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 themost revenue• Original model : Mirrlees (1971)• Saez’s goal: to clearly show that there is a
simple link between optimal tax formulas andelasticities 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 optimaltax schedules

1. HIGH INCOME OPTIMAL TAXRATESBase specifications• Maximisation of a utility function 𝑢 = 𝑢(𝑐, 𝑧)
𝑑𝑢𝑑𝑢
Where 𝑢𝑐 => 0 , 𝑢𝑧 =< 0 , ( 𝑧 = 𝑤𝑙),𝑑𝑐
𝑑𝑧

according to the constraint 𝑐 = 𝑧 1 − 𝜏 + 𝑅

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

Elasticity concepts𝑑(1−𝜏)• Uncompensated elasticity of earnings ∶ 𝜁 𝑢 = 𝑑𝑧:𝑧1−𝜏
(uncompensated, because it does not compensate for achange in income)• Income effects (= the marginal propensity to earn out of𝑑𝑧
non-labour income): 𝜂 = 1 − 𝜏≤ 0, since leisure is𝑑𝑅
assumed not to be an inferior good.
• Compensated elasticity of earnings :𝑑𝑧
𝜁 𝑐 = 1−𝜏𝑢 = 𝑐𝑠𝑡 : (purely substitution effects since it𝑧 𝑑 1−𝜏
compensates for a change income)• Slutsky equation: 𝜁 𝑐 = 𝜁 𝑢 − 𝜂 ≥ 0

Deriving the high income optimal taxrate• Government sets top marginal rate τ forincomes above 𝑧• Population with income above 𝑧 normalised to1• ℎ(𝑧) : density of earnings distribution at
optimum tax regime• Consider a small increase dτ in the top tax rate
τ for incomes above 𝑧

High income tax rate perturbation

Decomposing the change in total taxes
paid• Total taxes paid at income 𝑧 above 𝑧 = Marginal
rate for incomes above 𝑧 × Income above 𝑧 +Total taxes paid at income 𝑧• ⟹ 𝑇 𝑧 = 𝜏 𝑧 − 𝑧 + 𝑇(𝑧 )• ⟹ 𝑑𝑇 𝑧 = 𝑧 − 𝑧 𝑑𝜏 + 𝜏𝑑𝑧• ⟹∞𝑑𝑇 𝑧 ℎ 𝑧 𝑑𝑧 = 𝑀 + 𝐵𝑧

• The total taxes paid therefore changes due to twothings : a mechanical effect and behaviouralresponses

The mechanical effect• Mechanical effect: The increase in tax receipts
if there were no behavioural responses.
• Taxpayer with income 𝑧 > 𝑧 pays 𝑧 − 𝑧 𝑑𝜏 inadditional taxes.
• Summing over population with 𝑧 > 𝑧 , wehave total mechanical effect on tax receipts:
𝑀 = 𝑧𝑚 − 𝑧 𝑑𝜏Behavioural responses• As 𝑧 = 𝑧 1 − 𝜏, 𝑅 , therefore with total differential:𝜕𝑧𝜕𝑧𝑑𝑧 = −𝑑𝜏 + 𝑑𝑅
𝜕 1−𝜏
𝜕𝑅

•
• Let’s express this in terms of income effect anduncompensated elasticity :• 𝜂 = 1−𝜏𝑢• 𝜁 =𝑑𝑧𝜕𝑧𝜂⇒=𝑑𝑅𝜕𝑅(1−𝜏)𝑑𝑧 𝑑(1−𝜏)
𝑧1−𝜏

𝜕𝑧
⇒−
𝜕 1−𝜏

𝜁𝑢𝑧=1−𝜏• And as 𝑑𝑅 = 𝑧𝑑𝜏 (overall increase in virtual income),•𝑑𝜏𝑢Therefore: 𝑑𝑧 = −(𝜁 𝑧 − 𝜂𝑧): reduction in1−𝜏

individual z’s earnings due to behavioural responses

Reduction in tax receipts due tobehaviour responses• As we saw, a reduction in earnings of dz implies areduction in tax receipts of τdz, for one individual.
• This implies total that the total reduction in tax
receipts is :
• 𝐵=∞
𝜏𝑑𝜏
𝑢− 𝜁 𝑧 − 𝜂𝑧ℎ 𝑧 𝑑𝑧𝑧1−𝜏𝜏𝑑𝜏= −(𝜁 𝑧𝑚 − 𝜂 𝑧)1−𝜏
• Where 𝜁 𝑢 is the weighted average of theuncompensated elasticity, and 𝜂 the average
income effect.
𝑢Obtaining the optimal tax rate
• Need to equalise the revenue effect (the sum of themechanical effect and behavioural response) to thewelfare effect.
• Compute welfare effect : Let 𝑔 = Marginal social utilityof money for top bracket tax payers divided bymarginal value of public funds for government. Thus
each additional dollar raised by government as a resultof tax reduces on average social welfare of the top
bracket by 𝑔.
• Hence the total welfare loss due to tax reform is 𝑔M.
• Revenue effect = Welfare effect ⇔ M+B = gM

Interpretation𝜏(1−𝑔)(𝑧𝑚 𝑧−1)
• Result:=1−𝜏𝜁 𝑢 𝑧𝑚 𝑧−𝜂• Decreasing function of 𝑔, 𝜁 𝑢 , and increasing in𝜂.
• When 𝑧 is close to the top, 𝑧𝑚 𝑧 tends to 1
⟹ 𝜏 tends to zero. This is because M isnegligible compared to B near the top.

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

Pareto distributions• Distributions with constant 𝑧𝑚 𝑧 ratio areexactly Pareto distributions.
• A Pareto distribution is such that:𝑎𝑃𝑟𝑜𝑏 𝐼𝑛𝑐𝑜𝑚𝑒 > 𝑧 = (𝑧 𝑧)𝑎𝑧𝑧𝑚𝑎= 𝑧𝑚 =⇒=.
𝑎−1𝑧𝑎−1• We have 𝐸 𝑍
For 𝑧𝑚 = 2, 𝑎 = 2.
• The higher a, the thinner is the tail of theincome distributionRewriting the optimal marginal tax asa limiting tax for high incomes••𝜏(1−𝑔)(𝑧𝑚 𝑧−1)
From
=
∶𝑢1−𝜏
𝜁 𝑧𝑚 𝑧−𝜂
1−𝑔𝑧𝑚𝑎⇒𝜏=
with=𝑢𝑐
1−𝑔+𝜁 + 𝜁 (𝑎−1)𝑧𝑎−1• Decreasing function of a : thinner tail• Role of elasticity effects vs income effects isvisible• 𝑔 = 0, 𝜁𝑢1= 𝜁 gives the Laffer rate 𝜏 =.
𝑐1+ 𝜁 𝑎𝑐

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

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

Initial specifications• 𝐻(𝑧) : Cumulated income distribution function
i.e. the number of people with earnings below z(total population normalised to 1)• ℎ(𝑧): Density of the income distribution at z, i.e.
the number of people earning z• ℎ 𝑧 : Virtual density : density of income
distribution at z that would exist if the tax
schedule were replaced by a linear tax scheduleat z.
• 𝑔 𝑧 : Social marginal value of consumption fortaxpayers with income z, at optimum

Formula for optimal tax rate at level 𝑧𝑇′(𝑧)=1 − 𝑇′(𝑧)
1×𝑐
𝜁 (𝑧)

1 − 𝐻(𝑧)𝑧ℎ(𝑧)∞𝑧

1 − 𝑔 𝑧 exp𝑧𝑧×1𝜁 𝑢 (𝑧 ′ )ℎ 𝑧
1 − 𝑐 ′ 𝑑𝑧′𝑑𝑧𝑧′𝜁 (𝑧 )
1 − 𝐻(𝑧)An increase in the marginal rate for[𝑧,𝑧 + 𝑑𝑧]

Mechanical effect net of welfareloss• Every taxpayer with income 𝑧 > 𝑧 pays 𝑑𝜏𝑑𝑧
additional taxes, which are valued
1 − 𝑔 𝑧 𝑑𝜏𝑑𝑧 by the government.
• Therefore overall mechanical effect net ofwelfare loss is:• 𝑀 = 𝑑𝜏𝑑𝑧∞
1−𝑔𝑧𝑧ℎ 𝑧 𝑑𝑧

Elasticity effect• Two components:• Direct compensated elasticity effect due toexogenous increase 𝑑𝜏
• Indirect effect due to the shift of the taxpayer on
the tax schedule by 𝑑𝑧, inducing an endogenous
additional change in marginal rates equal to𝑑𝑇 ′ = 𝑑𝑇′′𝑑𝑧
𝑐 𝑑𝜏+𝑑𝑇′𝑑𝑧 = 𝜁 𝑧.
1−𝑇′•
• Using virtual density and summing:•′𝑇⇒ 𝐸 = −𝜁 𝑐 𝑧ℎ 𝑧 𝑑𝜏𝑑𝑧′
1−𝑇Income effect• A taxpayer with income 𝑧 > 𝑧 pays −𝑑𝑅 = 𝑑𝜏𝑑𝑧
additional taxes• ⟹ Taxpayers above the bracket [𝑧,𝑧 + 𝑑𝑧] areinduced to work more through income effects, which
reinforce mechanical effect.
• Direct income effect 𝜂 𝑑𝑅 1 − 𝑇′
• Indirect elastic effect due to endogenous change inmarginal rates 𝑑𝑇 ′ = 𝑑𝑇′′𝑑𝑧
𝑑𝜏+𝑑𝑇 ′𝑑𝜏𝑑𝑧
𝑑𝑧 = −𝜁 𝑧−𝜂
.
′1−𝑇1−𝑇′𝑐•• Using virtual density and summing:• ⇒ 𝐼 = 𝑑𝜏𝑑𝑧∞𝑇′−𝜂ℎ 𝑧 𝑑𝑧𝑧
1−𝑇′Total effect of tax reform• Revenue effect = Welfare effect therefore 𝑀 + 𝐸 + 𝐼 = 0 giving
differential equation:𝑇′=1−𝑇′1 1−𝐻(𝑧)𝜁 𝑐 𝑧ℎ(𝑧)• ⇒∞1−𝑔 𝑧𝑧∞ℎ 𝑧𝑇′ℎ 𝑧
𝑑𝑧 + 𝑧 −𝜂𝑑𝑧
1−𝐻(𝑧)
1−𝑇′ 1−𝐻(𝑧)• By integration:•𝑇′(𝑧)=
1−𝑇′(𝑧)11−𝐻(𝑧)𝜁 𝑐 (𝑧) 𝑧ℎ(𝑧)∞1−𝑔 𝑧𝑧exp

𝑧 1𝑧 𝑧′𝜁 𝑢 (𝑧 ′ )1− 𝑐 ′
𝜁 (𝑧 )𝑑𝑧′ℎ 𝑧𝑑𝑧
1−𝐻(𝑧)

Formula for optimal tax rate at level 𝑧𝑇′(𝑧)=1 − 𝑇′(𝑧)
1×𝑐
𝜁 (𝑧)

1 − 𝐻(𝑧)𝑧ℎ(𝑧)∞𝑧

1 − 𝑔 𝑧 exp𝑧𝑧

×1𝜁 𝑢 (𝑧 ′ )ℎ 𝑧
1 − 𝑐 ′ 𝑑𝑧′𝑑𝑧𝑧′𝜁 (𝑧 )1 − 𝐻(𝑧)Interpretation• Three elements determine optimal tax rates at𝑧:
- shape of the income distribution :-1−𝐻(𝑧)𝑧ℎ(𝑧)1substitution/income effects : 𝑐 and𝜁 (𝑧)𝑧1𝜁 𝑢 (𝑧 ′ )exp 𝑧
1 − 𝑐 ′ 𝑑𝑧′𝑧′𝜁 (𝑧 )- and social marginal weights : 1 − 𝑔 𝑧Shape of income distribution•1−𝐻(𝑧)The shape of the income distribution:𝑧ℎ(𝑧)• The elastic distortion at 𝑧 induced by marginal rateincrease is proportional to income at that level times
number of people at that level: 𝑧ℎ(𝑧).
• Gain in tax receipts is proportional to the number ofpeople above 𝑧 : 1 − 𝐻(𝑧)• ⟹ Government should apply high marginal rates atlevels where the density of taxpayers is low comparedto the number of taxpayers with higher income

Further explanation• This is clearly the case at the bottom : 𝑧ℎ(𝑧) is
close to 0 while 1 − 𝐻(𝑧) is close to 1
• At the top, assuming a Pareto distribution of
1−𝐻(𝑧)
parameter a,= 1/𝑎𝑧ℎ(𝑧)• For U.S., a = 2 ⟹ 1/a = 0.5

1−𝐻(𝑧)Variations ofacross incomes𝑧ℎ(𝑧)Substitution and income effects• Behavioural effects enter the formula in twoways:• Compensated response from taxpayers(substitution effect) via compensated
1elasticity 𝑐𝜁 (𝑧)• Increase in the tax burden of taxpayers above𝑧 inducing them to work more (via
exponential term which is larger than 1)

Social marginal welfare weights• Represented by the term 1 − 𝑔 𝑧 .
• 𝑔 𝑧 : the relative value for the government ofan additional dollar of consumption at incomez.
• If 𝑔 𝑧 decreases with z, then the government
has redistributive tastes.

3. NUMERICAL SIMULATIONSMethodology• Aim : To simulate the importance of substitution vsincome effects and utilitarian vs Rawlsian social welfareweights• Two utility functions:• Type 1 : 𝑢 = log

𝑙 1+𝑘𝑐−1+𝑘, no income effects
𝑙 1+𝑘1+1+𝑘• Type 2 : 𝑢 = log 𝑐 − log, with incomeeffects.
• In both cases, constant compensated elasticity = 1/k• Use of the skill distribution as exogenous measure ofincome distribution

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

Results• In all four cases optimal rates are U-shaped:
close to actual tax schedules• High rates for low w correspond to phasingout of guaranteed income levels• Income effects increase rates• Higher compensated elasticity decreases rates• Rawlsian criterion leads to higher rates, butdifference between Rawlsian and utilitarian
decreases for higher incomesGENERAL CONCLUSIONS• Elasticity estimates from the empiricalliterature suggest that top marginal rates
should not be below 50% and can go as highas 80%.
• The elasticity method is fruitful as it precisely
divides the individual impact of the shape of
the income distribution, substitution andincome effects, and redistributive tastes on
the optimal marginal tax rate.