ECON8026 Advanced Macroeconomic Analysis
Assignment 1
Semester 1, 2023
Question 1
Sales Tax. Consider the standard consumer problem we have been studying, in which a
consumer has to choose consumption of two goods c1 and c2 which have prices (in terms of
money) P1 and P2 , respectively. These prices are prices before any applicable taxes. Many
states charge sales tax on some goods but not on others – for example, many states charge
sales tax on all goods except food and clothes. Suppose that good 1 carries a per-unit sales
tax, while good 2 has no sales tax. Use the variable t1 to denote this sales tax, where t1 is a
number between zero and one (so, for example, if the sales tax on good 1 were 15 percent,
we would have t1 = 0.15).
a. With sales tax t1 and consumer income Y , write down the budget constraint of the
consumer. Explain economically how/why this budget constraint differs from the standard
one we have been considering thus far.
b. Graphically describe how the imposition of the sales tax on good 1 alters the optimal
consumption choice (ie, how the optimal choice of each good is affected by a policy shift
from t1 = 0 to t1 > 0).
c. Suppose the consumer’s utility function is given by u(c1, c2) = log c1 +log c2. Using a
Lagrangian, solve algebraically for the consumer’s optimal choice of c1 and c2 as functions
of P1 , P1, t1 , and Y . Graphically show how, for this particular utility function, the optimal
choice changes due to the imposition of the sales tax on good 1.
Question 2
Interaction of Consumption Tax and Wage Tax. A basic idea of President Bush’s
economic advisers throughout his administration was to try to move the U.S. further away
from a system of investment taxes and more towards a system of consumption taxes. A
nationwide consumption tax would essentially be a national sales tax, a system that many
Western European countries have in place. Here, you will modify our basic consumption-
leisure model to include both a proportional wage tax (which we will now denote by tn,
where, as before, 0 ≤ tn < 1) as well as a proportional consumption tax (which we will
denote by tc, where, as before, 0 ≤ tc < 1). A proportional consumption tax means that
for every dollar on the price tags of items the consumer buys, the consumer must pay
(1 + tc) dollars. Throughout the following, suppose that economic policy has no effect on
wages or prices (that is, the nominal wage W and the price of consumption P are constant
throughout).
a. Construct the budget constraint in this modified version of the consumption-leisure
model. Briefly explain economically how this budget constraint differs from that in the
standard consumption-leisure model we have studied in class.
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b. Suppose currently the federal wage tax rate is 20 percent (tn = 0.20) while the
federal consumption tax rate is 0 percent (tc = 0), and that the Bush economic team is
considering proposing lowering the wage tax rate to 15 percent. However, they wish to
leave the representative agent’s optimal choice of consumption and leisure unaffected. Can
they simultaneously increase the consumption tax rate from its current zero percent to
achieve this goal? If so, compute the new associated consumption tax rate, and explain the
economic intuition. If not, explain mathematically as well as economically why not.
c. A tax policy is defined as a particular combination of tax rates. For example a
labor tax rate of 20 percent combined with a consumption tax rate of zero percent is one
particular tax policy. A labor tax rate of five percent combined with a consumption tax
rate of 10 percent is a different tax policy. Based on what you found in parts a and b above,
address the following statement: a government can use many different tax policies to induce
the same level of consumption by individuals.
d. Consider again the Bush proposal to lower the wage tax rate from 20 percent to
15 percent. This time, however, policy discussion is focused on trying to boost overall
consumption. Is it possible for this goal to be achieved if the consumption tax rate is raised
from its current zero percent?
e. Using a Lagrangian, derive the consumer’s consumption-leisure optimality condition
(for an arbitrary utility function) as a function of the real wage and the consumption and
labor tax rates.
Question 3
A Backward-Bending Aggregate Labor Supply Curve? Despite our use of the
backward-bending labor supply curve as arising from the representative agent’s preferences,
there is controversy in macroeconomics about whether this is a good representation. Specif-
ically, even though a backward-bending labor supply curve may be a good description of a
given individual’s decisions, it does not immediately follow that the representative agent’s
preferences should also feature a backward-bending labor supply curve. In this exercise you
will uncover for yourself this problem. For simplicity, assume that the labor tax rate is
t = 0 throughout all that follows.
a. Suppose the economy is made up of five individuals, person A, person B, person C,
person D, and person E, each of whom has the labor supply schedule given below. Using
the indicated wage rates, graph each individual’s labor supply curve as well as the aggregate
labor supply curve.
b. Now suppose that in this economy, the “usual” range of the nominal wage is be-
tween $10 and $45. Restricting attention to this range, is the aggregate labor supply curve
backward-bending?
c. At a theoretical level, if we want to use the representative-agent paradigm and restrict
attention to this usual range of the wage, does a backward-bending labor supply curve make
sense?
d. Explain qualitatively the relationship you find between the individuals’ labor supply
curves and the aggregate labor supply curve over the range $10 – $45. Especially address
the “backward-bending” nature of the curves.
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Table 1: Labor supply of individuals A-E
Nominal Wage Person A Person B Person C Person D Person E
W (hours) (hours) (hours) (hours) (hours)
$10 20 0 0 0 0
$15 25 15 0 0 0
$20 30 22 8 0 0
$25 33 27 15 5 0
$30 35 30 20 15 0
$35 37 32 25 20 6
$40 36 31 27 25 21
$45 35 30 26 28 30
$50 33 29 24 25 29
Question 4
Quasi-Linear Utility. In the static (i.e., one-period) consumption-leisure model, suppose
the representative consumer has the following utility function over consumption and leisure,
u(c, l) = ln(c) +A · l
where, as usual, c denotes consumption and l denotes leisure. In this utility function, ln()
is the natural log function, and A is a number (a constant) smaller than one that governs
how much utility the individual obtains from a given amount of leisure. Suppose the budget
constraint the individual faces is simply c = (1− t) ·w ·n, where t is the labor tax rate, w is
the the real hourly wage rate, and n is the number of hours the individual works. (Notice
that this budget constraint is expressed in real terms, rather than in nominal terms.)
a. Does this utility function display diminishing marginal utility in consumption? Briefly
explain.
b. Does this utility function display diminishing marginal utility in leisure? Briefly
explain.
c. The representative agent maximizes utility. For the given utility function, plot
the labor supply function (i.e., plot on the vertical axis w and on the horizontal axis the
optimal choice of labor), explaining the logic behind your plotted function. Also, how
would a decrease in the tax rate t affect the optimal amount of labor supply (i.e., increase
it, decrease it, or leave it unchanged)? Briefly explain your logic/derivation. (NOTE: If
you can solve this problem without setting up a Lagrangian, you may do so. )