Abstract. Functional classes of Lorenz curves are derived from a general-ization of a relative poverty notion. All these Lorenz curves compareindividual income to the average of all larger or all smaller incomes. Theparameters of the Lorenz curves are effectively computed from empiricalincome data by least square regressions. Best fits are analyzed and resultingfunctional Gini indices are compared to empirical Gini indices.

Key words: Differential equation, Lorenz curve, relative poverty measure

JEL classification: O15, C430, Q01

1. Introduction

Income distribution and poverty measurement have received considerableattention over the past. The poverty notion of the European Union, whichconsiders an individual to be poor when he falls short of 50% of the averageincome of the whole population, has earlier been shown to lead to the Paretoclass of Lorenz curves. This and similar curve types are shown here to allowfor an explicit relation between individual and average incomes.

Comparisons between individual and average incomes will be facilitatedby differential equations. The Pareto type Lorenz curves were earlier obtainedas solution of a linear differential equation. This approach is further devel-oped here arriving at other linear as well as non-linear differential equationswhich each result in a one-parametric class of Lorenz curves. However, thefocus is on the interpretation of the differential equations rather than onsolution techniques.

The parameters of the Lorenz curves will be computed by regression inthe usual sense of minimum square error fits. Since the regressions here are

Empirical Economics (2004) 29:697–704DOI 10.1007/s00181-004-0197-5

Note

Functional inequality measurement

Thomas Kampke

Forschungsinstitut fur anwendungsorientierte Wissensverarbeitung FAW,Helmholtzstr. 16, 89081 Ulm, Germany (e-mail: kaempke@faw.uni-ulm.de)

First version received: September 2002/Final version received: April 2003

not computable in closed form, approximate fits are computed by anenumeration scheme. It will turn out that a certain intermediate type ofLorenz curve leads to best fits. This curve type lies ’’between’’ individualsolutions of two differential equations that cannot be fulfilled simulta-neously.

The threefold purpose of this paper is made exlicit as follows. First, sev-eral classes of parametric Lorenz curves including the Lorenz curves of thePareto type are derived from differential equations. This may be surprisingand allows certain interpretations of function value. Second, the differetnialequations yield at least one new class of Lorenz curves. Third, several classesof Lorenz curves can be intimately related to each other.

The remainder of this paper is organized as follows. In Sect. 2, the dif-ferential equation approach is concisely introduced. The approach is extendedin Sect. 3 based on a convexity argument and its graphical interpretation. InSect. 4, three one-parametric Lorenz curves are fitted to empirical data fromthe World bank. Lorenz curves that compare individual incomes to squaredaverage incomes rather than to mere average incomes provide for the best fits.The short Sect. 5 concludes this work.

2. Framework

2.1. Differential equation approach

The cumulative income distribution of a nation is described by a Lorenz curveLðxÞ, x 2 ½0; 1�. The cumulative income is related to individual incomes by thederivative of the Lorenz curve. This can be seen by considering an individualat income rank x who is one with 100x% of the population earning less and100% – 100x% of the population earning more than him. This individualadds L0ðxÞ to the values of the Lorenz curve whenever the curve is absolutelycontinuous meaning that LðxÞ ¼

R x0 L0ðuÞdu; the latter is assumed throughout.

The total income of a nation is normalized to unity which means thatLð1Þ ¼ 1.

The foregoing consideration is related to the relative poverty notion ofthe European Union. According to this notion, an individual of somenation is considered to be poor if his income falls short of 50% of theaverage per capita income of that nation, comp. [EU], [FI]. Instead ofconsidering only the poorest, any individual’s income will be compared tothe average income of all richer individuals. Moreover, the actual fractionof individual vs. average income need not be 50% but some other, yetunknown value.

Comparing individual income to average income leads to a differentialequation, see [KPRad]. The rationale is as follows. Quantile x of the popu-lation receives its proportion LðxÞ of income. The remaining income 1� LðxÞis distributed among the remaining fraction of the population which is 1� x.The average income of all richer individuals thus equals

1�LðxÞ1�x . An individual

at income rank x is supposed to have a constant fraction 1a, a � 1, thereof.

This results in the differential equation

L0ðxÞ ¼ 1

a� 1� LðxÞ

1� x:

698 T. Kampke

All solutions of this differential equation that satisfy the normalizationconditions Lð0Þ ¼ 0 and Lð1Þ ¼ 1 are given by the Pareto distributionLaðxÞ ¼ 1� ð1� xÞ

1a.

Distributional inequality increases in the parameter a both in the sense ofLorenz dominance and Gini index. The first means that Pareto Lorenz curvesdo not intersect for different parameters except at the endpoints and the curvewith larger parameter lies below the curve with smaller parameter. The sec-ond means that the Gini index, whose value is given by 2

R 10 x� LðxÞdx is

increasing in the parameter. This can be seen from the values 2aaþ1� 1.

2.2. Related work

A plethora of work on Lorenz curves exists. The ordinary Pareto-type Lorenzcurve has been proposed, among others, by Rasche et al. [Ra], even in the

more general form LðxÞ ¼ ð1� ð1� xÞaÞ1b with 0 < a < 1 and b � 1, see [Ra].

These curve types apparently were motivated by insufficient curvature fea-tures of other popular types of Lorenz curves.

These other parametric Lorenz curves make use, among others, of theb-distribution as introduced in [McD]. Examples are LðxÞ ¼ x� #xcð1� xÞdand the Singh-Maddala distributions [SiMa]. Other multi-parametric incomedistributions are quadratic which means that the Lorenz curves are of theform LðxÞ ¼ 1

2 ðbxþ eþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimx2 þ nxþ e2p

Þ. The motivation of the latter appearsto be pure empirics, see [Da]. A survey on parametric Lorenz curves, withemphasis on b- and similar distributions, is provided in [ChoGri]. A variety ofparametric Lorenz curves is adopted from probability distribution functions,see [p. 294, RySl].

A variation of the Rasche curves like LðxÞ ¼ xað1� ð1� xÞbÞ, a > 0,0 < b � 1, and exponential curves such as LðxÞ ¼ ejx�1

ej�1 , j > 0, have also beenproposed, see [Che]. Such parametric Lorenz curves are occasionally com-plemented by non-parametric methods, such as quantile ratios, see [Gi], andkernel estimation. The lack of a-priori justification for a certain type of curveand simplicity issues motivate those approaches.

Poverty measures like poverty lines, degree of shortfall of poverty line(poverty gaps), the Lorenz family of inequality measures [Aa], and – aboveall – the Foster-Greer-Thorbecke FGT measures, see [FoGrTh], are notconsidered here since they focus only on the lower segments of Lorenz curves.Inequality indices such as generalized entropy measures and the Atkinsonindex, see for example [p. 3-4, Lit], are also not considered in this note. Ageneral survey of inequity measurement can be found in [Sil].

In the light of all existing work on parametric Lorenz curves, the presentwork does neither intend to bring into play ‘‘just another Lorenz curve’’ norto make comprehensive, statistical comparisons to other curves and measures.Instead, another way to viewing, deriving and relating Lorenz curves isproposed.

3. Extensions

Each Lorenz curve is convex. Its differentiability then implies that the tangentslopes at any point can be sandwiched by the slopes of two secants as

Functional inequality measurement 699

indicated in Fig. 1. The secant slopes are equal to per capita income of thepopulation above and below point x. The slopes are related by

LðxÞx� L0ðxÞ � 1� LðxÞ

1� x:

Requiring that only one inequality becomes an equation leads to the trivialsolution LðxÞ ¼ x. Milder forms of equations are obtained by admitting twoparameters a; b � 1, where a is as before, so that

LðxÞx� b ¼ L0ðxÞ ¼ 1

a� 1� LðxÞ

1� x:

While the right differential equation is solved as above by functions of thePareto type, the left differential equation is solved by LbðxÞ ¼ xb for b � 1.Instead of considering the poorest of all richest, this differential equationconsiders the richest of all poorest, stating that his income typically exceedsthe average of all lower incomes by a constant multiple.

Both foregoing differential equations cannot be fulfilled simultaneously. Arelaxation leaving out the differential leads to the pure functional equation

LðxÞx� b ¼ 1

a� 1� LðxÞ

1� xwith solution La;bðxÞ ¼ x

ab�ðab�1Þx. These functions are valid Lorenz curvessince they are increasing, convex, and normalized to La;bð0Þ ¼ 0 andLa;bð1Þ ¼ 1. Replacing the product ab by a single value m allows to denote thisclass of Lorenz curves as one-parametric curves LmðxÞ ¼ x=ðm� ðm� 1ÞxÞ.For obvious reasons, these are called intermediate Lorenz curves.

The differential equations with their individual solutions, thus, result frombreaking the symmetry of two simultaneously unsatisfiable differentialequations. The latter have the remarkable property of leading to a Lorenzcurve by algebraic manipulations only; no differentials are involved.

Though the intermediate functions are derived without differentials, theyare differentiable and fulfill the two homogeneous differential equations

m � ðLðxÞxÞ2 ¼ L0ðxÞ ¼ 1

m� ð1� LðxÞ

1� xÞ2:

The interpretation of these differential equations is that individual incomesare proportional both to the squared average of all smaller and all larger

Fig. 1. Tangent at x and two secants, eachthrough one of the endpoints of the Lorenzcurve L

700 T. Kampke

incomes. The proportinate constants are m and 1=m respectively. The inter-mediate functions also satisfy the two linear differential equations

mm� ðm� 1Þx �

LðxÞx¼ L0ðxÞ ¼ 1

m� ðm� 1Þx �1� LðxÞ1� x

:

The intermediate Lorenz curves thus allow to relate individual income togroup income by proportionate functions that both depend on the incomelevel. These proportionate functions are m

m�ðm�1Þx and 1m�ðm�1Þx. Numerical

values which ease the interpretation of these functions are computed in sec-tion 4. Distributional inequality according to the Lorenz curve classes LbðxÞand LmðxÞ is increasing in the parameters and the Gini indices are given by

1� 2bþ1 and 1� 2m lnm�ðm�1Þ

ðm�1Þ2 .

4. Application

The class of Lorenz curves from Sect. 2 as well as the symmetry related classesfrom Sect. 3 are used for empirical investigations. Therefore, the curves arefitted to data from support sets fðxi; yiÞji ¼ 1; . . . ; ng in the usual least squaressense. This amounts to three regression problems for each data set:

mina�1

Xn

i¼1ðLaðxiÞ � yiÞ2; where LaðxÞ ¼ 1� ð1� xÞ1=a

minb�1

Xn

i¼1ðLbðxiÞ � yiÞ2; where LbðxÞ ¼ xb

minm�1

Xn

i¼1ðLmðxiÞ � yiÞ2; where LmðxÞ ¼

xm� ðm� 1Þx :

Since none of the regression problems can be solved in closed form, each isapproximately solved by selecting a finite subset of parameter values andchoosing the one with smallest regression error.

The regressions were formed on the so-called world developmentindicators [W]. That information is sparse, as only n ¼ 6 data are given perincome distribution. Lorenz curves were fitted to data of 30 nations whichwere selected from more than 150 nations. The best fit values together withthe resulting Gini indices and the Gini index given by the World bank aregiven in the subsequent table.1 The World bank data are listed with theprecision as given in the data source.

The Lorenz curves of types La, Lb, and Lm explain the distributionalinequality best when choosing the stated parameter values for each nation. Torepeat the interpretation of the parameters, an individual at rank 0:4 isconsidered. This individual is characterized by 40% of the population havingless and 60% having more income. For parameter a ¼ 1:81 (Canada), theincome of the rank 0:4 individual approximately is 1=1:81 � 100% � 55% ofthe average income of the 60% population ranked above him. For parameter

1 MATLAB computations of the regressions and functional Gini indices were performed by DirkBank (FAW Ulm).

Functional inequality measurement 701

b ¼ 1:96, the income of the rank 0:4 individual approximately is1:96 � 100% ¼ 196% of the average income of the 40% population rankedbelow him.

According to the intermediate functions, the proportionate values varywith income. The income of the 0:4 individual approximately is1=ðm� ðm� 1ÞxÞ � 100% ¼ 1=ð2:62� 1:62 � 0:4Þ � 100% � 51% of the aver-age of all larger incomes and it is m=ðm� ðm� 1ÞxÞ � 100% ¼ 2:62=ð2:62� 1:62 � 0:4Þ � 100% � 132% of the average of all smaller incomes.While the difference in comparison to the upper segment is small (51% in-stead of 55%), the difference to the lower segment is significant (132% insteadof 196%).

A consistent winner in terms of the regression error is the intermediatefunction type. (Regression errors are not listed here). This best fit finding isthe same for the very uneven as well as for the more even distributions. Thefunctions of the Pareto type consistently show the worst fit. While, in ten-dency, the regression errors increase with inequality for the Lorenz curves of

Table 1. Best fit regression parameters, corresponding Gini indices and Gini indices computed bythe World bank. Bold face Gini index indicates least absolute difference to the World bank Giniindex for each nation

Nation a b m Gini indices World BankGini index

La Lb Lm

Austria 1.54 1.61 2.01 0.213 0.234 0.229 0.231Brazil 3.60 4.65 7.53 0.565 0.646 0.599 0.600Can 1.81 1.96 2.62 0.288 0.324 0.311 0.315China 2.24 2.57 3.70 0.383 0.440 0.413 0.403Czech Rep. 1.62 1.69 2.17 0.234 0.257 0.253 0.254Denmark 1.59 1.66 2.11 0.228 0.248 0.244 0.247Finland 1.63 1.71 2.20 0.240 0.262 0.257 0.256France 1.86 2.02 2.73 0.301 0.338 0.324 0.327Germany 1.70 1.81 2.36 0.259 0.288 0.279 0.300Gr. Britain 1.99 2.20 3.06 0.331 0.375 0.358 0.361Greece 1.86 2.02 2.74 0.301 0.338 0.325 0.327Hungary 1.81 1.94 2.61 0.288 0.320 0.310 0.308India 2.14 2.31 3.36 0.363 0.396 0.385 0.378Italy 1.67 1.77 2.30 0.251 0.278 0.271 0.273Japan 1.61 1.67 2.15 0.234 0.251 0.250 0.249Korean Rep. 1.81 1.96 2.63 0.288 0.324 0.313 0.316Mexico 3.05 3.77 5.91 0.506 0.581 0.536 0.537Netherlands 1.85 2.00 2.71 0.298 0.333 0.322 0.326Nigeria 2.82 3.40 5.26 0.476 0.545 0.507 0.506Norway 1.61 1.69 2.16 0.234 0.257 0.252 0.258Poland 1.88 2.03 2.77 0.306 0.340 0.328 0.329Portugal 2.00 2.20 3.07 0.333 0.375 0.359 0.356Russian Fed. 2.68 3.18 4.85 0.456 0.521 0.486 0.487S. Africa 3.56 4.64 7.42 0.561 0.647 0.590 0.593Slovakia 1.45 1.48 1.80 0.184 0.194 0.194 0.195Spain 1.85 2.01 2.72 0.298 0.335 0.323 0.325Sweden 1.59 1.68 2.13 0.228 0.254 0.247 0.250Switzerland 1.86 2.03 2.74 0.301 0.340 0.325 0.331USA 2.14 2.43 3.44 0.363 0.417 0.392 0.408Venezuela 2.64 3.15 4.77 0.451 0.518 0.482 0.488

702 T. Kampke

the Pareto type and for type Lb, this dependency is less expressed for theintermediate Lorenz curves Lm.

Bold face values of Gini indices in table 1 indicate least absolute differenceto the Gini indices given by the World bank. In 25 of the 30 cases, theminimum regression error curve is also best approximating the Gini index. Inthe remaining five cases, the best approximation of the Gini index is attainedby functions of type Lb. Again, functions of the Pareto type are worst inempirical sense.

Gini indices for the Pareto curves fall short of the World Bank Giniindices in all cases. Gini indices of the other Lorenz curves fluctuate aroundthe World Bank Gini indices.

5. Conclusion

Lorenz curves were derived on the base of comparisons between individualincome and certain group incomes. Empirical best fits lead to the conclusion,that modified curves rather than the original, Pareto type curves gave bestresults.

It remains to be investigated how findings for individual nations carryover to the income distribution of regions of nations and even to the worldincome distribution.

References

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