in the second period is 0.
Assume that the interest rate is R.
(a) What does our model predict the individual should do in terms of saving behavior. (b)Assume that the government introduces a social security system that taxes first period in-
come at rate τ and pays people a benefit of b in the second period of life. Assume that the
benefit b satisfies the following:
b = (1 + R)τY1
Give an interpretation of what this equation means. If this equation holds, how will the social security system influence the consumption choices C1 and C2? How will it affect the
amount that individuals save? Explain your answer.
3. Consider the two period model of consumption that we described in class. Suppose the per period utility function takes the following form:
U(Ci ) =
1 σ
i − , 1 − σ
where σ ≥ 0.1 The marginal utility is Ur (Ci ) = C−σ . The total utility becomes
C1−σ
1−σ
1 − 1 + β C2 − 1
1 − σ 1 − σ
(a) As in class, denote incomes, price level and interest rate by Y1, Y2, P, and R. Write down the optimality condition of this consumer. Use the optimality condition to
solve for the ratio C2/C1 as a function of β, R, and σ.
(b) The elasticity of intertemporal substitution is used to measure the responsiveness of consumption growth to a change in interest rate R. It is defined as the percent change in
consumption growth per percent increase in the interest rate, i.e.,
e = d log(C2/C1) .
dR
The definition of e uses the fact that consumption growth (C2 − C1)/C1 can be approximated by log(C2/C1). Use the optimality condition in part(a) to prove that e = 1/σ when R is small. (Hint:
in your proof, use the approximation that log(1 + R) = R when R is small.)
(c) As σ → ∞, the elasticity of intertemporal substitution approaches 0. What happens to C2/C1
when e = 0?
