ECON8026 Advanced Macroeconomic Analysis
Assignment 2
Semester 1, 2023
Question 1
The Wealth Effect on Consumption. Consider the two-period consumption-savings
model we have been developing in class.
a. As in class, maintain the simplifying assumption that A0 = 0 . Show graphically
how a rise in the period-1 nominal price of consumption can lead to a decrease in optimal
consumption in period 1.
b. Now suppose that A0 6= 0 . Show graphically how a decrease in A0 can lead to a
decrease in optimal consumption in period 1.
c. The two effects you analyzed in parts a and b work through seemingly different
channels. Actually, they are usefully thought of as operating through the same broadly-
defined channel. Explain this broadly-defined channel.
Question 2
Taxes on Interest Earnings. In our two-period consumption-savings model (with no
leisure), suppose positive interest income in period 2 is taxed at the rate ts , where 0 < ts <
1. That is, if interest income in period 2 is positive, then the government takes a fraction
ts of the interest income, while if interest income in period 2 is non-positive, then there is
no tax. As in class, make the simplifying assumption that the individual has zero initial
wealth (i.e., A0 = 0). Also suppose that the interest tax has no effect on the nominal price
level in either period.
a. In this modified version of the model, algebraically express the period-1 budget
constraint and the period-2 budget constraint of the individual.
b. Using your period-1 and period-2 budget constraints from part a, derive the individ-
ual’s lifetime budget constraint (LBC). (Hint: Is the slope of this LBC continuous?)
c. Recall our assumption (based on empirical evidence) that the aggregate private
savings function is an increasing function of the real interest rate. Suppose that at the
representative agent’s current optimal choice, he is choosing to consume exactly his real
labor income in period 1.
(i) At his current optimal choice, is his marginal rate of substitution between present con-
sumption and future consumption equal to (one plus) the real interest rate? Explain
why or why not.
(ii) President Bush, as part of his first-term economic agenda, lowered the tax rate on
interest income from savings (one part of this package was eliminating the tax on
dividends – but there are other elements of this idea in his tax package as well). Part
1
of the rationale is that it will encourage individuals to save more. In this example,
would a decrease in the tax rate ts encourage the representative agent to save more
in period 1? Explain why or why not?
Question 3
Optimal Choice in the Consumption-Savings Model with Credit Constraints:
A Numerical Analysis. Consider our usual two-period consumption-savings model. Let
preferences of the representative consumer be described by the utility function
u(c1, c2) =
√
c1 + β
√
c2
where c1 denotes consumption in period one and c2 denotes consumption in period two.
The parameter β is known as the subjective discount factor and measures the consumer’s
degree of impatience in the sense that the smaller is β, the higher the weight the consumer
assigns to present consumption relative to future consumption. Assume that β = 1/1.1. For
this particular utility specification, the marginal utility functions are given by u1(c1, c2) =
1
2
√
c1
and u2(c1, c2) =
β
2
√
c2
.
The representative household has initial real financial wealth (including interest) of
a0 = 1 The household earns y1 = 5 units of goods in period one and y2 = 10 units in period
two. The real interest rate paid on assets held from period one to period two equals 10%
(i.e., r1 = 0.1).
a. Calculate the equilibrium levels of consumption in periods one and two. (Hint: Set
up the Lagrangian and solve.)
b. Suppose now that lenders to this consumer impose credit constraints on the consumer.
Specifically, they impose the tightest possible credit constraint – the consumer is not allowed
to be in debt at the end of period one, which implies that the consumer’s real wealth at
the end of period one must be nonnegative (a1 ≤ 0) (Note: here, a1 is defined as being
exclusive of interest, in contrast to the definition of a0 above). What is the consumer’s
choice of period-one and period-two consumption under this credit constraint? Briefly
explain, either logically or graphically or both.
c. Does the credit constraint described in part b enhance or diminish welfare (i.e., does
it increase or decrease lifetime utility)? Specifically, find the level of utility under the credit
constraint and compare it to the level of utility obtained under no credit constraint.
Suppose now that the consumer experiences a temporary increase in real income in period
one to y1 = 9, with real income in period two unchanged.
d. Calculate the effect of this positive surprise in income on c1 and c2 , supposing that
there is no credit constraint on the consumer.
e. Finally, suppose that the credit constraint described in part b is back in place. Will
it be binding? That is, will it affect the consumer’s choices?
Question 4
Intertemporal Consumption-Labor Model – Numerical Look. Consider the in-
tertemporal consumption-labor model. Suppose the lifetime utility function is given by
2
v(B1c1, l1, B2c2, l2) = u(B1c1, l1) + u(B2c2, l2), which is a slight modification of the utility
function presented in Chapter 5. The modification is that preference shifters B1 and B2
enter the lifetime utility function, with B1 the preference shifter in period one and B2 the
preference shifter in period two. In each of the two periods the function u takes the form
u(Btct, lt) = 2
√
Btct + 2
√
lt
. Note the t subscripts –t = 1, 2 depending on which period we are considering. Labor
tax rates, real wages, the real interest rate between period one and period two, and the
preference realizations are given by: t1 = 0.15, t2 = 0.2, w1 = 0.2, w2 = 0.25, r = 0.15, B1 =
1, B2 = 1.2. Finally, the initial assets of the consumer are zero.
a. Construct the marginal rate of substitution functions between consumption and
leisure in each of period one and period two (Hint: these expressions will be functions of
consumption and leisure – you are not being asked to solve for any numerical values yet).
How does the preference shifter affect this intratemporal margin?
b. Construct the marginal rate of substitution function between period-one consumption
and period-two consumption. (Hint: Again, you are not being asked to solve for any
numerical values yet.) How do the preference shifters affect this intertemporal margin?
c. Using the expressions you developed in parts a and b along with the lifetime budget
constraint (expressed in real terms. . . ) and the given numerical values, solve numerically
for the optimal choices of consumption in each of the two periods and of leisure in the two
periods. (Hint: You need to set up and solve the appropriate Lagrangian.) (Note: the
computations here are messy and the final answers do not necessarily work out “nicely.”
To preserve some numerical accuracy, carry out your computations to at least four decimal
places.)
d. Based on your answer in part c, how much (in real terms) does the consumer save in
period one? What is the asset position that the consumer begins period two with?
e. Suppose B2 were instead higher, at 1.6. How are your solutions in parts c and d
affected? Provide brief interpretation in terms of “consumer confidence.”