Lecture 2: Elasticity of taxable income€¦ · where = @logH=@log(1 ˝) is the elasticity of...

Lecture 2: Elasticity of taxable income

Michael Smart

Michael Smart (UToronto) Lecture 2: Elasticity of taxable income 1 / 22

Introduction

Excess burden of income taxesWhat is the excess burden of taxes on personal income?

Standard model:

max U(C,H) subject to C = (1− τ)H

where H is hours worked.

The expenditure function is

e(1− τ,u) = min C − (1− τ)H s.t. U(C,H) ≥ u

where−e1(1− τ,u) = H(1− τ,u)

is compensated labour supply, and

EB(τ) = e(1− τ,u)− e(1,u)− τH(1− τ,u)

Standard model:

where−e1(1− τ,u) = H(1− τ,u)

EB(τ) = e(1− τ,u)− e(1,u)− τH(1− τ,u)

Standard model:

where−e1(1− τ,u) = H(1− τ,u)

EB(τ) = e(1− τ,u)− e(1,u)− τH(1− τ,u)

Standard model:

where−e1(1− τ,u) = H(1− τ,u)

EB(τ) = e(1− τ,u)− e(1,u)− τH(1− τ,u)

Marginal excess burden is therefore

∂EB∂τ

= −e1(1− τ,u)− H(1− τ,u) + τ∂H

∂1− τ= τH1(1− τ,u)

1−t−δ

∆ EB=t H∆H(w)

Figure : Marginal excess burden and compensated labour supply

Define government revenue R(τ) = τH(1− τ,u), so

∂R∂τ

= H(1− τ,u)− τH1(1− τ,u)

Marginal excess burden per dollar of additional revenue is

µ =∂EB/∂τ∂R/∂τ

H − τH1=

τ1−τ ε

1− τ1−τ ε

where ε = ∂ log H/∂ log(1− τ) is the elasticity of labour supply.

But the compensated elasticity of hours worked is usually estimated tobe in [0,0.2] for prime age males.

With ε = 0.1,

τ = 0.4 =⇒ µ ≈ 0.07 and τ = 0.5 =⇒ µ ≈ 0.11!

∂R∂τ

= H(1− τ,u)− τH1(1− τ,u)

H − τH1=

τ1−τ ε

1− τ1−τ ε

With ε = 0.1,

τ = 0.4 =⇒ µ ≈ 0.07 and τ = 0.5 =⇒ µ ≈ 0.11!

∂R∂τ

= H(1− τ,u)− τH1(1− τ,u)

H − τH1=

τ1−τ ε

1− τ1−τ ε

With ε = 0.1,

τ = 0.4 =⇒ µ ≈ 0.07 and τ = 0.5 =⇒ µ ≈ 0.11!

In reality, tax avoidance involves more than just labour supply: Shifts to

1 Tax-deductible expenditures (business use of home andentertainment, health insurance, charitable giving)

2 Tax-advantaged compensation (self-employment, stock optionsand pension, payments in kind)

3 Saving in tax-advantaged forms (dividends and capital gains, realestate)

4 Other forms of tax avoidance and evasion.

Modelling tax avoidance (Feldstein, 1999)A taxpayer has economic income Z but spends D on tax-deductibleconsumption, and C on taxable consumption. Utility is V (C,D,Z ) andbudget constraint is

C + D = (1− τ)Z + τD

orC = (1− τ)(Z − D)

Now taxpayer solves

e(1− τ,u) = min C − (1− τ)(Z − D) s.t. V (C,D,Z ) ≥ u

where −e1 = Y = Z − D is taxable income.

Our previous analysis shows MEB per dollar of revenue is

1−τ γ

1− τ1−τ γ

where γ = ∂ log Y/∂ log(1− τ) is the elasticity of taxable income.

C + D = (1− τ)Z + τD

orC = (1− τ)(Z − D)

Now taxpayer solves

1−τ γ

1− τ1−τ γ

where γ = ∂ log Y/∂ log(1− τ) is the elasticity of taxable income.

C + D = (1− τ)Z + τD

orC = (1− τ)(Z − D)

Now taxpayer solves

1−τ γ

1− τ1−τ γ

where γ = ∂ log Y/∂ log(1− τ) is the elasticity of taxable income.Michael Smart (UToronto) Lecture 2: Elasticity of taxable income 7 / 22

Pure avoidance and optimal tax policy (Chetty, 2009)In the Feldstein model, the optimal tax rate is inversely proportional tothe elasticity of taxable income Y , which may be much larger than theelasticity of economic income Z – especially for high incometaxpayers. But is pure avoidance really so bad?

In Chetty (2009), part of avoidance cost is a transfer to other agents,which has different welfare consequences.

Let taxpayer solve

V (1− τ) = (1− τ)(Z − D)−G(Z ) + [D − H(D)]

where D is income sheltered from tax, and H(D) is cost of sheltering.

Taxpayer FOCs:

G′(Z ∗) = 1− τH ′(D∗) = τ

and the envelope theorem implies

V ′(1− τ) = Y ∗ = Z ∗ − D∗

Let taxpayer solve

V (1− τ) = (1− τ)(Z − D)−G(Z ) + [D − H(D)]

Taxpayer FOCs:

G′(Z ∗) = 1− τH ′(D∗) = τ

V ′(1− τ) = Y ∗ = Z ∗ − D∗

Let taxpayer solve

V (1− τ) = (1− τ)(Z − D)−G(Z ) + [D − H(D)]

Taxpayer FOCs:

G′(Z ∗) = 1− τH ′(D∗) = τ

V ′(1− τ) = Y ∗ = Z ∗ − D∗Michael Smart (UToronto) Lecture 2: Elasticity of taxable income 8 / 22

Pure avoidance and optimal tax policyWe wish to generalize our excess burden concept to incorporatetransfers among agents.

Define social welfare as the sum of taxpayer utility and tax revenue,and a fraction α ∈ [0,1] of sheltering cost H(D):

W (τ) = V (1− τ) + αH(D∗) + τ(Z ∗ − D∗)

(Why?)

Marginal welfare cost of taxation is

−∂W∂τ

= V ′(1− τ) + αH ′∂D∗

∂1− τ− Y ∗ + τ

∂Y ∗

∂1− τ

∂D∗

∂1− τ+

∂Y ∗

∂1− τ

∂Z ∗

∂1− τ+ (1− α)

∂Y ∗

∂1− τ

1− τ[αZ · ε+ (1− α)Y · γ]

W (τ) = V (1− τ) + αH(D∗) + τ(Z ∗ − D∗)

(Why?)

−∂W∂τ

= V ′(1− τ) + αH ′∂D∗

∂1− τ− Y ∗ + τ

∂Y ∗

∂1− τ

∂D∗

∂1− τ+

∂Y ∗

∂1− τ

∂Z ∗

∂1− τ+ (1− α)

∂Y ∗

∂1− τ

1− τ[αZ · ε+ (1− α)Y · γ]

W (τ) = V (1− τ) + αH(D∗) + τ(Z ∗ − D∗)

(Why?)

−∂W∂τ

= V ′(1− τ) + αH ′∂D∗

∂1− τ− Y ∗ + τ

∂Y ∗

∂1− τ

∂D∗

∂1− τ+

∂Y ∗

∂1− τ

∂Z ∗

∂1− τ+ (1− α)

∂Y ∗

∂1− τ

1− τ[αZ · ε+ (1− α)Y · γ]

W (τ) = V (1− τ) + αH(D∗) + τ(Z ∗ − D∗)

(Why?)

−∂W∂τ

= V ′(1− τ) + αH ′∂D∗

∂1− τ− Y ∗ + τ

∂Y ∗

∂1− τ

∂D∗

∂1− τ+

∂Y ∗

∂1− τ

∂Z ∗

∂1− τ+ (1− α)

∂Y ∗

∂1− τ

1− τ[αZ · ε+ (1− α)Y · γ]

W (τ) = V (1− τ) + αH(D∗) + τ(Z ∗ − D∗)

(Why?)

−∂W∂τ

= V ′(1− τ) + αH ′∂D∗

∂1− τ− Y ∗ + τ

∂Y ∗

∂1− τ

∂D∗

∂1− τ+

∂Y ∗

∂1− τ

∂Z ∗

∂1− τ+ (1− α)

∂Y ∗

∂1− τ

1− τ[αZ · ε+ (1− α)Y · γ]

Pure avoidance and optimal tax policyIn general, welfare cost of atxation depnds on a weighted average ofelasticity of economic income and elasticity of taxable income.

If sheltering has only deadweight costs,

α = 0 =⇒ −∂W∂τ

1− τY · γ

If sheltering has only transfer costs,

α = 1 =⇒ −∂W∂τ

1− τZ · ε

Examples of transfer costs of shifting:

shifting to other tax bases (e.g. corporate income)spillovers from shifting activities (e.g. charity?)penalties for tax evasion

α = 0 =⇒ −∂W∂τ

1− τY · γ

α = 1 =⇒ −∂W∂τ

1− τZ · ε

α = 0 =⇒ −∂W∂τ

1− τY · γ

α = 1 =⇒ −∂W∂τ

1− τZ · ε

Estimating the ETI from tax reforms

We want to know how much taxable income responds to changes intax rates.

A natural way to estimate is to examine actual responses to real-worldtax reforms.

Estimating the ETI informs us about two separate but relatedquestions:

If we raise tax rates by 1 per cent, how much will revenue go up?If we raise $1 more in revenue, how much will excess burden goup?

Estimating the ETI from tax reforms

We want to know how much taxable income responds to changes intax rates.

A natural way to estimate is to examine actual responses to real-worldtax reforms.

Estimating the ETI informs us about two separate but relatedquestions:

If we raise tax rates by 1 per cent, how much will revenue go up?If we raise $1 more in revenue, how much will excess burden goup?

Estimating the ETI

We want to estimateYit = Y 0

it (1− τit )γ

orlog Yit = log Y 0

it + γ log(1− τit )

where Y 0it is potential (zero-tax) income.

Problem: Y 0it is unobserved.

Assume that differences among taxpayers are uncorrelated with taxchanges, and income grows at a common rate over time (“paralleltrends”):

log Yit = αi + βt + γ log(1− τit ) + εit (1)

Estimating the ETI

We want to estimateYit = Y 0

it (1− τit )γ

orlog Yit = log Y 0

it + γ log(1− τit )

where Y 0it is potential (zero-tax) income.

Problem: Y 0it is unobserved.

Assume that differences among taxpayers are uncorrelated with taxchanges, and income grows at a common rate over time (“paralleltrends”):

log Yit = αi + βt + γ log(1− τit ) + εit (1)

This model admits a difference-in-difference (DD) estimator of γ, usingevidence on tax reforms that affect people in different tax bracketsdifferently.

If we consider only two periods and two tax brackets, this simplifiesthings. Let t ∈ {0,1} and let the tax bracket (i.e. MTR) of individual i bebi ∈ {l ,h}.

Differencing over time,

∆ log Yit = ∆βt + γ∆ log(1− τbi t ) + ∆εit (2)

This still depends on unobservable economic changes. Averaging overindividuals in brackets l ,h and differencing gives the DD estimator

γ̂ =∆log Yh −∆log Yl

∆ log(1− τh)−∆ log(1− τl)

Since reforms tend to change τh more than τl , h and l serve astreatment and control.

This still depends on unobservable economic changes.

Averaging overindividuals in brackets l ,h and differencing gives the DD estimator

The DD estimator and quasi-experimental methods

Estimating causal effects from time series data is hard because ofunobserved temporal variation.

To deal with this we’d like to be able to follow a control groupunaffected by tax changes but exposed to the same economicenvironment, and compare their differences.

Randomized trials are rare in public finance. Next best is aquasi-experiment: a policy reform that, by its nature, creates a“treatment” group who are affected by a policy change, and a “control”group who are not. But, assumptions are strong:

parallel trends: treatment and control would evolve in the sameway on average, in the absence of treatment; andadditivity: treatment effect is additive and homogeneous, so it canbe estimated from mean differences in treatment and controloutcomes.

Evidence from US TRA86

Feldstein (1995) estimates (2) using tax return data around the 1986US Tax Reform Act.

TRA86 was a “tax-flattening” reform: top MTR was 50% in 1985, butonly 28% in 1988 – a quasi-experiment for ETI.

Data from NBER–Michigan tax panel suggest a treatment group andtwo possible controls:

bracket 1985 MTR observationshighest 49–50 57high 42–45 197medium 22–38 3538

Dependent variable is taxable income minus capital gains.

Evidence from US TRA86

Feldstein (1995) estimates (2) using tax return data around the 1986US Tax Reform Act.

TRA86 was a “tax-flattening” reform: top MTR was 50% in 1985, butonly 28% in 1988 – a quasi-experiment for ETI.

Data from NBER–Michigan tax panel suggest a treatment group andtwo possible controls:

bracket 1985 MTR observationshighest 49–50 57high 42–45 197medium 22–38 3538

Dependent variable is taxable income minus capital gains.

Estimates using “high” as the control group:

bracket ∆ log(1− τ) ∆ log TIhighest (49-50) +42.2 +44.8high (42-45) +25.6 +20.3difference +16.6 +24.5

The corresponding elasticity estimate for this case is therefore24.5/16.6 ≈ 1.48. All estimated elasticities exceed one, implying muchstronger responses to taxation than previously estimated.

This suggests tax revenues decrease if the tax rate rises above about50%.

Estimates using “high” as the control group:

bracket ∆ log(1− τ) ∆ log TIhighest (49-50) +42.2 +44.8high (42-45) +25.6 +20.3difference +16.6 +24.5

The corresponding elasticity estimate for this case is therefore24.5/16.6 ≈ 1.48. All estimated elasticities exceed one, implying muchstronger responses to taxation than previously estimated.

This suggests tax revenues decrease if the tax rate rises above about50%.

Problems with the methodologyTax rate endogeneity

The structural model is

log Yit = αi + βt + γ log(1− τt (Yit )) + uit

Note that τt (Yit ) is endogenous, because of rate progressivity.Feldstein proxies it with 1985 tax bracket (i.e. dummies for initial taxbracket bi times dummies for time period t).

Subsequent literature using panel data instruments with τt (Yi0). Thispicks up variation in laws but not in behaviour. A more preciseinstrument, useful in many contexts.

Problems with the methodologyIncome endogeneityThe reduced form model is

log Yit = αi + βt + γ̂ log(1− τt (Yi0)) + uit

Even with an instrument τt (Yi0), income changes maybe correlatedwith initial income, so that “parallel trends” assumption violated.

Cases:

increase in inequality: If rich are getting richer anyway, thenE(uitYi0) > 0Mean reversion in income: If income changes are transitory, thenE(uitYi0) < 0

For a tax flattening reform, #1 biases γ̂ upwards, #2 downwards. Neteffect is uncertain.

For evidence on this, the graph from Saez, Slemrod, and Giertz (2010):

Cases:

increase in inequality: If rich are getting richer anyway, thenE(uitYi0) > 0

Mean reversion in income: If income changes are transitory, thenE(uitYi0) < 0

Cases:

For evidence on this, the graph from Saez, Slemrod, and Giertz (2010):Michael Smart (UToronto) Lecture 2: Elasticity of taxable income 18 / 22

Solutions to the endogeneity problemTo deal with mean reversion,

1 include rich controls for initial income: income class dummies, orpolynomials in Yi0 (Auten and Carroll, 1999). This absorbs muchof variation in tax rate changes, making identification harder.

2 instrument for τt (Yit ) with lags τt (Yit−p), τt (Yit−p−1), . . ., for p > 1.Weber (2013): This increases ETI substantially for US data.

3 Use repeated cross-section data (Saez, Slemrod and Giertz,2010)

To deal with inequality trends,

analyze several tax reforms, some “tax flattening” and someprogressivity-increasing. Inequality trends bias estimates inopposite directions and may wash out overall.

On balance, estimated ETI is smaller – a range of [0.2,0.6] for US datain 1980s and 1990s. Estimates in the literature are quite sensitive towhich reforms studied, which data, which controls, and so on.

Other issues with the ETIIntertemporal shifting

Taxpayers may respond to reforms in the short run by shifting incomeover time to the low-tax period. The effect on present value revenuesand hence on true µ is correspondingly smaller.

This might be visible in SSG’s aggregate data around 1993 taxincrease. Goolsbee (JPE, 2000) argues a big part of effect of thereform was in timing of stock realizations by top corporateexecutives.

Cross-base shifting

Estimated ETI may reflect shifting between tax bases rather than pureavoidance/evasion responses.

e.g. taxpayers can avoid personal taxes by sheltering incomeinside corporations, or vice versa. Then the net effect on revenuesand on µ depends on τp − τc , not τp – we overestimate the MCPFusing the Feldstein approach.

Other issues with the ETIIntertemporal shifting

Taxpayers may respond to reforms in the short run by shifting incomeover time to the low-tax period. The effect on present value revenuesand hence on true µ is correspondingly smaller.

This might be visible in SSG’s aggregate data around 1993 taxincrease. Goolsbee (JPE, 2000) argues a big part of effect of thereform was in timing of stock realizations by top corporateexecutives.

Cross-base shifting

Estimated ETI may reflect shifting between tax bases rather than pureavoidance/evasion responses.

e.g. taxpayers can avoid personal taxes by sheltering incomeinside corporations, or vice versa. Then the net effect on revenuesand on µ depends on τp − τc , not τp – we overestimate the MCPFusing the Feldstein approach.

Other issues with the ETICross-base shifting: Examples

Much of response to TRA86 may be conversion of smallbusinesses from “C” corporations to “S” corporations(flow-through entities).Gordon and Slemrod (2000): Much of variation in incomes ofhigh-income US taxpayers since 1965 can be explained bychanges in the corporate tax rate.Silamaa and Veall (JPubE, 2001) examine the 1988 tax flatteningin Canada. They estimate elasticity of 1.3 for self-employmentincome, but almost zero for other income. Self-employed are mostlikely to be shifting from corporate form (but also may have othermargins of substitution).Milligan and Smart (2014) estimate two-thirds of response toprovincial tax changes in Canada is due to income shifting toAlberta by the richest 0.1% of taxpayers.

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