Journal of Economic Perspectives—Volume 25, Number 4—Fall 2011—Pages 165–190

The Case for a Progressive Tax: From Basic Research to Policy Recommendations†

Peter Diamond and Emmanuel Saez

hhee ffairair ddistributionistribution ooff tthehe ttaxax bburdenurden hhasas llongong bbeeneen a ccentralentral iissuessue iinn ppolicy-olicy- mmaking.aking. A largelarge academicacademic literatureliterature hashas developeddeveloped modelsmodels ofof optimaloptimal taxtax T ttheoryheory ttoo castcast lightlight onon thethe problemproblem ofof optimaloptimal taxtax progressivity.progressivity. InIn thisthis ppaper,aper, wewe exploreexplore thethe pathpath fromfrom basicbasic researchresearch resultsresults inin optimaloptimal taxtax theorytheory toto fformulatingormulating ppolicyolicy rrecommendations.ecommendations. MModelsodels inin optimaloptimal taxtax theorytheory typicallytypically positposit thatthat thethe taxtax systemsystem shouldshould maximizemaximize a ssocialocial welfarewelfare functionfunction subjectsubject toto a governmentgovernment budgetbudget constraint,constraint, takingtaking intointo accountaccount tthathat individualsindividuals respondrespond toto taxestaxes andand transfers.transfers. SocialSocial welfarewelfare isis largerlarger whenwhen resourcesresources aarere moremore equallyequally distributed,distributed, butbut redistributiveredistributive taxestaxes andand transferstransfers cancan negativelynegatively aaffectffect incentivesincentives toto work,work, save,save, andand earnearn incomeincome inin thethe fi rrstst pplace.lace. TThishis ccreatesreates thethe clas-clas- ssicalical ttrade-offrade-off bbetweenetween eequityquity aandnd eeffiffi ciencyciency whichwhich isis atat thethe corecore ofof thethe optimaloptimal incomeincome ttaxax problem.problem. InIn general,general, optimaloptimal taxtax analysesanalyses maximizemaximize socialsocial welfarewelfare asas a functionfunction ofof iindividualndividual utilities—theutilities—the sumsum ofof utilitiesutilities inin thethe utilitarianutilitarian case.case. TheThe marginalmarginal weightweight forfor a ggiveniven ppersonerson iinn tthehe ssocialocial wwelfareelfare ffunctionunction mmeasureseasures tthehe vvaluealue ooff aann aadditionaldditional ddollarollar ooff cconsumptiononsumption expressedexpressed inin termsterms ofof publicpublic funds.funds. SuchSuch welfarewelfare weightsweights dependdepend onon tthehe levellevel ofof redistributionredistribution andand areare decreasingdecreasing withwith incomeincome wheneverwhenever societysociety valuesvalues mmoreore eequalityquality ooff iincome.ncome. TTherefore,herefore, ooptimalptimal iincomencome ttaxax ttheoryheory iiss fi rrstst a nnormativeormative ttheoryheory thatthat showsshows howhow a socialsocial welfarewelfare objectiveobjective combinescombines withwith constraintsconstraints arisingarising fromfrom llimitsimits oonn rresourcesesources aandnd bbehavioralehavioral rresponsesesponses ttoo ttaxationaxation iinn oorderrder ttoo dderiveerive sspecifipecifi c

■ Peter Diamond is Professor Emeritus of Economics, Massachusetts Institute of Tech- nology, Cambridge Massachusetts. Emmanuel Saez is Professor of Economics, University of California, Berkeley, California. Their e-mail addresses are 〈 [email protected]〉 and 〈[email protected]〉, respectively. † There is an Appendix at the end of this article. To access an additional online Appendix, visit http:// www.aeaweb.org/articles.php?doi=10.1257/jep.25.4.165. doi=10.1257/jep.25.4.165 166 Journal of Economic Perspectives

ttaxax ppolicyolicy rrecommendations.ecommendations. InIn addition,addition, optimaloptimal incomeincome taxtax theorytheory cancan bbee uusedsed ttoo eevaluatevaluate currentcurrent policiespolicies andand suggestsuggest avenuesavenues forfor reform.reform. UnderstandingUnderstanding whatwhat wouldwould bbee goodgood policy,policy, ifif implemented,implemented, isis a keykey stepstep inin makingmaking policypolicy recommendations.recommendations. WWhenhen ddoneone wwell,ell, mmovingoving ffromrom mmathematicalathematical rresults,esults, ttheorems,heorems, oorr ccalculatedalculated eexamplesxamples toto policypolicy recommendationsrecommendations isis a subtlesubtle process.process. TheThe naturenature ofof a modelmodel isis ttoo bebe a limitedlimited picturepicture ofof reality.reality. ThisThis hashas twotwo implications.implications. First,First, a modelmodel maymay bebe ggoodood forfor oneone questionquestion andand badbad forfor another,another, dependingdepending onon thethe robustnessrobustness ofof thethe aanswersnswers ttoo tthehe iinaccuraciesnaccuracies ooff tthehe mmodel,odel, wwhichhich wwillill nnaturallyaturally vvaryary wwithith tthehe qquestion.uestion. SSecond,econd, tractabilitytractability concernsconcerns implyimply thatthat ssimultaneousimultaneous cconsiderationonsideration ofof mmultipleultiple mmodelsodels isis appropriateappropriate sincesince differentdifferent aspectsaspects ofof realityreality cancan bebe usefullyusefully highlightedhighlighted iinn ddifferentifferent models;models; hencehence ourour reliancereliance onon tryingtrying toto drawdraw inferencesinferences simultaneouslysimultaneously ffromrom mmultipleultiple mmodels.odels. IInn oourur vview,iew, a ttheoreticalheoretical rresultesult ccanan bbee ffruitfullyruitfully uusedsed aass ppartart ooff fformingorming a ppolicyolicy rrecommendationecommendation oonlynly iiff tthreehree cconditionsonditions aarere mmet.et. FFirst,irst, tthehe rresultesult sshouldhould bbee bbasedased oonn aann eeconomicconomic mmechanismechanism tthathat iiss eempiricallympirically rrelevantelevant aandnd fi rrstst oorderrder ttoo tthehe pproblemroblem aatt hand.hand. Second,Second, thethe resultresult shouldshould bebe reasonablyreasonably robustrobust toto changeschanges inin thethe modelingmodeling aassumptions.ssumptions. IInn pparticular,articular, ppeopleeople hhaveave vveryery hheterogeneouseterogeneous ttastes,astes, aandnd ttherehere aarere mmanyany ddeparturesepartures ffromrom tthehe rrationalational mmodel,odel, eespeciallyspecially iinn tthehe rrealmealm ooff iintertemporalntertemporal cchoice.hoice. TTherefore,herefore, wwee sshouldhould vviewiew wwithith ssuspicionuspicion rresultsesults tthathat ddependepend ccriticallyritically oonn vveryery sstrongtrong hhomogeneityomogeneity oorr rrationalityationality aassumptions.ssumptions. DDerivingeriving ooptimalptimal taxtax formulasformulas asas a functionfunction ooff a ffewew eempiricallympirically eestimablestimable ““suffisuffi cientcient statistics”statistics” isis a naturalnatural wayway toto aapproachpproach tthosehose fi rrstst ttwowo cconditions.onditions. TThird,hird, tthehe ttaxax ppolicyolicy pprescriptionrescription nneedseeds ttoo bbee iimplementable—mplementable— tthathat iis,s, tthehe ttaxax ppolicyolicy nneedseeds ttoo bbee ssociallyocially aacceptablecceptable aandnd nnotot ttoooo ccomplexomplex rrelativeelative ttoo tthehe mmodelingodeling ooff ttaxax aadministrationdministration aandnd iindividualndividual rresponsesesponses ttoo ttaxax llaw.aw. BByy ssociallyocially aacceptable,cceptable, wwee ddoo nnotot mmeanean ttoo llimitimit tthehe cchoicehoice ttoo ccurrentlyurrently ppoliticallyolitically pplausiblelausible ppolicyolicy ooptions.ptions. RRather,ather, wwee mmeanean ttherehere sshouldhould nnotot bbee vveryery wwidelyidely hheldeld nnormativeormative vviewsiews tthathat mmakeake ssuchuch ppoliciesolicies sseemeem iimplausiblemplausible aandnd iinappropriatenappropriate aatt pprettyretty mmuchuch aallll ttimes.imes. FForor eexample,xample, a ppolicyolicy pprescriptionrescription ssuchuch aass ttaxingaxing hheighteight ((MankiwMankiw aandnd WWeinzierl,einzierl, 22010)010) iiss oobviouslybviously nnotot ssociallyocially aacceptablecceptable bbecauseecause iitt vviolatesiolates ccertainertain hhorizontalorizontal eequityquity cconcernsoncerns tthathat ddoo nnotot aappearppear iinn bbasicasic mmodels.odels. TThehe ccomplexityomplexity cconstraintonstraint ccanan aalsolso bbee aann iissuessue wwhenhen optimaloptimal taxestaxes dependdepend inin a complexcomplex wayway onon thethe fullfull historyhistory ofof earningsearnings andand cconsumption,onsumption, aass iinn ssomeome rrecentecent ppath-breakingath-breaking ppapersapers oonn ooptimalptimal ddynamicynamic ttaxation.axation. WWee obtainobtain threethree policypolicy recommendationsrecommendations fromfrom basicbasic researchresearch thatthat wewe believebelieve ccanan ssatisfyatisfy thesethese threethree criteriacriteria reasonablyreasonably well.well. First,First, veryvery highhigh earnersearners shouldshould bebe ssubjectubject toto highhigh andand risingrising marginalmarginal taxtax ratesrates onon earnings.earnings. InIn particular,particular, wewe discussdiscuss wwhyhy tthehe ffamousamous zzeroero mmarginalarginal ttaxax rrateate aatt tthehe ttopop ooff tthehe eearningsarnings ddistributionistribution iiss nnotot ppolicyolicy relevant.relevant. Second,Second, thethe earningsearnings ofof low-incomelow-income familiesfamilies shouldshould bebe subsidized,subsidized, aandnd tthosehose ssubsidiesubsidies sshouldhould tthenhen bbee pphasedhased ooutut wwithith hhighigh iimplicitmplicit mmarginalarginal ttaxax rrates.ates. TThishis rresultesult ffollowsollows becausebecause laborlabor supplysupply responsesresponses ofof lowlow earnersearners areare concentratedconcentrated aalonglong thethe marginmargin ofof whetherwhether toto participateparticipate inin laborlabor marketsmarkets atat allall (the(the extensiveextensive aass oopposedpposed toto thethe intensiveintensive margin).margin). TheseThese twotwo resultsresults combinedcombined implyimply thatthat thethe ooptimalptimal pprofirofi lele ofof transferstransfers andand taxestaxes isis highlyhighly nonlinearnonlinear andand cannotcannot bebe wellwell approx-approx- iimatedmated byby a fl atat taxtax alongalong withwith lumplump sumsum “demogrants.”“demogrants.” Third,Third, wewe argueargue thatthat capitalcapital iincomencome shouldshould bebe taxed.taxed. WeWe willwill reviewreview certaincertain theoreticaltheoretical results—inresults—in particular,particular, Peter Diamond and Emmanuel Saez 167

tthosehose ofof AAtkinsontkinson aandnd SStiglitztiglitz ((1976),1976), CChamleyhamley ((1986),1986), aandnd JJuddudd ((1985)—implying1985)—implying nnoo ccapitalapital incomeincome taxestaxes andand argueargue thatthat thesethese fi ndingsndings areare notnot robustrobust enoughenough toto bbee policypolicy relevant.relevant. InIn thethe end,end, persuasivepersuasive argumentsarguments forfor taxingtaxing capitalcapital incomeincome areare tthathat therethere areare diffidiffi cultiesculties inin practicepractice inin distinguishingdistinguishing betweenbetween capitalcapital andand laborlabor iincomes,ncomes, thatthat borrowingborrowing constraintsconstraints makemake fullfull reliancereliance onon laborlabor taxestaxes lessless effieffi cient,cient, aandnd tthathat ssavingsavings rratesates aarere hheterogeneous.eterogeneous. TThehe rremainderemainder ooff tthehe ppaperaper iiss oorganizedrganized aass ffollows:ollows: FFirst,irst, wwee cconsideronsider tthehe ttaxa-axa- ttionion ooff vveryery hhighigh eearners,arners, ssecond,econd, thethe taxationtaxation ofof lowlow earners,earners, andand third,third, thethe taxationtaxation ooff capitalcapital income.income. WeWe concludeconclude withwith a discussiondiscussion ofof methodology,methodology, contrastingcontrasting ooptimalptimal ttaxax aandnd mmechanismechanism ddesignesign ((“new“new ddynamicynamic ppublicublic fi nance”)nance”) approaches.approaches. InIn aann aappendix,ppendix, wewe contrastcontrast ourour lessonslessons fromfrom optimaloptimal taxtax theorytheory wwithith tthosehose ooff MMankiw,ankiw, WWeinzierl,einzierl, aandnd YaganYagan (2009),(2009), recentlyrecently publishedpublished inin thisthis journal.journal.

Recommendation 1: Very high earnings should be subject to rising marginal rates and higher rates than current U.S. policy for top earners.

TThehe ssharehare ooff ttotalotal iincomencome ggoingoing ttoo tthehe ttopop 1 ppercentercent ooff iincomencome eearnersarners ((thosethose wwithith aannualnnual iincomencome aabovebove aaboutbout $$400,000400,000 iinn 22007)007) hhasas iincreasedncreased ddramaticallyramatically ffromrom 9 percentpercent inin 19701970 toto 23.523.5 percentpercent inin 2007,2007, thethe highesthighest levellevel onon recordrecord sincesince 19281928 aandnd muchmuch higherhigher thanthan inin EuropeanEuropean countriescountries oror JapanJapan todaytoday (Piketty(Piketty andand Saez,Saez, 22003;003; Atkinson,Atkinson, Piketty,Piketty, andand Saez,Saez, 2011).2011). AlthoughAlthough thethe averageaverage federalfederal individualindividual iincomencome ttaxax rrateate ooff ttopop ppercentileercentile ttaxax fi lerslers waswas 22.422.4 percent,percent, thethe toptop percentilepercentile paidpaid 440.40.4 percentpercent ofof totaltotal federalfederal individualindividual incomeincome taxestaxes inin 20072007 (IRS,(IRS, 2009a).2009a). There-There- ffore,ore, tthehe ttaxationaxation ooff vveryery hhighigh eearnersarners iiss a ccentralentral aaspectspect ooff tthehe ttaxax ppolicyolicy ddebateebate nnotot oonlynly forfor eequityquity rreasonseasons bbutut aalsolso fforor rrevenueevenue rraising.aising. FForor eexample,xample, ssettingetting aasideside bbehav-ehav- iioraloral responsesresponses forfor a mmoment,oment, iincreasingncreasing tthehe aaverageverage federalfederal incomeincome ttaxax rrateate oonn tthehe ttopop ppercentileercentile ffromrom 222.42.4 ppercentercent ((asas ooff 22007)007) ttoo 229.49.4 ppercentercent wwouldould rraiseaise rrevenueevenue bbyy 1 ppercentageercentage ppointoint ooff GGDP.DP.1 IIndeed,ndeed, eveneven increasingincreasing thethe averageaverage federalfederal incomeincome taxtax rrateate ooff tthehe ttopop ppercentileercentile ttoo 443.53.5 ppercent,ercent, wwhichhich wwouldould bbee ssuffiuffi cientcient toto raiseraise revenuerevenue bbyy 3 percentagepercentage pointspoints ofof GDP,GDP, wouldwould stillstill leaveleave thethe after-taxafter-tax incomeincome shareshare ofof thethe toptop ppercentileercentile moremore thanthan twicetwice asas hhighigh aass iinn 11970.970.2 OOff course,course, increasingincreasing upperupper incomeincome ttaxax rratesates ccanan ddiscourageiscourage eeconomicconomic aactivityctivity tthroughhrough bbehavioralehavioral rresponses,esponses, aandnd hhenceence

1 In 2007, the top percentile of income earners paid $450 billion in federal individual taxes (IRS, 2009a), or 3.2 percent of the $14,078 billion in GDP for 2007. Hence, increasing the average tax rate on the top percentile from 22.4 to 29.4 percent would raise $141 billion or 1 percent of GDP. 2 The average federal individual tax rate paid by the top percentile was 25.7 percent in 1970 (Piketty and Saez, 2007) and 22.4 percent in 2007 (IRS, 2009a). The overall average federal individual tax rate was 12.5 percent in 1970 and 12.7 percent in 2007. The pre-tax income share for the top percentile of tax fi lers was 9 percent in 1970 and 23.5 percent in 2007. Hence, the top 1 percent after-tax income share in 1970 was 7.6 percent = 9% × (1 – .257)/(1 – .125), and in 2007 it was 20.9 percent = 23.5% × (1 – .224)/ (1 – .127) and, with a tax rate of 43.5 percent on the top percentile (which would increase the average tax rate to 17.7 percent), would have been 16.1 percent = 23.5% × (1 – .435)/(1 – .177). 168 Journal of Economic Perspectives

ppotentiallyotentially reducereduce taxtax ccollections,ollections, creatingcreating thethe sstandardtandard eequity-effiquity-effi ciencyciency trade-offtrade-off ddiscussediscussed iinn tthehe iintroduction.ntroduction.

The Optimal Top Marginal Tax Rate FForor tthehe UU.S..S. eeconomy,conomy, tthehe ccurrenturrent ttopop iincomencome mmarginalarginal ttaxax rrateate oonn eearningsarnings iiss aaboutbout 442.52.5 ppercent,ercent,3 ccombiningombining tthehe ttopop ffederalederal mmarginalarginal iincomencome ttaxax bbracketracket ooff 3355 ppercentercent wwithith tthehe MMedicareedicare ttaxax aandnd aaverageverage sstatetate ttaxesaxes oonn iincomencome aandnd ssales.ales.4 AsAs sshownhown iinn SSaezaez ((2001),2001), tthehe ooptimalptimal ttopop mmarginalarginal ttaxax rrateate iiss sstraightforwardtraightforward ttoo dderive.erive. DDenoteenote tthehe ttaxax rrateate iinn tthehe ttopop bbracketracket bbyy τ. FFigureigure 1 sshowshows hhowow tthehe ooptimalptimal ttaxax rrateate iiss dderived.erived. TThehe hhorizontalorizontal aaxisxis ooff tthehe fi guregure sshowshows ppre-taxre-tax iincome,ncome, wwhilehile tthehe vverticalertical aaxisxis sshowshows ddisposableisposable iincome.ncome. TThehe ooriginalriginal ttopop ttaxax bbracketracket iiss sshownhown bbyy tthehe ssolidolid lline.ine. AAss ddepicted,epicted, cconsideronsider a ttaxax rreformeform wwhichhich iincreasesncreases τ byby ΔΔττ aabovebove tthehe iincomencome llevelevel z * . TToo eevaluatevaluate tthishis cchangehange wwee nneedeed ttoo cconsideronsider tthehe eeffectsffects oonn rrevenueevenue aandnd ssocialocial wwelfare.elfare. IIgnoringgnoring bbehavioralehavioral rresponsesesponses aatt fi rrst,st, tthishis rreformeform mmechanicallyechanically rraisesaises aadditionaldditional rrevenueevenue bbyy aann aamountmount eequalqual ttoo tthehe cchangehange iinn tthehe ttaxax rrateate (ΔΔττ) mmultipliedultiplied bbyy thethe numbernumber ofof peoplepeople toto whomwhom thethe higherhigher raterate appliesapplies (N * ) mmultipliedultiplied bbyy tthehe aamountmount byby whichwhich thethe averageaverage incomeincome ofof thisthis groupgroup ( z m) iiss aabovebove tthehe ccut-offut-off iincomencome * * * llevelevel (z ) soso tthathat tthehe aadditionaldditional rrevenueevenue iiss ΔΔττ N [ z m – z ]].. AAss wwee sshallhall ssee,ee, tthehe ttopop ttailail ooff tthehe iincomencome ddistributionistribution iiss ccloselylosely aapproximatedpproximated bbyy a PParetoareto ddistributionistribution ccharacter-haracter- iizedzed bbyy a ppowerower llawaw ddensityensity ooff tthehe fformorm C/ z 1+a wwherehere a > 1 iiss tthehe PParetoareto pparameter.arameter. * * SSuchuch ddistributionsistributions hhaveave tthehe kkeyey ppropertyroperty tthathat tthehe rratioatio z m / z iiss tthehe ssameame fforor aallll z iinn thethe toptop tailtail andand equalequal toto a//((a – 11).). FForor tthehe UU.S..S. eeconomy,conomy, tthehe ccutoffutoff fforor tthehe ttopop ppercentileercentile ooff ttaxax fi llersers iiss aapproximatelypproximately $$400,000,400,000, aandnd tthehe aaverageverage iincomencome fforor tthishis * ggrouproup iiss aapproximatelypproximately $$1.21.2 mmillion,illion, ssoo tthathat z m / z = 3 aandnd hhenceence a = 11.5..5. RRaisingaising thethe taxtax raterate onon thethe toptop percentilepercentile obviouslyobviously reducesreduces thethe utilityutility ofof high-high- iincomencome ttaxax fi lers.lers. IfIf wewe denotedenote byby g tthehe ssocialocial mmarginalarginal vvaluealue ooff $$11 ooff cconsumptiononsumption fforor toptop incomeincome earnersearners (measured(measured relativerelative toto governmentgovernment revenue),revenue), thethe directdirect wwelfareelfare costcost isis g multipliedmultiplied byby thethe changechange inin taxtax revenuerevenue collected.collected.5 BecauseBecause thethe ggovernmentovernment vvaluesalues rredistribution,edistribution, thethe ssocialocial mmarginalarginal vvaluealue ooff cconsumptiononsumption fforor ttop-op- bbracketracket taxtax fi lerslers isis smallsmall relativerelative toto thatthat ofof thethe averageaverage personperson inin thethe economy,economy, andand ssoo g iiss ssmallmall aandnd asas a fi rstrst approximationapproximation cancan bebe ignored.ignored. A utilitarianutilitarian socialsocial welfarewelfare ccriterionriterion wwithith mmarginalarginal uutilitytility ooff cconsumptiononsumption ddecliningeclining ttoo zzero,ero, tthehe mmostost ccommonlyommonly

3 This top marginal tax rate is much higher than the current average tax rate among top 1 percent earners mentioned above because of deductions and especially lower tax rates that apply to realized capital gains. 4 The top tax rate τ is 42.5 percent for ordinary labor income when combining the top federal individual tax rate of 35 percent, uncapped Medicare taxes of 2.9 percent, and an average combined state top income tax rate of 5.86 percent and average sales tax rate of 2.32 percent. The average across states is computed using state weights equal to the fraction of fi lers with adjusted gross income above $200,000 that reside in the state as of 2007 (IRS, 2009a). The 2.32 percent average sales tax rate is estimated as 40 percent of the average nominal sales tax rate across states (as the average sales tax base is about 40 percent of total personal consumption.) As the 1.45 percent employer Medicare tax is deductible for both federal and state income taxes, and state income taxes are deductible for federal income taxes, we have ((1 – .35) × (1 – .0586) – .0145)/(1.0145 × 1.0232) = .575, and hence τ = 42.5 percent. 5 Formally, g is the weighted average of social marginal weights on top earners, with weights proportional to income in the top bracket. The Case for a Progressive Tax: From Basic Research to Policy Recommendations 169

Figure 1 Optimal Top Tax Rate Derivation

Disposable Top bracket: slope 1 – τ above z* income c = z – T(z) Reform: slope 1 – τ – Δτ above z*

Mechanical tax increase: Δτ[z – z*]

z* – T(z*) Behavioral response tax loss: τΔz = –Δτezτ/(1 – τ)

0 z* z Pre-tax income z

Source: The authors. Notes: The fi gure depicts the derivation of the optimal top tax rate τ * = 1/(1 + ae) by considering a small reform around the optimum which increases the top marginal tax rate τ by Δτ above z * . A taxpayer with income z mechanically pays Δτ[z – z * ] extra taxes but, by defi nition of the elasticity e of earnings with respect to the net-of-tax rate 1 – τ, also reduces his income by Δz = e z Δτ/(1 – τ) leading to a loss in tax revenue equal to Δτ e zτ/(1 – τ). Summing across all top bracket taxpayers and denoting by z m the average income * * * above z and a = z m /(z m – z )), we obtain the revenue maximizing tax rate τ = 1/(1 + ae). This is the optimum tax rate when the government sets zero marginal welfare weights on top income earners. uusedsed sspecifipecifi cationcation inin optimaloptimal taxtax models,models, hashas thisthis implication.implication. ForFor example,example, ifif thethe ssocialocial vvaluealue ooff uutilitytility iiss llogarithmicogarithmic iinn cconsumption,onsumption, tthenhen ssocialocial mmarginalarginal wwelfareelfare wweightseights aarere iinverselynversely pproportionalroportional toto cconsumption.onsumption. IInn tthathat ccase,ase, tthehe ssocialocial mmarginalarginal uutilitytility atat thethe $1,364,000$1,364,000 averageaverage incomeincome ofof thethe toptop 1 percentpercent inin 20072007 (Piketty(Piketty andand SSaez,aez, 22003)003) iiss oonlynly 33.9.9 ppercentercent ooff tthehe ssocialocial mmarginalarginal uutilitytility ooff tthehe mmedianedian ffamily,amily, wwithith iincomencome $$52,70052,700 ((U.S.U.S. CCensusensus BBureau,ureau, 22009).009). BBehavioralehavioral responsesresponses cancan bbee ccapturedaptured bbyy tthehe eelasticitylasticity e ooff rreportedeported iincomencome wwithith rrespectespect toto tthehe nnet-of-taxet-of-tax rrateate 1 – τ. ByBy defidefi nition,nition, e measuresmeasures thethe percentpercent increaseincrease inin 6 aaverageverage reportedreported incomeincome z m whenwhen thethe net-of-taxnet-of-tax raterate increasesincreases byby 1 percent.percent. AtAt tthehe ooptimum,ptimum, tthehe mmarginalarginal ggainain ffromrom iincreasingncreasing ttaxax rrevenueevenue withwith nono behavioralbehavioral rresponseesponse andand thethe marginalmarginal lossloss fromfrom thethe behavioralbehavioral reactionreaction mustmust bebe equalequal toto eacheach

6 Formally, this elasticity is an income-weighted average of the individual elasticities across the N * top bracket tax fi lers. It is also a mix of income and substitution effects as the reform creates both income and substitution effects in the top bracket. Saez (2001) provides an exact decomposition. 170 Journal of Economic Perspectives

oother.ther. IIgnoringgnoring thethe socialsocial valuevalue ofof marginalmarginal consumptionconsumption ofof toptop earners,earners, thethe optimaloptimal ttopop ttaxax rrateate τ * iiss ggiveniven bbyy tthehe fformulaormula

τ * = 1/(1 + ae).

The optimal top tax rate τ * is the tax rate that maximizes tax revenue from top bracket taxpayers.7 Since the goal of the marginal rates on very high incomes is to get revenue in order to hold down taxes on lower earners, this equation does not depend on the total revenue needs of the government. Any top tax rate above τ * would be (second-best) Pareto ineffi cient as reducing tax rates at the top would both increase tax revenue and the welfare of top earners. AAnn increaseincrease inin thethe marginalmarginal taxtax raterate onlyonly atat a singlesingle incomeincome levellevel inin thethe upperupper ttailail iincreasesncreases thethe deadweightdeadweight burdenburden (decreases(decreases revenuerevenue becausebecause ofof reducedreduced earn-earn- iings)ngs) atat thatthat incomeincome levellevel butbut raisesraises revenuerevenue fromfrom allall thosethose withwith higherhigher earningsearnings wwithoutithout alteringaltering theirtheir marginalmarginal taxtax rates.rates. TheThe optimaloptimal taxtax raterate balancesbalances thesethese twotwo eeffects—theffects—the increasedincreased deadweightdeadweight burdenburden atat thethe incomeincome levellevel andand thethe increasedincreased rrevenueevenue fromfrom allall hhigherigher llevels.evels. τ * iiss ddecreasingecreasing withwith thethe elasticityelasticity e ((whichwhich affectsaffects thethe ddeadweighteadweight burden)burden) andand thethe ParetoPareto parameterparameter a, whichwhich measuresmeasures thethe thinnessthinness ofof tthehe toptop ofof thethe incomeincome distributiondistribution andand soso thethe ratioratio ofof thosethose aboveabove a taxtax levellevel toto thethe iincomencome ooff tthosehose aatt tthehe ttaxax llevel.evel. * * TThehe solidsolid lineline inin FigureFigure 2 depictsdepicts thethe empiricalempirical ratioratio a = z m //(( z m – z ) wwithith z rranginganging fromfrom $0$0 toto $1,000,000$1,000,000 inin annualannual incomeincome usingusing U.S.U.S. taxtax returnreturn micro-datamicro-data fforor 22005.005. WWee uusese ““adjustedadjusted grossgross income”income” fromfrom ttaxax rreturnseturns asas ourour incomeincome defidefi nition.nition. TThehe ccentralentral fi ndingnding isis thatthat a iiss eextremelyxtremely stablestable forfor z * aabovebove $300,000$300,000 (and(and aroundaround 11.5)..5). TThehe eexcellentxcellent PParetoareto fi t ofof thethe toptop tailtail ofof thethe distributiondistribution hashas beenbeen wellwell knownknown fforor overover a centurycentury sincesince thethe pioneeringpioneering workwork ofof ParetoPareto (1896)(1896) andand verifiverifi eded inin manymany ccountriesountries aandnd mmanyany pperiods,eriods, aass ssummarizedummarized iinn AAtkinson,tkinson, PPiketty,iketty, aandnd SSaezaez ((2011).2011). IIff wwee aassumessume tthathat tthehe eelasticitylasticity e iiss rroughlyoughly cconstantonstant aacrosscross eearnersarners aatt tthehe ttopop ooff tthehe distribution,distribution, thethe formulaformula τ = 1/(11/(1 + ae) showsshows thatthat thethe optimaloptimal toptop taxtax raterate isis iindependentndependent ofof z * wwithinithin thethe toptop tailtail (and(and isis alsoalso thethe asymptoticasymptotic optimaloptimal marginalmarginal ttaxax raterate comingcoming outout ofof thethe standardstandard nonlinearnonlinear optimaloptimal taxtax modelmodel ofof Mirrlees,Mirrlees, 11971).971). ThatThat is,is, thethe optimaloptimal marginalmarginal taxtax raterate isis approximatelyapproximately thethe samesame overover thethe rrangeange ofof veryvery highhigh incomesincomes wherewhere thethe distributiondistribution isis ParetoPareto andand thethe marginalmarginal socialsocial wweighteight onon consumptionconsumption isis small.small.8 TThishis makesmakes thethe optimaloptimal taxtax formulaformula quitequite generalgeneral aandnd useful.useful.

7 If a positive social weight g > 0 is set on top earners’ marginal consumption, then the optimal rate is τ = (1 – g)/(1 – g + ae) < τ * . With plausible weights that are small relative to the weight on an average earner, the optimal tax does not change much. 8 If the elasticity e does not vary by income level, then the Pareto parameter a does not vary with τ. If the elasticity varies by income, the Pareto parameter a might depend on the top tax rate τ. The formula τ * = 1/(1 + ae) is still valid in that case, but determining τ * would require knowing how a varies with τ. Peter Diamond and Emmanuel Saez 171

Figure 2 Empirical Pareto Coeffi cients in the United States, 2005

2.5 * * a = zm/(zm – z ) with zm = E(z | z > z ) α = z*h/(z*)/(1 – H(z*))

2

1.5 Empirical Pareto coefficient

1 0 200,000 400,000 600,000 800,000 1,000,000 z* = Adjusted gross income (current 2005 $)

Source: The authors using public use tax return data. * * Notes: The fi gure depicts in solid line the ratio a = z m /( z m – z ) with z ranging from $0 to $1,000,000 * annual income and z m the average income above z using U.S. tax return micro data for 2005. Income is defi ned as Adjusted Gross Income reported on tax returns and is expressed in current 2005 dollars. Vertical lines depict the 90th percentile ($99,200) and 99th percentile ($350,500) nominal thresholds as of 2005. The ratio a is equal to one at z * = 0, and is almost constant above the 99th percentile and slightly below 1.5, showing that the top of the distribution is extremely well approximated by a Pareto distribution for purposes of implementing the optimal top tax rate formula τ * = 1/(1 + ae). Denoting by h(z) the density and by H(z) the cumulative distribution function of the income distribution, the fi gure also displays in dotted line the ratio α( z * ) = z * h( z * )/(1 – H( z * )), which is also approximately constant, around 1.5, above the top percentile. A decreasing (or constant) α(z) combined with a decreasing G(z) and a constant e(z) implies that the optimal marginal tax rate T ′(z) = [1 – G(z)]/[1 – G(z) + α(z) e(z)] increases with z.

The Tax Elasticity of Top Incomes TThehe keykey remainingremaining empiricalempirical ingredientingredient toto implementimplement thethe formulaformula forfor thethe ooptimalptimal taxtax raterate isis thethe elasticityelasticity e ofof toptop incomesincomes withwith respectrespect toto thethe net-of-taxnet-of-tax rrate.ate. WithWith thethe ParetoPareto parameterparameter a = 11.5.5 iiff e = ..25,25, a mmid-rangeid-range estimateestimate fromfrom thethe eempiricalmpirical lliterature,iterature, tthenhen τ * = 11/(1/(1 + 11.5.5 × ..25)25) = 7733 ppercent,ercent, ssubstantiallyubstantially hhigherigher tthanhan thethe currentcurrent 42.542.5 percentpercent toptop U.S.U.S. marginalmarginal taxtax raterate (combining(combining allall taxes).taxes).9

9 Using g * of .04, the optimal tax rate decreases by about 1 percentage point. 172 Journal of Economic Perspectives

TThehe currentcurrent rate,rate, τ = 42.542.5 percent,percent, wouldwould bebe optimaloptimal onlyonly ifif thethe elasticityelasticity e wwereere eextremelyxtremely hhigh,igh, eequalqual ttoo 00.9..9.1100 BBeforeefore turningturning toto empiricalempirical estimates,estimates, wewe reviewreview somesome ofof thethe interpretationinterpretation iissuesssues thatthat arisearise whenwhen movingmoving beyondbeyond thethe simplestsimplest versionversion ofof thethe MirrleesMirrlees (1971)(1971) mmodel.odel. IInn tthehe MMirrleesirrlees mmodel,odel, ttherehere isis a singlesingle taxtax onon eacheach individual.individual. WithWith manymany ttaxes,axes, fforor eexample,xample, iinn mmanyany pperiods,eriods, tthehe kkeyey mmeasureeasure iiss tthehe rresponseesponse ooff tthehe ppresentresent ddiscountediscounted valuevalue ofof allall taxes,taxes, notnot thethe responseresponse ofof revenuerevenue inin a singlesingle year.year. ThisThis oobservationbservation mattersmatters ggiveniven ssignifiignifi cantcant controlcontrol byby somesome peoplepeople overover thethe timingtiming ofof ttaxesaxes aandnd ooverver tthehe fformsorms iinn wwhichhich iincomencome mmightight bbee rreceived.eceived. AAlso,lso, bbecauseecause tthehe bbasicasic MMirrleesirrlees mmodelodel hhasas nnoo ttax-deductibleax-deductible charitablecharitable giving,giving, a tax-inducedtax-induced changechange inin ttaxableaxable incomeincome iinvolvesnvolves oonlynly ddistortionsistortions ffromrom rreducededuced eearnings.arnings. HHowever,owever, wwhenhen aann iincreasencrease inin marginalmarginal taxtax rratesates lleadseads ttoo aann iincreasencrease iinn ccharitableharitable ggiving,iving, tthehe ggainain ttoo tthehe rrecipientsecipients needsneeds toto bbee iincorporatedncorporated inin thethe effieffi ciencyciency measuremeasure (Saez,(Saez, 2004).2004). OtherOther ttaxax ddeductionseductions aarere mmoreore ddiffiiffi cultcult toto consider.consider. InIn thethe MirrleesMirrlees model,model, compensationcompensation eequalsquals thethe marginalmarginal product.product. InIn bargainingbargaining settingssettings oror withwith asymmetricasymmetric informa-informa- ttion,ion, ppeopleeople mmayay nnotot rreceiveeceive ttheirheir mmarginalarginal pproducts.roducts. TThus,hus, eeffortffort iiss rrespondingesponding ttoo a ppricerice thatthat isis hhigherigher oorr llowerower tthanhan mmarginalarginal pproduct,roduct, aandnd tthehe ttaxax rrateate iitselftself mmayay aaffectffect tthehe ggapap bbetweenetween ccompensationompensation aandnd mmarginalarginal pproduct.roduct. TThehe largelarge literatureliterature usingusing taxtax reformsreforms toto estimateestimate thethe elasticityelasticity relevantrelevant forfor thethe ooptimalptimal taxtax formulaformula hashas focusedfocused primarilyprimarily onon thethe responseresponse ofof reportedreported income,income, eithereither ““adjustedadjusted grossgross income”income” oror “taxable“taxable income,”income,” toto net-of-taxnet-of-tax rates.rates. Saez,Saez, Slemrod,Slemrod, andand GGiertziertz (forthcoming)(forthcoming) offeroffer a recentrecent survey,survey, whilewhile SlemrodSlemrod (2000)(2000) lookslooks atat studiesstudies ffocusingocusing onon thethe rich.rich. TheThe behavioralbehavioral elasticityelasticity isis duedue toto realreal economiceconomic responsesresponses ssuchuch asas laborlabor supply,supply, businessbusiness creation,creation, oror savingssavings decisions,decisions, butbut alsoalso taxtax avoidanceavoidance aandnd eevasionvasion rresponses.esponses. A nnumberumber ooff sstudiestudies hhaveave sshownhown llargearge aandnd qquickuick rresponsesesponses ooff rreportedeported incomesincomes alongalong thethe taxtax avoidanceavoidance marginmargin atat thethe toptop ofof thethe distribution,distribution, butbut nnoo compellingcompelling studystudy toto datedate hashas shownshown substantialsubstantial responsesresponses alongalong thethe realreal economiceconomic rresponsesesponses mmarginargin aamongmong ttopop eearners.arners. FForor eexample,xample, iinn tthehe UUnitednited SStates,tates, rrealizedealized ccapitalapital ggainsains ssurgedurged iinn 11986986 iinn aanticipationnticipation ooff tthehe iincreasencrease iinn tthehe ccapitalapital ggainsains ttaxax rrateate afterafter thethe TaxTax ReformReform ActAct ofof 19861986 (Auerbach,(Auerbach, 1988).1988). Similarly,Similarly, exercisesexercises ofof stockstock ooptionsptions surgedsurged inin 19921992 beforebefore thethe 19931993 toptop raterate increaseincrease tooktook placeplace (Goolsbee,(Goolsbee, 22000).000). TThehe TTaxax RReformeform AActct ooff 11986986 aalsolso lleded ttoo a sshifthift ffromrom ccorporateorporate ttoo iindividualndividual iincomencome aass iitt bbecameecame mmoreore aadvantageousdvantageous ttoo bbee oorganizedrganized aass a bbusinessusiness ttaxedaxed ssolelyolely aatt thethe individualindividual levellevel ratherrather thanthan asas a corporationcorporation taxedtaxed fi rrstst aatt tthehe ccorporateorporate llevelevel ((Slemrod,Slemrod, 1996;1996; GordonGordon andand Slemrod,Slemrod, 2000).2000). TheThe paperpaper GruberGruber andand SaezSaez (2002)(2002) isis ooftenften citedcited forfor itsits substantialsubstantial taxabletaxable incomeincome elasticityelasticity estimateestimate (e = 00.57).57) aatt tthehe ttopop ooff tthehe ddistribution.istribution. HHowever,owever, iitsts aauthorsuthors aalsolso ffoundound a ssmallmall eelasticitylasticity (e = 00.17).17) fforor iincomencome beforebefore anyany deductions,deductions, eveneven atat thethe toptop ofof thethe distributiondistribution (Table(Table 9,9, p.p. 24).24). WWhenhen a ttaxax ssystemystem ooffersffers ttaxax aavoidancevoidance oorr eevasionvasion oopportunities,pportunities, tthehe ttaxax bbasease iinn a givengiven yearyear isis quitequite sensitivesensitive toto taxtax rates,rates, soso thethe elasticityelasticity e iiss llarge,arge, aandnd tthehe ooptimalptimal ttopop taxtax raterate isis correspondinglycorrespondingly low.low. TwoTwo importantimportant qualifiqualifi cationscations mustmust bebe made.made.

10 Alternatively, if the elasticity is e = .25, then τ = 42.5 percent is optimal only if the marginal consump- tion of very high-income earners is highly valued, with g =.72. The Case for a Progressive Tax: From Basic Research to Policy Recommendations 173

FFirst,irst, asas mentionedmentioned above,above, manymany ofof thethe taxtax avoidanceavoidance channelschannels suchsuch asas retimingretiming oorr iincomencome sshiftinghifting produceproduce changeschanges inin taxtax revenuerevenue inin otherother periodsperiods oror otherother taxtax bbases—calledases—called “tax“tax externalities”—andexternalities”—and hencehence dodo notnot decreasedecrease thethe optimaloptimal taxtax rate.rate. SSaez,aez, Slemrod,Slemrod, aandnd GGiertziertz ((forthcoming)forthcoming) pproviderovide fformulasormulas sshowinghowing hhowow tthehe ooptimalptimal ttopop ttaxax rrateate sshouldhould bbee mmodifiodifi eded inin suchsuch cases.cases. Second,Second, andand mostmost important,important, thethe ttaxax aavoidancevoidance oror evasionevasion componentcomponent ofof thethe elasticityelasticity e iiss nnotot aann iimmutablemmutable param-param- eeterter andand cancan bebe reducedreduced throughthrough basebase broadeningbroadening andand taxtax enforcementenforcement (Slemrod(Slemrod aandnd Kopczuk,Kopczuk, 2002;2002; Kopczuk,Kopczuk, 2005).2005). Thus,Thus, thethe distinctiondistinction betweenbetween realreal responsesresponses aandnd ttaxax aavoidancevoidance responsesresponses isis criticalcritical forfor taxtax policy.policy. AsAs anan illustrationillustration usingusing thethe ddifferentifferent elasticityelasticity estimatesestimates ofof GruberGruber andand SaezSaez (2002)(2002) forfor high-incomehigh-income earnersearners mmentionedentioned above,above, thethe optimaloptimal toptop taxtax raterate usingusing thethe currentcurrent taxabletaxable incomeincome basebase ((andand ignoringignoring taxtax externalities)externalities) wouldwould bebe τ * = 11/(1/(1 + 11.5.5 × 00.57).57) = 5544 p percent,ercent, wwhilehile thethe optimaloptimal toptop taxtax raterate usingusing a broaderbroader incomeincome basebase withwith nono deductionsdeductions wwouldould bebe τ * = 11/(1/(1 + 11.5.5 × 00.17).17) = 8800 percent.percent. TakingTaking asas fi xedxed statestate andand payrollpayroll ttaxax rrates,ates, suchsuch ratesrates correspondcorrespond toto toptop federalfederal incomeincome taxtax ratesrates equalequal toto 4848 andand 7766 ppercent,ercent, respectively.respectively. AlthoughAlthough considerableconsiderable uncertaintyuncertainty remainsremains inin thethe esti-esti- mmationation ofof thethe long-runlong-run behavioralbehavioral responsesresponses toto toptop taxtax ratesrates (Saez,(Saez, Slemrod,Slemrod, andand GGiertz,iertz, forthcoming),forthcoming), thethe eelasticitylasticity e = 00.57.57 isis a conservativeconservative upperupper boundbound estimateestimate ooff tthehe ddistortionistortion ooff ttopop UU.S..S. ttaxax rrates.ates. TTherefore,herefore, tthehe ccasease fforor hhigherigher rratesates aatt tthehe ttopop aappearsppears rrobustobust iinn tthehe ccontextontext ooff tthishis mmodel.odel.

Link with the Zero Top Rate Result * * FFormally,ormally, z m / z reachesreaches 1 whenwhen z reachesreaches thethe levellevel ofof incomeincome ofof thethe singlesingle * * hhighestighest incomeincome earner,earner, inin whichwhich casecase a = z m //(( z m – z ) isis infiinfi nite,nite, andand indeedindeed τ = 11/(1/(1 + ae) = 0,0, whichwhich isis thethe famousfamous zerozero toptop raterate resultresult fi rstrst demonstrateddemonstrated byby SSadkaadka ((1976)1976) aandnd SSeadeeade (1977).(1977). However,However, noticenotice thatthat thisthis resultresult appliesapplies onlyonly toto thethe vveryery ttopop iincomencome eearner;arner; itsits lacklack ofof widerwider applicabilityapplicability cancan bebe verifiverifi eded empiricallyempirically uusingsing taxtax data.data.1111 IfIf oneone makesmakes thethe reasonablereasonable assumptionassumption thatthat thethe levellevel ofof toptop earn-earn- iingsngs iiss nnotot kknownnown iinn aadvance,dvance, aandnd iinsteadnstead cconsideronsider hhavingaving ppotentialotential eearningsarnings ddrawnrawn rrandomlyandomly ffromrom aann uunderlyingnderlying PParetoareto ddistributionistribution tthenhen ((asas wwee sshowhow iinn tthehe AAppendixppendix aavailablevailable onlineonline withwith thisthis paperpaper atat 〈hhttp://e-jep.orgttp://e-jep.org〉)),, withwith thethe budgetbudget constraintconstraint ssatisfiatisfi eded inin expectation,expectation, thethe formula,formula, τ * = 1/(11/(1 + ae)),, remainsremains thethe naturalnatural optimumoptimum ttaxax rate.rate. ThisThis fi ndingnding impliesimplies thatthat thethe zerozero toptop raterate resultresult andand itsits corollarycorollary thatthat mmarginalarginal ttaxax rratesates sshouldhould ddeclineecline aatt tthehe ttopop hhaveave nnoo ppolicyolicy rrelevance,elevance, a vviewiew tthathat wwee bbelieveelieve iiss wwidelyidely ssharedhared amongamong ppublicublic fi nancenance economists.economists.1122

11 * If, for example, the second-highest income is only one-half of the highest earner then z m / z = 2 * * (and hence a = 2) when z is just above the second-highest earner, so that convergence of z m / z to one really happens only between the top and second-highest earner. The IRS publishes statistics on the top 400 taxpayers (IRS, 2009b). In 2007, the threshold to be a top 400 taxpayer was $138.8m and the average income of top 400 taxpayers was $344.8m so that a = 1.67 at z * = $138.8m, very close to the value of 1.5 at the top percentile threshold, and still very far from the infi nite value it takes at the very top income. 12 With a known fi nite distribution, the marginal tax rate at the top is zero, but the average tax rate between the highest and second-highest earners is so large that highest earner gets no additional utility from being more productive than the next-highest earner. 174 Journal of Economic Perspectives

Should Marginal Tax Rates Rise with Income? AAssumingssuming awayaway incomeincome effectseffects onon laborlabor supply,supply, thethe optimaloptimal marginalmarginal taxtax raterate fformulaormula aatt aanyny iincomencome llevelevel ((applyingapplying ttoo tthehe ccombinationombination ooff aallll ttaxes)axes) ttakesakes a fformorm tthathat cancan bebe expressedexpressed directlydirectly asas a functionfunction ofof thethe incomeincome distributiondistribution asas followsfollows ((Diamond,Diamond, 11998):998):

T ′(z) = [1 – G(z)]/[1 – G(z) + α(z) e(z)] where e(z) is the elasticity of incomes with respect to the net-of-tax rate at income level z, G(z) is the average social marginal welfare weight across individuals with income above z, and α(z) = (zh(z))/(1 – H(z)) with h(z) the density of taxpayers at income level z and H(z) the fraction of individuals with income below z.13 The expression α(z) refl ects the ratio of the total income of those affected by the marginal tax rate at z relative to the numbers of people at higher income levels. A derivation of the optimal formula is presented in an appendix available with this paper at 〈http://e-jep.org〉. FForor ParetoPareto distributions,distributions, α(z) iiss cconstantonstant andand equalequal toto thethe ParetoPareto parameter.parameter. HHowever,owever, tthehe empiricalempirical U.S.U.S. incomeincome distributiondistribution isis notnot a ParetoPareto distributiondistribution atat lowerlower iincomencome llevels.evels. TThehe α(z) ttermerm iiss ddepictedepicted inin dotteddotted lineline onon FigureFigure 2 forfor thethe empiricalempirical 22005005 UU.S..S. iincomencome ddistribution.istribution. IItt iiss iinverselynversely UU-shaped,-shaped, rreachingeaching a mmaximumaximum ooff 22.17.17 aatt z = $135,000,$135,000, thenthen decreasingdecreasing andand stayingstaying approximatelyapproximately constantconstant aroundaround 1.51.5 aabovebove z = $400,000.$400,000. BecauseBecause socialsocial welfarewelfare weightsweights areare lowerlower forfor higherhigher incomes,incomes, G(z) decreasesdecreases withwith z. Therefore,Therefore, assumingassuming a constantconstant elasticityelasticity e acrossacross incomeincome ggroups,roups, thethe formulaformula impliesimplies thatthat thethe optimaloptimal marginalmarginal taxtax ratesrates shouldshould increaseincrease wwithith iincomencome iinn tthehe uupperpper ppartart ooff tthehe ddistribution.istribution. TThishis rresultesult wwasas ttheoreticallyheoretically eestab-stab- llishedished byby DiamondDiamond (1998)(1998) andand conficonfi rmedrmed byby allall subsequentsubsequent simulationssimulations thatthat useuse a PParetoareto distributiondistribution atat thethe toptop asas inin SaezSaez (2001)(2001) oror Mankiw,Mankiw, Weinzierl,Weinzierl, andand YaganYagan ((2009).2009). Quantitatively,Quantitatively, thisthis increaseincrease isis substantial.substantial. ForFor example,example, assumingassuming againagain aann eelasticitylasticity e = ..2525 aandnd tthathat G(z) = 00.5.5 aatt z = $$100,000,100,000, correspondingcorresponding toto thethe toptop ddecileecile tthresholdhreshold wwherehere α = 2.05,2.05, wewe wwouldould hhaveave T ′ = 4949 ppercentercent aatt tthishis iincome,ncome, wwellell bbelowelow tthehe vvaluealue ooff 7733 ppercentercent fforor tthehe ttopop ppercentileercentile aass ccalculatedalculated above.above. IInn tthehe ccurrenturrent taxtax systemsystem withwith manymany taxtax avoidanceavoidance opportunitiesopportunities atat thethe higherhigher eend,nd, asas discusseddiscussed above,above, thethe elasticityelasticity e iiss llikelyikely toto bebe higherhigher forfor toptop earnersearners thanthan fforor mmiddleiddle iincomes,ncomes, ppossiblyossibly leadingleading toto decreasingdecreasing marginalmarginal taxtax ratesrates atat thethe toptop ((GruberGruber andand Saez,Saez, 2002).2002). However,However, thethe naturalnatural policypolicy responseresponse shouldshould bebe toto closeclose ttaxax aavoidancevoidance opportunities,opportunities, inin whichwhich casecase thethe assumptionassumption ofof constantconstant elasticitieselasticities mmightight bbee a rreasonableeasonable bbenchmark.enchmark.

13 Technically, Saez (2001) shows that h(z) is the density of incomes when the nonlinear tax system is linearized at z. Saez (2001) also shows that a similar but more complex formula can be obtained with income effects that is quantitatively close to the equation above. Peter Diamond and Emmanuel Saez 175

Additional Considerations TToo ssomeome rreaders,eaders, proposingproposing marginalmarginal incomeincome taxtax ratesrates onon thethe toptop percentilepercentile ooff eearners,arners, alongalong withwith a broadenedbroadened taxtax base,base, inin a rangerange fromfrom 4848 toto 7676 percentpercent maymay sseemeem iimplausiblymplausibly hhigh.igh. OOnene wwayay ttoo jjudgeudge hhowow sseriouslyeriously ttoo ttakeake ssuchuch nnumbersumbers iiss ttoo considerconsider whetherwhether elementselements leftleft outout inin thethe derivationderivation pushpush forfor a signifisignifi cantlycantly ddifferentifferent answer.answer. TTwowo kkeyey oomittedmitted eelementslements aarere tthehe ppresenceresence ofof capitalcapital incomeincome andand a llonger-runonger-run ddynamicynamic pperspective.erspective. DDoesoes tthehe ppresenceresence ooff ccapitalapital iincomencome mmeanean tthathat eearningsarnings sshouldhould bbee ttaxedaxed ssignif-ignif- iicantlycantly ddifferently?ifferently? WhenWhen wwee ddiscussiscuss ttaxationaxation ooff ccapitalapital iincomencome iinn a llaterater ssection,ection, wwee nnoteote thatthat thethe abilityability toto convertconvert somesome laborlabor incomeincome intointo capitalcapital incomeincome isis a reasonreason fforor llimitingimiting tthehe ddifferenceifference bbetweenetween ttaxax rratesates oonn tthehe ttwowo ttypesypes ooff iincome—thatncome—that iis,s, aann aargumentrgument forfor taxingtaxing capitalcapital income.income. Plausibly,Plausibly, itit isis alsoalso anan argumentargument forfor a somewhatsomewhat llowerower laborlabor incomeincome tax,tax, assumingassuming thatthat laborlabor incomeincome shouldshould bebe taxedtaxed moremore heavilyheavily tthanhan ccapitalapital iincome.ncome. PPerhapserhaps mostmost critically,critically, doesdoes anan estimateestimate basedbased onon a singlesingle periodperiod modelmodel stillstill aapplypply wwhenhen rrecognizingecognizing thatthat ppeopleeople eearnarn aandnd ppayay iincomencome ttaxesaxes yyearear aafterfter yyear?ear? FFirst,irst, eearlierarlier ddecisionsecisions ssuchuch aass eeducationducation aandnd ccareerareer cchoiceshoices aaffectffect llaterater eearningsarnings oopportu-pportu- nnities.ities. ItIt isis conceivableconceivable thatthat a moremore progressiveprogressive taxtax systemsystem couldcould reducereduce incentivesincentives toto aaccumulateccumulate hhumanuman ccapitalapital iinn tthehe fi rstrst place.place. TheThe logiclogic ofof thethe equity–effiequity–effi ciencyciency trade-trade- ooffff wouldwould stillstill carrycarry through,through, butbut thethe elasticityelasticity e shouldshould reflrefl ectect notnot onlyonly short-runshort-run llaborabor supplysupply responsesresponses butbut alsoalso long-runlong-run responsesresponses throughthrough educationeducation andand careercareer cchoices.hoices. WhileWhile therethere isis a sizablesizable multiperiodmultiperiod optimaloptimal taxtax literatureliterature usingusing life-cyclelife-cycle mmodelsodels andand generatinggenerating insights,insights, wewe unfortunatelyunfortunately havehave littlelittle compellingcompelling empiricalempirical eevidencevidence ttoo aassessssess wwhetherhether ttaxesaxes aaffectffect eearningsarnings tthroughhrough tthosehose long-runlong-run channels.channels. SSecond,econd, therethere isis signifisignifi cantcant uncertaintyuncertainty inin futurefuture earnings.earnings. SuchSuch uncertaintyuncertainty ggivesives anan insuranceinsurance rolerole forfor earningsearnings taxationtaxation and,and, asas wewe shallshall see,see, alsoalso hashas conse-conse- qquencesuences forfor thethe taxationtaxation ofof savings.savings.1144 HHowever,owever, thethe applicabilityapplicability ofof resultsresults forfor policypolicy sseemseems uunclearnclear ttoo uus.s.

Recommendation 2: Tax (and transfer) policy toward low earners should include subsidization of earnings and should phase out the subsidization at a relatively high rate.

TTransfersransfers areare naturallynaturally integratedintegrated withwith taxestaxes inin anan optimaloptimal taxtax problem.problem. SuchSuch ttransfersransfers ooftenften ttakeake tthehe ggeneraleneral fformorm ooff a mmaximumaximum bbenefienefi t fforor tthosehose wwithith nnoo iincome,ncome, wwhichhich iiss pphasedhased ooutut aatt hhighigh rratesates aass eearningsarnings iincrease.ncrease. FForor eexample,xample, iinn tthehe UUnitednited SStates,tates, TTANFANF ((TemporaryTemporary AAidid ttoo NNeedyeedy FFamilies)amilies) aandnd SSNAPNAP ((SupplementalSupplemental NNutritionutrition AAssistancessistance PProgram,rogram, fformerlyormerly kknownnown aass FFoodood SStamps)tamps) ooperateperate iinn tthishis wway.ay. A ggrowingrowing ffractionraction ofof means-testedmeans-tested transferstransfers isis nownow administeredadministered throughthrough refundablerefundable taxtax creditscredits

14 The “new dynamic public fi nance” analyzes such settings using mechanism design. The new dynamic public fi nance has made recent progress on the optimal labor income taxation in the dynamic context. See Farhi and Werning (2011). 176 Journal of Economic Perspectives

ssuchuch aass tthehe EEITCITC ((EarnedEarned IIncomencome TTaxax CCredit)redit) oorr tthehe CChildhild TTaxax CCredit.redit. SSuchuch pprogramsrograms aarere typicallytypically fi rstrst phasedphased inin andand thenthen phasedphased outout withwith earningsearnings soso thatthat benefibenefi tsts areare cconcentratedoncentrated oonn llow-incomeow-income wworkingorking ffamiliesamilies iinsteadnstead ooff tthosehose wwithith nnoo eearnings.arnings. MManyany sstudiestudies hhaveave ffoundound ccompellingompelling eevidencevidence ooff ssubstantialubstantial llaborabor ssupplyupply rresponsesesponses ttoo ttrans-rans- ffersers alongalong thethe extensiveextensive marginmargin ofof whetherwhether oror notnot toto work.work. ForFor example,example, thethe EITCEITC eexpansionsxpansions hhaveave eencouragedncouraged llaborabor fforceorce pparticipationarticipation ooff UU.S..S. ssingleingle mmothersothers ((Meyer,Meyer, 22010).010). HHowever,owever, ttherehere iiss mmuchuch llessess ccompellingompelling eevidencevidence ooff bbehavioralehavioral rresponsesesponses aalonglong tthehe iintensiventensive mmargin—thatargin—that iis,s, hhoursours ooff wworkork oonn tthehe jjob—forob—for llower-incomeower-income eearners.arners. AAss wwee sshallhall ssee,ee, tthesehese ffactsacts pplaylay a ccriticalritical rroleole iinn tthehe ooptimalptimal pprofirofi lele ofof transfers.transfers.

Intensive Elasticities IInn thethe MirrleesMirrlees (1971)(1971) model,model, behavioralbehavioral responsesresponses taketake placeplace onlyonly throughthrough tthehe iintensiventensive mmarginargin ooff tthehe nnumberumber ooff hhoursours wworked.orked. IInn tthathat ccontext,ontext, iitt iiss ooptimalptimal ttoo pproviderovide incomeincome toto thosethose withwith nono earnings,earnings, whichwhich isis thenthen phasedphased outout withwith earnings,earnings, ppossiblyossibly atat a highhigh rate—whichrate—which actsacts asas anan implicitimplicit taxtax (see(see thethe onlineonline appendixappendix withwith tthishis ppaperaper aatt 〈hhttp://e-jep.orgttp://e-jep.org〉 forfor a derivation).derivation). TheThe intuitionintuition isis thatthat a highhigh phase-phase- ooutut rrateate aallowsllows tthehe ggovernmentovernment ttoo ttargetarget ttransfersransfers ttoo tthehe mmostost ddisadvantagedisadvantaged ffamilies.amilies. A highhigh phase-outphase-out raterate doesdoes reducereduce earningsearnings forfor low-incomelow-income families,families, becausebecause theythey rreduceeduce hourshours worked.worked. However,However, becausebecause earningsearnings ofof thosethose inin thethe phase-outphase-out areare ssmallmall ttoo sstarttart wwith,ith, tthishis eelasticitylasticity aappliespplies toto a lowlow incomeincome base.base. Therefore,Therefore, increasingincreasing tthehe mmaximumaximum bbenefienefi t (to(to tthosehose wwithith nnoo eearnings)arnings) aandnd iincreasingncreasing tthehe pphase-outhase-out rrateate iiss desirabledesirable forfor redistribution,redistribution, andand thethe behavioralbehavioral responsesresponses createcreate modestmodest fi scalscal ccostsosts rrelativeelative toto tthehe rredistributiveedistributive gains,gains, asas longlong asas tthehe pphase-outhase-out raterate isis nnotot ttoooo hhigh.igh. HHence,ence, tthehe MMirrleesirrlees mmodelodel ooff ooptimalptimal iincomencome ttaxationaxation ggeneratesenerates a ttraditionalraditional wwelfareelfare programprogram wwherehere bbenefienefi tsts areare concentratedconcentrated onon non-earnersnon-earners withwith highhigh phase-phase- ooutut rratesates oonn llow-incomeow-income wworkers.orkers.

Extensive Elasticities HHowever,owever, thethe optimalityoptimality ofof traditionaltraditional welfarewelfare withwith a highhigh phase-outphase-out raterate ddependsepends criticallycritically onon thethe absenceabsence ofof laborlabor supplysupply responsesresponses alongalong thethe extensiveextensive mmargin,argin, thatthat is,is, whetherwhether oror notnot toto work.work. IfIf laborlabor supplysupply responsesresponses areare concentratedconcentrated aalonglong tthehe eextensivextensive mmargin,argin, tthenhen iitt iiss ooptimalptimal ttoo ggiveive hhigherigher ttransfersransfers ttoo llow-incomeow-income wworkersorkers thanthan nonworkers,nonworkers, whichwhich amountsamounts toto a negativenegative phase-outphase-out rate,rate, asas withwith thethe ccurrenturrent EEarnedarned IIncomencome TTaxax CCreditredit ((Diamond,Diamond, 11980;980; SSaez,aez, 22002a).002a). TToo seesee this,this, supposesuppose thethe governmentgovernment startsstarts fromfrom a transfertransfer schemescheme withwith a posi-posi- ttiveive pphase-outhase-out rate—thatrate—that is,is, thethe transfertransfer isis graduallygradually reducedreduced asas earnedearned incomeincome rrises—andises—and introducesintroduces a smallsmall additionaladditional in-workin-work benefibenefi t forfor low-incomelow-income workers.workers. IIgnoringgnoring behavioralbehavioral responses,responses, suchsuch a reformreform isis desirabledesirable ifif thethe governmentgovernment valuesvalues rredistributionedistribution toto llow-incomeow-income eearners.arners. IIff bbehavioralehavioral rresponsesesponses aarere ssolelyolely aalonglong tthehe eextensivextensive margin,margin, thisthis reformreform inducesinduces somesome nnonworkersonworkers toto sstarttart wworkingorking toto ttakeake aadvantagedvantage ofof thethe in-workin-work benefibenefi t.t. However,However, becausebecause wewe startstart fromfrom a situationsituation withwith a positivepositive phase-outphase-out rate,rate, thisthis behavioralbehavioral responseresponse increasesincreases taxtax revenuerevenue asas low-low- iincomencome workersworkers stillstill endend upup receivingreceiving a smallersmaller transfertransfer thanthan nonworkers.nonworkers. Hence,Hence, wwithith thethe availabilityavailability ofof a desirabledesirable redistributionredistribution andand a gaingain inin revenuerevenue fromfrom thethe The Case for a Progressive Tax: From Basic Research to Policy Recommendations 177

bbehavioralehavioral response,response, a positivepositive phase-outphase-out raterate isis notnot optimaloptimal (we(we provideprovide a moremore ddetailedetailed ggraphicalraphical derivationderivation inin thethe onlineonline appendix.)appendix.) IInn practice,practice, bothboth extensiveextensive andand intensiveintensive elasticitieselasticities areare present.present. AnAn intensiveintensive mmarginargin responseresponse wouldwould induceinduce slightlyslightly higherhigher earnersearners toto reducereduce laborlabor supplysupply toto ttakeake advantageadvantage ofof thethe in-workin-work benefibenefi t,t, reducingreducing taxtax revenue.revenue. Therefore,Therefore, thethe govern-govern- mmentent hashas toto trade-offtrade-off thethe twotwo effects.effects. If,If, asas empiricalempirical studiesstudies show,show, thethe extensiveextensive eelasticitylasticity ofof choosingchoosing whetherwhether toto participateparticipate inin thethe laborlabor marketmarket isis largelarge forfor thosethose wwithith lowlow incomesincomes relativerelative toto thethe intensiveintensive elasticityelasticity ofof choosingchoosing howhow manymany hourshours toto wwork,ork, initiallyinitially lowlow (or(or eveneven negative)negative) phase-outphase-out ratesrates combinedcombined withwith highhigh positivepositive pphase-outhase-out ratesrates furtherfurther upup thethe distributiondistribution wouldwould bebe thethe optimaloptimal profiprofi le.le. IInn rrecentecent decadesdecades inin mostmost high-incomehigh-income countries,countries, a concernconcern arosearose thatthat tradi-tradi- ttionalional welfarewelfare programsprograms overlyoverly discourageddiscouraged work,work, andand therethere hashas beenbeen a markedmarked sshifthift towardtoward loweringlowering thethe marginalmarginal taxtax raterate atat thethe bottombottom throughthrough a combinationcombination oof:f: aa)) iintroductionntroduction andand tthenhen eexpansionxpansion ofof iin-workn-work bbenefienefi tsts suchsuch asas thethe EarnedEarned IIncomencome TaxTax CreditCredit inin thethe UnitedUnited States;States; b)b) reductionreduction ofof thethe statutorystatutory phase-outphase-out rratesates iinn ttransferransfer pprogramsrograms fforor eearnedarned iincome,ncome, aass uundernder tthehe UU.S..S. wwelfareelfare rreform;eform; aandnd cc)) reductionreduction ofof payrollpayroll taxestaxes forfor low-incomelow-income earners,earners, asas inin thethe recentrecent U.S.U.S. MakingMaking WWorkork PayPay credit.credit. ThoseThose reformsreforms areare consistentconsistent withwith thethe logiclogic ofof optimaloptimal taxationtaxation wwee hhaveave ooutlined,utlined, asas ttheyhey bbothoth eencouragencourage laborlabor forceforce participationparticipation andand provideprovide ttransfersransfers ttoo llow-incomeow-income wworkers,orkers, sseeneen aass a ddeservingeserving ggroup.roup.

Recommendation 3: Capital income should be taxed.

WWithith thethe standardstandard modelmodel forfor staticstatic laborlabor supplysupply decisions,decisions, thethe simplicitysimplicity ofof a oone-periodne-period modelmodel aandnd tthehe eextensivextensive eempiricalmpirical lliteratureiterature oonn llaborabor ssupplyupply eelasticities,lasticities, iitt iiss ppossibleossible ttoo pproviderovide uusefulseful qquantitativeuantitative analysisanalysis ofof optimaloptimal marginalmarginal taxtax rates.rates. IInn ccontrast,ontrast, thethe literatureliterature onon savingsaving behaviorbehavior seessees a widewide varietyvariety ofof basicbasic behaviors,behaviors, mmoreore widelywidely varyingvarying elasticityelasticity estimates,estimates, andand a complexitycomplexity thatthat comescomes fromfrom thethe iimportancemportance ooff tthehe ffutureuture fforor ddecisionsecisions aaffectedffected bbyy ccapitalapital iincomencome ttaxation.axation. TThus,hus, wwee llimitimit ourour discussiondiscussion toto a singlesingle qualitativequalitative recommendation:recommendation: capitalcapital incomeincome shouldshould bbee subjectsubject toto signifisignifi cantcant taxation.taxation. ThisThis conclusionconclusion isis importantimportant inin lightlight ofof repeatedrepeated ccallsalls fforor nnotot ttaxingaxing ccapitalapital iincome.ncome. AAcademiccademic aargumentsrguments aagainstgainst capitalcapital incomeincome taxationtaxation typicallytypically drawdraw onon oneone oror bbothoth ooff ttwowo ttheoreticalheoretical analyses:analyses: (1)(1) thethe theoremtheorem thatthat thethe optimumoptimum hashas nono asymp-asymp- ttoticotic llong-runong-run taxationtaxation ofof capitalcapital incomeincome inin ChamleyChamley (1986)(1986) andand JuddJudd (1985);(1985); andand ((2)2) tthehe ttheoremheorem tthathat tthehe ooptimumptimum hhasas nnoo ttaxationaxation ooff ccapitalapital iincomencome iinn AAtkinsontkinson aandnd StiglitzStiglitz (1976).(1976).1155 ForFor lengthierlengthier discussiondiscussion ofof thesethese arguments,arguments, seesee BanksBanks andand

15 The aggregate effi ciency theorem in Diamond and Mirrlees (1971) is sometimes cited as support for not taxing capital income. Taxes on transactions between households and fi rms (that do not vary with the particular fi rm) do not interfere with production effi ciency. While taxing all capital income of households will generally change the level of savings, and so investment, it does not move the economy inside the production possibility frontier. Thus, the aggregate effi ciency theorem, that the optimum is on the production frontier, has no direct implications relative to taxing the capital income of households. 178 Journal of Economic Perspectives

DDiamondiamond ((2010).2010). WWee aaddressddress eeachach ooff tthesehese iinn tturn.urn. WWee tthenhen aaddressddress ffourour aargumentsrguments fforor ppositiveositive taxationtaxation ofof capitalcapital income:income: thethe diffidiffi cultyculty ofof distinguishingdistinguishing betweenbetween ccapitalapital aandnd llaborabor iincomes;ncomes; tthehe ppositiveositive ccorrelationorrelation bbetweenetween earningsearnings opportunitiesopportunities aandnd ssavingsavings propensities;propensities; thethe rolerole ofof capitalcapital incomeincome taxestaxes inin easingeasing thethe taxtax burdenburden oonn thosethose whowho areare borrowingborrowing constrained;constrained; andand thethe rolerole ofof discouragingdiscouraging savingssavings inin eencouragingncouraging llaterater llaborabor ssupplyupply inin thethe presencepresence ofof uncertainuncertain futurefuture wagewage rates.rates.

Chamley and Judd IInn thethe modelsmodels aanalyzednalyzed iinn CChamleyhamley ((1986)1986) aandnd JJuddudd ((1985),1985), wwithith iinfinfi nitely-livednitely-lived aagents,gents, aann aasymptoticallysymptotically zzeroero ttaxax oonn ccapitalapital iincomencome iiss ooptimal.ptimal. IInn oorderrder ttoo eevaluatevaluate tthehe rrelevanceelevance ooff tthishis rresultesult fforor ppolicyolicy ppurposes,urposes, oonene nneedseeds ttoo uunderstandnderstand tthehe llogicogic ooff tthehe rresult,esult, aandnd pparticularlyarticularly iitsts rrobustnessobustness ttoo kkeyey aassumptions.ssumptions. AAss ppointedointed ooutut iinn JJuddudd ((1999),1999), tthehe llogicogic fforor tthehe rresultesult iiss sstraightforward.traightforward. A cconstantonstant ccapitalapital iincomencome ttaxax rrateate ccreatesreates a ggrowingrowing ttaxax wwedgeedge bbetweenetween ccurrenturrent cconsumptiononsumption aandnd ffutureuture cconsumptiononsumption aass tthehe hhorizonorizon ggrows.rows. WWithith iinterestnterest rrateate r aandnd nnoo ccapitalapital iincomencome ttaxes,axes, a ddollarollar ttodayoday iiss wworthorth ((11 + r)T aafterfter T yyears.ears. IIff aann iinvestornvestor iiss ssubjectubject ttoo aann aannualnnual ttaxax aatt rrateate τ oonn ccapitalapital iincome,ncome, tthenhen tthehe iinvestornvestor ccanan cconvertonvert oonene uunitnit ooff cconsumptiononsumption ttodayoday iintonto oonlynly ((11 + ((11 – τ)r)T uunitsnits aafterfter T yyears.ears. HHence,ence, tthehe ttaxax wwedgeedge 1 – ((11 + ((11 – τ)r)T/ ((11 + r)T growsgrows wwithith T.1166 ForFor example,example, withwith r = ..0505 aandnd τ = 3300 ppercent,ercent, tthehe ttaxax wwedgeedge iiss a modestmodest 13.413.4 percentpercent whenwhen T = 10,10, butbut isis a substantialsubstantial 43.843.8 percentpercent whenwhen T = 40.40. IInn oorderrder ttoo aavoidvoid ttaxax ccompoundingompounding tthathat ggrowsrows wwithoutithout llimitimit aass tthehe hhorizonorizon eextends,xtends, tthehe ooptimalptimal aaverageverage rrateate mmustust ggoo ttoo zzero,ero, aalthoughlthough nnoo iindividualndividual ttaxax rrateate nneedseeds ttoo bbee zzero.ero. TTherefore,herefore, tthehe rresultesult rrelieselies ccriticallyritically oonn tthehe aassumptionssumption tthathat iindividualsndividuals mmakeake cconsistentonsistent rrationalational ddecisionsecisions aaboutbout ssavingsavings bbehaviorehavior aacrosscross vveryery llongong hhorizons,orizons, aass iinn tthehe sstandardtandard iintertemporalntertemporal mmodel.odel. WWhenhen aagentsgents hhaveave llongong hhorizons,orizons, mmodelingodeling ttheirheir ccurrenturrent ddecisionecision mmakingaking uusingsing aann iinfinfi nniteite hhorizonorizon mmodelodel ccanan bbee mmathemati-athemati- ccallyally mmoreore ttractableractable wwhilehile ddoingoing llittleittle vviolenceiolence ttoo cconclusionsonclusions tthathat rrelateelate ttoo ccurrenturrent bbehavior.ehavior. IInn ccontrast,ontrast, ssubstitutingubstituting aann iinfinfi nite-horizonnite-horizon ddecisionmakerecisionmaker fforor a ssequenceequence ooff fi nnite-horizonite-horizon ddecisionmakersecisionmakers ccanan mmakeake a llargearge ddifferenceifference iinn tthehe aasymptoticsymptotic pposi-osi- ttionion ooff tthehe eeconomy.conomy. IInn aann ooverlappingverlapping ggenerationsenerations mmodelodel wwithith nnoo bbequestsequests aandnd ssoo nnoo ddynasticynastic llinkage,inkage, tthehe ooptimalptimal ccapitalapital iincomencome ttaxax iiss ggenerallyenerally nnotot zzero,ero, eevenven iinn tthehe llong-runong-run ((Diamond,Diamond, 11973;973; AAtkinsontkinson aandnd SSandmo,andmo, 11980).980). TThus,hus, tthehe sstrongtrong aasymptoticsymptotic zzeroero ttaxax rresultesult ooff CChamleyhamley ((1986)1986) aandnd JJuddudd ((1985)1985) rrequiresequires tthathat rrationalational iintertemporalntertemporal ddecisionecision mmakingaking nnotot oonlynly hholdsolds fforor eentirentire llifetimes,ifetimes, bbutut eextendsxtends aacrosscross ddynasties.ynasties. BBothoth aassumptionsssumptions hhaveave bbeeneen hheavilyeavily cchallengedhallenged iinn tthehe eempiricalmpirical lliterature.iterature. FFirst,irst, tthehe rrecentecent behavioralbehavioral economicseconomics literatureliterature hashas castcast muchmuch doubtdoubt onon thethe sstandardtandard modelmodel ofof intertemporalintertemporal decisiondecision makingmaking forfor a signifisignifi cantcant fractionfraction ofof tthehe ppopulation.opulation. A growinggrowing bodybody ofof empiricalempirical workwork showsshows thatthat savingssavings decisionsdecisions areare

16 While interest income and dividends are taxed in this compounding way, the same is not true for capital gains that are taxed on a realization basis. Nor is it true for tax-favored retirement saving, such as IRA or 401(k) accounts. Peter Diamond and Emmanuel Saez 179

hheavilyeavily iinflnfl uenceduenced byby psychologicalpsychological elementselements (such(such asas self-control)self-control) oror minorminor trans-trans- aactionction ccostsosts ((likelike tthehe ddefaultefault eeffectsffects iinn eemployer-sponsoredmployer-sponsored 4401(k)01(k) pplans).lans). SSecond,econd, eempiricalmpirical aanalysesnalyses ooff ggiftsifts aandnd bbequests,equests, wwhilehile cclearlylearly sshowinghowing cconcernsoncerns aaboutbout hheirs,eirs, aarere nnotot ssupportiveupportive ooff tthehe rrigorousigorous vversionersion ooff tthehe ddynastyynasty mmodelodel rrequiredequired fforor tthehe CChamley–Juddhamley–Judd rresult.esult. PPeopleeople lleaveeave bbequestsequests fforor mmanyany rreasons:easons: uunintendednintended bbequestsequests ddueue ttoo llackack ooff aannuitizationnnuitization oorr ttoo lloveove ooff wwealthealth aaccumulationccumulation pperer sse;e; iintendedntended bbequestsequests aarisingrising ooutut ooff bbargainingargaining wwithith hheirs,eirs, ““warmwarm gglow”low” ppreferences,references, oorr aaltruism.ltruism. TThehe ooptimalptimal ttaxax ttreatmentreatment ooff bbequestsequests ddependsepends hheavilyeavily oonn tthehe mmecha-echa- nnismism bbehindehind bbequestsequests ((CremerCremer aandnd PPestieau,estieau, 22006,006, pproviderovide a ssurvey).urvey). FForor eexample,xample, uunintendednintended bbequestsequests sshouldhould bbee ttaxedaxed hheavilyeavily bbecauseecause ttheyhey ddoo nnotot aaffectffect ddonorsonors aandnd iinheritancesnheritances iinducenduce ddoneesonees ttoo wworkork llessess tthroughhrough iincomencome eeffects.ffects. IInn ccontrast,ontrast, iiff bbequestsequests aarere aaltruisticltruistic aandnd tthehe ssocialocial pplannerlanner ttakesakes iintonto aaccountccount bbothoth pparents’arents’ aandnd kkids’ids’ wwelfareelfare ((asas oopposedpposed ttoo pparents’arents’ oonlynly iinn tthehe ttraditionalraditional ddynasticynastic mmodel),odel), tthenhen iitt ccanan bbee ddesirableesirable ttoo ssubsidizeubsidize bbequests,equests, eespeciallyspecially aamongmong tthehe ppooroor ((FarhiFarhi aandnd WWerning,erning, 22010).010). TThehe ddynasticynastic mmodelodel rreflefl ectsects sspecialpecial fformsorms ooff bbothoth aaltruismltruism aandnd tthehe ssocialocial wwelfareelfare ffunctionunction aandnd hhenceence llikelyikely ccapturesaptures oonlynly oonene aaspectspect ooff bbequestequest bbehavior.ehavior. RRejectingejecting thethe policypolicy relevancerelevance ofof thethe zerozero taxationtaxation resultresult doesdoes notnot removeremove thethe rrelevanceelevance ofof tthehe ccompoundingompounding ofof ccapitalapital iincomencome ttaxesaxes nnotedoted aabove.bove. TThishis cconcernoncern aaddsdds supportsupport toto thethe casecase forfor tax-favoredtax-favored retirementretirement savingssavings accountsaccounts comingcoming fromfrom cconcernoncern ooff iinadequatenadequate ssavingsavings bbyy ssomeome ((becausebecause ssavingaving fforor rretirementetirement iinvolvesnvolves llongong hhorizons).orizons). Conversely,Conversely, thethe presencepresence ofof suchsuch accountsaccounts supportssupports higherhigher taxationtaxation ofof ccapitalapital iincomencome tthanhan wwithoutithout ssuchuch a ssavingsavings ooption.ption. AAnothernother sstraightforwardtraightforward cconclusiononclusion ccomingoming ooutut ooff tthehe CChamley–Juddhamley–Judd mmodelodel iiss tthathat iitt iiss bbetteretter ttoo ttaxax eexistingxisting wwealthealth rratherather tthanhan ffutureuture ccapitalapital iincomencome bbecauseecause a ttaxax oonn ccurrenturrent wwealthealth iiss llump-sum,ump-sum, wwhilehile a ttaxax oonn ffutureuture ccapitalapital iincomencome ddistortsistorts iintertemporalntertemporal cchoices.hoices. WWhilehile tthehe aasymptoticsymptotic zzeroero ccapitalapital iincomencome ttaxax rresultesult hhasas ddrawnrawn ggreatreat aattention,ttention, tthehe iinitialnitial rresultesult iiss llargelyargely iignoredgnored fforor ppolicyolicy ppurposes,urposes, aalthoughlthough tthehe ssameame pperspective,erspective, cclearlylearly sstatedtated iinn tthehe lliterature,iterature, lliesies bbehindehind aargumentsrguments fforor sswitchingwitching ffromrom iincomencome ttaxationaxation ttoo cconsumptiononsumption ttaxationaxation iinn ooverlappingverlapping ggenerationeneration mmodelsodels aass a wwayay ttoo ttransferransfer wwealthealth aawayway ffromrom oolderlder ccohortsohorts aatt tthehe ttimeime ooff ttaxax iimplementationmplementation wwithith llittleittle iinn tthehe wwayay ooff ddistortingistorting iincentivesncentives ((Auerbach,Auerbach, KKotlikoff,otlikoff, aandnd SSkinner,kinner, 11983).983). HHowever,owever, ttaxingaxing iinitialnitial wwealthealth aass mmuchuch aass tthehe aavailablevailable ttaxax ttoolsools aallowllow ((whetherwhether aass a wwealthealth ttaxax oorr a ccapitalapital iincomencome ttax)ax) sstrainstrains tthehe rrelevanceelevance ooff tthehe aassumptionssumption tthathat tthehe ggovernmentovernment iiss ccommittedommitted ttoo a ppolicyolicy tthathat tthishis ttaxationaxation ooff wwealthealth wwillill nnotot bbee rrepeated.epeated. WWithoutithout a ccredibleredible ccommitmentommitment ((whichwhich mmayay nnotot bbee ppossible),ossible), cconfionfi sscatorycatory wwealthealth ttaxationaxation wwouldould aadverselydversely aaffectffect ssavingaving bbehaviorehavior aandnd hhaveave sseriouserious eeffiffi cciencyiency c costsosts bbecauseecause ooff cconcernsoncerns tthathat ssuchuch ttaxationaxation wwillill rreturn.eturn. IInn sshort,hort, wwee ddoo nnotot bbelieveelieve tthathat tthehe mmodelingodeling aassumptionsssumptions bbehindehind tthehe CChamleyhamley aandnd JJuddudd rresultsesults aarere sstrongtrong eenoughnough ttoo ssupportupport ddrawingrawing ppolicyolicy llessonsessons aaboutbout tthehe aappropriateppropriate ttaxationaxation ooff ccapital.apital.

Atkinson and Stiglitz IInn a two-periodtwo-period modelmodel withwith oneone periodperiod ofof work,work, thethe Atkinson–StiglitzAtkinson–Stiglitz theoremtheorem ((1976)1976) statesstates thatthat whenwhen thethe availableavailable taxtax toolstools includeinclude nonlinearnonlinear earningsearnings taxes,taxes, ddifferentialifferential taxationtaxation ofof fi rst-rst- andand second-periodsecond-period consumptionconsumption isis notnot optimaloptimal ifif twotwo 180 Journal of Economic Perspectives

kkeyey cconditionsonditions areare satisfisatisfi ed:ed: 1)1) allall consumersconsumers havehave preferencespreferences thatthat areare separableseparable bbetweenetween consumptionconsumption andand labor;labor; andand 2)2) allall consumersconsumers havehave thethe samesame ssubutilityubutility ffunctionunction ofof consumption.consumption. TheThe underlyingunderlying logiclogic behindbehind thethe resultresult startsstarts withwith thethe oobservationbservation tthathat tthehe iincentivencentive ttoo eearnarn ccomesomes ffromrom tthehe uutilitytility aachievablechievable ffromrom cconsumptiononsumption ppurchasesurchases wwithith aafter-taxfter-tax eearnings.arnings. WWithith sseparableeparable ppreferencesreferences aandnd tthehe ssameame ssubutilitiesubutilities forfor eeveryone,veryone, ddifferentialifferential consumptionconsumption taxationtaxation cannotcannot accom-accom- pplishlish aanyny ddistinctionistinction amongamong thosethose withwith differentdifferent earningsearnings abilitiesabilities beyondbeyond whatwhat isis aalreadylready accomplishableaccomplishable byby tthehe eearningsarnings ttax,ax, bbutut wwouldould hhaveave aann aaddeddded eeffiffi ciencyciency costcost ffromrom ddistortingistorting spendingspending choices.choices. ThusThus thethe useuse ofof distortingdistorting taxestaxes onon consumptionconsumption iiss a moremore costlycostly wayway ofof providingproviding thethe incentivesincentives forfor thethe “optimal”“optimal” earningsearnings patternpattern iinn eequilibrium.quilibrium.1177 InIn thisthis two-periodtwo-period model,model, differentialdifferential consumptionconsumption taxationtaxation isis thethe ssameame tthinghing aass ccapitalapital ttaxation.axation. WWhilehile tthehe AAtkinson–Stiglitztkinson–Stiglitz ttheoremheorem rrequiresequires aann aabsencebsence ooff a ssystematicystematic ppatternattern bbetweenetween earningsearnings abilitiesabilities andand savingssavings propensities,propensities, therethere appearsappears toto bebe a positivepositive ccorrelationorrelation betweenbetween laborlabor skillskill levellevel (wage(wage rate)rate) andand savingssavings propensities.propensities. WithWith thisthis pplausiblelausible assumption,assumption, implyingimplying thatthat thosethose withwith higherhigher earningsearnings abilitiesabilities savesave moremore ooutut ooff aanyny ggiveniven income,income, thenthen taxationtaxation ofof savingsaving helpshelps withwith thethe equity–effiequity–effi ciencyciency ttradeoffradeoff byby beingbeing a ssourceource ooff iindirectndirect eevidencevidence aboutabout whowho hashas higherhigher earningsearnings abili-abili- ttiesies aandnd tthushus ccontributesontributes ttoo mmoreore eeffiffi cientcient redistributiveredistributive taxationtaxation (Saez,(Saez, 2002b).2002b).1188 TThehe dimensionalitydimensionality ofof workerworker typestypes (relative(relative toto taxtax tools)tools) mattersmatters inin modelsmodels ofof ccapitalapital incomeincome taxation.taxation. ThisThis pointpoint cancan bebe broughtbrought outout byby contrastingcontrasting thethe analysisanalysis ooff thethe taxationtaxation ofof capitalcapital incomeincome inin a modelmodel withwith twotwo typestypes ofof workersworkers inin DiamondDiamond ((2003)2003) withwith thatthat inin a modelmodel withwith fourfour typestypes ofof workersworkers inin DiamondDiamond andand SpinnewijnSpinnewijn ((forthcoming).forthcoming). BothBoth paperspapers useuse two-periodtwo-period modelsmodels andand aassumessume aadditivedditive pprefer-refer- eences,nces, withwith workersworkers varyingvarying inin bothboth skillskill andand discountdiscount factor.factor. WithWith twotwo types,types, inin thethe ooptimum,ptimum, tthehe hhighigh eearnerarner hhasas nnoo mmarginalarginal ttaxes.axes. IInn ccontrast,ontrast, withwith moremore typestypes ofof wworkersorkers andand diversediverse discountdiscount ratesrates atat eacheach earningsearnings level,level, thethe optimumoptimum hashas taxa-taxa- ttionion ofof savingssavings ofof highhigh earnersearners andand subsidizationsubsidization ofof savingssavings ofof lowlow earners.earners. TheThe uunderlyingnderlying logiclogic comescomes fromfrom tthehe iincentivencentive compatibilitycompatibility constraints,constraints, sincesince high-high- ddiscountiscount typestypes areare moremore wwillingilling ttoo wworkork tthanhan llow-discountow-discount ttypesypes ggiveniven tthehe ssameame sskillkill aandnd ssavingsavings ttaxes.axes. RRecognizingecognizing tthehe rrelevanceelevance aandnd iimportancemportance ooff hheterogeneityeterogeneity iinn

17 Laroque (2005) and Kaplow (2006) provide an elegant and straightforward proof of this point. They show that one can always move to a system of nondistorting consumer taxes coupled with an appropriate modifi cation of the earned income tax and generate more government revenue while leaving every consumer with the same utility and the same labor supply. 18 Banks and Diamond (2010) review evidence on the relationship between savings and skill levels as well as psychological evidence on discount factors. Empirical studies of savings behavior mostly fi nd that those with higher lifetime incomes do save more, but that the full pattern of savings requires consider- able complexity in the underlying model (including uncertainties about earnings and medical expenses, asset tested programs, differential availability of savings vehicles, and bequest motives) to be consistent with the different aspects of savings at different ages. Thus the higher savings rates are consistent with the preference assumption of Saez (2002b), but not, by themselves, a basis for necessarily having the discount rate pattern that Saez assumes, since these other factors are also present. Golosov, Troshkin, Tsyvinski, and Weinzierl (2009) propose a calibration exercise. The Case for a Progressive Tax: From Basic Research to Policy Recommendations 181

ppreferencesreferences withinwithin andand acrossacross earningsearnings levels,levels, wewe rejectreject thethe directdirect policypolicy relevancerelevance ooff tthehe AAtkinson–Stiglitztkinson–Stiglitz ttheorem.heorem.

Distinguishing between Capital and Labor Incomes A straightforwardstraightforward argumentargument forfor taxingtaxing capitalcapital isis thatthat itit isis oftenoften diffidiffi cultcult toto ddistinguishistinguish betweenbetween capitalcapital andand laborlabor incomes.incomes. ForFor example,example, peoplepeople spendingspending ttimeime toto managemanage theirtheir investmentinvestment portfoliosportfolios areare convertingconverting laborlabor timetime intointo antici-antici- ppatedated capitalcapital income.income. InIn smallsmall businesses,businesses, profiprofi tsts arisearise bothboth fromfrom thethe laborlabor ofof oownerswners andand returnsreturns onon assetsassets soso that,that, toto somesome degree,degree, individualsindividuals cancan convertconvert llaborabor incomeincome intointo capitalcapital income.income. ForFor example,example, afterafter thethe 19931993 FinnishFinnish taxtax reformreform ttoo a dualdual incomeincome taxtax withwith a lowerlower raterate onon capitalcapital income,income, therethere werewere signifisignifi cantcant sshiftshifts ofof laborlabor incomeincome toto capitalcapital incomeincome amongamong thethe self-employedself-employed (Pirttilä(Pirttilä andand SSelin,elin, 2011).2011). InIn thethe UnitedUnited States,States, GordonGordon andand MacKie-MasonMacKie-Mason (1995)(1995) andand GordonGordon aandnd SlemrodSlemrod (2000)(2000) havehave foundfound incomeincome shiftingshifting betweenbetween thethe corporatecorporate taxtax basebase aandnd thethe individualindividual taxtax basebase drivendriven byby taxtax differentials.differentials. TheThe existenceexistence ofof taxtax differ-differ- eentialsntials betweenbetween laborlabor andand capitalcapital alsoalso createscreates pressurepressure toto extendextend thethe mostmost favorablefavorable ttaxax treatmenttreatment toto a widerwider setset ofof incomes.incomes. ForFor example,example, inin thethe UnitedUnited States,States, compen-compen- ssationation ofof privateprivate equityequity andand hedgehedge fundfund managersmanagers inin thethe formform ofof a shareshare ofof profiprofi tsts ggeneratedenerated onon behalfbehalf ofof clientsclients isis consideredconsidered realizedrealized capitalcapital gains,gains, althoughalthough itit isis cconceptuallyonceptually laborlabor income.income. TThehe ddiffiiffi cultiesculties inin tellingtelling apartapart laborlabor andand capitalcapital incomeincome areare perhapsperhaps thethe sstrongesttrongest rreasoneason wwhyhy ggovernmentsovernments wwouldould bbee rreluctanteluctant ttoo ccompletelyompletely eexemptxempt ccapitalapital iincomencome andand taxtax onlyonly laborlabor income.income. ChristiansenChristiansen andand TuomalaTuomala (2008)(2008) examineexamine a mmodelodel wwithith costlycostly (but(but legal)legal) conversionconversion ofof laborlabor incomeincome intointo capitalcapital income.income. DDespiteespite preferencespreferences thatthat wouldwould resultresult inin anan optimaloptimal zerozero taxtax onon capitalcapital incomeincome inin tthehe aabsencebsence ooff tthehe aabilitybility ttoo sshifthift iincome,ncome, ttheyhey fi ndnd a positivepositive optimaloptimal taxtax onon capitalcapital iincome.ncome. Similarly,Similarly, thethe Chamley–JuddChamley–Judd resultresult ofof zerozero capitalcapital incomeincome taxationtaxation doesdoes nnotot holdhold inin a modelmodel withwith anan inabilityinability toto distinguishdistinguish betweenbetween entrepreneurialentrepreneurial laborlabor iincomencome aandnd ccapitalapital iincomencome iinn tthehe ssameame bbasicasic mmodelodel ((Reis,Reis, 22007).007).

Borrowing Constraints TThehe modelsmodels discusseddiscussed aboveabove hadhad perfectperfect capitalcapital markets—includingmarkets—including anan aabsencebsence ofof bborrowingorrowing constraints.constraints.1199 BButut borrowingborrowing constraintsconstraints areare relevantrelevant forfor taxtax ppolicy,olicy, providingproviding anotheranother reasonreason forfor positivepositive capitalcapital incomeincome taxation.taxation. SinceSince capitalcapital iincomencome taxestaxes fallfall onon thosethose whowho areare notnot borrowingborrowing constrainedconstrained (because(because theythey havehave ccapital),apital), raisingraising revenuerevenue fromfrom a capitalcapital incomeincome taxtax allowsallows forfor a lowerlower earnedearned incomeincome ttax,ax, includingincluding thethe taxtax onon thosethose whowho areare soso constrained—allowingconstrained—allowing forfor anan effieffi ciencyciency ggainain wwhenhen taxestaxes areare collected.collected. ForFor example,example, AiyagariAiyagari (1995)(1995) andand ChamleyChamley (2001)(2001) cconsideronsider borrowing-constrainedborrowing-constrained agentsagents inin anan uncertaintyuncertainty settingsetting inin anan infiinfi nitely-livednitely-lived

19 Zeldes (1989) shows that, contrary to the predictions of the consumption-smoothing model with no liquidity constraints, consumption paths track predictable changes in income for low-wealth groups. 182 Journal of Economic Perspectives

aagentgent modelmodel andand showshow thatthat capitalcapital incomeincome taxationtaxation isis desirabledesirable whenwhen consumptionconsumption iiss ppositivelyositively ccorrelatedorrelated wwithith ssavings.avings.2200

Uncertain Future Earnings UUncertaintyncertainty aboutabout futurefuture earningsearnings opportunitiesopportunities isis largelarge andand ppervasiveervasive ((BanksBanks aandnd DDiamond,iamond, 22010).010). WWhenhen ssomeome cconsumptiononsumption ddecisionsecisions aarere ttakenaken bbeforeefore eearningsarnings uuncertaintiesncertainties areare resolved,resolved, thethe AtkinsonAtkinson andand StiglitzStiglitz (1976)(1976) resultresult doesdoes notnot holdhold aand,nd, iinn a ttwo-periodwo-period model,model, second-periodsecond-period consumptionconsumption shouldshould bebe taxedtaxed atat thethe mmarginargin relativerelative toto fi rst-periodrst-period consumption.consumption. TheThe underlyingunderlying logiclogic ofof thisthis resultresult isis tthathat welfarewelfare isis enhancedenhanced byby providingproviding insuranceinsurance aboutabout futurefuture earningsearnings opportuni-opportuni- ttiesies tthroughhrough thethe taxtax system.system. WhenWhen leisureleisure isis a normalnormal good,good, moremore savings,savings, ceterisceteris pparibus,aribus, willwill tendtend toto reducereduce workwork laterlater on.on. Thus,Thus, discouragingdiscouraging savingssavings enhancesenhances tthehe aabilitybility toto provideprovide insuranceinsurance againstagainst futurefuture poorpoor laborlabor marketmarket possibilities.possibilities. TheThe aadvantagedvantage ofof ddiscouragingiscouraging savingssavings iiss ppresentresent iinn mmodelsodels wwithith llongeronger ttimeime hhorizonsorizons aass wwell.ell. TThehe eextentxtent ooff iinsurancensurance iiss llimitedimited bbyy mmoraloral hhazardazard cconcerns.oncerns. TThehe literatureliterature makingmaking thisthis pointpoint hashas twotwo strands.strands. First,First, thethe optimaloptimal taxtax strandstrand cconsidersonsiders optimaloptimal linearlinear taxationtaxation ofof capitalcapital incomeincome alongalong withwith optimaloptimal nonlinearnonlinear eearningsarnings taxes.taxes. ProvidedProvided anan individual’sindividual’s planplan withwith lessless futurefuture workwork isis accompaniedaccompanied bbyy moremore savings,savings, introducingintroducing suchsuch taxationtaxation raisesraises welfarewelfare (Mirrlees(Mirrlees andand Diamond,Diamond, 11982;982; DiamondDiamond andand Mirrlees,Mirrlees, 2000).2000). A secondsecond strandstrand commonlycommonly calledcalled “the“the newnew ddynamicynamic ppublicublic fi nance”nance” (Golosov,(Golosov, Kocherlakota,Kocherlakota, andand Tsyvinski,Tsyvinski, 2003)2003) hashas mademade uuncertainncertain futurefuture earningsearnings opportunitiesopportunities a centralcentral concern,concern, anan elementelement largelylargely llackingacking inin thethe optimaloptimal taxtax approach.approach. ItIt usesuses thethe mechanismmechanism designdesign approachapproach ofof ssocialocial wwelfareelfare ooptimizationptimization wwithith tthehe ggovernmentovernment ccontrollingontrolling iindividualndividual cconsumptiononsumption aandnd labor,labor, subjectsubject toto incentiveincentive compatibilitycompatibility constraintsconstraints andand aggregateaggregate resources.resources. WWithith aadditivedditive ppreferences,references, a robustrobust fi ndingnding ofof thisthis literatureliterature isis thethe “Inverse“Inverse EulerEuler EEquation,”quation,” whichwhich impliesimplies thatthat inin thethe absenceabsence ofof restrictions,restrictions, anan individualindividual wouldwould wwantant toto savesave moremore thanthan calledcalled forfor byby thethe sociallysocially optimaloptimal plan.plan. ToTo implementimplement suchsuch aann aallocationllocation oneone needsneeds toto havehave a “wedge”“wedge” reflrefl ectingecting implicitimplicit marginalmarginal taxationtaxation ofof ffutureuture cconsumptiononsumption rrelativeelative ttoo eearlierarlier cconsumption,onsumption, aandnd ssoo aann iimplicitmplicit mmarginalarginal ttaxax oonn ssavingsavings oror capitalcapital income.income. InIn thisthis way,way, makingmaking itit lessless attractiveattractive forfor someonesomeone withwith hhigherigher futurefuture eearningsarnings sskillskills ttoo iimitatemitate ssomeoneomeone wwithith llowerower eearningsarnings sskillskills iimprovesmproves tthehe eequity–effiquity–effi ciencyciency tradeoff.tradeoff. 2211 TThehe mechanismmechanism ddesignesign aapproachpproach generatesgenerates thethe allocationallocation thatthat isis optimal,optimal, wwhichhich iiss tthenhen ssupplementedupplemented bbyy aanalysisnalysis ofof wwaysays ttoo iimplementmplement ssuchuch aann ooptimum,ptimum,

20 This correlation is always positive in the Aiyagari (1995) model with independent and identically distributed labor income, but Chamley (2001) shows that the correlation can be negative theoretically. 21 The “Inverse Euler equation” is that the reciprocal (inverse) of the marginal utility of consumption is equal to the expectation of the reciprocal (inverse) of the future marginal utility of consumption— that is, 1/u ′( c1 ) = E {1/u ′( c2 )}. In a certainty model, the Inverse Euler equation and the familiar Euler equation are the same. However, with uncertainty, the marginal utility of present consumption is less than the expected marginal utility of future consumption when the Inverse Euler equation holds. The Inverse Euler condition comes from optimally balancing the incentives for today’s work coming from additional compensation today with anticipated changes in future resources as a consequence of today’s additional earnings, because the inverse of marginal utility is the resource cost of increasing utility. Peter Diamond and Emmanuel Saez 183

ssometimesometimes usingusing familiarfamiliar taxtax tools.tools. InIn MirrleesMirrlees andand DiamondDiamond (1982)(1982) andand DiamondDiamond aandnd MMirleesirlees ((1986),1986), thethe authorsauthors implementimplement thisthis approachapproach throughthrough thethe adjustmentadjustment ooff rretirementetirement bbenefienefi tsts asas a ffunctionunction ofof tthehe aagege aatt rretirementetirement iinn a ssettingetting wwherehere tthehe aalternativeslternatives areare a particularparticular jobjob oror nono workwork atat allall andand therethere isis uncertaintyuncertainty aboutabout tthehe aabilitybility toto holdhold thethe job.job. ImplicitlyImplicitly taxingtaxing bothboth workwork andand savingssavings allowsallows forfor moremore rredistributionedistribution toto thosethose whowho shouldshould retireretire earlyearly byby ddiscouragingiscouraging savingssavings donedone inin oorderrder toto taketake advantageadvantage ofof anan earlyearly retirementretirement pension.pension. TheThe implicitimplicit taxtax comescomes ffromrom a bbenefienefi t levellevel thatthat growsgrows atat lessless thanthan anan actuariallyactuarially fairfair raterate withwith continuedcontinued wwork.ork. GolosovGolosov andand TsyvinskiTsyvinski (2006)(2006) studystudy optimaloptimal disabilitydisability insuranceinsurance andand recog-recog- nnizeize a rolerole forfor anan assetasset test,test, asas isis widespreadwidespread inin programsprograms forfor thethe poor.poor. However,However, iinn a many-periodmany-period modelmodel withwith a richrich stochasticstochastic dynamicdynamic patternpattern ofof wagewage rates,rates, fullfull iimplementationmplementation ofof a mechanismmechanism designdesign optimumoptimum callscalls forfor a complex,complex, sophisticatedsophisticated ttaxax sstructure.tructure. WWhenhen ttherehere ccanan bbee aalternativelternative wwaysays ttoo iimplementmplement a mmechanismechanism ddesignesign ooptimum,ptimum, wwithoutithout ffurtherurther rresearchesearch itit iiss nnotot cclearlear wwhichhich aapproachpproach sshedsheds llightight oonn hhowow ttoo llevyevy ttaxesaxes iinn mmoreore rrealisticealistic ssettingsettings wwithith llimitedimited ttaxax ttools.ools.2222 TThehe bbottomottom llineine iiss tthathat uuncertainncertain ffutureuture eearningsarnings oopportunitiespportunities aarguergue aagainstgainst zzeroero ttaxationaxation ooff ccapitalapital iincome,ncome, aass ddoo ssavingsavings ppreferencereference hheterogeneity,eterogeneity, llimitedimited ddistinctionsistinctions bbetweenetween ccapitalapital aandnd llaborabor iincomes,ncomes, aandnd bborrowingorrowing cconstraints.onstraints. IItt iiss ttruerue tthathat tthesehese aargumentsrguments aarere bbasedased oonn llife-cycleife-cycle aanalyses,nalyses, aandnd tthathat tthehe eempiricalmpirical lliteratureiterature fi ndsnds tthathat tthehe llife-cycleife-cycle aapproach,pproach, wwhilehile hhelpful,elpful, iiss llimitedimited iinn iitsts ssuccessuccess iinn eexplainingxplaining ssavingsavings bbehavior.ehavior. TThehe bbeliefelief tthathat mmanyany ppeopleeople ddoo nnotot ssaveave eenoughnough fforor ttheirheir oownwn rretirementsetirements ooftenften lleadseads ttoo ppoliciesolicies ttoo eencouragencourage ssavings,avings, pparticularlyarticularly rretirementetirement ssavings.avings. TThehe mmostost wwidelyidely eemployedmployed mmethodethod iiss ssomeome fformorm ooff fforcedorced ssavingaving tthroughhrough mmandatoryandatory ccontributionsontributions ttoo a rretirementetirement ssystem.ystem. TThishis iiss ooftenften ccomplementedomplemented wwithith a ccombinationombination ooff ttaxingaxing ccapitalapital iincomencome aandnd hhavingaving ttax-favoredax-favored rretirementetirement ssavingsavings ((includingincluding ssomeome ssubsidies)ubsidies) ttargetedargeted ttoo tthosehose lliableiable ttoo ssaveave ttoooo llittle.ittle.

Public Finance Methodology

IIff wwee wwereere hhelpingelping toto setset taxtax policy,policy, wwee wwouldould nneedeed ttoo rreacheach cconcreteoncrete conclusionsconclusions oonn ttaxax bbasesases andand taxtax rates.rates. InIn ourour rolerole asas partpart ofof thethe generalgeneral discussiondiscussion ofof taxationtaxation tthathat maymay iinflnfl uenceuence thethe tax-settingtax-setting process,process, wewe looklook toto informinform thinkingthinking aboutabout taxestaxes wwithoutithout necessarilynecessarily gettinggetting toto a concreteconcrete recommendation.recommendation. InIn decidingdeciding whatwhat issuesissues ttoo ppromulgateromulgate aandnd wwhathat ssupportiveupportive aargumentsrguments ttoo pputut fforth,orth, wwee ddrawraw oonn ppartsarts ooff tthehe ooptimalptimal ttaxax lliterature.iterature. WWee aalsolso rrecognizeecognize a rroleole fforor ttheoreticalheoretical aanalysesnalyses iinn rrebuttingebutting aargumentsrguments tthathat ddoo nnotot sseemeem ttoo bbee a ggoodood bbasisasis fforor mmakingaking ttaxax ppolicy.olicy. TThishis aapproach,pproach,

22 An example of a complex implementation is derived in Kocherlakota (2005). It calls for the taxes in any period to depend on the full history of earnings up to that period and has linear capital income tax rates that have a regressive relationship to contemporaneous earnings, and which collect no revenue in aggre- gate. This implementation discourages savings by making the return to savings stochastic even though the rate of return on investment is determinate. And the regressivity of the tax rate is designed to discourage savings by providing a higher return when marginal utility is lower. Werning (2011) proposes a tax imple- mentation with a progressive capital income tax that can be made independent of past earnings shocks. 184 Journal of Economic Perspectives

ddrawingrawing onon multiplemultiple researchresearch sourcessources forfor partialpartial insights,insights, seemsseems appropriateappropriate givengiven tthehe ccomplexityomplexity ofof issuesissues thatthat areare relevantrelevant forfor goodgood taxtax policy,policy, muchmuch lessless thethe eveneven rrichericher setset ofof issuesissues thatthat wouldwould alsoalso recognizerecognize thethe rolerole ofof argumentsarguments inin a complexcomplex ppoliticalolitical pprocess.rocess. AAss a ggoodood mmodelodel fforor aaddressingddressing tthehe mmanyany iissuesssues tthathat mmatteratter fforor ggoodood ttaxax ppolicy,olicy, wwee tthinkhink ooff tthehe MMeadeeade RReporteport ((Meade,Meade, 11978).978). CChapterhapter 2 ooff tthehe RReport,eport, ““TheThe CCharac-harac- tteristicseristics ooff a GGoodood TTaxax SStructure,”tructure,” iiss ddividedivided iintonto ssixix ssections:ections: IIncentivesncentives aandnd eeconomicconomic eeffiffi cciency;iency; DDistributionalistributional eeffects;ffects; IInternationalnternational aaspects;spects; SSimplicityimplicity aandnd ccostsosts ooff aadmin-dmin- iistrationstration aandnd ccompliance;ompliance; FFlexibilitylexibility aandnd sstability;tability; aandnd TTransitionalransitional pproblems.roblems. TToo cconsideronsider ddirectirect ttaxationaxation iinn tthehe UUnitednited KKingdom,ingdom, tthehe MMeadeeade CCommitteeommittee eexaminedxamined eeachach ooff tthesehese iissuesssues aandnd tthenhen ccombinedombined tthehe iinsightsnsights iintonto a ppolicyolicy rrecommendation.ecommendation. IItt sseemseems ttoo uuss tthathat eeconomicconomic aanalysisnalysis nneedseeds ttoo pproceedroceed iinn a ssimilarimilar ffashion.ashion. OOptimalityptimality aanalysesnalyses ofof ttaxationaxation havehave fl ourishedourished inin twotwo (mostly)(mostly) separateseparate rresearchesearch communities.communities. TheThe publicpublic economicseconomics communitycommunity hashas bbeeneen aactivelyctively doingdoing ooptimalptimal ttaxax aanalysesnalyses sincesince thethe midmid 1960s,1960s, whilewhile thethe macromacro community,community, underunder thethe bbanneranner ofof “new“new dynamicdynamic publicpublic fi nance,”nance,” hashas beenbeen activeactive sincesince thethe midmid 1980s.1980s. TheThe sstandardtandard optimaloptimal taxtax analysisanalysis beginsbegins withwith a setset ofof allowableallowable taxtax structuresstructures andand opti-opti- mmizesizes thethe taxtax ratesrates and/orand/or taxtax basesbases inin thethe allowableallowable structure.structure. InIn contrast,contrast, thethe mmacroacro aanalystsnalysts useuse a mechanismmechanism designdesign approach,approach, whichwhich beginsbegins byby derivingderiving eacheach iindividual’sndividual’s mmarginalarginal ratesrates ofof substitutionsubstitution consistentconsistent withwith thethe individual’sindividual’s optimizedoptimized cconsumptiononsumption andand llaborabor aallocation—thellocation—the bestbest ppossibleossible aallocationllocation thatthat iiss cconsistentonsistent wwithith aagentsgents rrevealingevealing theirtheir underlyingunderlying “types.”“types.” TheThe nextnext stepstep isis toto fi ndnd a taxationtaxation mmechanismechanism tthathat ccanan iimplementmplement tthishis aallocation.llocation. TThishis aapproachpproach rrulesules ooutut ttaxesaxes tthathat aarere mmodeledodeled aass rrequiringequiring iinformationnformation tthathat tthehe ggovernmentovernment iiss aassumedssumed nnotot ttoo hhave.ave. A ddrawbackrawback ooff tthehe mmechanismechanism ddesignesign aapproachpproach iiss tthathat iitt aallows—indeed,llows—indeed, itit ooftenften pprescribes—complexrescribes—complex taxtax structuresstructures thatthat areare quitequite uunlikenlike aanyny eexistingxisting ppublicublic ppolicies.olicies. FForor eexample,xample, tthehe lliteratureiterature ttypicallyypically pproceedsroceeds oonn tthehe assumptionassumption thatthat individualsindividuals choosechoose fromfrom thethe allowableallowable setset ofof completecomplete lifetimelifetime cconsumptiononsumption aandnd eearningsarnings pplans.lans. IItt tthenhen dderiveserives ooptimalptimal ttaxax mmechanismsechanisms tthathat mmakeake ttaxesaxes contingentcontingent onon everyevery observableobservable variablevariable thatthat maymay bebe correlatedcorrelated withwith thethe keykey uunobservablenobservable variablevariable (as(as inin optimaloptimal contractingcontracting theory,theory, Holmstrom,Holmstrom, 1979).1979). TheThe ttaxax facedfaced byby a personperson underunder thisthis approachapproach mightmight typicallytypically dependdepend onon thethe completecomplete eearningsarnings andand consumptionconsumption historyhistory ooff tthathat pperson,erson, eevenven wwithoutithout rrecognitionecognition ooff ootherther oobservablebservable variables.variables. JustJust asas thethe recognitionrecognition ofof complexitycomplexity shouldshould limitlimit allowableallowable ttools,ools, ttherehere iiss a ssimilarimilar rroleole fforor ppublicublic pperceptionserceptions ooff ttaxax ffairness.airness. 2233 AAlso,lso, aanalystsnalysts uusingsing tthehe ttwowo aapproachespproaches ssometimesometimes ddifferiffer iinn hhowow ttheyhey aapproachpproach ppolicyolicy iimplications.mplications. WWhilehile tthehe ppublicublic eeconomicsconomics ccommunityommunity llooksooks fforor llessonsessons fforor

23 A model (for example a game-theoretic equilibrium) that may be perfectly sensible with a small number of sophisticated agents may not be helpful for a large population with limitations in attention to long-term consequences, limited information about tax structures, and limited payoff possibilities. More concretely, legislators, tax administrators, and taxpayers have limited abilities to design, enforce, and comply with complex tax structures. We think that model tractability makes it appropriate to assume rather than derive plausible conditions when one thinks the two approaches would lead to the same central conclusion, even though, of course, some other conclusions would not carry over. The Case for a Progressive Tax: From Basic Research to Policy Recommendations 185

ddiverseiverse settings,settings, inin KocherlakotaKocherlakota (2010,(2010, p.p. 1),1), whichwhich providesprovides a comprehensivecomprehensive ttreatmentreatment ooff tthehe nnewew ddynamicynamic ppublicublic fi nance,nance, thethe authorauthor states:states: “The“The goalgoal ofof thisthis bbookook isis toto fi guregure outout atat leastleast somesome characteristicscharacteristics ofof thethe bestbest possiblepossible taxtax system.”system.” WWhilehile tthehe ppublicublic eeconomicsconomics ccommunityommunity ddrawsraws oonn mmultipleultiple mmodels,odels, sseekingeeking iinsights,nsights, nnotot ppreciserecise aanswers,nswers, KKocherlakotaocherlakota ((2010,2010, pp.. 44)) ssays:ays: ““TheThe uultimateltimate ggoaloal ooff tthehe NNDPFDPF [[newnew ddynamicynamic ppublicublic fi nance]nance] isis toto provideprovide relativelyrelatively preciseprecise recommendationsrecommendations asas ttoo wwhathat ttaxesaxes sshouldhould bbe.”e.” WWhilehile llessonsessons fromfrom mmechanismechanism ddesignesign havehave addedadded toto ourour understandingunderstanding ofof ttaxation,axation, thisthis mmethodologicalethodological nnarrownessarrowness rrejectsejects aanalysesnalyses tthathat mmightight bbee oonn ppointoint fforor a governmentgovernment consideringconsidering a limitedlimited taxtax reform.reform. A limitedlimited approachapproach toto taxtax reformreform mmayay rreflefl ectect politicalpolitical feasibility,feasibility, a valuevalue placedplaced onon historicalhistorical continuity,continuity, oror limitslimits inin aacceptablecceptable complexitycomplexity andand rrecord-keepingecord-keeping requirements.requirements. Therefore,Therefore, inin ourour vview,iew, llimitedimited ttaxax rreformeform aanalysisnalysis ccanan iinformnform rrelevantelevant ppolicyolicy qquestionsuestions aandnd hhenceence sshouldhould nnotot bbee rrejectedejected oonn mmethodologicalethodological pprinciples.rinciples. IInn ourour view,view, thethe modelsmodels availableavailable forfor analysis,analysis, likelike muchmuch ofof thethe underlyingunderlying ttheory,heory, remainremain limitedlimited andand stillstill tootoo farfar fromfrom realityreality toto proceedproceed inin anyany otherother fashionfashion tthanhan tthathat ffollowedollowed bbyy tthehe MMeadeeade CCommittee.ommittee. TThus,hus, wwee hhaveave iidentifidentifi eded basicbasic researchresearch fi nndingsdings thatthat wewe fi ndnd relevantrelevant inin thinkingthinking aboutabout practicalpractical taxtax setting,setting, andand alsoalso basicbasic rresearchesearch fi ndingsndings thatthat othersothers maymay fi ndnd relevantrelevant butbut wewe dodo not.not. InIn thethe latterlatter category,category, wwee iincludenclude aargumentsrguments fforor hhighigh iimplicitmplicit mmarginalarginal ttaxax rratesates oonn llowow eearnersarners iinn mmodelsodels wwithith onlyonly anan intensiveintensive marginmargin (because(because thethe extensiveextensive marginmargin isis soso importantimportant forfor llowow earners);earners); a zerozero optimaloptimal taxtax raterate atat a knownknown toptop ofof thethe earningsearnings distributiondistribution ((becausebecause thethe toptop isis notnot known);known); thethe lowlow andand decreasingdecreasing marginalmarginal taxtax raterate onon veryvery hhighigh eearnersarners tthathat ccomesomes ffromrom ssimulationsimulations uusingsing tthehe llognormalognormal ddistributionistribution ooff sskillskills ((becausebecause thethe ParetoPareto distributiondistribution isis wellwell documenteddocumented toto bebe a bbetteretter fi t);t); zerozero taxationtaxation ooff ccapitalapital incomeincome basedbased onon thethe aggregateaggregate effieffi ciencyciency resultresult (because(because thethe theoremtheorem ddoesoes notnot havehave thatthat implication);implication); zerozero taxationtaxation ofof capitalcapital incomeincome asymptoticallyasymptotically ((becausebecause bequestbequest behaviorbehavior doesdoes notnot conformconform withwith whatwhat isis neededneeded forfor thisthis descrip-descrip- ttionion ooff tthehe aasymptoticsymptotic ppositionosition ooff tthehe eeconomy);conomy); aandnd zzeroero ttaxationaxation ooff ccapitalapital iincomencome bbasedased onon tthehe AAtkinson–Stiglitztkinson–Stiglitz theoremtheorem (because(because savingssavings ratesrates areare notnot uuniformniform iinn tthehe ppopulation).opulation).

■ We are grateful to Henry Aaron, Alan Auerbach, James Poterba, Ivan Werning, Joel Yellin, and the editors for helpful comments and discussions.

References

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Appendix

CComparisonomparison withwith Mankiw,Mankiw, Weinzierl,Weinzierl, andand YaganYagan (2009)(2009)

IInn thisthis journal,journal, Mankiw,Mankiw, Weinzierl,Weinzierl, andand YaganYagan (2009)(2009)1 ppresentresent eighteight lessonslessons tthathat theythey drawdraw fromfrom thethe optimaloptimal taxtax literature.literature. OurOur paperpaper agreesagrees withwith somesome ofof theirtheir llessonsessons butbut alsoalso drawsdraws somesome veryvery differentdifferent conclusions.conclusions. InIn thisthis appendix,appendix, wewe discussdiscuss ssomeome ooff tthehe ddiscrepanciesiscrepancies bbetweenetween ourour interpretations,interpretations, followingfollowing thethe orderorder ofof thethe eeightight llessonsessons ttheyhey ppresent.resent.

Lesson 1: Optimal Marginal Tax Rate Schedules Depend on the Distribution of Ability: We agree.

Lesson 2: The Optimal Marginal Tax Schedule Could Decline at High Incomes: Major disagreement. MMankiw,ankiw, WWeinzierl,einzierl, andand YaganYagan basebase thisthis lessonlesson onon twotwo arguments:arguments: First,First, theythey ppresentresent tthehe zzeroero ttopop mmarginalarginal ttaxax rrateate rresult,esult, wwhich,hich, ccombinedombined wwithith ppositiveositive mmarginalarginal ttaxax rratesates bbelowelow tthehe ttop,op, iimpliesmplies tthathat tthehe ttaxax rrateate sshouldhould ddeclineecline aass iitt aapproachespproaches tthehe ttop.op. SSecond,econd, ttheyhey ddiscussiscuss nnumericalumerical ssimulationsimulations uusingsing llog-normalog-normal sskillkill ddistributionsistributions tthathat sshowhow mmodestodest rratesates tthathat ssometimesometimes ddecreaseecrease iinn tthehe uupperpper partpart ofof thethe distribu-distribu- ttion.ion. TTheyhey ddismissismiss tthehe rresultsesults tthathat uusese PParetoareto ddistributionsistributions aandnd wwhichhich oobtainbtain highhigh ttaxax rratesates onon upperupper incomesincomes onon twotwo grounds:grounds: First,First, theythey claimclaim thatthat oneone cannotcannot inferinfer tthehe aabilitybility distributiondistribution withoutwithout makingmaking undulyunduly strongstrong assumptions.assumptions. Second,Second, theythey eexaminexamine thethe rightright tailtail ofof thethe wagewage densitydensity distributiondistribution usingusing CurrentCurrent PopulationPopulation SSurveyurvey (CPS)(CPS) datadata (Mankiw,(Mankiw, Weinzierl,Weinzierl, andand Yagan,Yagan, FigureFigure 1,1, p.p. 154)154) andand concludeconclude tthathat iitt iiss nnotot ppossibleossible ttoo ddistinguishistinguish PParetoareto vversusersus llog-normalog-normal ddistributionsistributions ffromrom ssuchuch ddata.ata. WeWe fi ndnd bothboth ofof tthosehose aargumentsrguments iinvalid—thenvalid—the zerozero atat tthehe ttopop iiss nnotot rrelevantelevant fforor ppolicy,olicy, aass ddiscussediscussed aboveabove andand inin thethe onlineonline appendix,appendix, andand thethe evidenceevidence isis stronglystrongly ssupportiveupportive ooff tthehe PParetoareto ddistribution.istribution. FForor aanyny ddistributionistribution wwithith a tthinnerhinner ttopop ttailail tthanhan tthehe PParetoareto ddistribution,istribution, ssuchuch aass * a log-normallog-normal ddistribution,istribution, tthehe pparameterarameter a = z m//( ( z m – z ) divergesdiverges toto infiinfi nnity.ity. TThehe ttestest ooff PParetoareto vversusersus llognormalognormal rrightight ttailsails ppresentedresented bbyy MMankiw,ankiw, WWeinzierleinzierl aandnd YYaganagan ((2009)2009) iinn ttheirheir FFigureigure 1 llacksacks ppowerower fforor ttwowo rreasons.easons. FFirst,irst, iitt uusesses CCPSPS ddataata tthathat iiss tthinhin aandnd toptop codedcoded inin thethe upperupper partpart ofof thethe distribution.distribution. Indeed,Indeed, theirtheir graphgraph coverscovers a rangerange ooff eearningsarnings ffromrom $$80,00080,000 ttoo $$150,000150,000 ((forfor ffull-timeull-time aandnd ffull-yearull-year iindividualsndividuals wworkingorking 22,000,000 hhoursours pperer yyear).ear). SSecond,econd, iitt pplotslots ddensityensity fi tsts thatthat areare inherentlyinherently imprecise.imprecise. AsAs wwee mmadeade cclearlear iinn tthehe ooptimalptimal ttopop ttaxax rrateate dderivation,erivation, tthehe sstatistictatistic ooff ccentralentral iinterestnterest iiss * a = z m//( ( z m – z ) asas depicteddepicted oonn FFigureigure 2 iinn oourur ppaper.aper. TThishis sstatistictatistic iiss a mmuchuch mmoreore ppreciserecise wayway toto estimateestimate thethe relevantrelevant shapeshape thanthan a densitydensity fi t.t. UsingUsing individualindividual taxtax rreturneturn ddataata wwhichhich hhaveave hhighigh ssamplingampling aatt tthehe ttopop aandnd nnoo ttopop ccoding,oding, wwee sshowhow tthathat tthehe

1 Mankiw, N. Gregory, Matthew C. Weinzierl, and Danny Yagan. 2009. “Optimal Taxation in Theory and Practice.” Journal of Economic Perspectives, 23(4): 147–74. The Case for a Progressive Tax: From Basic Research to Policy Recommendations 189

sstatistictatistic a rremainsemains eextremelyxtremely sstabletable ooverver a vveryery llargearge rrangeange ooff iincomes,ncomes, wwhichhich iiss mmuchuch bbroaderroader tthanhan tthehe rrangeange cconsideredonsidered bbyy MMankiw,ankiw, WWeinzierl,einzierl, aandnd YYagan.agan. FFurthermore,urthermore, asas ourour derivationderivation hashas mademade clear,clear, tthehe ooptimalptimal ttopop rrateate dderivationerivation wwee hhaveave pproposedroposed ddoesoes nnotot rrequireequire uunusuallynusually sstrongtrong aassumptionsssumptions iinn ttermserms ooff hhomo-omo- ggeneityeneity ofof preferencespreferences inin thethe populationpopulation oror functionalfunctional formform assumptions.assumptions. Mankiw,Mankiw, WWeinzierl,einzierl, andand YaganYagan citecite thethe simulationsimulation resultsresults ofof SaezSaez (2001),(2001), whichwhich byby necessitynecessity rrequireequire makingmaking ffunctionalunctional formform aassumptions,ssumptions, bbutut ffailail ttoo nnoteote tthathat tthehe ggeneraleneral ttheo-heo- rreticaletical taxtax raterate formulaformula τ = 11/(1/(1 + ae) isis muchmuch moremore generalgeneral thanthan thethe numericalnumerical iillustration.llustration. TheThe virtuevirtue ofof thethe formulaformula τ = 11/(1/(1 + ae) isis preciselyprecisely thatthat itit dependsdepends oonlynly oonn eestimablestimable ssuffiuffi cientcient statistics.statistics.

LLessonesson 33:: A FFlatlat TTax,ax, withwith a UniversalUniversal Lump-SumLump-Sum TTransfer,ransfer, CCouldould BBee CCloselose ttoo OOptimal:ptimal: MMajorajor ddisagreement.isagreement. TThehe aanalysisnalysis wwee ppresentedresented sshowedhowed tthat,hat, aatt tthehe bbottom,ottom, ttransfersransfers iinitiallynitially iincreasencrease wwithith eearningsarnings toto preservepreserve incentivesincentives toto participateparticipate inin thethe laborlabor forceforce andand thenthen areare pphasedhased ooutut wwithith iincomencome aatt a hhighigh rrate.ate. MMankiw,ankiw, WWeinzierl,einzierl, aandnd YYaganagan ddoo nnotot ddiscussiscuss tthehe pparticipationarticipation margin.margin. AfterAfter transferstransfers areare phasedphased out,out, marginalmarginal taxtax ratesrates shouldshould bbee llowerower fforor tthehe bbroadroad mmiddleiddle cclasslass aandnd tthenhen rriseise iinn tthehe uupperpper iincomencome ggroupsroups ddueue ttoo ddecliningeclining mmarginalarginal wwelfareelfare wweightseights aandnd tthehe PParetoareto sshapehape ooff tthehe iincomencome ddistributionistribution ttowardoward thethe toptop butbut notnot lowerlower down.down. Therefore,Therefore, thethe optimaloptimal systemsystem appearsappears quitequite ddifferentifferent ffromrom tthehe fl atat taxtax withwith a universaluniversal lump-sumlump-sum transfertransfer thatthat theythey advocate.advocate.

LLessonesson 44:: TThehe OOptimalptimal EExtentxtent ooff RRedistributionedistribution RRisesises wwithith WWageage IInequality:nequality: WWee aagree.gree.

LLessonesson 55:: TTaxesaxes SShouldhould DependDepend onon PersonalPersonal CharacteristicsCharacteristics asas WellWell asas Income:Income: SSomeome ddisagreement.isagreement. WWhilehile a modelmodel ignoringignoring bothboth issuesissues ofof complexitycomplexity andand socialsocial acceptabilityacceptability wwouldould reachreach tthishis cconclusiononclusion fforor mmanyany oobservablebservable ccharacteristics,haracteristics, wwee tthinkhink tthathat tthesehese ttwowo issuesissues shouldshould getget duedue respect.respect. InIn practice,practice, taxestaxes andand transferstransfers dependdepend signifisignifi - ccantlyantly onon onlyonly fewfew characteristicscharacteristics (besides(besides income),income), andand thosethose characteristics,characteristics, suchsuch aass ffamilyamily sstructuretructure oorr ddisabilityisability sstatus,tatus, aarere rrelatedelated ttoo nneed.eed.

LLessonesson 66:: OOnlynly FFinalinal GGoodsoods OOughtught ttoo bbee TTaxed,axed, aandnd TTypicallyypically TTheyhey OOughtught ttoo BBee TTaxedaxed UUniformly:niformly: SomeSome ddisagreement.isagreement. LLimitingimiting vvariationariation inin commoditycommodity ((oror VVAT)AT) ttaxesaxes iiss aappropriate,ppropriate, bbutut ssomeome vvaria-aria- ttionion seemsseems wwellell jjustifiustifi ed,ed, althoughalthough tootoo muchmuch variationvariation seemsseems toto bebe presentpresent inin somesome ssystems.ystems. WWee ddisagreeisagree withwith theirtheir inferenceinference thatthat thisthis lineline ofof argumentargument supportssupports notnot ttaxingaxing capitalcapital income;income; inin ffact,act, tthishis ddisagreementisagreement isis a ccentralentral ppartart ooff oourur ppresentation.resentation.

LLessonesson 77:: CCapitalapital IncomeIncome OOughtught ttoo BBee UUntaxed,ntaxed, aatt LLeasteast iinn EExpectation:xpectation: MMajorajor ddisagreement.isagreement. MMankiw,ankiw, WWeinzierl,einzierl, andand YaganYagan invokeinvoke threethree argumentsarguments forfor zerozero capitalcapital iincomencome ttaxes,axes, iincludingncluding twotwo thatthat wewe havehave addressedaddressed inin thethe texttext andand foundfound notnot 190 Journal of Economic Perspectives

ppolicyolicy relevant.relevant. TheyThey claimclaim thatthat thethe Diamond–MirrleesDiamond–Mirrlees aggregateaggregate effieffi ciencyciency resultresult iimpliesmplies tthathat capitalcapital incomeincome shouldshould notnot bebe taxed.taxed. WeWe explainedexplained thatthat thethe theoremtheorem ddoesoes nnotot hhaveave tthishis iimplicationmplication iinn ffootnoteootnote 115.5.

LLessonesson 88:: IInn SStochastictochastic DDynamicynamic EEconomies,conomies, OOptimalptimal TTaxax PPolicyolicy RRequiresequires IIncreasedncreased SSophistication:ophistication: SSomeome ddisagreement.isagreement. WWee agreeagree thatthat stochasticstochastic elementselements callcall forfor moremore sophisticatedsophisticated analysisanalysis andand jjustifyustify moremore sophisticatedsophisticated structures.structures. However,However, wewe disagreedisagree withwith theirtheir emphasisemphasis oonn tthehe ooptimalityptimality ofof a regressiveregressive interactioninteraction betweenbetween capitalcapital incomeincome taxationtaxation andand llaborabor iincome,ncome, aandnd wwherehere capitalcapital incomeincome taxationtaxation raisesraises nono revenuerevenue inin expectationexpectation ((referredreferred toto inin thethe titletitle ofof LessonLesson 7).7). AsAs discusseddiscussed inin footnotefootnote 21,21, theythey focusfocus onon oonene iimplementationmplementation ofof thethe mechanismmechanism designdesign optimumoptimum withwith thisthis propertyproperty butbut iignoregnore thethe presencepresence ofof a differentdifferent implementationimplementation ofof thethe samesame optimumoptimum thatthat hashas ppositiveositive taxationtaxation ofof capitalcapital (Werning,(Werning, 2010).2010). ItIt isis notnot clearclear howhow toto drawdraw inferencesinferences ffromrom ddifferentifferent iimplementationsmplementations ooff a ffullull mmechanismechanism ddesignesign ooptimumptimum wwhenhen llimitedimited ccomplexityomplexity iimpliesmplies tthathat tthehe ffullull ooptimumptimum iiss nnotot bbeingeing iimplemented.mplemented. Journal of Economic Perspectives—Volume 25, Number 4—Fall 2011—Pages 1–6

Online Appendix: Derivations of Optimal Tax Formulas

Derivation of the optimal marginal tax rate at income level z in the Mirrlees model (Figure A1)

Figure A1 depicts the optimal marginal tax rate derivation at income level z. Again, the horizontal axis in Figure A1 shows pre-tax income, while the vertical axis shows disposable income. Consider a situation in which the marginal tax rate is increased by in the band from z to z z, but left unchanged anywhere Δτ + Δ else. The tax reform has three effects. First, the mechanical tax increase, leaving aside behavioral responses, will be the gap between the solid and dashed lines, shown by the vertical arrow equal to z. The total mechanical tax increase is Δτ Δ M z[1 – H(z)] as 1 – H(z) is the fraction of individuals above z. Second, this Δ = Δτ Δ tax increase creates a social welfare cost of W – z[1 – H(z)]G(z) as G(z) is Δ = Δτ Δ defined as the average social marginal welfare weight for individuals with income above z. Third, there is a behavioral response to the tax change. Those in the income range from z to z z have a behavioral response to the higher marginal tax rates, + Δ shown by the horizontal line pointing left. Assuming away income effects, this is the only behavioral response; those with income levels above z z face no change in + Δ marginal tax rates and hence have no behavioral response. The h(z) z taxpayers Δ in the band reduce their income by z – ez/(1 – T (z)) where e is the elasticity δ = Δτ ′ of earnings z with respect to the net-of-tax rate 1 – T . This response leads to a tax ′ loss equal to B –h(z)ez T (z)/(1– T (z)) z . At the optimum, the three effects Δ = ′ ′ Δ Δτ cancel out so that M W B 0. After introducing (z) zh(z)/(1 – H(z)), Δ + Δ + Δ = α = this leads to the optimal tax formula presented in the main text:

A1) T (z)/(1 – T (z)) (1/e)(1 – G(z))(1 – H(z))/(zh(z)), ′ ′ = or T (z) [1 – G(z)]/[1 – G(z) (z)e]. ′ = + α 2 Journal of Economic Perspectives

Figure A1 Derivation of the Optimal Marginal Tax Rate at Income Level z in the Mirrlees Model

Disposable income c z –T(z) Small band (z, z z): slope 1 – T (z) = + ∆ ′ Reform: slope 1 – T (z) – ′ ∆τ Mechanical tax increase: z [1 – H(z)] ∆τ∆ Social welfare effect: – z [1 – H(z)]G(z) ∆τ∆

z ∆τ∆

Behavioral response: z – ez/(1 – T (z)) δ = ∆τ ′ Tax loss: T (z) zh(z) z ′ δ ∆ – h(z)ezT (z)/(1 – T (z)) z = ′ ′ ∆ ∆τ

0 z z z Pre-tax income z +∆ Notes: The figure depicts the optimal marginal tax rate derivation at income level z by considering a small reform around the optimum, whereby the marginal tax rate in the small band (z, z z) is increased + Δ by . This reform mechanically increases taxes by z for all taxpayers above the small band, Δτ Δτ Δ leading to a mechanical tax increase z[1 – H(z)] and a social welfare cost of – z[1 – H(z)]G(z). Δτ Δ Δτ Δ Assuming away income effects, the only behavioral response is a substitution effect in the small band: the h(z) z taxpayers in the band reduce their income by z – ez/(1 – T (z)) leading to a Δ δ = Δτ ′ tax loss equal to –h(z)ez T (z)/(1 – T (z)) z . At the optimum, the three effects cancel out leading ′ ′ Δ Δτ to the optimal tax formula T (z)/(1 – T (z)) (1/e)(1 – G(z))(1 – H(z))/(zh(z)), or equivalently T (z) ′ ′ = ′ [1 – G(z)]/[1 – G(z) (z)e] after introducing (z) zh(z)/(1 – H(z)). = + α α =

Derivation of the optimal marginal tax rate at the bottom in the Mirrlees model (Figure A2)

For expositional simplicity, let us consider a discrete version of the Mirrlees (1971) model developed in Piketty (1997) and Saez (2002). As illustrated on Figure A2, suppose that low-ability individuals can choose either to work and earn z1 or not work and earn zero. The government offers a transfer c to those not working phased out at rate so that those working receive 0 τ1 on net c (1 – )z c . In words, nonworkers would keep a fraction 1 – of 1 = τ1 1 + 0 τ1 their earnings should they work and earn z . Therefore, increasing discour- 1 τ1 ages some low­-income workers from working. Let us denote by H0 the fraction of nonworkers in the economy and by e –(1 – )/H H / (1 – ) the elasticity 0 = τ1 0 Δ 0 Δ τ1 Peter Diamond and Emmanuel Saez 3

Figure A2 Optimal Bottom Marginal Tax Rate with Only Intensive Labor Supply Responses

Disposable

income Reform: Increase 1 by 1 and c0 by c0 z1 1 c τ ∆τ ∆ = ∆τ 1) Mechanical scal cost: – H c – H z ∆M = 0 ∆ 0 = 0 1∆τ 1 2) Welfare effect: W g H c g H z ∆ = 0 0 ∆ 0 = 0 0 1 ∆τ1 3) Fiscal cost due to behavioral responses: B – H z – e H /(1 – )z ∆ = ∆ 0 τ1 1 = ∆τ1 0 0 τ1 τ1 1

c c 0 + ∆ 0 Optimal phase-out rate : τ1 M W B 0 ∆ + ∆ + ∆ = c0 1/(1 – 1) (g0 – 1)/e0 Slope 1 – τ τ = τ1

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0 z1 Earnings z

Notes: The figure depicts the derivation of the optimal marginal tax rate at the bottom in the discrete Mirrlees (1971) model with labor supply responses along the intensive margin only. Let H0 be the fraction of the population not working. This is a function of 1 – , the net-of-tax rate at the bottom, τ1 with elasticity e0. We consider a small reform around the optimum where the government increases the maximum transfer by c by increasing the phase-out rate by leaving the tax schedule unchanged Δ 0 Δτ1 for those with income above z1, this creates three effects which cancel out at the optimum. At the optimum, we have /(1 – ) (g – 1)/e or (g – 1)/(g – 1 e ). Under standard redistributive τ1 τ1 = 0 0 τ1 = 0 0 + 0 preferences, g is large implying that is large. 0 τ1 of nonworkers H with respect to the net-of-tax rate 1– , where the minus sign is 0 τ1 used so that e 0. 0 > Suppose now that the government increases both the maximum transfer by c and the phase-out rate by leaving the tax schedule unchanged for those Δ 0 Δτ1 with income equal to or above z so that c z as depicted on Figure A2. The 1 Δ 0 = 1 Δτ1 fiscal cost is –H c but the welfare benefit isH g c where g is the social welfare 0Δ 0, 0 0 Δ 0 0 weight on nonworkers. If the government values redistribution, then g 1 and g 0 > 0 is potentially large as nonworkers are the most disadvantaged. Because behavioral responses take place along the intensive margin only in the Mirrlees model, with no income change above z1, the labor supply of those above z1 is not affected by the reform. By definition ofe , a number H e H /(1 – ) of low-income 0 Δ 0 = Δτ1 0 0 τ1 4 Journal of Economic Perspectives

workers stop working, creating a revenue loss of z H c H e /(1 – ). τ1 1 Δ 0 = Δ 0 0 0 τ1 τ1 At the optimum, the three effects sum to zero leading to the optimal bottom rate formula:

A2) /(1 – ) (g – 1)/e τ1 τ1 = 0 0 or (g – 1)/(g – 1 e ). τ1 = 0 0 + 0

Because g is large, will also be large. For example, if g 3 and e 0.5 (an 0 τ1 0 = 0 = elasticity in the mid range of empirical estimates), then 2/2.5 80 percent— τ1 = = a very high phase-out rate. Formula (A2) is the optimal marginal tax rate at zero earnings in the standard Mirrlees (1971) model when there is a fraction of indi- viduals who do not work, which is the most realistic case (this result does not seem to have been noticed in the literature). As is well known since Seade (1977), the optimal bottom tax rate is zero when everybody works and bottom earnings are strictly positive, but this case is not practically relevant.

Derivation of the optimal bottom marginal tax rate with extensive labor supply responses (Figure A3)

Consider now a model where behavioral responses of low- and mid-income earners take place through the extensive elasticity only—i.e., whether or not to work—and that earnings when working do not respond to marginal tax rates. As depicted on Figure A3, suppose the government starts from a transfer scheme with a positive phase-out rate and introduces an additional small in-work benefit τ1 c that increases net transfers to low-income workers earning z . Let h be the Δ 1 1 1 fraction of low-income workers with earnings z1. Let us denote by e1 the elasticity of h with respect to the participation net-of-tax rate 1 – , so that e (1 – 1 τ1 1 = )/h h / (1 – ). The reform has again three effects. First, the reform has a τ1 1Δ 1 Δ τ1 mechanical fiscal cost M –h c for the government. Second, it generates a Δ = 1 Δ 1 social welfare gain, W g h c where g is the marginal social welfare weight on Δ = 1 1 Δ 1 1 low-income workers with earnings z1. Third, there is a tax revenue gain due to behav- ioral responses B z h e /(1 – )h c . If g 1, then W M 0. Δ = τ1 1Δ 1 = 1 τ1 τ1 1 Δ 1 1 > Δ + Δ > In that case, if 0, then B 0 implying that 0 cannot be optimal. The τ1 > Δ > τ1 > optimal is such that M W B 0 implying that τ1 Δ + Δ + Δ =

A3) /(1 – ) (1 – g )/e τ1 τ1 = 1 1 or

(1 – g )/(1 – g e ). τ1 = 1 1 + 1 Peter Diamond and Emmanuel Saez 5

Figure A3 Optimal Bottom Marginal Tax Rate with Extensive Labor Supply Responses

Starting from a positive phase-out rate 0: Disposable τ1 > income 1) Increasing tranfers by c at z is desirable for redistribution: ∆ 1 1 c net effect (g – 1)h c 0 if g 1 1 1∆ 1 > 1 > 2) Participation response saves government revenue z h e /(1 – )h c 0 τ1 1 ∆ 1 = 1τ1 τ1 1∆ 1 > Win-win reform . . . if intensive response is small

c c 1 + ∆ 1 c1

c0 Optimal phase-out rate : τ1 (g – 1)h c e /(1 – )h c 0 1 1∆ 1 + 1τ1 τ1 1∆ 1 = Slope 1 – τ1 /(1 – ) (1 – g )/e 0 if g 1 τ1 τ1 = 1 1 < 1 >

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0 z1 z2 Earnings z

Notes: The figure depicts the derivation of the optimal marginal tax rate at the bottom in the discrete model with labor supply responses along the extensive margin only. Starting with a positive phase-out rate 0, the government introduces a small in-work benefit c . Let h be the fraction of low-income τ1 > Δ 1 1 workers with earnings z1, and let e1 be the elasticity of h1 with respect to the participation net-of-tax rate 1 – . The reform has three standard effects: mechanical fiscal cost M –h c , social welfare τ1 Δ = 1 Δ 1 gain, W g h c , and tax revenue gain due to behavioral responses B z h e ( /(1 – )) Δ = 1 1Δ 1 Δ = τ1 1 Δ 1 = 1 τ1 τ1 × h c . If g 1, then W M 0. If 0, then B 0 implying that 0 cannot be optimal. 1 Δ 1 1 > Δ + Δ > τ1 > Δ > τ1 > The optimal is such that M W B 0 implying that /(1 – ) (1 – g )/e . τ1 Δ + Δ + Δ = τ1 τ1 = 1 1

Derivation of the optimal asymptotic marginal tax rate with a random finite population

Actual distributions are both bounded and with a finite population that becomes progressively sparser in the upper tail. Moreover, the government does not know the exact realization of the earnings distribution when setting tax policy. Hence, the (known) bounded and finite model of optimal taxation does not seem practically useful for thinking about tax rates on top earners. A more realistic scenario is that the government knows the distribution of real- ized earnings (conditional on a given tax policy). A simple way to model this is to assume that individual skills n are drawn from a known Pareto distribution that is unbounded and with density f(n). Any finite draw will generate a distribution that 6 Journal of Economic Perspectives

is both bounded and finite. We retain the key assumption that individuals know their skill n before making their labor supply decision. As the government does not know the exact draw ex ante, a natural objective of the government is to maximize expected social welfare SWF G(u(n))f(n)dn. = ∫ The budget constraint for any particular draw is not met with an ex ante tax func- tion. However, if the population is large enough, the actual budget is close to the expected budget as long as the share of income accruing to the very top earners is not too large. Hence, it is natural to assume that the budget constraint needs to hold in expectation T(z)f(n)dn 0. Small fluctuations in debt from repeated ∫ ≥ realizations over time would justify use of a similar approximation. This replace- ment of the actual budget constraint by the expected budget constraint is the key point that generalizes the Mirrlees (1971) model to finite populations. Therefore, we are exactly back to the Mirrlees (1971) model, and hence the optimal tax system is given by the standard formulas. This in particular implies that the optimal top tax rate ​*​ ​ 1/(1 ae) τ = + continues to apply with a the Pareto parameter of the expected earnings distribu- tion. More concretely and coming back to our derivation presented in the text, recall that the optimal constant tax rate above ​z*​ ​ is given by formula ​ *​ ​ 1/(1 ae) τ = + with a z /z​*​ /(​ z /z​*​ ​ – 1) and z is the average income above ​z*​ ​. Obviously, z (and = m m m m hence a) are not defined if ​z*​ ​ is above the actual realized top. However, if the actual finite draw is unknown to the government when ​ *​ ​ is set, the government should τ * * * naturally replace zm/z​​ ​ by Ezm/z​​ ​, i.e., the expected average income above ​z​ ​ divided by z​​*​. Given the very close fit of the Pareto distribution up to the very top of the distribution (something that can actually be verified with actual rich lists that have been compiled by the press in a number of cases), the natural assumption is that * Ezm/z​​ ​ never converges to one.

Appendix Reference

Piketty, Thomas. 1997. “La Redistribution Fiscale face au Chomage.” Revue Francaise d’Economie, 12(1): 157–201.