Lower Tax For Minimum Wage Earners

Transcription

1 Lower Tx For Minimum Wge Erners Jim Jin nd Felix FitzRoy School of Economics nd Finnce Online Discussion Pper Series issn X info: School of Economics nd Finnce Discussion Pper No Fe 2017 (revised 17 Fe 2017) JEL Clssifiction: H20, D60 Keywords: flt tx, income redistriution, mximin, Preto improvement

2 LOWER TAX FOR MINIMUM WAGE EARNERS Felix FitzRoy* Jim Jin** School of Economics nd Finnce University of St Andrews JEL Clssifiction Numer: H20, D60 Key Words: flt tx, income redistriution, mximin, Preto improvement Ferury 2017 Astrct: We show tht minimum wge erners should py lower tx thn high erners. Though intuitive, this ide is not supported y the existing literture. The optiml mximin tx curve nd two-nd txes re usully decresing. Since decresing mrginl txes would e unpopulr, y continuity flt tx seems to e superior to incresing mrginl txes nd should e second est solution. However, using simple utility function nd generl income distriution, we find tht lowering the mrginl tx for minimum wge erners not only domintes the optiml flt tx under mximin, ut lso mke everyone etter off. We thnk Ttin Dmjnovic nd Dvid Ulph for their helpful comment nd discussion. The usul disclimer pplies. * School of Economics nd Finnce, University of St Andrews, St Andrews KY16 9AL, Scotlnd, UK, Tel. (01334) , Fx: (01334) , e-mil: frf@st-ndrews.c.uk. ** Corresponding uthor: School of Economics nd Finnce, University of St Andrews, St Andrews KY16 9AL, Scotlnd, UK, Tel. (01334) , Fx: (01334) , e-mil: jyj@st-ndrews.c.uk.

3 1. Introduction In mny countries no income tx is levied on the lowest ernings up to certin level, politiclly populr mesure to help the poor. As income inequlity rises, the threshold for tx exemption hs recently een rised to 11,000 in the U.K. Though widely ccepted, this mesure is not supported y the existing tx literture, lthough incresing mrginl txes (IMT) re common in developed economies, usully in the form of piecewise liner tx systems. In this pper we rgue tht indeed minimum wge erners should e txed t lower rte thn high erners. Income redistriution through progressive txtion ppers more desirle s inequlity hs risen to record heights (Stiglitz, 2012; Piketty, 2014; Atkinson, 2015). As the proportion of low wge erners increses, so does the importnce of setting pproprite tx rtes for them to ensure work incentives nd reduce inequlity. However the literture provides little justifiction for uniformly IMT nd there is no cler-cut result for the optiml tx structure. With sutle differences in socil welfre nd income distriution functions, the optiml mrginl tx curves cn e U-shped (Dimond (1998), Sez (2001)), or inverse U-shped (see Sdk (1976), Sede (1977), Tuoml (1984), Hindricks et l, 2006, Knur nd Tuoml (1994), Bodwy et l (2000), Trkiinen nd Tuoml (2007), Hshimzde nd Myles (2007), Kplow (2008) nd see good survey y Jcos (2013)) 1. Although inverse U-shped curves suggest lower txes for the lowest erners, they lso imply decresing tx rtes for top erners which re politiclly infesile. This politicl concern nd dministrtive costs my render the continuous tx curves too fr removed from the tx enefit systems oserved in prctice to e useful guide for 1 One exception is Aerge nd Colomino s (2013) numericl simultion, sed on Norwegin dt, which yields the optiml IMT. Their result relies on declining elsticity of lour supply. 1

4 policy (Choné nd Lroque 2005, p.396). Dimond nd Sez (2011) cll for prgmtic reserch on tx policy. The opposite extreme to the continuous tx curve is flt tx nd the next step is two-nd txes. If we restrict tx rtes to e non-decresing on high income, n inverse U-shped optiml tx curve might e reduced to incresing two-nd txes. Then one my expect IMT to e justified s first pproximtion to the originl optiml tx curve. In fct this is wht Sheshinski (1989) initilly found when he exmined two-nd txes under utilitrin nd mximin ojectives. However Slemrod et l (1994) lter showed tht this is not correct nd the optiml two-nd txes should e decresing 2. Slnie (2003), Hindricks nd Myles (2006) otin similr results in simple two-clss economy. Therefore lower txes for minimum wge erners re not esily justified. Nonetheless, s we mentioned erlier, decresing mrginl txes re often viewed s unfir nd politiclly uncceptle. They lso imply high mrginl tx rtes for the lowest income erners nd the resulting poverty trp. In contrst flt tx is politiclly more cceptle nd y continuity it seems to e closer to the optimum thn IMT. Hence Mnkiw et l (2009) suggest tht A flt tx, with universl lump-sum trnsfer, could e close to optiml. It cn sve dministrtive costs nd eliminte incentive distortions ssocited with progressive txes. Severl Estern Europen countries hve moved in this direction. Atkinson (1995) provides comprehensive tretment of flt tx. Finlly, ut not lest importnt, n optiml flt tx is esy to find. Piketty nd Sez (2012) otin the optiml flt tx given generl welfre function nd relistic informtion, though they do not necessrily dvocte its implementtion 3. 2 Bodwy nd Jcquet (2008) lso show the optiml mrginl tx curve is declining under mximin. 3 They rgue for higher tx for top incomes (pproximtely the top 5%) sed on decresing mrginl utility of income. Jcos (2013) lso rgues ginst flt tx for vrious resons not discussed here. Forml proof tht flt tx is worse thn IMT is presented in FitzRoy nd Jin (2009), on prts of which this pper is sed. 2

5 In contrst we rgue tht ny flt tx is Preto inferior to two-nd txes with lower rte for minimum wge erners. This result seems to contrdict the literture nd suggests non-monotonicity of the optiml tx structure. In two-group exmple Slemrod et l (1994) grphiclly demonstrte tht flt tx cn e worse thn IMT for oth groups. In other words, moving from flt tx to IMT my e Preto improvement. This seems surprising to mny economists. In his critique of rdicl income redistriution through txtion, Mnkiw (2013) wrote: As fr s I know, no one hs proposed ny credile policy intervention to del with rising inequlity tht will mke everyone, including those t the very top, etter off. The exmple of Slemrod et l (1994) indictes tht such possiility my exist under flt tx s dvocted y Mnkiw et l. (2009). This issue hs not een further discussed in the literture. This pper seeks to fill this gp nd show tht Preto improvement from flt tx is not only possile, ut rises under generl conditions, including the most unfvourle ssumption of identicl elsticity of lour supply nd mrginl utility of income. We find tht given n optiml flt tx under mximin it is lwys desirle to lower the tx rte for minimum wge erners. Moreover, given ny optiml flt tx tx reduction on minimum wge erners y their proportion in the popultion must e Preto improvement. This result lso holds when there is voluntry nd involuntry unemployment, or more thn two tx nds. The model is developed in the next section. Section 3 shows tht lowering tx for minimum wge erners domintes the optiml flt tx under mximin. In Section 4 we demonstrte tht such simple tx reduction cn enefit everyone. Section 5 concludes nd proofs re in ppendices. 2. The Model 3

6 We ssume tht popultion, normlized to unity, consists of continuum of households, whose wge is denoted y w, nd is distriuted on [, ], where 0 < <, nd represents the minimum wge. The wge distriution follows cumultive nd density functions F(w) nd f(w). A household s pre-tx ernings y re proportionl to its wge w nd lour supply x, i.e., y = wx. The government oserves ernings nd imposes two tx rtes, t1 nd t2, when ernings re elow or ove threshold y. If one s ernings do not exceed y, his fter-tx ernings re wx(1 t1). Otherwise they re wx(1 t2) + y (t2 t1). The tx revenue is eqully distriuted s sic income B to everyone, fter fixed pulic expenditure P is pid. The unemployed re not included in the unit popultion nd their welfre will e discussed in Section 5. We ssume everyone s utility function is m x /(1 +1/), where m is income nd is the elsticity of lour supply. Similr utility functions re used in the literture (e.g. Atkinson 1995, Bodwy nd Jcquet 2008). Our identicl elsticity of lour supply for the entire popultion is oviously less fvourle for IMT thn empiriclly plusile declining elsticity, used y Aerge nd Colomino (2013). Given t1, t2 nd y, the utility function for different erners cn e written s: V1 = wx( t1) 11/ ε x + B for wx y (1) 11/ V2 = wx(1 t2) + y (t2 t1) 11/ ε x + B for wx > y (2) 11/ Every household chooses its lour supply to mximize its utility. When t1 < t2, we define w1 nd w2 y (1 t1) w1 = y nd (1 t2) w2 = y. Then households cn e divided into three groups. Those with w w1 will choose lour supply x = w (1 t1). Those with w > w2 choose x = w (1 t2). The remining ones with w1 < w w2 choose 4

7 x = y /w nd just ern y, i.e., unching. The tx revenues generted from these three groups re t1(1 t1) w1 1 w dw, t2(1 t2) w2 w dw (t2 t1)[1 F(w2)] y w nd t1 y 2 w1 dw respectively. So the totl tx revenue is equl to: R = t1(1 t1) w1 1 w dw + t2(1 t2) w2 w dw + {t1[1 F(w1)] t2[1 F(w2)]} y (3) We ssume the fixed pulic expenditure P is less thn the revenue, so it does not ffect our tx decision nd the sic income B cn e treted s R (minus P). We sustitute the optiml lour supply w (1 t1) into the utility function (1), nd let w =, to otin the utility of minimum wge erners s the mximin ojective function. u = (1 t1) + B (4) 1 Under mximin ojective, we mximize the utility of minimum wge erners. We ssume F() > 0, so minimum wge erners hve positive mss nd the mximin is resonle socil ojective. Also prt time employees often ern less thn full time minimum wge erners. Comining them together we hve significnt frction F() in most countries. In UK nd USA for instnce, this frction is pproximtely 20% nd 30% respectively 4. As minimum wges rise in these countries, so will the proportion of minimum wge erners nd the importnce of the relevnt tx policy. In the next section we show the optiml flt tx under mximin is lwys inferior to two-nd IMT. 4 In UK the minimum wge ws 6.08 per hour in 2011, implying nnul ernings of 12,000, exceeding 20% British income ( The US verge minimum wge is out $7.5/hour, or nnul ernings $15,000, more thn 30% Americns 2010 income ( 2010"). 5

8 3. Mximin To justify lower tx for minimum wge erners, we first find the optiml flt tx under mximin. Piketty nd Sez (2012) find n optiml flt tx under generl welfre function, which is weighted sum of individul welfre. It hs simple form of (1 g )/(1 + g ), where the elsticity of ggregte ernings is equivlent to our individul elsticity of lour supply, nd g is the rtio of the verge income weighted y individul socil welfre weights to the verge income of the popultion. When t1 = t2, the tx revenue in (3) reduces to t(1 t) simplicity, we let E w f ( ) w f ( ) w dw. For w dw, where the integrl includes the mss t. So E is the totl nd lso verge ernings of the popultion without tx. Then we cn write the mximin ojective function (4) s (1 t) /(1 + ) + t(1 t) E. Its first-order derivtive with respect to t is (1 t) {[1 (1 + )t]e (1 t)}, which is positive if nd only if t is smller thn the expression t* given elow. Hence we otin the optiml flt tx under mximin. E Proposition 1: The optiml flt tx under mximin is t* = ( 1 ) E. The result is specil cse of Piketty nd Sez (2012), where /E represents their g s ll the socil welfre weight is given to minimum wge erners (they use ctul ernings, ut the rtio is sme under flt tx). The optiml flt tx t* flls with /E s the gp etween the poor nd rich is smller, nd flls with s higher elsticity implies more efficiency loss due to txtion. The next question is whether lowering tx for minimum wge erners cn mke them etter off thn the optiml flt tx. The nswer seems to e no from the existing 6

9 literture. With low productivity, minimum wge erners my not gin very much from lower tx, ut lose more in sic income B due to the loss of tx revenue from the rest of the popultion. However we cn show tht minimum wge erners cn lwys enefit from lower tx rte elow the optiml tx t*. It suffices to show this is true when we let the threshold income y = (1 t1). It ensures tht minimum wge erners do not fce the high tx when choosing optiml lour supply. Given y = (1 t1) nd our definition of w1 implies w1 =. The only people fcing the low tx nd not unching re minimum wge erners. If t2 > t1, we hve w2 >. Strting from t*, if we lower t1, oth y nd w2 increse. Individuls with w etween nd w2 will e unching, erning just y. We cn show tht when t1 = t2 = t*, u/t1 < 0 (see Appendix A). Hence there lwys exist some IMT with t1 < t2 = t*, tht give higher vlue of (4), i.e. enefit minimum wge erners. Proposition 2: Given y = (1 t1), lowering t1 elow the optiml flt tx t* enefit minimum wge erners. Thus the optiml flt tx under mximin is dominted y two-nd IMT with lower tx for minimum wge erners. Relisticlly, however, the government is concerned not only with the welleing of minimum wge erners, ut lso the society s whole. The mximin optiml flt tx t* is not the one dvocted y Mnkiw et l. (2009). To justify lower tx for minimum wge erners, we need to compre it with n optiml flt tx in more generl sense nd use more roust criterion to demonstrte its superiority. In the next section we will show tht strting from ny optiml flt tx, lower tx rte for minimum wge erners cn mke everyone etter off. 4. Preto dominnce 7

10 The mximin ojective is not politiclly fesile, so the question is whether some tx reduction for minimum wge erners cn mke everyone etter off thn under flt tx. If so, we would hve strong cse for such tx reduction. We will show tht reduction of t1 is indeed Preto improvement over ny flt tx. The flt tx need not e t*, ut ny optiml tx under generl socil welfre function. We follow Piketty nd Sez (2012) to construct generl socil welfre function. Let s(w) ( 0) e the socil welfre weight ttched to households with wge w, suject to s ( w) dw = 1. Following (1) nd (2), under flt tx t, household s net utility is u(w) = w (1 t) /(1 + ) + B, where B is equl to t(1 t) E P. Hence the generl socil welfre function cn e written s s ( w) u( w) dw = 1 (1 t) 1 w s( w) f ( ) w dw + t(1 t) E P (5) As we mentioned erlier, our elsticity of lour supply plys the sme role s the elsticity of ggregte ernings in Piketty nd Sez (2012). Differentiting the welfre function (5) with respect to t, we see tht its vlue increses with tx rte t if nd only if t < [E tx. Letting w s( w) f ( ) w s( w) f ( ) w dw ]/[(1 + )E the sme form s in Piketty nd Sez (2012): w s( w) f ( ) w dw ], which is the optiml flt w dw /E g, the optiml tx mximizing (5) cn e written in 1 g Proposition 3: The generl optiml flt txtˆ =. 1 g The welfre weight s(w) should e non-incresing with w. If s(w) = 1, we hve the utilitrin welfre function. Thus g = 1 nd the optiml flt tx is zero. If s(w) = 0 except for w =, we hve g = /E, tˆ ecomes the previous optiml flt tx t* under 8

11 mximin. In generl the vlue of g lies etween 1 nd /E, nd tˆ is etween zero nd t*. Conversely, ny flt tx within this rnge cn e viewed s n optiml tx under certin socil welfre function ssocited with non-incresing socil welfre weight function s(w). Given generl optiml flt tx tˆ, we show tht tx reduction for minimum wge erners cn provide Preto improvement. We lredy know tht reduction in t1 enefits minimum wge erners. Then we consider the next income group, i.e. individuls with wge etween nd w2. They only work up to the threshold ernings y = (1 t1), i.e. unching. Their lour supply is (1 t1) /w. When t1 flls, they py lower tx rte nd gin from incresed lour supply due to higher y. Sustituting these vriles into (1), we get their utility function. U1 = ( t1) ( ) (1 t1) + B (6) 1 w Finlly we consider high income erners with w > w2. Their lour supply is not ffected y t1. When t1 flls nd y = (1 t1) rises, they will enefit from lower infr-mrginl tx rte nd higher tx threshold y. Sustituting y = (1 t1) into (2), we cn write their utility function s: U2 = w (1 t2) + (t2 t1)(1 t1) + B (7) 1 Differentiting (6) nd (7) with respect to t1, we find tht oth derivtives re negtive when t1 = t2 = tˆ (see Appendix B). Hence (6) nd (7) rise when t1 flls nd we hve Preto improvement over the optiml flt tx tˆ. Proposition 4: Lowering t1 is Preto improvement over the optiml flt txtˆ. 9

12 Our result generlize the insight from the exmple of Slemrod et l (1994) tht ny flt tx is Preto inferior to some IMT. This outcome depends on the ssumption of F() > 0, i.e. positive mss of minimum wge erners. This my seem surprising, ut the intuitive explntion is strightforwrd. When we lower the tx rte for minimum wge erners, the high income group will not chnge their lour supply. The other two groups, the unching group nd minimum wge erners, will increse their lour supply, which is (1 t1) /w nd (1 t1) respectively. Everyone enefits from lower tx ut receives less income trnsfer, i.e. the sic income B. If minimum wge erners hve positive mss, their incresed lour supply will increse the totl output, which ensures the totl enefit exceeds the reduction in tx revenue. As everyone enefits t lest from tx reduction of y t1, nd loses n equl mount of sic income, no one cn e worse off. This intuitive rgument pplies not only in our model with specific utility function, ut lso in the generl cse. Although everyone enefit from such tx reduction, they my not enefit eqully. One my expect tht minimum wge erners enefit the most since they re the min trget of tx reduction. This is not necessrily true s higher wge erners lso py less tx even though they my not djust their lour supply very much. In fct, if we compre the derivtives of the utility respect to t1 in Appendices A nd B, it is esy to see tht U1/t1 < u/t1 lwys nd U2/t1 < u/t1 given t1 < t2. Hence these two groups will enefit more thn minimum wge erners do when t1 flls. Proposition 5: Higher income erners enefit more from the tx reduction on minimum wge erners. This should reduce ny politicl resistnce ginst the tx reduction. This finding cn e compred with Sdk (1976), nd Sede (1977), who show tht the tx rte on the 10

13 top ernings should e zero. While the zero tx rte only pplies to one individul in their cse, our tx reduction should e pplied to positive mss, nd thus cn ring in significnt improvement. So fr we hve only shown the positive impct of mrginl tx reduction for minimum wge erners. Generlly this impct my chnge when the tx reduction ecomes significnt. For prcticl policy recommendtion, it is desirle to know if this positive effect cn e sustined for discrete chnge in the tx rte. Hence we will show tht this is gurnteed for tx reduction y the mount of F() tˆ. We first consider the impct of reduction in t1 y F() tˆ on minimum wge erners. In Appendix A when we keep t2 = tˆ nd let t1 fll from tˆ to [1 F()] tˆ, (A3) implies tht u/t1 = m(1 t1) tˆ [F(w2) F()] < 0. So u rises s t1 flls, nd minimum wge erners re etter off with the tx reduction. Similrly, the derivtives of U1 nd U2 respect to t1 re oth negtive when t1 flls from tˆ to [1 F()] tˆ. Hence (6) nd (7) must rise when t1 flls y F() tˆ. Therefore we otin: Proposition 6: Reducing the tx rte on minimum wge erners from the optiml flt tx tˆ y F() tˆ is gurnteed to e Preto improvement. Our finding leds to simple policy recommendtion: the tx rte on minimum wge erners should e lower thn the next tx rte y proportion of F(), which is their proportion in the popultion. The group, including minimum wge erners nd most prt-timers, is roughly 20% in the UK (2011) nd 30% in the USA (2010). Their full time ernings re roughly 12,000 nd $15,000, with mrginl tx rtes 20% (on income from 11,000 to 35,000) nd 15% (up to $34,000) respectively. In the UK, ntionl insurnce contriutions of 12% on incomes over 8,000 rise the effective tx rte to 32% on income over 11,000. Our result suggests tht everyone will e etter off if the tx rtes 11

14 up to 12,000 nd $15,000 fll to 16% nd 10.5%. Tx reduction of F() tˆ is not the limit for Preto improvement. Even though further reduction my not enefit everyone, it cn e Preto improvement over the flt tx. In ddition to the positive effect on employment, we my justify zero tx for low income erners, such s the income-tx free llownce of 11,000 in U.K. We see from the proof tht the Preto improving tx reduction needs not e from the optiml tx tˆ, ut ny flt tx. Furthermore, it my lso e from ny multi-nd txes. We my ssume tht eyond y, there re other income thresholds, ove which other tx rtes pply. Then the tx revenue R in (3) remins the sme except for the third term, t2(1 t2) w 2 w dw ecomes multiple integrtions with different tx nds. However these new terms will not e ffected y t1. So when we clculte B/t1, the result remins the sme. Hence the tx reduction for minimum wge erners is still Preto improving under generl multi-nd tx system. Finlly we rgue tht the tx reduction for minimum wge erners remins Preto improvement when there is unemployment, either voluntry or not. So fr we hve not explicitly considered unemployment, while the fixed pulic expenditure my include the unemployment enefit. If the unemployment is involuntry, the tx reduction will not ffect it. The voluntrily unemployed my choose to work fter the tx reduction. Given the sme unemployment enefit nd Preto improvement for ll employed, when the unemployment flls, the extr tx revenue cn e used to increse the unemployment enefit. So the tx reduction is still Preto improvement. 5. Conclusions Our results suggest tht lowering the tx rte for minimum wge erners lwys domintes ny flt tx under mximin. Moreover tx reduction for minimum wge 12

15 erners cn enefit everyone, nd hence cn e justified under ny socil welfre function. This result seems intuitive ut to the est of our knowledge it hs not een reported in the literture. Of course our results depend on sic income, quite different from existing enefit systems, nd income txes re generlly progressive. However, there is growing interest in sic income, often in conjunction with flt tx. Perhps of more relevnce, totl (direct nd indirect) verge tx pid in the UK is roughly similr percentge of gross income cross the distriution. The welfre gin from the tx reduction my e smll in our model, ut is likely to e greter in more relistic cses. For instnce we do not consider decresing mrginl utility of income s Piketty nd Sez (2012) do, which provides further socil justifiction for progressive txes. Also, differing from Aerge nd Columino (2013), we ssume n identicl elsticity of lour supply for the entire popultion. In relity its vlue for full-time nd high income erners is much lower thn tht for low income erners. This mens less efficiency loss due to txtion on high erners. In prticulr, we do not consider the crucil prticiption decision of low income erners, which genertes the poverty trp under high implicit mrginl tx rtes s enefits re phsed out. A lower tx for minimum wge erners will reduce the disincentives to work, nd generte potentilly more welth for the society. These fctors my further justify the tx reduction, ut re omitted from our tx model for simplicity nd trctility. Tx reduction for low erners nd its impct ecome more significnt with lrge F(), which hs incresed in mny developed countries fter the recent finncil crisis nd provides dditionl support for tx reform. 13

16 References Aerge, R. nd U. Colomino, 2013, Using Microeconometric Model of Household Lour Supply to Design Optiml Income Txes, Scndinvin Journl of Economics, vol. 115, pp Atkinson, A. B., Pulic Economics in Action: The Bsic Income/Flt Tx Proposl, Clrendon Press, Oxford, 1995 Bodwy, R, K. Cuff, nd M. Mrchnd, 2000, Optiml Income Txtion with Qusi-Liner Preferences Revisited, Journl of Pulic Economic Theory, Vol. 2, pp Bodwy, R, nd L. Jcquet, 2008, Optiml Mrginl nd Averge Income Txtion under Mximin, Journl of Economic Theory, vol. 143, pp Chone, P., nd G. Lroque, 2005, Optiml Incentives for Lour Force Prticiption, Journl of Pulic Economics, vol. 89(2-3), pp Dimond, P. A., 1998, Optiml Income Txtion: An Exmple with U shped Pttern of Optiml Mrginl Rtes, Americn Economic Review, vol. 88(1), pp Dimond, P. A., nd Sez, E., 2011, The Cse for Progressive Tx: From Bsic Reserch to Policy Recommendtions, The Journl of Economic Perspectives, Vol. 25, No. 4 (Fll 2011), pp FitzRoy, F. nd J. Jin, 2009, Are Two Txes etter thn One? IZA dp Hshimzde, N. nd G. Myles, 2007, Structure of the Optiml Income Txtion in the Qusi- Liner Model, Interntionl Journl of Economic Theory, vol. 3, pp Hindricks, J., E. Lehmnn nd A. Prmentier, 2006, Optiml Income Txtion nd the Shpe of Averge Tx Rtes, Economic Bulletin, vol. 8, pp. 1-6 Hindricks, J. nd G. Myles, Intermedite Pulic Economics, MIT Press, London, 2006 Jcos, B., 2013, From Optiml Tx Theory to Applied Tx Policy, CES-Info Working Pper, #4151. Knur, R. nd M. Tuoml, 1994, Inherent Inequlity nd the Optiml Grdution of Mrginl Tx Rtes, Scndinvin Journl of Economics, vol. 96, pp Kplow, L., The Theory of Txtion nd Pulic Economics, Princeton University Press, Princeton nd Oxford, 2008 Mnkiw, N. G., M. Weinzierl & D. Ygn, 2009, Optiml txtion in Theory nd Prctice, NBER Working Pper No Mnkiw, N. G., 2013, Defending the One Percent, Journl of Economic Perspective, 27(3): pp Piketty,T. nd E. Sez, 2012, Optiml Lour Income Txtion, NBER working pper Sdk, E., 1976, On Income Distriution, Incentive Effects nd Optiml Income Txtion, Review of Economic Studies, vol. 43, pp Sez, E., 2001, Using Elsticities to Derive Optiml Income Tx Rtes, Review of Economic Studies, vol. 68, pp Slnié, B., The Economics of Txtion, MIT Press, 2003 Sede, J., 1977, On the Shpe of Optiml Income Schedules, Journl of Pulic Economics, vol. 7, pp Sheshinski, E., 1989, Note on the Shpe of the Optimum Income Tx Schedule, Journl of Pulic Economics, vol. 40, pp Slemrod, J., S. Yitzhki, J. Myshr, nd M. Lundholm, 1994, The Optiml Two Brcket Liner Income Tx, Journl of Pulic Economics, vol. 53, pp Stiglitz, J., 1982, Self-selection nd Preto Efficient Txtion, Journl of Pulic Economics, vol. 17, pp Trkiinen, R. nd M. Tuoml, 2007, On Optiml Income Txtion with Heterogeneous Work Preference, Interntionl Journl of Economic Theory, vol. 3, pp Tuoml, M., 1984, On the Optiml Income Txtion: Some Further Numericl Results, Journl of Pulic Economics, vol. 23, pp

17 Powered y TCPDF ( Appendix A, Proof of Proposition 2: Strting from t*, we show u/t1 < 0. Given t1 t2 = t*nd y = (1 t1), i.e., w1 =, (3) implies the tx revenue cn e written s: R = t1(1 t1) + t2(1 t2) w 2 w dw t2(1 t1) [1 F(w2)] (A1) As w 2 (1 t2) = (1 t1), we hve R/w2 = t2(1 t2) w 2 f(w2) + t2(1 t1) f(w2) = 0. So we ignore indirect impcts vi w2 when differentite R with respect to t1. B = (1 t1) {1 (1 + )t1 + t2[1 F(w2)]} (A2) t 1 We sustitute these results when differentiting u in (4) nd otin u = (1 t1) {t2[1 F(w2)] t1} (A3) t 1 When t2 = t1 = t*, u/t1 < 0 s t2f(w2) = t*f() > 0. Appendix B, Proof of Proposition 4: We first differentite (6) with respect to t1, nd get U t 1 1 = (1 t1) [1 + w ) ] + B t 1 (B1) Sustituting B/t1 in (A2) into (B1), we otin U t 1 1 = (1 t1) {(1 w ) )(1 t1) + t1 t2[1 F(w2)]} (B2) When t2 = t1 =tˆ, (B2) is negtive if [1 (/w) ](1 tˆ ) + tˆ F() > 0. As < w, this inequlity holds. So U1 must rise s t1 flls. Differentiting (7) with respect to t1 we get U t 1 2 = (1 t1) [1 t1 + (t2 t1)] + B t 1 (B3) Plugging (A2) into (B3), we otin U2/t1 = (1 t1) t2f(w2) < 0. 15