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1. You are invited to a small tropical island nation to become a tax policy consultant, with particular
responsibility for analyzing tax-induced distortions in labor supply. Since it is winter in
Cambridge, you accept. The republic's finance minister faxes you a copy of the current tax
schedule, defined over total daily (labor plus non-labor) income, which is:
After arriving in the republic, you discover it is smaller than you thought. The Complete Population
Survey contains only four observations. Undaunted, you proceed with your proposal to estimate
a linear hours-of-work model for the republic's population:
where h denotes daily hours, yv is virtual income, and w is the household's after-tax wage rate.
The data you receive are shown below:
Household Non-labor Income Hours Worked Total Pretax Income
1 0 15.15 45.45
2 0 10.00 60.00
3 10 8.25 92.50
4 20 7.25 92.50.
Problem
T(y) =
0 if y < 60
.5*(y - 60) if y > 60.
h = α + βyv + γw

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(a) Use these data to estimate (α, β, γ). You should be able to solve for the parameters exactly (i.e.,
with no error terms in the hours equation) using three data points. (Hint: Remember to check
whether each household is on a linear segment of the budget set, or at a "kink point.")
(b) Compute the total amount of revenue currently collected by the tax system, and find the lump
sum tax (equal across all households) that would be needed to raise the same amount of
revenue. For household 1, find the equivalent variation of shifting to this tax.
2. The next summer, you are invited by the government of a small state in one of the
large South American nations to help in designing a tax reform. Eager for a some
Andean skiing, you accept. Upon arrival, you discover that the economy consists of
two types of individuals, with each type accounting for half of the population. Jello is
the numeraire consumption good in the economy. The first type of individual has a
wage of 1 jello packet per unit of labor supplied, and the second has a wage of 2 jello
packets per unit of labor. Individuals of each type have an endowment of one unit of
labor. Each individual also has a lump sum income of one packet of jello.
Individuals of both types have preferences over three goods: leisure (1-L), jello (J), and
coconuts (C). These preferences are described by a utility function:
U = log J + log C + log (1-L).
There is a domestic production technology that turns one package of jello into one unit
of a general consumption good. The process can be reversed.

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The current tax system is as follows. Taxable income, which is measured in units of jello, is
defined as labor earnings plus lump sum income. The marginal tax rate on the first 1.5 jellos
of income is 25 percent, while the marginal tax rate is 50 percent on all income above this
level.
(a) You begin your analysis of the tax system by trying to determine the optimal labor supply
behavior of the two types of individuals in this economy. Find the labor supply,
consumption of jello, and consumption of coconuts for individuals with a wage of 1, and for
those with a wage of 2.
(b) What is per capita tax revenue from the current tax code?
(c) The domestic retail lobby has proposed a tax deduction for purchases of the general
consumption good. If all expenditures on this good were excluded from taxable income,
describe the new tax schedule. How would per capita revenue change if this policy were
adopted? Explain the factors that contribute to any revenue change that you identify.
(d) The retail lobby has evaluated the potential revenue cost of the proposal in (c) by
multiplying outlays on the general consumption good under the current (no deduction) tax
system by estimates of marginal tax rates for consumers of types 1 and 2. (This is known
as "static revenue estimation.") Do your calculations offer any insights on the validity or
limitations of this approach?
3. The following table provides data on the number of single tax filers in various Adjusted Gross
Income (AGI ) categories for 2008 in the United States (the data are based on actual U.S. data with
some interpolation). These data are also available in a STATA dataset (.dta file) on the course’s .
The data table excludes tax returns with AGI of less than $5000 and of more than $100,000.

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Adjusted Gross Income (AGI) Level (000s) Number of Returns (millions)
5-10 8.853
10-15 6.913
15-20 5.973
20-25 4.906
25-30 4.306
30-35 3.450
35-40 3.521
40-45 2.700
45-50 2.172
50-55 1.500
55-60 1.350
60-65 1.300
65-70 1.150
70-75 0.936
75-80 0.800
80-85 0.650
85-90 0.375
90-95 0.240
95-100 0.157

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The 2008 personal income tax schedule for single individuals applies a 10% tax rate when
taxable is below $8025, a 15% marginal rate on taxable income between $8025 and
$32,550, 25% for taxable income between $32550 and $78850, and 28% for taxable
income above $78850. To convert AGI to taxable income, assume that all single taxpayers
claim a standard deduction of $5450, and one dependent allowance of $3500, so Taxable
Income = AGI - 8950.
(a) Using STATA or another package of your choice, estimate a quadratic relating the
number of returns in each income "bin" to the first three powers of AGI at the midpoint
of the bin. Then add an indicator variable for the bin (or bins) in which you might
expect to see "bunching" of taxpayers. Draw a diagram of the budget that corresponds
to the tax rate schedule and graphically or otherwise explain your rationale for
searching for particular bunching patterns. Can you reject the null hypothesis of "no
bunching"? How do your results change if you use a third-order polynomial to fit the
income distribution?
(b) What difficulties do you encounter in trying to assess the degree of bunching using
aggregate cell data such as that in this table? Under what conditions might aggregate
data of this form be reasonably helpful in testing for bunching?

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4. Consider an individual with preferences over consumption in two periods given by:
V(C1, C2) = log C1 + [1/(1+δ)] log C2.
This individual faces a wage income tax, so that the interest rate at which she can
borrow and lend is simply r. Assume that her wage income in periods 1 and 2 is fixed
at Y1 and Y2, and that her labor income tax rate in period 1 is 1 while that in period
2 is 2. Assume that 1 > 2, and that both taxes are linear. Further assume that
the individual has access to a “tax avoidance technology” that permits wage income to
be shifted from period 1 to period 2. If the individual chooses to shift A dollars from
period 1 to period 2, where A is between 0 and Y1, her taxable income in period 1
will be Y1 – A and that in period 2 will be Y2 + A. Using the tax avoidance
technology is costly; the cost of shifting A dollars is (A). This cost can be viewed as
the legal and administrative fees associated with tax avoidance, and it must be paid in
period 1.
(a) Find the lifetime budget constraint for this individual, recognizing both the impact of
tax avoidance on income net of taxes, and the cost of tax avoidance.
(b) Now obtain first order conditions for the optimal choice of A. Does the optimal level
of A depend on the utility function? Explain why or why not.
(c) Consider the case in which (A) = A2, and assume that r = 0. Obtain a formula
for A as a function of the tax rates in the two periods, and compute the elasticity of
tax avoidance (A) with respect to (1- 1). Briefly describe the implications for the
impact of tax changes on revenue.

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Solutions
1. (a) Note that household two is at the kink induced by the exemption threshold. We will
ignore it in estimating the model. Household 1 has total income less than the
exemption amount and therefore will not be taxed. The pre-tax wages are
The after-tax wages equal:
Virtual income is the post-tax income that the individual would get if his earnings
were equal to zero was allowed to stay on the "virtual" linear schedule. For household
1, its virtual income is y1 =0. Household v three is taxed. Its total after-tax income is
y3 = 92.5 − 0.5 ∗ 32.5 = 76.25. Virtual income is then y3 = v T v y3 − h3w3 = 76.25
− 5 × 8.25 = 35. For household four we get that the virtual income is y4 = 40. The
data for “estimation” is then

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Conveniently, we don’t even need to use anything we learned in metrics, because the system
has as many parameters as observations. The solution is given by
Hence, labor supply for household not at the kink is given by:
h = α + βyv + γw = 15 − 0.2yv +0.05wT
Finally, we verify that household two’s behavior is consistent with the model. With a virtual
income of zero and after-tax wage of 6, household two would like to work
h = 15 − 0.2 × 0+0.05 × 6 = 15.3
hours, which would result in an income of 91.8, well in excess of the exemption amount.
However, with virtual income of 30 and after-tax wage of 3 household two would like to
work
h = 15 − 0.2 × 30 + 0.05 × 3=9.15
hours, which would result in an income of 54.9, less than the exemption amount. Thus
household two’s behavior is consistent with the estimated model.

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(b) Compute the total amount of revenue currently collected by the tax system, and find the
lump sum tax (equal across all households) that would be needed to raise the same
amount of revenue. For household 1, f ind the equivalent variation of shifting to this tax.
Total revenue is T (y3)+ T (y4)=2 × 0.5 × (92.5 − 60) = 32.5. The lump sum tax required
to raise the same amount of revenue is simply 32.5/4=8.125. Recall that the equivalent
variation is the amount of money an individual would be willing to pay to avoid a
policy change. Since the only difference between the nonlinear income tax and the lump
sum tax for household one is the lump sum, the individual would be willing to pay an
amount equal to the lump sum to avoid it, i.e. 8.125.
2. (a) Let us guess (and later verify) that consumer 1 will work less than 0.5 units and
hence face the 25% marginal tax rate. Consumer 1 solves:

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(b) Revenue from type 1 is:
Revenue from type 2 is:
The per capita tax revenue is
(c) Taxable income is now 1+ wl − C . Let us guess (and later verify) that consumer 1 will face the
25% marginal tax rate. Consumer 1 solves:

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From the FOC’s and the B.C., we get L =income 1+ wl − C<13, J =12and C =23. And thus verify our
conjecture: taxable .
Let us guess (and later verify) that consumer 2 now also faces the 25% marginal tax rate (since his
taxable income drops in consumption). Consumer 2 solves:
From the FOC’s and the B.C., we get L =income 1+ wl − C<12, J =34and C =1 . And thus verify our
conjecture: taxable .
Revenue from type 1 is:
Revenue from type 2 is:
The per capita tax revenue is 24 which is a drop of 50% triggered by
1. A substitution towards the cheaper deductible consumption good

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2. A switch of type 2 individuals towards the lowest tax bracket (higher deductions reduce taxable
income)
On the other hand, because of the lower marginal rate, the high-skilled worker works more hours
(counterbalancing the other 2 effects somewhat). Finally, we also have income effects resulting from
lower taxes.
(d) The static revenue cost is
The actual revenue cost was -0.208, which is slightly less than the static estimate. In this case the
static approximation is reasonable, but in general this may not be the case. The static estimate picks up
only the mechanical revenue loss, and does not include the effects from behavioral responses.
3 (a) We have the following tax schedule:

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Hence the marginal rates jump at four “kink” points:
• at AGI =8, 950 from 0 to 10%.
• at AGI = 16, 975 from 10 to 15%.
• at AGI = 32, 550 + 8, 950 = 41, 500 from 15 to 25%.
• at AGI = 78, 850 + 8, 950 = 87, 800 from 25% to 28%.
We will thus test for bunching at those 4 kink points by adding variables called kink1, kink2, kink3
andkink4 for each bins in which these 4 points fall.
Figures (1) and () illustrate that the fit of the quadratic is decent while the cubic seems to perform
even a little better. Table (1) shows that the only kink were we can reject the null of “no bunching”
is the first kink. This is not surprising since this is the kink between paying taxes and no taxes. The
level of significance drops from 0.01 to 0.05 when we move from the quadratic (column (1)) to the
cubic specification (column (2)).

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Table 1: Quadratic and Cubic regressions
(1) (2)
nbrret nbrret
AGI -0.1827*** -0.2796***
(0.0128) (0.0303)
AGI squared 0.0010*** 0.0030***
(0.0001) (0.0006)
kink1 1.4865*** 0.8453**
(0.3561) (0.3236)
kink2 0.1820 0.0706
(0.3055) (0.2267)
Kink3 -0.0310 0.1406
(0.2795) (0.2114)
kink4 -0.0121 -0.0831
(0.2885) (0.2127)
AGI cube -0.0000***
(0.0000)
Constant 8.6800*** 9.9410***
(0.3220) (0.4436)
N 19 19
Standard errors in parentheses
* p<0.10 ,** p<0.05 ,*** p<0.01

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(b) Identifying bunching with aggregate data is difficult because (i) the bins are relatively
large, (ii) the number of observations is limited (iii) data on taxable income are coarse, (iv)
non-convexities of other social programs (e.g. EITC, means-tested benefits, ...) are omitted
and (v) the number of kinkpoints is relatively large compared to the number of bins.
Aggregate data might be helpful if (i) kink points fell on the boundaries of the cells, (ii)
we had data on taxable income instead of AGI and/or if (iii) the bins were finer.
4 (a)The lifetime budget constraint is:
The optimal level of revenue shifting does not depend on the utility function because it
does not affect utility directly. The optimal level of shifting is chosen to maximize the
present value of wealth (1 − τ1)(Y1 − A)+ (1−τ2) 1+r (Y2 + A) − β(A) which yields the
following FOC:
(b)
(c) From the FOC, we get:
The elasticity of tax avoidance is then given by:

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Revenues are equal to:
Then:
which has an ambiguous sign. Indeed, raising the tax in the early period:
1. Has a mechanic positive effect on tax revenues (at constant reported income)
2. Has a negative effect by incentivizing to shift income from the early high-tax period to
the late low-tax period.
The relative importance of the 2nd effect will depend on how much tax evasion is
occurring which depends on the marginal cost (seeγ) and benefit (see τ1 − τ2 ) of
evading. But:
which has an positive sign. Indeed, raising the tax in the early period:
1. Has a mechanic positive effect on tax revenues (at constant reported income)
2. Has a positive effect by disincentivizing to shift income from the early high-tax period
to the late low-tax period.