ECON8026 Advanced Macroeconomic Analysis
Assignment 3
Semester 1, 2023
Question 1
Government and Credit Constraints in the Two-Period Economy. Consider again
our usual two-period consumption-savings model, augmented with a government sector.
Each consumer has preferences described by the utility function
u(c1, c2) = ln c1 + ln c2
where c1 is consumption in period one, and c2 is consumption in period two. The associated
marginal utility functions are
u1(c1, c2) =
1
c1
and u2(c1, c2) =
1
c2
Suppose that both households and the government start with zero initial assets (i.e., A0 = 0
and b0 = 0), and that the real interest rate is always 10 percent. Assume that government
purchases in the first period are one (g1 = 1) and in the second period are 9.9 (g1 = 9.9). In
the first period, the government levies lump-sum taxes in the amount of 8 (t1 = 8). Finally,
the real incomes of the consumer in the two periods are y1 = 9 and y2 = 23.1.
a. What are lump-sum taxes in period two (t2), given the above information?
b. Compute the optimal level of consumption in periods one and two, as well as national
savings in period one.
c. Consider a tax cut in the first period of 1 unit, with government purchases left
unchanged. What is the change in national savings in period one? Provide intuition for the
result you obtain.
d. Now suppose again that t1 = 8 and also that credit constraints on the consumer are
in place, with lenders stipulating that consumers cannot be in debt at the end of period one
(i.e., the credit constraint again takes the form a1 ≥ 0 ). Will this credit constraint affect
consumers’ optimal decisions? Explain why or why not. Is this credit constraint welfare
enhancing, welfare-diminishing, or welfare-neutral?
e. Now with the credit constraint described above in place, consider again the tax cut
of 1 unit in the first period, with no change in government purchases. (That is, t1 falls from
8 units to 7 units.) What is the change in national savings in period one that arises due to
the tax cut? Provide economic intuition for the result you obtain.
Question 2
Habit Persistence in Consumption. An increasingly common utility function used in
macroeconomic applications is one in which period-t utility depends not only on period-t
1
consumption but also on consumption in periods earlier than period t. This idea is known
as “habit persistence,” which is meant to indicate that consumers become “habituated”
to previous levels of consumption. To simplify things, let’s suppose only period-(t − 1)
consumption enters the period-t utility function. Thus, we can write the instantaneous
utility function as u(c1, c2). When a consumer arrives in period t, ct−1 of course cannot be
changed (because it happened in the past).
a. In a model in which stocks (modeled in the way we introduced them in class) can be
traded every period, how is the pricing equation for St (the nominal stock price) altered due
to the assumption of habit persistence? Consumption in which periods affects the period-t
stock price under habit persistence? To answer this, derive the pricing equation using a
Lagrangian and compare its properties to the standard model’s pricing equation developed
in class. Without habit persistence (i.e., our baseline model in class), consumption in which
periods affects the stock price in period t?
b. Based on your solution in part a and the pattern you notice there, if the instantaneous
utility function were u(ct, ct−1, ct−2) (that is, two lags of consumption appear, meaning that
period t utility depends on consumption in periods t, t−1, and t−2), consumption in which
periods would affect the period-t stock price? No need to derive the result very formally
here, just draw an analogy with what you found above.
Question 3
“Hyperbolic Impatience” and Stock Prices. In this problem you will study a slight
extension of the infinite-period economy from Chapter 8. Specifically, suppose the repre-
sentative consumer has a lifetime utility function given by
u(ct) + γβu(ct+1) + γβ
2u(ct+2) + γβ
3u(ct+3) + ...
in which, as usual, u(.) is the consumer’s utility function in any period and β is a number
between zero and one that measures the “normal” degree of consumer impatience. The
number γ (the Greek letter “gamma,” which is the new feature of the analysis
here) is also a number between zero and one, and it measures an “additional”
degree of consumer impatience, but one that ONLY applies between period t
and period t+1.1 This latter aspect is reflected in the fact that the factor γ is
NOT successively raised to higher and higher powers as the summation grows.
The rest of the framework is exactly as studied in Chapter 8: at−1 is the representative
consumer’s holdings of stock at the beginning of period t, the nominal price of each unit of
stock during period t is St , and the nominal dividend payment (per unit of stock) during
period t is Dt. Finally, the representative consumer’s consumption during period t is ct
and the nominal price of consumption during period t is Pt . As usual, analogous notation
describes all these variables in periods t+ 1, t+ 2, etc.
1The idea here, which goes under the name “hyperbolic impatience”, is that in the “very short run” (i.e.,
between period t and period t + 1), individuals’ degree of impatience may be different from their degree of
impatience in the “slightly longer short run” (i.e., between period t + 1 and period t + 2, say). “Hyperbolic
impatience” is a phenomenon that routinely recurs in laboratory experiments in experimental economics
and psychology, and has many farreaching economic, financial, policy, and societal implications.
2
The Lagrangian for the representative consumer’s utility-maximization problem (start-
ing from the perspective of the beginning of period t) is
u(ct) + γβu(ct+1) + γβ
2u(ct+2) + γβ
3u(ct+3) + ...
+ λt
[
Yt + (St +Dt)at−1 − Ptct − Stat
]
+ γβλt+1
[
Yt+1 + (St+1 +Dt+1)at − Pt+1ct+1 − St+1at+1
]
+ γβ2λt+2
[
Yt+2 + (St+2 +Dt+2)at+1 − Pt+2ct+2 − St+2at+2
]
+ γβ3λt+3
[
Yt+3 + (St+3 +Dt+3)at+2 − Pt+3ct+3 − St+3at+3
]
+ ...
a. Compute the first-order conditions of the Lagrangian above with respect to both at
and at+1 . (Note: There is no need to compute first-order conditions with respect to any
other variables.)
b. Using the first-order conditions you computed in part a, construct two distinct stock-
pricing equations, one for the price of stock in period t , and one for the price of stock in
period t + 1 . Your final expressions should be of the form St = ... and St+1 = ... (Note:
It’s fine if your expressions here contain Lagrange multipliers in them.)
For the remainder of this problem, suppose it is known that Dt+1 = Dt+2, and
that St+1 = St+2, and that λt = λt+1 = λt+2
c. Does the above information necessarily imply that the economy is in a steady-state?
Briefly and carefully explain why or why not; your response should make clear what the
definition of a “steady state” is. (Note: To address this question, it’s possible, though not
necessary, that you may need to compute other first-order conditions besides the ones you
have already computed above.)
d. Based on the above information and your stock-price expressions from part b, can
you conclude that the period-t stock price (St) is higher than St+1, lower than St+1, equal
to St+1, or is it impossible to determine? Briefly and carefully explain the economics (i.e.,
the economic reasoning, not simply the mathematics) of your finding.
Now also suppose that the utility function in every period is u(c) = ln c, and also that the
real interest rate is zero in every period.
e. Based on the utility function given, the fact that r = 0, and the basic setup of
the problem described above, construct two marginal rates of substitution (MRS): the
MRS between period-t consumption and period-t+ 1 consumption, and the MRS between
period-t+ 1 consumption and period-t+ 2 consumption.
f. Based on the two MRS functions you computed in part e and on the fact that r = 0
in every period, determine which of the following two consumption growth rates
ct+1
ct
OR
ct+2
ct+1
3
is larger. That is, is the consumption growth rate between period t and period t + 1 (the
fraction on the left) expected to be larger than, smaller than, or equal to the consumption
growth rate between period t + 1 and period t + 2 (the fraction on the right), or is it
impossible to determine? Carefully explain your logic, and briefly explain the economics
(i.e., the economic reasoning, not simply the mathematics) of your finding.