THE THEORY OF OPTIMAL TAXATION: NEW DEVELOPMENTS AND POLICY RELEVANCE

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1 Aprl 2009 THE THEORY OF OPTIMAL TAXATION: NEW DEVELOPMENTS AND POLICY RELEVANCE by Peter Brch Sørensen Abstract: The theory of optmal taxaton has often been crtczed for beng of lttle practcal polcy relevance, due to a lack of robust theoretcal results. Ths paper argues that recent advances n optmal tax theory has made that theory easer to apply and may help to explan some current trends n nternatonal tax polcy. Coverng the taxaton of labour ncome and captal ncome as well as ndrect taxaton, the paper also llustrates how some of the key results n optmal tax theory may be derved n a smple, heurstc manner. Keywords: Optmal taxaton, unform taxaton, tax neutralty JEL classfcaton: H2, H24, H25 Address for correspondence: Peter Brch Sørensen Department of Economcs, Unversty of Copenhagen Studestraede 6, 455 Copenhagen K, Denmark E-mal: pbs@econ.ku.dk

2 THE THEORY OF OPTIMAL TAXATION: NEW DEVELOPMENTS AND POLICY RELEVANCE Peter Brch Sørensen Unversty of Copenhagen, EPRU and CESfo In the tradton establshed by the classcal poltcal economsts, normatve analyss of tax polcy tended to follow a prncples-orented approach accordng to whch a good tax system should satsfy certan desrable crtera. For example, Lord Overstone, who served as Presdent of Brtan s Royal Statstcal Socety from 85 to 853, thought that a tax should be productve, computable, dvsble, frugal, non-nterferent, unannoyant, equal, popular, and uncorruptve (see the dscusson by O Bren, 2009). The classcal economsts rarely dscussed the trade-offs between the varous goals of tax polcy. In partcular, they dd not pay much attenton to the trade-off between redstrbuton and economc effcency, snce they typcally ruled out redstrbutve progressve taxaton as a matter of prncple, seeng t as a fundamental threat to property rghts. The denouncement of any devaton from proportonal taxaton was vvdly expressed by McCulloch (845) who argued that The moment you abandon the cardnal prncple of exactng from all ndvduals the same proporton of ther ncome or of ther property, you are at sea wthout rudder or compass, and there s no amount of nustce and folly you may not commt (see Creedy, 2009, p. 2). Followng the neoclasscal margnalst revoluton n economc theory, Edgeworth (897) argued that taxaton should nvolve an equal margnal sacrfce (of utlty) for each ndvdual taxpayer n order to mnmse the aggregate utlty loss mposed on taxpayers. When combned wth the neoclasscal assumpton of declnng margnal utlty of ncome, ths utltaran prncple of equal margnal (.e. mnmum total) sacrfce dd provde a ratonale for progressve ncome taxaton. Ths paper s a revsed and extended verson of an nvted plenary lecture gven at the 20 th Scentfc Meetng of the Socetà talana d economa pubblca (SIEP) n Pava on September, Wthout mplcatng her n any remanng shortcomngs, I thank my dscussant Lsa Grazzn for her comments on my lecture. I am also grateful to Claus Thustrup Krener and Etenne Lehmann for valuable comments on an earler draft of the paper. 2

3 Edgeworth was aware that redstrbutve taxaton nvolves a trade-off between equty and effcency, but the development of a rgorous coherent framework for analysng ths trade-off had to awat the semnal work by Mrrlees (97). However, much of the optmal tax lterature buldng on Mrrleess contrbuton has been hghly techncal and abstract, and for many years ths body of theory seemed to offer few robust results. For these reasons many polcy makers have tended to see the theory of optmal taxaton as beng of lttle practcal relevance. In ths paper I shall argue that recent advances n optmal tax theory has made that theory easer to apply and may help to explan some current trends n nternatonal tax polcy. I shall also llustrate how some of the key results n optmal tax theory may be derved n a smple, heurstc manner. The frst part of the paper deals wth the theory of optmal taxaton of labour ncome. In Part 2 I focus on optmal ndrect taxaton, whle Part 3 dscusses the optmal taxaton of ncome from captal. Part 4 summarses the man conclusons.. Optmal taxaton of labour ncome The Mrrlees model In the canoncal model of optmal ncome taxaton set up by Mrrlees (97) consumers are assumed to maxmse a utlty functon of the general form subect to the budget constrant U U C, L, () C wl TwL, (2) where C s consumpton, L s labour supply, w s the real wage, and T(wL) s a non-lnear taxtransfer schedule. The soluton to the consumer s problem yelds hs ndrect utlty functon V(w). In the Mrrlees-model the pre-tax real wage rates are treated as exogenous and taken to reflect the dfferent non-observable ablty levels of ndvdual taxpayers. Wth wage rates beng dstrbuted over the nterval ww,, 0 ww, Mrrlees assumed that the benevolent polcy maker wshes to maxmse an ndvdualstc Bergson-Samuelson welfare functon of the form w w, ' 0, '' 0, W V w f w dw (3) 3

4 Here f w ndcates the densty of taxpayers earnng the wage rate w, and the (numercal) magntude of the second dervatve '' reflects the strength of the polcy maker s preference for equty. The maxmsaton of (3) takes place subect to the constrant that the government must rase an exogenous amount of revenue R: w w. T wl w f w dw R (4) The soluton to the above optmal tax problem s techncally demandng and does not yeld very clear-cut results regardng the shape of the optmal ncome tax schedule. To get a feel for the lkely shape of ths schedule, Mrrlees carred out smulatons assumng Cobb-Douglas utlty functons, a classcal utltaran socal welfare functon (wth ' and '' 0 ) and a log-normal wage dstrbuton. On these assumptons he found that the optmal tax schedule was approxmately lnear, wth an exempton level below whch postve net transfers are payable. Had ths early result been robust, t would have had great practcal polcy relevance, snce a lnear labour ncome tax s farly smple to admnster. In partcular, because a lnear ncome tax features a constant margnal tax rate, t does not requre nformaton on ndvdual ncomes, snce t can be mplemented as a proportonal payroll tax combned wth a flat transfer to all taxpayers. However, subsequent work by Tuomala (984) and others revealed that the nearoptmalty of a lnear ncome tax s not a robust result once one allows for plausble respecfcatons of utlty functons and of the shape of the wage dstrbuton. Atknson and Stgltz (980, ch. 3) also found that the optmal tax schedule devates substantally from lnearty when the socal planner has more egaltaran preferences than those mpled by classcal utltaransm. Wth these dscouragng fndngs, t seemed for a whle that optmal tax theory could offer lttle gudance on ncome tax desgn. But buldng on earler contrbutons by Revesz (989), Pketty (997), Damond (998), and Roberts (2000), Saez (200, 2002a) showed how a formula for the optmal margnal tax rate at every ncome level can be derved n terms of the relevant elastctes of taxable ncome and the propertes of the wage dstrbuton. Snce these parameters can n prncple be observed or estmated emprcally, the work of Saez has greatly enhanced the practcal applcablty of optmal ncome tax theory. Another mportant contrbuton by Saez (2002a) was the explct allowance for tax dstortons to the extensve as well as the ntensve margn of labour supply. Ths s hghly relevant snce many emprcal studes have shown that labour supply s often more elastc on the extensve margn (where workers decde whether or not to 4

5 partcpate n the labour market) than on the ntensve margn (where people make decsons on ther hours of work, gven that they are already employed). The Saez (2002a) model Gven ts mportance, t s worth restatng the key contrbuton of Saez (2002a) n a heurstc, ntutve manner. 2 Followng Pketty (997), Saez assumes that employed workers allocate themselves across a range of dfferent occupatons nvolvng dfferent levels of effort and ncome. Workers can decde to partcpate or to stay outsde the labour market. In the latter case they receve the publc transfer S 0, whereas a worker employed n ob category earns the after-tax ncome c. As the net gan c S0 from employment n occupaton ncreases, more workers move from non-employment nto ths occupaton. Ths s the extensve labour supply response. Once employed, workers can move one step up or down the ob ladder by adustng ther effort. In case of an ncrease n the gap c c between the net ncomes obtanable n ob categores + and, respectvely, some workers prevously employed n occupaton wll therefore be nduced to upgrade themselves to the hgherearnng occupaton + by ncreasng ther effort. Smlarly, an ncrease n the net earnngs dfferental c c wll spur some workers to upgrade themselves from ob category - to category. These movements between ob categores nvolvng dfferent earnngs levels represent the ntensve labour supply response. Wth these assumptons, the fracton h of the work force employed n occupaton can be specfed as where c z T z 0 h h c S, c c, c c, (5) s the dfference between the pre-tax earnngs z n occupaton and the net tax Tz payable on that ncome. By assumng that labour supply decsons depend only on the dfferences between the net ncomes obtanable n dfferent labour market states, equaton (5) mplctly abstracts from ncome effects (snce otherwse labour supply would also depend on the level of ncome). In recent years most emprcal studes have ndeed tended to fnd rather small ncome effects on ndvdual labour supply. Thus there s a good case for consderng the benchmark stuaton wth zero ncome effects. From (5) we can defne the partcpaton elastcty 2 The present heurstc exposton follows the perturbaton method adopted n the man text of Saez s artcle but s slghtly more elaborate. The rgorous formal proofs of Saez s results may be found n the appendx to hs paper. 5

6 and the ntensve labour supply elastcty h c S c S h 0 h c c c c h 0,. (6) (7) As shown by Saez (2002a, p. 070), z zz z T' z, where T' z s the elastcty of z taxable ncome wth respect to one mnus the margnal tax rate, estmated n numerous recent emprcal studes. Optmal taxaton n the Saez model Consder now the welfare effects of ncreasng the tax lablty by the (small) amount dt for all of the occupatons,, 2,... up to the hghest-earnng occupaton J. Measured n money metrc unts, the mpact on the welfare of an ndvdual n one of these occupatons equals the drect mpact on dsposable ncome, dt, snce any change n dsposable ncome stemmng from a change n labour supply has no frst-order effect welfare, assumng that the taxpayer has optmsed hs/her labour supply n the ntal equlbrum (ths s ust an applcaton of the Envelope Theorem). Suppose now that the polcy maker s evaluaton of the margnal socal utlty of net ncome for ndvduals n occupaton s gc. The socal evaluaton of the mpact of the tax ncrease on aggregate prvate welfare s then gven by J. dw g c h dt (8) Aganst ths prvate welfare loss one must balance the rse n publc revenue. Abstractng from behavoural changes, the drect ( mechancal ) revenue gan s dm J h dt. (9) However, publc revenue also changes because of the tax-nduced changes n labour supply. The tax hke reduces the net ncome dfferental c c z T z T by the amount dt, whereas t has no mpact on the ncome dfferental c c between any other two neghbourng occupatons. Wth dh denotng the change n h resultng from the ntensve labour supply response, the revenue loss mpled by the change n behavour on the ntensve margn ( db ) s 6

7 hdt T T c c db dh T T where the last equalty n (0) follows from (7). The tax hke also reduces the net ncome gan c S by the amount dt for all persons n ob category and above. If 0, (0) e dh ndcates the change n h stemmng from labour supply responses on the extensve margn, the drop n revenue occurrng as a result of lower labour force partcpaton becomes J J ht S e 0 0, (2) e db dh T S dt c where we have used the defnton (6) to arrve at the last equalty. Under the optmal tax polcy, the tax burden on any occupaton from and upwards s ncreased up to the pont where the socal valuaton of the resultng prvate utlty loss s ust matched by the government s net revenue gan,.e. untl e 0. Insertng (8) through (2) nto (3), we get the optmal tax rule S dw dm db db (3) J T T T S 0 h g. (4) c c h c S 0 0 The margnal tax rate at the ncome level z may be defned as from whch t follows that T T m z z, (5) z T z T c c m T T m. z z z z m c c Further, the partcpaton tax rate measurng the ncrease n net taxes mposed when a person moves from non-employment to nto employment may be defned as t T S (6) 0, (7) z mplyng that T S0 T S0 t c S z T S t 0 0. (8) 7

8 Substtutng (6) and (8) nto (4), we may wrte the optmal tax rule as J m t h g. (9) m h t Implcatons of the optmal tax rule Equaton (9) has a number of mplcatons for tax polcy: ) The optmal margnal tax rate at the ncome level z s lower the hgher the ntensve labour supply elastcty ( ) and the larger the number of taxpayers ( h ) at that ncome level. Ths s ntutve, snce the effcency loss from a rse n the margnal tax rate wll be greater the more taxpayers who are affected by t and the stronger ther labour supply responds to a change n the net gan from addtonal effort. 2) Snce a rse n the margnal tax rate at ncome level z reduces the net labour ncome of all taxpayers above that earnngs level, t nduces some of them to ext the labour market. The strength of ths extensve labour supply response s larger the hgher the partcpaton elastctes,,,..., J, and the resultng loss n net publc revenue s greater the hgher are the ntal partcpaton tax rates t of the affected groups and the greater the number of people n these groups. The labour supply response at the extensve margn therefore reduces the optmal margnal tax rate at ncome level z to a larger extent the hgher are the values of, t and h above that ncome level. 3) Because a hgher margnal tax rate at ncome level z cuts nto the dsposable ncome of all taxpayers above that level, the optmal margnal tax rate s lower the hgher s the socal valuaton of ncome for taxpayers above the ncome level consdered;.e. the larger the values of the welfare weghts g,,,..., J, and the greater the number of people carryng these weghts ( h ). Whle the factors mentoned n ) and 2) reflect how concerns about economc effcency shape the optmal tax schedule, the parameters n 3) obvously reflect equty concerns. However, note from (9) that allowng for labour supply responses at the extensve margn (an effcency concern) s equvalent to attachng a hgher socal welfare weght to groups wth hgh partcpaton elastctes and/or groups wth hgh partcpaton tax rates. One further pont s worth emphaszng: to apply formula (9), no assumptons about ndvdual preferences are needed. The polcy maker only has to specfy hs/her dstrbutonal 8

9 value udgements n terms of the socal welfare weghts parameters,, t and h. 3 g and to obtan estmates of the To llustrate the applcablty of formula (9), we may consder some nstructve specal cases. As already mentoned, the analyss above abstracts from ncome effects on labour supply snce such effects have typcally been estmated to be small. Wth zero ncome effects a lump-sum transfer to an ndvdual taxpayer has no mpact on on hs/her labour supply and hence does not generate any ndrect change n publc revenue va ths channel. A margnal euro of publc funds wll then be valued exactly as much as an addtonal euro dstrbuted evenly across all taxpayers, mplyng that J hg. (20) 0 Tradtonally the optmal tax lterature has abstracted from labour supply responses on the extensve margn, mplctly assumng 0. It then follows from (4) that the optmal margnal tax rate at the bottom of the ncome ladder s T T c c h h J J J h g h h0 hg h0g0, (2) where ndvduals outsde the labour market have been categorzed as group zero (ndcated by subscrpt 0), whle the lowest-pad workers are categorzed as group. By defnton, we have J h whch may be nserted nto (2) along wth (20) to gve 0 T g h T 0 c c h If polcy makers have a strong preference for redstrbuton, the value of the socal welfare weght g 0 wll tend to be far greater than one, snce the weghted average value of. (22) g across all taxpayers s unty (cf. (20)). 4 Accordng to (22) the margnal tax rate at the bottom of the earnngs dstrbuton wll then be hgh. Ths s a well-known mplcaton of the Mrrleesan model of optmal ncome taxaton when the least productve ndvduals are outsde the labour force. In other words, the 3 One dffculty s that estmaton of the elastctes and has to be based on the ndvdual labour supples actually observed whereas formula (9) requres knowledge of the elastctes that would preval under the optmal tax system where labour supply mght dffer from ts current level. 4 If polcy makers beleve that people outsde the labour market are unemployed mostly because they are lazy, t s concevable that the welfare weght g could be smaller than one, but ths possblty s dsregarded here. 0 9

10 much-debated poverty traps caused by the phase-out of socal transfers as people belongng to the poorest secton of the populaton move from unemployment nto employment are n fact part of an optmal polcy when the labour supply of the rest of the work force does not respond at the extensve margn. However, suppose nstead that labour supply responds only at the extensve margn,.e. 0,. Ths alternatve benchmark case s of some nterest snce emprcal studes tend to fnd that labour supply s ndeed much more elastc at the extensve than at the ntensve margn, at least for the low-sklled and for females (see Heckmann, 993). Wth 0 t follows from (9) that optmal tax polcy requres t t g Thus the optmal partcpaton tax rate for ncome group s lower the hgher the partcpaton elastcty of that group (effcency concern) and the hgher the socal valuaton of ncome for members of the group (equty concern). Snce S0 T0, we have. (23) TS0 TT0 t TT0 TT0 t. z z t z T T c c 0 0 (24) Accordng to (23) and (24) the optmal margnal tax rate at the bottom s therefore gven by T T0 g. c c 0 Gven that the weghted average value of the socal welfare weghts s unty (see (20)), a polcy preference for redstrbuton wll almost surely mply g except n the extreme Rawlsan case where polcy makers care only about the poorest group (so that g0 and g 0 for all ). Thus (25) mples that, when labour supply only responds at the extensve margn, the margnal tax rate on the lowest-pad workers should generally be negatve. Ths polcy could be mplemented by grantng a suffcently large Earned Income Tax Credt whch s phased out wth rsng levels of labour ncome (to reflect the fact that g vares negatvely wth ncome). The result n (25) s n stark contrast to the more conventonal result reported n (22), and t hghlghts the mportance of allowng for labour supply responses at the extensve margn. (25) 0

11 Saez (2002a) apples the general formula (9) to smulate the optmal tax schedule, usng data on the U.S. wage dstrbuton plus alternatve assumptons about labour supply elastctes and the government s tastes for redstrbuton. The latter are specfed as g gc, 0, pc where the parameter measures the strength of the preference for redstrbuton, and p s the margnal value of publc funds, calbrated to satsfy (20). For plausble values of the ntensve labour supply elastctes, Saez fnds that t takes farly hgh partcpaton elastctes to ratonalze negatve margnal tax rates at the bottom, especally f the preference for redstrbuton s strong. However, wth realstc partcpaton elastctes, the lowest-pad workers should face rather low margnal tax rates n order not to dscourage ther partcpaton, and ths could stll provde a role for some form of Earned Income Tax Credt (EITC). The new focus n optmal tax theory on the mportance of the extensve margn of labour supply thus offers a ratonale for the recent trend n many OECD countres towards the ntroducton of varous n-work benefts (such as an EITC) that are ntended to make work pay. (26) An alternatve applcaton of optmal tax theory: dervng mplct socal welfare weghts One potental obstacle to the applcablty of optmal tax theory s that polcy makers may not be able or wllng to explctly specfy the socal valuaton weghts g n the optmal tax formula (9). However, by turnng the formula on ts head, the researcher can ask: what are the magntudes of the socal welfare weghts for the varous ncome groups that would make the exstng tax system optmal, gven realstc assumptons on labour supply elastctes and the dstrbuton of pre-tax earnngs? Havng estmated the mplct socal welfare weghts emboded n the current tax-transfer system, the researcher may then confront polcy makers and ask them: do these socal welfare weghts provde a reasonable representaton of your actual dstrbutonal preferences? If the answer s negatve, perhaps polcy makers can be nduced to reconsder whether the exstng tax-transfer system represents a ratonal trade-off between equty and effcency. For example, suppose the researcher could pont out that the socal welfare weghts mpled by the current tax system are not monotonously decreasng wth the taxpayer s level of ncome. Presumably the polcy maker would fnd t hard to defend such a system and would therefore be wllng to consder proposals for reformng t.

12 To llustrate how such a procedure mght work, note that equaton (4) may be nverted to gve (usng S0 T0 c0): g T T T T J 0 J J J J J, cj c0 cj cj (27) J T T0 T T T T 0 g h g. (28) c c0 c c h c c 0 Equaton (27) may be used to estmate the mplct socal welfare weght for the top ncome group (group J), gven estmates of the labour supply elastctes J and J plus data on the net tax payments TJ, TJ and T0 and the dsposable ncomes cj, cj and c0. Once ths has been done, one can apply (28) to calculate the mplct welfare weghts for groups J-, J-2, etc. n a recursve manner, all the way down to group, usng data on ncomes and tax payments plus estmated elastctes. The welfare weght g 0 may then fnally be calculated from (20). Such an exercse has recently been undertaken by Spadaro (2008) for seven western European welfare states. Spadaro uses data from the tax-beneft calculator EUROMOD whch groups the populaton nto ten ncome decles. Interestngly, he fnds that the mplct socal welfare weghts are non-monotonc n all the countres consdered. 5 Optmal taxaton n mperfect labour markets The bulk of the lterature on optmal ncome taxaton abstracts from labour market dstortons other than those caused by the tax system. In compettve labour markets, a swtch from proportonal to progressve ncome taxaton ncreases the deadweght loss from taxaton, except under hghly mplausble assumptons regardng the ncome effects of taxaton (see Sandmo, 983). But once one allows for non-tax dstortons due to labour market mperfectons, some amount of tax progressvty can actually be defended on pure effcency grounds (see Sørensen, 999). For example, n unonsed labour markets unon wage settng tends to generate nvoluntary unemployment. If the government rases the margnal tax rate whle holdng the average tax rate constant (e.g. by rasng the personal exempton level), unons can be motvated to moderate ther wage clams, thus pavng 5 Implct socal welfare weghts may also be derved from the condton that the margnal socal cost of rasng an addtonal euro of tax revenue should be equalzed across all ncome groups under the optmal tax system. Applyng ths method to Dansh data, Petersen (2007) lkewse fnds that the Dansh tax-transfer system mples non-monotonc socal welfare weghts, assgnng hgher welfare weghts to the mddle ncome groups than to the poorest groups. 2

13 the way for lower unemployment. The reason s that, wth a hgher margnal tax rate, t becomes less costly for unons to buy more obs through wage moderaton, snce a gven fall n the pre-tax wage rate wll now lead to a smaller drop n the after-tax wage. Stronger tax progressvty also generates wage moderaton and lower nvoluntary unemployment n effcency wage models where employers pay wages above the market-clearng level as a means of nducng hgher productvty of ther workers. The explanaton s that a rse n the margnal tax rate reduces the effectveness of a hgh (pre-tax) wage rate as an nstrument for encouragng hgh labour productvty, gven that workers care about after-tax rather than pre-tax wages. These observatons do not mply that tax progressvty s a free lunch n mperfect labour markets, snce hgher margnal tax rates also have dstortonary effects, e.g. by nducng unons to bargan for fewer workng hours, and by reducng work efforts n an effcency wage settng. However, even when these tax dstortons are accounted for, the numercal analyss n Sørensen (999) suggests that a substantal degree of tax progressvty can be ratonalsed on pure effcency grounds, especally when unemployment benefts are generous. A challengng task s to analyse optmal taxaton when the tax system must serve the goal of redstrbutng ncome whle at the same tme accountng for non-tax labour market frctons. Ths problem was recently addressed by Boone and Bovenberg (2004) and Hungerbühler et al. (2006) for an economy wth labour market frctons stemmng from mperfect nformaton that gves rse to ob search. When an employer wth a ob vacancy has been matched wth an unemployed ob seeker, the wage bargan between the two partes determnes the dstrbuton of the rent from the ob match. 6 The worker s share of the rent represents the return to hs ob search effort, and the employer s share s the return to hs nvestment n searchng for a worker. To generate an effcent labour market equlbrum, the employer s share of the surplus from the match should correspond to the ncrease n the probablty of a match occurrng when he posts an extra vacancy. Ths effcency condton s known as the Hosos condton (Hosos, 990). However, f the worker s barganng power s too strong (weak), hs share of the rent wll be larger (smaller) than the effcent share, thus weakenng (strengthenng) the ncentve for employers to post vacances and thereby generatng too much (lttle) labour market slack and hence too much (lttle) unemployment. Hungerbühler et al. (2006) analyse the optmal non-lnear tax-transfer system n the benchmark case where the Hosos condton s met so that the labour market equlbrum would be 6 The rent arses from the fact that nether party can mmedately and costlessly fnd another match. 3

14 effcent n the absence of tax. The ntroducton of a postve margnal labour ncome tax rate for redstrbutve purposes nduces frms and workers to bargan for lower gross wages, because t rases the cost to the employer of provdng the worker wth any gven ncrease n the after-tax wage whle at the same tme reducng the cost to the worker of concedng more proft to the employer by acceptng a lower pre-tax wage rate. Hence the optmal tax system nvolves a level of wages and unemployment below the effcent level when the Hosos condton s met, snce t s optmal for the government to accept ths labour market dstorton n return for the redstrbutve gan from progressve taxaton. Indeed, snce progressvty mples that the average tax rate ncreases wth ncome, and snce a rse n the average (as opposed to the margnal) tax rate generates upward wage pressure n the Nash barganng set-up consdered by Hungerbühler et al., they fnd that t s optmal to set margnal tax rates above the average tax rates to moderate wages. Smulatng ther model for plausble parameter values, they show that the optmal margnal tax rates n ther economy wth ob search frctons tend to be substantally hgher than the optmal tax rates n the tradtonal Mrrleesan optmal tax model wth a compettve labour market. The study of optmal taxaton n mperfect labour market s stll n ts nfancy, but the work dscussed n ths secton suggests that t may generate new nterestng nsghts wth mportant mplcatons for tax polcy. 4

15 2. Optmal ndrect taxaton Does optmal tax theory offer any useful gudance for ndrect taxaton? Ths s the topc addressed n the present part of the paper. It s well known that a unform ndrect ad valorem tax on all goods and servces s equvalent to a proportonal tax on labour ncome. 7 From a theoretcal perspectve, the man ssue s thus whether there s a case for ntroducng a system of dfferentated commodty taxes? And f so, whch commodtes should bear the hghest (lowest) rates of tax? There are two strands of optmal tax lterature dealng wth ths ssue. The Ramsey approach to ndrect taxaton The frst one, buldng on the classcal contrbuton by Ramsey (927), assumes that the government has to rase some gven amount of revenue va ndrect taxes. When one abstracts from consumer heterogenety, the optmal tax problem bols down to rasng the requred amount of revenue n a manner that mnmses the total deadweght loss. Ths leads to the famous Ramsey rule whch says that ndrect taxes should be desgned so as to cause an equ-proportonate reducton n the compensated demands for all commodtes. Thus the optmal ndrect tax system seeks to avod dstortng the quanttatve pattern of consumpton, but snce the own prce and cross prce elastctes wll generally dffer across commodtes, commodty taxes (as a percentage of the total consumer prce) should generally be dfferentated to nduce the same relatve reducton n all quanttes demanded. 7 We can be a lttle more precse here. Suppose the consumer s lfe cycle s dvded nto two perods, ndcated by subscrpts and 2. Let C, W, and T denote consumpton, wage ncome, rent ncome ( pure proft ) and transfer ncome n perod, respectvely, and suppose the consumer receves an nhertance I n perod and leaves a bequest B at the end of perod 2. If S s the savng undertaken n perod, r s the nterest rate, and t s the unform ndrect tax on consumpton, assumed constant over tme, the consumer s budget constrants n the two perods are: Perod : S W T I t C, Perod 2: t C B r S W T Elmnatng S and consoldatng, one obtans the lfetme budget constrant: C W T B t, r r r t C 2 W T 2 2 I. Ths shows that a unform ad valorem commodty tax leved at the rate t s equvalent to a proportonal tax leved at the B t/ t. rate on the sum of wages, rents, transfers and the present value of net bequests receved I r 5

16 The well-known nverse elastcty rule statng that the optmal commodty tax rates are nversely related to the (compensated) own-prce elastcty of demand s a specal case of the Ramsey rule, holdng only when the cross prce elastctes of demand for the taxed commodtes are zero. The nverse elastcty rule essentally seeks to mnmse tax dstortons to labour supply, for when cross prce elastctes n commodty demands are zero, the only way a tax-nduced rse n the consumer prce can reduce the demand for some commodty s by causng substtuton from materal consumpton towards lesure. In such a settng a low own prce elastcty of demand means that a commodty tax has lttle dscouragng effect on labour supply. The optmal tax rule dscovered by Corlett and Hague (953) s another specal case of the Ramsey rule derved n a context wth only two taxed commodtes plus (untaxed) lesure. The Corlett-Hague rule states that the commodty whch s more complementary to (less substtutable for) lesure should carry a relatvely hgh tax rate to offset the tendency of the tax system to nduce substtuton towards lesure. Thus unform taxaton s optmal only n the specal case where both commodtes are equally substtutable for (complementary to) lesure. Agan, we see that optmal ndrect tax desgn seeks to mnmse the tax dstortons to labour supply that nevtably occur when only commodtes (but not lesure) can be taxed. More generally, f utlty s generated by consumpton of the goods bundle x0, x, x2,..., x n, where x 0 s untaxed lesure, t can be shown (see Sandmo (974) or Sadka (977)) that unform commodty taxaton s optmal f the utlty functon takes the form 0,,..., n 0,,..., n U x x x U x v x x (29) where the subutlty functon vx,..., x n s homothetc. In other words, unform commodty taxaton s optmal f preferences are separable n lesure and commodtes so that all commodtes are equally substtutable for lesure - and f all commodtes have the same ncome elastcty of demand. The ntuton s that when ncome elastctes are dentcal, the equ-proportonate reducton of all compensated commodty demands prescrbed by the Ramsey rule also requres the same relatve reducton of the uncompensated demands for all commodtes. Snce a unform advalorem tax on all commodtes s equvalent to a proportonal labour ncome tax, t wll ndeed generate the same relatve fall n the consumpton of all commodtes when they all have the same ncome elastcty and are all equally substtutable for lesure (labour). The classcal Ramsey rule focuses on the pure effcency aspects of ndrect taxaton by abstractng from consumer heterogenety. Damond (975) showed how the Ramsey rule s 6

17 modfed n a world of heterogeneous consumers where polcy makers trade off effcency aganst ther redstrbutonal goals. Hs analyss ndcates that whle effcency may call for relatvely hgh commodty tax rates on lesure complements, concerns about equty call for relatvely low tax rates on commodtes that wegh heavly n the budgets of low-ncome famles. The Mrrleesan approach to ndrect taxaton and the Atknson-Stgltz theorem Although Damond (op.ct.) ntroduced equty concerns nto the theory of optmal ndrect taxaton, he dd not account for the role that a non-lnear ncome tax could play as a means of achevng the polcy maker s dstrbutonal goals. Another strand of lterature on ndrect taxaton whch mght be termed the Mrrleesan approach asks the queston: whch (f any) commodty taxes should supplement the ncome tax n order to attan an optmal trade-off between equty and effcency when the government has to rase a gven amount of total revenue? Ths lterature assumes that consumers have dfferent abltes, reflected n ther wage rates. The government cannot observe ndvdual wage rates, but t observes ndvdual ncomes and s hence able to levy a non-lnear ncome tax. At the same tme, the government does not observe the ndvdual taxpayer s consumpton of a partcular good, so commodty taxes must be mpersonal and hence lnear. Note that snce unform commodty taxaton s equvalent to a proportonal wage ncome tax, the ssue for ndrect tax polcy s whether there s any need for dfferentated commodty taxes when the government can levy a personal tax on labour ncome. The benchmark result n ths lne of research was establshed early on by Atknson and Stgltz (976) who showed that f preferences are weakly separable n lesure and all other goods taken together, t s noptmal to dfferentate taxes across commodtes when the government optmses the non-lnear labour ncome tax. Weak separablty of preferences mples that the utlty functon takes the form (29), but the Atknson-Stgltz theorem does not requre that the subutlty functon vx,..., x n be homothetc. The ntuton behnd the Atknson-Stgltz theorem may be explaned as follows. The nonlnear ncome tax s deployed to acheve the optmal amount of redstrbuton. If the tax system becomes too progressve (relatve to the optmum), an ndvdual wth a hgh wage rate wll choose to work less so as to mmc the ncome level of an ndvdual wth a lower wage rate. In that case the two persons wll pay the same amount of ncome tax and have the same dsposable ncome, but the person wth the hgher wage rate wll enoy more lesure. However, the government obvously cannot use the ncome tax to acheve a further redstrbuton of welfare from hgh-ablty to low- 7

18 ablty ndvduals when the former persons mmc the ncomes of the latter. If dfferentated commodty taxes could reduce the ncentve for mmckng, the scope for redstrbuton va the ncome tax would ncrease. But wth separablty between goods and lesure, the government cannot use dfferentated commodty taxes to mpose a hgher tax burden on hgh-ablty persons by explotng any relatonshp between lesure and the consumpton of taxable commodtes, snce the hgh-ablty ndvduals choose the same commodty bundle as the low-ablty persons whom they mmc, even though they consume more lesure. Hence dfferentated commodty taxes cannot play any useful role by relaxng the non-mmckng constrants that lmt the government s ablty to redstrbute ncome. Dfferentated taxaton wll only ntroduce dstortons n commodty choces and hence t s not optmal. Atknson and Stgltz proved ther theorem by assumng that the government optmses the non-lnear ncome tax schedule. More recently, Laroque (2005) and Kaplow (2006) have shown that as long as preferences are weakly separable n lesure and commodtes, dfferentated commodty taxes are undesrable even when the non-lnear ncome tax schedule s not optmal. Snce ths strong result s not so well known, and snce Kaplow s proof strengthens the ntuton for the Atknson-Stgltz theorem, t s worth provdng a sketch of hs proof. 8 Suppose each consumer has a utlty functon of the separable form (29), wth lesure x 0 beng equal to E L, where E s the tme endowment and L s labour tme. An ndvdual wth the pre-tax wage rate w (reflectng hs exogenous productvty) maxmses the utlty functon (29) subect to the budget constrant where n p tx wltwl, (30) p s the (fxed) producer prce of commodty, t s the tax rate on that commodty, and T wl s the consumer s labour ncome tax bll, gven the non-lnear tax schedule T. Wth t denotng the vector of commodty tax rates, we may now ntroduce the ndrect subutlty functon V t, TwL, wl, defned as the value of subutlty,..., n v x x maxmsed over all the commodtes x,..., x n, where the commodty tax vector t, the ncome tax schedule TwL, and pre-tax ncome wl are taken as gven. Note that the ndrect subutlty depends only on ncome, wl, but not on labour supply L. Ths s a consequence of the separable utlty functon (29) whch 8 The sketch offered here s a condensed verson of the heurstc exposton n Kaplow (2008, pp ). The rgorous mathematcal proof s gven n Kaplow (2006). Workng ndependently of Kaplow, Laroque (2005) also provded a rgorous proof of the redundancy of ndrect taxaton under the assumptons stated above. 8

19 mples that the margnal rate of substtuton between any two commodtes s ndependent of the amount of labour. Startng from an ntal stuaton wth dfferentated commodty taxaton (dfferent ad valorem tax rates) where there exsts, such that p t / p t p / p * * move towards a unform commodty tax vector t* satsfyng, consder now a p t / p t p / p for all,. By gong through the followng steps, Kaplow (2006) shows that ths reform allows the government to generate a Pareto mprovement. Frst, suppose the government ntroduces a new nonlnear * ncome tax schedule T wl whch has the property that, at all ncome levels, the utlty of taxpayers wll be the same after the reform as before the reform, provded they do not change ther labour supply. If the pre-reform tax schedule s TwL, the post-reform tax schedule wll thus * * UE satsfy V t, T wl, wl V t, T wl, wl for all wl, snce n ths case all taxpayers wll then also enoy the same total utlty L, V before and after the reform, assumng that ther labour supply L does not change. And n fact no taxpayer wll have any ncentve to change hs/her labour supply. The reason s that the adustment of the ncome tax schedule s undertaken for all ncome levels, so when a taxpayer wth wage rate w vares hs labour supply, thereby varyng wl, he wll fnd that the equalty Therefore, f * * V t, T wl, wl V t, T wl, wl holds for all choces of L. o L was the optmal labour supply supply that maxmsed total utlty U E L, V before the reform, t must stll be the optmal labour supply after the reform, gven that the maxmum attanable subutlty V s unchanged for any choce of L. The next step n the proof s to show that the move towards unform commodty taxaton combned wth the utlty-preservng change n the ncome tax schedule wll generate an ncrease n government revenue. Suppose for a moment that taxpayers do not respond to the change n relatve commodty prces, contnung to consume the same amounts of each ndvdual commodty as before. Wth an unchanged labour supply n all ncome groups, the utlty-preservng change n the ncome tax schedule would then mply that the change n ndrect tax payments would be exactly offset by the change n ncome tax payments so as to leave real dsposable ncomes unchanged at all ncome levels. As a consequence, total tax revenue would also be unchanged. In realty, consumers would of course react to the commodty tax reform and the ensung change n relatve consumer prces by substtutng towards those commodtes that have become relatvely 9

20 cheaper. Snce ths substtuton can only make consumers better off, t allows a further ncrease n the ncome tax bll at each ncome level wthout reducng consumer utlty below the pre-reform level. At the same tme, the substtuton across commodtes does not change the total revenue from ndrect taxaton, snce all commodtes bear the same ad valorem tax rate after the reform. The utlty-preservng reform of drect and ndrect taxaton must therefore ncrease total publc revenue, allowng the government to create a Pareto mprovement by dstrbutng the extra revenue to all taxpayers. Although popular and mportant as a benchmark case, the assumpton that preferences are separable n lesure and commodtes s of course rather restrctve. Chrstansen (984) analysed whch commodty taxes should supplement an optmal nonlnear ncome tax when preferences are not separable. He found that a commodty should be taxed (subsdsed) f t s postvely (negatvely) related to lesure n the sense that more (less) of the good s consumed f more lesure s obtaned at constant ncome. Ths result has the same flavour as the Corlett-Hague rule: the ndrect tax system should dscourage the purchase of commodtes that tend to be consumed ontly wth lesure. The ntuton for Chrstansen s result follows drectly from the ntuton underlyng the Atknson-Stgltz theorem: by taxng complements to lesure, the government makes t less attractve for the more productve ndvduals (who must sacrfce less lesure to earn a gven amount of ncome) to mmc the ncome of the less productve ndvduals, so n ths way commodty taxes relax the ncentve-compatblty constrants that restrct the government s ablty to redstrbute ncome va the ncome tax. Saez (2002b) has extended Chrstansen s analyss to a settng wth heterogeneous consumer tastes, showng that the optmal nonlnear labour ncome tax should be supplemented not only by excses on commodtes that are consumed ontly wth lesure, but also by taxes on commodtes for whch hgh-ncome earners tend to have a relatvely strong taste. Further arguments for unform ndrect taxaton The emprcal work by Brownng and Meghr (99) as well as casual observaton strongly ndcates that some commodtes are better substtutes for lesure than others,.e. consumer preferences are n fact not weakly separable n lesure and commodtes. Ths would seem to call for a system of dfferentated commodty taxes wth the characterstcs suggested by Chrstansen (984) and Saez (2002b). A number of practcal arguments nevertheless speak n favour of unform taxaton. 20

21 Frst of all, the government may smply lack the sold evdence on the compensated cross-prce elastctes wth lesure that would be needed to mplement the optmal dfferentaton of commodty taxes. Second, a commodty tax system dfferentated accordng to the prncples of optmal tax theory would requre frequent changes n tax rates n response to changes n tastes and technologes. Ths would ntroduce a potentally welfare-reducng element of rsk and uncertanty nto the tax system. Thrd, a unform VAT s easer to admnster and less susceptble to fraud than a VAT system wth several dfferentated rates, snce a unform VAT does not requre any borderlnes to be drawn between dfferent categores of goods. Fourth, acceptance of dfferentated taxaton as a general prncple mght nvte specal nterest groups to lobby for low tax rates on partcular economc actvtes, so adherence to a prncple of unformty may provde a stronger bulwark aganst wasteful lobbysm. All of ths suggests that unformty should be the gudng prncple for ndrect taxaton, except where consumpton of specfc commodtes generates obvous externaltes that need to be corrected through hgh excses. 9 However, recent contrbutons to optmal tax theory by Kleven, Rchter and Sørensen (2000) 0 and Kleven (2004), allowng for home producton along wth market producton, have ponted to some specfc areas where devatons from unform taxaton may be warranted. Optmal ndrect taxaton wth household producton: a smple example To llustrate the mportance of home producton for optmal taxaton, suppose the representatve h consumer produces servces wthn the household ( S ) subect to the concave household producton functon h S hh, h' 0, h'' 0, (3) where H denotes hours spent workng n the home. Suppose further that servces can also be purchased n the market place so that total servce consumpton (S) s the sum of servces bought n m the market ( S ) and servces produced at home: m h S S S. (32) If the consumer spends L hours workng n the market and hs total tme endowment s E=, hs consumpton of lesure ( ) wll be L H. (33) 9 Sandmo (975) provdes the classcal treatment of optmal commodty taxaton n the presence of externaltes. 0 The paper by Kleven et al. (2000) bult on the earler work by Sandmo (990). 2

22 Apart from lesure and servces, the consumer also consumes goods (G). For concreteness, let utlty be gven by U U, v G, S G S, 0, 0. (34) Thus preferences are weakly separable n lesure and commodtes, and the sub-utlty functon G S s homothetc. As mentoned earler, wth such preferences unform commodty taxaton would be optmal n the absence of home producton, but as we shall see, ths concluson no longer holds when home producton s allowed for. Wthout loss of generalty we may assume that t takes one unt of market work to produce one unt of each of the two commodtes G and S. Choosng lesure as the numerare good (.e. normalsng the wage rate at unty) and assumng that labour s the only factor of producton, the producer prces of G and S wll then both be equal to one. Wth t G and t S denotng the unt commodty taxes mposed on G and S, respectvely, the consumer s budget constrant therefore becomes m PGPS L, P t, P t, (35) G S G G S S where P G and P S are consumer prces, and L wl s the consumer s market ncome. Usng (3) through (33), we may rewrte (35) as PGPS Y, Y Ph H H. (36) G S S The varable Y s the consumer s full ncome, consstng of hs potental market ncome (=) plus the proft from home producton, Ph S H H. Accordng to (36) ths full ncome may be spent on acqurng commodtes or on buyng lesure (by abstanng from work). The consumer maxmses utlty (34) subect to the budget constrant (36). The soluton to ths problem can be shown to mply that Y, P G Y, P S Y, (37) G h' S S Ph S '' S Ph' H H H t, H' 0. (38) Accordng to (38) the consumer engages n home producton untl the resultng margnal savng on servces bought n the market, Ph' S H, equals the margnal opportunty cost of workng at home rather than n the market ( w ). Snce P t, we see that ths behavour mples that the S tme spent on home producton rses wth the tax rate mposed on servces delvered from the S 22

23 h market. Usng the results n (38) and recallng that S hh t S n the followng way: S S S S, we note that full ncome depends on h Y t t h H t H t, Y' S. (39) Insertng the solutons n (37) nto the drect utlty functon (34), we obtan the ndrect utlty functon V c tg ts Y ts, (40) where c s a constant dependng solely on the taste parameters and. We may now prove that, startng from an ntal stuaton wth unform taxaton ( t G t S ), the government can ncrease welfare by movng towards a stuaton where t G t S. We do so by showng that f the government rases t G and lowers t S n a way that mantans constant utlty, t wll earn addtonal revenue. Clearly the government wll then be able to rase consumer welfare by transferrng the addtonal revenue back to the consumer. Dfferentatng (40) and recallng from (39) that Y' S that Note that snce m h S S S Y h PS S Y dv 0 dtg dts. and h, one fnds Y PS S (cf. (37)), the numerator on the rght-hand (4) sde of (4) equals PS S m, so as long as the consumer buys some servces from the market m ( S 0), the mantenance of a constant utlty level wll ndeed requre a rse n t G when t S s lowered. Consder now how ths utlty-preservng perturbaton of tax rates wll affect government revenue. Usng (32) and nsertng the consumer s optmal demands for G and S stated n (37), we may wrte total revenue as m h t G t S RtGGtSS tggts SS YtStShHtS, tg ts whch may be dfferentated to gve t Y Y t h t dr dt 2 G 2 S H ' dts, t t t (42) (43) 23

24 where we have used (3), (38), (39) and the fact that t G t S t ntally. Substtutng (4) nto (43), we obtan: t h dr S H ' ts dts 0 for dts 0. t (44) Thus government revenue does n fact ncrease, enablng the government to rase consumer welfare by recyclng the extra revenue. Note that n the absence of home producton we would have h S H' 0, so n that case there would be no revenue gan and hence no welfare gan by devatng from unform taxaton, accordng to (44). Absent home producton, our model thus reproduces the standard result that unform commodty taxaton s optmal when preferences are separable n lesure and commodtes and utlty s homothetc n commodtes. But once home producton s allowed for, t follows from our analyss that commodtes whch can be produced at home as well as n the market economy should be taxed more lghtly than commodtes whch cannot be produced wthn the household. The practcal mportance of ths result s that t s easy to thnk of commodtes that would be canddates for reduced taxaton under ths prncple. For example, housng repar and repar of other consumer durables, chld care, cleanng and wndow-cleanng, garden care, cookng etc. are all consumer servces that can ether be produced at home or be delvered from the market. Accordng to the analyss above such servces should be taxed more lghtly than, say, manufactured goods that cannot realstcally be produced wthn the household. Kleven, Rchter and Sørensen (2000) show that ths concluson wll almost surely hold also when preferences are not separable and homothetc. In fact they fnd that even f the consumer servces whch can be produced at home as well as n the market mght be complementary to lesure, t may stll be optmal to mpose a relatvely low tax rate on these servces. The ntuton for ths modfcaton to the Corlett-Hague rule s that a hgh tax on complements to lesure s not an effcent way of stmulatng tax-dscouraged labour supply to the market f such a tax nduces a shft from market producton to home producton. The pont s that the optmal tax system must mnmse the dstortonary substtuton away from market actvtes towards untaxed actvtes. Taxes should dstort the pattern of market actvty as lttle as possble, and ths calls for lenent taxaton of those There are two reasons for the rse n revenue. Frst, even f home producton were unchanged, the fall n S t reduces the revenue loss caused by the fact that servces produced n the home cannot be taxed. Ths effect s captured by the frst term n the square bracket n (44). In addton, the fall n S t nduces consumers to substtute market-produced servces for home producton, thereby ncreasng the tax base. Ths s captured by the last term n the square bracket n (44). 24