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

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 : 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.

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

1. HIGH INCOME OPTIMAL TAX RATES

Base specifications • Maximisation of a utility function 𝑣 = 𝑣(𝑑, 𝑨) 𝑒𝑣 𝑒𝑣 Where 𝑣 𝑑 = 𝑒𝑑 > 0 , 𝑣 𝑨 = 𝑒𝑨 < 0 , ( 𝑨 = 𝑥𝑚 ), according to the constraint 𝑑 = 𝑨 1 − 𝜐 + 𝑆 Where • τ 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 “virtual” linear schedule

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 ‘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 ‘consume’ less leisure, i.e. I work more.

Elasticity concepts • Uncompensated elasticity of earnings ∶ 𝜂 𝑣 = 𝑒𝑨 𝑒(1−𝜐) : 𝑨 1−𝜐 ( uncompensated, because it does not compensate for a change in income) • Income effects (= the marginal propensity to earn out of 𝑒𝑨 non-labour income): 𝜃 = 1 − 𝜐 𝑒𝑆 ≤ 0 , since leisure is assumed not to be an inferior good. • Compensated elasticity of earnings : 𝜂 𝑑 = 1−𝜐 𝑒𝑨 𝑒 1−𝜐 𝑣 = 𝑑𝑡𝑢 : (purely substitution effects since it 𝑨 compensates for a change income) • Slutsky equation: 𝜂 𝑑 = 𝜂 𝑣 − 𝜃 ≥ 0

Deriving the high income optimal tax rate • Government sets top marginal rate τ for incomes above 𝑨 • Population with income above 𝑨 normalised to 1 • ℎ(𝑨) : density of earnings distribution at optimum tax regime • Consider a small increase dτ in the top tax rate τ for incomes above 𝑨

High income tax rate perturbation

Decomposing the change in total taxes paid • Total taxes paid at income 𝑨 above 𝑨 = Marginal rate for incomes above 𝑨 × Income above 𝑨 + Total taxes paid at income 𝑨 • ⟹ 𝑈 𝑨 = 𝜐 𝑨 − 𝑨 + 𝑈(𝑨 ) • ⟹ 𝑒𝑈 𝑨 = 𝑨 − 𝑨 𝑒𝜐 + 𝜐𝑒𝑨 ∞ • ⟹ 𝑒𝑈 𝑨 ℎ 𝑨 𝑒𝑨 = 𝑁 + 𝐶 𝑨 • The total taxes paid therefore changes due to two things : a mechanical effect and behavioural responses

The mechanical effect • Mechanical effect : The increase in tax receipts if there were no behavioural responses. • Taxpayer with income 𝑨 > 𝑨 pays 𝑨 − 𝑨 𝑒𝜐 in additional taxes. • Summing over population with 𝑨 > 𝑨 , we have total mechanical effect on tax receipts: 𝑁 = 𝑨 𝑛 − 𝑨 𝑒𝜐

Behavioural responses • As 𝑨 = 𝑨 1 − 𝜐, 𝑆 , therefore with total differential: 𝜖𝑨 𝜖𝑨 • 𝑒𝑨 = − 𝜖 1−𝜐 𝑒𝜐 + 𝜖𝑆 𝑒𝑆 • Let’s express this in terms of income effect and uncompensated elasticity : 𝑒𝑨 𝜖𝑨 𝜃 • 𝜃 = 1 − 𝜐 𝑒𝑆 ⇒ 𝜖𝑆 = (1−𝜐) 𝜂 𝑣 𝑨 • 𝜂 𝑣 = 𝑒𝑨 𝜖𝑨 𝑒(1−𝜐) ⇒ − 𝜖 1−𝜐 = 1−𝜐 𝑨 1−𝜐 • And as 𝑒𝑆 = 𝑨 𝑒𝜐 (overall increase in virtual income), 𝑒𝜐 • Therefore: 𝑒𝑨 = −(𝜂 𝑣 𝑨 − 𝜃𝑨 ) 1−𝜐 : reduction in individual z’s earnings due to behavioural responses

Reduction in tax receipts due to behaviour responses • As we saw, a reduction in earnings of dz implies a reduction in tax receipts of τdz , for one individual. • This implies total that the total reduction in tax receipts is : ∞ 𝜐𝑒𝜐 − 𝜂 𝑣 𝑨 − 𝜃𝑨 • 𝐶 = 1−𝜐 ℎ 𝑨 𝑒𝑨 𝑨 = −(𝜂 𝑣 𝑨 𝑛 − 𝜃 𝑨 ) 𝜐𝑒𝜐 1 − 𝜐 • Where 𝜂 𝑣 is the weighted average of the uncompensated elasticity, and 𝜃 the average income effect.

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 of 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 of 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

Interpretation 𝜐 (1−𝑕 )(𝑨 𝑛 𝑨 −1) 1−𝜐 = • Result: 𝑣 𝑨 𝑛 𝑨 −𝜃 𝜂 • Decreasing function of 𝑕 , 𝜂 𝑣 , and increasing in 𝜃 . • When 𝑨 is close to the top, 𝑨 𝑛 𝑨 tends to 1 ⟹ 𝜐 tends to zero. This is because M is negligible compared to B near the top.

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

Pareto distributions • Distributions with constant 𝑨 𝑛 𝑨 ratio are exactly Pareto distributions. • A Pareto distribution is such that: 𝑏 𝑄𝑠𝑝𝑐 𝐽𝑜𝑑𝑝𝑛𝑓 > 𝑨 = (𝑨 𝑨) • We have 𝐹 𝑎 = 𝑨 𝑛 = 𝑏𝑨 𝑏−1 ⇒ 𝑨 𝑛 𝑏 𝑨 = 𝑏−1 . For 𝑨 𝑛 = 2 , 𝑏 = 2 . • The higher a , the thinner is the tail of the income distribution

Rewriting the optimal marginal tax as a limiting tax for high incomes • From 𝜐 1−𝜐 = (1−𝑕 )(𝑨 𝑛 𝑨 −1) ∶ 𝑣 𝑨 𝑛 𝑨 −𝜃 𝜂 1−𝑕 𝑑 (𝑏−1) with 𝑨 𝑛 𝑏 • ⇒ 𝜐 = 𝑨 = 𝑏−1 𝑣 + 𝜂 +𝜂 1−𝑕 • Decreasing function of a : thinner tail • Role of elasticity effects vs income effects is visible • 𝑕 = 0, 𝜂 𝑣 = 𝜂 𝑑 gives the Laffer rate 𝜐 = 1 𝑑 𝑏 . 1+ 𝜂

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

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

Initial specifications • 𝐼(𝑨) : Cumulated income distribution function i.e. the number of people with earnings below z (total population normalised to 1) • ℎ(𝑨) : Density of the income distribution at z , i.e. the number of people earning z 𝑨 : Virtual density : density of income • ℎ 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

Formula for optimal tax rate at level 𝑨 𝑈′(𝑨 ) 1 − 𝑈′(𝑨 ) = 1 𝜂 𝑑 (𝑨 ) × 1 − 𝐼(𝑨 ) × (𝑨 ) 𝑨 ℎ ∞ 𝑨 𝑨′ 1 − 𝜂 𝑣 (𝑨 ′ ) exp 1 ℎ 𝑨 1 − 𝑕 𝑨 𝜂 𝑑 (𝑨 ′ ) 𝑒𝑨′ 1 − 𝐼(𝑨 ) 𝑒𝑨 𝑨 𝑨

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

Mechanical effect net of welfare loss • Every taxpayer with income 𝑨 > 𝑨 pays 𝑒𝜐𝑒𝑨 additional taxes, which are valued 1 − 𝑕 𝑨 𝑒𝜐𝑒𝑨 by the government. • Therefore overall mechanical effect net of welfare loss is: ∞ • 𝑁 = 𝑒𝜐𝑒𝑨 1 − 𝑕 𝑨 ℎ 𝑨 𝑒𝑨 𝑨

Elasticity effect • Two components: • Direct compensated elasticity effect due to exogenous increase 𝑒𝜐 • 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 𝑒𝑈 ′ = 𝑒𝑈′′𝑒𝑨 • 𝑒𝑨 = 𝜂 𝑑 𝑨 𝑒𝜐+𝑒𝑈′ 1−𝑈′ . • Using virtual density and summing: 𝑈 ′ 𝑨 𝑒𝜐𝑒𝑨 • ⇒ 𝐹 = −𝜂 𝑑 𝑨 1−𝑈 ′ ℎ

Income effect • A taxpayer with income 𝑨 > 𝑨 pays −𝑒𝑆 = 𝑒𝜐𝑒𝑨 additional taxes • ⟹ Taxpayers above the bracket [𝑨 , 𝑨 + 𝑒𝑨 ] are induced to work more through income effects , which reinforce mechanical effect. • Direct income effect 𝜃 𝑒𝑆 1 − 𝑈′ • Indirect elastic effect due to endogenous change in marginal rates 𝑒𝑈 ′ = 𝑒𝑈′′𝑒𝑨 𝑒𝜐+𝑒𝑈 ′ 𝑒𝜐𝑒𝑨 • 𝑒𝑨 = −𝜂 𝑑 𝑨 1−𝑈 ′ − 𝜃 1−𝑈′ . • Using virtual density and summing: ∞ 𝑈′ 𝑨 𝑒𝑨 • ⇒ 𝐽 = 𝑒𝜐𝑒𝑨 −𝜃 1−𝑈′ ℎ 𝑨