Frank Cowell: UB Public Economics Optimal Tax Design June 2005 Public Economics: University of...

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Optimal Tax Design

June 2005 June 2005

Public Economics: University of Barcelona Public Economics: University of Barcelona

Frank CowellFrank Cowell

http://darp.lse.ac.uk/ubhttp://darp.lse.ac.uk/ub

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Purpose of tax design

The issue of design is fundamental to public economicsThe issue of design is fundamental to public economics Move from what we would like to achieve…Move from what we would like to achieve… ……to what we can actually implementto what we can actually implement Plenty of examples of this issue:Plenty of examples of this issue:

Public-good provisionPublic-good provision RegulationRegulation Social insuranceSocial insurance Optimal taxation – see below.Optimal taxation – see below.

Important to be clear what the purpose of the tax design Important to be clear what the purpose of the tax design problem is.problem is.

A brief review of the elements of the problem. …A brief review of the elements of the problem. …

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Components of the problem ObjectivesObjectives

Could be an attempt to satisfy a particular objective Could be an attempt to satisfy a particular objective function or class of functionsfunction or class of functions

Could be a characterisation of policies that achieve Could be a characterisation of policies that achieve some broad objectives.some broad objectives.

Scope for policyScope for policy Methods of interventionMethods of intervention ConstraintsConstraints Informational problemsInformational problems

Available toolsAvailable tools The tax baseThe tax base Direct and indirect taxationDirect and indirect taxation

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Overview...DesignIssues

General labour model

Generalisations

Optimal Income Taxation

Why this kind of problem is set up

“linear” labour model

Education model

•Objectives•Scope for policy•Informational issues•Available tools

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Specific objectives?

The objectives of the tax design could include:The objectives of the tax design could include:

1.1. Bergson-Samuelson welfare maximisation Bergson-Samuelson welfare maximisation

2.2. Overall concern for efficiencyOverall concern for efficiency

3.3. Overall concern for reduction of inequality Overall concern for reduction of inequality of outcome.of outcome.

4.4. Inequality of opportunityInequality of opportunity

5.5. Poverty, horizontal inequity...Poverty, horizontal inequity... More than one of the above may be relevant.More than one of the above may be relevant.

Could be a class of functionsCould be a class of functions

Could be incorporated in objective #1

Could be incorporated in objective #1

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Implementation of objectives

What is domain of the SWF?What is domain of the SWF? Incomes?Incomes? Individual utilities?Individual utilities?

What social entities? What social entities? IndividualsIndividuals FamiliesFamilies Household units?Household units?

Need a model of cardinal, comparable utility

Need a model of cardinal, comparable utility

Welfarist approach usually founded on this basisWelfarist approach usually founded on this basis

Data is often on this basis…Data is often on this basis…

…or this…or this

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Generalisations


Types of intervention. The tax base “linear” labour

model

Education model


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Scope for policy

What is potentially achievable?What is potentially achievable?

We need to do this before we can examine We need to do this before we can examine specific policy tools and their associated specific policy tools and their associated constraints.constraints.

If we have in mind income redistribution it is If we have in mind income redistribution it is appropriate to look at the determinants of incomeappropriate to look at the determinants of income

Do this within the context of an elementary Do this within the context of an elementary microeconomic model.microeconomic model.

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Take the standard microeconomic model of a Take the standard microeconomic model of a person’s total income in a market economyperson’s total income in a market economy

Composed of resources valued at their Composed of resources valued at their market prices:market prices:

incomeincome

endowment of good iendowment of good i

Non-market incomeNon-market income

Does this mean public policy has to be limited toDoes this mean public policy has to be limited to

1.1. redistributing resources, orredistributing resources, or2.2. manipulating prices?manipulating prices?

There could be other forms of incomeThere could be other forms of income

The Composition of Income

price of good iprice of good i

.

Problems with 1 and 2 above are also importantProblems with 1 and 2 above are also important

And there may be other types of interventionAnd there may be other types of intervention

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Problems with redistributing resources:

The lump-sum tax issue:The lump-sum tax issue: Special information – such as personal characteristics Special information – such as personal characteristics Political problems of implementationPolitical problems of implementation

Non-transferabilityNon-transferability Fixed resourcesFixed resources Inalienability of certain rights – No slavery Inalienability of certain rights – No slavery

Ways of getting round these problems?Ways of getting round these problems? Could redistribute the purchasing power generated by the Could redistribute the purchasing power generated by the

resource?resource? Or modify the supply of “co-operant factors”?Or modify the supply of “co-operant factors”?

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Problems with price manipulation

Identification of commoditiesIdentification of commodities The boundary problemThe boundary problem Artificial definition of a good or service on which a tax Artificial definition of a good or service on which a tax

is to be levied. is to be levied. ComplexityComplexity

Proliferation of implied pricing structuresProliferation of implied pricing structures Informational problemsInformational problems

Uncertainty leads to wrong price signals? Uncertainty leads to wrong price signals? Misinformation leads to wrong price signals?Misinformation leads to wrong price signals? May even be missing marketsMay even be missing markets

Need to focus on economics of informationNeed to focus on economics of information

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Generalisations


Fundamental theoretical issues in design problem


Education model


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Informational issues in microeconomics There are two key types of informational problem:There are two key types of informational problem: Both can be relevant to policy design.Both can be relevant to policy design. Hidden actionHidden action: :

Regulation and optimal contracts.Regulation and optimal contracts. Moral hazard in social insuranceMoral hazard in social insurance Compliance issues.Compliance issues.

Hidden informationHidden information: : Problems of “tailoring” tax rates.Problems of “tailoring” tax rates. Adverse selection in social insurance.Adverse selection in social insurance. Focus on this issue hereFocus on this issue here

But the “information issue” is quite deep:But the “information issue” is quite deep: There is connection with discussion of social welfareThere is connection with discussion of social welfare A fundamental relationship with the “Arrow” problemA fundamental relationship with the “Arrow” problem

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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

Similar to the concept of constitutionSimilar to the concept of constitution But maps from set of preference profiles to But maps from set of preference profiles to set of social set of social

statesstates Given a particular set of preferences for the populationGiven a particular set of preferences for the population Picks out the preferred social statePicks out the preferred social state

Not surprising to find result similar to ArrowNot surprising to find result similar to Arrow Introduce weak conditions on the Social-choice functionIntroduce weak conditions on the Social-choice function There’s no SCF that satisfies all of themThere’s no SCF that satisfies all of them

But key point concerns the But key point concerns the implementationimplementation issue issue

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Implementation

Is the social-choice function consistent with private Is the social-choice function consistent with private economic behaviour?economic behaviour?

Yes if the social state picked out by the SCF corresponds Yes if the social state picked out by the SCF corresponds to an equilibriumto an equilibrium

Problem becomes finding an appropriate mechanismProblem becomes finding an appropriate mechanism mechanism can be thought of as a kind of cut-down gamemechanism can be thought of as a kind of cut-down game to be interesting the game is one of imperfect informationto be interesting the game is one of imperfect information is the desired social state an equilibrium of the game?is the desired social state an equilibrium of the game?

There is a wide range of possible mechanismsThere is a wide range of possible mechanisms Focus on a type that is useful for expositional purposes...Focus on a type that is useful for expositional purposes...

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Direct mechanisms

Map from collection of preferences to statesMap from collection of preferences to states Involves a very simple game.Involves a very simple game. The game is “show me your utility function”The game is “show me your utility function” Enables us to focus directly on the informational aspects of Enables us to focus directly on the informational aspects of

implementationimplementation

Here the SCF is the mechanism itselfHere the SCF is the mechanism itself An SCF that encourages misrepresentation may be of An SCF that encourages misrepresentation may be of

limited uselimited use Is truthful implementation possible?Is truthful implementation possible?

Will people announce their true attributes?Will people announce their true attributes? Will it be a dominant strategy to do so?Will it be a dominant strategy to do so?

Introduce another key resultIntroduce another key result

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Gibbard-Satterthwaite result Can be stated in a variety of ways.Can be stated in a variety of ways. A standard versions is: A standard versions is:

If the set of social states contains at least three elements;If the set of social states contains at least three elements; ...and the social choice function is defined for the all logically ...and the social choice function is defined for the all logically

possible preference profiles...possible preference profiles... ...and the SCF is truthfully implementable in dominant strategies... ...and the SCF is truthfully implementable in dominant strategies... ...then the SCF must be dictatorial...then the SCF 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 Underlies issues of public-good provision, regulation, tax designUnderlies issues of public-good provision, regulation, tax design

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Generalisations


What practical options available to achieve the objectives?


Education model


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Informational issues in taxation

What distinguishes taxation from highway What distinguishes taxation from highway robbery?robbery? Taxation principlesTaxation principles Appropriate informationAppropriate information

What information is/should be available? What information is/should be available? AttributesAttributes BehaviourBehaviour

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Available tools

Availability determined by a variety of Availability determined by a variety of considerations.considerations.

Fundamental economic constraints Fundamental economic constraints

Institutional constraints. These may come from:Institutional constraints. These may come from: Legal restrictionsLegal restrictions Administrative considerationsAdministrative considerations Historical precedentHistorical precedent

But each of these institutional aspects may really But each of these institutional aspects may really follow from the economics.follow from the economics.

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We focus here on the taxation of individuals rather than We focus here on the taxation of individuals rather than corporations or other entities.corporations or other entities.

An approach to the individual tax-base might begin with An approach to the individual tax-base might begin with an examination of the individual’s budget constraint:an examination of the individual’s budget constraint:

The Tax Base

expenditureexpenditure consumption of good iconsumption of good i

number of goodsnumber of goods

So taxation might be based on consumption of specific So taxation might be based on consumption of specific goods or on some concept of income or expenditure goods or on some concept of income or expenditure

We will see that using the above as an elementary We will see that using the above as an elementary method of classifying taxes can be misleading method of classifying taxes can be misleading

First take a closer look at income:First take a closer look at income:

incomeincome

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It is tempting to think of the distinction between It is tempting to think of the distinction between different types of tax in terms of the budget different types of tax in terms of the budget constraint:constraint:

A fundamental difference?

Indirect taxes here?Indirect taxes here? Direct taxes

here?Direct taxes here?

This misses the pointThis misses the point

Any tax on RHS can be converted to tax on LHSAny tax on RHS can be converted to tax on LHS

Real question is about informationReal question is about information

.

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Information again

The government and its agencies must work with The government and its agencies must work with imperfect information.imperfect information.

To model taxes appropriately need to take this To model taxes appropriately need to take this into account. into account.

Information imposes specific constraints on tax Information imposes specific constraints on tax designdesign

In a typical market economy there are two main In a typical market economy there are two main types of information:types of information: About individualsAbout individuals About transactionsAbout transactions

IncomeTotal expenditureAge, marital status?

IncomeTotal expenditureAge, marital status?

Expenditure by product categoryExpenditure by industryInput and output quantities

Expenditure by product categoryExpenditure by industryInput and output quantities

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Fundamental constraints... Public budget constraintsPublic budget constraints

ExampleExample: In simple redistribution sum of net receipts : In simple redistribution sum of net receipts (taxes (taxes cash subsidies) must be zero cash subsidies) must be zero

Participation constraintsParticipation constraints ExampleExample: Labour supply: Labour supply

Incentive-compatibility (self-selection) constraintsIncentive-compatibility (self-selection) constraints ExampleExample: Differential subsidies for specific : Differential subsidies for specific

commoditiescommodities

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Design basics: summary

Objectives follow on logically from our Objectives follow on logically from our discussion in previous lectures.discussion in previous lectures.

Beware of oversimplifying assumptions Beware of oversimplifying assumptions about the tax base.about the tax base.

Information plays a key role.Information plays a key role.

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Generalisations


What types of tax formula used in theory and practice?

“Linear” labour model

Education model

•Tax schedules•Outline of problem•The solution

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Income tax – example of design problem Standard types of taxStandard types of tax

simple examples simple examples integration with income supportintegration with income support

General issues of how to set up an optimisation General issues of how to set up an optimisation problemproblem

Solution of optimal tax problem:Solution of optimal tax problem: Solution of the general tax design problemSolution of the general tax design problem Solution of the special “linear” caseSolution of the special “linear” case Alternative models of optimal income taxAlternative models of optimal income tax

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Income tax – notation

y – y – taxable income taxable income c – c – disposable income (“consumption”)disposable income (“consumption”) TT((·)·) – – tax scheduletax schedule cc((·)·) – – disposable income scheduledisposable income schedule – – marginal tax ratemarginal tax rate yy00 – – exemption-level incomeexemption-level income

BB – lumpsum benefit/guaranteed income – lumpsum benefit/guaranteed income

0 10 1

c(y) = y – T(y)c(y) = y – T(y)

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Income space

y

c=c(y)

pre-tax income

disp

osab

le in

com

e

no-in

terve

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line

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The simple income tax

y

c=c(y)

1-

y0

Exemption levelExemption level

Marginal retention rateMarginal retention rate

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...extended to Negative Income Tax

y

c=c(y)

1-

By0

B = y0B = y0

Incomes subsidised through NIT

Incomes subsidised through NIT

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How to generalise this approach…?

Other functional forms of the income taxOther functional forms of the income tax

Administrative complexity of ITAdministrative complexity of IT

Interaction with other contingent taxes and Interaction with other contingent taxes and benefits.benefits.

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Increasing marginal tax rate

y

c(y)

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Example 1 UK: UK:

piecewise linear taxpiecewise linear tax

stepwise jumps in MTRstepwise jumps in MTR

compare with contingent tax/benefit modelcompare with contingent tax/benefit model

Germany: Germany:

linearly increasing marginal tax ratelinearly increasing marginal tax rate

quadratic tax and disposable income schedulesquadratic tax and disposable income schedules

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Germany 1981-1985, single person (§32a Einkommensteuergesetz): Germany 1981-1985, single person (§32a Einkommensteuergesetz): up to 4,212: up to 4,212: T T = 0 = 0 4,213 to 18,000: 4,213 to 18,000: TT = 0.22 = 0.22y y – 926 – 926 18,001 to 59,999: 18,001 to 59,999: TT = 3.05 = 3.05 z z44 – 73.76 – 73.76 zz33 + 695 + 695 z z22 + 2,200 + 2,200 zz + 3,034 + 3,034 zz = = yy/10,000 - 18,000; /10,000 - 18,000; 60,000 to 129,999: 60,000 to 129,999: TT = 0.09 = 0.09zz4 4 – 5.45– 5.45zz3 3 + 88.13 + 88.13 zz22 + 5,040 + 5,040 zz + 20,018 + 20,018

zz = = yy/10,000 - 60,000; /10,000 - 60,000; from 130,000: from 130,000: TT = 0.56 = 0.56 yy – 14,837 – 14,837 (units: DM)(units: DM)

Example 2

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

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Interaction with income support

y

c(y)

B

y1y2y00

Untaxed income support

Untaxed income support

“Clawback” of support“Clawback” of support

Tax-payments kick in with benefitsTax-payments kick in with benefits

Straight income tax at constant marginal rate

Straight income tax at constant marginal rate

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The approach to IT – summary

The “linear” form may be a reasonable approximation to The “linear” form may be a reasonable approximation to some practical casessome practical cases

We may also see an appealing intuitive argument for We may also see an appealing intuitive argument for linearity as simplificationlinearity as simplification

““Income tax” may need to be interpreted fairly broadlyIncome tax” may need to be interpreted fairly broadly

Interaction amongst various forms of government Interaction amongst various forms of government intervention is important for an appropriate model intervention is important for an appropriate model

This may lead to nonlinearity in the effective scheduleThis may lead to nonlinearity in the effective schedule

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Generalisations


Basic ingredients of OIT analysis.

“Linear” labour model

Education model


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Basic Ingredients of An Optimal Income Tax model A distribution of abilitiesA distribution of abilities

Individuals’ behaviourIndividuals’ behaviour

Social-welfare functionSocial-welfare function

Feasibility ConstraintFeasibility Constraint

Restriction on types of Restriction on types of functional formfunctional form

What resources are potentially available for redistribution?

What resources are potentially available for redistribution?

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Distribution of Ability...

Assume... Assume...

a single source of earning power – “ability” a single source of earning power – “ability”

ability is fully reflected in the (potential) wage ability is fully reflected in the (potential) wage ww

So ability is effectively measured by So ability is effectively measured by ww..

the distribution the distribution FF of of ww is observable is observable

individual values of individual values of ww are are notnot observable by the observable by the tax authoritytax authority

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Can we infer the distribution of ability? Practical approachPractical approach Select relevant group or groups in population.Select relevant group or groups in population.

male manual workers?male manual workers?

Choose appropriate earnings concept. Choose appropriate earnings concept. Full time earnings?Full time earnings?

Divide earnings by hours to get wages.Divide earnings by hours to get wages. Use parametric model to capture shape of Use parametric model to capture shape of

distribution.distribution. Lognormal?Lognormal?

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Distribution: example

0 y

f(y

)

NES 2000

Lognormal (5.7,0.13)

Example from UK 2000Example from UK 2000

Gives distribution of Gives distribution of y=why=wh for full-time male manual workers for full-time male manual workers

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Basic Ingredients of An Optimal Income Tax model (2) A distribution of abilitiesA distribution of abilities





In what ways do we assume that people will respond to the tax authority’s instruments ?

In what ways do we assume that people will respond to the tax authority’s instruments ?

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The individual's problem

Individual’s utility is determined by disposable income Individual’s utility is determined by disposable income (consumption) (consumption) cc and leisure. and leisure.

So the optimisation problem can be writtenSo the optimisation problem can be written

maxmaxhh U U((cc,,hh) )

subject to subject to c = y – Tc = y – T((yy))

and and y = why = wh

This yields maximised utility as a function of ability This yields maximised utility as a function of ability (wage):(wage):

((ww))

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A Characterisation of Tastes

Introduce a definition to capture the shape of Introduce a definition to capture the shape of individual preferencesindividual preferences

Normalised MRSNormalised MRS

The following restriction on “regularity” of The following restriction on “regularity” of preferences is important for clean-cut resultspreferences is important for clean-cut resultsThe way slope of indifference

curve changes with abilityThe way slope of indifference curve changes with ability

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A representation of preferences

(co

nsu

mp

tio

n)

(leisure)

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Indifference curve in (h,c)-space

h

c(c

on

sum

pti

on

)

(hours worked)

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Contour translated to (y,c)-space

c

y

(co

nsu

mp

tio

n =

net

in

com

e)

(gross income)

slope = q

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The regularity condition

y

c

low w

high w

Illustrates the qw < 0 property

Ensures “single-crossing” of ICs for different ability groups

Illustrates the qw < 0 property

Ensures “single-crossing” of ICs for different ability groups

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Individual's problem: points to note

Incorporates standard assumptionsIncorporates standard assumptions Same basic model as in earlier lecturesSame basic model as in earlier lectures Consistent with the modelConsistent with the model

y = why = wh or with the modelor with the model

y = wh + Iy = wh + I

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How to represent the objectives of the optimisation problem?

How to represent the objectives of the optimisation problem?

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The Government’s Objective...

Social evaluation functionSocial evaluation function

Take a standard version of the SWFTake a standard version of the SWF

Assume additive separabilityAssume additive separability

Take “weighted average” over typesTake “weighted average” over typesMaximised utility of a w-type personMaximised utility of a w-type person

Proportion of w-type persons in the populationProportion of w-type persons in the population

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Real-world restrictions on government and the design problem

Real-world restrictions on government and the design problem

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Main types of constraint

The Government’s budget constraintThe Government’s budget constraint

Incentive compatibilityIncentive compatibility

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Government’s Budget Constraint

disposable income in populationdisposable income in population

Earnings in populationEarnings in population

Net revenue requirementNet revenue requirement

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Two Ability Levels

y

c

low w

high w

Incentive compatibility problem:

Original and disposable income must increase with ability

Incentive compatibility problem:

Original and disposable income must increase with ability

c(y)

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Restrictions on form...

It may make sense to consider cases where It may make sense to consider cases where marginal tax-rate is everywhere constant marginal tax-rate is everywhere constant (like the NIT model earlier):(like the NIT model earlier):

Administrative costs of general modelAdministrative costs of general model informational problemsinformational problems “ “fairness” arguments [?]fairness” arguments [?]

Pre-1985 Germany?Pre-1985 Germany?

Cf the US “flat tax” discussionCf the US “flat tax” discussion

Lack of detail about tailsLack of detail about tails

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Review: ingredients of OIT model

A distribution of abilitiesA distribution of abilities Ability is measured by potential wage Ability is measured by potential wage ww.. Distribution Distribution FF of of ww is observable is observable Individual values of Individual values of ww are not observable are not observable

Individuals’ behaviourIndividuals’ behaviour maxmaxhh U U((cc,,hh), subject to ), subject to c = y – Tc = y – T((yy) and ) and y = why = wh Gives utility as function of ability Gives utility as function of ability ((ww))

Social-welfare functionSocial-welfare function Assume individualistic additively separable SWFAssume individualistic additively separable SWF WW = = uu( ( ((ww) ) ff((ww) d) dww

Feasibility constraintsFeasibility constraints The Government’s budget constraintThe Government’s budget constraint Incentive compatibilityIncentive compatibility

Restriction on types of functional formRestriction on types of functional form (Piecewise) linear schedule?(Piecewise) linear schedule?

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Generalisations


Characterising a general optimal income tax. “Linear” labour

model

Education model


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The general OIT model

No preconditions on form of income taxNo preconditions on form of income tax Use results from economics of informationUse results from economics of information

IC condition required for “sensible” resultsIC condition required for “sensible” results However, no consideration of administrative complexity However, no consideration of administrative complexity

Use a general variational approach to give the solutionUse a general variational approach to give the solution Has similarities with techniques used for optimal growthHas similarities with techniques used for optimal growth Terminal conditions can be importantTerminal conditions can be important

Illustrate the variational approach diagrammaticallyIllustrate the variational approach diagrammatically Use the disposable income schedule Use the disposable income schedule cc((••))

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The general tax-design problem

c(y)

y

Variation in general tax scheduleVariation in general tax schedule

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Individual optimisation Begin with the way each tax-payer is assumed to act.Begin with the way each tax-payer is assumed to act. The optimisation problem can be writtenThe optimisation problem can be written

Disposable incomeDisposable income

workwork The function The function cc((··) is chosen by the government) is chosen by the government Define normalised MRSDefine normalised MRS

Slope of disposable income functionSlope of disposable income function

The solution is of the form The solution is of the form ((ww) := max) := maxhh U U((cc((whwh), ), hh) )

The first-order condition isThe first-order condition is

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Contour in (y,c)-space

c

y

slope = q

Proportional to workProportional to work

Disposable income schedule c(•)Disposable income schedule c(•)

optimised incomeoptimised income

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A regularity condition

y

c

low w

high w

If the property qw < 0 holds this ensures “single-crossing” of ICs for different ability groups

If the property qw < 0 holds this ensures “single-crossing” of ICs for different ability groups

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Incentive compatibility condition

y

c

low w

high w

Design of c must ensure that utility and income increase with ability

Design of c must ensure that utility and income increase with ability

c(y)

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Incentive compatibility The IC condition means that high ability people should The IC condition means that high ability people should

not have an incentive to “masquerade” as low ability.not have an incentive to “masquerade” as low ability. This requires maximised utility This requires maximised utility ((ww) to increase in ) to increase in ww. . By differentiation of the solution function we haveBy differentiation of the solution function we have

Optimised value of hOptimised value of h

If If cc((·) is monotonic and differentiable everywhere ·) is monotonic and differentiable everywhere then this becomesthen this becomes

But if these conditions are violated, problems arise…But if these conditions are violated, problems arise…

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Fundamental design problem

It may seem odd that the IC condition be violated in actual It may seem odd that the IC condition be violated in actual designdesign

But it can happen by accident:But it 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 effectively.generated by the desire to “target” support more effectively. A well-meant gross inefficiency?A well-meant gross inefficiency?

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

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“Notch problem” / “poverty trap”

c(y)

yy0

Withdrawal of benefit hereWithdrawal of benefit here

Discontinuous non-monotonic c(·) Discontinuous non-monotonic c(·)

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Violation of IC condition

c(y)

yy0

low w

Where high w “should” be

Where high w would choose

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Government: maximisation Uses a constrained maximum methodUses a constrained maximum method

But there is a constraint at each ability level from But there is a constraint at each ability level from wwminmin to to wwmaxmax. .

Similar to maximisation over time.Similar to maximisation over time.

Choose Choose cc((··) to max) to max

subject tosubject to

and, at each ability level:and, at each ability level:

disposable income in populationdisposable income in population

Earnings in populationEarnings in populationNet revenue

requirementNet revenue requirement

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Government: maximisation Introduce a Lagrange multiplier Introduce a Lagrange multiplier for the budget constraint for the budget constraint

and a multiplier and a multiplier ((ww) for the incentive compatibility ) for the incentive compatibility constraint at each ability level.constraint at each ability level.

Then, on rearranging, the Lagrangean isThen, on rearranging, the Lagrangean is

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General Model: Characterisation of Marginal Tax Rate

Lagrange multiplier for incentive-compatibility constraint

Lagrange multiplier for incentive-compatibility constraint

Lagrange multiplier for Government budget constraint

Lagrange multiplier for Government budget constraint

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Interpreting the FOC

Can be used to give us an impression of the shape Can be used to give us an impression of the shape of the solutionof the solution

But an explicit form for the OIT is usually not But an explicit form for the OIT is usually not possiblepossible

Some key resultsSome key results

First for the overall shapeFirst for the overall shape Second for what happens at each end of the ability Second for what happens at each end of the ability

range…range…

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Main result 1 Mirrlees 1971:Mirrlees 1971: The optimal marginal tax rate must be The optimal marginal tax rate must be

greater than or equal to 0 and less than 1 greater than or equal to 0 and less than 1 http://darp.lse.ac.uk/papersdb/Mirrlees_(REStud_71).pdfhttp://darp.lse.ac.uk/papersdb/Mirrlees_(REStud_71).pdf

The condition “The condition “0” means that in trying to raise tax it 0” means that in trying to raise tax it never makes sense to introduce a distortionary labour never makes sense to introduce a distortionary labour subsidy subsidy − − see Tuomala (1990) see Tuomala (1990)

The condition “<1” follows fromThe condition “<1” follows from

Agent monotonicity impliesAgent monotonicity implies

So So it is immediate that T'(y) < 1. For the lower extreme of the distribution need to

look at “bunching”…

@@

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Bunching: w'<w''<w'''

w'

w'''w''c

y

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No bunching: w'<w''<w'''

y

c

w'

w'''

w''

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Main result 2 Seade 1977,Ebert 1992.Seade 1977,Ebert 1992. The optimal marginal tax rate: The optimal marginal tax rate:

is 0 on the highest income is 0 on the highest income is 0 on the lowest income if there is no bunchingis 0 on the lowest income if there is no bunching is positive on the lowest income if there is bunchingis positive on the lowest income if there is bunching

For bottom of distribution see Tuomala (1990) For bottom of distribution see Tuomala (1990) For top of distribution note the FOC:For top of distribution note the FOC:

At At wwmaxmax IC constraint becomes irrelevant; so IC constraint becomes irrelevant; so ((wwmaxmax) = 0.) = 0.

Therefore Therefore T'(yymaxmax) = ) = T'(wwmax max hh((wwmaxmax)) = 0)) = 0

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Problems of the general model (1)

There appear to be commonsense general resultsThere appear to be commonsense general results And clear-cut results for the extremes,And clear-cut results for the extremes, But little guidance on the structure for the majority of the But little guidance on the structure for the majority of the

workforce.workforce. Some broad principles can be adduced from the first order Some broad principles can be adduced from the first order

conditions. conditions. But you cannot get further without an explicit modelBut you cannot get further without an explicit model

On the distribution of On the distribution of ww On individual preferencesOn individual preferences On the SWFOn the SWF

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The results for the extremes are not robustThe results for the extremes are not robust Should have low or decreasing tax rates close to the top of Should have low or decreasing tax rates close to the top of

the income distribution?the income distribution? This does not seem to be the case from simulation study This does not seem to be the case from simulation study

Tuomala: J. Pub. Econ 1984Tuomala: J. Pub. Econ 1984 Part of the problem arises from assumed Part of the problem arises from assumed FF((••) of ) of ww

convenient to assume that support of the distribution convenient to assume that support of the distribution FF is finite is finite But this means an artificial assumption about known “endpoints” But this means an artificial assumption about known “endpoints”

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Most applied models assume something like lognormal Most applied models assume something like lognormal or Paretoor Pareto Support is unbounded above.Support is unbounded above. No “maximum” income No “maximum” income

If you rework the model with a distribution that is If you rework the model with a distribution that is “open-ended” at the top things appear very different.“open-ended” at the top things appear very different. Diamond (1998) uses Pareto.Diamond (1998) uses Pareto. Gets high marginal tax rates where ability follows a Pareto Gets high marginal tax rates where ability follows a Pareto

distribution distribution http://darp.lse.ac.uk/papersdb/Diamond_(AER98).pdf Saez (2001) is a general extension of the Mirrlees results Saez (2001) is a general extension of the Mirrlees results

http://darp.lse.ac.uk/papersdb/Saez_(REStud01).pdf

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Cannot get stronger results on tax rates analytically.Cannot get stronger results on tax rates analytically. Can do this for special cases Can do this for special cases Example Salanie model with quasi-linear preferencesExample Salanie model with quasi-linear preferences Or could use simulation in a numerical modelOr could use simulation in a numerical model

But to do this you need to implement a specific model But to do this you need to implement a specific model which can be:which can be: Computationally messyComputationally messy Sensitive to specific assumptions made about labour supply and Sensitive to specific assumptions made about labour supply and

abilityability

It may make sense to imposeIt may make sense to impose more structure more structure a prioria priori on the on the tax functiontax function

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Generalisations


A “cut-down” version of the labour-leisure problem


Education model

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Approach 2: The linear model

Same behavioural assumptions as beforeSame behavioural assumptions as before Same objectivesSame objectives Restriction to linear (affine) tax functions: two parametersRestriction to linear (affine) tax functions: two parameters First analysed by Sheshinski (1972) First analysed by Sheshinski (1972)

http://darp.lse.ac.uk/papersdb/Sheshinski_(REStud_72).pdfhttp://darp.lse.ac.uk/papersdb/Sheshinski_(REStud_72).pdf

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Simplified version is much more tractable analyticallySimplified version is much more tractable analytically

No longer choosing a general tax/disposable income No longer choosing a general tax/disposable income schedule schedule cc((••))

Instead, just a two-parameter model.Instead, just a two-parameter model.

Disposable income isDisposable income is

Linear Model: outline

Marginal tax rateMarginal tax rate

Pre-tax incomePre-tax income

Minimum disposable income

Minimum disposable income

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Arguments for “linear” model

Relatively easy to interpret parametersRelatively easy to interpret parameters as uniform marginal tax rateas uniform marginal tax rate BB as minimum income, or… as minimum income, or… B /B / as exemption rate as exemption rate

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 constraint…Sidesteps the incentive compatibility constraint…

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Incentive compatibility resolved

y

c

low w

high w

Original and disposable income will increase with abilityOriginal and disposable income will increase with ability

c(y)

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In effect this makes the issue a one-variable problem…In effect this makes the issue a one-variable problem…

Given that the IC condition vanishes, there is only one Given that the IC condition vanishes, there is only one constraintconstraint

The Government Budget constraint:The Government Budget constraint:

Linear Model: Constraint

Tax raised on working populationTax raised on working population

Minimum guaranteed income for allMinimum guaranteed income for all

Extra revenue requirementExtra revenue requirement

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Take the linear income tax...

y

c(y)

1-

B

NoteMarginal tax-rate is constant: If B>0 average tax-rate –B/y is everywhere rising with income

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Higher B needs higher

y

c(y)

B+B

1-

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The constrained optimisation problem can be set up as The constrained optimisation problem can be set up as the Lagrangean:the Lagrangean:

Linear Model: Lagrangean


Lagrange multiplierLagrange multiplier

Maximise Lagrangean by choice of tax instruments Maximise Lagrangean by choice of tax instruments and and BB

This can be done using classical optimisation methods.This can be done using classical optimisation methods.

Government budget constraintGovernment budget constraint

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Linear Model: FOC (1)

Consider the social value of $1 lump-sum income.Consider the social value of $1 lump-sum income. This is defined as:This is defined as:

Maximised utilityMaximised utility

Differentiating the Lagrangean with respect to Differentiating the Lagrangean with respect to BB::Average social value of $1 should be 1Average social value of $1 should be 1

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Again the formula can be used to give guidance on Again the formula can be used to give guidance on policy…policy…

Differentiating the Lagrangean with respect to Differentiating the Lagrangean with respect to and and rearranging we get:rearranging we get:

Linear Model: FOC (2)

Compensated labour-supply elasticityCompensated labour-supply elasticity

Optimal marginal tax rateOptimal marginal tax rate

Covariance of social marginal valuation and incomeCovariance of social marginal valuation and income

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Outcomes from the linear model

If If R R = 0 then = 0 then B B > 0 > 0

Implies progressive taxation. Implies progressive taxation.

FOC cannot be solved to give an explicit formulaFOC cannot be solved to give an explicit formula

The covariance and the elasticities will themselves be functions of The covariance and the elasticities will themselves be functions of ..

However the “natural” restriction imposed by linearity However the “natural” restriction imposed by linearity makes construction of simulation easiermakes construction of simulation easier

Better behaved at special points of the distribution Better behaved at special points of the distribution

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Components of simulation

Structure of ability (wage) distributionStructure of ability (wage) distribution

Empirically determined?Empirically determined?

Individual preferencesIndividual preferences

Determines labour supply responsesDetermines labour supply responses

Social welfare functionSocial welfare function

Use evidence from social surveys etc? Use evidence from social surveys etc?

Government budget constraintGovernment budget constraint

Experiment with alternative assumptionsExperiment with alternative assumptions

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John Broome suggested a great simplification for OIT.John Broome suggested a great simplification for OIT. * = 58.6% !! * = 58.6% !!

http://darp.lse.ac.uk/papersdb/Broome_(REStud_75).pdfhttp://darp.lse.ac.uk/papersdb/Broome_(REStud_75).pdf

The basis for this astounding claim?The basis for this astounding claim? When we spot that the tax rate is in fact 2 – When we spot that the tax rate is in fact 2 – 2 the 2 the

remark is not so outlandishremark is not so outlandish Rather it serves as a useful lesson in applied modelling Rather it serves as a useful lesson in applied modelling

Broome’s revelation

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Take standard Cobb-Douglas preferences:Take standard Cobb-Douglas preferences:

Broome’s model… Make the (empirically relevant?) assumption that no-one has Make the (empirically relevant?) assumption that no-one has

ability less than 0.7071 times the average:ability less than 0.7071 times the average:

““Rawlsian” max-min social welfare:Rawlsian” max-min social welfare:

Balanced budget:Balanced budget:

But in UK 2000: 1 Average wage was £10.53 /

hour2 Min wage was £4.10!

But in UK 2000: 1 Average wage was £10.53 /

hour2 Min wage was £4.10!

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The Stern simulation model

Stern’s (1976) model is less tongue-in-cheek.Stern’s (1976) model is less tongue-in-cheek. But can be taken as a generalisation of Broome.But can be taken as a generalisation of Broome. Also based on a linear OITAlso based on a linear OIT Ingredients are:Ingredients are:

Lognormal abilityLognormal ability Isoelastic utilityIsoelastic utility Isoelastic social welfareIsoelastic social welfare A variety of assumptions about the government budget constraintA variety of assumptions about the government budget constraint

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Representation of ability distribution

Simple two parameter distribution Simple two parameter distribution ((ww; ; mm, , ss22 ) ) First parameter First parameter m m is log of the medianis log of the median The second parameter The second parameter ss22 is itself an inequality index – the is itself an inequality index – the

variance of log income. variance of log income.

Support is [0, Support is [0, )) Not a bad approximation to empirical distributionsNot a bad approximation to empirical distributions

Particularly for manual workersParticularly for manual workers Stern assumed Stern assumed s s = 0.39 (same as Mirrlees)= 0.39 (same as Mirrlees) In this case less than 2% of the population have less than 0.7071 In this case less than 2% of the population have less than 0.7071 × ×

mean (Broome)mean (Broome)

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The lognormal distribution

0 1 2 3 40

f(w)

w

—(w; 0, 0.25 )

…(w; 0, 1.0 )

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Becomes the Broome model in the case Becomes the Broome model in the case =1=1

Take an empirically relevant version of household Take an empirically relevant version of household utility:utility:

Isoelastic utility

hours workedhours worked

consumptionconsumption

elasticity ofSubstitution ( 0)elasticity ofSubstitution ( 0)

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Labour Supply and Income...

Define Define w Could have backward-bending labour supply if <1

Could have backward-bending labour supply if <1

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Resulting Labour Supply and Income… (Broome case)

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Standard SWF

Take additive form of Bergson-Samuelson SWF: Take additive form of Bergson-Samuelson SWF:

W = uu(()) dF()

= uu(()) ff(()) d

Use the iso-elastic form of the (social) Use the iso-elastic form of the (social) uu-function:-function:

1 – – 1 u() = ————, 1 –

Bentham corresponds to the case Bentham corresponds to the case Max-min (“Rawls”) corresponds to the case Max-min (“Rawls”) corresponds to the case

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Stern's Optimal Income Tax Rates

0.2 36.20.4 22.30.6 17.00.8 14.11.0 12.7

Notes: • Calculations are for a purely redistributive tax: i.e. R = 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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“Linear” model: assessment

Solution to problem becomes much more transparentSolution to problem becomes much more transparent

But exact tax formulas are still elusive.But exact tax formulas are still elusive.

Optimal tax rates are very sensitive to precise assumptions Optimal tax rates are very sensitive to precise assumptions about about labour-supply elasticity.labour-supply elasticity.

Distribution of abilityDistribution of ability

Inequality aversionInequality aversion

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Generalisations


An alternative focus on human capital “linear” labour

model

Education model

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Approach 3: Alternative Income Determination

Most OIT models focus on just one area of personal Most OIT models focus on just one area of personal decision makingdecision making

Casual discussion of policy suggest that other economic Casual discussion of policy suggest that other economic incentives may be relevantincentives may be relevant

What about the long-run determination of earning power?What about the long-run determination of earning power?

Need a model of investment.Need a model of investment.

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Components of Atkinson’s human capital model

Given structure of ability distributionGiven structure of ability distribution Individuals maximise lifetime disposable incomeIndividuals maximise lifetime disposable income Essentially investment modelEssentially investment model

Based on Becker (and Mincer) human capital modelBased on Becker (and Mincer) human capital model Schooling only, not experienceSchooling only, not experience

Conventional social welfare functionConventional social welfare function Government budget constraint of zero net revenueGovernment budget constraint of zero net revenue

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Notation in Atkinson’s human capital model

ww - exogenously given ability - exogenously given ability yy - pretax income - pretax income SS - years of schooling - years of schooling LL - length of working life - length of working life rr - interest (discount) rate - interest (discount) rate cc - disposable income - disposable income

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Life Cycle in the Atkinson Model

age

earn

ings

S L+S

y=wS

t

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Atkinson’s Becker-type approachPareto distribution of ability

Pareto distribution of ability

pretax income determined by Becker schooling model

pretax income determined by Becker schooling model

choose schooling to maximise discounted lifetime consumption

choose schooling to maximise discounted lifetime consumption

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Atkinson’s human capital model: optimised schooling Disposable income is Disposable income is c = B +c = B + [1- [1-] ] yy Define a critical ability level in terms of tax Define a critical ability level in terms of tax

parameters parameters

For medium/high ability schooling increases with abilityFor medium/high ability schooling increases with ability For low ability it’s not worth investing in educationFor low ability it’s not worth investing in education

Ability type Ability type ww chooses optimal schooling as chooses optimal schooling as

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Atkinson’s human capital model: optimised utility

Substitute optimal Substitute optimal SS into formula for into formula for discounted lifetime consumption to get:discounted lifetime consumption to get:

Gives relationship between ability and Gives relationship between ability and utilityutility

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Atkinson model: social objectives and constraints

Maximise additively separable SWF as Maximise additively separable SWF as before. before.

Government budget constraint becomesGovernment budget constraint becomes

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Atkinson’s human-capital model:

pretax and disposable incometaxable income will become more unequal the more progressive is the tax

taxable income will become more unequal the more progressive is the tax

disposable income will have the same inequality as ability!disposable income will have the same inequality as ability!

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0

0.02

0.04

0.06

0.08

0.1

0.12

150 200 250 300 350 400

ability

sch

ooli

ng 0 (No tax)

0.05 (low prog)0.10 (medium)0.15 (high)

Ability-Schooling Relationship for values of w0 = rB/[1 – ]

In a high-progression model the able invest a lot in education In a high-progression model the able invest a lot in education This pays for the income supplements for the less able This pays for the income supplements for the less able

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Atkinson’s “Becker” model: optimal marginal tax rates

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Education model: assessment

Key to model is investment response to anticipated taxKey to model is investment response to anticipated tax

In simple model schooling chosen increases when tax In simple model schooling chosen increases when tax progression is increased.progression is increased.

Result can appear to offset effect on Result can appear to offset effect on currentcurrent income income

But target is distribution of lifetime utility.But target is distribution of lifetime utility.

Result of low optimal marginal rates depends crucially on Result of low optimal marginal rates depends crucially on appropriateness of the precise investment modelappropriateness of the precise investment model

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Generalisations


What if we combine insights from the two main branches of optimal taxation?


Education model

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More general tax issues

Should we rely on direct or indirect taxation?Should we rely on direct or indirect taxation?

Is there much to be gained by combining the two Is there much to be gained by combining the two branches of theory?branches of theory?

Can a unified optimising model be developed?Can a unified optimising model be developed?

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Direct versus Indirect Taxation Issues

1.1. Nonlinear commodity taxation? Nonlinear commodity taxation?

2.2. Informational requirements.Informational requirements.

3.3. Participation and incentive compatibility Participation and incentive compatibility constraints.constraints.

4.4. Direct versus indirect tax progressivity.Direct versus indirect tax progressivity.

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1 Nonlinear commodity taxation?

Should consider the issue of proportional versus Should consider the issue of proportional versus nonlinear taxation of commodities. nonlinear taxation of commodities.

““Nonlinear” includes affine functions (like the so-Nonlinear” includes affine functions (like the so-called linear income tax function). called linear income tax function).

The argument is whether each commodity should The argument is whether each commodity should be “repriced”, perhaps not in a proportional be “repriced”, perhaps not in a proportional fashion.fashion.

Similar argument is applied in other areas: tariffs Similar argument is applied in other areas: tariffs for output of state-owned industries, price support for output of state-owned industries, price support schemesschemes

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2 Informational requirements

Recall the main differences between the two types of tax:Recall the main differences between the two types of tax: Not the formal tax base (income versus expenditure) but the Not the formal tax base (income versus expenditure) but the

informational baseinformational base.. Direct tax authority can know details of personal resources.Direct tax authority can know details of personal resources. Indirect tax authority can know structure of production and Indirect tax authority can know structure of production and

transactionstransactions

Informational requirements may preclude extensive Informational requirements may preclude extensive application of nonlinear commodity taxes.application of nonlinear commodity taxes.

To see this consider problem of nonlinear pricing of To see this consider problem of nonlinear pricing of consumer goods.consumer goods. Can work for water, gas, electricityCan work for water, gas, electricity But for food? Clothes?But for food? Clothes?

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3 Participation and Incentive Compatibility Constraints

ICC issues are central to nonlinear income tax designICC issues are central to nonlinear income tax design Same difficulty can arise with nonlinear pricing schemes: Same difficulty can arise with nonlinear pricing schemes:

Some groups may choose the “wrong contract”Some groups may choose the “wrong contract” Arises both in private and public sectorArises both in private and public sector

Difficulties usually disappear if you impose the regularity Difficulties usually disappear if you impose the regularity conditions implied by linearityconditions implied by linearity

Supports the strong case for considering linear commodity Supports the strong case for considering linear commodity taxestaxes

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4 Direct versus Indirect Tax Progressivity

Can measure progressivity in a number of waysCan measure progressivity in a number of ways

A standard method is to compute the implied tax rates that A standard method is to compute the implied tax rates that emerge from actual expenditure decisionsemerge from actual expenditure decisions

Can do this for the definitions of “direct” and “indirect” Can do this for the definitions of “direct” and “indirect” taxes in the UKtaxes in the UK

In practice indirect taxes are more regressive than direct In practice indirect taxes are more regressive than direct taxes.taxes.

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0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Bottom10th

2nd 3nd 4th 5th 6th 7th 8th 9th Top10th

Direct

Indirect

Implied average tax rates in Economic Trends. UK 1994

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Integrating direct and indirect taxation: consumer’s problem

so the budget constraint is:so the budget constraint is:

Total disposable income is given byTotal disposable income is given by

.

Assume there is no lump sum income (Assume there is no lump sum income (II=0)=0)

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Integrating direct and indirect taxation: government’s problem

First order conditions yieldFirst order conditions yield

Government budget Government budget constraint isconstraint is

. Given the generality of the problem we should Given the generality of the problem we should

reduce the number of degrees of freedomreduce the number of degrees of freedom

otherwise you’ll get lump sum taxation again!otherwise you’ll get lump sum taxation again!

Use this to give general guidance on tax structure.Use this to give general guidance on tax structure.

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Policy rules

Commodity taxes should be zero if preferences are Commodity taxes should be zero if preferences are weakly separable in leisure and other goodsweakly separable in leisure and other goods

Tax on good Tax on good ii should be higher if the MRS between should be higher if the MRS between good good ii and labour increases. and labour increases.

Focus tax on goods for which the most able have the Focus tax on goods for which the most able have the strongest preference.strongest preference.

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Conclusions Direct versus indirectDirect versus indirect

Distinction between the two is essentially an issue Distinction between the two is essentially an issue of information.of information.

Big differences in terms of distributional effect.Big differences in terms of distributional effect.

Uniform commodity taxationUniform commodity taxation No compelling case within the context of the modelNo compelling case within the context of the model

There may be a case if you appeal to other factorsThere may be a case if you appeal to other factors

““Flat tax”Flat tax” Argument as for uniform commodity taxationArgument as for uniform commodity taxation

Frank Cowell: UB Public Economics Optimal Tax Design June 2005 Public Economics: University of...

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