OPTIMAL MARGINAL TAX RATES FOR LOW INCOMES: POSITIVE, NEGATIVE, OR ZERO?

Transcription

1 OPTIMAL MARGINAL TAX RATES FOR LOW INCOMES: POSITIVE, NEGATIVE, OR ZERO? BY STEFAN HOMBURG DISCUSSION PAPER NO. 255 MAY 2002 ISSN ABSTRACT: Previous studies ave sown tat te optimal marginal tax rate at te bottom of te inome distribution may be positive, negative, or even zero. Tis paper reexamines tis problem in a unified framework and tries to evaluate te arguments. It turns out tat te ase for positive marginal tax rates is quite strong, wereas negative marginal tax rates seem to be based on a misoneption. KEYWORDS: Optimal inome taxation, ontrat teory, inentives, welfare. JEL-CLASSIFICATION: H2, D3

2 2. INTRODUCTION Designing inome support programs is still an important matter of onern. Te teory of nonlinear inome taxation, initiated by Mirrlees (97), approaes tis problem as follows: It onsiders an abstrat tax sedule, embraing bot positive tax liabilities in te ordinary sense as well as transfer payments wi are oneived of as negative taxes. Te analysis aims at araterizing te qualitative properties of su inome tax sedules. Sine negative taxes are transfer payments, it yields reommendations regarding te optimal design of inome support programs as a by-produt. However, tese reommendations are anyting but lear-ut. Tey an be ategorized into tree groups. Te first group, wi inludes Mirrlees original paper, states tat optimal marginal inome tax rates at te bottom of te inome distribution sould be stritly positive. In fat, Mirrlees as proven tat marginal inome tax rates are always non-negative. But is simulations produed marginal tax rates wi are stritly positive at te bottom and dereasing on teir entire domain, ontraditing te austomed ideal of tax rate graduation. Optimal marginal tax rates at zero inome varied between 2% and 50% in is study. Tese results agree wit te ig impliit tax rates of many ontemporary welfare programs. Te seond group, inluding Seade (977, 982), argues tat te optimal marginal tax rate sould be zero at te bottom of te inome distribution in order to leave labor supplies of te least skilled persons undistorted. Sine te same result olds undisputedly at te top of te inome distribution, Seade suggested tat optimal tax sedules typially take an S-form, wit marginal tax rates inreasing from zero to some maximum and ten dereasing again to zero. His no-distortion-at-te-bottom result armonizes wit te basi exemptions integrated in most inome tax sedules. Te tird group ontends tat marginal tax rates at te bottom may well be stritly negative. Tese results go bak to Diamond (980) wo used te Mirrlees model, too, but made one important ange in te assumptions: Instead of onsidering persons tat an adjust teir labor supplies smootly, Diamond assumed fixed working ours, implying tat people are onfronted wit binary oies only. Negative marginal tax rates araterize te well-known U. S. earned inome tax redit (EITC). Tus, te results obtained so far exaust te entire range of possibilities and leave te poliymaker elpless. Te objetive of tis paper is to disuss tese inonsistent outomes in a unified framework and to evaluate tem. Te framework used is a finite variant of te standard model wi assumes an unountable infinity of taxpayers. Wit infinitely many taxpayers, eonomi reasoning is liable to beome replaed by purely matematial operations wi do not tell anyting about te eonomi fores at work. By ontrast, te finite model allows deriving all results referred to above in a simple and luid fasion. Moreover, some limit araterizations will be provided wi larify te relationsip between te optimal tax formulae of te standard model and teir finite ounterparts. Setion 2 desribes te basi model. Setion 3 reports on a number of establised teorems. Te main setions 4 to 6 are devoted to repliating te findings of te positive, zero, and negative marginal tax rates, respetively. Setion 7 onludes.

3 3 2. THE MODEL Tere are finitely many persons wo ave different skills and tus earn different wage rates per our, w, were = 0,...H and 0 w 0 < w <... < w H. In order to derive limit properties later on, it is onvenient to assume a limit probability distribution funtion F wi possesses a ontinuous density f>0 on its support [w 0, w H ]. Te atual skill distribution of te finite eonomy is given by probability masses f 0 = F(0) > 0 and f = F(w ) F(w ) for all > 0. Ea person wit wage rate w (person, for sort) ooses some onsumption and some labor supply su tat te ommodity bundle (, ) belongs to te uniform onsumption spae = o+ [0; max ], were max > 0. A person s gross inome is denoted as y = w. Te uniform utility funtion u: is ontinuous, stritly monotonially inreasing in, stritly monotonially dereasing in and stritly onave on its entire domain. At least in te interior, it is twie ontinuously differentiable wit partial derivatives u (, ) > 0 and u (, ) < 0 and a negative definite Hessian. Moreover, onsumption is a gross substitute for leisure (meaning tat in te absene of taxes, onsumption is an inreasing funtion of te wage rate), and leisure is non-inferior. An alloation is a vetor (, ) =0...H in H+. Te soial objetive, wi an be interpreted as an expeted utility representation of a man oosing te optimal tax sedule beind a veil of ignorane, reads () max! u(, )f. (, ) H = 0...H +! "$#%'&()%*$#+,"-/.0)&2+,3&4- f. Assuming a linear tenology, aggregate (per 5768"9;:)6%<>=?:48?:@92A y f. Wit an exogenous tax revenue g > 0, also defined in per apita terms, an alloation must satisfy te resoure onstraint H = 0 (2) ( y )f g. H = 0 As te differenes y = T(y ) are in fat tax payments, tis inequality represents te government s budget onstraint. A tax sedule is a mapping B o+ C wi assoiates a tax payment T(y) wit every inome and tus onfronts te persons wit legal oies (, y). Sine wage rates are exogenous, te preferene ordering over ommodity bundles (, D ) indues a preferene ordering over pairs (, y) of onsumption and inome wi depends on te respetive wage rate. Some person k aepts te pair ( k, y k ) intended for im only if no oter pair (, y ) exists wi e prefers stritly. Oterwise person k will mimi person by oosing instead of k and y /w k instead of E k. Hene, any feasible alloation must satisfy te resoure onstraint and te self-seletion onstraints (3) u( k, F k ) u(, y /w k ) for all k and were y /w k G A self-seletion onstraint is alled downward if k > and upward if k < ; it is alled adjaent if k = ±. A seond-best optimum maximizes te soial objetive () subjet to te resoure onstraint (2) and te self-seletion onstraints (3). max.

4 e f i i d d 4 Tis model is idential to te standard model introdued by Mirrlees (97), exept tat tere are finitely many tax payers. Te standard model an be obtained as a limiting ase in te following way. Wit w 0 and w H fixed and H inreasing, onsider an equidistant partition of te interval [w 0, w H ], i. e. a finite set of wage rates w 0 < w <...< w H su tat w w H I 0 for all > 0. Te eonomy is ompletely desribed by te partition, te limit distribution funtion, te utility funtion, and te required tax revenue. A sequene of inreasingly fine eonomies is a sequene wit te property JLKMN OPQR"STUS7NWVPYXZSTP[LV2\"S$]^4_`_`S%TSQ"OSNba e- tween adjaent skill levels beome smaller and smaller, and so do te probability masses f, wereas te ratios f / [F(w ) F(w )]/(w w ) onverge to te density f for all > 0. Tus, te limit distribution beomes approximated by an inreasing sequene of step funtions. 3. PRELIMINARY RESULTS Te following lemma simplifies te problem onsiderably sine te bulk of te self-seletion onstraints is eliminated and te remaining onstraints are mostly equalities rater tan weak inequalities. LEMMA : Te original maximization problem () to (3) is equivalent to te transformed maximization problem: (4) max! (, ) H 0...H + = s.t. i) ii) iii) H = 0 u( y (y, H = 0 y u(, ) f ) = u( ) f = g,, y w for all > 0. ). for all > 0, Tis lemma, proven in te appendix, is easy to understand. It says, firstly, tat a seond-best optimum is always prodution effiient in te sense tat te resoure onstraint olds wit equality. Seondly, all adjaent downward self-seletion onstraints are binding beause te optimization meanism redistributes as mu as possible from top to bottom. At te optimum, every person > 0 is indifferent between te pair (, y ) intended for im and te pair (, y ) intended for is left-and neigbor. Tis is referred to as te ain property of seond-best alloations. Finally, inome is monotonially inreasing in te wage rate. Sine te final inequalities are weak, it may our tat persons wit distint wage rates ave te same inome. Tis is usually referred to as buning. In te absene of distortionary taxes, ea person s marginal rate of substitution ug /u equals te wage rate w at any interior solution. A positive marginal tax rate implies tat te marginal rate of substitution falls sort of te wage rate and vie versa. Terefore impliit marginal tax rates an be defined as follows: + u (, ) u (, y w ) (5) L = + and R = u (, )w u (, y w )w

5 5 Te variable L denotes te impliit marginal tax rate of person, wereas R denotes te impliit marginal tax rate of person + if te latter as te same inome as te former. Tis ours wen person + mimis person or wen tere is buning at inome y. Te two impliit marginal tax rates play te entral role in our model. Tere is an important relationsip between tem. To derive it, substitute j = y/w in te utility funtion and selet some u from te utility funtion s range. Te equation u(, y/w) = u defines an impliit funtion (y) wit te derivative u ((y), y w) (6) (y) = k > 0. u ((y), y w) w Tis derivative represents te marginal rate of substitution in -y-spae. From definitions (5) it is lear tat te impliit marginal tax rates L and R equal (y ), evaluated at w and w +, respetively. Te mean value teorem implies existene of a wage rate w between w and w + su tat d (y ) (7) R + L = σ (w w ) > 0 were σ : = 0 dw >. Te positive sign of follows from te gross substitutability assumption. Tus, indifferene urves in -y-spae beome flatter if te wage rate inreases, see figure. Considering a tax sedule T(y) = y (y) wit slope T (y) = (y) it follows at one tat te sedule beomes steeper at a ertain inome level if te wage rate inreases. Tis is a result of fundamental importane: R will always exeed L at any positive inome level; at zero inome te two are idential by definition. w w + y y FIGURE Indifferene Curves in y--spae. LEMMA 2: Any seond-best optimum an be supported by a ontinuous tax sedule wi as left-derivatives L and rigt-derivatives R at all inomes y > 0 ( < H). An example of su a tax sedule for an eonomy wit four persons is depited in figure 2. Te kinks at inome levels y and y 2 are points of non-differentiability. For instane, te leftderivative at inome y equals L and te rigt-derivative equals R, and as R > L, te kink is upward. Te tax sedule is differentiable at zero inome beause ere, only a rigt-derivative exists. A formal proof of lemma 2 an be found in Homburg (2002).

6 Ä 6 T(y) T(y 0 ) y y 2 y 3 y FIGURE 2 Optimal Tax Sedule. Before onluding tis setion we would like to relate te impliit marginal tax rates to te onept of te deadweigt loss wi plays su an important role in te teory of ommodity taxation. Assume tat person < H as stritly positive onsumption and inome and faes a marginal tax rate L = 0 at te outset. Introduing a small distortion means reduing onsumption and leisure in su a way as to leave te resoure onstraint unaffeted. Tus, onsider a number lnmpoqsrut4v"wxoyzy%{ }2otwru~t)}42};t4 ƒ ) = u( ' 2 ) and differentiate wit respet ˆ ˆ $ Šˆ) Œ( $ˆ4Ž" W LŒ( " " ˆ)Œ2 Œ; $ ˆ)Œ2 4Œ;ˆ2š" (8) U'(0) = u (, œ ) L. Wit a lump-sum tax, utility falls by u (, ). Terefore, L = u L /u represents te marginal deadweigt loss, expressed as a mark-up. As usual, te deadweigt loss vanises if L = 0. Te variable R as an analogous interpretation. Let Û( ) = u( ž, (y Ÿ`2 + ) denote te utility of person + if e mimis person. Differentiating again yields (9) Û'(0) = u (, y /w + ) R. Sine a lump-sum tax diminises utility by u (, y /w + ), R is te marginal deadweigt loss imposed on person + if te latter mimis person, expressed again as a mark-up. From inequality (7) one infers tat a distortion imposed on person arms te potential mimiker + more tan it arms person imself. Te intuition beind tis result is obvious: If person + aepted te new pair ( > y ` 4 + ), e would suffer from te same derease in 7 ª"«/ x 4 ª²±«³ " µ«z ª ¹ "º4» ª"º(¼²½± ;ª ¾4 + extra units of leisure, wereas person À7Á;Â"à w extra units. For lak of a better term, R L is alled te differential deadweigt loss. To reiterate, te differential deadweigt loss is stritly positive at any stritly positive inome, indiating tat a distortion at inome y arms person + more tan it arms person. In partiular, te differential deadweigt loss is stritly positive if L itself vanises, i.e. if person s oie is not distorted at te margin. In te following setion it will beome lear tat te differential deadweigt loss is te driving fore beind te optimality of positive marginal tax rates in te finite model, but not so in te ontinuum model.

7 Ú Ý Þ Þ Ü Ù Ý Ý à á ß Ü Ý Ý Ü Ù 7 4. POSITIVE MARGINAL RATES To avoid tenial problems wi ave already been dealt wit in te literature, we assume tat tere is no buning at te optimum, i.e. y > y for all > 0, and tat individual optima are interior. Consequently, te final onstraints in te transformed optimization problem (4) drop out and all remaining onstraints are equalities. Moreover, it is easy to sow tat te resoure onstraint and te adjaent self-seletion onstraints are linearly independent. Tis allows using a onventional Lagrangean approa witout apology, sine te familiar firstorder onditions are neessarily fulfilled at an optimum, irrespetive of te problem s onvexity properties (Bertsekas 999, proposition 3..), and tus an be used to araterize te ÅzÆÇ;ÈÉÊ4ÆË"ÅzÌÍ!ËÉ4ÎÆÏÈ"ÐÊ(Ë"Ñ$ÅÓÒ"ÔÏÆÕ ÖÎÊ4Ð7ØÅ Ô%Ë"Ï for te resoure onstraint and te adjaent selfseletion onstraints, respetively, te Lagrangean funtion reads: H (0) = u(, )f + λ ( y g) f + µ ( u(, ) u(, y w ) ) = 0 H = 0 All sadow pries are stritly positive at an optimum beause any slak in a onstraint would allow inreasing te soial objetive. Differentiating wit respet to and Û, () = u = u ( (,, )f )f and rearranging terms yields λ f + λ w + µ f u + µ ( u, (, H = ) µ + ) µ u (, y + w u (, y + + (2) ( f + µ ) u (, ) = µ u (, y w ) + λ f, + u (, ) + u (, y w ) (3) ( f + µ ) = µ à λ f. + w w + w ) = 0, + w ) w + = 0, For =âãä4å"æ³ä)æç`è é³ê(ë"ìí;î"ïê(ë"ð + vanis, and dividing te two equations gives L H = 0. Tis is te familiar result tat no distortion sould be imposed at te very top of te inome distribution. For every person <H, adding te two equations and substituting L and R from (5) yields (4) (f + ) u (, ñ ) L = + u (, y /w + ) R. Tis equation is interesting beause it exibits tree impats of distortionary taxes: Firstly, taxing person redues individual utility by u L and aggregate utility by f u L. Tis is an effiieny ost. Seondly, te redution in s utility tigtens te self-seletion onstraint wi prevents from mimiking. Terefore, u L represents a redistributive ost. Finally, slak emerges in te self-seletion onstraint wi prevents + from mimiking beause te tax lowers te potential mimiker s utility, too. Multiplying tis slak by te sadow prie of te subsequent self-seletion onstraint, one obtains te redistributive gain + u R. An optimal inome tax sedule must balane te effiieny and redistributive ost on te one and and te redistributive gain on te oter..

8 ú 8 Equation (4) implies tat L and R ave te same sign but does not elp to determine it. To do so, use (2) to òóóô'òóõö;õ4óõ + u (, y /w + ) by (f + ) u (, ø ) ù, subtrat and add f L on te left-and side of (4) and solve for L to obtain (f + µ )u(, ) λf (5) L = (R L ) > 0. λf ûýü"þ ÿ ÿnÿ ÿ þ"þ ÿ%þ 2ü"þ þ xþ 2ÿ + u > 0 and R L as already been sown to be stritly positive. Tus, te optimal marginal rate is te produt of two effets: A redistribution effet wi equals te ratio on te rigt-and side. Tis ratio represents te soial rate of return of inreasing person s onsumption. Su an inrease osts, raises aggregate utility by f u and allows additional redistribution toward persons below, wi is wort u. An inentive effet R L, orresponding to te differential deadweigt loss defined in te preeding setion. Imposing a distortion on person arms te potential mimiker + more tan it arms person imself and terefore enables additional redistribution. Te total effet is te produt of tese partial effets. It is stritly positive sine te government wises to redistribute from top to bottom and a positive distortion enables it to do so. Equation (5), wi is te prinipal result for te finite eonomy, states tat te leftderivative of an optimal tax sedule must be stritly positive at every inome y > 0, exept at te igest inome were it vanises. Sine R > L, te same is true for te rigt-derivatives, and if all inomes appen to be stritly positive, te araterization is omplete. However, it may our tat y 0 vanises, in wi ase te above derivation does not apply. As buning was assumed absent, y will be stritly positive, and from te ain property one knows tat person is indifferent between te pairs (, y ) and ( 0, 0). Moving along te indifferene urve of person from te former pair to te latter, te marginal rate of substitution inreases, implying R 0 > L. Terefore, if te smallest inome appens to vanis, te marginal tax rate at te bottom of te inome distribution equals T (0) = R 0 > L > 0, were te equality is lear from figure 2 and te last inequality follows from (5). In tis ase te tax sedule is even differentiable at te bottom. If, on te oter and, te smallest inome is stritly positive, wi presumes tat te smallest wage rate is stritly positive, ten te tax sedule is nondifferentiable at te bottom, but te left- and rigt-derivatives are bot stritly positive. In te finite eonomy, te sign of te marginal tax rate equals te sign of te differential deadweigt loss under any redistributive motive. It is a remarkable fat tat tis inentive effet goes astray as te eonomies beome inreasingly fine: If w + approaes w, te rigt-derivative R approaes te left-derivative L, as is lear from definitions (5). However, tis does not imply tat te marginal tax rates onverge to zero sine formula (5) sows tat te redistribution effet diverges: As f beomes arbitrarily small in te limit, inreasing onsumption of some person > 0 osts almost noting, but doing so as a finite value even in te limit beause it enables redistributing more to people below. In order to determine te tax rate s limit beavior, use (7) to express R L as (w + w ) = substitute into (5) and!" #%$"&'")(*,+*-$&./#$0"& 2&34" (*56*+# 7 " 8#9 ution effet:

9 i : 9 (6) L (f = + µ )u(, λ f δ ) λf σ µ uσ λf. Te limit expression is idential wit Mirrlees (97) equation (27) or Seade s (977) equation (9), wi were derived using te maximum priniple and partial integration. Te present derivation makes use of te fat tat f onverges to zero wereas f ; <>=?A@BCEDB>FHG=IG JB density f wen te eonomies beome inreasingly fine. It orroborates Seade s intuition tat te marginal tax rate as only a redistributive funtion and not an inentive funtion in te ontinuum model. Hene te eonomi stories beind te positive marginal rate in te finite model on te one and and in te ontinuum model on te oter are quite different. 5. ZERO MARGINAL RATES Te main result of te preeding setion onfirms te previous findings of Stiglitz (982) and Guesnerie and Seade (982) wo sowed tat te marginal tax rate at te bottom of te inome distribution will always be stritly positive in te finite model, provided tat te tax system wises to redistribute toward te lower skilled. At te same time it runs ounter to Seade s (982) onlusion tat te marginal tax rate vanises at te bottom of te inome distribution under ertain assumptions. Essentially, 0 and y 0 are stritly positive in is model (implying tat w 0 is stritly positive), and buning is exluded. Under tese premises, equation (5KMLNO PQSR N TMUVT8QN WYX[Z\]^])LVQ_L`PNabUT8 d>v 0 vanises: Preventing person 0 from mimiking is left-and neigbor as no soial value beause te latter does not exist. Substituting tis and R L = efg (5) and eliminating f yields 0 0 u(, ) λ 0 (7) L 0 = σ δ 0, λ j)k^lm)monl-p8j l-qm r>smr-tur-vw p8r-xyljz{k jzjwr}j%zw)~rsm jxyjw l>s Seade assumed. At te bottom of te inome distribution, te inentive effet onverges to zero again but te redistribution effet remains finite, giving rise to a vanising ombined effet. In te preeding setion it was sown tat te redistribution effet diverges at any skill level exeeding w 0 as it enables additional redistribution to persons below at virtually no ost. At skill level w 0, were tere are no su persons, te argument beomes void, and tus undistortionary taxation is optimal. In te ontinuum model, te no distortion at te bottom and at te top results follow from te transversality onditions wi state tat te ƒ ˆ ŠŒ Ž Œ)Ž Ž) H ends of te inome distribution. In te finite model, tese š 8œ- ž Ÿ- 8žœ> %š š } E >œ 0 = 0 ª H+ = 0, respetively. As persons numbered and H+ do not exist, tese multipliers are not present in te Lagrangean (0), implying tat te orresponding marginal tax rates vanis. Despite te seeming symmetry of te arguments, te no distortion at te top result is relatively robust wereas its ounterpart, regarding te marginal tax rate at te bottom, is not. In deriving te latter one must assume tat te persons wit te lowest skill level ave a stritly positive inome at te optimum. Tis presupposes w 0 > 0 but te latter assumption is not suffiient: At te bottom of te optimal inome distribution, tere will normally be an interval of nonworkers even if all skill levels were stritly positive. Aordingly, all simulation studies

10 0 inluding Mirrlees (97), Stern (976), Tuomala (990), or Kanbur and Tuomala (994) arrive at stritly positive tax rates at te bottom. Moreover, one must keep in mind tat (4) is a loal result. Even if it olds, marginal tax rates will be stritly positive in any rigt-and neigborood of te smallest inome. 6. NEGATIVE MARGINAL RATES Te standard model of optimal inome taxation presumes tat people an vary teir effort ontinuously over some interval [0, «max ]. Tis is referred to as an intensive oie. Diamond (980) argued tat at least for some people te only relevant oie may be weter or not to work at all (extensive oie). Te following analysis takes up tis suggestion and assumes {0, } for all, were is a stritly positive number. Peraps in order to keep is ontinuum model differentiable, Diamond also assumed tat workers differ in te disutility of work. As te finite model does not rely on differentiability assumptions, su a premise is not required ere and will not be made: We would like to study in isolation te effets resulting from substituting extensive for intensive oies. Diamond s model of extensive oies makes it neessary to assume w 0 = 0, sine mimiking would oterwise present no problem and a first-best tax sedule ould be implemented. It is important to keep in mind tat wit intensive oies exluded, a person wit wage rate w > 0 an only pretend to be totally unable to earn positive inome, tus oosing = 0 and mimiking person 0; e annot mimi oter persons wit distint positive wage rates. As a result, te self-seletion onstraints (3) must be replaed by te onditions (8) u( k, k ) u(, 0) for all k and. Finding a seond-best tax sedule means maximizing () subjet to (2) and (8) under te assumptions w 0 = 0 and ± {0, ² } for all, but wit all oter assumptions introdued in setion 2 unanged. Every seond-best tax sedule as te following tree features: Firstly, all nonworkers enjoy te same onsumption. Tis follows diretly from te preeding self-seletion onstraints: If te government offered two pairs wit distint onsumption quantities and zero inome, everybody would oose te pair wit te iger onsumption. As person 0 as zero inome, all nonworkers reeive 0. Seondly, all workers enjoy te same onsumption, say. For, assume tere exist two workers k and su tat te first as a iger onsumption at te optimum. Sine an optimum must satisfy te self-seletion onstraints (8), we ave u( k, ³ ) > u(, ) u( 0, 0). Redistributing some onsumption from k to in aordane wit te resoure onstraint will not violate te self-seletions onstraints and will ertainly inrease te soial objetive beause person k ad a lower marginal utility of onsumption at te outset. Tirdly, workers are indifferent between working and beoming unemployed; te respetive self-seletion onstraints old as equalities: (9) u(, µ ) = u( 0, 0). Tis is easy to see: At any feasible alloation, workers onsumption must exeed nonworkers onsumption 0 sine oterwise everybody would prefer to be unemployed. As a

11 onsequene, workers ave a lower marginal utility of onsumption tan nonworkers. Tis is obvious in te ase of an additive utility funtion but also true if leisure is non-inferior (Homburg 2002, lemma 2). Hene, redistributing onsumption from workers to nonworkers inreases te soial objetive, and te government will do so as mu as possible, equation (9) araterizing te state were no more redistribution is possible witout violating te selfseletion onstraints. To summarize, a seond-best optimum is araterized by two onsumption levels only, one for te workers and one for te nonworkers, and by te remarkable fat tat all persons enjoy te same utility. As people annot vary teir oies loally, te impliit marginal tax rates ease to be meaningful, but optima an still be araterized using disrete marginal tax rates T(y ) T(y ) (20) m =, y y wi are defined for all y > y. Beause workers net inomes are all idential wereas gross inomes differ aording to te different wage rates, it follows immediately tat for any two adjaent persons and bot working, te disrete marginal tax rate m equals one undred per ent. For any two adjaent persons bot not working, te marginal rate is not defined. Te interesting marginal tax rate at te bottom of te inome distribution is enountered were y > 0 and y = 0. PROPOSITION: At any seond-best optimum, te disrete marginal tax rate m is non-negative for y > 0 and y = 0. PROOF: Sine is a worker and is a nonworker, T(y ) = y and T(y ) = 0. Tus, if te marginal rate m = (y + 0 )/y is stritly negative at te alleged optimum, 0 exeeds y. Switing to = 0 and = 0 yields an output surplus beause te fall in aggregate onsumption exeeds te fall in aggregate output. Tis move is feasible and does not ange person s utility beause of (9). Te emerging surplus an be used to make all persons better off, ontraditing te optimality of te original alloation. Hene, m 0. Q.E.D. At te optimum, nonworkers are indexed 0... and workers are indexed...h. Te separating index > 0 always exists sine w 0 = 0 implies existene of at least one zero inome, wereas g > 0 implies existene of at least one stritly positive inome. Te proposition says tat it does not pay to bribe a nonworker to join te labor fore if te required inrease in onsumption exeeds te resulting inrease in output. In te non-generi ase, m = 0 and it is immaterial weter or not person works; te anges in aggregate onsumption and aggregate output just balane. Generially, m will be stritly positive. Terefore, te optimality of non-negative marginal tax rates still olds if labor supply deisions are extensive rater tan intensive. Tis is at variane wit Diamond s (980) result. To ontrast Diamond s findings wit te above proposition, assume a Cobb-Douglas utility funtion u(, )=[(500 )] 0.4, fixed working ours ¹ = 250, and a per apita tax revenue g = 00. Te tables display five ourly wage rates distributed uniformly; f equals 20 per ent for all. Table I sows te optimal distribution of onsumption (net inome) wi was alulated subjet to a given distribution of gross inome. Utility u = 94.5 is te same for all persons. Te marginal tax rate equals 6 per ent at te bottom and one undred per ent oterwise.

12 2 Compared wit te least skilled person, person reeives an extra onsumption amounting to = 055. Te bottom marginal tax rate is stritly negative beause tis extra onsumption exeeds is gross inome. Diamond, optimizing only wit respet to onsumption in is teoretial derivation, presented a similar example. TABLE I NEGATIVE MARGINAL TAX RATE w º y T m % % % % Te important message of te proposition above is tat negative marginal tax rate an never be optimal. Table I does not desribe te full optimum. Te full optimum, resulting from optimizing over onsumption and labor supply, is sown in table II, were person no longer joins te labor fore. All utilities ave risen from 94.5 to 95.0, implying tat tis move represents a Pareto-improvement. Te eonomi explanation is obvious: If person beomes unemployed, an output surplus emerges wi an be used to make all persons (inluding person imself) better off. Every rational taxpayer or welfare reipient would prefer te alloation depited in table II to te negative marginal tax ase depited in table I. TABLE II OPTIMAL INCOME TAX SCHEDULE» w y T m % % % Introduing a negative marginal tax rate su as te U. S. earned inome tax redit (EITC) means moving from table II to table I. Wile tis move makes all persons worse off, output and employment inrease. Tis suggests tat in evaluating su a reform measure one must arefully distinguis welfare effets on te one and and output and employment effets on te oter (Browning, 995). Most empirial analyses of te EITC (like Hotz and Solz, 200) onentrate solely on te latter, wi gives rise to a bias in favor of te EITC. Keynes one oined te famous prase tat one an always ensure full employment by letting te unemployed dig someting into te eart and dig it out tereafter, paying tem an inome for tis useless ativity. To do so would ertainly not be a sensible poliy.

13 3 Using a variant of Diamond s (980) model, Saez (999) defended te optimality of negative marginal tax rates wit te following argument: Starting from a situation wit equal transfers to low inome workers and te unemployed, inreasing te transfers to te low inome workers is benefiial from a pure redistributive point of view... Tis also enourages some of te unemployed to join te labor fore at zero fisal ost as transfers are initially equal for te two groups. As a result, it is unambiguously welfare enaning to provide a larger transfer to low inome workers tan to te unemployed. (Saez 999, p. 2). Tis is learly wrong. Assume transfers to persons 0 and are equal at te outset. Introduing a stritly negative marginal tax rate means offering person an extra onsumption, relative to te unemployed person 0, wi exeeds te extra output resulting from is effort. If person aepts and starts working, tis indues a stritly positive fisal ost. Te additional ost neessitates reduing oter persons utilities. As (9) implies tat all utilities are idential at te optimum, a Pareto deterioration results. From te taxpayers perspetive, it is better to aept and to finane a ertain degree of voluntary unemployment tan to figt tis unemployment using negative marginal tax rates. To be sure, tis reasoning does not rejet negative taxes (wage subsidies) in general; it only rejets negative marginal tax rates. From table II, wi desribes te full optimum, one infers tat te person wit te smallest positive inome reeives a wage subsidy amounting to 25. Te marginal tax rate equals 47 per ent at te bottom. T(y) T(y 0 ) y min y H y FIGURE 3 Limit Tax Sedule. Wen te eonomies beome inreasingly fine, te tax sedule onverges to a limit su as depited in figure 3. Te limit sedule onsists of an isolated point representing te transfer to te unemployed and a straigt line, starting at some minimum observable inome y min. Te tax T(y min ) may be positive or negative, but must exeed T(y 0 ). Te slope of te line equals unity. Marginal tax rates of one undred per ent are an unrealisti feature, of ourse, triggered by te assumption tat people annot find part time jobs or oterwise redue teir efforts, wi itself is unrealisti. 7. CONCLUSION Tis paper examined te sign of optimal marginal tax rates at te bottom of te inome distribution. Te framework used was te finite ounterpart of te standard model of te optimal inome tax. Our results sustain te view tat te marginal inome tax rate sould be stritly

14 4 positive at te bottom beause it as an inentive funtion in te finite model and at least a redistributive funtion in te ontinuum model. Zero marginal rates are more or less teoretial artifats: In te finite model, te marginal rate will never vanis at te bottom, wereas in te ontinuum model it will do so only under stringent assumptions, and just for a null set of persons. Considering extensive instead of intensive labor supply oies does not ange tese results: Zero marginal rates are still non-generi, and stritly negative rates are never optimal. Tese onlusions ave been derived, of ourse, witin te usual framework and may be upset one different assumptions beome introdued. However, te premises of te usual framework are quite general and onvining. Negative marginal tax rates su as te EITC seem to stem from a multilateral illusion, were te poor believe tey profit from te subsidy wile te ri are appy about te seeming relief from taxation wi tey tink will aompany te fall in unemployment. Tus tere is unanimous politial support, and as long as eonomists endorse tis illusion, te EITC will ave a brigt future. But te analysis onduted in setion 6 sowed tat induing low-skilled people to join te labor fore by means of negative marginal tax rates results in a Pareto deterioration. From tis perspetive, taxpayers are better advised to aept a ertain degree of voluntary unemployment. Faulty of Business Administration and Eonomis, University of Hannover, Königsworter Platz, D-3067 Hannover, Germany. APPENDIX PROOF OF LEMMA : Propositions 2a), 4, and b) in Homburg (2002) sow tat every solution of te original maximization problem must be prodution effiient and must satisfy te ain property and te monotoniity property. Hene, te onstraints i) to iii) of te transformed optimization problem are indeed neessary for an optimum. It remains to be sown tat all alloations satisfying tese onstraints belong to te feasible set defined by (2) and (3). To do so we make use of Homburg s (2002) agent monotoniity: If a person is indifferent between two distint pairs (, ȳ ) and (_, _y), were te former ontains more inome tan te latter, ten every person wit a iger wage rate stritly prefers (, ȳ ) and every person wit a lower wage rate (tat an rea bot pairs) stritly prefers (_, _y). Tis follows from te gross substitutability assumption in te text. Te binding self-seletion onstraint u(, ¼ ) = u(, y /w ) states tat person is indifferent between (, y ) and (, y ). For all numbers i =..., te preeding self-seletion onstraints assert tat person i is indifferent between ( i, y i ) and ( i, y i ). Now, agent monotoniity implies tat person prefers every ( i, y i ) to every ( i, y i ) beause ompared to person i e as a iger wage rate and beause te former pairs ontain more inome tan te latter; te preferene is even strit if any two pairs under onsideration are distint. From te transitivity of te preferene ordering it follows tat person prefers te pair intended for im to all pairs intended for low-skilled persons. Terefore, all downward self-seletion onstraints are satisfied. As a perfetly analogous argument olds for te up-

15 5 ward self-seletion onstraints, every alloation satisfying i) to iii) belongs to te feasible set. i.e. satisfies all onstraints of te original problem. Q.E.D. REFERENCES Bertsekas, Dimitri P. (999): Nonlinear Programming. Belmont: Atena Sientifi. Browning, E. K. (995) Effets on te Earned Inome Tax Credit on Inome and Welfare. National Tax Journal, 4, Diamond, P. A. (980): Inome Taxation wit Fixed Hours of Work, Journal of Publi Eonomis, 3, 0 0. Guesnerie, R. and J. Seade (982): Nonlinear Priing in a Finite Eonomy, Journal of Publi Eonomis, 7, Homburg, St. (2002): Te Optimal Inome Tax: Restatement and Extensions, Disussion Paper, University of Hannover. Hotz, V. J. and J. K. Solz (200): Te Earned Inome Tax Credit. NBER Working Paper Mirrlees, J. A. (97): An Exploration in te Teory of Optimum Inome Taxation, Review of Eonomi Studies, 38, Kanbur, R. and M. Tuomala (994): Inerent Inequality and te Optimal Graduation of Marginal Tax Rates, Sandinavian Journal of Eonomis, 96, Mirrlees, J. A. (97): An Exploration in te Teory of Optimum Inome Taxation, Review of Eonomi Studies, 38, Saez, E. (999): Optimal Inome Transfer Programs: Intensive versus Extensive Labor Supply Responses, NBER Working Paper Seade, J. K. (977) On te Sape of Optimal Tax Sedules, Journal of Publi Eonomis, 7, Seade, J. K. (982): On te Sign of te Optimum Marginal Inome Tax, Review of Eonomi Studies, 49, Stern, N. (976): On te Speifiation of Models of Optimum Inome Taxation, Journal of Publi Eonomis, 6, Stiglitz, J. E. (982): Self-seletion and Pareto Effiient Taxation, Journal of Publi Eonomis, 7, Tuomala, M. (984): On te Optimal Inome Taxation, Journal of Publi Eonomis, 23,