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Using elasticities to derive optimal income tax rates Emannuel Saez - - PowerPoint PPT Presentation

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Using elasticities to derive optimal income tax rates Emannuel Saez (2001) Tax and transfer policies M2 PPD Nicholas McSpedden-Brown Introduction How much progressivity should there be in tax schedules? equity-efficiency trade-off :

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SLIDE 1

Using elasticities to derive

  • ptimal income tax rates

Emannuel Saez (2001)

Tax and transfer policies M2 PPD Nicholas McSpedden-Brown

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Introduction

  • How much progressivity should there be in tax

schedules? ⟹ equity-efficiency trade-off : redistribution vs incentives

  • Optimal tax rate: Tax rate that collects the

most revenue

  • Original model : Mirrlees (1971)
  • Saez’s goal: to clearly show that there is a

simple link between optimal tax formulas and elasticities of earnings.

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Plan

  • 1. Optimal marginal tax rate for top

incomes

  • 2. General non-linear optimal tax

rates for any tax bracket.

  • 3. Numerical simulations of optimal

tax schedules

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SLIDE 4
  • 1. HIGH INCOME OPTIMAL TAX

RATES

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Base specifications

  • Maximisation of a utility function 𝑣 = 𝑣(𝑑, 𝑨)

Where 𝑣𝑑 =

𝑒𝑣 𝑒𝑑 > 0 , 𝑣𝑨 = 𝑒𝑣 𝑒𝑨 < 0 , ( 𝑨 = π‘₯π‘š),

according to the constraint 𝑑 = 𝑨 1 βˆ’ 𝜐 + 𝑆 Where

  • Ο„ is the top marginal tax rate on
  • R is virtual (non-labour) income : this is the post-tax income

and individual would get if he supplied zero labour and was allowed to stay on the β€œvirtual” linear schedule

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SLIDE 6

For those who failed/skipped/forgot Micro 101…

  • Substitution effect : If the price of a good increases

relative to another, then people will consume relatively more of the other good.

  • ⟹ If the tax rate goes up, leisure becomes more

attractive because the β€˜price’ paid for it (after-tax income forgone by not working) has fallen.

  • Income effect : If total income is reduced, then people

will cut back on the consumption of all goods that are not essential (i.e. normal goods).

  • ⟹ If the tax rate goes up, I have less income, and

therefore I β€˜consume’ less leisure, i.e. I work more.

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Elasticity concepts

  • Uncompensated elasticity of earnings ∢ πœ‚π‘£ = 𝑒𝑨

𝑨 𝑒(1βˆ’πœ) 1βˆ’πœ

: (uncompensated, because it does not compensate for a change in income)

  • Income effects (= the marginal propensity to earn out of

non-labour income): πœƒ = 1 βˆ’ 𝜐

𝑒𝑨 𝑒𝑆 ≀ 0, since leisure is

assumed not to be an inferior good.

  • Compensated elasticity of earnings :

πœ‚π‘‘ = 1βˆ’πœ

𝑨 𝑒𝑨 𝑒 1βˆ’πœ 𝑣 = 𝑑𝑑𝑒 : (purely substitution effects since it

compensates for a change income)

  • Slutsky equation: πœ‚π‘‘ = πœ‚π‘£ βˆ’ πœƒ β‰₯ 0
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Deriving the high income optimal tax rate

  • Government sets top marginal rate Ο„ for

incomes above 𝑨

  • Population with income above 𝑨 normalised to

1

  • β„Ž(𝑨) : density of earnings distribution at
  • ptimum tax regime
  • Consider a small increase dΟ„ in the top tax rate

Ο„ for incomes above 𝑨

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High income tax rate perturbation

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Decomposing the change in total taxes paid

  • Total taxes paid at income 𝑨 above 𝑨 = Marginal

rate for incomes above 𝑨 Γ— Income above 𝑨 + Total taxes paid at income 𝑨

  • ⟹ π‘ˆ 𝑨 = 𝜐 𝑨 βˆ’ 𝑨 + π‘ˆ(𝑨 )
  • ⟹ π‘’π‘ˆ 𝑨 = 𝑨 βˆ’ 𝑨 π‘’πœ + πœπ‘’π‘¨
  • ⟹

π‘’π‘ˆ 𝑨 β„Ž 𝑨 𝑒𝑨 = 𝑁 + 𝐢

∞ 𝑨

  • The total taxes paid therefore changes due to two

things : a mechanical effect and behavioural responses

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The mechanical effect

  • Mechanical effect: The increase in tax receipts

if there were no behavioural responses.

  • Taxpayer with income 𝑨 > 𝑨 pays 𝑨 βˆ’ 𝑨 π‘’πœ in

additional taxes.

  • Summing over population with 𝑨 > 𝑨 , we

have total mechanical effect on tax receipts: 𝑁 = 𝑨𝑛 βˆ’ 𝑨 π‘’πœ

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Behavioural responses

  • As 𝑨 = 𝑨 1 βˆ’ 𝜐, 𝑆 , therefore with total differential:
  • 𝑒𝑨 = βˆ’

πœ–π‘¨ πœ– 1βˆ’πœ π‘’πœ + πœ–π‘¨ πœ–π‘† 𝑒𝑆

  • Let’s express this in terms of income effect and

uncompensated elasticity :

  • πœƒ = 1 βˆ’ 𝜐

𝑒𝑨 𝑒𝑆 β‡’ πœ–π‘¨ πœ–π‘† = πœƒ (1βˆ’πœ)

  • πœ‚π‘£ = 𝑒𝑨

𝑨 𝑒(1βˆ’πœ) 1βˆ’πœ

β‡’ βˆ’

πœ–π‘¨ πœ– 1βˆ’πœ = πœ‚π‘£π‘¨ 1βˆ’πœ

  • And as 𝑒𝑆 = 𝑨 π‘’πœ (overall increase in virtual income),
  • Therefore: 𝑒𝑨 = βˆ’(πœ‚π‘£π‘¨ βˆ’ πœƒπ‘¨ )

π‘’πœ 1βˆ’πœ : reduction in

individual z’s earnings due to behavioural responses

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Reduction in tax receipts due to behaviour responses

  • As we saw, a reduction in earnings of dz implies a

reduction in tax receipts of Ο„dz, for one individual.

  • This implies total that the total reduction in tax

receipts is :

  • 𝐢 =

βˆ’ πœ‚π‘£π‘¨ βˆ’ πœƒπ‘¨

πœπ‘’πœ 1βˆ’πœ β„Ž 𝑨 𝑒𝑨 ∞ 𝑨

= βˆ’(πœ‚ 𝑣𝑨𝑛 βˆ’ πœƒ 𝑨 ) πœπ‘’πœ 1 βˆ’ 𝜐

  • Where πœ‚ 𝑣 is the weighted average of the

uncompensated elasticity, and πœƒ the average income effect.

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Obtaining the optimal tax rate

  • Need to equalise the revenue effect (the sum of the

mechanical effect and behavioural response) to the welfare effect.

  • Compute welfare effect : Let 𝑕 = Marginal social utility
  • f money for top bracket tax payers divided by

marginal value of public funds for government. Thus each additional dollar raised by government as a result

  • f tax reduces on average social welfare of the top

bracket by 𝑕 .

  • Hence the total welfare loss due to tax reform is 𝑕 M.
  • Revenue effect = Welfare effect ⇔ M+B = gM
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Interpretation

  • Result:

𝜐 1βˆ’πœ = (1βˆ’π‘• )(𝑨𝑛 𝑨 βˆ’1) πœ‚ 𝑣𝑨𝑛 𝑨 βˆ’πœƒ

  • Decreasing function of 𝑕 , πœ‚ 𝑣, and increasing in

πœƒ .

  • When 𝑨 is close to the top, 𝑨𝑛 𝑨

tends to 1 ⟹ 𝜐 tends to zero. This is because M is negligible compared to B near the top.

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𝑨𝑛 𝑨 for the U.S. in 1992/93 : Constant for high incomes ⟹ Zero top result has no practical interest

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Pareto distributions

  • Distributions with constant 𝑨𝑛 𝑨

ratio are exactly Pareto distributions.

  • A Pareto distribution is such that:

𝑄𝑠𝑝𝑐 π½π‘œπ‘‘π‘π‘›π‘“ > 𝑨 = (𝑨 𝑨)

𝑏

  • We have 𝐹 π‘Ž = 𝑨𝑛 = 𝑏𝑨

π‘βˆ’1 β‡’ 𝑨𝑛 𝑨 = 𝑏 π‘βˆ’1 .

For 𝑨𝑛 = 2, 𝑏 = 2.

  • The higher a, the thinner is the tail of the

income distribution

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Rewriting the optimal marginal tax as a limiting tax for high incomes

  • From 𝜐

1βˆ’πœ = (1βˆ’π‘• )(𝑨𝑛 𝑨 βˆ’1) πœ‚ 𝑣𝑨𝑛 𝑨 βˆ’πœƒ

∢

  • β‡’ 𝜐 =

1βˆ’π‘• 1βˆ’π‘• +πœ‚ 𝑣+ πœ‚ 𝑑(π‘βˆ’1) with 𝑨𝑛 𝑨 = 𝑏 π‘βˆ’1

  • Decreasing function of a : thinner tail
  • Role of elasticity effects vs income effects is

visible

  • 𝑕 = 0, πœ‚ 𝑣 = πœ‚ 𝑑gives the Laffer rate 𝜐 =

1 1+ πœ‚ 𝑑𝑏 .

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Optimal tax rates for high earners (using asymptotic rate formula)

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2.OPTIMAL NON-LINEAR INCOME TAX RATES FOR ANY TAX BRACKET

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Initial specifications

  • 𝐼(𝑨) : Cumulated income distribution function

i.e. the number of people with earnings below z (total population normalised to 1)

  • β„Ž(𝑨): Density of the income distribution at z, i.e.

the number of people earning z

  • β„Ž

𝑨 : Virtual density : density of income distribution at z that would exist if the tax schedule were replaced by a linear tax schedule at z.

  • 𝑕 𝑨 : Social marginal value of consumption for

taxpayers with income z, at optimum

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Formula for optimal tax rate at level 𝑨

π‘ˆβ€²(𝑨 ) 1 βˆ’ π‘ˆβ€²(𝑨 ) = 1 πœ‚π‘‘(𝑨 ) Γ— 1 βˆ’ 𝐼(𝑨 ) 𝑨 β„Ž (𝑨 ) Γ— 1 βˆ’ 𝑕 𝑨 exp 1 𝑨′ 1 βˆ’ πœ‚π‘£(𝑨′) πœ‚π‘‘(𝑨′) 𝑒𝑨′

𝑨 𝑨

β„Ž 𝑨 1 βˆ’ 𝐼(𝑨 ) 𝑒𝑨

∞ 𝑨

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An increase in the marginal rate for [𝑨 ,𝑨 + 𝑒𝑨 ]

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Mechanical effect net of welfare loss

  • Every taxpayer with income 𝑨 > 𝑨 pays π‘’πœπ‘’π‘¨

additional taxes, which are valued 1 βˆ’ 𝑕 𝑨 π‘’πœπ‘’π‘¨ by the government.

  • Therefore overall mechanical effect net of

welfare loss is:

  • 𝑁 = π‘’πœπ‘’π‘¨

1 βˆ’ 𝑕 𝑨 β„Ž 𝑨 𝑒𝑨

∞ 𝑨

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SLIDE 25

Elasticity effect

  • Two components:
  • Direct compensated elasticity effect due to

exogenous increase π‘’πœ

  • Indirect effect due to the shift of the taxpayer on

the tax schedule by 𝑒𝑨, inducing an endogenous additional change in marginal rates equal to π‘’π‘ˆβ€² = π‘’π‘ˆβ€²β€²π‘’π‘¨

  • 𝑒𝑨 = πœ‚π‘‘π‘¨ π‘’πœ+π‘’π‘ˆβ€²

1βˆ’π‘ˆβ€² .

  • Using virtual density and summing:
  • β‡’ 𝐹 = βˆ’πœ‚π‘‘π‘¨

π‘ˆβ€² 1βˆ’π‘ˆβ€² β„Ž

𝑨 π‘’πœπ‘’π‘¨

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Income effect

  • A taxpayer with income 𝑨 > 𝑨 pays βˆ’π‘’π‘† = π‘’πœπ‘’π‘¨

additional taxes

  • ⟹ Taxpayers above the bracket [𝑨 ,𝑨 + 𝑒𝑨 ] are

induced to work more through income effects, which reinforce mechanical effect.

  • Direct income effect πœƒ 𝑒𝑆

1 βˆ’ π‘ˆβ€²

  • Indirect elastic effect due to endogenous change in

marginal rates π‘’π‘ˆβ€² = π‘’π‘ˆβ€²β€²π‘’π‘¨

  • 𝑒𝑨 = βˆ’πœ‚π‘‘π‘¨

π‘’πœ+π‘’π‘ˆβ€² 1βˆ’π‘ˆβ€² βˆ’ πœƒ π‘’πœπ‘’π‘¨ 1βˆ’π‘ˆβ€² .

  • Using virtual density and summing:
  • β‡’ 𝐽 = π‘’πœπ‘’π‘¨

βˆ’πœƒ

π‘ˆβ€² 1βˆ’π‘ˆβ€² β„Ž

𝑨 𝑒𝑨

∞ 𝑨

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Total effect of tax reform

  • Revenue effect = Welfare effect therefore 𝑁 + 𝐹 + 𝐽 = 0 giving

differential equation:

  • β‡’

π‘ˆβ€² 1βˆ’π‘ˆβ€² = 1 πœ‚π‘‘ 1βˆ’πΌ(𝑨 ) 𝑨 β„Ž (𝑨 )

1 βˆ’ 𝑕 𝑨

β„Ž 𝑨 1βˆ’πΌ(𝑨 ) 𝑒𝑨 +

βˆ’πœƒ

π‘ˆβ€² 1βˆ’π‘ˆβ€² β„Ž 𝑨 1βˆ’πΌ(𝑨 ) 𝑒𝑨 ∞ 𝑨 ∞ 𝑨

  • By integration:
  • π‘ˆβ€²(𝑨 )

1βˆ’π‘ˆβ€²(𝑨 ) = 1 πœ‚π‘‘(𝑨 ) 1βˆ’πΌ(𝑨 ) 𝑨 β„Ž (𝑨 )

1 βˆ’ 𝑕 𝑨 exp

1 𝑨′ 1 βˆ’ πœ‚π‘£(𝑨′) πœ‚π‘‘(𝑨′) 𝑒𝑨′ 𝑨 𝑨 β„Ž 𝑨 1βˆ’πΌ(𝑨 ) 𝑒𝑨 ∞ 𝑨

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Formula for optimal tax rate at level 𝑨

π‘ˆβ€²(𝑨 ) 1 βˆ’ π‘ˆβ€²(𝑨 ) = 1 πœ‚π‘‘(𝑨 ) Γ— 1 βˆ’ 𝐼(𝑨 ) 𝑨 β„Ž (𝑨 ) Γ— 1 βˆ’ 𝑕 𝑨 exp 1 𝑨′ 1 βˆ’ πœ‚π‘£(𝑨′) πœ‚π‘‘(𝑨′) 𝑒𝑨′

𝑨 𝑨

β„Ž 𝑨 1 βˆ’ 𝐼(𝑨 ) 𝑒𝑨

∞ 𝑨

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Interpretation

  • Three elements determine optimal tax rates at

𝑨 :

  • shape of the income distribution :

1βˆ’πΌ(𝑨 ) 𝑨 β„Ž (𝑨 )

  • substitution/income effects :

1 πœ‚π‘‘(𝑨 ) and

exp

1 𝑨′ 1 βˆ’ πœ‚π‘£(𝑨′) πœ‚π‘‘(𝑨′) 𝑒𝑨′ 𝑨 𝑨

  • and social marginal weights : 1 βˆ’ 𝑕 𝑨
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Shape of income distribution

  • The shape of the income distribution:

1βˆ’πΌ(𝑨 ) 𝑨 β„Ž (𝑨 )

  • The elastic distortion at 𝑨 induced by marginal rate

increase is proportional to income at that level times number of people at that level: 𝑨 β„Ž(𝑨 ).

  • Gain in tax receipts is proportional to the number of

people above 𝑨 : 1 βˆ’ 𝐼(𝑨 )

  • ⟹ Government should apply high marginal rates at

levels where the density of taxpayers is low compared to the number of taxpayers with higher income

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Further explanation

  • This is clearly the case at the bottom : 𝑨 β„Ž(𝑨 ) is

close to 0 while 1 βˆ’ 𝐼(𝑨 ) is close to 1

  • At the top, assuming a Pareto distribution of

parameter a, 1βˆ’πΌ(𝑨 )

𝑨 β„Ž(𝑨 ) = 1/𝑏

  • For U.S., a = 2 ⟹ 1/a = 0.5
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Variations of

1βˆ’πΌ(𝑨) π‘¨β„Ž(𝑨) across incomes

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Substitution and income effects

  • Behavioural effects enter the formula in two

ways:

  • Compensated response from taxpayers

(substitution effect) via compensated elasticity

1 πœ‚π‘‘(𝑨 )

  • Increase in the tax burden of taxpayers above

𝑨 inducing them to work more (via exponential term which is larger than 1)

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Social marginal welfare weights

  • Represented by the term 1 βˆ’ 𝑕 𝑨

.

  • 𝑕 𝑨 : the relative value for the government of

an additional dollar of consumption at income z.

  • If 𝑕 𝑨 decreases with z, then the government

has redistributive tastes.

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SLIDE 35
  • 3. NUMERICAL SIMULATIONS
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Methodology

  • Aim : To simulate the importance of substitution vs

income effects and utilitarian vs Rawlsian social welfare weights

  • Two utility functions:
  • Type 1 : 𝑣 = log 𝑑 βˆ’

π‘š1+𝑙 1+𝑙 , no income effects

  • Type 2 : 𝑣 = log 𝑑 βˆ’ log 1 +

π‘š1+𝑙 1+𝑙 , with income

effects.

  • In both cases, constant compensated elasticity = 1/k
  • Use of the skill distribution as exogenous measure of

income distribution

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Results : optimal non-linear & linear rates according to wage income

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Results

  • In all four cases optimal rates are U-shaped:

close to actual tax schedules

  • High rates for low w correspond to phasing-
  • ut of guaranteed income levels
  • Income effects increase rates
  • Higher compensated elasticity decreases rates
  • Rawlsian criterion leads to higher rates, but

difference between Rawlsian and utilitarian decreases for higher incomes

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GENERAL CONCLUSIONS

  • Elasticity estimates from the empirical

literature suggest that top marginal rates should not be below 50% and can go as high as 80%.

  • The elasticity method is fruitful as it precisely

divides the individual impact of the shape of the income distribution, substitution and income effects, and redistributive tastes on the optimal marginal tax rate.