The Optimal Labor Income Tax Schedule for Brazil

Transcription

1 The Optimal Labor Icome Tax Schedule for Brazil Thiago Neves Pereira Abstract We compute the optimal o-liear icome tax schedule for Brazil, followig the lead of Mirrlees (97) ad Saez (2). Our first cotributio is to take ito accout the o-liearity that is preset i the curret labor icome tax system whe we back up the distributio of skills from the labor icome data. We use a separable specificatio for prefereces ad cosider differet levels of risk aversio. By varyig the level of risk aversio we capture both variatios i the prefereces for redistributio ad variatios i the elasticity of labor supply. We compare the welfare gais associated with chagig the curret tax system. We also cosider the loss i efficiecy that arises as a cosequece of our restrictig ourselves to affie icome tax schedules. JEL Classificatio: H, H2, H3 Keywords: Optimal No-liear Taxatio, Distributio of Skills, Affie Tax System Itroductio I this paper we aalyze ad propose some optios of icome tax reform. Our attetio is o idirect (cosumptio taxes) ad icome taxatio. Our first target is the simulatios of optimal icome taxatio i Brazil. Our mai refereces are Mirrlees (97) ad Saez (2). Our first departure from Saez (2) is the approach we use to create the skill distributio. I Saez (2), the distributio of skills is backed up from the empirical distributio of icome by approximatig the labor icome tax schedule by a liear oe. That is, give a assumed utility fuctio defied over labor ad cosumptio oe derives the level of skills that is compatible with the observed taxable icome, uder this liear tax schedule. We, istead, allow for the o-liearities that characterize the Brazilia icome tax schedule. Ph.D Cadidate. Graduate School of Ecoomics (EPGE) at Fudação Getulio Vargas/Rio de Jaeiro- Brazil, tpereira@fgvmail.br. This is joit work with Carlos da Costa.

2 We derive optimal icome taxes for differet values of risk aversio uder a separable specificatio for prefereces. By varyig the degree of risk aversio we capture both chages i the willigess to redistribute icome ad the elasticity of labor supply. observe that the optimal tax schedule is sesitive to the risk aversio ad labor supply elasticity. I our simulatios, we cosider a risk aversio parameter equal, smaller ad bigger tha those cosidered by Mirrlees (97) ad Saez (2). Our labor supply elasticity parameter has a smaller value tha Saez (2). 2 With these parameters, the compesated ad ucompesated elasticities ad the icome effects vary across the skill distributio. As we see below, the compesated elasticity has a importat role i the optimal icome taxatio formula. We show that the margial tax rate is icreasig i the risk aversio ad labor supply elasticity parameters. The secod goal of this paper is propose ad aalyze a ew simple icome tax schedule i Brazil. This ew tax schedule is completely defied by a costat margial tax rate ad a cash trasfer per head. The idea is to fid a feasible ew tax schedule that is more efficiet tha the curret tax schedules. Other importat characteristic i the ew schedule is to be easy to implemet. Additioally, we wat to ivestigate the effects of cash trasfer program i the labor supply, cosumptio ad utility. The paper is orgaized as follows. I the sectio 2 the geeral model is itroduced. First we preset the bechmark model ad after the Mirrlees model. I the sectio 3 we explai our data ad the methodology. The sectio 4 is dedicated to our results. We split this sectio i four parts. I the first part we preset the results to bechmark model. I the secod part we preset optimal icome taxatio results i Brazil. I the third part we preset the optimal taxatio i Uited State accordig to our procedure. I the forth part we preset the results to the ew tax schedule i Brazil. Sectio 5 is devoted to cocludig remarks. I the Appedix we show the complete derivatio of our model. Chetty (26) uses labor supply elasticities to boud risk aversio. 2 He cosidered the labor supply elasticity equal to 3 ad 5 i his simulatios We 2

3 2 Model 2. The Bechmark Model Followig Mirrlees (97), cosider a two good model i which a cotiuum measure oe of idividuals have idetical prefereces defied over cosumptio, c, ad effort, l, represeted by u(c, l). u(.) is a smooth strictly cocave fuctio, strictly icreasig i c ad decreasig i l. Idividuals differ with respect to their productivity,, which we shall also refer to as their skill. The cross-sectioal distributio of productivity is F (.) with associated desity f(.). A idividual with productivity eeds to put effort l = z/ to geerate output z whereas a idividual with productivity eeds to put effort l = z/ to geerate the same output. Because we cosider a competitive ecoomy, z is also a idividual s earigs, z = l, ad we shall use the terms output ad earigs iterchageably. We may represet the prefereces of a idividual of productivity i the cosumptio earigs space as U(c, z; ) = u(c, z/). We shall make extesive use of this latter represetatio whe we derive the optimal icome tax schedule. For ow, however, ote that give ay labor icome tax schedule T (.) ote that a idividuals budget costrait is c z T (z). Now, cosider a idividual who is earig z, ad payig a total of T (z) i taxes. If T (z) is differetiable at z let τ = T (z) be the margial tax rate ad R = τz T (z) be the virtual icome. 3 We may use this liear approximatio of a type- idividual s budget costrait aroud his optimal choice, z, c z( τ) + R, to defie the followig problem: max U(c, z/) () c,z s.t. c = z( τ) + R. (2) The solutio to this problem defies z( τ, R, ) ad c( τ, R, ). Followig Saez (2) we further defie the compesated ad ucompesated price elasticities as well as the icome elasticity of earigs through ζ c = τ z z ( τ), u 3 If the tax schedule is ot affie, the τ is itself a fuctio of z. 3

4 ζ u = τ z z ( τ), ad respectively. η = ( τ) z R, The Slutsky equatio, ζ c = ζ u η, (3) relates the three elasticities. The compesated elasticities is always o-egative. I all the discussio that follows we assume that leisure is ot a iferior good. The fuctioal forms that we use i the umeric exercises will also imply this property. 2.2 The Mirrlees Model For sake of completeess, i this sectio we derive the optimal icome taxatio schedule, followig Mirrlees (97). We, the, relate the expressios foud herei to those i Saez (2). The mai advatage of his represetatio is to relate optimal tax formulae to well kow empirical parameters. Our focus o Mirrlees (97) is due to the fact that, although Saez (2) provides a iterestig way of derivig the expressios for the optimal tax, his approach is ot suited for the actual computatioal implemetatio. As discussed before, a idividual s cosumptio as a fuctio of his or her earigs is c(z) = z T (z), where T (z) is the tax fuctio. The govermet chooses a tax schedule to maximize a give social welfare fuctio G(u) subject to raisig eough resources to meet a exogeously give expediture level. Mirrlees (97) tackled this very complex problem by solvig the primal mechaism desig program of maximizig W = ˆ G(u())f()d, where G(.), is a icreasig ad cocave fuctio of utility u() = U(c(), z(); ). This maximizatio is subject to a resource costrait of the form ˆ (z() c()) f()d E, (4) 4

5 where E is the govermet expeditures, ad a icetive compatibility costrait, arg max U(c(m), z(m), ). (5) m Recall that f() is the desity associated with the cumulative distributio of skill F (), [, ], where >. Uder sigle-crossig, which obtais i our case if cosumptio is ormal, 4 the local ecessary coditios U c (c(), z(), )c () + U z (c(), z(), )z () = (6) ad z () may substitute for the global costrait (5). As it turs, it is sometimes easier to work i the (u, z) space by defiig fuctio ψ implicitly through The, usig the evelope coditio U(ψ(u(), z(), ), z(), ) = u(). U (ψ(u(), z(), ), z(), ) = u (), to substitute for (6), we have completely elimiated c() from the program. This will be useful for us to provide a heuristic examiatio of the optimal tax formulae. Toward this ed let us specialize to the case of separable iso-elastic prefereces ad igore for the momet the mootoicity coditio. For lack of a better ame, we call this problem without the mootoicity costrait, a relaxed problem. Let where h(l) = l γ /γ ad ν(c) = c ρ ( ρ). u(c, l) = ν(c) h(l) Next, defie θ = γ ad let φ(θ) is the desity iduced by f(). Abusig otatio somewhat, let u(θ) = v(ψ(u(θ), z(θ), θ)) θz(θ) γ /γ, 4 Normality of cosumptio is sufficiet, ot ecessary. 5

6 the usig the evelope coditio 5 u (θ) = z(θ) γ /γ Now, the Lagragia associated with the plaer s problem is ˆ θ { u(θ)φ(θ) + µ(θ) [ u (θ) + z(θ) γ /γ ] +λ [z(θ) ψ(u(θ), z(θ), θ) E] φ(θ) dθ. } Itegratig the secod term by parts we get ˆ θ { u(θ)φ(θ) µ (θ)u(θ) + µ(θ) } γ z(θ)γ + λ [z(θ) ψ(u(θ), z(θ), θ) E] φ(θ) dθ It is the straightforward to see that which, from µ( θ) = µ() = implies µ (θ) = φ(θ) λφ(θ)ψ u (u(θ), z(θ), θ), λ = {ˆ θ c(θ) ρ φ(θ)dθ}, µ( θ)u( θ) + µ()u() ad µ(θ) = ˆ θ c( θ) ρ {ˆ θ c(θ) ρ φ(θ)dθ } φ( θ)d θ Next, ote that ca be writte z(θ) γ φ(θ) ˆ θ {ˆ θ } µ(θ) λφ(θ) z(θ)γ = λ { θz(θ)γ c(θ) ρ c(ˆθ) ρ φ(ˆθ)dˆθ c( θ) ρ } φ( θ)d θ = θz(θ)γ c(θ) ρ 5 Alteratively we could use Milgrom ad Segal (22) to write the icetive compatibility costrait as u(θ) = u( θ) + γ ˆ θ θ θz( θ) γ d θ. 6

7 This expressio fully describes how margial tax rates are determied i this simple settig. A full derivatio for the case with o-separable utility is foud i Mirrlees (97, 976). I the appedix, we discuss the problem that arises whe the solutio to this relaxed problem is ot mootoic. This expressio is the oe used i our computatioal exercise. Yet it coveys little about the forces at work. We shall, the, preset (ad show the derivatio i the appedix) a more ituitive expressio foud i both Diamod (998) ad Saez (2). We maitai the separability assumptio U lc =, ad write T (l) T (l) = ( + ζu ) ζ c ( ) F () u c () f() ˆ [ ] ( ) u c (m) G (u m ) f(m) dm. (7) λ F () The subscripts represet the fact that a parameter is computed at the skill level. There are also two trasversality coditios, µ() = µ( ) =, which allows us to derive the expressio for λ, λ = G (u m )f(m)dm u c(m) f(m)dm. Terms ζ c ad ζ u are respectively the compesated ad ucompesated elasticities of eared icome. Equatio (7) ca be rewritte as where T (l) T = A()B()C()D(), (8) (l) A() = ( + ζu ) ζ c, B() = F () f(), C() = u c (), ad D() = ˆ [ ] ( ) u c (m) G (u m ) f(m) dm. λ F () The A() term expresses labor supply resposes i ucompesated ad compesated elasticities. Keepig others thigs costat, the margial tax rate is decreasig i ζ c ad ζ u. As described by Tuomala (26), the greater icome effect i absolute compared to 7

8 ucompesated effect implies i a higher margial tax rate. As the elasticity varies across populatio, it is importat to kow how the elasticity varies with the wage rate. The secod term, B() is kow as the iverse hazard ratio. It tells us how the shape of the distributio of skills affects the optimal margial tax at the level. Whe we icrease the margial tax rate at the some, we collect more reveue from idividuals whose productive is above. Sice these idividuals are F ( ) i umber, the higher this term the largest the gai from icreasig the margial tax rate. The very same icrease results i a lower output of idividuals of skill. Accordig to equatio (7), the margial tax rate is higher whe f() is lower ad f() is smaller. Note that, if we raised the margial rates o very low earigs, we would substatially raise the tax reveue, for most of the taxpayers have earigs higher tha this low level. Moreover, the higher margial rates at the bottom have very margial effects to this group due to the low value of. The third term, C(), reflects both icome effects, ad differeces i the social value of icome. The higher the icome effects (i absolute value), the higher is the margial tax rate. I this kid of model, whe there is the icome effects, the govermet is cocered with the icome iequality amog the idividuals, whilst i the quasi-liear case, the govermet is worried about the utility iequality. The fourth term, D(), icorporates distributioal cocers.the itegral i m measures the social welfare gai from slightly icreasig the margial tax rate at ad distributig as a poll subsidy to those below. The itegral icreases i util a skill level ad decrease after. The turig poit depeds o the Lagrage multiplier, λ, also kow as margial social cost of public fuds. Sice the itegral affects the margial tax rate positively, this meas that the rage over which the latter icreases also stretches further. Therefore, more tax reveue leads to a less progressive tax structure. The ituitio is that the lower is the reveue requiremet, the more the govermet ca afford to support the poor by a geerous poll subsidy, recoupig at least part of this by a patter of risig the margial tax rate o the better off. The exact patter of this term i the equatio (7) follows as rises depeds o the social welfare fuctio ad the shape of the skill distributio. So the shape of the skill distributio is also importat here. Moreover it is obvious i the itegral term i the equatio (7) that the fuctioal form of u c has the importat role i the determiig the shape of the schedule. 8

9 Note that if we specialize (7) to the prefereces we have used, T (z()) T (z()) = + ζu c() ρ ζ c f() ˆ which is quite close to the expressio we foud. [ c(m) ρ ˆ ] c() ρ f()d f(m)dm, 3 Methodology 3. Icome Data I this paper, oe of our objective is to fid the optimal icome tax schedule for Brazil. Our first step is to cosider the empirical icome data of Natioal Household Sample Survey (PNAD 6 ) i 26. As every survey, PNAD is subjected to errors i its collectio or i differet way. Tryig to elimiate part of these troubles, we have refied the origial icome sample. I the first procedure, we have chose idividuals that work betwee 3 ad 6 hours per week. Additioally, we also have chose idividuals with icome more or equal tha R$ (oe hudred reais) ad less or equal tha R$2, per moth. After these procedures, we have 33, 8 observatios i our sample which we use to create distributio of skills. I 26, the Brazilia s icome tax system had three margial tax rate. The first icome margial tax rate was % for idividuals icome betwee R$, per moth. The secod group facig a margial tax rate of 5% icluded idividuals with icome above R$, ad up to R$2, Fially, the idividuals with icomes above R$2, 52.8 faced a margial tax rate equal to 27.5%. However, i our metric of taxatio, besides the icome tax, we also cosider the idirect levy. The idirect levy or cosumptio tax has the same icidece o all groups. Accordig to Pereira (28), the idirect tax rate was 6.97% i 26. Usig the equivalece betwee taxes 7, the effective margial tax rate for each group is respectively 5.25%, 27.97% ad 38.56%. We use these values to create the distributio of skills ad to solve the bechmark model. 6 Collected by Pesquisa Nacioal por Amostra de Domicilios (PNAD) (26). 7 We ca rewrite the equatio p( + t c)c ( t l )z as pc ( τ)z, where τ = ( t l) (+t c). 9

10 Figure : Empirical Wage Icome Distributio ad Kiks 3 x 3 Kiks 2.5 R$ (9.79%) 2 f(z).5 R$ (8.6%).5 2, 4, 6, 8,, 2, Earigs (z) R$ Studet Versio of MATLAB

11 The empirical icome distributio 8 is preseted i Figure. We also plot the empirical icome distributio with the kiks of curret tax system. Accordig to our empirical icome distributio, 8.6% of taxpayers are i the first group, that pay a margial tax rate equal to 5.25%. I the secod group we have 9.63% of taxpayers that pay 27.97% ad i the last group, we have oly 9.2% of taxpayers that pay a margial tax rate equal to 38.56%. The modal value is equal to R$35., the miimum wage i 26. If a idividual had earigs above R$3,, he or she would be i the top 5% of earig distributio ad if a idividual had earigs above R$7, reais, he or she would be i the top % of earig distributio. 3.2 Distributio of Skills We caot directly apply equatio (4) i Saez (2) because the earigs distributio is affected by taxatio. We, thus, eed to back up the uderlyig distributio of skills from the distributio of icome ad the curret tax schedule. That is, if we are to use the equatio (7), we have to create a exogeous distributio of skills that replicates the empirical earigs distributio. After this procedure, we ca use the approximated distributio of skill to perform our umerical simulatios. 9 Give the icome distributio, the curret tax schedule ad the utility fuctio we ca fid the empirical distributio of skills, as i Saez (2). Cotrary to Saez (2) i that we take ito accout the o-liearity of the curret labor icome tax schedule. To illustrate the procedure let us focus o a very coveiet specificatio of prefereces, u(c, l) = l c + γ l( l). (9) Assume that a idividual has productivity ad faces a liear icome tax schedule, T (z) = τ z. The idividual s maximizatio problem is to maximize (9) subject to c z( τ). It is ot hard to see that z() = /(γ + ). If we ca observe the distributio of z, which is just the cross-sectioal distributio of labor icome i the society, we ca, defie (z) = z(γ + ), the iverse of z() ad recover the distributio of. 8 The empirical icome distributio was estimated by Gaussia kerel desity distributio. 9 Note also that give the empirical earigs distributio ad the tax schedule, whe we chage the utility fuctio parameters, we also geerate a differet distributio of skills. I Figure 2 we plot the empirical ad approximated distributio of skills accordig to the risk aversio ad labor supply elasticity parameters.

12 Figure 2: Empirical ad Approximated Skill Distributio γ =.25 ad ρ =.5 3 x 3 γ =.25 ad ρ = x x 8 f() f() 5 x 6 γ =.25 ad ρ = 2 f() γ = 2 ad ρ = x 3 γ = 2 ad ρ = x Approximated Empirical x 5 γ = 2 ad ρ = x 7 f() f() f() 2

13 Roughly speakig, this is the procedure adopted by Saez (2) i his semial paper. This procedure works fie provided that the icome tax schedule is reasoably well approximated by a liear tax fuctio. The problem here is how to defie what a reasoable approximatio is i this cotext. To try ad hadle this issue i the simplest possible maer, we recogize the oliearity that is preset i the Brazilia icome tax schedule whe we recover the distributio of skills. What makes this task feasible is the use of a liear approximatio alog each idividual s choice. R(z), Let us get back to our previous example, ad recall our defiitio of virtual icome, R(z) = τz T (z). where we have ow made explicit its depedece o z. First assume that the tax system is progressive i the sese of o-decreasig margial tax rates ad ote that the curret tax system is piecewise liear with a fiite umber of kiks. Let z i, i =,...,, z i < z j for i < j, be the icome levels where there is a kik i the icome tax schedule. Defie τ i as the margial tax rate that applies to all idividuals earig z such that z i < z < z i+ ad let us first focus o a idividual that is ot choosig i a kik of the tax system. I this case, we have that R(z) = R i for all z (z i, z i+ ), where z i is the highest icome level that is less tha z ad for which there is a kik. That is z i < z, ad z < z j for all j > i. Let us the get back to a idividual s maximizatio problem. We ca split it ito two parts. First, for a idividual with productivity ad for each pair (τ i, R i ) defie ẑ i () as the solutio for this problem. So the solutio for the idividual s problem is It is ow easy to see that the affie fuctio geeralizes i for all icome levels z (z i, z i ). ẑ() = ( τ i) γr i (γ + )( τ i ). i(z) = (γ + )z + γ τ i R i 3

14 Next defie z i () through: z i () = z i z i () = ẑ i () z i () = z i if, ẑ i () z i if, z i < ẑ i () z i ifẑ i () > z i This will defie V (τ i, R i ; ) l(z i ()( τ i ) + R i ) + γ l( z i ()/) The, the secod stage is simply that of choosig u() max {V (τ i, R i ; )} N i i=. Whe there are cocave kiks i the budget set defied o the Z C space, the we have that z i () = z i for all idividuals such that V (τ i, R i ; ) = V (τ i, R i ; ). If this is the case, defie, the set N i { [, ]; V (τ i, R i ; ) = V (τ i, R i ; )}, () the set of all types that are buched at z i. Cocave kiks i the budget set meas that these sets are ot sigletos, i.e. the presece of a cocave kik leads to a mass poit i all z i s. The mai difficulty that this situatio imposes o us is that, i priciple, we caot distiguish these idividuals from observig their icomes. A simple way for us to deal with this issue would be to assume that the desity is flat alog these itervals. Alteratively, we could use a splie to try to complete the empirical distributio. There are two reasos why we believe that this should ot pose too much of a problem for us. First it should however have little impact o the tax schedule we derive sice we use a parametrized distributio that we shall adjust to this empirical distributio of skills. Secod, as it turs we do ot observe these buchig poits i the data. The Parametrized Distributio We choose the Geeralized Pareto (GP) distributio to fit the empirical distributio of skills. The GP distributio fits well at the upper tail. Previous papers has used the log-ormal distributio, despite the fact that it approximates very poor the empirical distributios at the top ad bottom tails. 4

15 The parameters of Geeralized Pareto distributio are chose so that the skill distributio replicates the empirical earigs distributio. The GP has three parameters, σ, κ ad µ. Specifically we choose these parameters to as much as possible recreate the icome distributio ad the empirical distributio of skills ad to match the empirical ratio expediture/gdp. The GP probability desity fuctio is give by f( κ, σ, µ) = ( ) ( + σ ) κ( µ) κ σ where the Geeralized Pareto distributio with the tail idex (shape) parameter κ, scale parameter σ ad threshold (locatio) parameter, µ, evaluated at the values i. If κ = ad µ =, the GP distributio is equivalet to the expoetial distributio. If κ > ad µ = σ/κ, the GP distributio is equivalet to the Pareto distributio. We plot the empirical ad approximated skill distributio i the Figure 2 to all parametrizatio that we cosider i this paper. I the appedix B, i the Figure we plot oe case to make easier to observe the fit betwee empirical ad approximated skill distributio. I the Figure 3 we observe that the hazard ratio is very high at the bottom of icome distributio because f() is close to zero while F () is close to oe. At the top, the hazard ratio coverge to levels that deped o the parameters of risk aversio ad the elasticity of labor supply. I the appedix B, we plot the hazard ratio fuctio to 99.9% idividuals i Figure 3. It is easier to observe that the hazard ratio is almost stable for earigs above R$9, reais. Holdig the elasticity of labor supply costat the hazard ratio at the top is lower whe risk aversio is smaller ad it is bigger whe risk aversio is higher. I all cases cosidered, the hazard ratio is L-shaped while i Saez (2), the hazard ratio is U-shaped. Accordig to equatio (7), the optimal margial tax rate is icreasig i the hazard ratio. Thus, we expect the higher hazard ratio (risk aversio) to iduce a higher optimal margial tax rate. Saez (2) foud a hazard ratio aroud.5 while we foud a rage betwee.45 ad.. It is easy to observe that, at the top, the hazard ratio has a smaller variace whe γ = 2 tha i the case where γ =.25. () Covex Kiks Before closig this sectio, a word is due o covex kiks i the budget sets, which is iduced by regressivity i the icome tax schedule. Now, let z j be a poit where the tax system iduces a covex kik i the agets budget set. The issue ow is 5

16 Figure 3: Hazard Ratio - 99% of Idividuals ( F()) / (f()) γ =.25 ρ =.5 ρ = ρ = 2 ( F()) / (f()).4, 2, 3, 4, 5, 6, 7, 8, 9, Earigs (z) R$ γ = 2.4, 2, 3, 4, 5, 6, 7, 8, 9, Earigs (z) R$ that there will be a productivity level such that ay idividual with productivity will be idifferet betwee to levels of earig z ad z where z < z i < z. Nobody would be observed choosig betwee these two values. It should be oted that this observatio does ot create ay difficulty for our procedure sice the set of agets that are idifferet betwee two levels of earigs is of measure zero. 6 Studet Versio of MATLAB

17 4 Results I our simulatio, we calibrate our model alog the lies of Mirrlees (97) ad Saez (2) to represet the Brazilia ecoomy i 26 The govermet s expeditures E are set at 24.9% of output. It is importat to remember that we cosider both cosumptio ad labor icome taxes rates i our simulatios. The utility fuctio is of form u(c, l) = c ρ ρ lγ γ, where ρ is the parameter of risk aversio ad γ is the parameter of labour supply. I our simulatios we use the followig values for risk aversio, ρ = {.5,, 2} ad for the parameter of labor supply, γ = {.25, 2}. We cosider two social welfare criteria. The first is the Utilitaria criterio, where G(u) = u, respectively G (u) =. The secod criteria is the Rawlsia criterio, where G (u) =. 4. The Bechmark Model The bechmark model is the oe that results from applyig the curret tax system to the distributio of skills that results from solvig problem (). The margial taxes rate, τ, are the values described i the last sectio. The virtual icome is the empirical values i 26. Ideally the model would exactly replicate the Brazilia ecoomy i 26. The approximatios we use, however make it diverge somewhat from the actual data. 2 Table summarizes some statistics for the data ad the bechmark model, like value per percetile, the Expediture/GDP ratio, mea, variace, etc. It also displays the empirical ad the bechmark earigs accordig to label supply elasticity ad risk aversio parameters. For example, i the percetile 5, the idividual ears R$5 reais, while i the bechmark model with γ =.25, the earigs are R$292.49, R$ ad R$59.92, respectively to ρ =.5, ρ = ad ρ = 2. The large differece i the stadard deviatio is explaied by the peeks i the empirical distributio. As the approximated distributio is smoother, the values are spread followig a parametric distributio. I cotrast, the empirical has may peeks alog the distributio. There are cluster poits i the empirical distributio, that could be explaied because the wages are discotiuous. Whe ρ = the utility fuctio has the form u(c, l) = l(c) lγ. γ is directly associated with the Frisch γ elasticity of labor supply η f. Whe γ =.25 the Frisch elasticity is equal to 4, η f = 4 ad η f = whe γ = 2. 2 For example, i the bechmark model, idividuals do buch at kiks. 7

18 Table : Statistic Details: Empirical ad Approximated Earigs (R$) Percetile Empirical γ =.25 γ = 2 ρ =.5 ρ = ρ = 2 ρ =.5 ρ = ρ = , 85, 348., 823.7, , , 643.3, , 2, , , , , , , 9, , , , 77. 8, 85, 5 8, , 28, , , , , , E M ea , 65.24, , 7.4 St.Deviatio, 522 4, 679 4, 456 4, 4 4, 322 4, 372 4, 28 Source: Elaborated by Authors * E = Expediture/GDP 8

19 Figure 4: Earigs versus Cumulative Fuctio Earigs (z) R$ 2,,5, 5 γ =.25: 9% of idividuals ro =.5 ro = ro = 2 Empirical F( idividuals ) Earigs (z) R$ γ =.25: 9% 99% of idividuals, 8, 6, 4, 2, F( idividuals ) 2, γ = 2: 9% of idividuals, γ = 2: 9% 99% of idividuals Earigs (z) R$,5, 5 Earigs (z) R$ 8, 6, 4, 2, F( idividuals ) F( idividuals ) I Figure 4 we plot the empirical ad the approximated (bechmark) earigs associated with differet parameters for the cumulative fuctio of idividuals. The graphics i the left side are plotted to 9% of idividuals that are sorted by earigs (skills). I the right side we have the idividuals that belog to the rage betwee the percetile 9% ad 99% of earigs. Figure 4 ad the Table suggest that model with ρ = 2 fits the empirical earigs Studet Versio of MATLAB better tha the models whe ρ =.5 ad ρ =. I geeral, our approximatio fits the empirical earigs very well. The exceptio could be made after the percetile 98%. The bechmark earigs grow faster tha the empirical earigs after the 98% of sample. The approximated earigs values are higher tha the empirical earigs. We explai the "bad" fit at the bottom due to the behavior of empirical earigs distributio at the bottom. The 9

20 empirical earigs data have a icrease i the desity fuctio at the bottom before the model (see Figure i appedix B). We ca also observe that there is buchig i the kiks i our bechmark (approximated) data, which is cotrast with the actual data. Figure 5: Compesated Elasticity, Ucompesated Elasticity ad Icome Effect - 99% of Idividuals γ =.25 ad ρ =.5 γ =.25 ad ρ = γ =.25 ad ρ = Icome Effect Compesated Elas. Ucompesated Elas ,, Earigs R$ 5,, Earigs R$ 5,, Earigs R$ γ = 2 ad ρ =.5.6 γ = 2 ad ρ =.2 γ = 2 ad ρ = ,, Earigs R$ ,, Earigs R$ ,, Earigs R$ Other importat result is the directio ad magitude of the ucompesated ad compesated elasticities ad the icome effect. Figure 5 we plot the directio ad magitude of each elasticity to 99% of populatio. The compesated elasticity is decreasig i ρ ad is always positive. The estimated icome effect depeds o risk aversio ad labor supply elasticities. Keepig the labor supply elasticity costat, the resultig icome effect is Studet Versio of MATLAB icreasig, i absolute values, i earigs (skills). I Figure 2 i appedix B we plot the elasticities to 99.9% of populatio. 2

21 Accordig to equatio (7), the optimal margial tax rate is decreasig i compesated elasticity ad icreasig i the ucompesated elasticity. The ucompesated elasticity is decreasig i risk aversio ad it chages its sigal from positive to egative. The term A() i equatio (8) may be writte as ζ + ζu c ζ. Because ζ u is always smaller c tha, ζ > ζu c ζ. Therefore, the compesated elasticity s effect domiates the term +ζu c ζ. c Although the sig of ζ u chages from positive to egative whe we icrease risk aversio, the ucompesated elasticity s effect could oly atteuate or itesify the effects of compesated elasticity o term A() but ever chage its sig. 4.2 The Optimal Taxatio Model I this sectio, we preset our simulatios for the optimal o-liear tax schedule for Brazil. As we said before, we cosider differet values for ρ ad γ. These utility fuctios geerate a compesated elasticity that varies with earigs. Table 2 summarizes the welfare measure accordig to differet parameters for the utility fuctio ad differet social welfare criteria. As expected, at the optimal schedule there are some idividuals that are better off ad idividuals that are worse off tha i the bechmark case. 3 The gais are located at the bottom ad at the top ad the loses i the ceter of distributio. As we cosider the Geeralized Pareto (GP) Distributio, the umber of idividuals at the bottom is very large whe compared to umber of idividuals at the top. Oe readig is that a optimal tax schedule icreases the utility to the poor idividuals. A possible readig of our result is that the social welfare fuctio implicit i the Govermet s decisio places more weight o richer idividuals tha what is implied by a utilitaria SWF. This is ot i cotradictio with Mattos (28) sice we are allowig for a direct desire for redistributio through the cocavity of the idividuals utility fuctios. The results for the optimal margial rates i our simulatios are plotted i Figure 6 for mothly earigs i reais (R$). Withi this rage of earigs we fid about 99% of idividuals. I Appedix B we plot the same picture for 99.9% of our populatio (see Figure 4). 3 This should be the case if we cosider the fidigs i Mattos (28) which shows that the Brazilia tax system is Paretia. However, because he cosiders differet fuctioal forms ad data sets, it is ot obvious that his result would apply to our settig. 2

22 Table 2: Utility: Bechmark ad Optimal Utilitaria criterio Rawlsia criterio γ =.25 γ = 2 γ =.25 γ = 2 Bechmark Optimal Bechmark Optimal Bechmark Optimal Bechmark Optimal ρ = ρ = ρ = Source: Elaborated by Authors 22

23 Figure 6: Optimal Margial Tax Rate - 99% of Idividuals Utilitaria criterio, γ =.25 Rawlsia criterio, γ =.25 Margial Tax Rate ρ =.5 ρ = ρ = 2 2, 4, 6, 8, Earigs (z) R$ Margial Tax Rate , 4, 6, 8, Earigs (z) R$ Utilitaria criterio, γ = 2 Rawlsia criterio, γ = 2 Margial Tax Rate Margial Tax Rate , 4, 6, 8, Earigs (z) R$ 2, 4, 6, 8, Earigs (z) R$ The rage of values of skill distributio is bouded at the bottom, i.e., [, ], with > ad the distributio is ubouded distributio at the top. Our values for the Margial Tax Rate (MTR) at the bottom of distributio are smaller tha those i Saez (2) ad Tuomala (26). As expected, the level of the optimal margial rate depeds o the risk aversio ad labor supply parameters. I all situatios, the optimal tax schedule has the form of iverted U-shape for the Utilitaria criterio. We verify that the optimal tax schedule Studet leads Versio to icreasig of MATLAB earigs, z () >. It is the ecessary ad sufficiet coditio for idividual secod-order coditios explaied i details i Mirrlees (97). Whe γ =.25, the optimal margial rate is icreasig i earigs for most of the rage. It falls dramatically at the top ad is equal to for the most productive idividual. This (zero) margial tax rate at the top is ecessary for efficiecy as show by Sadka (976) ad Seade (977). We do ot focus o this result that depeds o the bouded- 23

24 ess of the skills distributio. We believe this is ot a good represetatio of the relevat distributio of skills. The maximum optimal margial rate is aroud 43%, 68% ad 98% respectively to the risk aversio parameters equal to ρ =.5, ρ = ad ρ = 2. Whe γ = 2, the margial tax rates is very close to the case γ =.25. However, the margial tax rate is a little higher tha i the first case, the exceptio is i the case ρ = 2. The maximum margial tax rate for ρ =.5, ρ = ad ρ = 2 are, respectively, 54%, 75% ad 96%. The Rawlsia criterio yields higher margial rates at low earigs. However, the results whe ρ = 2 are a bit surprisig. Idepedetly of the labor supply parameter, the margial rate i the Rawlsia criterio falls i the begi ad subsequet icreases before the R$, reais. The values of MTR reach about 97% ad keep costat util the R$9, reais, whe the MTR starts to decrease ad coverges to zero. To the others values of risk aversio, the MTR is mootoe decreasig i the earigs ad coverges to the margial tax rate equal to zero at the top. The margial tax rate does ot have the U-shape foud elsewhere. Two forces seem to be at play here. First, the hazard ratio i the our paper has the L-shape istead of the U-shape foud i Saez (2). Secod, we do ot have buchig at the bottom. Whe z () < i the relaxed problem, oe must chage the iitial coditio. Whe we chage this iitial coditio, the margial rate at the bottom becomes higher. As we do ot have this problem, the iitial MTR is zero at the bottom. This procedure is described i great detail i Mirrlees (97). The high margial tax rates for those at the top of the distributio i the case ρ = 2 are due to the fact that the compesated elasticity (ζ c ) is small ad close to zero. As we ca see i equatio (7), the compesated elasticity has a importat role i the determiatio of optimal margial rate. Other importat poit is the hazard ratio, that is icreasig i the risk aversio. With a large value of risk aversio, the govermet has great icetives to smooth cosumptio ad, cosequetly, icrease the distributio of earigs. Thus, the high value for the margial tax rate to support the redistributive goals of the govermet. This effect is apparet i Figure 7. I Figure 7 we plot cosumptio agaist earigs for various levels of ρ ad γ. Trasfers ca be iferred by the distace to the 45 degrees lie. The higher the risk aversio the 24

25 Figure 7: Optimal Cosumptio: Utilitaria criterio - 9% of Idividuals γ =.25 ad ρ =.5 γ =.25 ad ρ = γ =.25 ad ρ = 2 Cosumptio o Cosumptio.5. 5 Cosumptio Earigs R$ 5..5 Earigs R$ 5..5 Earigs R$ γ = 2 ad ρ =.5.6 γ = 2 ad ρ = γ = 2 ad ρ = 2 Cosumptio Cosumptio Cosumptio Earigs R$ 5..5 Earigs R$ 5..5 Earigs R$ lower the cosumptio iequality. Figure 7 represets 9% of idividuals. Note that the level of guarateed cosumptio, i.e., the itersectio with the vertical axis is icreasig i risk aversio. This illustrates the redistributive goals of the govermet. There are some differeces i the results obtaied here ad those obtaied elsewhere. Studet Versio of MATLAB Brazil is a coutry characterized by a very high level of icome iequality, which is reflected i a very disperse distributio of skills. The lower margial tax rates foud at the bottom of the may be explaied by the distributio of skills. The other importat poit is the hazard ratio ( F ())/(f()). Due to the Geeralized Pareto distributio, the hazard ratio has L-shape. However, the it does ot ted to zero like i log-ormal distributio. Mirrlees (97) foud a lower margial tax rates tha we did. The reasos for that are the utility fuctio ad the log-ormal distributios of skills cosidered by him. Saez (2) foud a high rates at top like us. He cosidered a compesated elasticities equal to 25

26 ζ c =.25 ad ζ c =.5. The other importat poit is the hazard ratio, that has a U-shape. The shape of hazard ratio is resposible to the U-shape i the margial tax rate i his paper. 4.3 The Optimal Margial Tax Rate i Uited State I this subsectio we simulate the optimal margial tax rate i Uited State (US). There are may articles that aalyze the optimal margial tax rate i US, some examples are Tuomala (99), Saez (2) ad Tuomala (26). Like Brazilia case described above ad cotrary to Saez (2), we take accout the o-liearity of the Uited State labor icome tax schedule. We recogize the o-liearity that is preset i the US icome tax schedule whe we recover the distributio of skills. I preset paper, the labor icome tax schedule is composed by icome ad cosumptio taxes schedules. To simulate the US margial tax rate we cosider the pael study of icome dyamic (PSID) data i 27. We use the head s icome from wages ad salaries i We also drop all head that declare with o wife, husbad or first-year cohabitor ad idividuals with icome equal to zero. I 26 the icome tax schedule to head of household filig status was % o the yearly icome betwee $ $, 75, 5% o $, 75 $4, 5, 25% o icome betwee $4, 5 $6,, 28% o $6, $7, 65, the idividuals pay 33% o the icome $7, 65 $336, 55 ad 35% o the icome over $336, 55. To cosumptio tax, we cosider the mea amog US states of the sale tax i 26, where the values was 5.%. As we drop the idividuals with icome equal to zero, whe we recover the distributios of skills we have the rage of skills, [, ], with >. I our simulatios, we cosider oly the case where γ = 2. To compare the optimal tax rate i Brazil ad i Uited State, we calibrate the govermet expediture (E) to be aroud.249 of product. We also cosider a Geeralized Pareto distributio to fit the empirical distributio of skills. To the Utilitaria criterio, we fid the iverted U-shape to MTR i US. The Figure 8 we plot the earigs versus MTR i US to 99% of idividuals. Likewise the previews simulatios, the MTR i US is icreasig i the risk aversio. As we said above, the MTR is icreasig i the risk aversio because the idividuals do ot tolerate the icome iequality amog them. The MTR i US is a little lower compared to Brazil i all situatio. Whe ρ =.5 ad 4 The variable idex is ER

27 Figure 8: Optimal Margial Tax Rate i US - 99% of Idividuals Utilitaria criterio, γ = 2 Rawlsia criterio, γ = 2 Margial Tax Rate ρ =.5.2 ρ = ρ = 2 $ $2, $4, $6, $8, Earigs (z) yearly Margial Tax Rate $ $2, $4, $6, $8, Earigs (z) yearly ρ =, the MTR begis to fall after the $, dollars ad keeps almost costat betwee $, ad $8,. To ρ = 2 the MTR is icreasig i the earigs ad it is costat aroud the rate 84%. I this case, the MTR i US is very similar to Brazil, because it is stable at the top taxpayers. To the Rawlsia criterio, as expected, the MTR is higher compared to the Utilitaria criterio. As i Brazilia case, the largest differece betwee two criteria is at low earigs. At the bottom, ours values are closed to Saez (2), however we do ot observe the icrease i MTR like him. Our margial tax rate is decreasig toward. Compared to Saez (2), the values at the top are similar to his article. I the logarithm case, (ρ = ), the value of MTR at the top belogs to the rage of values foud to him, i.e. betwee.6 ad.8, though we cosider a differet compesated elasticity. Ours simulatios differ completely from Saez (2) at the bottom. We do ot fid the fall i the begi of the earigs. As we said above, our distributio of skills is bouded to a lower boud, >. For this reaso, we do ot observe the fall of MTR i the begi of earigs. The other importat poit is the empirical hazard ratio fuctio, see Figure 6 i appedix B. Our hazard ratio has the L-shape while Saez Studet (2) Versio has the of MATLAB U-shape. As we show i equatio (7), the hazard ratio has a very importat role to defie the shape of optimal margial tax rate. Figure 7, i the appedix B, we plot the margial tax rate for yearly earigs betwee 27

28 $ ad $2,,. I this rage we have aroud 99.9% of our sample. I appedix B we also plot the distributio of earigs i 26, see Figure Affie Tax Schedule Proposal I this sectio we compute the optimal affie tax schedule for Brazil. The idea here is to see if we ca get close to the optimum with a simple tax schedule. I fact, i a static settig, this schedule is equivalet to a poll tax (or subsidy) ad a cosumptio tax, which is a very appealig system i practice. As we saw i the previous sectio, at the bottom, the optimal tax schedule is characterized by a cash trasfer to idividuals with low earigs. This trasfer depeds o the redistributive goals of the govermet that guaratee a cosumptio level to the idividuals. Grossly speakig, a cash trasfer program could have the same fuctio at the bottom. The most famous cash trasfer program i Brazil is the Bolsa Família Program. The Bolsa Família is coditioal to the family s structure 5. Our proposal cosists i a simplificatio of the umber of margial tax rate ad the icorporatio of ay kid of the cash trasfer program to the affie tax schedule. This way, the taxatio ad the trasfers would be made by the same istitutio. The affie tax schedule is composed by a pair (t, B), where t is the margial tax rate ad B is the trasfer. The trasfers are fiaced by the tax. The idividual s problem may the be writte max c,z U(c, z/) (2) s.t. c = z( t) + B. As i the problem describes by equatio (), the solutio is defied by z( t, B, ) ad c( t, B, ). Beyod fiacig the program the govermet must also fiace the other expeditures, E. Therefore the govermet s budget costrait is (z m c m B) f(m)dm E, where B is idepedet of. As we wat to compare the curret tax schedule to the affie tax schedule, we have chose the same level of govermet expediture, E, that we cosider i the previews sectios. 5 For details o Bolsa Família, Program (29) 28

29 Figure 9: Idividual Utility - Bechmark ad Affie Schedule : 99% of Idividuals γ =.25 ad ρ =.5 Affie Bechmark Optimal γ =.25 ad ρ = γ =.25 ad ρ = 2 Utility Utility Utility F( idividuals ) γ = 2 ad ρ = F( idividuals ) γ = 2 ad ρ = F( idividuals ) γ = 2 ad ρ = 2 Utility Utility Utility F( idividuals ) F( idividuals ) F( idividuals ) 29 Studet Versio of MATLAB

30 To fid the affie tax schedule, we defie a vector of margial taxes rates, t, ad a vector of cash trasfers, B. I the first step, we start with a big rage of values to t ad B. I the secod step we refie ad reduce the rage of t ad B. We repeat this procedure util to fid the global maximizer. I this sectio we cosider oly the Utilitaria criterio. Observe that, for each pair (ρ, γ) of the parameters of risk aversio ad labor supply elasticity, we have to look for the global maximizer poit i a differet rage of t ad B. Figure 9 shows who is better off ad who is worse off if we substitute the affie tax schedule for the bechmark tax schedule. We also plot the utility i the optimal tax schedule. We plot idividual utility i the y-axis ad the cumulative fuctio of umber of idividuals i x-axis. The figure is plotted to 99% of idividuals. It is easy to see that the affie schedule provides a icrease i the utility for a large fractio of the populatio. The utility icreases for at least 45% of the idividuals i all simulatios. We ca observe that i some situatios the affie tax schedule provides a utility close to the optimal schedule. I Table 3 we summarize the aggregate utility to the bechmark ad to the affie schedules models. Accordig to our results, except whe ρ =.5, the affie tax schedule leads to a icrease i the social welfare i the Utilitaria criterio. Table 3: Aggregate Utility: Bechmark ad Affie Tax Schedule γ =.25 γ = 2 Bechmark Affie Schedule Bechmark Affie Schedule ρ = ρ = ρ = Source: Elaborated by Authors We explai the result whe ρ =.5 due to the low risk aversio. It implies i a greater earigs iequality amog the idividuals compared to the case whe ρ = ad ρ = 2. Although the aggregate utility is smaller i the affie schedule tha i the bechmark schedule, lookig at Figure 9 we observe a icrease i the utility for idividuals with low earigs. However, this icrease is ot eough to compesate for the losses imposed o such a large fractio of idividuals. As the optimal tax schedule is extremely difficult to implemet i reality, we have to 3

31 look for some alteratives to the optimal schedule. I the sectio we preset oe of this alteratives. We show that is possible to fid a feasible affie schedule that is better tha the curret oe uder the Utilitaria metric. 5 Coclusios I preset paper we ivestigate alteratives ways to the curret tax schedule i Brazil. I the first part of this paper, we back the distributio of skills for Brazil usig the distributio of icome ad the tax schedule for the year 26 for various parameters for idividuals prefereces. Our approach is similar to Mirrlees (97) ad Saez (2). We differ from Saez (2) i the way we create the empirical distributio of skills. Saez (2) approximates the curret tax system i the US usig a flat tax to create the skill distributio whilst we take ito accout the o-liearities of the Brazilia system. I the secod part, we derive optimal tax schedules for differet prefereces for distributio. As the optimal taxatio is difficult to implemet i reality, we also calculate the optimal affie tax schedule. This allows us to focus o the effect of cash trasfers programs, like Bolsa Família, i labor supply ad utility. The idea is to fid a simple tax system which is better tha the curret system ad verify how close we get to the optimal tax system. Our results of optimal taxatio are i lie with others papers i the literature i some dimesios. However, we fid iverted U-shaped margial tax rate fuctio istead of U- shaped, which is i cotrast with Saez (2). The fact that we ivestigate optimal policy i a developig coutry with a very high iequality i earigs may be the uderlyig reaso for our fidigs. 3

32 Refereces Chetty, Raj, A Boud o Risk Aversio Usig Labor Supply Elasticities, Workig Paper 267, NBER 26. Diamod, Peter A, Optimal Icome Taxatio: A Example with a U-Shaped Patter of Optimal Margial Tax Rates, America Ecoomic Review, March 998, 88 (), Mattos, Eliso, The Revealed Social Welfare Fuctio: USA X Brazil, Brazilia Review of Ecoometrics, 28, 28 (2), Milgrom, Paul ad Ilya Segal, Evelope Theorems for Arbitrary Choice Set, Ecoometrica, 22, 7 (2), Mirrlees, James A., A Exploratio i the Theory of Optimal Icome Taxatio, Review of Ecoomic Studies, 97, 38, , Optimal Tax Theory: A Sythesis, Joural of Public Ecoomics, 976, 6, Pereira, Thiago, Tax Reform: Theory ad Proposal to Brazil. PhD dissertatio, Graduate School of Ecoomics (EPGE) at Fudação Getulio Vargas, Praia de Botafogo, Rio de Jaeiro-RJ, Brazil October 28. Pesquisa Nacioal por Amostra de Domicilios (PNAD) Pesquisa Nacioal por Amostra de Domicilios (PNAD), Techical Report, Istituto Brasileiro de Geografia e Estatística 26. Program, Bolsa Família, Miistry of Social Developmet, 29. Sadka, Efraim, O Icome Distributio, Icetive Effects ad Optimal Icome Taxatio, Review of Ecoomic Studies, Jue 976, 43 (2), Saez, Emmauel, Usig Elasticities to Derive Optimal Icome Tax Rates, Review of Ecoomic Studies, 2, 68, Seade, Jesus, O the Shape of Optimal Tax Schedules, Joural of Public Ecoomics, 977, 7 (2), Tuomala, Matti, Optimal Icome Tax ad Redistributio, Oxford Uiversity Press, Claredo Press,

33 , O The Shape of Optimal o-liear Icome Tax Schedule, Tampere Ecoomic Workig Papers Net Series, April 26, 49,

34 A Appedix I this appedix we derive equatio (7) ad the equatios of the differetial system that we solve umerically i the preset paper. The equatios were foud with blid Hamiltoia optimizatio ad uder u lc = assumptio. Followig Mirrlees (97), we cosider that u() is a state variable, l() is the cotrol variable ad c() is determied implicitly as a fuctio of u() ad l() from equatio u() = u(c(), l()). Others examples are Sadka (976), Seade (977), Tuomala (99) ad Diamod (998). The govermet s problem is described by equatio (3). The terms λ ad µ() are respectivily the Lagrage multiplier of resources costrait ad the multiplier of icetives costrait. E is the govermet expediture.to obtai the icetive costrait we derive u(c(), l()) ito, i.e., du d = lu l. H = ˆ [ˆ ] G(u()f()d + λ (l() c()) f()d E [ ] du(c(), l() +µ() d To solve the problem we derive the eq.(3) ito u ad l. The first order coditio ito u() is equatio (3) dµ d = H ] [G u = (u()) λuc f() (4) The first order coditio ito l is give by: [ λ + u ] l f () + µ() ϕ l u c = (5) where ϕ(u, l) = lu l. Itegratig the equatio (4) from to ifiity ad we also cosider the trasversality coditio, µ() = µ( ) =. The we have a ew equatio that is give by: µ() = ˆ [ G (u(m)) λ ] f(m)dm (6) u c (m) Now we substitute the equatio (6) ito the equatio (5) ad we have equatio (7): [ λ + u ] l f () = ϕ l u c ˆ [ ] λ u c (m) G (u(m)) f(m)dm (7) 34