Review of Economic Studies (2006) 73, /06/ $02.00 c 2006 The Review of Economic Studies Limited.

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1 Revew of Economc Studes (2006) 73, /06/ $02.00 c 2006 The Revew of Economc Studes Lmted Far Income Tax MARC FLEURBAEY CNRS-CERSES, Pars and IDEP and FRANÇOIS MANIQUET Unversty of Louvan-La-Neuve and CORE Frst verson receved February 2003; fnal verson accepted February 2005 (Eds.) In a model where agents have unequal sklls and heterogeneous preferences over consumpton and lesure, we look for the optmal tax on the bass of effcency and farness prncples and under ncentvecompatblty constrants. The farness prncples consdered here are: (1) a weak verson of the Pgou Dalton transfer prncple; (2) a condton precludng redstrbuton when all agents have the same sklls. Wth such prncples we construct and justfy specfc socal preferences and derve a smple crteron for the evaluaton of ncome tax schedules. Namely, the lower the greatest average tax rate over the range of low ncomes, the better. We show that, as a consequence, the optmal tax should gve the greatest subsdes to the workng poor (the agents havng the lowest skll and choosng the largest labour tme). 1. INTRODUCTION Farness s a key concept n redstrbuton ssues. In ths paper, we study how partcular requrements of farness can shed lght on the desgn of the optmal ncome tax schedule. We consder a populaton of heterogeneous ndvduals (or households), who dffer n two respects. Frst, they have unequal sklls and, therefore, unequal earnng abltes. Second, they dffer n terms of ther preferences about consumpton and lesure and, as a consequence, typcally make dfferent labour tme choces. Both knds of dfferences generate ncome nequaltes. We study how to justfy and compute a redstrbuton ncome tax n ths context. Redstrbuton through an ncome tax usually entals dstortons of ncentves, but the resultng effcency loss has to be weghed aganst potental mprovements n the farness of the dstrbuton of resources. We address ths effcency equty trade-off here by constructng socal preferences whch obey the standard Pareto prncple n addton to farness condtons. Two farness requrements are ntroduced below. Brefly, the frst requrement, a qualfcaton of the Pgou Dalton prncple, states that transfers reducng ncome nequaltes are acceptable, provded they are performed between agents havng dentcal preferences and dentcal labour tme. Thanks to ths provso, ths requrement (contrary to the usual Pgou Dalton transfer prncple whch apples to all ncome nequaltes) s stll justfed f we consder that ncomes should not necessarly be equalzed among agents havng dfferent labour tme or, more generally, dfferent wllngness to work. The second farness requrement s that the lasser-fare (that s, the absence of redstrbuton) should be the socal optmum n the hypothetcal case when all agents have equal earnng abltes. The underlyng dea s that ncome nequaltes would then reflect free choces from dfferent preferences on an dentcal budget set, and that such choces ought to be respected. These two requrements, together wth the Pareto prncple and ancllary condtons of nformatonal parsmony and separablty (the dea that ndfferent agents should not nfluence socal 55

2 56 REVIEW OF ECONOMIC STUDIES preferences), lead us to sngle out a partcular knd of socal preferences. These socal preferences measure ndvdual well-beng n terms of what we call equvalent wage (see Secton 2). For any gven ndvdual, her equvalent wage, relatve to a partcular ndfference curve, s the hypothetcal wage rate whch would enable her to reach ths ndfference curve f she could freely choose her labour tme at ths wage rate. Ths partcular measure of well-beng, whch s nduced by the farness condtons, does not requre any other nformaton about ndvduals than ther ordnal non-comparable preferences about ther own consumpton lesure bundles. It s then shown that, under some rchness assumptons about the dstrbuton of characterstcs n the populaton, such socal preferences yeld a very smple crteron for the welfare comparson of tax schedules. Ths crteron s the maxmal average tax rate over low ncomes (.e. ncomes below the mnmum wage). Ths crteron can be used for the comparson of any par of tax schedules, no matter how far from the optmum, but t can also be used to seek the optmal tax schedule. As far as the optmal tax s concerned, the man result s that those ndvduals who have the lowest earnng ablty but work full tme, namely, the hardworkng poor, wll be granted the greatest subsdy (.e. the smallest tax) of the whole populaton. The lterature on optmal taxaton has focused mostly on socal objectves defned n terms of welfarst (typcally, utltaran) socal welfare functons, based on nterpersonal comparsons of utlty. It has obtaned valuable nsghts nto the lkely shape of the optmal tax, as can be grasped from the outstandng works of Mrrlees (1971), Atknson (1973, 1995), Sadka (1976), Seade (1977) Tuomala (1990), Ebert (1992), and Damond (1998), among many others. Many results depend on the partcular choce of ndvdual utlty functon and socal welfare functon. The socal margnal utlty of an ndvdual s ncome may thus reflect varous personal characterstcs (ndvdual utlty) and ethcal values emboded n the socal welfare functon, ncludng, potentally, farness requrements. But, apart from the mportant relatonshp between nequalty averson and (Schur-)concavty of the socal welfare functon, the lnk between farness requrements and features of the socal welfare functon are not usually made explct. In contrast, our approach starts from requrements of farness, and derves socal preferences on ths bass. Ths lterature has tradtonally assumed that agents dffer only n one dmenson (typcally, ther earnng ablty). Several authors (Choné and Laroque, 2001, Boadway, Marchand, Pesteau and Raconero, 2002) have recently examned optmal taxaton under the assumpton that agents may be heterogeneous n two dmensons, ther consumpton lesure preferences and ther earnng ablty, or skll. They mmedately face a conceptual dffculty: there s no clear way to defne the objectve of a utltaran planner, as summng utlty levels of agents havng dfferent preferences requres a partcular choce of utlty functons. It seems therefore necessary to mpose what Choné and Laroque (2001) approprately call an ethcal assumpton. Boadway et al. (2002) consder a whole span of possble weghts for varous utlty functons. In ths paper we show that the relatve weght of agents havng dfferent preferences does not need to be determned by assumpton, but can be derved from farness condtons. An addtonal notorous dffculty of mult-dmensonal screenng s the mpossblty to derve smple solutons due to wdespread bunchng. 1 We are however able to descrbe some basc features of the optmal tax and to obtan a smple crteron for the comparson of taxes. Ths recent lterature suggests that, wth double heterogenety, negatve margnal ncome tax rates are more lkely to be obtaned than f agents dffer wth respect to one parameter only. Our results go n the same drecton. In Choné and Laroque (2001), however, the focus s on labour partcpaton, so that agents work ether zero or one unt, whereas we consder the whole nterval. In addton, ther socal objectve gves absolute prorty to agents wth the smallest ncome, so that negatve tax rates may obtan for hgh ncomes (and only for specal dstrbutons), whereas 1. See, e.g. Armstrong (1996) or Rochet and Choné (1998).

3 FLEURBAEY & MANIQUET FAIR INCOME TAX 57 our socal objectve gves prorty to the workng poor, and non-postve tax rates are obtaned on low ncomes (for all dstrbutons). In Boadway et al. (2002), negatve margnal rates are obtaned on low ncomes and n a closer way to ours, snce they arse n the case when the weghts assgned to agents wth a hgh averson to work are lower than those assgned to agents wth a low averson to work. But ther framework has only four types of agents, whereas our result s obtaned for an unlmted doman. 2 Our work also bulds on prevous studes of the same model (wth unequal earnng abltes and heterogeneous preferences) whch dealt wth frst-best allocatons (Fleurbaey and Manquet, 1996a, 1999) or wth lnear tax (Bossert, Fleurbaey and Van de Gaer, 1999), or focused on dfferent farness concepts (Fleurbaey and Manquet, 2005). The paper s organzed as follows. Secton 2 ntroduces the model and the concept of socal preferences. Secton 3 contans the axomatc analyss and derves socal preferences. Secton 4 develops the analyss of taxaton. Concludng remarks are offered n the last secton. 2. THE MODEL There are two goods, labour and consumpton. 3 A bundle for agent s a par z = (l,c ), where l s labour and c consumpton. The agents consumpton set X s defned by the condtons 0 l 1 and c 0. The populaton contans n 2 agents. Agents have two characterstcs, ther personal preferences over the consumpton set and ther personal skll. For any agent = 1,...,n, personal preferences are denoted R, and z R z (resp. z P z, z I z ) means that bundle z s weakly preferred (resp. strctly preferred, ndfferent) to bundle z. We assume that ndvdual preferences are contnuous, convex and monotonc. 4 The margnal productvty of labour s assumed to be fxed, as n a constant returns to scale technology. Agent s earnng ablty s measured by her productvty or wage rate, denoted w, and s measured n consumpton unts, so that w 0 s agent s producton when workng l = 1 and, for any l,w l s the agent s pre-tax ncome (earnngs). Fgure 1 dsplays the consumpton set, wth typcal ndfference curves, and earnngs as a functon of labour tme. As llustrated on the fgure, an agent s consumpton c may dffer from her earnngs w l. Ths s a typcal consequence of redstrbuton. An allocaton s a collecton z = (z 1,...,z n ). Socal preferences wll allow us to compare allocatons n terms of farness and effcency. Socal preferences wll be formalzed as a complete orderng over all allocatons n X n, and wll be denoted R, wth asymmetrc and symmetrc components P and I, respectvely. In other words, zrz means that z s at least as good as z, zpz means that t s strctly better, and ziz that they are equvalent. Socal preferences may depend on the populaton profle of characterstcs (R 1,...,R n ) and (w 1,...,w n ). Formally, they are a mappng from the set of populaton profles to the set of complete orderngs over allocatons. For the sake of smplcty, we do not ntroduce addtonal no- 2. Another branch of the lterature sometmes obtans smlar results by studyng socal objectves dsregardng ndvdual lesure consumpton preferences and focusng on ncome mantenance. See Besley and Coate (1995) for a synthess. Here we retan a concern for effcency va the Pareto prncple, so that the socal preferences obtaned respect ndvdual preferences. 3. Introducng several consumpton goods would not change the analyss much f prces were assumed to be fxed. The case of varable consumpton prces would requre a specfc analyss. See Fleurbaey and Manquet (1996b, 2001) for exploratons of the problem of far dvson of consumpton goods. 4. Preferences are monotonc f l l and c > c mples that (l,c )P (l,c ). Our analyss could be easly extended to the larger doman of preferences whch are strctly monotonc n c, but not necessarly monotonc n l. Assumng only local non-sataton, on the other hand, would requre a more radcal revson of the analyss (see footnote 5 below).

4 58 REVIEW OF ECONOMIC STUDIES FIGURE 1 tatons for these notons. The doman of economes for whch we want socal preferences to be defned contans all economes obeyng the above condtons. 3. FAIR SOCIAL PREFERENCES 3.1. Farness requrements The man ethcal requrement we wll mpose on socal preferences, n ths paper, s derved from the Pgou Dalton transfer prncple. Tradtonally, however, ths prncple was appled to all ncome nequaltes. Ths entals that no dstncton s made between two agents wth the same ncome but very dfferent wage rates and dfferent amounts of labour. We wll be more cautous here, and apply t only to agents wth dentcal labour. In addton, we wll also restrct t to agents wth dentcal preferences. There are two reasons for ths addtonal restrcton. Frst, applyng the Pgou Dalton prncple to agents wth dfferent preferences would clash wth the Pareto prncple (to be defned more precsely below), as proved by Fleurbaey and Trannoy (2003). Second, when two agents have dentcal preferences one can more easly argue that they deserve to obtan smlar ncomes, whereas ths s much less clear n the case of dfferent preferences, as work dsutlty may dffer. Ths gves us the followng requrement: 5 Transfer prncple. If z and z are two allocatons, and and j are two agents wth dentcal preferences, such that l = l j = l = l j, and for some δ>0, c δ = c > c j = c j + δ, whereas for all other agents k, z k = z k, then zrz. Fgure 2 llustrates the transfer. The axom may sound too weak wth the restrcton l = l j f one thnks that an agent wth hgher skll and dentcal preferences s lkely to work more n ordnary crcumstances (lke those of taxaton descrbed n the next secton). But recall that, at 5. The transfer prncple makes sense only when preferences are strctly monotonc n c. Otherwse, a transfer mght fal to ncrease the recever s satsfacton.

5 FLEURBAEY & MANIQUET FAIR INCOME TAX 59 FIGURE 2 the stage of the constructon of socal preferences, we are only tryng to fnd smple cases where our moral ntuton s strong about how to mprove the allocaton. And we are not restrcted to consder allocatons that are lkely to occur under specfc nsttutons, snce socal preferences must rank all allocatons. What ths axom says s smply that f, by whatever means, two agents wth dentcal preferences and the same labour tme happened to have dfferent consumptons, then reducng ths nequalty would be socally acceptable. Independently of whether such a stuaton s lkely or unlkely to occur (t s actually very common, n real lfe, for people who work full tme), t s qute useful to consder t n order to put mnmal constrants on socal preferences. Another possble objecton s that f two agents have the same preferences, same labour but dfferent productvty, t may seem normal that the more productve consumes more, whereas Pgou Dalton transfers tend to elmnate nequalty. In effect, the above axom s justfed only when agents cannot be held responsble for ther dfferental productvty. Ths rases n partcular the ssue of whether the low-sklled may be consdered to have responsbly chosen ther lower productvty, or nstead have suffered from varous handcaps whch have prevented them from acqurng hgher sklls. The Transfer Prncple axom s consstent wth the latter vew. We leave for future research the study of a rcher model n whch agents could be held partally responsble for ther wage rate, va ther educatonal or occupatonal choces. The second farness requrement we ntroduce has to do wth provdng opportuntes and respectng ndvdual preferences. Although reducng ncome nequaltes s a generous goal, t s not obvous how to deal wth agents who choose poverty out of a budget set whch contans better ncome opportuntes. In partcular, when all agents have the same wage rate, t can be argued that there s no need for redstrbuton, as they all have access to the same labour consumpton bundles (Dworkn, 1981). Any ncome dfference s then a matter of personal preferences. A lasser-fare allocaton z s such that for every agent, z s the best for R over the budget set defned by c w l. The followng requrement says that a lasser-fare allocaton, 6 n ths partcular case of unform earnng ablty, s (one of) the best among all feasble allocatons. 6. There may be several lasser-fare allocatons f preferences are not strctly convex. But all lasser-fare allocatons, n a gven economy, gve agents the same satsfacton.

6 60 REVIEW OF ECONOMIC STUDIES FIGURE 3 Lasser-fare. If all agents have the same wage rate w, then for any lasser-fare allocaton z and any allocaton z such that c w l, one has z Rz. A lasser-fare allocaton n a two-agent equal-skll economy s llustrated n Fgure 3. Both agents have the same budget. Agent, on the fgure, may choose to have more lesure and less consumpton, and the axom of Lasser-Fare declares ths to be unproblematc. One sees that ths prncple s acceptable f ndvdual preferences are fully respectable, but should be treated wth cauton f some ndvdual preferences are nfluenced by questonable socal factors (e.g. apparent lazness may be due to dscouragement and socal stgma; workaholsm may be due to socal pressure). The other requrements are basc condtons derved from the theory of socal choce. Frst, we want socal preferences to obey the standard Pareto condton. Ths condton s essental n order to take account of effcency consderatons. Socal preferences satsfyng the Pareto condton wll never lead to the selecton of neffcent allocatons. In ths way we are preserved aganst excessve consequences of farness requrements, such as equalty obtaned through levellngdown devces. Weak pareto. If z and z are such that for all, z P z, then zpz. Second, we want our socal preferences to use mnmal nformaton about ndvdual preferences, n the sprt of Arrow s (1951) condton of ndependence of rrelevant alternatves. Arrow s condton s, however, much too restrctve, and leads to the unpalatable results of hs mpossblty theorem. Arrow s ndependence of rrelevant alternatves requres socal preferences over two allocatons to depend only on ndvdual preferences over these two allocatons. Ths condton makes t mpossble, for nstance, to check that two agents have the same preferences, or that an allocaton s a lasser-fare allocaton, etc. For extensve dscussons of how excessve Arrow s ndependence s, see Fleurbaey and Manquet (1996b, 2001) and Fleurbaey, Suzumura and Tadenuma (2003). We wll nstead follow Hansson (1973) and Pazner (1979) who have proposed a weaker condton stll consstent wth the dea that nformaton needed to make socal choces should be as parsmonous as possble. That condton requres socal preferences over two allocatons to depend only on ndvdual ndfference curves at these two allocatons. More formally, t requres socal preferences over two allocatons to be the same n two dfferent

7 FLEURBAEY & MANIQUET FAIR INCOME TAX 61 profles of preferences when agents ndfference curves through the bundles they are assgned n these allocatons are the same. Hansson ndependence. Let z and z be two allocatons, and R, R be the socal orderngs for two profles (R 1,...,R n ) and (R 1,...,R n ), respectvely. If for all, and all q X, z I q z I q z I q z I q, then zrz zr z. Fnally, we want our socal preferences to have a separable structure, as s usual n the lterature on socal ndex numbers. The ntuton for separablty requrements s that agents who are not concerned by a socal decson need not be gven any say n t. Ths s not only appealng because t smplfes the structure of socal preferences, but also because t can be related to a standard concepton of democracy, mplyng that unconcerned populatons need not ntervene n socal decsons. Ths s often called the subsdarty prncple. We retan the followng condton, requrng socal preferences over two allocatons to be unchanged f an agent recevng the same bundle n both allocatons s removed from the economy. Separablty. Let z and z be two allocatons, and an agent such that z = z. Then zrz z R z, where z = (z 1,...,z 1, z +1,...,z n ), and R s the socal preference orderng for the economy wth reduced populaton {1,..., 1, + 1,..., n} Socal preferences The farness condtons ntroduced above do not convey a strong averson to nequalty. Actually, the only redstrbutve condton here s the Transfer Prncple, whch, n the above weak formulaton, s compatble wth any degree of nequalty averson, ncludng zero. Nonetheless, the combnaton of all the propertes entals an nfnte averson to nequalty, and forces socal preferences to rely on the maxmn crteron. Moreover, the maxmn crteron needs to be appled to a precse evaluaton of ndvdual stuatons, as stated n the followng theorem. Theorem 1. Let socal preferences satsfy Transfer Prncple, Lasser-Fare, Weak Pareto, Hansson Independence and Separablty. For any allocatons z, z, one has z P z f one of the followng condtons holds: () z P (0,0) and z R (0,0) for all, and mn W (z )>mn W (z ), where W (z ) = max{w R + l, z R (l,wl)}; () z P (0,0) for all and (0,0) P z for some. When z R (0,0), the set {w R + l, z R (l,wl)} s not empty (t contans at least 0), and by monotoncty and contnuty of preferences, t s compact, so that ts maxmum s well

8 62 REVIEW OF ECONOMIC STUDIES FIGURE 4 defned. The computaton of W (z ) s llustrated n Fgure 4. Concretely, W (z ) s the wage rate whch would enable agent to reach the same satsfacton as n z, f she were allowed to choose her labour tme freely, at ths wage rate: What wage rate would gve you the same satsfacton as your current stuaton? Of course, we cannot thnk of usng ths queston as a practcal devce for assessng ndvduals stuatons. Frst, they may have a hard tme workng out what the true answer s. Second, they would have ncentves to msrepresent ther stuaton. The next secton wll examne how ths knd of measure can be practcally mplemented. Another nterpretaton of W (z ) relates t more drectly to the axom of Lasser-Fare. Consder an agent who s ndfferent between z and the bundle z she would choose n a lasserfare allocaton that would be socally optmal f all agents had an equal wage rate w. Then W (z ) = w. In other words, W (z ) s the hypothetcal common wage rate whch would render ths agent ndfferent between z and an optmal allocaton. 7 The functon W (z ) s a partcular utlty representaton of agent s preferences (for a part of the consumpton set). It makes t possble to compare the stuatons of ndvduals who have dentcal or dfferent preferences, on the bass of ther current ndfference curves. In addton, the socal preferences descrbed n Theorem 1 gve absolute prorty to agents wth the lowest W (z ). In ths way, ths result suggests a soluton to the problem of weghtng dfferent utlty functons, mentoned n the ntroducton. By gvng prorty to the worst-off, such socal preferences also escape Mrrlees crtcsm of utltaran socal welfare functons. Mrrlees (1974), ndeed, proved that utltaran frst-best allocatons had to dsplay the property that hgh-sklled agents envy lowsklled agents, that s, the former are assgned bundles on lower ndfference curves than the latter. 8 In contrast, a frst-best allocaton maxmzng mn W (z ) would have the property that all agents have the same W (z ). Consequently, two agents havng the same preferences would be assgned bundles on the same ndfference curve, ndependently of ther sklls, and no one would envy the other. The proof of the theorem s n the Appendx. We provde the ntuton for t here (the rest of ths secton may be skpped wthout any problem for understandng the rest of the paper). Let us 7. Ths concept s closely related to the Equal Wage Equvalent frst-best allocaton rule characterzed on dfferent grounds n Fleurbaey and Manquet (1999). 8. Choné and Laroque (2001) generalze the crtcsm to the case where agents also dffer n terms of ther preferences, and use t as a justfcaton for adoptng socal preferences of the maxmn knd.

9 FLEURBAEY & MANIQUET FAIR INCOME TAX 63 FIGURE 5 frst show how the combnaton of Weak Pareto, Transfer Prncple and Hansson Independence entals an nfnte averson to nequalty. Consder two agents and j wth dentcal preferences R 0, and two allocatons z and z such that z P 0 z P 0 z j P 0 z j. The related ndfference curves are shown n Fgure 5, and one sees n ths partcular example that the axom of Transfer Prncple cannot drectly ental that z s preferable to z, because agent s loss of consumpton between z and z s much greater than agent j s gan, and also because ther labour tmes dffer. By Hansson Independence, socal preferences over z and z can only depend on the ndfference curves through those allocatons, so that they must concde wth what they would be f the dotted ndfference curves represented n Fgure 5 were also part of agents and j s preferences. In ths partcular case, one can construct ntermedate allocatons such as z 1, z 2, z 3, z 4 n the fgure. By Weak Pareto, z 1 Pz. By Transfer Prncple, z 2 Rz 1. By Weak Pareto agan, z 3 Pz 2. By Transfer Prncple agan, z 4 Rz 3. Fnally, Weak Pareto mples zpz 4, so, by transtvty, one can conclude that zpz. Snce ths knd of constructon can be done even when the gan s very small for j whle s loss s huge, one then obtans an nfnte nequalty averson regardng ndfference curves of agents wth dentcal preferences. The second central part of the argument conssts n provng that the maxmn has to be appled to W (z ). The crucal axoms are now Lasser-Fare and Separablty. Let us llustrate the proof n the case of two agents and j and two allocatons z and z such that z k = z k for all k, j, and W (z )>W (z )>W j (z j )>W j (z j ). We need to conclude that z s better than z. Introduce two new agents, a and b, whose dentcal wage rate w s such that W (z )>w>w j (z j ), and whose preferences are R a = R and R b = R j. Let z denote a lasser-fare allocaton for the two-agent economy formed by a and b, and (z a, z b ) be another allocaton whch s feasble but neffcent n ths two-agent economy, and such that Fgure 6 llustrates these allocatons. W (z )>W a (z a )>w>w b (z b )>W j (z j ).

10 64 REVIEW OF ECONOMIC STUDIES FIGURE 6 Let R {a,b}, R {a,b,, j} and R {, j} denote the socal preferences for the economes wth populaton {a,b}, {a,b,, j} and {, j}, respectvely. By Lasser-Fare and Weak Pareto, a lasser-fare allocaton s strctly better than any neffcent feasble allocaton, so z P {a,b} (z a, z b ). Therefore, by Separablty, t must necessarly be the case that (z a, z b, z,z j ) P {a,b,, j} (z a, z b, z, z j ). By the above argument producng an nfnte nequalty averson among agents wth dentcal preferences (from Transfer Prncple and Hansson Independence), one also sees that, by reducng the nequalty between agents a and, and between agents b and j, As a consequence, by transtvty one has from whch Separablty entals that (z a, z b, z,z j ) P {a,b,, j} (z a, z b, z, z j ) (z a, z b, z, z j ) P {a,b,, j} (z a, z b, z,z j ). (z a, z b, z,z j ) P {a,b,, j} (z a, z b, z, z j ), (z, z j ) R {, j} (z, z j ). We would have obtaned the desred strct preference (z, z j ) P {, j} (z, z j ) by referrng, n the prevous stages of ths argument, to another allocaton (z, z j ) Pareto-domnatng z, nstead of z tself. Then, from Separablty agan, one can fnally derve the concluson that zpz n the ntal economy. From ths ntutve proof, one sees that t s the combnaton of Transfer Prncple and Hansson Independence whch leads to focusng on the worst-off, and that t s the combnaton of Lasser-Fare and Separablty whch sngles out W (z ) as the proper measure of ndvdual stuatons.

11 FLEURBAEY & MANIQUET FAIR INCOME TAX 65 Ths theorem does not gve a full characterzaton of socal preferences, because t does not say how to compare allocatons for whch mn W (z ) = mn W (z ). But for the purpose of evaluatng taxes and fndng the optmal tax, the descrpton gven n the theorem s suffcent to yeld precse results, as we wll show n the next secton. Moreover, the theorem does not say how to defne the socal rankng wthn the subset of allocatons such that (0,0) P z for some, but t says that such allocatons are low n the socal rankng and agan that s suffcent for the purpose of tax applcatons. As an addtonal llustraton of ths result, let us brefly examne how other knds of socal preferences fare wth respect to the axoms. In order to smplfy the dscusson, we restrct our attenton to how socal preferences rank allocatons z such that z R (0,0) for all. Frst, consder socal preferences based on W (z ) nstead of mn W (z ): zrz W (z ) W (z ). Such socal preferences volate Transfer Prncple and Lasser-Fare. Socal preferences based on the medan W (z ) would, n addton, volate Separablty. Now, consder socal preferences smlar to those retaned n Choné and Laroque (2001), and based on lexmn C (z ), 9 where C (z ) = max{c R + z R (0,c)}. Such socal preferences satsfy all our axoms except Lasser-Fare. Consder socal preferences based on lexmn V (z ), where V (z ) = max{t R l, z R (l,t + w l)}. These socal preferences satsfy all our axoms except Transfer Prncple. As a fnal example, consder utltaran socal preferences based on U (z ), where U s an exogenously gven utlty functon representng R. Such socal preferences requre more nformaton (the U functons) than the socal preferences studed n ths paper, and therefore do not ft exactly n our framework. One can nonetheless examne whether they satsfy some of our axoms. They fully satsfy Weak Pareto and Separablty. They also satsfy Transfer Prncple when the utlty functons are concave n c (and when two agents wth dentcal preferences also have dentcal utlty functons). They do not satsfy Lasser-Fare, except on the subdoman of utlty functons whch are quas-lnear n c, and do not satsfy Hansson Independence on any reasonable doman. 4. TAX REDISTRIBUTION 4.1. Settng In ths secton, we examne the ssue of devsng the redstrbuton system under ncentvecompatblty constrants and wth the objectve of achevng the best possble consequences accordng to the above socal preferences. As s standard n the second-best context, whose formalsm dates back to Mrrlees (1971), we assume that only earned ncome y = w l s observed, so that redstrbuton s made va a tax functon τ(y ). Ths tax s a subsdy when τ(y )<0. Indvduals are free to choose ther labour tme n the budget set modfed by the tax schedule. The government s assumed to know the dstrbuton of types (preferences, earnng abltes) n the populaton but gnores the characterstcs of any partcular agent. Snce t s easy to forecast the behavour of any gven type of agent under a tax schedule, knowng the dstrbuton of 9. Lexmn s the lexcographc extenson of maxmn (when the smallest value s equal, one looks at the second smallest value, and so on).

12 66 REVIEW OF ECONOMIC STUDIES FIGURE 7 types enables the government to forecast the socal consequences of any tax functon. It may then evaluate or choose a tax functon n vew of the foreseen socal consequences. Under ths knd of redstrbuton, agent s budget set s defned by (see Fgure 7(a)): B(τ,w ) ={(l,c) X c w l τ(w l)}. Notce that τ(0) s the mnmum ncome granted to agents wth no earnngs. It s convenent to focus on the earnngs consumpton space, n whch the budget s defned by (see Fgure 7(b)): B(τ,w ) ={(y,c) [0,w ] R + c y τ(y)}. We retan the same notaton for the two sets snce no confuson s possble. Smlarly, n our fgures z wll smultaneously denote the bundle (l,c ) n one space and the bundle (y,c ) = (w l,c ) n the other space. In the earnngs consumpton space, one can defne ndvdual preferences R over earnngs consumpton bundles, and they are derved from ordnary preferences over labour consumpton bundles va (y,c) R (y,c ) ( ) y,c w R ( y w,c The fact that all agents are submtted to the same constrant c y τ(y) mples that for any par of agents, j, when chooses (y,c ) n B(τ,w ) and j chooses (y j,c j ) n B(τ,w j ), one must have (y,c )R (y j,c j ) or y j >w. Conversely, 10 any allocaton z satsfyng ). for all, j, (y,c ) R (y j,c j ) or y j >w (self-selecton constrants) s ncentve-compatble and can be obtaned by lettng every agent choose her best bundle n a budget set B(τ,w ) for some well-chosen tax functon τ. Ths tax functon must be such that y τ(y) les nowhere above the envelope curve of the ndfference curves of the populaton n the (y,c)-space, and ntersects ths envelope curve at all ponts (y,c ) for = 1,...,n. By monotoncty of ndvdual preferences, we may restrct attenton to tax functons τ such that y τ(y) s non-decreasng. 10. See, e.g. Stgltz (1987, pp ) or Boadway and Keen (2000, pp ).

13 FLEURBAEY & MANIQUET FAIR INCOME TAX 67 FIGURE 8 An allocaton s feasble f t satsfes n c A tax functon τ s feasble f t satsfes =1 n y. =1 n τ(w l ) 0 =1 when all agents choose ther labour tme by maxmzng ther satsfacton over ther budget set. Consder an ncentve-compatble allocaton z. By the assumptons made on ndvdual preferences, the envelope curve of the agents ndfference curves n (y, c)-space, at z, s then the graph of a non-decreasng, non-negatve functon f defned on an nterval S(z) [0,max w ]. Let τ be a tax functon yeldng the allocaton z. It s called mnmal when y τ(y) = f (y) for all y S(z), or equvalently when any tax functon τ whch yelds the same ncentve-compatble allocaton z s such that τ (y) τ(y) for all y S(z). Concretely, when a tax τ s not mnmal, one can devse tax cuts whch have no consequence on the agents behavour and on tax recepts (because no agent has earnngs n the range of the tax cuts). Fgure 8 llustrates ths, wth a mnmal tax τ and a non-mnmal tax ˆτ. When max w / S(z), then there s y such that lm y y,y<y f (y) =+ (see Fgure 8). In ths case, by conventon we let any correspondng mnmal tax have τ(y) = (or equvalently y τ(y) =+ ) for all y y. When z R (0,0) for all, then 0 S(z), so on the nterval [0,max w ] there s only one mnmal tax τ correspondng to z. 11 In the followng, we explore the evaluaton of taxes for the class of socal preferences hghlghted n Theorem 1. Ths means that an ncentve-compatble allocaton z s socally preferred to another ncentve-compatble allocaton z whenever mn W (z )>mn W (z ). The way W (z ) s computed n the earnngs consumpton space s llustrated n Fgure The defnton of τ(y) for y > max w does not matter. By conventon, for all tax functons consdered n ths paper, we let τ(y) = τ(max w ) for all y > max w.

14 68 REVIEW OF ECONOMIC STUDIES FIGURE Two agents As an ntroductory analyss, consder the case of a two-agent populaton {1, 2}. Assume that w 1 <w 2. As a consequence, agent 2 s budget set always contans agent 1 s one. And f agent 1 s labour tme s postve at the lasser-fare allocaton z, necessarly W 1 (z1 )<W 2(z2 ) snce W (z ) w for = 1,2, wth equalty W (z ) = w when the agent has a postve labour tme. (If an agent s so averse to labour that l = 0, then W (z ) equals the margnal rate of substtuton at (0,0), whch s greater than or equal to w.) If the agents have the same preferences R 1 = R 2, then the optmal tax s the one whch maxmzes the satsfacton of agent 1 (snce agent 2 s budget set contans agent 1 s one, n the case of dentcal preferences one has W 2 (z 2 ) W 1 (z 1 ) n any ncentve-compatble allocaton). Ths result extends mmedately to a larger populaton: When all agents have the same preferences, an optmal tax s one whch, among the feasble tax functons, maxmzes the satsfacton of the agents wth the lowest wage rate. In the general case when the agents may have the same or dfferent preferences (assumng that agent 1 has a postve labour tme at the lasser-fare allocaton), then ether the optmal tax acheves an allocaton such that W 1 (z 1 ) = W 2 (z 2 ), or t maxmzes the satsfacton of agent 1 over the set of feasble taxes. The argument for ths fact s the followng. Startng from the lasserfare z where W 1 (z1 )<W 2(z2 ), one redstrbutes from agent 2 to agent 1, and ths ncreases W 1 (z 1 ) and decreases W 2 (z 2 ), followng the second-best Pareto fronter. When one reaches the equalty W 1 (z 1 ) = W 2 (z 2 ), redstrbuton has to stop, snce, by Pareto-effcency, there s no other allocaton wth a greater mn W (z ). But an alternatve possblty s that the ncentvecompatblty constrant (y 2,c 2 )R2 (y 1,c 1 ) puts a lmt on redstrbuton, whch occurs when the pont maxmzng agent 1 s satsfacton s reached. Then, the nequalty W 1 (z 1 )<W 2 (z 2 ) remans at the optmal tax. Fgure 10 llustrates these two possbltes. In (a), the optmal allocaton has W 1 (z 1 ) = W 2 (z 2 ). The fact that t does not maxmze the satsfacton of agent 1 s transparent n ths example because agent 2 s self-selecton constrant s not bndng note that the allocaton s then frst-best effcent. In (b), the optmal allocaton maxmzes the satsfacton of agent 1 and W 1 (z 1 )<W 2 (z 2 ).

15 FLEURBAEY & MANIQUET FAIR INCOME TAX 69 FIGURE General populaton Let us now turn to the case of a larger populaton. The computaton of the optmal tax s qute complex n general, n partcular because the populaton s heterogeneous n two dmensons, preferences and earnng ablty. 12 We wll, however, be able to derve some conclusons about, frst, the part of the tax schedule whch should be the focus of the socal planner and, second, some features of the optmal tax. The man dffculty n such an analyss comes from the theoretcal possblty of observng rankng reversals, wth hgh-sklled agents earnng lower ncomes than low-sklled agents. In the standard settng wth agents dfferng only n the skll dmenson, ths s usually excluded by the Spence Mrrlees sngle-crossng assumpton. In the current mult-dmensonal settng, t would be exceedngly artfcal to exclude such reversals, snce agents wth slghtly dfferent wages may obvously have qute dfferent preferences, and t would be questonable to assume that hghsklled agents are always more hardworkng than low-sklled agents. Fortunately, t appears that the real dffculty does not le wth ndvdual reversals, that s, wth the fact that some hghsklled agent may earn less than some low-sklled agent. For our purposes, we only need to exclude the possblty of observng gaps n the dstrbuton of earnngs of low-sklled agents, wth such gaps flled only wth hgh-sklled agents. That s, we need to exclude the possblty of havng, say, a successon of ntervals [0, y 1 ], (y 1, y 2 ), [y 2,w], such that agents wth wage rate w earn only ncomes n the ntervals [0, y 1 ] and [y 2,w], whereas n the earnngs nterval (y 1, y 2 ) one only fnds agents wth skll w >w.excludng ths possblty s qute natural. Ths can be done by assumng that whenever some hgh-sklled agents are ready to have earnngs n some ntermedate nterval (y 1, y 2 ), there are also low-sklled agents wth locally smlar preferences n the (y,c)-space who are wllng to earn smlar levels of ncome. Formally, let uc((y,c ),w, R ) denote the closed upper contour set for R at (y,c ): uc((y,c ),w, R ) = { (y,c) [0,w ] R + (y,c)r (y,c ) }. The assumpton that we ntroduce says that a hgh-sklled agent, when contemplatng low earnngs, always fnds low-sklled agents who have locally smlar preferences n the (y, c)-space. Let w m = mn w. We assume throughout ths secton that w m > Actually, snce the set of ndvdual preferences s tself nfntely mult-dmensonal, ths s a problem of screenng wth, a pror, nfntely many dmensons of heterogenety (but the populaton s fnte n our model). The fact that the complexty of the mult-dmensonal screenng problem ncreases wth the number of dmensons s shown n Matthews and Moore (1987).

16 70 REVIEW OF ECONOMIC STUDIES FIGURE 11 Assumpton (Low-Skll Dversty). For every agent, and every (y,c) such that y w m, there s an agent j such that w j = w m and uc((y,c),w j, R j ) uc((y,c),w, R ). Fgure 11 llustrates ths confguraton. The ncluson of upper contour sets means that whenever agent chooses (y,c ) n a budget set, there s a low-sklled agent j who s wllng to choose the same bundle (y,c ) from the same budget set (for another bundle, t may be another lowsklled agent). Ths assumpton s of course rather strong for small populatons. As explaned above, however, what s needed for the results below s only that there be no gap n the dstrbuton of earnngs for low-sklled agents. More precsely, the consequence of Low-Skll Dversty that s used below s that for all ncentve-compatble and feasble allocatons, the envelope curve n (y,c)-space of the ndfference curves of low-sklled agents concdes over the nterval [0,w m ] of earnngs wth the envelope curve of the whole populaton. Ths weaker assumpton s qute natural for large populatons, and Low-Skll Dversty s probably the smplest assumpton on the prmtves of the model whch guarantees that t wll be satsfed. The frst result n ths secton has to do wth translatng the abstract objectve of maxmzng mn W (z ) nto a more concrete objectve about the part of the agents budget set whch should be maxmzed. Theorem 2. Consder two ncentve-compatble allocatons z and z obtanable wth two mnmal tax functons τ and τ, respectvely, such that τ(0)<0 and τ (0) 0. If socal preferences satsfy Transfer Prncple, Lasser-Fare, Weak Pareto, Hansson Independence and Separablty, then z s socally preferred to z whenever the maxmal average tax rate over low ncomes y [0,w m ] s smaller n z: τ(y) τ (y) max < max. 0 y w m y 0 y w m y The proof of ths result (see the Appendx) goes by showng that ths nequalty on tax rates entals that mn W (z )>mn W (z ),

17 FLEURBAEY & MANIQUET FAIR INCOME TAX 71 FIGURE 12 so one may apply Theorem 1 to conclude that z s socally preferred. Note that τ(0)<0 mples that z P (0,0) for all. The prorty of the worst-off n socal preferences, combned wth the assumpton of Low-Skll Dversty, s the key factor that leads to focusng on earnngs n the range [0,w m ]. The measure of ndvdual stuatons by W (z ), on the other hand, s the key ngredent for takng the average tax rate τ(y)/y as the relevant token. Indeed, consder on Fgure 12 that the graph of y τ(y) over the range [0,w m ] concdes (by Low-Skll Dversty and the assumpton that the tax functon s mnmal) wth the envelope curve of low-sklled agents ndfference curves. As shown n the fgure, the smallest value of W (z ) for the low-sklled agents s then found by lookng for the ray that s tangent to ths porton of the graph, and therefore equals y τ(y) W m = w m mn. 0 y w m y It turns out that ths s actually the smallest value of W (z ) over the whole populaton. The concluson of Theorem 2 mmedately follows. Ths result has three features whch deserve some comments. Frst, ths result does not only provde nformaton about the optmal tax, sayng that t must mnmze τ(y) max 0 y w m y, but also gves a crteron for the assessment of suboptmal taxes. Gven the fact that poltcal constrants and dsagreements often make the computaton of the optmal tax look lke an ethereal exercse, t s qute useful to be able to say somethng about realstc taxes and pecemeal reforms n an mperfect world. Second, t provdes a very smple crteron for the observer who wants to compare taxes. The applcaton of the crteron requres no nformaton about the populaton characterstcs, except the value of w m, whch, n practce, may be thought to concde wth the legal mnmum wage. 13 Therefore, there s no need to measure W (z ) for every ndvdual, nor even to estmate the 13. Except, perhaps, when there s more than frctonal unemployment. See below.

18 72 REVIEW OF ECONOMIC STUDIES FIGURE 13 The U.S. reform dstrbuton of W (z ) over the populaton. A smple examnaton of the tax schedule yelds a defnte answer. Thrd, the content of the crteron tself s qute ntutve. It says that the focus should be on the maxmum average tax rate τ(y)/y over low earnngs. Near the optmal tax, low earnngs wll actually be subsdzed, that s, τ(y) wll be negatve over ths range. Then, the crteron means that the smallest average rate of subsdy should be as hgh as possble, over ths range. Interestngly, when w m tends to zero, the crteron bols down to comparng the value of the mnmum ncome (or demogrant) τ(0), and advocates that t should be as hgh as possble. It may be useful here to llustrate how the smple comparson crteron provded n the above result can be appled. The next fgure presents the 2000 budget set for a lone parent wth two chldren n the U.S. 14 Net ncome s computed ncludng ncome tax, socal securty contrbutons, food stamps and Temporary Assstance to Needy Famles (TANF), a scheme whch replaced the Ad to Famles wth Dependent Chldren (AFDC) programme n Snce the TANF s temporary (t has a fve-year lmt), t s also relevant to look at the budget set after wthdrawal of TANF. Ths s drawn on the fgure wth a dotted lne. An approxmate representaton of a 1986 (pre-reform) budget s also provded, n order to assess the mpact of the reform. The reform has had a postve mpact accordng to the crteron provded n Theorem 2, as shown by the dotted rays from the orgns. The concluson remans even when wthdrawal of TANF s consdered. In the followng theorem, we provde more nformaton about the optmal tax. Theorem 3. Assume that there exsts a feasble (not necessarly ncentve-compatble) allocaton z such that z P (0,0) for all. If z s an optmal (ncentve-compatble) allocaton for socal preferences satsfyng Transfer Prncple, Lasser-Fare, Weak Pareto, Hansson Independence and Separablty, then t can be obtaned wth a tax functon τ whch, among all feasble tax functons, maxmzes the net ncome of the hardworkng poor, w m τ(w m ), under the constrants that 14. In ths paper, we do not deal wth the ssue of unequal household szes. Theorem 2 does however apply to any subpopulaton of households of a certan knd. The case of lone parents wth chldren s probably the most relevant f one wants to focus on the subgroup of the populaton whch s the worst-off n all respects. 15. The TANF programme s managed at the State level. Fgures correspondng to Florda are retaned n Fgure 8.

19 FLEURBAEY & MANIQUET FAIR INCOME TAX 73 FIGURE 14 τ(y) τ(w m) y w m for all y (0,w m ], τ(y) τ(w m ) for all y, τ(0) 0. The ntal assumpton made n the theorem smply excludes the case when the zero allocaton s effcent and there s therefore no nterestng possblty of redstrbuton. The three constrants lsted at the end of the statement mean, respectvely, that the average tax rate on low ncomes s always lower than at w m, that the tax (subsdy) s the smallest (largest) at w m, and that the tax (subsdy) s non-postve (non-negatve) at 0. Ths result does not say that every optmal tax must satsfy these constrants, but t says, qute relevantly for the socal planner, that there s no problem,.e. no welfare loss, n restrctng attenton to taxes satsfyng those constrants, when lookng for the optmal allocaton. Ths result shows how the socal preferences defned n ths paper lead to focusng on the hardworkng poor, who should get, n the optmal allocaton, the greatest absolute amount of subsdy, among the whole populaton. However, the taxes computed for those wth a lower ncome than w m also matter, as those agents must obtan at least as great a rate of subsdy as the hardworkng poor. Theorem 3 s llustrated n Fgure 14. From the pont (w m,w m τ (w m )) one can construct the hatched area delmted by an upper lne of slope 1 and a lower boundary made of the ray to the orgn (on the left) and a flat lne (on the rght). Now, Theorem 3 says that computng the optmal tax may, wthout welfare loss, be done by maxmzng the second coordnate of the pont (w m,w m τ (w m )) under the constrant that the ncome functon y τ (y) s located n the correspondng hatched area. It s useless to consder ncome functons whch le outsde ths area. Interestngly, the shape of ths area mples that the margnal tax rate over ncomes below w m s, on average, non-postve. As explaned above, when there are agents wth almost zero earnng ablty, our results bol down to a smple maxmzaton of the mnmum ncome. The case of a zero w m can be related to productve dsabltes but also to nvoluntary unemployment. Snce unemployment may be vewed as nullfyng the agents earnng ablty, ths result should best be nterpreted

20 74 REVIEW OF ECONOMIC STUDIES as suggestng that the focus of redstrbutve polces should shft from the hardworkng poor to the low-ncome households when the extent of unemployment s large, and especally when long-term unemployment s a sgnfcant phenomenon. Then, for nstance, the assessment of the welfare reform n the U.S., as llustrated n Fgure 13, would be much less postve snce the mnmum ncome has been reduced (and the temporarness of TANF would appear qute questonable n ths context). On the other hand, physcal dsabltes and unemployment are more or less observable characterstcs, whch may elct specal polces toward those affected by such condtons, as can be wtnessed n many countres. 16 If ths s the case, then the above result should apply to the rest of the populaton, and the relevant value of w m s then lkely to be the mnmum legal hourly wage. Nonetheless when unemployment takes the form of constraned part tme jobs (a less easly observable form than ordnary unemployment), ths should also be tackled by consderng t as a reducton of the agents earnng ablty. 5. CONCLUSION In ths paper, we have examned how two farness prncples, a weak verson of the Pgou Dalton transfer prncple and a lasser-fare prncple for equal-skll economes, sngle out partcular socal preferences and a partcular measure of ndvdual stuatons. Such socal preferences grant absolute prorty to the worst-off, n the maxmn fashon. Ths result 17 mght contrbute to lendng more respectablty to the maxmn crteron, whch s sometmes crtczed for ts extreme averson to nequalty. 18 The measure of ndvdual stuatons obtaned here s the tax-free wage rate whch would enable an agent to mantan her current satsfacton. Ths may be vewed as a specal money-metrc utlty representaton of ndvdual preferences. The choce of ths measure, however, dd not rely on ntrospecton or a phlosophcal examnaton of human well-beng. It derved from the farness prncples (especally the lasser-fare prncple), and the analyss dd not requre any other nformatonal nput about ndvdual welfare than ordnal non-comparable preferences. The famous mpossblty of socal choce (Arrow s theorem) was avoded by weakenng Arrow s axom of Independence of Irrelevant Alternatves n order to take account of the shape of ndvdual ndfference curves at the allocatons under consderaton. It must be stressed that we do not consder W as the only reasonable measure of ndvdual stuatons. In Fleurbaey and Manquet (2005), alternatve socal preferences, usng dfferent measures, are defended on the bass of other ethcal prncples. Our purpose s not to defend a sngle vew of socal welfare, but to clarfy the lnk between farness prncples and concrete polcy evaluatons. It s a legtmate exercse of economc analyss to examne the consequences of varous value judgments, whether or not they are shared by the theorst (Samuelson, 1947, p. 220). The second part of the paper studed the mplcatons of such socal preferences for the evaluaton of ncome tax schedules, under ncentve constrants due to the unobservablty of sklls and the possblty for agents to freely choose ther labour tme n ther budget set. The man result was the dscovery of a smple crteron for the comparson of tax schedules, based on the smallest average subsdy (or greatest average tax rate) for low ncomes. Another result was that the average margnal tax rate for low ncomes should optmally be non-postve, and that the hardworkng poor should receve maxmal subsdes, under the constrant that lower ncomes 16. Observaton of dsabltes and nvoluntary unemployment s, however, mperfect. For an analyss of optmal taxaton under mperfect taggng, see Salané (2002). 17. Smlar dervatons of the maxmn crteron have also been obtaned n dfferent contexts by Fleurbaey (2001) and Manquet and Sprumont (2004). 18. It has always been, however, one of the promnent crtera n the lterature of optmal taxaton. See, e.g. Atknson (1973, 1995) and, more recently, Choné and Laroque (2001).