MSc Public Economics 2011/12

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24 October 2011. MSc Public Economics 2011/12 Policy Design: Income Tax Frank A. Cowell. Overview. Policy Design: Income Tax. Design principles. Roots in social choice and asymmetric information. Simple model. Generalisations. Interpretations. - PowerPoint PPT Presentation

Transcript of MSc Public Economics 2011/12

  • MSc Public Economics 2011/12 Design: Income Tax

    Frank A. Cowell

    24 October 2011

  • Overview...Design principlesSimple modelGeneralisationsInterpretations

    Policy Design: Income Tax Roots in social choice and asymmetric information

  • Social values: the Arrow problemUses weak assumptions about preferences/valuesWell-defined individual orderings over social statesWell-defined social ordering over social statesUses a general notion of social preferencesThe constitutionA map from set of preference profiles to social preferenceAlso weak assumptions about the constitutionUniversal DomainPareto UnanimityIndependence of Irrelevant AlternativesNon-DictatorshipTheres no constitution that does all fourExcept in cases where there are less than three social states

  • Social choice functionA social state: q QIndividual hs evaluation of the state vh(q)A given population is indexed by h = 1,2,, nhA reduced-form utility function vh().A profile: [v1, v2, , vh, ]An ordered list of utility functionsSet of all profiles: VA social choice function G: VQFor a particular profile q = G(v1, v2, , vh, )Argument is a utility function not a utility levelPicks exactly one chosen element from Q

  • ImplementationIs the SCF consistent with private economic behaviour?Yes if the q picked out by G is also the equilibrium of an appropriate economic gameImplementation problem: find/design an appropriate mechanismMechanism is a partially specified game of imperfect informationrules of game are fixedstrategy sets are specifiedpreferences for the game are not yet specifiedPlug preferences into the mechanism: Does the mechanism have an equilibrium?Does the equilibrium correspond to the desired social state q?If so, the social state is implementableThere is a wide range of possible mechanismsExample: the market as a mechanismGiven the distribution of resources and the technologythe market maps preferences into prices.The prices then determine the allocation

  • ManipulationConsider outcomes from a direct mechanism in two cases:If all, including h, tell the truth about preferences: q = G(v1,, vh, , )If h misrepresents his preferences but others tell the truth: q = G(v1,, vh, , )How does the person really feel about q and q? If vh(q) > vh(q) then there is an incentive to misrepresent informationIf h realises this we say that G is manipulable.

  • Gibbard-Satterthwaite resultResult on SCF G can be stated in several ways (Gibbard 1973, Satterthwaite, 1975 )A standard version is: If the set of social states Q contains at least three elements;and G is defined for all logically possible preference profilesand G is truthfully implementable in dominant strategies... then G must be dictatorialClosely related to the Arrow theoremHas profound implications for public economicsMisinformation may be endemic to the design problemMay only get truth-telling mechanisms in special cases

  • Overview...Design principlesSimple modelGeneralisationsInterpretations

    Policy Design: Income Tax Preferences, incomes, ability and the government

    Analogy with contract theory

  • The design problemThe government needs to raise revenueand it may want to redistribute resourcesTo do this it uses the tax systempersonal income taxand income-based subsidiesBase it on ability to payincome rather than wealthability reflected in productivityTax authority may have limited informationwho have the high ability to pay?what impact on individuals willingness to produce output?Whats the right way to construct the tax schedule?

  • Model elements A two-commodity modelleisure (i.e. the opposite of effort) consumption a basket of all other goodssimilar to optimal contracts (Bolton and Dewatripont 2005)Income comes only from workindividuals are paid according to their marginal productworkers differ according to their abilityIndividuals derive utility from:their leisuretheir disposable income (consumption)Government / tax agencyhas to raise a fixed amount of revenue Kseeks to maximise social welfarewhere social welfare is a function of individual utilities

  • Modelling preferencesIndividuals preferencesu = y(z) + yu : utility level z : efforty : income receivedy() : decreasing, strictly concave, function Special shape of utility functionquasi-linear formzero-income effecty(z) gives the disutility of effort in monetary unitsIndividual does not have to workreservation utility level u requires y(z) + y u

  • Ability and income Individuals work (give up leisure) to get consumptionIndividuals differ in talent (ability) t higher ability people produce more and may thus earn moreindividual of type t works an amount zproduces output q = tzbut individual does not necessarily get to keep this output?Disposable income determined by tax authorityintervention via taxes and transfersfixes a relationship between individuals output and income(net) income tax on type t is implicitly given by q y Preferences can be expressed in terms of q, y for type t utility is given by y(z) + yequivalently: y(q / t) + yA closer look at utility

  • The utility functionincreasingpreferencey1 zu = y(z) + yyz(z) < 0u uuu = y(q/t) + yyz(q/t) < 0increasingpreferencey qu Preferences over leisure and income Indifference curves Reservation utilityTransform into (leisure, output) space

  • The single-crossing conditionincreasingpreferenceyqtype btype aPreferences over leisure and output High talent qa = taza Low talent qb = tbzb Those with different talent (ability) will have different sloped indifference curves in this diagram

  • A full-information solution?Consider argument based on the analysis of contractsFull information: owner can fully exploit any managerPays the minimum amount necessaryChooses their effortSame basic story here Can impose lump-sum taxChooses agents effort no distortion But the full-information solution may be unattractiveInformational requirements are demandingPerhaps violation of individuals privacy?So look at second-best case

  • Two typesStart with the case closest to the optimal contract modelExactly two skill typesta > tb proportion of a-types is pvalues of ta , tb and p are common knowledgeFrom contract design we can write down the outcomeessentially all we need to do is rework notationBut let us examine the model in detail:

  • Second-best: two typesThe governments budget constraintp[qa - ya] + [1-p][qb - yb ] Kwhere qh - yh is the amount raised in tax from agent hParticipation constraint for the b type: yb + y(zb) ubhave to offer at least as much as available elsewhereIncentive-compatibility constraint for the a type:ya + y(qa/ta) yb + y(qb/ta)must be no worse off than if it behaved like a b-typeimplies (qb, yb) < (qa, ya) The government seeks to maximise standard SWFp z(y(za) + ya) + [1-p] z(y(zb) + yb) where z is increasing and concave

  • Two types: model We can use a standard Lagrangean approachgovernment chooses (q, y) pairs for each typesubject to three constraintsConstraints are:government budget constraintparticipation constraint (for b-types)incentive-compatibility constraint (for a-types)Choose qa, qb, ya, yb to max p z(y(qa/ta) + ya) + [1-p] z(y(qb/tb) + yb) + k [p[qa - ya] + [1-p][qb - yb ] - K]+ l [yb + y(qb/tb) - ub]+ m [ya + y(qa/ta) - yb - y(qb/ta)]where k, l, m are Lagrange multipliers for the constraints

  • Two types: methodDifferentiate with respect to qa, qb, ya, yb to get FOCs:pzu(ua)yz(za)/ta + kp + myz(za)/ta 0[1-p]zu(ub)yz(zb)/tb + k [1-p] + lyz(zb)/tb - myz(qb/ta)/ta 0pzu(ua) - kp + m 0[1-p]zu(ub) - k[1-p] + l - m 0For an interior solution, where qa, qb, ya, yb are positivepzu(ua)yz(za)/ta + kp + myz(za)/ta = 0[1-p]zu(ub)yz(zb)/tb + k [1-p] + lyz(zb)/tb - myz(qb/ta)/ta = 0pzu(ua) - kp + m = 0[1-p]zu(ub) - k[1-p] + l - m = 0Manipulating these gives the main resultsFor example, from first and third condition:[kp - m ] yz(za)/ta + kp + myz(za)/ta = 0kp yz(za)/ta + kp = 0

  • Two types: solutionFrom first-order conditions we get:- yz(qa/ta) = ta- yz(qb/tb) = tb + kp/[1-p], where k := yz(qb/tb) - [tb/ta] yz(qb/ta) < 0Also, all the Lagrange multipliers are positiveso the associated constraints are bindingfollows from standard adverse selection modelResults are as for optimum-contracts model:MRSa = MRTaMRSb < MRTb Interpretationno distortion at the top (for type ta)no surplus at the bottom (for type tb)determine the menu of (q,y)-choices offered by tax agency

  • Two ability types: tax designyqa types reservation utilityb types reservation utilityb types (q,y)incentive-compatibility constrainta types (q,y)menu of (q,y) offered by tax authority Analysis determines (q,y) combinations at two pointsIf a tax schedule T() is to be designed where y = q T(q) then it must be consistent with these two points

  • Overview...Design principlesSimple modelGeneralisationsInterpretations

    Policy Design: Income Tax Moving beyond the two-ability model

  • A small generalisationWith three types problem becomes a bit more interestingSimilar structure to previous caseta > tb > tcproportions of each type in the population are pa, pb, pcWe now have one more constraint to worry aboutParticipation constraint for c type: yc + y(qc/tc) ucIC constraint for b type: yb + y(qb/tb) yc + y(qc/tb) IC constraint for a type: ya + y(qa/ta) yb + y(qb/ta)But this is enough to complete the model specificationthe two IC constraints also imply ya + y(qa/ta) yc + y(qc/tb)so no-one has incentive to misrepresent as lower ability

  • Three typesMethodology is same as two-ability modelset up LagrangeanLagrange multipliers for budget constraint, participation constraint and two IC constraintsmaximise with r