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MSc Public Economics 2011/12

http://darp.lse.ac.uk/ec426/

Policy Design: Income TaxPolicy Design: Income Tax

Frank A. Cowell Frank A. Cowell

24 October 2011

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Overview...

Design principles

Simple model

Generalisations

Interpretations

Policy Design: Income Tax

Roots in social choice and asymmetric information

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Social values: the Arrow problem

Uses weak assumptions about preferences/valuesUses weak assumptions about preferences/values Well-defined individual orderings over social statesWell-defined individual orderings over social states Well-defined social ordering over social statesWell-defined social ordering over social states

Uses a general notion of social preferencesUses a general notion of social preferences The The constitutionconstitution A map from set of preference profiles to social preferenceA map from set of preference profiles to social preference

Also weak assumptions about the constitutionAlso weak assumptions about the constitution Universal DomainUniversal Domain Pareto UnanimityPareto Unanimity Independence of Irrelevant AlternativesIndependence of Irrelevant Alternatives Non-DictatorshipNon-Dictatorship

There’s no constitution that does all fourThere’s no constitution that does all four Except in cases where there are less than three social statesExcept in cases where there are less than three social states

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Social choice function

A social state: A social state: Individual Individual hh’s evaluation of the state ’s evaluation of the state vvhh(())

A given population is indexed by A given population is indexed by h h = 1,2,…, = 1,2,…, nnhh

A “reduced-form” utility function A “reduced-form” utility function vvhh(().).

A profile: [A profile: [vv11, , vv22, …, , …, vvhh, …, … ] ] An ordered list of utility functionsAn ordered list of utility functions Set of all profiles: Set of all profiles: VV

A social choice function A social choice function : : VV→→ For a particular profile For a particular profile ((vv11, , vv22, …, , …, vvhh, …, … ) ) Argument is a utility function not a utility levelArgument is a utility function not a utility level Picks exactly one chosen element from Picks exactly one chosen element from

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Implementation Is the SCF consistent with private economic behaviour?Is the SCF consistent with private economic behaviour?

Yes if the Yes if the picked out by picked out by is also… is also… … … the equilibrium of an appropriate economic gamethe equilibrium of an appropriate economic game

Implementation problem: find/design an appropriate Implementation problem: find/design an appropriate mechanismmechanism Mechanism is a partially specified game of imperfect information…Mechanism is a partially specified game of imperfect information… rules of game are fixedrules of game are fixed strategy sets are specifiedstrategy sets are specified preferences for the game are not yet specifiedpreferences for the game are not yet specified

Plug preferences into the mechanism:Plug preferences into the mechanism: Does the mechanism have an equilibrium?Does the mechanism have an equilibrium? Does the equilibrium correspond to the desired social state Does the equilibrium correspond to the desired social state ?? If so, the social state is implementableIf so, the social state is implementable

There is a wide range of possible mechanismsThere is a wide range of possible mechanisms Example: the market as a mechanismExample: the market as a mechanism Given the distribution of resources and the technology…Given the distribution of resources and the technology… ……the market maps preferences into prices.the market maps preferences into prices. The prices then determine the allocationThe prices then determine the allocation

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Manipulation

Consider outcomes from a “direct” mechanism in two cases:Consider outcomes from a “direct” mechanism in two cases: If all, including If all, including hh, tell the truth about preferences: , tell the truth about preferences:

((vv11,…, ,…, vvhh, …,, …, ))

If If hh misrepresents his preferences but others tell the truth: misrepresents his preferences but others tell the truth: ((vv11,…, ,…, vvhh, …,, …, ))

How does the person “really” feel about How does the person “really” feel about and and ? ? If If vvhh(() > ) > vvhh(() then there is an incentive to misrepresent ) then there is an incentive to misrepresent

informationinformation If If hh realises this we say that realises this we say that is is manipulablemanipulable. .

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Gibbard-Satterthwaite result Result on SCF Result on SCF can can be stated in several ways be stated in several ways

((Gibbard 1973, , Satterthwaite, 1975 ))

A standard version is: A standard version is: If the set of social states If the set of social states contains at least three elements; contains at least three elements; and and is defined is defined for all logically possible preference profilesfor all logically possible preference profiles and and is truthfully implementable in dominant strategies... is truthfully implementable in dominant strategies... then then must be dictatorial must be dictatorial

Closely related to the Arrow theoremClosely related to the Arrow theorem Has profound implications for public economicsHas profound implications for public economics

Misinformation may be endemic to the design problemMisinformation may be endemic to the design problem May only get truth-telling mechanisms in special casesMay only get truth-telling mechanisms in special cases

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Overview...

Design principles

Simple model

Generalisations

Interpretations

Policy Design: Income Tax

Preferences, incomes, ability and the government

Analogy with contract theory

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The design problem

The government needs to raise revenue…The government needs to raise revenue… ……and it may want to redistribute resourcesand it may want to redistribute resources To do this it uses the tax systemTo do this it uses the tax system

personal income tax…personal income tax… ……and income-based subsidiesand income-based subsidies

Base it on “ability to pay”Base it on “ability to pay” income rather than wealthincome rather than wealth ability reflected in productivityability reflected in productivity

Tax authority may have limited informationTax authority may have limited information who have the high ability to pay?who have the high ability to pay? what impact on individuals’ willingness to produce output?what impact on individuals’ willingness to produce output?

What’s the right way to construct the tax schedule?What’s the right way to construct the tax schedule?

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Model elements A two-commodity modelA two-commodity model

leisure (i.e. the opposite of effort) leisure (i.e. the opposite of effort) consumption – a basket of all other goodsconsumption – a basket of all other goods similar to optimal contracts (similar to optimal contracts (Bolton and Dewatripont 2005)Bolton and Dewatripont 2005)

Income comes only from workIncome comes only from work individuals are paid according to their marginal productindividuals are paid according to their marginal product workers differ according to their abilityworkers differ according to their ability

Individuals derive utility from:Individuals derive utility from: their leisuretheir leisure their disposable income (consumption)their disposable income (consumption)

Government / tax agencyGovernment / tax agency has to raise a fixed amount of revenue has to raise a fixed amount of revenue KK seeks to maximise social welfare…seeks to maximise social welfare… ……where social welfare is a function of individual utilitieswhere social welfare is a function of individual utilities

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Modelling preferences

Individual’s preferencesIndividual’s preferences = = zz + + yy : utility level : utility level z z : effort: effort y y : income received: income received : decreasing, strictly concave, function : decreasing, strictly concave, function

Special shape of utility functionSpecial shape of utility function quasi-linear formquasi-linear form zero-income effectzero-income effect zz gives the disutility of effort in monetary units gives the disutility of effort in monetary units

Individual does not have to workIndividual does not have to work reservation utility level reservation utility level requires requires zz + + y y ≥≥

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Ability and income

Individuals work (give up leisure) to get consumptionIndividuals work (give up leisure) to get consumption Individuals differ in talent (ability) Individuals differ in talent (ability)

higher ability people produce more and may thus earn morehigher ability people produce more and may thus earn more individual of type individual of type works an amount works an amount zz produces output produces output qq = = zz but individual does not necessarily get to keep this output?but individual does not necessarily get to keep this output?

Disposable income determined by tax authorityDisposable income determined by tax authority intervention via taxes and transfersintervention via taxes and transfers fixes a relationship between individual’s output and incomefixes a relationship between individual’s output and income (net) income tax on type (net) income tax on type is implicitly given by is implicitly given by qq − − yy

Preferences can be expressed in terms of Preferences can be expressed in terms of q,q, yy for type for type utility is given by utility is given by zz + + yy equivalently: equivalently: q q // + + yy

A closer look at utility

A closer look at utility

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The utility function

increasing

preference

y

1– z

= = zz + + yy zzzz < 0 < 0

≥≥

= = q/q/ + + yy zzq/q/ < 0 < 0

increasingpreference

y

q

Preferences over leisure and income Indifference curves

Reservation utility

Transform into (leisure, output) space

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The single-crossing condition

increasingpreference

y

q

type b

type a

Preferences over leisure and output

High talent

qqaa = = aazzaa

Low talent

qqbb = = bbzzbb

Those with different talent (ability) will have different sloped indifference curves in this diagram

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A full-information solution?

Consider argument based on the analysis of contractsConsider argument based on the analysis of contracts Full information: owner can fully exploit any managerFull information: owner can fully exploit any manager

Pays the minimum amount necessaryPays the minimum amount necessary ““Chooses” their effortChooses” their effort

Same basic story here Same basic story here Can impose lump-sum taxCan impose lump-sum tax ““Chooses” agents’ effort Chooses” agents’ effort —— no distortion no distortion

But the full-information solution may be unattractiveBut the full-information solution may be unattractive Informational requirements are demandingInformational requirements are demanding Perhaps violation of individuals’ privacy?Perhaps violation of individuals’ privacy? So look at second-best case…So look at second-best case…

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Two types

Start with the case closest to the optimal contract modelStart with the case closest to the optimal contract model Exactly two skill typesExactly two skill types

a a > > b b

proportion of proportion of aa-types is -types is values of values of a a , , b b and and are common knowledge are common knowledge

From contract design we can write down the outcomeFrom contract design we can write down the outcome essentially all we need to do is rework notationessentially all we need to do is rework notation

But let us examine the model in detail:But let us examine the model in detail:

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Second-best: two types

The government’s budget constraintThe government’s budget constraint [[qqaayyaa] + [1] + [1][][qqbbyybb] ] ≥ ≥ KK wherewhere q qhhyyhh is the amount raised in tax from agent is the amount raised in tax from agent hh

Participation constraint for the Participation constraint for the bb type: type: yybb + + zzbb ≥ ≥ bb

have to offer at least as much as available elsewherehave to offer at least as much as available elsewhere

Incentive-compatibility constraint for the Incentive-compatibility constraint for the aa type: type: yyaa + + qqaa//aa ≥ ≥ yybb + + qqbb//aa must be no worse off than if it behaved like a must be no worse off than if it behaved like a bb-type-type implies implies qqbb,, yybbqqaa,, yyaa

The government seeks to maximise standard SWFThe government seeks to maximise standard SWF zzaa + + yyaa) + [1) + [1]]zzbb + + yybb) ) where where is increasing and concave is increasing and concave

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Two types: model We can use a standard Lagrangean approachWe can use a standard Lagrangean approach

government chooses (government chooses (qq, , yy) pairs for each type) pairs for each type ……subject to three constraintssubject to three constraints

Constraints are:Constraints are: government budget constraintgovernment budget constraint participation constraint (for participation constraint (for bb-types)-types) incentive-compatibility constraint (for incentive-compatibility constraint (for aa-types)-types)

Choose Choose qqaaqqbbyyaayybb to max to max qqaa//aa + + yyaa) + [1) + [1]]qqbb//bb + + yybb) )

+ + [ [[[qqaayyaa] + [1] + [1][][qqbbyybb] ] KK]]+ + [ [yybb + + qqbb//bb bb]]+ + [ [yyaa + + qqaa//aa yybb qqbb//aa]]

where where are Lagrange multipliers for the constraintsare Lagrange multipliers for the constraints

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Two types: solution From first-order conditions we get:From first-order conditions we get:

zzqqaa//aa = = aa

zzqqbb//bb = = bb + + kk[1[1], ], where where k k :=:= zzqqbb//bb [ [bb//aa] ] zzqqbb//aa

Also, all the Lagrange multipliers are positiveAlso, all the Lagrange multipliers are positive so the associated constraints are bindingso the associated constraints are binding follows from standard adverse selection modelfollows from standard adverse selection model

Results are as for optimum-contracts model:Results are as for optimum-contracts model: MRSMRSaa = MRT = MRTaa

MRSMRSbb << MRT MRTb b

InterpretationInterpretation no distortion at the top (for type no distortion at the top (for type aa)) no surplus at the bottom (for type no surplus at the bottom (for type bb)) determine the “menu” of (determine the “menu” of (qq,,yy)-choices offered by tax agency)-choices offered by tax agency

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Two ability types: tax designy

q

q aq b

y a

y b

a type’s reservation utility

b type’s reservation utility

b type’s (q,y)

incentive-compatibility constrainta type’s (q,y)

menu of (q,y) offered by tax authority

Analysis determines (q,y) combinations at two points

If a tax schedule T(∙) is to be designed where y = q −T(q) …

…then it must be consistent with these two points

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Overview...

Design principles

Simple model

Generalisations

Interpretations

Policy Design: Income Tax

Moving beyond the two-ability model

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A small generalisation

With three types problem becomes a bit more With three types problem becomes a bit more interestinginteresting Similar structure to previous caseSimilar structure to previous case aa > > bb > > cc

proportions of each type in the population are proportions of each type in the population are aa, , bb, , cc

We now have one more constraint to worry aboutWe now have one more constraint to worry about1.1. Participation constraint for Participation constraint for cc type: type: yycc + + qqcc//cc≥ ≥ cc

2.2. IC constraint for IC constraint for bb type: type: yybb + + qqbb//bb ≥ ≥ yycc + + qqcc//bb 3.3. IC constraint for IC constraint for aa type: type: yyaa + + qqaa//aa ≥ ≥ yybb + + qqbb//aa

But this is enough to complete the model specificationBut this is enough to complete the model specification the two IC constraints also imply the two IC constraints also imply yyaa + + qqaa//aa ≥ ≥ yycc + +

qqcc//bb so no-one has incentive to misrepresent as lower abilityso no-one has incentive to misrepresent as lower ability

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Three types

Methodology is same as two-ability modelMethodology is same as two-ability model set up Lagrangeanset up Lagrangean Lagrange multipliers for budget constraint, participation constraint and Lagrange multipliers for budget constraint, participation constraint and

two IC constraintstwo IC constraints maximise with respect to maximise with respect to qqaa,y,yaaqqbb,y,ybbqqcc,y,ycc

Outcome essentially as before :Outcome essentially as before : MRSMRSaa = MRT = MRTaa

MRSMRSbb << MRT MRTb b

MRSMRScc << MRT MRTcc

Again, no distortion at the top and the participation constraint Again, no distortion at the top and the participation constraint binding at the bottombinding at the bottom determines determines q,yq,y-combinations at exactly three points-combinations at exactly three points tax schedule must be consistent with these pointstax schedule must be consistent with these points

A stepping stone to a much more interesting model…A stepping stone to a much more interesting model…

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A richer model: N+1 types

The multi-type case follows immediately from the The multi-type case follows immediately from the three-type casethree-type case

Take Take N N + l types+ l types 00

< < 11 < < 22

< … < < … < NN (note the required change in notation)(note the required change in notation) proportion of type proportion of type jj is is jj

this distribution is common knowledgethis distribution is common knowledge

Budget constraint and SWF are nowBudget constraint and SWF are now jj jj[[qqjjyyjj] ] ≥ ≥ KK jj jjzzjj + + yyjj) ) where sum is from 0 to where sum is from 0 to NN

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N+1 types: behavioural constraints

Participation constraintParticipation constraint is relevant for lowestis relevant for lowest type type j j = 0 = 0 form is as before:form is as before: yy00 + + zz00 ≥ ≥ 00

Incentive-compatibility constraint Incentive-compatibility constraint applies where applies where j j > 0> 0 jj must be no worse off than if it behaved as the type below ( must be no worse off than if it behaved as the type below (jj1)1) yyjj + + qqjj//jj ≥ ≥ yyjj11 + + qqjj11 //jj.. implies implies qqjj11,, yyjj11qqjj,, yyjj and and jj≥≥jj11

From previous cases we know the methodologyFrom previous cases we know the methodology (and can probably guess the outcome)(and can probably guess the outcome)

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N+1 types: solution

Lagrangean is only slightly modified from beforeLagrangean is only slightly modified from before Choose {(Choose {(qqjjyyj j )} to max)} to max jj=0 =0 jj qqj j jj + + yyjj) )

+ + [ [jj jj[[qqjjyyjj] ] KK]]

+ + [ [yy00 + + zz00 00]]+ + jj=1 =1 jj [ [yyjj + + qqjj//jj yyjj11 qqjj11 //jj]]

where there are now where there are now NN incentive-compatibility Lagrange multipliers incentive-compatibility Lagrange multipliers

And we get the result, as beforeAnd we get the result, as before MRSMRSNN = MRT = MRTNN

MRSMRSNN−−11 << MRT MRTNN−−1 1

…… MRSMRS11 << MRT MRT1 1

MRSMRS00 << MRT MRT0 0

Now the tax schedule is determined at Now the tax schedule is determined at NN+1 points+1 points

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A continuum of types One more step is required in generalisationOne more step is required in generalisation Tax agency is faced with a continuum of taxpayersTax agency is faced with a continuum of taxpayers

common assumptioncommon assumption allows for general specification of ability distributionallows for general specification of ability distribution

This can be reasoned from the case with This can be reasoned from the case with N N + 1 types+ 1 types allow allow NN

From previous cases we know From previous cases we know form of the participation constraintform of the participation constraint form that IC constraint must takeform that IC constraint must take an outline of the outcomean outline of the outcome

Can proceed by analogy with previous analysis…Can proceed by analogy with previous analysis…

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The continuum model Continuous ability Continuous ability

bounded support [bounded support [ densitydensity f f(())

Utility for talent Utility for talent as before as beforeyy(() + ) + q q(())

Participation constraint isParticipation constraint is) ≥) ≥

Incentive compatibility requiresIncentive compatibility requiresdd) /d) /d≥≥

SWF isSWF is∫∫ (() ) ff d d

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Output and disposable income under the optimal tax

y

q

q_

q_

_

_

45°

Lowest type’s indifference curveLowest type’s output and incomeIntermediate type’s indifference curve, output and incomeHighest type’s indifference curveHighest type’s output and incomeMenu offered by tax authority

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Continuum model: results

Incentive compatibility implies dIncentive compatibility implies dyy /d /dqq >> 0 0 optimal marginal tax rate < 100% (optimal marginal tax rate < 100% (Mirrlees 1971))

No distortion at top implies dNo distortion at top implies dyy /d /dqq = 1 = 1 zero optimal marginal tax rate! (zero optimal marginal tax rate! (Seade 1977)) but does not generalise to incomes close to top (but does not generalise to incomes close to top (Tuomala 1984)) does not hold if there is no “topmost income” (does not hold if there is no “topmost income” (Diamond 1998 ) )

May be 0 on the lowest incomeMay be 0 on the lowest income depends on distribution of ability there (depends on distribution of ability there (Ebert 1992))

Explicit form for the optimal income tax requiresExplicit form for the optimal income tax requires specification of distribution specification of distribution ff((∙∙)) specification of individual preferences specification of individual preferences ((∙∙)) specification of social preferences specification of social preferences ( (∙∙)) specification of required revenue specification of required revenue KK ((Saez 2001, , Brewer et al. 2010, , Mankiw 2009))

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Overview...

Design basics

Simple model

Generalisations

Interpretations

Design: Taxation

Apply design rules to practical policy….

Plus a “cut-down” version of the OIT problem

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Application of design principles The second-best method provides some pointersThe second-best method provides some pointers

but is not a prescriptive formulabut is not a prescriptive formula explicit form of OIT usually not possibleexplicit form of OIT usually not possible ( (Salanié 2003Salanié 2003)) model is necessarily over-simplifiedmodel is necessarily over-simplified exact second-best formula might be administratively complexexact second-best formula might be administratively complex

Simple schemes may be worth consideringSimple schemes may be worth considering roughly correspond to actual practiceroughly correspond to actual practice illustrate good/bad designillustrate good/bad design

Consider affine (linear) tax systemConsider affine (linear) tax system benefit benefit BB payable to all (guaranteed minimum income) payable to all (guaranteed minimum income) all gross income (output) taxable at the same marginal rate all gross income (output) taxable at the same marginal rate t…t… ……constant marginal retention rate: dconstant marginal retention rate: dy y /d/dq q = 1 = 1 tt

Effectively a negative income tax scheme:Effectively a negative income tax scheme: (net) income related to output thus: (net) income related to output thus: yy = = BB + [1 + [1 tt] ] qq so so yy > > qq if if qq < < B / t B / t … and vice versa … and vice versa

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1t

A simple tax-benefit system

y

q

Low-income type’s indiff curveLow-income type’s output, incomeHigh-income type’s indiff curveHighest type’s output and income

Constant marginal retention rate

Guaranteed minimum income B

B

Implied attainable set

“Linear” income tax system ensures that incentive-compatibility constraint is satisfied

Analysed by Sheshinski (1972)

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Violations of design principles?

The IC condition be violated in actual designThe IC condition be violated in actual design This can happen by accident:This can happen by accident:

interaction between income support and income tax.interaction between income support and income tax. generated by the desire to “target” support more effectivelygenerated by the desire to “target” support more effectively a well-meant inefficiency?a well-meant inefficiency?

Commonly known asCommonly known as the “notch problem” (US)the “notch problem” (US) the “poverty trap” (UK)the “poverty trap” (UK)

Simple exampleSimple example suppose some of the benefit is intended for lowest types onlysuppose some of the benefit is intended for lowest types only an amount an amount BB00 is withdrawn after a given output level is withdrawn after a given output level relationship between relationship between y y and and qq no longer continuous and no longer continuous and

monotonicmonotonic

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A badly designed tax-benefit system

y

q

Low-income type’s indiff curveLow type’s output and incomeHigh-income type’s indiff curve

High type’s intended output and income

Menu offered to low income groups

Withdrawal of benefit B0

q aq b

y a

y b

Implied attainable set

High type’s utility-maximising choice

B0

The notch violates IC…

…causes a-types to masquerade as b-types

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Neglected design issues? Administrative complexity Administrative complexity Example 1. UK today (Example 1. UK today (Mirrlees et al 2011)) Example 2. Germany Example 2. Germany 1981-19851981-1985: :

linearly increasing marginal tax ratelinearly increasing marginal tax rate quadratic tax and disposable income schedulesquadratic tax and disposable income schedules

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

0 20000 40000 60000 80000 100000 120000 140000

rates for single person (§32a Einkommensteuergesetz); units DM:

income x up to 4,212: T = 0 4,213 to 18,000: T = 0.22x – 926 18,001 to 59,999: T =  3.05 X4 – 73.76 X3 + 695 X2 + 2,200 X + 3,034 where X = x/10,000 – 18,000; 60,000 to 129,999: T = 0.09X4 – 5.45X3 + 88.13 X2 + 5,040 X + 20,018 where X = x/10,000 – 60,000; from 130,000:    T = 0.56 x – 14,837

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Arguments for “linear” model

Relatively easy to interpret parametersRelatively easy to interpret parameters Pragmatic: Pragmatic:

Approximates several countries’ tax systemsApproximates several countries’ tax systems Example Example –– piecewise linear tax in UK piecewise linear tax in UK

Sidesteps the incentive compatibility constraintSidesteps the incentive compatibility constraint Simplified version is more tractable analyticallySimplified version is more tractable analytically

Not choosing a general tax/disposable income scheduleNot choosing a general tax/disposable income schedule Given Given t, B t, B and the government budget constraint… and the government budget constraint… ……in effect we have a single-parameter problemin effect we have a single-parameter problem See See Kaplow (2008), pp 58-63 Kaplow (2008), pp 58-63

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Linear model: Lagrangean Social welfare is a function of individual utilitySocial welfare is a function of individual utility Individual utility is maximised subject to budget constraint Individual utility is maximised subject to budget constraint

Determined by individual abilityDetermined by individual ability Tax parameters Tax parameters BB and and tt

Optimisation problem: choose Optimisation problem: choose BB and and tt to max social welfare to max social welfare Subject to government budget constraint Subject to government budget constraint

From maximised Lagrangean get a messy result involving From maximised Lagrangean get a messy result involving the the covariance of social marginal valuation and incomecovariance of social marginal valuation and income the compensated labour-supply elasticitythe compensated labour-supply elasticity

If If K K = 0 then = 0 then B B > 0 > 0 No explicit general formula?No explicit general formula?

FOC cannot be solved to give FOC cannot be solved to give tt covariance and elasticities will themselves be functions of covariance and elasticities will themselves be functions of tt

And in some cases you get a clear-cut result…And in some cases you get a clear-cut result…

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John Broome’s revelation

Broome (1975) suggested a great simplification. suggested a great simplification. Optimal income tax rate should be 58.6% !! Optimal income tax rate should be 58.6% !!

The basis for this astounding claim?The basis for this astounding claim? Tax rate is in fact 2 – Tax rate is in fact 2 – 2; follows from a simple model2; follows from a simple model

Rather it is a useful lesson in applied modellingRather it is a useful lesson in applied modelling He makes conventional assumptionsHe makes conventional assumptions

no-one has ability less than 0.707 times the averageno-one has ability less than 0.707 times the average Cobb-Douglas preferences:Cobb-Douglas preferences: ““Rawlsian” max-min social welfareRawlsian” max-min social welfare Balanced budget: pure redistributionBalanced budget: pure redistribution

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A simulation model

Stern’s (1976) model of linear OIT model of linear OIT can be taken as a generalisation of Broomecan be taken as a generalisation of Broome simulation uses standard ingredients: simulation uses standard ingredients:

Lognormal abilityLognormal ability ……more on this belowmore on this below

Isoelastic individual utilityIsoelastic individual utility elasticity of substitution elasticity of substitution

Isoelastic social welfareIsoelastic social welfare W = (()) dF()

1 – – 1 () = ———— , 1 – inequality aversion inequality aversion

Variety of assumptions about government budget constraintVariety of assumptions about government budget constraint

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Lognormal ability

0 1 2 3 40

f(w)

w

—(w; 0, 0.25 )

…(w; 0, 1.0 )

Two parameter distribution Two parameter distribution ((ww; ; mm, , ss22 ) ) m m is log of the medianis log of the median ss22 is the variance of log income is the variance of log income support is [0, support is [0, ))

Approximation to empirical Approximation to empirical distributionsdistributions Particularly manual workersParticularly manual workers Stern took Stern took s s = 0.39 (same as = 0.39 (same as

Mirrlees 1971)) In this case less than 2% of the In this case less than 2% of the

population have less than 0.707 population have less than 0.707 × mean (× mean (Broome 1975 ))

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Stern's Optimal Tax Rates

0.2 36.20.4 22.30.6 17.00.8 14.11.0 12.7

• Calculations are for a purely redistributive tax: i.e. K = 0• Broome case corresponds to bottom right corner. But he assumed that there was no-one below 70.71% of the median.

62.747.738.933.129.1

92.683.975.668.262.1

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Summary

Could we have “full information” taxation?Could we have “full information” taxation? OIT is a standard second-best problemOIT is a standard second-best problem Elementary version a reworking of the contract modelElementary version a reworking of the contract model Can be extended to general ability distributionCan be extended to general ability distribution Provides simple rules of thumb for good designProvides simple rules of thumb for good design In practice these may be violated by well-meaning In practice these may be violated by well-meaning

policiespolicies

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References (1) Bolton, P. and Dewatripont, M. (2005) Bolton, P. and Dewatripont, M. (2005) Contract TheoryContract Theory, The MIT Press, pp , The MIT Press, pp

62-67.62-67. **Brewer, M., Saez, E. and Shephard, A. (2010) “Means-testing and Tax “Means-testing and Tax

Rates on Earnings,” Rates on Earnings,” in Dimensions of Tax Design: The Mirrlees Reviewin Dimensions of Tax Design: The Mirrlees Review, , Oxford University Press, Chapter 2, pp 90-164 Oxford University Press, Chapter 2, pp 90-164

Broome, J. (1975) “An important theorem on income tax,” “An important theorem on income tax,” Review of Review of Economic StudiesEconomic Studies, , 4242, 649-652, 649-652

Diamond, P.A. (1998) “Optimal Income taxation: an example with a U- “Optimal Income taxation: an example with a U-Shaped pattern of optimal marginal tax rates,” Shaped pattern of optimal marginal tax rates,” American Economic ReviewAmerican Economic Review, , 8888, 83-95, 83-95

Ebert, U. (1992) “A re-examination of the optimal non-linear income tax,” “A re-examination of the optimal non-linear income tax,” Journal of Public EconomicsJournal of Public Economics, , 4949, 47-73, 47-73

Gibbard, A. (1973) “Manipulation of voting schemes: a general result,” “Manipulation of voting schemes: a general result,” EconometricaEconometrica, , 4141, 587-60, 587-60

*Kaplow, L. (2008) *Kaplow, L. (2008) The Theory of Taxation and Public EconomicsThe Theory of Taxation and Public Economics, , Princeton University PressPrinceton University Press

*Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009) “Optimal Taxation in *Mankiw, N.G., Weinzierl, M. and Yagan, D. (2009) “Optimal Taxation in Theory and Practice,” Theory and Practice,” Journal of Economic PerspectivesJournal of Economic Perspectives, , 2323, 147-174 , 147-174

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References (2)

Mirrlees, J. A. (1971) “An exploration in the theory of the optimal income tax,” Mirrlees, J. A. (1971) “An exploration in the theory of the optimal income tax,” Review of Economic StudiesReview of Economic Studies, , 3838, 135-208, 135-208

Mirrlees, J. A. et al (2011) “The Mirrlees Review: Conclusions and Mirrlees, J. A. et al (2011) “The Mirrlees Review: Conclusions and Recommendations for Reform,” Recommendations for Reform,” Fiscal StudiesFiscal Studies, , 3232, 331–359, 331–359

Saez, E. (2001) “Using elasticities to derive optimal income tax rates,” Saez, E. (2001) “Using elasticities to derive optimal income tax rates,” Review of Review of Economic StudiesEconomic Studies, , 6868,205-22,205-22

*Salanié, B. (2003) *Salanié, B. (2003) The Economics of TaxationThe Economics of Taxation, MIT Press, pp 59-61, 79-109, MIT Press, pp 59-61, 79-109 Satterthwaite, M. A. (1975) “Strategy-proofness and Arrow's conditions, Satterthwaite, M. A. (1975) “Strategy-proofness and Arrow's conditions, Journal of Journal of

Economic TheoryEconomic Theory, , 1010, 187-217, 187-217 Seade, J. (1977) “On the shape of optimal tax schedules,” Seade, J. (1977) “On the shape of optimal tax schedules,” Journal of Public Journal of Public

EconomicsEconomics, , 77, 203-23, 203-23 Sheshinski, E. (1972) “The optimal linear income tax,” Sheshinski, E. (1972) “The optimal linear income tax,” Review of Economic StudiesReview of Economic Studies, , 3939, 297-302, 297-302

Stern, N. (1976) “On the specification of models of optimum income taxation” Stern, N. (1976) “On the specification of models of optimum income taxation” Journal of Public EconomicsJournal of Public Economics, , 66,123-162,123-162

Tuomala, M. (1984) “On the Optimal Income Taxation: Some Further Numerical Tuomala, M. (1984) “On the Optimal Income Taxation: Some Further Numerical Results,” Results,” Journal of Public EconomicsJournal of Public Economics, , 2323, 351-366, 351-366