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Econ 131
Spring 2015
Emmanuel Saez
Problem Set 2 Solution
DUE DATE: March 11
1. True False Statements/Questions
Explain your answer fully based on what was discussed in class (no more than 10 lines per
question), since all the credit is based on the explanation.
a) The corporate income tax is highly progressive.
True: High income people pay a much larger fraction of their income in corporate income
taxes than low and middle income earners. See CBO statistics discussed in class. This is because
corporate taxes fall mostly on capital income earners and capital income is highly concentrated
among high income earners.
b) If incomes are fixed, then it is optimal for the government to fully redistribute incomes.
Partly true: this is true if the government maximizes a utilitarian social welfare function
where everybody has the same utility function and marginal utility decreases with consumption.
However, in practice, most people views of justice differ from the utilitarian criterion and hence
they would not necessarily support 100% redistribution, even with fixed incomes.
c) Suppose the elasticity of average earnings with respect to one minus the tax rate is equal
to 1. Can it be desirable to set the tax rate at 60% if the government cares a lot about
redistribution?
False: with e = 1, the tax revenue maximizing rate is τ ∗ = 1/(1 + e) = 50% (see class notes).
It is never optimal to set the tax rate above τ ∗ . If the tax rate were above the maximizing
revenue rate, decreasing the tax rate would make taxpayers happier and increase tax revenue,
a win-win situation [a Pareto improvement].
d) Suppose the elasticity of reported incomes of high income earners with respect to one
minus the top marginal tax rate is large because high income earners can exploit tax loopholes
to avoid taxes when top tax rates are high. The government should not impose a high tax rate
on high income earners.
True in a narrow sense if the government does not change tax avoidance opportunities as the
revenue maximizing tax rate at the top is given by τ = 1/(1 + a · e) (see class notes). However,
by changing the definition of taxable income (broader base, fewer deductions, elimination of tax
favored income items), it is possible to reduce the elasticity of taxable income and then increase
the top tax rate.
e) The EITC has a positive effect on labor force participation but reduces hours of work
conditional on working.
Partly true: Yes, the EITC encourages labor force participation. Conditional on working,
the EITC reduces hours of work in the plateau and phase-out ranges and has ambiguous effects
in the phase-in region. See class notes.
f) If labor supply responses to taxes and transfers are concentrated along the extensive
margin (whether or not to work), it is desirable to provide benefits to those with no earnings
but then phase-out these benefits quickly as earnings increase.
False, when labor supply are concentrated along the extensive margin, high benefits and
a quick phase-out rate make people stop working completely. Then it is better to have a low
benefit for those with no earnings and subsidies for low earners (such as an EITC program).
g) Evidence from the Israeli Kibbutz shows that redistribution reduces incentives to acquire
an education.
Partly true based on the Abramitzky-Lavy study discussed in class
2. Optimal Income Taxation
Suppose that individuals have the following utility function on consumption c and labor l:
U (c, l) = c − l2 /2
Let us assume that the maximum amount L of labor an individual can supply is large enough
that the constraint l ≤ L never binds. Leisure is defined as L − l. All individuals have the same
wage rate w. If an individual works l and has wage w, he earns wl.
a) Assume that there is no tax and that individuals have no other income but their labor
income. Solve for the labor supply l as a function of w that maximizes the utility of the
individual.
Max wl − l2 /2 implies l = w
b) The government imposes a linear tax on labor income at rate 0 < t < 1.
Show graphically in a consumption-leisure diagram how introducing the linear tax on labor
income modifies the budget constraint of the individual.
Solve for the labor supply l as a function of w and t that maximizes the utility of the
individual. What is the effect of increasing t on l?
l = w(1 − t) increasing t reduces l
c) Suppose that in addition to the linear tax, the government is transferring a fixed amount
R > 0 to every individual. That amount R is independent of labor supply choices.
Show graphically how the budget constraint is affected by the introduction of R.
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Solve for the labor supply l as a function of w, t, and R that maximizes the utility of the
individual. What is the effect of increasing R on l?
l = w(1 − t) increasing R has no effect on l. No income effects.
d) Assume that R = 0 and t > 0 as in b). Compute income taxes collected by the government
per individual as a function of t and w. Draw a graph of taxes collected as a function of t. What
is the tax rate t∗ maximizing tax revenue?
Taxes twl = tw2 (1 − t). Maximized at t∗ = 0.5 (Laffer rate)
e) Suppose that the government has set the tax rate t larger than t∗ (defined in d)). Can we
conclude, no matter what the redistributive tastes of the government are, that it is desirable to
decrease t? Make sure to explain your answer.
Yes, decreasing taxes increases government revenue and increases the utility of individuals.
Win-win situation when you are on the wrong side of the Laffer curve.
f) Suppose now that, wage rate w differs across individuals, but that all individuals have
the same utility function as defined above. What is in this case the income tax rate maximizing
tax revenue? Is it larger or smaller than the t∗ that you obtained in d)?
R
It’s the same. Taxes t(1 − t) w2 f (w)dw
g) Suppose that, the utility function is now given by:
U (c, l) = c − lk+1 /(k + 1)
where k > 0 is a fixed parameter. Solve for the labor supply l as a function of w, t, and k
that maximizes the utility of the individual. How is k related to the elasticity of labor supply
with respect to the net-of-tax wage rate w(1 − t)?
Solve for the tax rate maximizing tax revenue in that situation as a function of k.
l = w1/k (1 − t)1/k
1/k is the elasticity of labor supply with respect to the net-of-tax wage rate w(1 − t).
Taxes w1+1/k t(1 − t)1+1/k maximized for t∗ = k/(k + 1).
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3. Labor supply and benefits
Pam is a single mother with two kids eligible for welfare: she receives $4,000 a year in
benefits and also receives health insurance for her family through Medicaid that she values at
$2,000 a year. If she works, she earns $20 per hour and can work up to 2,000 hours per year.
She gets taxed at 50% for every hour worked. In addition, she loses her Medicaid coverage if
she earns more than $20,000.
a) Draw Pam‘s opportunity set in the earnings-consumption space (carefully labeling the
slope of the budget constraint, intercept points, and any discontinuities) and write down the
opportunity set equation. How much earnings does Pam need to pay as much in taxes as she
receives in transfers?
30,000
25,000
ty
ni
rtu
t
Se
po
Op
Consumption
20,000
15,000
Break even point
(12,000 ; 12,000)
10,000
45
lin
e
5000
0
0
5000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
Earnings
b) Are there any hours of work that she definitely wont choose? Why or why not?
Since Pam looses Medicaid when she earns more than $20,000 her opportunity set displays a
discrete jump down. By earning one more dollar ($20,001) she goes from being able to consume
$16,000 to only $14,001. This is called a tax notch and there are no preferences that can justify
such a choice (it is strictly dominated). The exact earnings interval that she would never choose
is [20,001, 24,000]. At any point of that interval she could reach at least the same consumption
with less hours of work.
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c) Is loosing Medicaid past $20,000 likely to produce a substitution effect, an income effect,
or both? Justify your answer.
The discrete jump down of the opportunity set will only produce an income effect: the
opportunity cost of leisure stays constant (the slope of the Opportunity set isn’t changed) but
the person is now poorer by $2,000.
Congress is concerned that there is too high a tax on work for welfare recipients and passes
a new law. Under the new law benefits are only $3,000 but the tax rate is reduced to 25%
Medicaid rules stay exactly the same.
d) Draw and write the equation of Pam‘s new opportunity set. What it the point where
Pam pays as much in taxes as she receives in transfers?
35,000
t
ty
30,000
Se
i
un
t
or
pp
O
25,000
Consumption
Break even point
(20,000 ; 20,000)
20,000
15,000
10,000
45
lin
e
5000
0
0
5000
10,000
15,000
20,000
25,000
30,000
35,000
40,000
Earnings
e) Discuss how the new tax and transfer system affects her labor choice compared to the
previous one. Discuss separately what would happen for a low earning Pam and a high earning
Pam.
With the new tax system a low earning Pam would certainly work more. The reduction in
benefits from $4,000 to $3,000 generates an income effect and pushes her to work - in the same
time the reduction in the tax rate from 50For a high earning Pam the effect is ambiguous. The
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substitution effect due to the tax rate decrease pushes her to work, however she now has a wider
opportunity set due to the extra income that does not get taxed. This produces an income
effect pushing her to work less.
f) How might this law affect the total number of people on welfare? Given your knowledge
from the class, is the extensive margin of work an important one?
The law would push people on welfare and with zero hours of work to work. Not working
only pays $3,000 now. In addition for every hour of work a person now keeps 75
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