Optimal taxation and public provision for poverty reduction

Transcription

1 Int Tax Publc Fnance : Optmal taxaton and publc provson for poverty reducton Rav Kanbur 1 Tuul Paukker 2,3 Jukka Prttlä 4,5 Matt Tuomala 5 Publshed onlne: 27 Aprl 2017 The Authors Ths artcle s an open access publcaton Abstract The exstng lterature on optmal taxaton typcally assumes there exsts a capacty to mplement complex tax schemes, whch s not necessarly the case for many developng countres. We examne the determnants of optmal redstrbutve polces n the context of a developng country that can only mplement lnear tax polces due to admnstratve reasons. Further, the reducton of poverty s typcally the expressed goal of such countres, and ths feature s also taken nto account n our model. We derve the optmalty condtons for lnear ncome taxaton, commodty taxaton, and publc provson of prvate and publc goods for the poverty mnmzaton case and compare the results to those derved under a general welfarst obectve functon. We also study the mplcatons of nformalty on optmal redstrbutve polces for such countres. The exercse reveals non-trval dfferences n optmal tax rules under the dfferent assumptons. B Tuul Paukker tuul.paukker@vatt.f Rav Kanbur sk145@cornell.edu Jukka Prttlä ukka@wder.unu.edu Matt Tuomala matt.tuomala@uta.f 1 Cornell Unversty, Ithaca, Y, USA 2 VATT Insttute for Economc Research, Helsnk, Fnland 3 Aalto Unversty, Helsnk, Fnland 4 UU-WIDER, Helsnk, Fnland 5 Unversty of Tampere, Tampere, Fnland

2 Optmal taxaton and publc provson for poverty reducton 65 Keywords Redstrbuton Income taxaton Commodty taxaton Publc good provson Poverty JEL Classfcaton H21 H40 O12 1 Introducton Hgh levels of wthn-country nequalty n many otherwse successful developng countres have become a key polcy concern n global development debate. Whle some countres have very unequal nherent dstrbutons e.g., due to hstorcal land ownershp arrangements, n others the fruts of economc growth have been unequally shared. o matter what the underlyng reason for the hgh nequalty, often the only drect way for governments to affect the dstrbuton of ncome s va redstrbutve tax and transfer systems. Clearly, publc spendng on socal servces also has an mpact on the dstrbuton of well-beng, although some of the effects such as skll-enhancng mpacts from educatonal nvestment only materalze over a longer tme horzon. Reflectng the desre to reduce poverty and nequalty, redstrbutve transfer systems have, ndeed, prolferated n many developng countres. Startng from Latn Amerca, they are now spreadng to low-ncome countres, ncludng those n Sub- Saharan Afrca. 1 In low-ncome countres, n partcular, redstrbutve arrangements va transfers are stll at an early stage, and they often consst of solated, donor-drven programs. There s an urgent and well-recognzed need to move away from scattered programs to more comprehensve tax-beneft systems. Ths paper examnes the optmal desgn of cash transfers, commodty taxes or subsdes, the provson of publc and prvate goods such as educaton and housng, and fnancng them by a lnear ncome tax. The paper also ncludes an analyss of optmal ncome taxaton n the presence of an nformal sector. The paper therefore provdes an overvew of many of the most relevant nstruments for redstrbutve polces that are needed for a system-wde analyss of socal protecton. We buld on the optmal ncome tax approach, whch s extensvely used n the developed country context 2, but much less appled for the desgn of redstrbutve systems n developng country crcumstances. Ths approach, ntated by Mrrlees 1971, allows for a rgorous treatment of effcency concerns e.g., the potentally harmful effect of dstortonary taxaton on employment and redstrbutve obectves. Achevng the government s redstrbutve obectves s constraned by lmted nformaton: the socal planner cannot drectly observe ndvduals ncome-earnng capacty, and therefore t needs to base ts tax and transfer polces on observable varables, such as gross ncome. The most general formulatons of optmal tax models apply nonlnear tax schedules, but n a developng country context, usng fully nonlnear taxes s rarely feasble. In ths paper, we therefore lmt the analyss to redstrbutve lnear ncome taxes, whch combne a 1 For a recent treatment and survey, see Barrentos See IFS and Mrrlees 2011 for an nfluental applcaton of optmal tax theory to polcy analyss for rch countres.

3 66 R. Kanbur et al. lump-sum transfer wth a proportonal ncome tax, and whch can be mplemented by wthholdng at source f necessary. Lnear ncome taxes are not very common n practce: less than 30 countres had flat tax rates for personal ncome n 2012, wth some concentraton n ex-sovet Eastern Europe Pechl It s noteworthy that even though flat taxes are not partcularly common n low-ncome countres, n many nstances n such countres the progressve ncome tax reaches only a small share of the populaton. Ths would ndcate that despte the exstence of a progressve ncome tax, these countres do not yet possess enough tax capacty to mplement well-functonng progressve ncome taxes. Ths s one motvaton for our nterest of modelng optmal lnear taxes. Pechl 2014 suggests that smplfcaton benefts can be especally relevant for developng countres. 3 In conventonal optmal taxaton models, the government s obectve functon s modeled as a socal welfare functon, whch depends drectly on ndvdual utltes. We depart from ths welfarst approach by presentng general non-welfarst tax rules, as nkanbur et al. 2006, and, n partcular, optmal tax and publc good provson rules when the government s assumed to mnmze poverty. We have chosen ths approach as t resembles well the tone of much of the polcy dscusson n developng countres, ncludng the Mllennum Development Goals MDGs and the new Sustanable Development Goals SDGs, where the obectve s explctly to reduce poverty rather than maxmze well-beng. 4 Smlarly, the dscusson regardng cash transfer systems s often couched especally n terms of poverty allevaton. Whle we do not necessarly want to advocate poverty mnmzaton over other socal obectves, we regard examnng ts mplcatons, and contrastng them wth tradtonal welfarstc approaches, useful. Usng non-welfarst obectves s, as such, nothng new n economcs. In fact, as Sen 1985 has argued, one can be crtcal of utltaransm for many reasons. ote also that the obectve of poverty mnmzaton s not at odds wth the restrcton of a lnear tax scheme that we mpose: a flat tax regme together wth a lump-sum ncome transfer component can acheve smlar amounts of redstrbuton toward the poor as a progressve tax system, f specfed sutably Keen et al. 2008; Pechl In all our analyss, we frst present welfarst tax rules whch are mostly already avalable n the lterature to provde a benchmark to examne how applyng poverty mnmzaton as an obectve changes the optmal tax and publc servce provson rules. We also deal wth some extensons to exstng models, whch are motvated by the developng country context, such as the case where publc provson affects the ndvduals ncome-earnng capacty, thus capturng albet n a very stylzed way possbltes to affect ther capabltes. An mportant feature to take nto account n tax analyss of developng countres s the presence of a large nformal sector, and we also examne the mplcatons of ths for optmal redstrbutve polces. Our paper s related to varous strands of earler lterature. Frst, Kanbur et al and Prttlä and Tuomala 2004 study optmal ncome tax and commodty tax rules, respectvely, from the poverty allevaton pont of vew, but ther papers buld on the nonlnear tax approach whch s not well suted to developng countres. Kanbur and 3 ote that t mght be reasonable for some countres to move to a progressve ncome tax system as ther tax capacty ncreases wth development; the study of such dynamcs s beyond the scope of ths paper. 4 In fact, the frst SDG s smply End poverty n all of ts forms everywhere.

4 Optmal taxaton and publc provson for poverty reducton 67 Keen 1989 do consder lnear ncome taxaton together wth poverty mnmzaton, but they do not produce optmal tax rules but focus on a tax reform perspectve, and provde tax rate smulatons. Others have consdered dfferent departures from the welfarst standard. For example, Fleurbaey and Manquet 2007 consder farness as an obectve of the tax-transfer system and ts mplcatons on optmal taxaton. Roemer et al employ a maxmn type of socal goal and characterze how well tax and transfer systems acheve the goal of equalty of opportunty. Second, our work s related to new contrbutons n behavoral publc fnance, whch address the stuaton where the behavoral bases of the ndvduals lead the socal planner to adopt a dfferent obectve functon than the ndvduals have; see Chetty 2015, Gerrtsen 2016, Farh and Gabax A thrd strand of lterature consders taxaton and development more generally, such as Gordon and L 2009,Keen 2009, 2012,Brd and Gendron 2007 and Besley and Persson Ths feld, whle clearly very relevant, has not concentrated much on the desgn of optmal redstrbutve systems. Fnally, optmal lnear ncome taxaton has been studed from the standard welfarst perspectve. We descrbe these models n Sect The most recent descrpton of lnear ncome tax models can be found n Pketty and Saez They also emphasze how lnear tax rules, whle analytcally more feasble, provde the same ntuton as the more complcated nonlnear models. The lnear tax rules, they argue, are robust to alternatve specfcatons 6, and examnng ths forms part of our motvaton: we study optmal lnear tax polces, n our understandng for the frst tme, from the poverty mnmzaton perspectve. The paper proceeds as follows. Secton 2 examnes optmal lnear ncome taxaton, whle Sect. 3 turns to optmal provson rules for publcly provded prvate and publc goods that are fnanced by such a lnear ncome tax. Secton 4 analyzes the combnaton of optmal lnear ncome taxes and commodty taxaton and asks under whch condtons one should use dfferentated commodty taxaton f the government s nterested n poverty mnmzaton and also has optmal cash transfers at ts dsposal. The queston of how optmal poverty-mnmzng ncome tax polces are altered n the presence of an nformal sector s examned n Sect. 5, whereas Sect. 6 presents a numercal llustraton of optmal ncome taxaton for poverty mnmzaton. Fnally, conclusons are provded n Sect Lnear ncome taxaton 2.1 Optmal lnear ncome taxaton under the welfarst obectve In ths secton, we gve an overvew of some of the models and results for optmal lnear ncome taxaton as they have been presented n the lterature. Many formulae for optmal taxaton were developed n the 1970s and 1980s see Dxt and Sandmo 1977; 5 Besley and Persson 2013 use a model wth groups that can dffer n ther ncome-earnng abltes. Ther analyss focuses, however, on explanng how economc development and tax capacty are nterrelated, and not on redstrbuton between ndvduals. 6 They also descrbe some mplcatons of departures from the welfarst standard n the optmal nonlnear tax model.

5 68 R. Kanbur et al. Tuomala 1985 and the survey by Tuomala 1990, and they are stll beng used, whereas Pketty and Saez 2013 offer fresh expressons of the tax rules. Our exposton manly follows that of Tuomala 1985, but Appendx 1 shows how the results relate to those n Pketty and Saez The government collects a lnear ncome tax τ, whch t uses to fnance a lumpsum transfer b, along wth other exogenous publc spendng R. The ndvduals dffer n ther ncome-earnng capacty w, and z denotes ndvdual labor ncome w L, where L represents hours worked. Consumpton equals c = 1 τz +b, where the superscrpt- refers to ndvduals. 7 There s a dscrete dstrbuton of ndvduals, whose heterogeneous preferences over consumpton and labor are captured by the utlty functon u c, z. The maxmzed subect to the ndvdual budget constrant value of ths utlty functon s captured by the ndrect utlty functon, whch s denoted by V 1 τ,b, and we refer to the net-of-tax rate as 1 τ = a. To smplfy notaton, subscrpt-a refers to the dervatve wth respect to the net-of-tax rate. The government has redstrbutve obectves represented by a Bergson Samuelson functon W V 1,...,V wth W > 0, W < 0. The government s problem s to choose the tax rate τ and transfer b so as to maxmze the socal welfare functon W V a, b under the budget constrant 1 a z = b+ R. 8 We denote the socal margnal utlty of ncome by β = W V Vb. All the mathematcal detals are presented n Appendx 1. There t s shown that the optmal tax rule s gven by τ 1 τ = 1 1 zβ, 1 ε z d z 1 τ d1 τ z where ε = s the elastcty of total ncome wth respect to the net-of-tax β rate, z s average ncome and zβ = z β welfare-weghted average ncome. Defne Ω = zβ z, so that I = 1 Ω s a normatve measure of nequalty or, equvalently, of the relatve dstorton arsng from the second-best tax system. Clearly Ω should vary between zero and unty. One would expect t to be a decreasng functon of τ gven the per capta revenue requrement g = R. There s a mnmum feasble level of τ for any gven postve g, and of course g must not be too large, or no equlbrum s possble. Hence any soluton must also satsfy τ>τ mn f the tax system s to be progressve. That s, f the tax does not rase suffcent revenue to fnance the nontransfer expendture, R, the shortfall must be made up by mposng a poll tax b < 0 on each ndvdual. One would also expect the elastcty of labor supply wth respect to the net-of-tax rate to be an ncreasng functon of τ t need not be. 1 Ω+ε 1 Ω We can rewrte 1 asτ = to llustrate the basc propertes of the optmal tax rate. Because ε 0 and 0 Ω<1, both the numerator and denomnator are nonnegatve. The optmal tax rate s thus between zero and one. The formula captures 7 We consder ncome here as the labor ncome of ndvduals, but consderng that our model s ntended especally for the poorer countres, agrcultural ncome could as well be ncluded n the concept of ncome. In Sect. 5 we dscuss the mplcatons of untaxed home consumpton n agrcultural producton. 8 Summaton s always over all ndvduals, whch s suppressed for smplfcaton.

6 Optmal taxaton and publc provson for poverty reducton 69 neatly the effcency-equty trade-off. τ decreases wth ε and Ω, and we have the followng general results: 1 In the extreme case where Ω = 1,.e., the government does not value redstrbuton at all, τ = 0 s optmal. We can call ths case lbertaran. Accordng to the lbertaran vew, the level of dsposable ncome s rrelevant rulng out both basc ncome b, and other publc expendtures, g, funded by the government. 2 If there s no nequalty, then agan Ω = 1 and τ = 0. There s no nterventon by the government. The nherent nequalty wll be fully reflected n the dsposable ncome. Furthermore, lump-sum taxaton s optmal; b = g or T = b. 3 We can call the case where Ω = 0 as Rawlsan or maxmn preferences. The government maxmzes tax revenue optmal τ = 1+ε 1 as t maxmzes the basc ncome b assumng the worst off ndvdual has zero labor ncome. In fact, maxmzng b can be regarded as a nonwelfarst case, whch s the focus n the next subsecton. 2.2 Optmal lnear ncome taxaton under non-welfarst obectves A non-welfarst government s one that follows a dfferent set of preferences than those employed by ndvduals themselves Kanbur et al Thus, nstead of maxmzng a functon of ndvdual utltes, the government has other, paternalstc obectves that go beyond utltes. A specal case taken up n more detal below s the obectve of mnmzng poverty n the socety. To be as general as possble, let us defne a socal evaluaton functon as n, e.g., Kanbur et al ass = Fc, z, whch the government maxmzes nstead of the socal welfare functon. Fc, z measures the socal value of consumpton c for a person wth ncome z and can be related to uc, z but s not restrcted to t. Followng Tuomala s model as above, gven the nstruments avalable, lnear ncome tax τ, lump-sum grant b and other expendture R the government thus maxmzes Faz + b, z subect to the budget constrant 1 a z b = R. Defne Fc z + aza + F zza Fc 1 + az b + F zz b F, 2 whch reflects the relatve mpact of taxes and transfers on the socal evaluaton functon. Usng ths defnton, and followng the same steps as n the prevous secton see Appendx, the optmal tax rate becomes: τ 1 τ = 1 1 F. 3 ε z The result resembles the welfarst tax rule n 1. In addton to labor supply consderatons va the term 1 ε, they both ental a term that measures the relatve benefts of taxes and transfers, n the welfarst case va welfare-weghted ncome, n the non-welfarst case va F, the relatve mpact on the socal evaluaton functon. ote that snce under non-welfarsm ndvduals are not necessarly at ther utlty optmum, the envelope condton does not apply and thus the behavoral responses za and z b are not cancelled

7 70 R. Kanbur et al. out n F. That s, the mpacts of tax changes on labor supply are not trval under non-welfarsm. The terms za F ca + F z n the numerator and zb F ca + F z n the denomnator of 2 capture these effects on the socal evaluaton functon. If taxaton had no behavoral mpacts za = z b = 0, t would affect the value of the socal evaluaton functon only by mechancally alterng ndvdual after-tax ncome. Fc z ote that n ths case, F = β Fc would be a more drect equvalent to zβ = z β. The same equvalence would be acheved also when F c a + F z = 0, that s, the socal margnal rate of substtuton between ncome and consumpton equals the prvate rate: F z F c = a = u z u c the latter s obtaned from the ndvdual s frst-order condton. In these cases, F would be a purely redstrbutve term, albet a non-welfarstc one. Paternalstc concerns addtonally enter the optmal tax rule va labor supply changes, captured by the response of z. In ths way, the tax rule n 3 can be decomposed, and ths decomposton s smlar n sprt to the correctve parts of the tax formulae n the new optmal tax lterature wth behavoral agents, such as Farh and Gabax 2015 and Gerrtsen The sgns and magntudes of F c and F z and thus of F depend on the specfc obectve of the government, that s, on the shape of F. Let us consder the specfc case of poverty mnmzaton below Specal case: poverty mnmzaton ow let us derve the optmal lnear tax results for a government whose obectve s to mnmze poverty n socety. The nstruments avalable to the government are the same, τ and b, and other exogenous expendture s R. ote frst that the revenue-maxmzng tax rate s n fact equvalent to the tax rate obtaned from a maxmn obectve functon, snce when the government only cares about the poverty consumpton of the poorest ndvdual, ts only goal s to maxmze redstrbuton to ths ndvdual,.e., maxmze tax revenue. Let us frst defne the obectve functon of the government explctly. Poverty s defned as deprvaton of ndvdual consumpton c relatve to some desred level c and measured wth a deprvaton ndex D c, c, such that D > 0 c [0, c and D = 0 otherwse, and D c < 0, D cc 0 c [0, c, asnprttlä and Tuomala A typcal example of such an ndex would be the P α famly of Foster Greer Thorbecke FGT poverty ndces. We dscuss the applcaton of FGT ndces n our model n Appendx 2. ote, however, that the choce of poverty ndex depends on the preferences of the government, whether they wsh to mnmze the total amount of deprvaton n the socety, or are for nstance concerned especally about the ncomes of the poorest of the poor. The socal evaluaton functon Fc, z becomes D c, c and the obectve functon s mn P = D c, c.owf c = D c and F z = 0, so Dc z F = D + az a =, 4 Dc 1 + az b and the optmal tax rule becomes:

8 Optmal taxaton and publc provson for poverty reducton 71 τ 1 τ = 1 1 D. 5 ε z Snce now F z = 0, the result s closer to 1 than 3 was, although part of the labor supply mpacts stll reman. Here D descrbes the relatve effcency of taxes and transfers n reducng deprvaton. Both the numerator and denomnator of D depend on D c, so the dfference n the relatve effcency of the two depends on za and z b.the more people react to taxes relatve to transfers by earnng less, the hgher s D and the lower should the tax rate be. In 1, the hgher s the socal value of ncome, the hgher s zβ and the lower should the tax rate be. Snce the form of the result s smlar n the welfarst and the poverty mnmzaton cases, the analyss could be also seen as a specal case of the argument n Saez and Stantcheva 2016, who derved generalzed socal welfare weghts and express the tax formulae n terms of those. 9 Here, the generalzed socal welfare weght would thus be derved from a poverty mnmzaton obectve. It could be close to a sutably defned welfarst crteron, and clearly t would be exactly the same only f the welfarst crteron would correspond to the chosen poverty mnmzaton obectve. We can also rewrte D, usng a = 1 τ, as: Dc 1+ 1 τ z z 1 τ z Dc 1+1 τz = b Dc z z +1 τ 1 τ Dc 1+1 τz = b Dc 1+ε z Dc 1+1 τz b. Thus the D n the optmal tax result 5 entals a further consderaton that depends on labor supply responses. It combnes paternalstc preferences how much poverty s reduced wth the behavoral responses to a tax system how much labor ncome ncreases when the take-home pay goes up. The latter effect tends to lower the optmal tax rate to nduce the poor to work more. Kanbur et al fnd a smlar result n ther nonlnear poverty-mnmzng tax model. Here, however, we are restrcted to lower the tax on everyone nstead of only the poorest ndvduals. To summarze, the non-welfarst tax rules dffer from the welfarst ones, dependng on the defnton of non-welfarsm n queston the F c and F z terms. However, when we take poverty mnmzaton as the specfc case of non-welfarsm, the tax rules are qute smlar to welfarst ones. The basc dfference s that equty s not consdered n welfare terms but n terms of poverty reducton effectveness. A more notable dfference arses from effcency consderatons. Wth lnear taxaton, takng nto account labor supply responses means that everybody s tax rate s affected, nstead of ust the target group s. If we want to nduce the poor to work more to reduce ther poverty, we need to lower everyone s tax rate. The welfarst lnear tax rule does not take ths nto account. It s not, however, possble to state that under poverty mnmzaton tax rates are optmally lower than under welfare maxmzaton, snce we cannot drectly compare the welfare and deprvaton terms. However, there s an addtonal effcency consderaton nvolved under poverty mnmzaton. onlnear tax rules of course make t possble to target lower tax rates on the poorer ndvduals, but n a developng country context wth 9 We are grateful to a referee for ths pont.

9 72 R. Kanbur et al. lower admnstratve capacty ths s not necessarly possble, and such consderatons affect everyone s tax rate. 3 Publc good provson wth lnear ncome taxes 3.1 Optmal publc provson under the welfarst obectve Let us frst extend the welfarst model of lnear taxaton to nclude the provson of pure publc goods. The government offers a unversal pure publc good G, whch enters ndvdual utltes n addton to the consumpton of prvate goods. The government s obectve functon s now W V a, b, G, whereas the budget constrant becomes 1 a z b πg = R where π s the producer prce of the publc good. The consumer prce of prvate consumpton s normalzed to 1. Let us now defne the margnal wllngness to pay for the publc good by the expresson σ = V G Vb and σ = as the welfare-weghted average margnal rate of substtuton between β σ β publc good and ncome for ndvdual. The rule for publc provson can then be wrtten as π = σ τ σ z b z G. 6 Ths publc good provson rule s a verson of a modfed Samuelson rule. It equates the relatve cost of provdng the publc good to the welfare-weghted sum of margnal rates of substtuton MRS. It also ncludes a revenue term, whch takes nto account the mpacts of publc good provson and ncome transfers on labor supply and thus tax revenue. Consder frst the case when labor supply does not depend on publc good provson and there are no ncome effects,.e., z G = z b = 0. Then we are left wth a more famlar rule that welfare-weghted aggregate MRS must equal the cost of the publc good. When we add ncome effects so that z b < 0, and snce σ s postve, then because of the second term n 6, the fnancng costs of the publc good are reduced. Lkewse, f labor supply and publc provson are postvely related, the fnancng costs of the publc good are reduced. 3.2 Optmal provson of publc goods under poverty mnmzaton ow consder a non-welfarst government nterested n mnmzng poverty. The publc good G whch t offers enters the deprvaton ndex separately from other, prvate consumpton x: D x, G, x, Ḡ. The government stll offers a lump-sum cash transfer b as well and fnances ts expenses wth the lnear ncome tax τ. Agan alternatve formulatons of the publc good provson rule can be wrtten. The frst s π = D τ D z b z G, 7 DG + D x az G Dx 1+az b whch can be compared wth Eq. 6. Here, D = captures the effcency of the publc good n reducng deprvaton relatve to the ncome trans-

10 Optmal taxaton and publc provson for poverty reducton 73 fer because D G, D x < 0, D > 0. Agan, f z G = z b = 0, the equaton reduces to π = D DG = Dx. Ths rule hghlghts a consderable dfference to the standard modfed Samuelson rules, reflectng nstead of a welfare-based MRS the drect poverty reducton mpact of the publc good. Wth z G = 0 and z b = 0, D also depends on the ndrect mpacts of the publc good va labor supply on consumpton. As prevously, the rght-hand sde ncludes a tax revenue term. Usng the same example as n the context of 6, f z G = 0 and z b < 0, the prce π of the publc good would be hgher than ts relatve effcency n elmnatng deprvaton. Here we have allowed the government to be drectly nterested n the consumpton of some pure publc good. But f the government s solely nterested n reducng ncome poverty, t mght not nclude such goods n the deprvaton measure. 10 However, suppose that ndvdual welfare does not drectly depend on the publc good provded but the publc good can have a productvty ncreasng mpact. An example could be publcly provded educaton servces that affect ndvduals productvty va the wage rate. We therefore suppose that the drect mpact of the publc good on deprvaton cancels out.e., D G = 0, whereas the wage rate becomes an ncreasng functon of G,.e., w G >0 denotng z = wgl. Ths means that the expresson for D s rewrtten as D Dx a w G L = + w L. 8 Dx 1 + aw L b Ths means that even f labor supply would not react to changes n publc good provson, such provson would stll be potentally desrable through ts mpact on the wage rate. In ths way, publc good provson can be nterpreted as ncreasng the capablty of the ndvduals to earn a lvng wage, whch serves as a poverty reducng tool, and whch can n some cases be a more effectve way to reduce poverty rather than drect cash transfers. The optmalty depends on the relatve strength of w G >0 versus the drect mpact of the transfers. An alternatve provson rule for the publc good, whch results from extendng the Pketty Saez approach, n the usual case where t also enters ndvduals utlty functon s D G + D x 1 τ z Dx dν G dν = π τ dz dg. 9 In the numerator of the left-hand sde, the frst term s the drect deprvaton effect of G and the second term captures the ndrect deprvaton effect, operatng va the labor supply mpacts of the publc good, whch affect the level of prvate consumpton, x. These mpacts are scaled by the poverty allevaton mpact of prvate consumpton tself the mpact of a cash transfer. The rght-hand sde reflects the costs of publc good provson: besdes the drect cost of the good there s an ndrect tax revenue effect 10 See also Appendx 2 for multdmensonal consderatons n poverty measurement.

11 74 R. Kanbur et al. operatng through labor supply. The condton s drectly comparable to the welfarst rule, gven n 39 n the Appendx, because even though the welfarst case reles on utltes, n the FOC for G no envelope condton s evoked. The only dfference between Eqs. 39 and 9 s that the utlty and welfare weght terms are exchanged for deprvaton terms. Consder fnally the provson of a quas-prvate good, such that n addton to the publcly provded amount, ndvduals can purchase top-up the good themselves as well. The good s denoted by s and ts total amount conssts of prvate purchases h and publc provson G: s = G + h. In addton to good s, ndvduals consume other prvate goods, denoted by x. The ndvdual budget constrant s thus c = x + ph = 1 τz + τ Z1 τ R πg, where p s the consumer prce of prvate purchases of the quas-prvate good. The producer prce of educaton n the prvate sector p or n the publc sector π can be equal, or one sector could have access to cheaper technology. Deprvaton s determned n terms of consumpton of x and s, sothe obectve functon s mn P = D x, s, x, s dν. In ths case, the provson rule s [ D x 1 τ z s G s p h G Dx dν ] s + D s G dν = π τ dz dg. 10 The result s analogous to the pure publc good result n 9, wth the dfference that now the mpact G has on poverty depends on whether publc provson fully dh ds crowds out prvate purchases of the good.e., dg = 1 dg = 0 or not.e., dh dg = 0 dg ds = 1. If there s full crowdng out, an ncrease n publc provson of G that s fully funded va a correspondng ncrease n the tax rate has no mpact on the consumpton of s and consequently no mpact on poverty. If there s no crowdng out, however, the FOC becomes [ ] D x 1 τ z s + D s dν = π τ dz Dx dν dg, 11 whch s the same as n the case of a pure publc good n Eq. 9. To summarze, the welfarst publc provson rule, when publc goods are fnanced wth lnear ncome taxes and supplemented wth lump-sum transfers, dffers from the standard modfed Samuelson rule. It equates a welfare-weghted sum of MRS to the margnal cost where tax revenue mpacts are taken nto account. Indrect effects of publc provson through labor supply decsons and thus prvate consumpton are ncorporated. The poverty-mnmzng publc provson rule, however, replaces the welfare-weghted sum of MRS wth the relatve margnal returns to deprvaton reducton. Here the MRS term measures how well publc good s translated to reduced poverty ncorporatng ndrect effects as well, relatve to prvate consumpton. Fnally, when the publc good has postve effects on productvty, ts provson can be desrable even f t would not have any drect mpact on poverty.

12 Optmal taxaton and publc provson for poverty reducton 75 4 Commodty taxaton wth lnear ncome taxes 4.1 Optmal commodty taxaton wth lnear ncome tax under the welfarst obectve Ths secton consders the possblty that the government also uses commodty taxaton subsdes to nfluence consumers welfare. We follow the modelng of Damond Unlke the analyss above, there are J consumer goods x nstead of ust two. Workng wth many goods s used to be able to more clearly descrbe the condtons under whch unform commodty taxaton occurs at the optmum. The government leves a tax t on the consumpton of good x, so that ts consumer prce s q = p +t, where p represents the producer prce a commodty subsdy would be reflected by t < 0. Letq denote the vector of all consumer prces. In addton, the government can use a lump-sum transfer, b. ote that n ths exposton, lesure s the untaxed numerare commodty. Alternatvely, one could also mply a lnear tax on labor supply as above and treat one of the consumpton goods as the untaxed numerare. However, choosng lesure as the numerare makes the exposton easer. Thus, the consumer s budget constrant s q x = z + b. The government maxmzes W V b, q subect to ts budget constrant t x b = R. It s useful to defne, followng Damond 1975, γ = β + λ t x b 12 as the net socal margnal utlty of ncome for person. Ths noton takes nto account the drect margnal socal gan, β, and the tax revenue mpact arsng from commodty demand changes. The rule for optmal commodty taxaton for good k s shown to be 1 x k t = 1 q λ covγ, xk. 13 The left-hand sde of the rule s the aggregate compensated change weghted by commodty taxes of good k when commodty prces are changed. The rght-hand sde refers to the covarance of the net margnal socal welfare of ncome and consumpton of the good n queston. The rule says that the consumpton of those goods whose demand s the greatest for people wth low net socal margnal value of ncome presumably, the rch should be dscouraged by the tax system. Lkewse the consumpton of goods such as necesstes should be encouraged by the tax system. The key polcy queston s whether or when unform commodty taxes are optmal, or, n other words, when would a lnear ncome tax combned wth an optmal demogrant be suffcent to reach the socety s dstrbutonal goals at the smallest cost. Deaton 1979 shows that weakly separable consumpton and lesure and lnear Engel curves are suffcent condtons for the optmalty of unform commodty taxes. These requrements are qute strngent and unlkely to hold n practce; however, the economc mportance they mply s unclear. If mplementng dfferentated commodty

13 76 R. Kanbur et al. taxaton entals sgnfcant admnstratve costs, they may easly outwegh the potental benefts of dstrbutonal goals and that s why economsts have typcally been qute skeptcal about non-unform commodty taxaton when appled to practcal tax polcy. 4.2 Optmal commodty taxaton wth lnear ncome tax under poverty mnmzaton Poverty could be measured n many ways when there are multple commodtes: the government may care about overall consumpton, the consumpton of some of the goods those that are n the basket used to measure poverty or then t cares about both the overall consumpton and the relatve share of dfferent knds of consumpton goods such as mert goods. We dscuss these measurement ssues n Appendx 2,but here we examne the smplest set-up where deprvaton only depends on dsposable ncome, c = z + b. Usng the consumer s budget constrant, ths s equal to the overall consumpton level, q x. The government thus mnmzes the sum of the poverty ndex D the budget constrant s the same as before. It s agan useful to defne q x, c, and γp = D x c q b + λ t x b 14 as the net poverty mpact of addtonal ncome for person. Ths noton takes nto account the drect mpact on poverty and the tax revenue mpact arsng from commodty demand changes. As shown n Appendx 1 secton Commodty taxaton, ths leads to an optmal tax rule as below: 1 t x k q = 1 λ 1 D c xk + 1 x k D c q q + 1 λ cov γ P, x k 15 In ths formulaton, the left-hand sde s the same as n the welfarst case and t reflects the aggregate compensated change n the demand of good k. The frst two terms n the square brackets at the rght-hand sde capture the mpacts of tax changes on poverty: the frst term s the drect mpact of the prce change keepng consumpton unaffected on measured poverty, whereas the second depends on the behavoral shft n consumpton. Multpled by the mnus sgn, the former term mples that the consumpton of the good should be encouraged, whereas f demand decreases when the prces ncrease, the latter term actually serves to dscourage consumpton. The last term on the rght reflects the same prncples as the covarance rule n Eq. 13, the correlaton of the net poverty mpact of ncome and the consumpton of the good n queston. That s, the covarance part of the tax rule moves the tax rule n the drecton of favorng goods that have hgh poverty reducton mpact on the poor.e., that the poor consume more..

14 Optmal taxaton and publc provson for poverty reducton 77 The key lesson to note from the optmal commodty tax rule n the poverty mnmzaton case s that the conventonal condtons for unform commodty tax to be optmal are not vald anymore. The reason s that even f demand was separable from labor supply, the frst term on the rght stll remans n the rule, and ts magntude clearly vares dependng on the quantty of good consumed. Thus, ncome transfers are not suffcent to allevate poverty when the government ams to mnmze poverty that depends on dsposable ncome. The ntuton s very smple: commodty tax changes have a drect effect on the purchasng power of the consumer, and these depend on the amount consumed. The extent of encouragng the consumpton of the goods s the greater, the larger s ther share of consumpton among the consumpton bundles of the poor. The result resembles that of Prttlä and Tuomala 2004, meanng that the ntuton from optmal nonlnear ncome taxaton under poverty mnmzaton carres over to lnear ncome taxaton. A formal proof s provded n Appendx 1. In sum, the rule for optmal commodty taxaton s changed when we shft from welfare maxmzaton to poverty mnmzaton. The welfarst rule reflects a farly straghtforward trade-off between effcency tax revenue and equty dstrbutonal mpacts. The poverty-mnmzng commodty tax rule brngs new terms; the nterrelatons of whch are not easy to dsentangle. It, however, also takes nto account effcency consderatons tax revenue through ndrect labor supply effects and equty drect mpact of the taxed good on poverty and ndrect mpact va labor supply effects. Most mportantly, the conventonal wsdom of when unform commodty taxaton s suffcent fals to hold n the poverty mnmzaton case. Thus, observed commodty subsdes n developng countres, such as fuel or food subsdes, can be consdered optmal gven the preference for poverty mnmzaton. 11 In practce, t would be wse to lmt the number of dfferentated commodty tax rates to a few essental categores such as fuel and food, n order to keep the admnstratve complexty at a mnmum. 5 Poverty mnmzaton n the presence of an nformal sector An mportant ssue for a developng country attemptng to collect taxes s the ssue of a large nformal sector. If part of tax revenue s lost due to tax evason n the nformal sector, whch s lkely to be the case n the less developed economes, then the ncome transfer s reduced and redstrbutve targets may not be met. In ths secton, we dscuss the mplcatons of nformalty for optmal redstrbutve polces for a government wshng to mnmze poverty. 12 The results can thus be contrasted to those obtaned n prevous sectons. 11 Keen 2014 uses a tax reform approach and examnes how much more effectve transfers need to be than dfferentated commodty subsdes n reachng the poor to acheve the same poverty reducton wth lower government outlays. 12 Such a socety mght also reflect poor admnstratve power and corrupton n the tax collectng authorty. otce, however, that consderng only the leakage of tax revenue n the model would only reduce the extent of poverty reducton acheved wth taxaton by lowerng the ncome transfer for everyone. The poverty reducton effcency of taxaton would thus be lowered, but there would be no dfferental effects across ndvduals.

15 78 R. Kanbur et al. Followng Kanbur 2015 and Kanbur and Keen 2014, nformal operators can be categorzed as those who should comply wth regulatons but llegally choose not to, and those who legally reman outsde regulaton, e.g., due to the smaller sze of operatons ether naturally or by adustng sze as a response to regulaton. For our purposes, however, t s enough to lump these categores nto one nformal sector, where t s possble to avod taxes at least to some extent. It s also possble for workers to work n both sectors, such that part of total ncome s declared for taxaton and part s evaded consder, e.g., supplementng offcal employment ncome wth street vendorng. ote also that especally n the case of agrculture, evason can also consst of home producton. In ths case, the reason for nformalty would be the small sze of the producng entty, such that they are naturally not lable for taxes. Producton for own consumpton s, however, stll relevant for the well-beng and measured poverty of the famly. In ths applcaton, we follow the approach poneered n Besley and Persson They work wth a model that fts nto the descrpton above, where part of the tax base evades taxes. We thus take nformalty as gven, and do not consder whether nformalty s natural, llegal or a response to taxaton. Furthermore, ths ntensve margn model what extent of ncome s earned n the nformal sector, they argue, yelds essentally smlar results as an extensve margn model whether to partcpate n the formal ob market. Consder the case of ncome taxaton. We can ncorporate nformalty nto the model by notng that people can shelter part e of ther labor ncome from taxaton. The extent of evason s assumed to ncrease when the tax rate goes up, and thus e a < 0. Income taxes are only pad from ncome z e. It s noteworthy that for a government wshng to mnmze ncome poverty, ths s n fact benefcal: dsposable ncomes rse. The more ths effect s concentrated among the poor who enter the deprvaton ndex, the better. Indvdual consumpton s now z τz e + b = e +az e +b. On the other hand, tax collectons are reduced: the budget constrant becomes 1 a z e = b+ R. Our formulaton follows that of Besley and Persson 2013, but we smplfy t n order to explctly consder the problem of optmal taxaton, whereas they focus on the ssue of nvestments n the state s fscal capacty we abstract from ths ssue here and take evason as gven. 13 The framework, however, ncely captures the essental trade-offs a government faces when there s tax evason. The government now mnmzes the Lagrangan L = D e + az e + b, c + λ1 a z e b R. The frst-order condton wth respect to the netof-tax rate s: e z Dc a + z e + a a e a z = λ e 1 a z a e a, Another dfference s that n ther orgnal formulaton, people face costs of evason. When the tax rate goes up, the relatve attractveness of tax evason ncreases, producng the same knd of effect e a < 0 we assume drectly here for brevty. These costs could be related to, e.g., Allngham Sandmo-type rsk of beng caught and facng sanctons. Also Slemrod s 1990 revew suggests that hgher tax rates tend to ncrease the supply of labor to the nformal sector.

16 Optmal taxaton and publc provson for poverty reducton 79 whereas, under the assumpton that there are no ncome effects n evason, the frstorder condton wth respect to b stays the same. From here, we can derve a rule for the optmal tax followng the same steps as n Sect. 2.2: τ 1 τ = 1 ε e 1 D e z e, 17 where now ε e s a tax elastcty of the net-of-evason tax base z e = z ē and D e represents the relatve mpact of taxes and transfers on the deprvaton ndex see Appendx 1 for further detal. The rule represents a trade-off between poverty reducton and effcency, both of whch are now altered by evason. There s a pressure toward lower tax rates, as now dstortons of taxaton are ncreased by evason behavor, so ε e >ε. Contrary to ths effect, D e s reduced compared to D because reducng taxes ncreasng a s now a less useful nstrument for poverty reducton, as part of < 0, people pay more taxes when tax rates are reduced, and therefore poverty n fact ncreases. D e thus works to ncrease tax rates. Therefore, an nterestng trade-off arses: nformalty ncreases the cost of rasng taxes, but t also means that hgher taxes are less harmful as those n the nformal sector do not need to pay them and they are stll enttled to the lump-sum transfer. 14 These countervalng forces have not been noted by the lterature before. The presence of nformalty therefore seems to gve rse to tax polcy rules that are far from trval. Future work could also look more deeply nto the ssue of the tax mx n the presence of nformalty. If ncome tax s more easly evaded than commodty taxaton, as Boadway et al suggest, ths could gve rse to polces that focus taxaton and redstrbuton on commodty taxes and subsdes, nstead of ncome taxes and lumpsum transfers. Slemrod and Glltzer 2014 have also suggested focusng on a tax systems approach and ncludng, among other thngs, evason behavor nto optmal taxaton analyss to obtan more useful prescrptons for actual tax polcy. Ths topc certanly deserves a more detaled analyss. the taxes have been evaded. As e a 6 A numercal llustraton To further llustrate the dfferences of tax rates under poverty mnmzaton and welfarsm, we provde a smple numercal smulaton. Here we concentrate on the specal case where there are no ncome effects on labor supply and the elastcty of labor supply wth respect to the net-of-tax wage rate s constant. If ε denotes ths elastcty, the quas-lnear ndrect utlty functon s gven by vw1 τ,b = b + [w1 τ]1+ε 1+ε,so that ε s constant. Lke most work on optmal nonlnear and lnear ncome taxaton, we use the lognormal dstrbuton lnn, mσ 2 to descrbe the dstrbuton of productvtes wth support [0, and parameters m and σ see Atchson and Brown The frst parameter, m, s the log of the medan wage. The second parameter, the varance 14 The dea that those n the nformal sector can stll receve transfers matches well wth realty: many of the cash transfer systems reach those wth lttle or no connecton to the formal sector.

17 80 R. Kanbur et al. of log wage σ 2, s tself an nequalty measure. As s well known, the lognormal dstrbuton fts reasonably well over a large part of the ncome range but dverges markedly at both tals. The Pareto dstrbuton n turn fts well at the upper tal. We also use the two-parameter verson of the Champernowne dstrbuton known also as the Fsk dstrbuton. Ths dstrbuton approaches asymptotcally a form of Pareto dstrbuton for large values of wages but t also has an nteror maxmum. In our smulatons, the revenue requrement s set to zero; thus, the system s purely redstrbutve. To llustrate the poverty-mnmzng tax formula n 3, we also need to specfy a measure of poverty. Typcally, poverty ndces consst of computng some average measure of deprvaton by settng ndvdual needs as defned above at the agreed upon poverty lne c. For ths purpose, we take a poverty ndex of the form developed by Foster et al They have proposed defnng a poverty ndex as the average of these poverty gaps across ndvduals rased to some power α. When α = 1, t s ust the proporton of unts below the poverty lne multpled by the average poverty gap. See Appendx 2 for more detals. We consder the cases where ether 30 or 40% of the populaton le below the poverty lne. The results from the smulaton of the optmal tax when the government mnmzes the poverty gap for the lognormal case are presented n Table 1. Results are shown for two dfferent values of labor supply elastcty ε, two dfferent values regardng ncome dsperson σ, and two values of the share of populaton below the poverty lne F w. The tax rates are hgh, above 60%, for all the combnatons of parameter values. 15 Comparng these results to the welfarst case s not straghtforward, as those depend on the chosen welfare functon. We adopt a constant relatve nequalty averson form of the welfare functon: the contrbuton to socal welfare of the th ndvdual s w 1 η 1 η, where η s the constant relatve nequalty averson coeffcent. Hence, the socal margnal value of ncome to an ndvdual wth wage rate w s proportonal to w η. Usng the property of the lognormal dstrbuton lnew s = sm + s 2 σ 2 2, we can τ calculate the optmal tax rate from the followng formula: 1 τ = 1 ε [1 e η1+εσ 2 ]. Or, usng the property of the lognormal dstrbuton that ln1 + cv 2 = σ 2, where cv 1 s the coeffcent of varaton, we can rewrte τ =. 1+ε/[1+cv 2 ] η1+ε A wde range of values for the nequalty averson parameter η have been employed n the lterature, varyng typcally from 0.5 to 2. ote that, as dscussed n Sect. 2.1,as η, socal preferences approach maxmn preferences, where the optmal tax rate s the same as the revenue-maxmzng tax rate, τ = 1+ε 1, whch does not depend on the orgnal ncome dstrbuton. aturally, f there s no regard for nequalty n the socety, η = 0 and τ = 0. Table 2 dsplays the welfarstc tax smulaton results for two dfferent values of labor supply elastcty ε, for two dfferent values of ncome dsperson σ, and for fve dfferent values of nequalty averson η. The smulaton results llustrate clearly that at conventonal nequalty averson levels, optmal welfarstc tax rates le well below the poverty-mnmzng rates. Only as nequalty averson becomes extremely hgh do the welfarstc rates approach the 15 The results are very smlar usng the Champernowne dstrbuton wth ncome dsperson parameters chosen so that nequalty s smlar n both cases, whch s not very surprsng as the dstrbutons only dffer at the top of the ncome schedule. These results are avalable upon request.

18 Optmal taxaton and publc provson for poverty reducton 81 Table 1 Smulated tax rates for poverty mnmzaton under dfferent values of ε, σ,andf w ε σ = 0.7 σ = 1.0 F w = 0.3 F w = 0.4 F w = 0.3 F w = Table 2 Smulated tax rates n the welfarstc case under dfferent values of ε, σ,andη ε σ = 0.7 η = 0.5 η = 1 η = 2 η ε σ = 1.0 η = 0.5 η = 1 η = 2 η poverty-mnmzng ones. Wth poverty mnmzaton as the socal obectve, optmal tax rates are close to the revenue-maxmzng maxmn rate. Another pont of comparson could be the welfarstc lnear tax smulatons of Stern Hs calculatons dffer from ours as he ncorporates ncome effects and a nonconstant elastcty of labor supply wth respect to the tax rate. 16 Wth the elastcty of substtuton between consumpton and lesure at 0.5 and ncome dsperson descrbed by σ = 0.39, as concern for nequalty rses from low to medum and hgh, he fnds tax rates rsng from 19 to 43 and 48%. The extreme maxmn result s 80%. These tax rates are also clearly lower than the poverty-mnmzng rates, except at very extreme values of nequalty averson. These numercal examples and Stern s 1976 results tend to suggest that the tax rates for the poverty mnmzaton case are lkely to be hgher than for many welfarst examples. The results compare to Kanbur et al. 1994, who also found that the nonlnear margnal tax rates on the poor are farly hgh under the poverty mnmzaton obectve. Both ther and our results are nterestng from the pont of vew that the analytcal formulae for the optmal tax rate nclude a term that, ceters parbus, encourages labor supply, but n computatonal results ts nfluence s offset, most lkely, by the need to mnmze the poverty gap. The hgher the poverty rate, the hgher the lumpsum grant fnanced by these taxes needs to be, n order to rase more people out of poverty. 16 Our smplfyng assumptons allow us to provde tax rates wth respect to the three parameters n Table 2.

19 82 R. Kanbur et al. 7 Concluson Ths paper examned optmal lnear ncome taxaton, publc provson of publc and prvate goods and the optmal combnaton of lnear ncome tax and commodty taxes when the government s am s to mnmze poverty. The lnear tax envronment was chosen because such taxes are more easly mplementable n a developng country context and snce optmal lnear tax rules are seen to provde smlar ntuton as the more complex nonlnear tax formulas. The results show that the lnear ncome tax ncludes addtonal components that work toward lowerng the margnal tax rate. Ths result arses from the goal to boost earnngs to reduce ncome poverty. Unlke n the optmal nonlnear ncome tax framework, ths lower margnal tax affects all taxpayers n the socety. However, the numercal smulatons offered suggest that ths mechansm s offset by the dstrbutve concerns and n practce the optmal tax rates for poverty mnmzaton appear hgh. Publc good provson n the optmal tax framework under poverty mnmzaton was shown to depend on the relatve effcency of publc provson versus ncome transfers n generatng poverty reductons. One partcular avenue where publc provson s useful s va ts potentally benefcal mpact on ndvduals earnngs capacty. Thus, publc provson can be desrable even f ts drect welfare effects were non-exstent. Perhaps more mportantly, poverty mnmzaton as an obectve changes completely the condtons under whch unform commodty taxaton s optmal. When the government s obectve s to mnmze poverty that depends on dsposable ncome, unform commodty taxaton s unlkely to be ever optmal: ths s because the commodty tax changes have frst-order effects on consumers budget va the drect mpact on the cost of lvng, and ths drect effect depends on the relatve mportance of dfferent goods n the overall consumpton bundle. Separablty n demand coupled wth lnear Engel curves s not suffcent to guarantee optmalty of unform commodty taxes. In realty, the admnstratve dffcultes of mplementng commodty taxaton wth many tax rates must, of course, be taken nto account, as well. We also examned the mplcatons of the presence of an nformal sector for optmal tax and transfer polces. The results revealed that when the government s concerned about ncome poverty, the presence of the nformal sector s, on the one hand, useful, as t reduces the poverty-ncreasng effect of hgher taxes but, on the other hand, t s also costly snce t s lkely to ncrease the elastcty of the tax base. Examnng the mplcatons of nformalty on the role of other nstruments of government polces s an mportant avenue for future work. Another strand of follow-up work should address the queston of complementary polces for redstrbuton, such as mnmum wages. It should be borne n mnd that dfferent polces mpose dfferent requrements on admnstratve capacty, 17 and 17 For example, Lee and Saez 2012 show how a mnmum wage polcy can usefully complement an optmal nonlnear ncome tax and transfer polcy under welfarst obectves. However, mposng mnmum wage regulaton mples that the government needs to be ether able to observe ndvdual wage rates, or has suffcent nsttutonal strength to rely on whstleblowers to denounce non-complyng employers, n order to enforce the legslaton.