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Design: Taxation

MICROECONOMICSMICROECONOMICSPrinciples and AnalysisPrinciples and Analysis

Frank Cowell Frank Cowell

Almost essential

Contract Design

Almost essential

Contract Design

PrerequisitesPrerequisites

September 2006September 2006

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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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A link with contract theory Base approach on the analysis of contractsBase approach on the analysis of contracts

close analogy with case of hidden characteristicsclose analogy with case of hidden characteristics owner hires manager…owner hires manager… ……but manager’s ability is unknown at time of hiring but manager’s ability is unknown at time of hiring

Ability here plays the role of unobservable typeAbility here plays the role of unobservable type ability may not be directly observable…ability may not be directly observable… ……but distribution of ability in the population is knownbut distribution of ability in the population is known

A progressive treatment:A progressive treatment: outline model componentsoutline model components use analogy with contracts to solve two-type caseuse analogy with contracts to solve two-type case proceed to large (finite) number of typesproceed to large (finite) number of types then extend to general continuous distributionthen extend to general continuous distribution

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Overview...

Design basics

Simple model

Generalisations

Interpretations

Design: Taxation

Preferences, incomes, ability and the government

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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

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 provide consumptionIndividuals work (give up leisure) to provide 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 qq and and 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 (1)

increasing

preference

y

1– z

Preferences over leisure and income

Indifference curves

= = zz + + yy zzzz < 0 < 0

Reservation utility

≥≥

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The utility function (2)

increasingpreference

y

q

Preferences over leisure and output

Indifference curves

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

Reservation utility

≥≥

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Indifference curves: pattern

All types have the same preferencesAll types have the same preferences Function Function is common knowledge is common knowledge

but utility level but utility level of type of type depends on effort depends on effort zz and and payment payment yy

value of value of may be information that is private to may be information that is private to individualindividual

Take indifference curves in (Take indifference curves in (q, yq, y) space) space = = qq + + yy slope of a given type’s indifference curve depends on slope of a given type’s indifference curve depends on

value of value of indifference curves of different types cross once onlyindifference curves of different types cross once only

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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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Similarity with contract model

The position of the Agent instead of a single Agent with known ex-ante probability

distribution of talents,… … a population of workers with known distribution of abilities.

The position of the Principal (designer) designer is the government acting as Principal. knows distribution of ability (common knowledge) the objective function is a standard SWF

One extra constraint the community has to raise a fixed amount K ≥ 0 the government imposes a tax drives a wedge between market income generated by worker and

the amount available to spend on other goods.

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Overview...

Design basics

Simple model

Generalisations

Interpretations

Design: Taxation

Analogy with contract theory

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A full-information solution?

Consider argument based on the analysis of contractsConsider argument based on the analysis of contracts Given full information owner can fully exploit any managerGiven full 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 Start with the case closest to the optimal contract modelmodel

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 From contract design we can write down the outcomeoutcome 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: method Differentiate with respect to Differentiate with respect to qqaaqqbbyyaayybb to get FOCs: to get FOCs:

aazzzzaa//aa + + + + zzzzaa//aa ≤≤ 0 0 [1[1]]bbzzzzbb//b b + + [1 [1] + ] + zzzzbb//b b zzqqbb//aa//a a ≤≤ 0 0 aa+ + ≤≤ 0 0 [1[1]]bb[1[1] + ] + ≤≤ 0 0

For an interior solution, where For an interior solution, where qqaaqqbbyyaayybb are all positive are all positive aazzzzaa//aa + + + + zzzzaa//aa == 0 0 [1[1]]bbzzzzbb//b b + + [1 [1] + ] + zzzzbb//b b zzqqbb//aa//a a == 0 0 aa+ + == 0 0 [1[1]]bb[1[1] + ] + == 0 0

Manipulating these gives the main resultsManipulating these gives the main results For example, from first and third condition:For example, from first and third condition: [[ ] ]zzzzaa//aa + + + + zzzzaa//aa == 0 0 zzzzaa//aa + + == 0 0

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Two types: solution Solving the FOC we get:Solving the FOC 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 basics

Simple model

Generalisations

Interpretations

Design: Taxation

Moving beyond the two-ability model

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A small generalisation

With three types problem becomes a bit more interestingWith three types problem becomes a bit more interesting 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 binding Again, no distortion at the top and the participation constraint binding at the bottomat 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 three typesThe multi-type case follows immediately from three types

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 like the type below ( must be no worse off than if it behaved like 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 Suppose the tax agency is faced with a continuum of Suppose the tax agency is faced with a continuum of

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

This case can be reasoned from the case with This case 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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Continuum model: optimisation Lagrangean isLagrangean is

⌠⌠ │ │ (()) ff d d ⌡ ⌡

⌠⌠++│ [ │ [ qq− − yy−− ff d d ⌡ ⌡

+ + [ [ − − ⌠⌠dd+ + │ │ ———— ff d d ⌡ ⌡ dd

where where yy(() + ) + q q(()) Lagrange multipliers areLagrange multipliers are

: government budget constraint: government budget constraint : participation constraint: participation constraint incentive-compatibility for type incentive-compatibility for type

Maximise Lagrangean with respect to Maximise Lagrangean with respect to qqand and yy for all for all [[

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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 curve

Highest type’s output and incomeMenu offered by tax authority

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Continuum model: results

Incentive compatibility implies Incentive compatibility implies ddyy /d /dqq >> 0 0 optimal marginal tax rate < 100%optimal marginal tax rate < 100%

No distortion at top impliesNo distortion at top implies ddyy /d /dqq = 1 = 1 zero optimal marginal tax rate!zero optimal marginal tax rate!

But explicit form for the optimal income tax requiresBut explicit 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

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icroeconomics

Microeconom

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Overview...

Design basics

Simple model

Generalisations

Interpretations

Design: Taxation

Applying design rules to practical policy

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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 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

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Violations of design principles?

Sometimes the IC condition be violated in actual designSometimes the 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 monotonic no longer continuous and monotonic

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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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Summary

Optimal income tax is a standard second-best Optimal income tax is a standard second-best problemproblem

Elementary version a reworking of the contract Elementary version a reworking of the contract modelmodel

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