On the Optmal Margnal Rate of Income Tax Gareth D Myles Insttute for Fscal Stues an Unversty of Exeter June 999 Abstract: The paper shows that n the quas-lnear moel of ncome taxaton, the optmal margnal rate of tax can be calculate wthout neeng to specfy the utlty of consumpton Ths result s use to nvestgate the qualtatve behavor of the margnal rate It s shown that every possble qualtatve outcome may be acheve by approprate selecton of the skll strbuton Therefore the moel oes not place any a pror restrctons on the behavor of the margnal rate an the constancy of fnngs n prevous smulaton analyss s a consequence of ther restrcte structures Keywors: Income taxaton, optmal, margnal rate JEL Classfcaton: 022, 323 Acknowlegements: Thanks are ue to Frank Page, John Weymark an an anonymous referee Corresponence: Department of Economcs, Unversty of Exeter, Exeter, EX4 4RJ, UK, emal GDMyles@exacuk, phone 0392 264487, fax 0392 263242
Introucton The analyss of nonlnear ncome taxaton poneere by Mrrlees (97) has characterze a number of propertes that the optmal tax must possess (see eg Myles (995) for a survey of these) The theoretcal results though o not answer all the questons that are rase about ncome taxaton The most hotly-ebate practcal ssue s the behavor of the margnal rate of tax, n partcular whether the optmal ncome tax shoul be progressve - a property that the tax systems of all evelope countres possess The theory has so far not fully resolve ths queston It s wellknown that the margnal tax rate shoul be zero for the hghest skll consumer, so the tax functon cannot be progressve everywhere But ths en-pont results proves no nformaton on the behavor of the tax scheule on the nteror of the skll strbuton Ths s a sgnfcant gap n our knowlege Gven the relatve paucty of theoretcal results, numercal smulatons have been use n orer to prove nsght nto the overall structure of the tax functon These have combne a log-normal strbuton of skll wth ether Cobb-Douglas (Mrrlees (97)) or CES utlty (Kanbur an Tuomala (994)) These specfcatons generate what wll be calle here the classcal optmal tax functon: as a functon of skll, the margnal rate of tax ether frst rses an then falls or s hghest ntally an then falls 2 Whch behavor obtans epens on the egree of equty n socal welfare an the stanar evaton of the skll strbuton (see Kanbur an Tuomala (994)) These qualtatve propertes of the tax functon have remane consstent throughout all smulaton results that have been reporte Such consstency woul be re-assurng, an suggestve that the ncome tax shoul always have these propertes, were t not for the very narrow range of specfcatons that have been use to generate the results For these results to have valuable polcy mplcatons t must be shown that they are robust to changes n specfcaton The reason that the range of specfcatons s so lmte can probably be foun n the computatonal ffcultes nvolve n solvng the moel These are not nsurmountable, but o suggest that an alternatve approach woul be better Consequently, ths paper aopts the approach of pursung a computatonally smpler moel 3 Ths s one by explotng an nterestng property of the quas-lnear moel of Weymark (986a, b 987): the optmal margnal rate of tax facng each consumer can For nstance the practcal relevance of the zero enpont result has been assesse by conserng ts mplcatons for consumers close to the top of the skll strbuton 2 Mrrless also conclue that the margnal tax rate was farly constant but other smulatons have snce sprove ths 3 An alternatve approach s followe n Saez (999)
2 be foun explctly as functon of consumers sklls an s nepenent of the utltyof-consumpton functon Ths makes t computatonally smple to assess the effect of varyng the skll strbuton upon the optmal margnal rates What the analyss shows s the followng result: any qualtatve structure for the optmal tax functon can be supporte by some skll strbuton Expresse alternatvely, except for the fact that the margnal rate cannot rse between the secon to hghest an hghest skll consumers, there are no a pror restrctons on the qualtatve propertes of the optmal tax functon So the structure of the classcal optmal tax functon s just a consequence of the restrcte set of smulaton specfcatons an oes not capture some eeper feature of optmal taxaton The moel use here assumes utlty s lnear n labor supply Damon (998) has alreay explote a lnear-n-consumpton moel to show tax rates may ncrease above the moal ncome for some skll strbutons It shoul be stresse that although the results are evelope here n the context of optmal ncome taxaton, they are also applcable to nonlnear ncentve schemes n general The same propertes of the quas-lnear moel can easly be explote n other contexts to prove smlar nsghts 2 Quas-lnearty an margnal tax rates Ths secton brefly ntrouces the moel of ncome taxaton wth quas-lnear utlty Lollver an Rochet (983) apple ths to a moel wth a contnuum of consumers The moel wth a fnte numbers of consumers on whch ths paper s base s analyze n etal n Weymark (986a, b 987) The bass of the moel s that utlty s quas-lnear n labor supply so U z ( x ) = u( x ) = u!, () s where x s consumpton of consumer, z s pre-tax ncome an s s the level of skll The margnal rate of substtuton for ( MRS ) s equal to u' s, so that t s consstent wth the requrements of agent monotoncty Wth a weghte utltaran welfare functon an a tax polcy that s purely restrbutve 4, the choce of an optmal tax functon s equvalent to the government choosng an allocaton {x, z } for each consumer =,", to solve the followng program 4 Ths assumpton s not necessary
3 Program : { } u( x ) x, z z max µ, = s subject to: () x = z = = ' () u( x ) u( x ), all, ', ' z z, s s where () s the buget constrant an () the ncentve compatblty constrants The general soluton for the contnuum verson of ths moel s gven n Lollver an Rochet (983) They prove a characterzaton of the optmal consumpton functon from whch the tax rate coul be nferre However t s more rect to work from the results of Weymark (986a, b) for the fnte case These show that Program s equvalent to max β x, Program 2: { } u( x ) x = where [ ] s = + + : λh s s = µ β, λ : =, an h= s no bunchng, the soluton to Program 2 s escrbe by ( x ) =, =, + s s an arbtrary number Wth β u' ", (2) As Weymark (986b) note, the tax functon s knke at the locaton of each consumer, so the margnal tax rate s not formally efne at these ponts However, t s possble to take the graent of the nfference curve as etermnng an mplct margnal rate of tax Dong ths, the margnal tax rate facng consumer ( MTR ) s MTR β : = MRS =, (3) s usng (2) Hence the margnal tax rate s efne by the skll strbuton an s nepenent of the utlty of consumpton In unertakng the calculatons reporte n Secton 3 the normalzaton of Weymark (986a, b) s aopte so = λ = For a weghte utltaran socal welfare functon, the weghts µ can be selecte to ensure that ths oes not mpose any atonal restrcton However, an unweghte utltaran socal welfare functon s use n the calculatons wth µ = for all The jont effect of ths choce an the
4 normalzaton s to place an atonal restrcton on the permssble strbutons of ablty Relaxng ths woul smply make t easer to fn ablty strbutons that generate the requre patterns of tax rates The reason for aoptng the normalzaton s to permt rect use of the necessary an suffcent conton for no-bunchng n Theorem 2 of Weymark (986b) Ths s the requrement that # β < β 2 <" < β an β 2 > 0 The results reporte n Table are for skll strbutons that satsfy ths conton whch justfes the use of the necessary conton n (2) 3 umercal results The numercal results reporte n ths secton are constructe for a fve-consumer economy The number of consumers was chosen as the mnmum necessary to exhbt a suffcently nterestng set of results Before escrbng the results n etal, t s worth scussng what these are amng to acheve The am of the paper, as note n the Introucton, s to nvestgate the qualtatve behavor of the margnal rate of tax In ths respect, after rankng consumers by ncome, a progressve tax system woul have a margnal rate whch ncrease from one consumer to the next More generally, startng wth the lowest ncome consumer (who s also the lowest skll va ncentve compatblty), qualtatvely the margnal tax rate can be ether hgher or lower for the next skll level The same s true n passng from the secon skll level to the thr Wth fve consumers an the fact that the margnal rate s zero at the top, eght possble qualtatve patterns can arse These are llustrate n Fgures an 2 where enotes an ncrease n the tax rate between consumers an enotes a ecrease (,,) (,,) Drecton of tax change (,,) (,,) (,,) (,,) (,,) (,,) Consumer Fgure : Changes n Tax Rate
5 Tax rate Tax rate Tax rate Tax rate 2 3 4 Tax rate Tax rate Tax rate Tax rate 5 6 7 8 Fgure 2: Qualtatve Patterns Out of the eght possbltes, exstng smulatons have foun the patterns 6, 7 an 8 - these are the classcal optmal tax functons escrbe n the Introucton Ths woul be a clear gue to polcy f t was correct that these were the only qualtatve forms of tax functon that coul arse In fact, what s now shown s that patterns - 5 can also arse Consequently, snce the moel can generate every qualtatve form of tax functon, no a pror restrctons can be place on t whatsoever Expresse alternatvely, the fact that only the classcal form has emerge n prevous smulatons s not a reflecton of somethng eeper but just a consequence of the assumptons Table reports the optmal margnal tax rates for fve fferent strbutons of ablty In each case, the skll level of consumer 5 s mple by the other four va the normalzaton rule It shoul be note that the relaxaton of the assumpton of a utltaran socal welfare functon coul only serve to enhance the conclusons As alreay note, these results show that any pattern of optmal tax rates that s theoretcally possble can be acheve for some strbuton of ablty The moel oes not place any restrcton on the pattern that emerges except that, snce the hgh ablty faces a zero rate an the rate must be nonnegatve, t cannot rse n gong from the secon hghest to hghest ablty
6 s s 2 s 3 s 4 s 5,, 2,, 3,, 4,, 5,, Ablty 65 75 4 242 Margnal tax rate (%) 08 29 35 425 0 Ablty 8 82 83 24 Margnal tax rate (%) 0063 0059 29 689 0 Ablty 9 92 93 22 Margnal tax rate (%) 0025 002 05 02 0 Ablty 65 75 05 2 292 Margnal tax rate (%) 08 35 2 94 0 Ablty 85 9 94 95 64 Margnal tax rate (%) 02 009 004 27 0 Table : Optmal Margnal Tax Rates 4 Conclusons The paper has consere at the qualtatve propertes of the optmal margnal rate of ncome taxaton base on the observaton that quas-lnearty of utlty allows ths to be etermne wthout specfyng the utlty of consumpton Lnke wth the fact that the number of consumers s fnte, ths allows a break from the restrctve set of strbutons an preferences aopte n prevous smulatons The results emonstrate that the moel s capable of generatng all qualtatve patterns of margnal tax rates The contnual emergence of the classcal tax functon n prevous stues can therefore be seen as just an artfact of ther restrctve structure The paper use a utltaran socal welfare functon but a more general socal welfare functon woul smply a another egree of freeom an make t easer to construct examples wth the propertes requre In concluson, the nference from prevous smulatons that the optmal tax functon may belong to a narrow class has been shown to be nval The moel oes not restrct the qualtatve structure n any way beyon that establshe n the exstng theoretcal results It certanly oes not prove any a pror restrcton on the structure of margnal rates snce these can behave n any way over the populaton
7 The man message of the paper must be that t smulatons are to prove any gue to polcy, they must be base on real ata - there are no general propertes watng to be scovere usng artfcal ata References Damon, PA (998) Optmal ncome taxaton: an example wth a U-shape pattern of optmal margnal tax rates, Amercan Economc Revew, 88, 83 95 Kanbur, SMR an M Tuomala (994) Inherent nequalty an the optmal grauaton of margnal tax rates, Scannavan Journal of Economcs, 96, 275-282 Lollver, S an J-C Rochet (983) Bunchng an secon-orer contons: a note on optmal tax theory Journal of Economc Theory, 3, 392 400 Mrrlees, JA (97) An exploraton n the theory of optmum ncome tax, Revew of Economc Stues, 38, 75-208 Myles, GD (995) Publc Economcs, Cambrge Unversty Press: Cambrge Saez, E (999) Usng elastctes to erve optmal ncome tax rules (Mmeo, MIT) Weymark, JA (986a) A reuce-form optmal ncome tax problem, Journal of Publc Economcs, 30, 99-27 Weymark, JA (986b) Bunchng propertes of optmal nonlnear ncome tax problems, Socal Choce an Welfare, 3, 23-232 Weymark, JA (987) Comparatve statcs propertes of optmal nonlnear ncome taxes, Econometrca, 55, 65-85