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Problems 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: 0

T(y) =

if y < 60

.5*(y - 60) if y > 60.

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: h = α + βyv + γw 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 1 2 3 4

Non-labor Income

Hours Worked 0 0 10 20

Total Pretax Income 15.15 10.00 8.25 7.25

45.45 60.00 92.50 92.50.

(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.

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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.

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?

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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 website. The data table excludes tax returns with AGI of less than $5000 and of more than $100,000.

Adjusted Gross Income (AGI) Level (000s) 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80 80-85 85-90 90-95 95-100

Number of Returns (millions) 8.853 6.913 5.973 4.906 4.306 3.450 3.521 2.700 2.172 1.500 1.350 1.300 1.150 0.936 0.800 0.650 0.375 0.240 0.157

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 Question 1 You are invited to a small tropical island nation to become a tax policy consultant, with particular respon sibility for analyzing tax-induced distortions in labor supply. Sensing the opportunity for an interesting vacation, 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

0 y < 60 0.5 (y − 60) y ≥ 60.

T (y) =

After arriving in the republic, you discover it is smaller than you thought. The Complete Population Survey (CPS) contains only four observations. Undaunted, you proceed with your proposal to estimate a linear hours-of-work model for the republic’s population: h = α + βy v + γw, where h denotes daily hours, y v is virtual income, and w is the household’s after-tax wage rate. The data you receive are shown in the table below.

Household 1 2 3 4

Non-labor income 0 0 10 20

Hours worked 15.15 10.00 8.25 7.25

Total pre-tax income 45.45 60.00 92.50 92.50

(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.)

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  1 = w 3=   ww 4 =

45.45 15.15 82.5 8.25= 72.5 7.25=

= 3 10 10

w 32

= 5 = 5

The after-tax wages equal: 

T=

w T3 =   w1T w 4=

3 w24

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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 y1v = 0. Household three is taxed. Its total after-tax income is y3 = 92.5 − 0.5 ∗ 32.5 = 76.25. Virtual income is then y3v = y3 − h3w3T = 76.25 − 5 × 8.25 = 35. For household four we get that the virtual income is y4v “estimation”  =for40.

The data is then     0 3  α   15.15   1 1 35 5 β = 8.25 .

1 40 5 γ 7.25 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    15  αβ  =  −0.2  . γ 0.05 Hence, labor supply for household not at the kink is given by: h = α + βy v + γw = 15 − 0.2y v + 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. (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.

Total revenue is T ( y 3 ) + T (y 4 ) = 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.

Question 2

J , C ,L

s.t. (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: 3 J + C ≤ (1 − τ )(Lw + e) = (L + 1) max log(J ) + log(C ) + 4log(1 − L )

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From the FOC’s and the B.C., we get L = 1 < 1 , J = C = 1 .

For type 2, guess that he will be facing the higher tax schedule (i.e. L > 1 1 be 3(1 ) 2+ ( L − ) =7 4 + L8. Consumer 2 solves: 4 +

13

2 2 ). Hence his after tax income will 4

max log(J ) + log(C ) + log(1 − L ) J , C ,L s.t.

J + C ≤ From the FOC’s and the B.C., we get L = (b)

3

>

1

,J = C =

5

8

.

8

4

7 + L 8

Revenue from type 1 is: 1

1

1

4(1 + 3 ) = 3

Revenue from type 2 is: 1

(c)

The per capita tax revenue is

12

5

(1 +

1

4

1 1

1

) + ( ∗ 2) = 2 2 8 2

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: J , C ,L

max log(J ) + log(C ) + log(1 − L )

s.t.

J +

4

3 C ≤

4

3 (L + 1)

From the FOC’s and the B.C., we get L = 1 , J = 1 and C = 2 .3 And thus verify our 2 3 conjecture: taxable 3 income 1 + wl − C < .

2 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: Jmax , C ,L

s.t.

log(J ) + log(C ) + log(1 − L ) J +

From the FOC’s and the B.C., we get L3 = income 1 + wl − C < .

2 Revenue from type 1 is:

1

,J =

3

4

3 C ≤

4

3 (2L + 1)

and C = 1 . And thus4 verify our conjecture: taxable 2

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1

1

2

1

4 (1 + 3 − 3 ) = 6

Revenue from type 2 is: 1

(1 + 1 − 1) =

4 The per capita tax revenue is24 5 which is a drop of 50% triggered by:

1

4

1. A substitution towards the cheaper deductible consumption good 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 (counter balancing the other 2 effects somewhat).

Finally, we also have income effects resulting from lower taxes. (d) The static revenue cost is

∆R = −

1

1

1

1

5

2 ×4 +2 ×2 8 1 1 +5 2 8 16 = − 7 = − = −0.219.

32 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.

Question 3 (a) We have the following tax schedule: T (A G I ) = τ T I = τ max(0, A G I − 8, 950) Hence the marginal rates jump at four “kink” points: • at A G I = 8, 950 from 0 to 10%.

• at A G I = 16, 975 from 10 to 15%.

• at A G I = 32, 550 + 8, 950 = 41, 500 from 15 to 25%.

• at A G I = 78, 850 + 8, 950 = 87, 800 from 25% to 28%.

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We will thus test for bunching at those 4 kink points by adding variables called kink1, kink2, kink3 and kink4 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)).

Figure 2: Cubic fit

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Table 1: Quadratic and Cubic regressions

(1) nbrret AGI AGI squared kink1 kink2 kink3 kink4

(2) nbrret

-0.1827*** (0.0128) 0.0010*** (0.0001) 1.4865*** (0.3561) 0.1820 (0.3055) -0.0310 (0.2795) -0.0121 (0.2885)

AGI cube Constant 8.6800*** Standard errors in parentheses (0.3220) * N p < 0.10 , * * p < 0.05 , * * * p < 0.01 19

-0.2796*** (0.0303) 0.0030*** (0.0006) 0.8453** (0.3236) 0.0706 (0.2267) 0.1406 (0.2114) -0.0831 (0.2127) -0.0000*** (0.0000) 9.9410*** (0.4436) 19

(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.

Question 4 (a) The lifetime budget constraint is: (b)

C 1+

C 1+r

2

+ β(A ) ≤ (1 − τ )(Y 1 − A1 ) +

1+r

(1 − τ )2 2

(Y + A )

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 − τ 1 )(Y 1 − A ) + ( 1 − τ 2 ) (Y +2 A ) − β(A ) which yields the following FOC: 1+r

β (A ) =

(1 − τ )2 − (1 − τ ) 1 1+r

(c) From the FOC, we get:

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A =

2γ

1 ((1 − τ2 ) − (1 − τ1 ))

The elasticity of tax avoidance is then given by: A

Revenues are equal to:

= −

(1

1 − τ1 − τ2 ) − (1 − τ1 )

R = τ1 (Y 1 − A ) + τ2 (Y 2 + A ) Then: ∂R τ −τ = Y 1 − 2A = Y 1 − ∂τ1 γ which has an ambiguous sign. Indeed, raising the tax in the early period:

1

2

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: ∂τ2

∂R

= Y 2 + 2A

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.

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