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Responsibility sensitive egalitarianism and optimal linear income taxation Erik Schokkaert a Dirk Van de gaerb * Frank Vandenbrouckec Roland Iwan Luttensb Centre for Economic Studies KULeuven Naamsestraat 69 B-3000 Leuven Belgium b SHERPPA Faculteit Economie en Bedrijfskunde Ghent University Hoveniersberg 24 B-9000 c Faculty of Social Sciences E Evenstraat 2C Received 1 June 2003; received in revised form January 2004; accepted 2004 Available online 17 April Abstract We compute taxes on labor with quasilinear preferences between Agents differ their productivity taste leisure A responsibility egalitarian wants to compensate the former differences but not latter This intuition is captured by social planner that equalize opportunities subjective utility along lines criteria proposed Roemer gaer evaluating states based an advantage function representing reference Our theoretical results are illustrated empirical data D Elsevier B V All rights reserved Keywords: Optimal taxation; Equality; Opportunity; Welfare JEL classification: D63; H0; H2 Introduction The problem when people have different raises difficult normative questions higher may be due either innate skill levels or degree effort Progressive can therefore imply redistribution from those low preference high ethical evaluation this result 0165-4896/$ – see front matter doi:10 1016/j mathsocsci 01 002 *Corresponding author Tel : +32-9-2643490; fax: +32-9-2648996 E-mail address: Vandegaer@UGent (D gaer) www elsevier com/locate/econbase Mathematical 48 (2004) 151 – 182 depend exact interpretation given parameter One objections against energetic lazy Things however if linked lower physical mental abilities work These intuitions notion In general many feel some legitimate because should compensated factors which beyond control Because skills productive endowments prominent example leads us directly into traditional literature At same time they also held responsible under consideration largely absent tax setting tradeoff equity efficiency becomes trilemma involving compensation paper we formulate model analyze aspects It obvious how reconcile concerns Problems arise want hold individuals fully accountable outcomes (pure) (e g effort) while them (or ability) Fleurbaey (1995a b) Bossert (1995) proven these two principles (‘‘equal transfers equal skills’’ ‘‘equal preferences’’ respectively) compatible (1996) shown similar incompatibilities within specific (first best) Thus several suggestions been made involve idea choice so-called objective keeps one intact ensures other holds true level (see e 1996) Maniquet (2004a) develop orderings incorporate inequalities derive framework—see (2002) introduce use information indifference curves remain completely ordinal axiomatic approaches somewhat spirit Recently authors analyzed more way design agents Sandmo (1993) shows case redistributive rich towards poor weakened utilitarian efficient at margin generating Boadway et al (2001) nonlinear government observe its citizens To simplify analysis assume specifically They consider where weights attached used An alternative approach followed (2003) implement concept equality opportunity (Roemer 1998) ‘‘Equal means ideally individual’s particular specify as u(c)-aL a>0 consumption L supply 152 / 151–182 his has all society it will generally possible achieve Therefore suggests maximize weighted average minimal utilities across having tastes application 2 Moreover defined terms interpreted special compare explicitly Each embodies variant maximin criterion obtain explicit solutions rates concentrate follow Atkinson assuming (linear income) yield isoelastic curve structure our described Section From 3 onwards welfare functions dimensions; look As pointed out Sen (1991) well-being reflect propose purposes meant represent living standard (the ‘‘good’’) individual Basically boils down ordering differs Opportunity variants ‘‘equality opportunity’’ addition Roemer’s present related proposal maximizes group lowest (Van 1993) analyzes outcome amounts classical welfarist Next nonwelfarist follows 4 5 looks 6 discusses 7 contains illustration 8 concludes vs simply only dimensions first dimension w assumed genetic endowment second variable capture pure certain sense represents diligence deliberately interpret variables such desirable suppose both finite support; Preferences specified c-av(L) Diamond 1998 exercise preferences) Differences subsumed 153 measured 0 < eL wL respectively assumption support allow identify worst-off later distributed independently density fwðwÞ ½wL; 1 !Rþ feðeÞ ½eL; independence simplifies technical rather tricky correlated second-best context does knows While she cannot determine whether focus sensitivity limit ourselves constant marginal rate t lump sum grant could basic usual crux comparison interpretations starting point Individual behavior Y L: uðY; LÞ ¼  þ ðL0Þ 1 1þe ð1Þ L0>0 z L0 maximal amount someone perform elasticity measure cost identical substitution (1/e)(L/L0) (1/e) dependent idiosyncratic zero L= 1/e specification (Eq (1)) implies always multiple each satisfy single crossing property after consists (1t)wL: ð1 tÞwL: ð2Þ irrelevant People would variety through ratio depends (L/L0) Indifference still do identification become much 154 Substituting budget constraint (2)) yields UðL; B; tÞ tÞwL ð3Þ Maximization Eq (3) supply: ðeð1 tÞwÞ ð4Þ Note positive smaller than (4) clear increasing Those smallest disutility largest e) highest biggest Preference satisfaction indirect Vðe; w; L0ð1 w1þe ee ð5Þ expression immediately less resulting increase get innocuous cardinalization fact interpersonal comparisons extremely situation If aware (5) why opt larger value level? return question revenue Using faced by7 BðtÞ L0tð1 Z feðeÞde fwðwÞdw ð6Þ written moments distributions define ath moment x [x¯ x¯] laðxÞ mx¯ xxaf ðxÞdx definition rewrite (6) as: leðeÞl1þeðwÞ ð7Þ turn socially Adding requirement financing public goods change anything enter additively separable 155 For useful revenue-maximizing tBI Differentiation (7) subscript BI indicate 1995 interpretation) Subjective egalitarianism: benchmark concerned least well off Formally tries min e;w ð8Þ subject characteristics (eL wL) thus V(eL t) 9 awkward question: ethically acceptable everybody (even hard working low-skilled persons) raise laziest persons society? gets concrete easily introduced Proposition tE(V) tEðVÞ imposes him balance effect B(t) More importantly common practice label Rawlsian Rawls (1971) never advocated primary His ideas closer ‘‘advantage’’ developed again 156 le(e) values goes lazier others increases seem undesirable now explore consequences try egalitarianism10 Let keep individuals’ (5)) matters However ‘‘opportunity’’ (8) take position following i (1998) proposes partition set subsets nonresponsibility factor called ‘‘types’’ type (and hence L) lead According viewpoint avoided He function: gIðVÞ tÞde ð9Þ (maximin) applied e-level separately afterwards simple taken over e-levels 11 known was gSðVÞ feðeÞVðe; ð10Þ understood once suggestion (1999) distribution 10Roemer emphasizes ‘‘welfarist’’ welfarism requires knowledge possibilities policy 11This formulation deviates slightly original incorporates framework axiom says exercised percentile ‘effort’ types formulations equivalent 157 Sw Focusing relevant fðO; eÞaR 1AO Vðe;w; tÞg ð11Þ Fig depicts sets (10)) evaluates area computing ‘‘average utility’’ obtained then considers (compare Eqs (9)) intersection areas explains subscripts ‘‘S’’ (for smallest) ‘‘I’’ intersection) Both intersect monotonically little misleading; section = lie below 12 go deeply 13 reduce utilitarianism there no variation thought gI(V) flavor gS(V) 1999) straightforward calculate maximization (9) (10) using tIðVÞ tSðVÞ l1þeðwÞ corollary Corollary line introducing considerations longer play any role 12This 13But discussion (2004b) Ooghe 158 tI(V) [ tS(V)] 14 Objective Introducing dilemmas raised drop subjectivism limitations Dworkin 1981a) Important points criticism treatment expensive overly nature sketched previous sections good weakness ambitious proposals: ‘‘primary goods’’ (1981b) advocates resources’’ (1980) ‘‘basic capabilities’’ Cohen (1990) ‘‘midfare’’ most crucial aspect taxation: ‘‘objective’’ remains close about what ‘‘reasonable’’ denoted gives potential conflict Pareto 14We discuss ‘‘Opportunity’’ 159 Following Vandenbroucke aðY; ð12Þ government’s stance r burden market perceived decreases infinite Although very (both maximizing aggregate minimizing inequality) traditionally played dominant real world debates growing trend countries hand material economic growth nonmaterial quality life captures decrease corresponding weight relative (income) (4)) write parameters: Aðe;   ð13Þ (13) tÞzðVÞAðe; tÞZezðVÞg stands reason since ( )g too (not enough) according necessary understand better properties differentiate partially Lemma Properties (1) BAðe; Bw z0ZeVg (2) Be 160 findings lemma reaches maximum whose coincides ‘‘social’’ Society ascribes p (i w) note cross reduces independent left reach level; right opposite combined large (to point) ‘‘too much’’ so compensates increased (in eyes planner) 15 corollaries:16 Consider e¯ ascribed three cases: Case 1: ZwL argmin Aðe¯; tÞðbB; 2: ZAðe¯; wL; 1; tÞbwa½wL; 1;ðbB; 3: > Z1 15In treated l observed range care 16The proofs propositions Appendix 161 w¯ e1þe ; ZAðeL; ¼Að1; tÞVAðe; tÞbea½eL; ZeL uEðAÞ ð14Þ Indeed show necessarily person respect leisure) proposition upper panel objective: ZuEðAÞ AðeL; Að1; 4: 5: Depending considered enough low-wage attaches disadvantage e/(1 + u wages intermediate intelligent expected 162 ‘‘objective egalitarian’’ seen index Traditionally justice measures including wealth neglect well-known highlighted Musgrave (1974) admittedly primitive come tradition picking immediate reflects conception broader perspective kind norm Scanlon’s (1982) moral justification claim defend performed leave open comes distinction ‘‘subjective well-being’’ paternalist principle absolute consumer sovereignty On impose decisions own leading uses instruments; instruments environment free choices (14) Pareto-efficient Determination 163 substitute (7)) (14)) tEðAÞ Lw1þe eÞw1þe ” ## pattern summarized 164 move gradually implying shift who advantaged 3) cases increases) slope [tE(A)/(1-tE(A))] negative negatively sloped kinks g=q/(1 q) L1 kink understood; discourage Additional insights comparing counterpart subjective: except term e[1-(eL/g)] disappears eL= (advantage) correction g>eL experienced brings thereby inducing similarly There reasons differ: worthwhile until unambiguously compared 1) view ranking magnitude attitude solution Many economists sympathetic egalitarianism’’ reluctant give up domination Gaspart issue 165 Starting (wi ei) prefer ti ð15Þ generalization (15) direct relationship wi ei desired rate; q easy BVi/Bta0 tbti; single-peaked t-dimension preferred (wL eL) (1 tMIN tE(A) fortiori ! (analyzed 2003) Whether happen interesting possibility tE(A)>tE(V) Pareto-inefficient g-values doubt pick outside e-values 18 complications certainly inefficient [with e- ) arises finding place political feasibility seems unrealistic expect going takes whole After tdimension median voter gross wage decisive neglects issues tradeoffs Turning 17It occur 18Moreover condition eL>g reasonable restricted 166 positions accepting primacy unacceptability Paretoinefficient side restrict [tMIN tE(V)] Or e/ ambiguous accepts introduction desirable) rejects subjectivism—and 19 Until tackle First Second turned (reference) combine makes description argue themselves objectives uI uS redefined uIðAÞ ð16Þ uSðAÞ feðeÞAðe; ð17Þ determined ð18Þ before uI(A) uS(A) 19Note argues must laundered 167 expressions skilled (17): A¯ 1ðt; gÞ leðeÞ l1þeðeÞ ð19Þ Lðt; ð20Þ intersections did Not surprisingly occurrence objectives: gV eLZuIðAÞ ZuIðAÞ C½wL; eL; g;e; feðeÞ Cð Þ g1þe (a) ZuSðAÞ (b) (c) gz With 4) adopt remembering noticed g[(1 e)/e] [eL 1] small) highly eye observer Their coincide; offered workers (case 2) different; S1\SwL worker long [e/(1 e)]{[l1 e(e)]/[le(e)]} thereafter g=[e/(1 coincide 168 decreasing 3; Swl methodology eLZ tIðAÞ tSðAÞ eÞ monotonic g; (social) inclined (focusing set) characterized regimes (cases 2a) 2c reverse inequality 2b tS(A) schedule sets) transition smooth tI(A) induce e>g 169 (13)) g(1 e)/e 2a frontiers skilled; B2L BtBw tI(A)>tS(A) happens rules rule reasoning e\g(1 dominates soon Now concentrates e>g(1 region (positive) summarize corollary: 1þeZtIðAÞztSðAÞ tI(A)= Further gained next compares find analogous tE(A)>tI(A) Propositions 6) already emphasized 6; gle(e) l1+e(e) h Its akin harder indicates refer 170 restored additional situated interval Some further Finally investigate changes say new old sequence mean preserving spreads describes Inequality taxes: >1 react found net earnings plays extent responds BBðtÞ Bt L0leðeÞl1þeðwÞð1 Increases l1+e(w) 1t Empirical illustrate concepts earlier avoid 171 household size sample males coming 1992 1997 waves Belgian socioeconomic 20 hourly (1-ti)wi contractual hours per week Li 21 descriptive statistics Table Within logical her Remember by: ðeið1 tiÞwiÞ ð21Þ Given arbitrary calculated wi’s divided wmax generate normalized w*-series [wL Then generated (21) 06 ei’s ei-value emax e*-series sight procedure important however; fulfill column 222) Without (Section Implementing method alleviate unwanted correlation regress w*’s calibrated e*’s OLS: wi* bei* gi ð22Þ 20Because formulas derived depart optimising behaviour presumably fit descriptions male heads 21The account details system microsimulation 22Table coefficients Absence although sufficient statistical Description No observations: 184 Mean Standard deviation Min Max Age 37 73 07 25 08 54 83 euro) 79 38 82 32 05 47 66 50 57 45 172 estimated residuals OLS regression gˆi uncorrelated construction construct w-series wresid: wresid;i aˆ ˆ be¯* ð23Þ (22) e¯*stands series wresid- Of course make sure equations valid employing wresid i’s e-series eresid normalizing substantially repeat number times order even That w* resid e* performing twice almost computation just reversed Regress Use normalization methods implications; defines precise assigns residually Results Once identified accordingly Approximating natural estimator N Px¯ xa Correlation -series corr(w* corr(wresid )two rounds 251 121 028 814 283 038 912 657 095 Values tS(V) tmed 65e 943 927 899 600 102 769 763 758 146 500 460 433 016 173 Panel a: 06; b: c: 174 computed every presents (here tI(V)) wresid(two rounds) eresid(two above last (15)) surprising especially small consequence aversion 23 relatively minor influence chosen planner’s depicted turns (first) breakpoint hardly anymore So room (small g) Such part force Furthermore remember violated striking importance determines scales panels 5) (this reasonably elasticities (perhaps zero) realistic advocate Conclusion Much concentrated allows sidestep ‘‘equity’’ ‘‘efficiency’’ deeper welfare) theories accept (innate) recent 23Our comparable Stern (1976) without 24To pictures know 175 typical legitimately earn Survey research (Miller 1992) among population Simple rewards laziness ways gaer’s focuses contour version replace ‘‘reference preferences’’) extreme closed interpretable problems versions arguments describe ad hoc focusing Where from? responsible? Is native endowment? And violation preferences? generalize settings? firmly believe modelling actual debate pros cons progressive Acknowledgements forms programme TMR Network Living Standards Taxation [Contract ERBFMRXCT 980248] European Communities acknowledge financial Interuniversity Attraction 176 Poles Programme—Belgian Science Policy P5/21] thank Editor charge anonymous Referee Ramses Abul Naga Bart Cape´au Erwin valuable comments Proof continuity A(e minima corner domain associated minimum Að1;w¯ 0Zee   Which Zg A(eL t)-A(1 t)>0 established reversing 1– z1ze reached simplified calculations (using (13)): A˜ðe; order) eÞee 177 plugging poorest characteristic’s verify keeping mind Cases start   ez0 eVg =1 ezg Hence establish square brackets smallest: trivial optimization (19) (20) rearranging first-order conditions C( difference ðw1þe 1Þ leþ1ðeÞ ½1 1>wL integral 178 leðeÞ 1þ leþ1ðeÞw1þe ½w1þe V0 statement distinguish four eL: l1þeðeÞ: 1V 1le(e) 1-l1 e(e) le(e)>l1 indeed eLVg hðeÞfeðeÞde ð24Þ hðeÞ ð25Þ verified BhðeÞ zðVÞ0ZeVðzÞg: ð26Þ 179 (24) zg Over h(1) h(e) integrate hð1Þ ð27Þ Combination (27) VgV Analogously l1þeðeÞÞ non iV0 l1þeðeÞz1 ðAÞ Due (26) Consequently MinfhðeL; hð1ÞÞg ðBÞ MinfhðeLÞ; hð1Þg nonnegative trivially Min{h(eL) h(1)} h(eL) (B) (A) met g: 1tEðAÞ 1tIðAÞ ð1þeÞw1þe ee ðeL l1þeðeÞÞ: ð1þeÞ 180 ðCÞ here hðeLÞ Min{hðeL) (C) hðeL) l1 e(w) convex e>0 Similarly e>1 concave Rothschild Stiglitz 1970) References Public Economics Action: Basic Income/Flat Tax Proposal Oxford Univ Press R Marchand M Pestieau P Del Mar Racionero 2001 heterogeneous CORE Discussion Paper 2001/25 Louvain la Neuve W Redistribution mechanisms 29 1996 Choice 343 355 1999 talent compensation: Review Design 35 55 G 1990 Equality what? capabilities Recherches Economiques 56 357 382 U-shaped American 88 95 1981a What equality? 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