Optimal Tax-Transfer Systems and Redistributive Policy Author(s): Johan Fellman, Markus Jantti, Peter J. Lambert Source: The Scandinavian Journal of Economics, Vol. 101, No. 1, (Mar., 1999), pp. 115-126 Published by: Blackwell Publishing on behalf of The Scandinavian Journal of Economics Stable URL: http://www.jstor.org/stable/3440571 Accessed: 25/06/2008 09:22 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=black. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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Scand. J of Economics 101(1), 115-126, 1999
OptimalTax-TransferSystemsand RedistributivePolicy* JohanFellman FIN-00101Helsinki,Finland SwedishSchoolof EconomicsandBusinessAdministration,
MarkusJdntti Abo AkademiUniversity,FIN-20500Abo, Finland
PeterJ Lambert Universityof York,HeslingtonYO1 5DD,England
Abstract In this paperwe develop "optimalyardsticks"to gauge the effectivenessof given tax and benefitpolicies in reducinginequality.We show thatthe conjunctionof the optimaltax and optimalbenefits policies constitutesthe optimaltax-and-benefitpolicy, given the tax and benefitbudgetsizes. A decompositionformulaenablestrendsin the inequalityimpactof taxes andbenefitsto be explainedin termsof changingpolicy effectivess(targeting)andbudgetsize effects.The analysisincorporatesa distributional for sensitivityanalysis, judgementparameter, andconcludeswith an examinationof the Finnishcase forthe period1971-1990. Keywords:Incometax;benefits;inequality JELclassification:D63
I. Introduction We developan approachto evaluatingthe designof the tax systemandsocial benefitscheme,by relatingthe redistributive propertiesof in-placesystems to the redistributionthat would have occurredunderan optimaldesign of taxesandbenefits. In new workwhich generalisesthatof Fellman(1976) andFei (1981), we identify the optimal tax and benefit policies in wide classes of possible policies, constrainedonly to raise a given amountin tax revenue and/or distributea given amountin total cash benefits.These optimalpolicies for the given budget size would maximize welfare in the distributionof disposablemoney income in the absenceof disincentiveeffects. The extent to which observedpolicies fall shortof this ideal, in reducinginequality,1is *Wewish to thanktwo anonymousrefereesby penetratingandinsightfulcomments. 1Othercriteriain termsof whichan idealcouldbe formulated,andagainstwhichthe shortfalls of observedpolicies could be measured,includethe minimisationof administrative costs and of aggregatedeadweightloss; see Sandmo(1976). Theseaspectsarenot consideredhere. ? The editors of the Scandinavian Journal of Economics 1999. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.
116 J. Fellman,M. Jadntti& P J. Lambert
measuredby new indices,in whicha distributional judgementparametercan be set to reflectalternativedegreesof inequalityaversionand to carryout sensitivityanalysis. In the Finnishcase, for the period1971-1990, benefitsarefoundnot to be very efficient in redistributingincome (standardizedby the OECD equivalence scale) across households,whereastax policies come much closer to effect of an optimalpattern. the inequality-reducing
II. Optimal Tax Policy Consider the before-tax income distribution,assumed given, with the distributionfunctionFx(x), densityfunctionfx(x), LorenzcurveLx(p) and mean y,x. We studya class of tax policies characterizedby the transformaand continuous tion g(X), where g(-) is non-negative,monotone-increasing withthe properties
g(x) < x(1)
x-Tyx
E(g(X))l
where g(x) is the post-taxincome associatedwith pre-taxincomex and r is the meantax, assumedgiven. We consider here (and throughoutthe paper) impact effects only, not allowing individualagents for example to adjusttheir labour supplies in anticipationof the particulartax policy in the class which may be applied. This assumptionis standardin the literature.We returnto this point in the concludingcomments. The polarcase: go(x)
a,
(2)
x>a
serves as a referenceor benchmarkfor what follows. Here for incomes x < a thereis no tax, but for incomesx > a the tax is x - a. It can be shown that there exists a unique value ao such that E(go(X)) = ,ux - r < ao (with
equalityif and only if Fx(ao) = 0), and that the after-taxLorenzcurve in this case is: [
Lo(p)
-=
x
P , Po
Lx(p),
(3)
u
YI
\'-x
T
Lx(po)+
,xx
? The editors of the Scandinavian Journal of Economics 1999.
-
T
(ppo)
P>Po
Optimaltax-transfersystemsand redistributivepolicy
117
wherepo = Fx(ao). It can be furthershownthatthe Lorenzcurve(3) is the highestfor the whole class of transformations (1).2 It is also higherthanthe pre-taxLorenzcurve Lx(p), by Fellman's(1976) theorem;althoughnot all membersof the class of policies underconsiderationare progressive,i.e., inequality-reducing. Followingthe Atkinson(1970) theorem,(3) therefore implies maximal social welfare in this class, and go(x) is in this sense optimal. The generalizedGini coefficientof Yitzhaki(1983) for income afterthis optimaltax policy is Go(v)- 1 - v(v - 1) (1 _ p)-2Lo(p) dp, Jo
v> 1
(4)
whichmay be expressedin termsof the originalLorenzcurveLx using (3). Herev is a distributional judgementparameter,increasesin whichconnotea more inequality-averse stanceon the partof the social decision-maker.The case v = 2 is thatof the ordinaryGinicoefficient. Now considerany actual(non-optimal)tax policy with meantax r and let Gx(v) and Gx-T(V) be the generalized Gini coefficients for pre- and post-
tax income, respectively.We proposeto measurethe effectivenessof this actualpolicy by the index: IT()=
Gx(v)
-
Gx(v)
-
T(V)
Go(v)
X 100
(5)
which recordsits inequality-reducing performanceas the percentageIT(V) of the maximumreductionthatcould have been achievedwith the sametax yield r. This is in contrastwith some existing approachesto the measurement of redistributiveeffect, namely those of Musgraveand Thin (1948), Pechmanand Okner(1974) and Blackorbyand Donaldson(1984), which expressactualinequalityreductionas a percentageof pre-taxinequalityor equality.3Ourindexthususes the optimaltax policy as a yardstick,whereas the others use the pre-tax distribution.In fact, the Pechmanand Okner construction,if not the othertwo, uses an implicit "optimal"yardstick,in essence comparingactual redistributionwith that occurringif all income unitsweregiventhe samepost-taxincome(i.e., it uses perfectredistribution, with zero net budget,as a reference).By confiningattentionto the class of tax policies which satisfy the governmentbudget constraintto assess the 2For proofs of these and all subsequentmathematicalassertions,see Fellman(1995) and Fellmanet al. (1996). 3Inthe firsttwo cases cited,the Gini coefficientis used,and in the last the Atkinsonindex,to measureinequality.See Lambert(1993, pp. 200-201) formoreon this. C?
The editors of the Scandinavian Journal of Economics 1999.
118 J Fellman,M.Jdntti& P J Lambert effectivenessof an actualtax, our index has, we think, more realism and directappeal.It also incorporatesthe distributional judgementparameterv whichcan be variedto carryout sensitivityanalysis.
III. Optimal Benefit Policy Consider the income Y with distributionand density functions Fy(y), fy(y), mean uy and Lorenzcurve Ly(p). We now study a whole class of benefitpolicies characterizedby a transformationh(Y), where h(-) is nonandcontinuouswith the properties negative,monotone-increasing f h(y) > y
E(h(Y))= zy++ p
where h(y) is the income, includingcash transferfrom government,associatedwith originalincome y. The scenariopursuedhere can applyas well to an incomes policy: in that case h(y) is income afterthe policy-induced increase. These properties indicate that no income decreases, that the internalorderof the incomesremainsthe same (a pointto whichwe return) and that all the policies raise the mean income to u,y + p, where p is the meanbenefit,takenas given. The polarcase which servesas a referenceor benchmarkfor whatfollows is: b
ho(y) { y
(7)
,b
i.e., all incomesbelow the level b are raisedup to b and all incomes above this level remain.It can be shownthatthereexists a uniquelevel bo suchthat E(ho(Y)) =
yy+ p
andfor whichthe Lorenzcurvefor incomeincludingbenefits:
L1(p)=
bo Y+P, PPqo
(where qo = Fy(bo)) is the highest for the whole class of transformations definedby (6) - and higher than Ly(p), thus engenderinghighest social ? The editors of the Scandinavian Journal of Economics 1999.
Optimaltax-transfersystemsand redistributivepolicy
119
welfare in the class (again,not all policies in the class (6) are inequalityreducing). The generalizedGini coefficient for income after this optimalbenefits policy is ,1
Gi(v) = 1 - v(v - 1) (1 - p)V-2 L (p) dp 0
(9)
whichmaybe expressedin termsof the originalLorenzcurveby using (8). As withtaxes,the effectivenessof anyactual(non-optimal)benefitspolicy with mean benefitp andpre- and post-benefitgeneralizedGini coefficients Gy(v) and GY+B(v),respectively,maybe measuredin indexformby IB(V) =
)
(V
)
Gy(v) - GI(v)
100
(10)
expressingthe performanceas a percentageof the maximuminequality reductionachievablefor the given budgetp. The index in (10) can also be used to assess the inequality-reducing performanceof an incomes policy h(Y), measuredagainst the optimal income policy ho(Y) for the same averageincreasep in people'sincomes.
IV. The Optimal Redistributive Tax-TransferPolicy We characterizeeach tax and transferpolicy by the mean tax r and mean benefitp where,we assume,p < r. The transformation of originalincomes can be performedin two steps,firstthe taxationwhichreducesmeanincome from,ux by an amountr, and then the distributionof cash benefitsso that the meanincreasesto Ux - r + p. In this situation,the optimal tax and the optimal benefit policies of SectionsII andIIIcan be joined to give a tax andtransferstrategy.Underthe assumptionthat both r and p are taken as given, the joint strategycan be proved optimal, and actual combined tax and transferprogramscan be gaugedagainstit for theirwelfare.4We startwith the taxation.The optimal tax policy Lorenzdominatesany othertax policy.Let Yobe post-taxincome underthe optimaltax policy andlet Ygbe post-taxincomeunderan arbitrary tax policy.AssumethatE(Yo)= E(Yg) anddenoteas abovethe correspond-
4An anonymous referee stressed that the rigorous assumption that both r and p are taken as given is necessary for the optimality. Under the weaker assumption that only the difference r - p is taken as given, perfect redistributionwill be attainable. ( The editorsof the ScandinavianJournalof Economics1999.
120 J Fellman,M. Jdntti& P. Lambert ing Lorenz curves Lo(p) and Lg(p). Under arbitrarytaxation the poorest part of the population (after taxes) is poorer than under the optimal taxation (no taxes paid). If, after taxation, we consider the benefit policy, then for an optimal income distribution, the optimal benefit policy must be performed. This means all benefits must go to the poor. Then the minimum income under the optimal taxation, bo (say), is greater than the minimum income under the arbitrarytaxation, bg. Consider the Lorenz curve after the benefit. Let the breaking points in (8) be qo and qg, respectively. Obviously qo > qg. Hence, Lo(p) > Lg(p) for p < qg and for p > qo. For qg - p P< qo the curved part in Lg(p) is convex and monotone and cannot intersect twice the linear part in Lo(p). Hence, Lo(p) > Lg(p) for all 0 < p S< 1. Consequently, if we join the optimal tax policy and the optimal benefit policy, the joint policy is optimal. 0 As shown in Fellman et al. (1996), if bo > ao then qo = 1 and bo creates the case r in which + equality. perfect optimal policy p, If, on ax the other hand, bo < ao then qo < po and the final Lorenz curve LD is defined by: LD(P) = bo
P < qo
P, Ux - T + P
bo ^X~-r+p bo
/x -Tr + p
(Lx(p) - Lx(qo)), ,x ^L(11) -T+p
qo + ,
qo +
x
x-r
+ ,
A /Ux-r
+ p + p
(Lx(po)-
(P-Po),
qo < p<
Po
Lx(qo))
P>Po
We have thus derived, in a short and straightforwardmanner, the result of Fei (1981): restricting attention to the case in which no taxpayer is also a benefit recipient and p - r, i.e., to pure redistributionof all tax revenue, the class of combined tax-transfer policies in which (11) is optimal is Fei's class of "equity-oriented fiscal programs"; moreover, (11) is Fei's "two-valued program" shown to be optimal in his Theorem 7 (whose proof is combinatorial). Our own analysis thus extends Fei's insight to the more general case of fiscal programs with a non-balanced budget, in which the mean excess tax revenue r - p > 0 can be devoted to publicly provided goods and services, repayment of debt, etc. (Note, in this regard, however, that our welfare function is defined over disposable incomes only.) We have shown that in the case of these more general fiscal programs, where the tax yield Trand benefit ? The editorsof the ScandinavianJournalof Economics1999.
Optimaltax-transfersystemsand redistributivepolicy
121
budgetp are both specified,the two-valuedprogramwith "floorvalue" b0o and "ceilingvalue"a0 (in Fei'sterminology)is also optimal.5 Finally,using the generalizedGini coefficientGD(V)for income afterthe optimaltax andbenefitsystem,namely GD(V)=1 - v(V - 1) (1 - p)-2LD(p)
dp
(12)
which is determinedby the originaldistributionLx accordingto (11), the inequality-reducing performanceof any actual(non-optimal)combinedtax andbenefitpolicy withmeantax r andmeanbenefitp can be assessed.Let Gx(v) - Gx- T+B(v) G)x Ir, B(v) = X^ Gx(v)- GD(v)
100
(13)
be the index, where Gx-T+B(v) is the generalizedGini coefficient for disposableincomeafterapplicationof the policy.
V. Empirical Illustration: Finland 1971-1990 We illustrateour methodsusing datafor Finlandfrom 1971 to 1990, drawn fromthe HouseholdBudgetSurveys(HBS), collectedby StatisticsFinland, in 1971, 1976, 1981, 1985 and 1990, comprisinga series of cross-sectional studieswhich are comparableover time. The income data in these surveys stem fromtax andotheradministrative registersandcan be consideredto be of high quality.The samplesize variesfrom 1296 in 1971 to 2897 in 1990. Werestrictthe sampleto thosehouseholdswithpositivedisposableincome. The base x for taxes includesall taxableincome, such as earnings,selfemploymentincome,capitalincome,work-relatedandtaxabletransfersand privatetransfers.Fromthis, we subtractdirecttaxes t to get the base for all non-taxablebenefits b. These are taken in this applicationto be the two majorbenefitschemesthat have remainednon-taxablethroughoutthe time periodcovered,namelychild allowancesandhousingsubsidies.Childallowances arepaidto householdsat a flatrateper each child underthe age of 16 (17 in 1990). From the third child onwards,the sum per child increases. Housing subsidieshave been means-testedthroughoutthe time periodand arethereforenegativelycorrelatedwiththe tax base. 5Inparticular,our analysisextendsFei'sTheorem4, in whichhe shows(for T = p) thateither ao = bo (the 'maximalrationalbudget' engenderingperfect equality)or a0 < bo. Fei also proves,in his Theorem5, thatao is decreasing,andbo increasing,in the commonvaluer = p; ourown analysisprovesthatmoregenerally,a0 is decreasingin r andbois increasingin p. ? The editors of the Scandinavian Journal of Economics 1999.
122 J Fellman,M. Jdntti & P J Lambert We standardizedthe income variables to be comparable across households of different sizes using the OECD equivalence scale, which assigns the weight of 1.0, 0.7 and 0.5 equivalent adults to the first and additional adults and children, respectively. Household disposable income per equivalent adult is equal to x - t + b. In Table 1 we show inter alia the effectiveness indices IT(V), IB(V) and IT,B(V)estimated from our data (along with some other statistics discussed below).6 The effectiveness of the actual tax system measured by our index, i.e., the inequality reduction of actual taxes relative to the optimal policy, declines from 1971 to 1981 and rises thereafter, thus having a U-shaped pattern over time. The inequality effectiveness of benefits declined between 1971 and 1990 - with the exception of 1981. The combined effectiveness of taxes and transfers followed the same U-shaped pattern as that of taxes alone. For instance, using v = 2.0, in 1990 taxes achieved a 17.7 percent reduction in the Gini coefficient on moving from pre-tax to post-tax (but prebenefit) income relative to the optimal tax policy. On moving from actual post-tax income to post-benefit income, the inequality of post-tax income is reduced by 4.3 percent relative to the optimal benefits. On the other hand, moving from pre-tax and pre-transfer income to disposable income would achieve a 15.2 percent reduction in inequality relative to the optimal combined tax and transfer policy. The inequality effectiveness of benefits is always smaller than that of taxes. This is unsurprising, as the actual tax schedule in Finland is progressive during the period covered by our data. However, the main benefit we study, the child allowance, depends only on the number of children in the household. The optimal tax schedule thus only increases, rather than introduces, progressivity, whereas the optimal benefit policy would redistribute child allowances heavily to the lower tail, thus greatly increasing the inequality reduction of the actual benefits.7 Our indices IT(V), IB(V) and IT,B(V) measure the effectiveness of tax and benefit policies relative to optimal yardsticks which are conditional on the budget sizes p and r. Consider the Pechman and Okner (1974) indices
6Thethresholdfor the optimaltax was calculatedby the followingsimpleprocedure.Wefixed the thresholdto be equal to the ith income unit'spre-taxincome,x(i) say, and collectedall incomeabovex(i) of the incomeunitsthathave higherincome.If the total tax thus collected was higherthanthe actuallycollectedamount,the thresholdwas set at x(i + 1). Thisprocedure was then repeateduntil the tax thresholdled to less taxes being collectedwhen the threshold was set at x(k). The optimalpost-taxincome is then x(i) for i < k and x(k) for i s> k. The benefitthresholdandpost-benefitincomedistributionwereanalogouslyestimated. 7Thecentralargumentfor this is thatthe tax rateis relatedto the individualmoney incomes andnot to the equivalentincomecalculatedfor thewholehousehold. ? The editors of the Scandinavian Journal of Economics 1999.
Table 1. Redistributive effectiveness of taxes and benefits in Finland, 197 coefficients Benefits
Taxes
v 1.5 ?
;i D
2.0
a o 0D
2.5 S?
Year 1971 1976 1981 1985 1990 1971 1976 1981 1985 1990 1971 1976 1981 1985 1990
Actual
Optimal
Maximum
Actual
Optimal
DTr() 8.8 8.2 7.0 9.9 12.5 7.8 7.8 6.8 9.1 11.5 7.2 7.5 7.1 8.6 11.0
IT(v) 17.3 12.9 11.2 15.0 17.5 18.3 14.0 12.4 15.5 17.7 19.7 15.1 14.3 16.2 18.6
PT(v) 0.51 0.63 0.63 0.66 0.71 0.43 0.56 0.55 0.58 0.65 0.37 0.50 0.49 0.53 0.59
DB(v) 1.7 1.8 3.1 2.6 1.4 1.5 1.5 2.8 2.5 0.9 1.5 1.6 2.6 2.7 0.7
IB(V) 14.3 13.0 17.9 11.8 8.0 10.5 10.0 13.7 9.7 4.3 9.3 8.8 11.3 9.2 3.1
Maxim
PB 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.2 0.2 0.2 0.1 0.1 0.2 0.2 0.2
0
09 0 Q, 'i 0
*
Source: Authors'calculationsfrom HBS data files. Notes: The reductionin inequality D is measuredas the percentage decline in the generalized Gini coeff inequalityreductionI is measured as the actual decline in pre-tax (transferor tax and transfer) income text, especially equations (5), (10) and (13) for exact definitions. The maximal decline P is measured optimalpolicy were implemented.These are related as D = I X P. Note that D and I are expressed as pe and I X P in the reportedfiguresare due to roundingerrors.
124 J Fellman,M. Jdntti& P . Lambert Dr =
(Gx- -Gx T)100 x 100 Gx
of inequalityimpact(herefor taxes). Thereis a simplerelationshipbetween our indices and those of Pechman and Okner (suitably generalizedfor v -: 2). It is as follows: Dr(v) = IT(V)PT(V)
(14a)
DB(V) - IB(V)PB(V)
(14b)
DT,B(V)- IT,B(V)PT,B(V)
(14c)
where the terms PTr() - [Gx(v) - Go(v)]/Gx(v), PB(v) [Gx(v)Gj(v)]/Gx(v) and PT,B(V)= [Gx(v) - GD(v)]/Gx(v) express in propor-
tionate terms the maximum inequality reduction that could have been achievedwith the givenbudgetsizes (by use of the optimalyardstick).8 Table1 also showsthe decompositionsin (14a)-(14c) for v = 1.5, 2, 2.5. In interpretingtime series, it should be kept in mind that these optimal yardstickschange from year to year. If, for example,the budget (r and p) were to be substantiallyincreased,then the inequalityimpactof taxes and benefitscouldbe improvedeven in the face of reducedeffectiveness.Indeed this happenedin Finlandbetween1971 and 1990,when,for v = 2, effectiveness fell from 16.8 to 15.2 percentwhilst the Pechmanand Okner(1974) indexrose from9.2 to 12.3 percent,concomitantwith a substantialincrease in the amountof benefits.This declinein the "targetingcomponent"IT,B(V) was thereforedominatedby the increasein the "size component"PT,B(V), accountingfor the overallincreasein the pureinequalityimpactof benefits DT,B(V).Ourindices thus have a useful role to play in explainingobserved inequalitytrendsovertime.
VI. Concluding Remarks We have demonstratedthe propertiesof optimaltax andbenefitpolicies and shownhow to gaugethe effectivenessof actual(non-optimal)tax andbenefit usingthe inequalityimpact policies,as well as combinedtax-benefit-systems, of optimalpolicy as a yardstick.This has resultedin new indicesfor income taxes which contrastmarkedlywith some existing indices of redistributive effect (progressivity),which eitherinvolveno optimalyardstickor at best a 8Wethankan anonymousrefereefor pointingout this importantand useful featureof our construction. ? The editors of the Scandinavian Journal of Economics 1999.
Optimaltax-transfersystemsand redistributivepolicy
125
very unrealistic one.9 In the case of benefit systems, our indices lend themselves directly to another use: to measure the inequality performance of an incomes policy. All of our indices incorporate an inequality aversion parameter, and can be used to assess the contribution of "targeting" to observed inequality trends, along with that of budget size. We illustrated this by an application to Finnish data (and showed, incidentally, that the findings were quite robust to changes in the assumed inequality aversion of the evaluator). We reiteratethat all of our constructed indices are impact measures, which take the pre-tax income distribution as exogenous to the choice of tax and benefit policies from classes which would have the given mean budget size (r or p). With more sophisticated modeling, for example of people's preferences over consumption and leisure or, more ambitiously, in a computable general equilibrium environment, one could in principle devise indices of policy effectiveness with superior welfare properties - but these would not be measurable from published income data. Another restrictive assumption of the mathematical modelling is that taxes and transfersdo not disturbthe ranking of households from poorest to richest by their living standards(equivalent incomes). Some lump-sum elements in the tax code (e.g. child allowances) can cause rerankingin equivalent income terms, as can benefits going to people on the basis of factors outwith the equivalence scale (e.g. single mothers, the handicapped, etc). By using the Lorenz dominance criterion, we have neglected any wider consideration of social needs; this is a direction for future theoretical research and more refined measurement.
References Atkinson, A. B. (1970), On the Measurementof Inequality,Journal of Economic Theory 2, 244-263. Blackorby,C. and Donaldson,D. (1984), Ethical Social Index Numbersand the Measurement of Effective Tax/Benefit Progressivity, Canadian Journal of Economics 17, 693-694. Fei, J. C. H. (1981), EquityOrientedFiscal Programs,Econometrica49, 869-881. Fellman, J. (1976), The Effect of Transformationson Lorenz Curves, Econometrica44, 823-824. Fellman,J. (1995), IntrinsicMathematicalPropertiesof Classes of Income Redistributive Policies, Swedish School of Economics and Business AdministrationWorkingPaper no. 306, Helsinki. Fellman, J., Jiintti, M. and Lambert, P. (1996), Optimal Tax-TransferSystems and RedistributivePolicy: The Finnish Experience, Swedish School of Economics and Business AdministrationWorkingPapersno. 324, Helsinki. 9Ourown optimalyardstickis, of course,not fully realistic(see the next paragraph).It serves as a benchmark,just as, for example,the 45? line of perfectequality,thoughunattainable,is takenroutinelyas the yardstickagainstwhichto measureinequalityusingthe Giniindex. ? The editors of the Scandinavian Journal of Economics 1999.
126 J Fellman,M. Jdntti& P J Lambert Lambert,P J. (1993), The Distribution and Redistributionof Income: A Mathematical Analysis, 2nd edition, ManchesterUniversityPress, Manchester. Musgrave, R. A. and Thin, T. (1948), Income Tax Progression, 1929-48, Journal of Political Economy56, 498-514. Pechman,J. A. and Okner,B. (1974), WhoBears the TaxBurden?,Brookings Institution, Washington,DC. Sandmo, A. (1976), Optimal Taxation: An Introductionto the Literature,Journal of Public Economics 6, 37-54. Yitzhaki, S. (1983), On an Extension of the Gini Index, InternationalEconomic Review 24, 617-28. Firstversion submittedAugust 1996; final version received February1998.
? The editors of the Scandinavian Journal of Economics 1999.
