Modeling Bivariate Lorenz Curves with Applications to Multidimensional Inequality in Well-Being

Transcription

1 Fifth meeting of the Society for the Study of Economic Inequality ECINEQ Bari, Italy, July 22-24, 213 Modeling Bivariate Lorenz Curves with Applications to Multidimensional Inequality in Well-Being José María Sarabia 1, Vanesa Jordá Department of Economics, University of Cantabria, 395-Santander, Spain Updated version 1th June 213 Abstract The extension of the univariate Lorenz curve to higher dimensions is not an obvious task. The three existing definitions were proposed by Taguchi (1972a,b), Arnold (1983) and Koshevoy and Mosler (1996), who introduced the concepts of Lorenz zonoid and Gini zonoid index. In this paper, using the definition proposed by Arnold (1983), closed expressions for the bivariate Lorenz curve are given, assuming different formulations of the underlying bivariate income distribution. We study a relevant type of models based on the class of bivariate distributions with given marginals described by Sarmanov and Lee (Lee, 1996; Sarmanov, 1966). This model presents several advantages, in particular the expression of the bivariate Lorenz curve can be easily interpreted as a convex linear combination of products of classical and concentrated Lorenz curves. A closed expression for the bivariate Gini index (Arnold, 1987) in terms of the classical and concentrated Gini indices of the marginal distributions is given. This index is specially useful, and can be decomposed in two factors, corresponding to the equality within and between variables. Specific models with Beta, GB1, Pareto, Gamma and lognormal marginal distributions are studied. Other alternative families of bivariate Lorenz curves are discussed. Some concepts of stochastic dominance are explored. Extensions to higher dimensions are included. Finally, an application to measurement multidimensional inequality in well-being is given. 1 Corresponding author. Tel.: ; Fax: address: sarabiaj@unican.es (J.M. Sarabia). 1

2 Key Words: bivariate Lorenz curve; Sarmanov-Lee distribution; bivariate Gini index; Stochastic dominance; well-being 1 Introduction The present economic research have stressed the importance of using more than one attribute in the study of economic inequality. The different works by Atkinson (23), Atkinson and Bourguignon (1982), Kolm (1977), Maasoumi (1986), Slottje (1987) and Tsui (1995, 1999) move in this direction. In the last decades academics interest in assessing country levels of wellbeing has shifted from income alone to a more comprehensive conception of such a process, which has an intrinsic multidimensional nature. In fact, there is nearly consensus that income is not an adequate indicator of well-being (see e.g. Sen (1988, 1989, 1999)), thus other factors need to be taken into account to evaluate this phenomenon effectively. Different approaches have been proposed in the literature to measure inequality in well-being, being the most satisfactory one the consideration multidimensional inequality measures since this methodology takes into account inequality within each dimension and the degree of association among them. However, as in the unidimensional case, these measures only offer overall conclusions about the evolution of well-being distribution, thus other statistical tools are needed to analyse the evolution of inequality in different parts of the distribution. In this context, the extension of the univariate Lorenz curve to higher dimensions is not an obvious task. The three existing definitions were proposed by Taguchi (1972a,b), Arnold (1983) and Koshevoy and Mosler (1996), who introduced the concepts of Lorenz zonoid and Gini zonoid index. In this paper, using the definition proposed by Arnold (1983), closed expressions for the bivariate Lorenz curve are given, assuming different formulations of the underlying bivariate income distribution. We study a relevant type of models based on the class of bivariate distributions with given marginals described by Sarmanov and Lee (Lee, 1996; Sarmanov, 1966). This model presents several advantages, in particular the expression of the bivariate Lorenz curve can be easily interpreted as a convex linear combination of products of classical and generalized Lorenz curves. A closed expression for the bivariate Gini index (Arnold, 1987) in terms of the classical and generalized Gini indices of the marginal distributions is given. This index is specially useful, and can be decomposed in two factors, corresponding to the equal- 2

3 ity within and between variables. Specific models with Beta, GB1, Pareto, Gamma and lognormal marginal distributions are studied. Other alternative families of bivariate Lorenz curves are discussed. Some concepts of stochastic dominance are explored. Extensions to higher dimensions are included. An application to measurement multidimensional inequality in well-being is given. The contents of the paper are the following. In Section 2 we present preliminary results including the definition of univariate Lorenz and concentration curves and a short review about the three different definitions of bivariate Lorenz curves proposed in the literature. In Section 3 we introduce the bivariate Sarmanov-Lee Lorez curve, where we obtain a nice closed expression, as well as a closed expression for the corresponding bivariate Gini index, according to the Arnold s (1987) definition. A decomposition of this index in two factors is given, which correspond to the equality within and between variables. Different models of bivariate Lorenz curves are presented in Section 4. Stochastic ordering are discussed in Section 5. Extension to higher dimensions are discussed in Section 6. An application to measurement multidimensional inequality in well-being is given in Section 7. Finally, some conclusions are given in Section 8. 2 Preliminary Results 2.1 Univariate Lorenz and Concentration Curves We denote the class of univariate distributions functions with positive finite expectations by L and denote by L + the class of all distributions in L with F () = corresponding to non-negative random variables. We use the following definition by Gastwirth (1971). Definition 1 The Lorenz curve L of a random variable X with cumulative distribution function F L is, L(u; F ) = u F 1 (y)dy 1 F 1 (y)dy = u F 1 (y)dy E[X], u 1, (1) 3

4 where F 1 (y) = sup{x : F (x) y}, y < 1, = sup{x : F (x) < 1}, y = 1, is the right continuous inverse distribution function or quantile function corresponding to F. Now, we present the concept of concentration curve introduced by Kakwani (1977). Let g(x) be a continuous function of x such that its first derivative exists and g(x). If the mean E F [g(x)] exits, then one can define L g (y; F ) = x g(x)df (x) E F [g(x)] where y = g(x) and f(x) and F (x) are respectively the probability density function (PDF) and the cumulative distribution function (CDF) of the random variable X. The implicit relation between L g (g(x); F ) and F (x) will be called the concentration curve of the function g(x). If η g (x) and η g (x) denote the elasticities of g(x) and g (x), the concentration curve for the function g(x) will lie above (below) the concentration curve for the function g (x) if η g (x) is less (greater) than η g (x) for all x. The concentration curve admits the simple explicit representation, L g (u; F ) = which will be used in the next Sections. 1 u g[f 1 (t)]dt, E F [g(x)] 2.2 The three different definitions of Bivariate Lorenz curve In this section we discuss briefly the three definitions proposed in the literature for constructing a bivariate Lorenz curve. These definitions were suggested by Taguchi (1972a,b), Arnold (1983, 1987) and Koshevoy and Mosler (1996). We consider the bivariate definition, which can be extend to higher dimensions. 4

5 2.2.1 The Taguchi s (1972) definition of bivariate Lorenz curve This definition was the first proposal in the literature of bivariate Lorenz curve. Definition 2 The Lorenz surface of F 12 is the set of points (s, t, L(s, t)) in R 3 + defined by, s = u v f 12 (x 1, x 2 )dx 1 dx 2, t = L(s, t; F 12 ) = u v u v x 1 f 12 (x 1, x 2 )dx 1 dx 2, E[X 1 ] x 2 f 12 (x 1, x 2 )dx 1 dx 2. E[X 2 ] This definition was investigated by Taguchi (1972a,b, 1988). The definition is not symmetric in (s, t), and its extension to higher dimensions does not look simple Arnold s (1983, 1987) definition of bivariate Lorenz curve The following definition was proposed by Arnold (1983, 1987) and it is an extension of (1) quite natural to higher dimensions. Let X = (X 1, X 2 ) be a bivariate random variable with bivariate probability distribution function F 12 on R 2 + having finite second and positive first moments. We denote by F i, i = 1, 2 the marginal CDF corresponding to X i, i = 1, 2 respectively. Definition 3 The Lorenz surface of F 12 is the graph of the function, L(u 1, u 2 ; F 12 ) = s 1 s 2 x 1 x 2 df 12 (x 1, x 2 ), (2) x 1 x 2 df 12 (x 1, x 2 ) where u 1 = s1 df 1 (x 1 ), u 2 = s2 df 2 (x 2 ), u, v 1. 5

6 The two-attribute Gini-Arnold index GA(F 12 ) is defined as, 1 GA(F 12 ) = 4 1 [u 1 u 2 L(u 1, u 2 ; F 12 )]du 1 du 2, (3) where the egalitarian surface is given by L (u 1, u 2 ; F ) = u 1 u 2. Previous definition has not been explored in detail in the literature. We highlight some of its properties: 1. The marginal Lorenz curves can be obtained as L(u 1 ; F 1 ) = L(u 1, ; F 12 ) and L(u 2 ; F 2 ) = L(, u 2 ; F 12 ). 2. The bivariate Lorenz curve does not depend on changes of scale in the marginals. 3. If F 12 is a product distribution function, then L(u 1, u 2 ; F 12 ) = L(u 1 ; F 1 )L(u 2 ; F 2 ), which is just the product of the marginal Lorenz curves. 4. We denote by F a the one-point distribution at a R 2 +, that is, the egalitarian distribution at a. Then, the egalitarian distribution has bivariate Lorenz curve L(u 1, u 2 ; F a ) = u 1 u In the case of a product distribution, the two-attribute Gini-Arnold defined in (3) can be written as, 1 GA(F 12 ) = [1 G(F 1 )][1 G(F 2 )]. The Arnold s Lorenz curve can be evaluated implicitly in some relevant bivariate families of income distributions. We consider the following model. Example 1 Let X = (X 1, X 2 ) be a bivariate second kind beta distribution (or inverted beta distribution) with joint PDF, x α x α f(x 1, x 2 ; α) = K(α 1, α 2, α 3 ) (1 + x 1 + x 2 ) α 1+α 2 +α 3, x 1, x 2, (4) where K(α 1, α 2, α 3 ) = Γ(α 1+α 2 +α 3 ) Γ(α 1 )Γ(α 2 )Γ(α 3 ), α i >, i = 1, 2, 3. The marginals are second kind beta distributions: X 1 B2(α 1, α 3 ) and X 2 B2(α 2, α 3 ), with 6

7 PDF f(x; p, q) = 1 B(p,q) xp 1 /(1 + x) p+q. The second kind beta distribution is very common in inequality analysis (see Chotikapanich et al, 27) and it is a subfamily of the GB2 distribution (McDonald, 1984). The joint CDFs of (4) is (see Balakrishnan and Lai, 29), F (x 1, x 2 ; α 1, α 2, α 3 ) = K(α 1, α 2, α 3 )x α 1 1 x α 2 2 H[x 1, x 2 ; α 1, α 2, α 3 ], (5) where K(α 1, α 2, α 3 ) = K(α 1,α 2,α 3 ) α 1 α 2 and H[x 1, x 2 ; α 1, α 2, α 3 ] = F 2 [α 1 + α 2 + α 2 ; α 1, α 2 ; α 1 + 1, α 2 + 1; x 1, x 2 ], where F 2 is the Appell s hypergeometric function of two variables, defined as (Bailey, 1935) Γ(β)Γ(β )Γ(γ β)γ(γ β ) Γ(γ)Γ(γ F 2 [α; β, β ; γ, γ ; x, y] ) = 1 1 u β 1 v β 1 (1 u) γ β 1 (1 v) γ β 1 (1 ux vy) α dudv. α The product moment of (4) is E[X 1 X 2 ] = 1 α 2, which exists if α (α 3 1)(α 3 2) 3 > 2. The bivariate Lorenz curve is defined implicitly by (u 1, u 2, L(u 1, u 2 ; F 12 )) where, L(u 1, u 2 ; F 12 ) = F [s 1, s 2 ; α 1 + 1, α 2 + 1, α 3 2], where u i = F i (s i ), i = 1, 2, α 3 > 2 and F [.,.;.,.,.] is defined in (5) The Koshevoy and Mosler (1996) Lorenz Curve The following definition of Lorenz curve was initially proposed by Koshevoy (1995), were this author identifies the suitable definition. Then, the rest of results were obtained by Koshevoy and Mosler (1996, 1997) and Mosler (22). Denote by L 2 + de set of all 2-dimensional non-negative random vectors X with finite positive marginal expectations. Let Ψ (2) denote the class of all measurable functions from R 2 + to [, 1]. Definition 4 Let X L 2 +. The Lorenz zonoid L(X) of the random vector X = (X 1, X 2 ) with distribution F 12 is defined as, {( ) } x1 ψ(x) L(X) = ψ(x)df 12 (x), E[X 1 ] df x2 ψ(x) 12(x), E[X 2 ] df 12(x) : ψ Ψ (2), {( = E[ψ(X)], E[X 1ψ(X)], E[X ) } 2ψ(X)] : ψ Ψ (2). E[X 1 ] E[X 2 ] 7

8 The Lorenz zonoid if a convex American football-shaped subset of the 3-dimensional unit cube that includes the points (,, ) and (1, 1, 1). Extension to higher dimensions is quite direct. However, the computation of these formulas in parametric income distributions is not easy. 3 The bivariate Sarmanov-Lee Lorenz Curve In this section we introduce the so-called bivariate Sarmanov-Lee Lorenz curve. As a previous step, we introduce an explicit expression for the Arnold s bivariate LC an the bivariate Sarmanov-Lee distribution. 3.1 Explicit expression for the Arnold s (1983) bivariate Lorenz curve Many of our results are based on a explicit expression of the bivariate Lorenz curve defined in (2). The bivariate Lorenz curve (2) admits the following simple explicit representation, Lemma 1 The bivariate Lorenz curve can be written in the explicit form, where L(u 1, u 2 ; F 12 ) = 1 E[X 1 X 2 ] u 1 u 2 A(x 1, x 2 )dx 1 dx 2, u 1, u 2 1, (6) A(x 1, x 2 ) = F 1 1 (x 1 )F2 1 (x 2 )f 12 (F1 1 (x 1 ), F2 1 (x 2 )) f 1 (F1 1 (x 1 ))f 2 (F2 1. (7) (x 2 )) Proof: The proof is direct making the change of variable (u 1, u 2 ) = (F 1 (x 1 ), F 2 (x 2 )) in (2). 3.2 The bivariate Sarmanov-Lee Distribution Let X = (X 1, X 2 ) be a bivariate Sarmanov-Lee (SL) distribution with joint PDF, f(x 1, x 2 ) = f 1 (x 1 )f 2 (x 2 ) {1 + wφ 1 (x 1 )φ 2 (x 2 )}, (8) 8

9 where f 1 (x) and f 2 (y) are the univariate PDF marginals, φ i (t), i = 1, 2 are bounded nonconstant functions such that, φ i (t)f i (t)dt =, i = 1, 2, and w is a real number which satisfies the condition 1+wφ 1 (x 1 )φ 2 (x 2 ) for all x 1 and x 2. We denote µ i = E[X i ] = tf i(t)dt, i = 1, 2, σi 2 = var[x i ] = (t µ i) 2 f i (t)dt, i = 1, 2 and ν i = E[X i φ i (X i )] = tφ i(t)f i (t)dt, i = 1, 2. Properties of this family has been explored by Lee (1996). Moments and regressions of this family can be easily obtained. The product moment is E[X 1 X 2 ] = µ 1 µ 2 + wν 1 ν 2, and the regression of X 2 on X 1 is given by The copula associated to (8) is, E[X 2 X 1 = x 1 ] = µ 2 + wν 2 φ 1 (x 1 ). C(u 1, u 2 ; w, φ) = u 1 u 2 + u 1 u 2 φ 1 (s 1 ) φ 2 (s 2 )ds 1 ds 2, where φ i (s i ) = φ i (F 1 i (s i ), i = 1, 2, and F i (x i ) are the CDF of X. The PDF of the copula associated to (8) is, c(u 1, u 2 ; w, φ) = C(u 1, u 2 ; w, φ) u 1 u 2 = 1 + wφ 1 (F 1 1 (u 1 ))φ 2 (F 1 2 (u 2 )). Note that (8) and its associated copula has two components: a first component corresponding to the marginal distributions and the second component which defines the structure of dependence, given by the parameter w and the functions φ i (u), i = 1, 2. These two components will be translated to the structure of the associated bivariate Lorenz curve, and the corresponding bivariate Gini index. In relation with other copulas, the Sarmanov-Lee copula has several advantages: its joint PDF and CDF are quite simple; the covariance structure in general is not limited and its different probabilistic features (moments, conditional distributions...) can be obtained in a explicit form. On the other hand, the SL distribution includes several relevant special cases including the classical Farlie-Gumbel-Morgenstern distribution, and the variations proposed by Huang and Kotz (1999) and Bairamov and Kotz (23). 9

10 3.3 The bivariate SL Lorenz Curve The bivariate SL Lorenz curve is obtained using (8) in definition (2) Theorem 1 Let X = (X 1, X 2 ) a bivariate Sarmanov-Lee distribution with joint PDF (8) with non-negative marginals satisfying E[X 1 ] <, E[X 2 ] < and E[X 1 X 2 ] <. Then, the bivariate Lorenz curve is given by, L SL (u 1, u 2 ; F 12 ) = πl(u 1 ; F 1 )L(u 2 ; F 2 ) + (1 π)l g1 (u 1 ; F 1 )L g2 (u 2 ; F 2 ), (9) where π = µ 1µ 2 E[X 1 X 2 ] = µ 1 µ 2 µ 1 µ 2 + wν 1 ν 2, and L(u i ; F i ), i = 1, 2 are the Lorenz curves of the marginal distribution X i, i = 1, 2 respectively, and L gi (u i ; F i ), i = 1, 2 represent the concentration curves of the random variables g i (X i ) = X i φ i (X i ), i = 1, 2, respectively. Proof: The function (7) for the Sarmanov-Lee distribution can be written of the form, A SL (x 1, x 2 ) = F 1 1 (x 1 )F 1 2 (x 2 ) { 1 + wφ 1 (F 1 1 (x 1 ))φ 2 (F 1 2 (x 2 )) }, and integrating in the domain (, u) (, v) we obtain, µ 1 µ 2 L(u 1 ; F 1 )L(u 2 ; F 2 ) + we F1 [g 1 (X 1 )]E F2 [g 2 (X 2 )]L g1 (u 1 ; F 1 )L g2 (u 2 ; F 2 ), and after normalized we obtain (9). The interpretation of (9) is quite direct: the bivariate Lorenz curve can be written as a convex linear combination of two components: (a) a first component corresponding to the product of the marginal Lorenz curves (marginal component) and a second component corresponding to the product of the concentration Lorenz curves (structure dependence component). 3.4 Bivariate Gini index The following result provides a convenient write of the two-attribute bivariate Gini defined in (3). This expression permits a simple decomposition of the equality in two factors: a first factor which represent the equality within variables and a second factor which represent the equality between variables. 1

11 Theorem 2 Let X = (X 1, X 2 ) be a bivariate Sarmanov-Lee distribution with bivariate Lorenz curve L(u, v; F 12 ). The two-attribute bivariate Gini index defined in (3) is given by, 1 G(F 12 ) = π[1 G(F 1 )] [1 G(F 2 )]+(1 π)[1 G g1 (F 1 )] [1 G g2 (F 2 )], (1) where G(F i ), i = 1, 2 are the Gini indices of the marginal Lorenz curves, and G gi (F i ), i = 1, 2 represent the concentration indices of the concentration Lorenz curves L gi (u i, F i ), i = 1, 2. Proof: The proof is direct using expression (9) and taking into account that G(F 12 ) = L(u, v; F 12)dudv. Then the overall equality (OE) given by 1 G(F 12 ) can be decomposed in two factors, OE = W E + BE, (11) where OE = 1 G(F 12 ), W E = π[1 G(F 1 )][1 G(F 2 )], BE = (1 π)[1 G g1 (F 1 )][1 G g2 (F 2 )], and WE represents the equality within variables and the second factor BE represent the equality between variables. Note that the decomposition is well defined (since OE 1 and W E 1 and in consequence BE 1) and BE includes the structure of dependence of the underlying bivariate income distribution through the functions g i, i = 1, 2. 4 Bivariate Lorenz curve Models In this section we consider several relevant models based on the previous methodology. 4.1 Bivariate Pareto Lorenz curve based on the FGM family Let X = (X 1, X 2 ) be a bivariate FGM with classical Pareto marginals and joint PDF, f 12 (x 1, x 2 ; α, σ) = f 1 (x 1 )f 2 (x 2 ){1 + w[1 2F 1 (x 1 )][1 2F 2 (x 2 )]}, (12) 11

12 where ( ) αi x F i (x i ) = 1, x i σ i, i = 1, 2, σ i f i (x i ) = α ( ) αi 1 i x, x i σ i, i = 1, 2 σ i σ i are the CDF and the PDF respectively, of the classical Pareto distributions (Arnold, 1983), with α i > 1, σ i >, i = 1, 2, 1 w 1 and φ i (x i ) = 1 2F i (x i ), i = 1, 2. Using (9) with g i (x i ) = x i [1 2F i (x i )], i = 1, 2 and after some computations, the bivariate Lorenz curve associated to (12) is, L F GM (u 1, u 2 ; F 12 ) = π w L(u 1 ; α 1 )L(u 2 ; α 2 ) + (1 π w )L g1 (u 1 ; α 1 )L g2 (u 2 ; α 2 ), (13) where L(u i ; α i ) = 1 (1 u i ) 1 1/α i, u 1, i = 1, 2, and and, L gi (u i ; α i ) = 1 (1 u i ) 1 1/α i [1 + 2(α i 1)u i ], u 1, i = 1, 2, The bivariate Gini index is given by, where π w = (2α 1 1)(2α 2 1) (2α 1 1)(2α 2 1) + w. G(α 1, α 2 ) = (3α 1 1)(3α 2 1)(2α 1 + 2α 2 3) + [h(α 1, α 2 )]w, (3α 1 1)(3α 2 1)[(1 2α 1 )(1 2α 2 ) + w] h(α 1, α 2 ) = 3 4α 2 1(α 2 1) 2 + (5 4α 2 )α 2 + α 1 (5 + α 2 (8α 2 7)). The FGM bivariate distribution has a correlation structure limited, that is ρ(x 1, X 2 ) 1, and in consequence, models (??) and (13) must be used 3 for income data configurations with moderate correlation. The following classes of models do not present this inconvenient. 12

13 4.2 Bivariate Sarmanov-Lee Lorenz curves with Beta and GB1 Marginals Let X i Be(a i, b i ), i = 1, 2 be two classical beta distributions with PDF, f i (x i ; a i, b i ) = xa i 1 i (1 x i ) b i 1, x i 1, i = 1, 2 B(a i, b i ) where B(a i, b i ) = Γ(a i )Γ(b i )/Γ(a i + b i ), for i = 1, 2. This distribution has been proposed as a model of income distribution by McDonald (1984) and more authors. If we consider the mixing functions φ i (x i ) = x i µ i, where µ i = E[X i ] = a i /(a i + b i ), i = 1, 2, the bivariate SL distribution is, f 12 (x 1, x 2 ) = f 1 (x 1 ; a 1, b 1 )f 2 (x 2 ; a 2, b 2 ) where w satisfies, (a 1 + b 1 )(a 2 + b 2 ) max{a 1 a 2, b 1 b 2 } { 1 + w ( x 1 a 1 a 1 + b 1 w (a 1 + b 1 )(a 2 + b 2 ) max{a 1 b 2, a 2 b 1 }. ) ( x 2 a 2 a 2 + b 2 (14) A good property of this family is that it can be expressed as a linear combination of products of univariate beta densities. The univariate Lorenz curve of the univariate classical beta distribution if given by (Sarabia, 28), L(u i ; F i ) = G Be(ai +1,b i )[G 1 Be(a i,b i ) (u i)], i = 1, 2, (15) z where G Be(a,b) (z) = (1/B(a, b)) t a 1 (1 t) b 1 dt represents the CDF of a classical Beta distribution. In consequence, the concentration curve can be written as, L gi (u i ; F i ) = E[X2 i ]G Be(ai +2,b i )(G 1 Be(a i,b i ) (u i)) E[X i ] 2 L(u i ; F i ), i = 1, 2. var[x i ] (16) Note that ν i = E Fi [X i φ i (X i )] = var[x i ], i = 1, 2. Finally, combining (15) with (16) in (9), we obtain the bivariate beta Lorenz curve. This model can be extended easily to the SL distribution with generalized beta of the first type (GB1) marginals, with PDF, f(x i ; a i, b i, p i ) = p ix a ip i 1 i (1 x p i B(a i, b i ) 13 i )b i 1, x i 1, i = 1, 2, )},

14 and mixing function, φ i (x i ) = x i µ i = x i Γ(a i + 1/p i )Γ(a i + b i ), i = 1, 2. Γ(a i + b i + 1/p i )Γ(a i ) 4.3 Bivariate SL Lorenz curves with Gamma Marginals Let X i G(α i, λ i ), i = 1, 2 be classical gamma distributions with PDF f i (x i ) = λ α i i x α i 1 i e λ ix i /Γ(α i ), where x i > and α i, λ i >, i = 1, 2. If we consider the mixing function φ i (x i ) = e x i L(1; α i, λ i ), where L(s, α, λ) = (1 + s/λ) α is the Laplace transform of a gamma distributions (Lee, 1996), we can construct the SL distribution with gamma marginals defined as, f(x 1, x 2 ) = f 1 (x 1 )f 2 (x 2 ) { 1 + w ( e x 1 L(1; α 1, λ 1 ) ) ( e x 2 L(1; α 2, λ 2 ) )}, (17) with x 1, x 2 >. Note that (17) can be written as a linear combination of products of univariate gamma distributions. Then, if we define, g i (x i ) = x i e x i x i L(1; α i, λ i ), i = 1, 2, the bivariate SL Lorenz curve with gamma marginals is, where and L G (u 1, u 2 ; F 12 ) = πl(u 1 ; F 1 )L(u 2 ; F 2 ) + (1 π)l g1 (u 1 ; F 1 )L g2 (u 2 ; F 2 ), L(u i ; F i ) = H αi +1[H 1 α i (u i )], i = 1, 2, L gi (u i ; F i ) = (1 + λ i) 1 H αi +1[(1 + 1 λ i )Hα 1 i (u i )] λ 1 i H αi +1[Hα 1 i (u i )] (1 + λ i ) 1 λ 1, i = 1, 2, i being H α (x) = x tα 1 e t dt/γ(α) the CDF of a classical gamma distribution. 4.4 Distributions with lognormal marginals The lognormal distribution plays an important role in the analysis of income and wealth data. There are several classes of bivariate distributions with lognormal distributions. Perhaps, the most natural definition of bivariate lognormal distribution is given in terms of a monotone marginal transformation of the classical bivariate normal distribution. Other alternatives have 14

15 been described by Sarabia et al (27). In these cases we can obtain a bivariate Lorenz curve using the Arnold s definition. If X i LN (µ Xi, σ 2 X i ), i = 1, 2 denote a lognromal distribution, it is possible to consider a SL distribution with lognormal marginals, where the mixing functions are defined by, φ i (x i ) = f i (x i ) = f i (x i ) f 2 i (x i )dx i ( ) 1 exp µ Xi + σ2 X i, i = 1, 2. 2σ Xi π 4 Now, if g i (x i ) = x i φ i (x i ) = x i f i (x i ) x i f 2 i (x i )dx i, i = 1, 2 we can use expression (9) for obtaining the corresponding bivariate Lorenz curve. 4.5 Other classes of bivariate Lorenz curves Other alternative families of bivariate Lorenz curves can be construted, including models with marginals specified in terms of univariate Lorenz curves (see Sarabia et al., 1999) and models based on mixture of distributions (see Sarabia et al, 25), which permits to incorporate heterogeneity factors in the inequality analysis. 5 Stochastic Orderings In this section we study some stochastic orderings related with the Lorenz curves previously defined. 5.1 The univariate case The usual Lorenz ordering is given in the following definition. Definition 5 Let X, Y L with CDFs F X and F Y and Lorenz curves L(u; F X ) and L(u; F Y ), respectively. Then, X is less than Y en the Lorenz order denoted by X L Y if L(u; F X ) L(u; F Y ), for all u [, 1]. A similar definition for concentration curves can also be considered. 15

16 5.2 The general case We denote by L k + the set of all k-dimensional nonnegative random vectors X and Y with finite marginal expectations, that is E[X i ] R ++. We define the following orderings (Marshall, Olkin and Arnold, 211) Definition 6 Let X, Y L k +, and we define the orders (i) X L Y if L(u; F X ) L(u; F Y ), [ ( )] [ ( )] X (ii) X L1 Y if E g 1,..., X k Y E[X 1 ] E[X k E g 1,..., Y k ] E[Y 1 ] E[Y k, for every continuous convex function g : R k R for which expectations ] exists, (iii) X L2 Y if k i=1 a ix i L k i=1 a iy i for every a R k (iv) X L3 Y if k i=1 b ix i L k i=1 b iy i for every b R k + (v) X L4 Y if X i L Y i, i = 1, 2,..., k It can be verified (Marshall, Olkin and Arnold, 211) that L1 L2 and L2 L L3 L4. We have the next theorem. Theorem 3 Let X, Y L 2 + with the same Sarmanov-Lee copula. Then, if X i L Y i, and X i Lgi Y i i = 1, 2, then X L Y. Proof: The proof is direct based on the expression of the bivariate Sarmanov-Lee Lorenz curve defined in (9). 6 Extensions to higher dimensions In this section we discuss briefly how to extend the results in previous sections to dimensions higher than two. First, we consider the general definition of Lorenz curve (2). Let X = (X 1,..., X m ) be a random vector in L m + with joint CDF F 12...m (x 1,..., x m ). The multivariate Arnold s Lorenz curve can be defined as, L(u; F 12...m ) = s 1... s m m x i df 12...m (x 1,..., x m ) [ m ], E X i i=1 i=1 16

17 where u i = s i df i (x i ), i = 1, 2,..., m and u 1,..., u m 1. The m-dimensional version of the Sarmanov-Lee distribution is defined as, { m } f(x 1,..., x m ) = f i (x i ) {1 + R φ1...φ mω m (x 1,..., x m )}, (18) where i=1 R φ1...φ mω m (x 1,..., x m ) = w i1 i 2 φ i1 (x i1 )φ i2 (x i2 ) = i 1 i 2 m 1 i 1 i 2 i 3 m m +w 12...m φ i (x i ), i=1 w i1 i 2 i 3 φ i1 (x i1 )φ i2 (x i2 )φ i3 (x i3 ) and Ω m = {w i1 i 2, w i1 i 2 i 3,..., w 12...m }, k 2 and m 3. The set of real numbers Ω m is such that 1+R φ1...φ mω m (x 1,..., x m ), (x 1,..., x m ) R m. Using expression (18), the m-variate Lorenz curve takes de form, m L(u; F 12...m ) = w L(u i ; F i ) + i=1 m +w 12...m L gi (u i ; F i ), i=1 i 1,...,i k w i1...i k i k j=i 1 L gj (u j ; F j ) i m j=i k +1 L(u j ; F j ) where g i (x i ) = x i φ i (x i ), i = 1, 2,..., m. Similar expression for the multivariate Gini is also possible. 7 Application: Multidimensional inequality in well-being In the last decades the interest of academics in well-being has shifted from income alone to a more comprehensive conception of such a process, which has an intrinsic multidimensional nature. In fact, there is nearly consensus 17

18 that income is not an adequate indicator of well-being (see e.g. Sen (1988, 1989, 1999)), thus other factors need to be taken into account to evaluate this phenomenon effectively. A whole array of indicators have been proposed in the literature to be considered in the measurement of well-being, including education (Morrison and Murtin, 212), health (Bourguignon and Morrison, 22; Becker et al., 25), security (Lawson-Remer et al., 29; Bilbao-Ubillos, 211), democracy (Cornwall and Gaventa, 29; environmental questions (Neumayer; 21; Briassoulis, 21) and distributional aspects (Alkire and Foster, 21; Hicks, 1997; Seth, 211). Note that the selection of dimensions is not only a technical choice, and it also lead to normative implications. It is beyond the scope of this paper to discuss in detail which indicators should be included in the assessment of well-being, hence we use the human development index (HDI) as a theoretical benchmark 2, since this approach allows us to compare and complement our results with previous studies. Therefore, in this work we assume that well-being evaluation at country level focuses on the three dimensions considered in the HDI: income (I), health (H) and educational attainment (E). These variables, placed on a scale of to 1, are transformed indicators of the original attributes, namely GNI per capita, life expectancy and a geometric average of expected years of schooling and mean years of schooling. Finally, the HDI is constructed using a geometric mean of the three transformed variables. Having reached this point, it is worth nothing that different approaches have been proposed to analyse multidimensional inequality in well-being. An intuitive procedure would be the construction of a composite index and then compute inequality measures of such an indicator (Pillarisetti, 1997; McGillivray y Pillarisetti, 24; Martnez, 212). This approach allows us to establish overall conclusions about the evolution of well-being although some problems can arise. In fact, it is supported that this methodology sweeps 2 The HDI has been highly criticised on the grounds of construction (Grimm et al., 28; Kelley, 1991), selection of variables (Srinivasan, 1994), arbitrary weighting scheme (McGillivray and White, 1993; Noorbakhsh, 1998), and redundancy with its components (Cahill, 25; McGillivray, 1991, McGillivray and White, 1993; Ravallion, 1997). These criticisms suggest that the HDI is not an ideal indicator of development. However, it should be emphasised that the conception of human development is complex, abstract and difficult to synthesise. Independently of its limitations, this indicator attracts a great amount of attention from the media and politicians due to its simplicity, transparency and capacity to capture the most important aspects of human development. 18

19 the multidimensional nature of well-being under the carpet (Decancq et al., 29, pp. 14). Moreover, even when it is accepted that the composite index should satisfy some qualitative properties which define it as a well-behaved index, there are a whole array of well-being indicators, thus implying that subjective judgements play an important role and the arbitrariness of this choice comes with critics and disagreements. To avoid making arbitrary decisions about the functional form of the index, the simplest alternative is to calculate inequality in each dimension independently (Hobin y Franses, 21; Neumayer, 23; World Development Report, 26; McGillivray and Markova, 21). Obviously, this method provides additional insights than a sole focus on income, thus allowing us to extract conclusions about the existing disparities within each dimension, which is the so called distribution sensitive inequality (Kolm, 1977). Note, however, that, when some indicators have worsened their inequality levels and others have reduced their disparities, it is not possible to draw integral conclusions of the evolution of inequality using the dimension-by-dimension approach. Furthermore, this methodology ignores the relationship between dimensions, more concretely the degree of association among them. It is well-known that the dependence between dimensions has a significant influence on the assessment of disparities (Decancq and Lugo, 212; Duclos, 26; 211; Kovacevic, 21; Seth, 211), which has been denominated association sensitive inequality (Atkinson and Bourguignon, 1982). To illustrate it, consider that we are interested in evaluating cross-country inequality based on two indicators of well-being, e.g. income and education. Suppose that we are considering two regions (Z A, Z B ) made up of two countries (C 1, C 2 ) and (C 3, C 4 ) respectively, which result in the following distributional matrix: Z A = [ ] ; Z B = [ where the jth column is the distribution of the jth indicator and the ith row includes the amount of attributes that the ith country has. Note that Z A is derived from Z B by switching the amounts of the second variable (education) between the countries considered. The Lorenz curves for both dimensions are plotted in Figure 1, showing that, income is less unequal than education. At this point, it is important to recall the interpretation of both curves. Naturally, in the case of income, this curve informs about the percentage of income owned by the k poorest 19 ]

20 1..8 Income Education 1..8 Region ZB Region ZA Figure 1: Lorenz curves of income and education (left) and concentration curves of education of regions Z A and Z B (right). percent of the population (k [, 1]). Instead, the interpretation for the education variable reveals the percentage of education possessed by the k least educated people. It is worth noting that this graph is exactly the same for the second group of countries, based on distributional matrix Z B given that both regions have analogous levels of within-dimension inequality. Therefore, a dimension-by-dimension approach would conclude that these distributions are equally unequal. However, distributional matrices Z A and Z B are not equivalent, therefore a feasible inequality analysis would offer different results for both regions, differences that come from variations in degrees of correlation between dimensions. As is supported by Duclos et al. (211), it is ethically accepted that inequality is higher in the region Z A since C 2 is relatively worse-off in terms of both attributes, whereas in Z B, C 4 has lower level of education and C 3 lower level of income. Therefore, not considering the patterns between dimensions would imply to abandon one of the principal motivations for measuring multidimensional inequality (Decancq and Lugo, 212). The proposed methodology accounts for this type of inequality using the concentration curves of each dimension. Let us illustrate this using the simple example of regions Z A and Z B. Figure 1 (right graph) plots the concentration curve of education with respect to the income component. In this case, concentration curves inform about the percentage of education possessed by the poorest k percent of population. Therefore, this curve gives information about the relationship between these variables. In fact, when the concen- 2

21 tration curve is concave indicates that a negative relationship exist between both attributes, thus resulting in a negative value of the concentration index (Kakwani, 1977). Conversely, convex curves are related with positive concentration index. Using (1), it is directly concluded that the region Z B is more equal than Z A, given that the first component is the same for both regions. Therefore, our methodology not only accounts for the two different types of inequality, it also quantifies the influence of each one in the overall multidimensional inequality. Note however that inequality measures summarize the state of well-being distribution at some point of time, thus letting international and intertemporal comparisons. At this point, it should be recalled that greater values of an specific inequality measure, e.g. the Gini index, does not imply that the whole distribution has worsened. It could be possible that poor countries have a more unequal situation whereas the wealthiest economies would be found more equally distributed. To draw some conclusions about these dynamics multidimensional stochastic dominance conditions have been developed in the literature (Atkinson and Bourguignon, 1982; Duclos et al., 211; Muller and Trannoy, 211). We also have presented some results for multidimensional Lorenz orderings (see Section 5) since it is considered as a preliminary task before the estimation of multidimensional inequality indices (Muller and Trannoy, 211). In fact, concluding stochastic dominance implies that the whole distribution is less unequal and thus calculating inequality measures gives an integral conclusion of distributional dynamics of well-being. However, if no dominance relationships can be observed, we only can obtain summarized information about the evolution of such a distribution in general terms. In that case, additional tools are needed to state which parts of the distribution are more equal and which ones have worsened in terms of inequality. In the unidimensional framework, Lorenz curves are used to shed light on this process, hence we use multidimensional extensions of these statistical measures to extract conclusions about how multidimensional inequality has evolved over the study period. Under the theoretical benchmark of the HDI, three different dimensions are considered. Unfortunately, even when m-dimensional Lorenz curves have been developed in Section 6, only bidimensional Lorenz curves can be plotted due to obvious limitations. It should be stated, however, that this fact does not restrict our analysis strongly, since compensations between health and education seems to have lack of interest in practice (Muller and Trannoy, 21

22 211). These authors give some arguments to assume that education and health are not dependent each other, based primarily on political reasons. Generally, policies targeted to health improvements and education programs are nationally implemented by different ministries. Such a division can also be observed at supranational level, having international organisations, such as the World Bank, different departments for each dimension. Assuming that both attributes are independent would imply that an increase of income would compensate bad health and low levels of education, but it is practically and politically less attractive that high educational levels would be used as a substitutes of bad health. Conversely, it is absolutely valid to assume that better educated people have more possibilities to enjoy a healthy life, so we also consider this case Data We use the most recent available data from International Human Development Indicators (UNDP, 212) on the HDI and its three components for the period with five years intervals. Income is represented by Gross National Income per capita measured in PPP 25 US dollars, to make incomes comparable across countries and over time. The second component is measured by life expectancy at birth, which is considered an indicator of the health level. The education index comprises two indicators, expected years of schooling and mean years of schooling, which are aggregated using with the geometric mean. This first educational variable indicates the number of years that a child of school entrance age can expect to receive if prevailing patterns of age-specific enrolment rates persist throughout the child s life (UNDP, 212), whereas the second represents average number of years of education received by people aged 25 and older, converted from education attainment levels using official durations of each level (Barro and Lee, 21). The HDI is computed by a geometric mean of the depicted components. Originally, our sample comprised only 15 countries, covering less than the 75 percent of global population. We had non-available data for 26 countries for one or more years before In order to offer comparable results across periods and to not restricting the sample considerably, missing values have been estimated. The estimation is based on two complementary 3 Including this assumption can lead to incongruent solutions given that it would imply that international aid to improve health conditions would be targeted to most educated economies, thus increasing inequality levels (Muller and Trannoy, 211). 22

23 methodologies which jointly offer feasible and consistent results according to the sample: piecewise cubic Hermite interpolating polynomial (PCHI) and the average rate of change, which is used when PCHI offers unfeasible estimations or out of range results. After this procedure, our dataset includes 132 countries whose indicators of income, health and education are available for eight points of time. Consequently, the sample covers over 9 percent of the world population during the whole period. Notwithstanding this large coverage, many African and Eastern European countries are not included in the sample due to the scarcity of data. Given that practically all absentees are developing countries, our estimates can be biased downward. Therefore the conclusions pointed out by our results should be cautiously interpreted. 7.2 Estimation methods Before describing the estimation procedure for the statistical tools developed in this paper, it is necessary to clarify what is actually measured when we refer to multidimensional inequality in well-being. Due to the scarcity of individual data on non-income dimensions for long periods of time and for most countries, we use country-level data which automatically implies that within country disparities are being ignored. This situation gives us two possible alternatives: we can consider the countries as units of observation (unweighted inequality) or we can assume that all the citizen of a given country has the same level of well-being, thus being individuals the subjects of our analysis (population weighted inequality) (Milanovic, 25). Both approaches are well-known from the literature on income inequality which also points out the advantages and shortcomings of each one. In this study we focus on unweighted inequality, thus implying that each country counts the same in the global distribution, due to at least three reasons. First of all, we are studying inequality in well-being which also considers education and health, therefore its distribution is strongly conditioned by public policies equally implemented all over the country. Then, the countries can be seen as a set of policies (Decancq et al., 29) whose effectiveness would contribute to that developing countries catch up the most developed ones. Secondly, weighted inequality is extremely sensitive to the performance of the most populous countries such as China and India. Last but not least, one of the principal objectives of this application is to compare our results with the classical approach of the evolution of income inequality and with previous studies. Consequently, less ambiguous conclusions can be extracted 23

24 considering unweighted inequality, given that population growth plays a crucial role in the evolution of weighted inequality measures (Firebaugh, 2). The theoretical bivariate distribution for modeling two of the components of the HDI is the Sarmanov-Lee distribution with classical beta marginals, studied in Section 4.2. Since our sample is made up of the three indicators of the HDI which are normalized variables, we consider that this model is the most satisfactory in this case given that it is bounded from to 1. Let X = (X 1, X 2 ) be a bivariate distribution with joint PDF (14)and let (x 11, x 21 ),..., (x 1n, x 2n ) a sample of n countries from (14). For the estimation of the parameters (a 1, b 1, a 2, b 2, w) we proceed in two steps: Estimation of the marginal distributions. We define, m i = 1 n n x ij, s 2 i = 1 n j=1 n (x ij m i ) 2, i = 1, 2 j=1 and then, the point estimates of the couples (a i, b i ), i = 1, 2 are, with i = 1, 2. â i = m i(m i m 2 i s 2 i ), s 2 i (19) ˆbi = (1 m i)(m i m 2 i s 2 i ), s 2 i (2) Estimation of the structure of dependence. The estimate of the w parameters is based on the simple relation ρ = wσ 1 σ 2. Then, if r denotes the sample linear correlation coefficient, and s i, i = 1, 2 the sample standard deviation of the marginal distributions X i, i = 1, 2, the point estimate of w is, ŵ = r s 1 s 2. (21) The Gini index of the marginals distributions can be obtained by the formula, G(Be(a i, b i )) = Γ(a i + b i )Γ(a i + 1)Γ(b 2 i + 1) 2 Γ(a i + b i + 1)Γ(a 2 i + 1)Γ(b i ), i = 1, 2. (22) π 24

25 7.3 Results In this section we present the results of applying the concepts developed in this paper to well-being data for the last three decades. Using the methodology described in the previous section we have estimated the parameters of the Sarmanov-Lee distribution considering the beta distribution for modelling marginal distributions (Table 1). As stated before, since we are considering the HDI as a benchmark, three dimensions of well-being are included in the analysis, thus leading to three different bidimensional distributions: income with education, income with health and, finally, education with health. As a result, we calculate three bivariate Gini indices (Table 2) which give us summarised information about how global inequality has evolved over de study period. Before analysing the evolution of multidimensional inequality, let us look at inequality patterns in each dimension independently. In line with previous studies (McGillivray y Markova, 21; Decancq, 211), it is observed that, the unidimensional Gini index for education decreases continuously during the entire period, pointing out a reduction of 35 percent over the last three decades. Such a convergence process has its origin in the increase of the mean years of schooling which have been doubled in the last 4 years, thanks to the efforts in education performed in developing countries, especially in Asia (WDR, 26; Morrison and Murtin, 212). Conversely, health inequality shows a positive trend in the nineties decade as a consequence of the effects of AIDS in Sub-Saharan Africa (Neumayer, 23; Becker et al., 25), effect partially offset by the decrease in infant mortality (Deaton, 24). Inequality in health falls during the rest of the period mainly due to the expansion of life expectancy in East and South Asia and the North of Africa (Goesling and Firebaugh, 24). Taking the study period as a whole, a process of convergence in health levels is concluded, which has lead to a decrease in inequality levels of 16 percent. Income inequality has received by far the more attention than the other dimensions. It is well-known that cross-country inequality has increased over the second half of the last century (see e.g. Pritchett, 1997; Milanovic, 25; WDR 21). In spite of the success of Asia which rapidly converged to the incomes of developed countries in the last 3 years, the failure of Africa in the eradication of poverty has lead to the increase of income disparities. Notwithstanding this trend, income inequality across countries has been reduced in 7 percent over the last 3 years, thanks to the convergence process that took 25

26 place in the last decade. Having reached this point is worth focusing on bivariate Gini index trends. It is concluded that the highest levels of inequality are found when considering education and income together, followed by education and health and, finally, the joint distribution of income and health is the least unequal among the relationships considered. Taking the period as a whole, a decreasing inequality pattern is observed for education and income components jointly, thus indicating that the reduction of inequality in education has offset the increase of income disparities in the last decades of the XX century. Note that the same dynamic is observed when considering health and education. In contrast, bivariate Gini index for health and income shows a positive trend from 1985 to 2, given that both indicators have experienced a process of divergence during that period. Bivariate Gini index can be decomposed in two components, using Equations (1) and (11). Overall equality (expressed as 1 minus the Bivariate Gini) is decomposed in terms of equality within each dimension (which is related with the concept of distribution sensitive inequality) and equality between dimensions (which is related with the concept of association sensitive inequality), thus allowing us to analyze the evolution each component in terms of global equality. From Table 3, it is pointed out that the evolution of global equality is mainly determined by the within-dimension distribution, while equality between dimensions represents a residual proportion which has been reduced over time in all cases except for education and income. Pervious results reveal that multidimensional inequality in well-being has decreased over time. Having reached this point it is important to recall that bivariate Gini indices only gives summarized information about the evolution of well-being distribution, thus being necessary bidimensional Lorenz curves to study if this conclusion can be extrapolated for the whole distribution. Figures in Table 4 shows bidimensional Lorenz curves for the three studied bidimensional distributions in 198 and 21. The decrease of inequality is also observed from figure in Table 4, since the curve in 21 lies above the curve in 198 almost completely. However, it is not possible to conclude that well-being distribution in 21 Lorenz dominates well-being distribution in 198 given that Lorenz curves in 21 lies below its analogous in 198 at the bottom of the distribution in all cases. Therefore, poorest, least educated and least healthy countries currently have more unequal distribution than 3 years ago in terms of well-being, whereas the rest of nations enjoy comparatively lower levels of well-being 26

27 inequality than in 198. Independently of the previous dynamics, the most relevant fact that should be noted is that these patterns cannot be concluded using inequality measures even the multidimensional ones, thus pointing out that multidimensional Lorenz curves are essential to analyse the internal dynamics of well-being distribution. 8 Conclusions In this paper, using the definition proposed by Arnold (1983), we have obtained closed expressions for the bivariate Lorenz curve, assuming different formulations of the underlying bivariate income distribution. We have studied a relevant type of models based on the class of bivariate distributions with given marginals described by Sarmanov and Lee (Lee, 1996; Sarmanov, 1966). The expression of the bivariate Lorenz curve can be easily interpreted as a convex linear combination of products of classical and concentrated Lorenz curves and we have obtained a closed expression for the bivariate Gini index (Arnold, 1987) in terms of the classical and concentrated Gini indices of the marginal distributions. This index can be decomposed in two factors, corresponding to the equality within and between variables. Several models with Beta, GB1, Pareto, Gamma and lognormal marginal distributions are studied and other alternative families of bivariate Lorenz curves are discussed. Some concepts of stochastic dominance are explored and extensions of the bivariate Lorenz curve to higher dimensions are included. The methodology developed in this paper has been used to study the evolution of global multidimensional inequality in well-being. Our results point out that bidimensional inequality has been reduced in all of the relationships considered. Our application also reveals that the consideration of the correlation degree among dimensions plays a crucial role in the determination well-being inequality. However, inequality measures only offer summarised information of the evolution well-being distribution, thus some internal dynamics can be masked. Concretely, it has been concluded that, in spite of the decrease of the bivariate Gini index, poorest, least educated and least healthy countries have more unequal distribution than 3 years ago. Therefore, multidimensional Lorenz curves are essential to analyse the internal dynamics of well-being distribution. 27