In this paper we use the distributions of order statistics to define functions with the appropriate properties and represent social preferences regarding income distributions. Following the approach of Yaari (1987, 1988), this allows constructing a set of social welfare functions from which the corresponding inequality indices are derived. The obtained measures incorporate diverse normative criteria, with different degrees of preference for equality. The generalized Gini coefficients and the family of indices proposed in Aaberge (2000) are obtained as particular cases. This approach shows that each of these families of indices characterizes the income distribution, but for a change of scale

Barcena-Martin, Elena Maria

Imedio-Olmedo, Luis J.

Parrado-Gallardo, Encarnacion Macarena

English

Parrado-Gallardo, Encarnación Macarena

Bárcena-Martín, Elena María

Repositorio Institucional Universidad de Málaga

INEQUALITY, WELFARE AND ORDER STATISTICSEncarnación M. Parrado GallardoElena Bárcena MartínLuis José Imedio OlmedoApplied Economics (Statistics and Econometrics) Faculty of EconomicsUniversity of MálagaSpain5th ECINEQ Meeting, Bari. July 2013Introduction Inequality measures             SWF (Yaari approach  (1987, 1988) Order statistics          Distribution F.          Preference F. SWF    Inequality MeasuresPreference functions Normative criteria  Inequality aversionDiverse normative criteria Different response to progressive transfersGeneralized GiniAaberge (2000)Other measures (intermediate positions wrt aversion)Introduction Use of order statistics is rare: Generalized Gini first-order statistics (Lambert 2001)         mean values of these statistics characterizes any income distribution with finite mean (Kleiber y Kotz 2002) Our proposal extends and generalizes this analysis Advantages of the use of order statistics:1. Alternative characterization of distributions (empirical)2. Provides diverse distributive criteria in assessing welfare and inequality 3. Clear interpretation of each measure in terms of the statistics computed from a random sample drawn from the population.  Identification of unbiased estimators of both the SWFs and their associated inequality indices.Inequality Common and intuitive way to assess inequality: weight the deviations between the income perceived by each individual and the mean income (or relative to the mean), using a weight function (value judgments):∫∫ λµ−µ=−=−∞ 1010dp)p())p(F(1)µ)ω(x)dF(x(xµ1IInequality Geometric interpretation from the Lorenz curve: Gini: w(x)=2F(x) o π(p)=2 Generalized Gini π(p)=n (n-1)(1-p)n-2 Aaberge (2000) π(p)=n pn-2, n≥21.p0  (p),'(p)  ,dp)p())p(Lp(I10<<λ=pipi−= ∫Welfare and Inequality Yaari approach (1987, 1988) YSWF is given by:	Yaari shows that         presents aversion to inequality if and only if 	ϕ is concave∫∫∫−−φ φ=φ=φ=+101'101Rdp)p(F)p()p(d)p(F))x(F(xd)F(W)F(WφWelfare and Inequality If µ is the mean income of the distribution and L(p) its Lorenz curve, the YSWF can be expressed as a social welfare function associated to a linear measure of inequality:where[ ])F(I1)F(W φφ −µ=)p()p(   , dp)p())p(Lp()F(I ''10φ−=pipi−= φφφ ∫relationship between SWF and inequality measure& relationship between weighting scheme of the Lorenz differences and distribution of preferenceWelfare and Inequality Fulfillment of the Principles of Transfers (Necessary and sufficient condition): PDPT: concavity of ϕ PPTS (given difference in ranks ): PDT (given difference in incomes):                                orstrictly increasing    0)p(''' >φ0  x,  ))x(F()x(F))x(F())x(F(2''''''''>>φφ−)x(F))x(F( '''φOnly the properties of its preference distribution matterDepends on the properties of its preference distribution  and on the shape of the income distributionOrder statistics Order statistics. DefinitionLet                   be a sample of size n, from a distribution F(.) , and dene the order statistics                              in the ascending order by,                                                           The variable that assigns the value at position k-th to each sample.  Distribution function of      , The mean values of the order statistics n21 X ..., ,X ,Xn:nn:2n:1 X... XX ≤≤≤n:kX )(F n:k ⋅( ) ( ) jnjnkjn:k )x(F1))x(Fjn)x(F −=−= ∑( ) ∫∫ −−−∞−==10kn1k10n:kn:k dp)p1(p)p(Fknk)x(xdFXE}n..., 2, ,1{k ∈,X n:k From the mean of the order statistics: If the distribution has a finite mean, the existence of the first moment of any order statistic is assured. It is important for those distributions, such as heavy-tailed income distributions, for which only a few potential moments exist, and therefore no characterization in terms of (ordinary) moments is feasible. It is interesting to analyze whether the distribution can be characterized by the moments of the order statistics.  Recurrence relation between the first moments of order statistics (David, 1981),XcEXE n:k ≤)nE(X)kE(X)k)E(X(n 1n:kn:1kn:k −+ =+−Order statistics Proposition 2.Let X be a random variable with finite mean and k(n) a positive integer, 1≤k(n) ≤n , the distribution F(.) is uniquely determined by the sequence .)}{E(X Nnn:k(n) ∈Order statisticsWelfare functions and inequality measures generated through mean values of order statistics. Order statistics          Distribution F.           Preference F.   SWF Inequality measuresWelfare functions and inequality measures generated through mean values of order statistics. First order statistics and generalized Gini coefficients.if x=F-1(p)( )nn:1 )x(F11)x(F −−=,p)(11(p))(FF(p) n1n:1n:1 −−==φ − 1p0 ≤≤ , n≥2Can be interpreted as distribution of preferences2.n  ,L(p)dpp)(11)µn(n)E(X(p)(p)dF(F)W102nn:110n:11n:1 ≥−−==φ= ∫∫ −−2n ,E(X))E(X1µ(F)W1(F)I n:1n:1n:1 ≥−=−= Inequality measure2n  ,dpp)L(p))(1(p1)n(n dp)p(L)p1()1n(n1(F)I102n102nn:1 ≥−−−=−−−= ∫∫−−Generalized Gini of order nWelfare functions and inequality measures generated through mean values of order statistics. First order statistics and generalized Gini coefficients.The absolute indices:if we take random samples of size n, n≥2, from the income distribution  and the welfare associated to each sample is identified with the minimum income, the mean value that is obtained when considering all possible samples of the given size is the welfare that the underlying SWF assigns to the generalized Ginicoefficient of parameter n, As a consequence of Proposition 2, we can ensure that any distribution  is characterized by the succession of SWFs                                   any F(.) is characterized by the sequence of the generalized absolute Gini coefficients ( )n:1nn:1n:1 XXE)F(Wµ)F(Iµ −=−=( )n:1n:1 XE)F(W ={ } { }nn:1nn:1 )E(X(F)W ≡Welfare functions and inequality measures generated through mean values of order statistics. General case The distribution function of the order statistics are increasing, but not necessarily concave over the whole range SWFs and indices of inequality that would not meet the PDPT. However, if for fixed sample size n,  we calculate consecutively the arithmetic mean of the functions                       we obtain a sequence of functions which have an appropriate behavior to be considered distributions of social preferences.nk11n:k ))}(F(F{ ≤≤− ⋅Welfare functions and inequality measures generated through mean values of order statistics. General case Definition. For each (n, k),       ,              , we consider the function Proposition 3. Each of the functions , defined in the interval  [0,1], shows the properties required of a distribution of social preferences (increasing and concave).2n ≥ n ..., 2, ,1k =n. ..., 2, 1,k   1,p0  ,(p))(FFk1(p)k1i1n:in:k =≤≤=φ ∑=−)}({ n:k ⋅φWelfare functions and inequality measures generated through mean values of order statistics. General caseAs a consequence,If the level of welfare assigned to any sample of n incomes from F(.)  is identified with the mean of their k lower incomes, the welfare of the population is the expectation of those values when considering all possible samples of size n.Inequality measures: ( ) n. .., 2, 1,k ,XXXk1E)p(d)p(F)F(W10n:kn:2n:1n:k1n:k =+⋅⋅⋅++=φ= ∫ −( ) 2n ,XXXEk11µ(F)W1(F)I n:kn:2n:1n:kn:k ≥+⋅⋅⋅++µ−=−=2n  ,dpp)(1L(p))p(pkn)kn((F)I101kn1kn:k ≥−−−= ∫ −−−Welfare functions and inequality measures generated through mean values of order statistics. General case The welfare loss due to inequality is measured by the corresponding absolute indices: Therefore, is an unbiased estimator of .   ( )+⋅⋅⋅++−=−= n:kn:2n:1nn:kn:k XXXk1XE)F(Wµ)F(Iµ( ) k/XXXX n:kn:2n:1n +⋅⋅⋅++−)F(µI n:kWelfare functions and inequality measures generated through mean values of order statistics. Particular cases For k=1 we get the Generalized Gini For k= n-1 we get the family of indices proposed by Aaberge (2000). For k=n, the SWF shows no aversion to inequality. It identifies the welfare of each income distribution with its average income, and the associated inequality index is zero for any distribution. This does not imply the absence of inequality, but that both the SWF and its corresponding index are indifferent to inequality.Some additional policy considerationsThe distribution of preferences reduce their concavityGiven nWhen k increasesThe SWFs show less aversion to inequality, from the corresponding to the Generalized Gini until indiferenceThe associated inequality measures assign less weight to the inequality corresponding to low incomes and greater weight to the inequality corresponding to high incomesSome additional policy considerationsThe Genelralized Gini indices and the indices of the family proposed by Aaberge (2000) weight local inequality through monotonic functions along the distribution so that the greater weight is assigned to one of its ends. However, the weights for the Lorenz differences for the indices of the family,                  ,  1<k<n-1 are not monotonic. This allows for measures with different attitudes in assessing inequality and welfare, as they pay more attention to different parts of the distribution.)n,k(n:k )}F(I{010 1Preference distributions1:52:53:54:55:50 1Weights funcions1:52:53:54:55:5 Proposition 4.  The indices of the family                  , satisfy the PPTS if and only if The index Ik:n(F), which is applied over the distribution function F, satisfies the DTP if and only if:Some additional policy considerations)n,k(n:k )}F(I{ ,2n ≥ nk1 ≤≤[ ] 0)1k(p)2n()p,k,n(T >−−−= 1) ,0(p∈for any0  x,  ))x(F()x(F))x(F1)(x(F)1k()x(F)2n(2'''>>−−−− Regarding the PPTS: The Gini index does not satisfy the PPST.  Generalized Gini indices satisfy the PPST (except the Gini index) Other indices exhibit a behavior opposite to the PPTS, as the indices of the family proposed by Aaberge. There are also indices  whose behavior with respect to this principle is not uniform.Some additional policy considerationsSome additional policy considerations Regarding the DTP: If an index has aversion towards inequality (φ’’ (p)<0) and its preference function has a non-negative positive third derivative , it will satisfy the PDT for any concave income distribution  This is the case of the Gini coefficient.  If an inequality measure satisfies the PDT in a certain range, any other measure with greater inequality aversion also verifies that principle on that interval and possibly on others of greater amplitude  In our case, the smaller k and the greater the inequality aversion of the index, the wider the set of income distributions for which the PDT is satisfied.Conclusions The use of order statistics in the definition of SWFs and indices of inequality provides a joint treatment of measures that share common features, but differ from and complement each other from the normative standpoint. The approach adopted allows us to proving that, given the mean income, certain families of indices characterize the income distribution, and provides a clear statistical interpretation to each SWF and its corresponding index of inequality.Conclusions The appropriate selection of various elements of the set of indices or welfare functions , permits applying very different distributional judgments when comparing levels of inequality or welfare associated with different income distributions  Hence, the conclusion in a particular application may be interesting either when a robust result is obtained, or if the outcome is different depending on the index considered, as the properties of the different measures are taken into account. Thank you

Inequality, welfare and order statistics

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Inequality, welfare and order statistics

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Repositorio Institucional Universidad de Málaga