This paper provides an elementary characterization of a family of multi-dimensional development indices that allows introducing distributive considerations. It consists of the generalized mean of the egalitarian equivalent values of the different dimensions. The key property that defines that family of indices is that of separability.multidimensional well-being, distribution sensitive indices, separability, human development.

Antonio Villar

English

Research Papers in Economics

 
 
    Working papers series 
   Department of Economics 
 
 
 
WP ECON 09.14 
 
 
Distribution Sensitive Multidimensional 
Development Indices 
 
 
 
Antonio Villar (U. Pablo de Olavide & IVIE) 
 
 
 
 
 
JEL Classification numbers: D63, I31, O15 
 
Keywords: multidimensional well-being, distribution sensitive indices, 
separability, human development  
 
 
 
 
http://www.upo.es/econ 
 
Distribution Sensitive Multidimensional 
Development Indices 
 
 
Antonio Villar 
 
Universidad Pablo de Olavide & Ivie 
 
 
 
 
 
 
 
Abstract 
 
  This paper provides an elementary characterization of a family of multi-
dimensional development indices that allows introducing distributive 
considerations. It consists of the generalized mean of the egalitarian equivalent 
values of the different dimensions. The key property that defines that family of 
indices is that of separability.  
 
 
 
Key words: multidimensional well-being, distribution sensitive indices, 
separability, human development. 
 
 JEL Classification Numbers: D63, I31, O15 
 
 
 
Acknowledgements: Thanks are due to Ricardo Martínez for helpful 
discussions. Financial support from Ministerio de Educación y Ciencia, under 
project SEJ2007-67734/ECON and Junta de Andalucía, under project 
SEJ2905, is gratefully acknowledged. 
     
 
Address for correspondence: Department of Economics, Pablo de Olavide 
University, Ctra. Utrera km. 1, 41013, Seville, Spain. 
  
E-mail: avillar@upo.es 
 
  
 
 
 
 
http://www.upo.es/econ 
 
1. Introduction 
 
The need of multidimensional indicators for the assessment of economic 
development is already well established. The recent report by Stiglitz, Sen and 
Fitoussi (2009) is one of the last attempts to transform such a need into an 
institutional commitment that should lead to a change in our national accounting 
systems. That report discusses the limits of the standard growth indicators and 
suggests some sensible ways of introducing additional variables that capture 
the relevant dimensions of economic development.  
Once those relevant dimensions have been identified and the variables that 
approximate them agreed upon, there is always the problem of how to 
aggregate this information into a single indicator. This paper aims at 
contributing to that discussion by providing a characterization of a well defined 
family of indices: the generalized means. Our approach responds to two main 
concerns. One is that of building indicators that are distribution sensitive, as we 
believe distributive considerations are part of the basic features of the economic 
performance. The other is to provide a theoretical support to those indicators in 
a relatively simple way.  
Let us take as reference the Human Development Index (HDI), perhaps the 
most popular multidimensional socio-economic indicator. It consists of the 
average of three normalized variables that approach the social achievements in 
health, education and material well-being (see UNDP (2009)). In spite of its 
popularity, the HDI has a number of well known shortcomings (e.g. Sagar & 
Najam  (1999)), among which there is the lack of theoretical justification of the 
additive structure and the absence of distributive considerations. Hicks (1997), 
Chakravarty (2003), Foster, López-Calva & Székely (2005), and Herrero, 
Martínez & Villar (2009), among others, are contributions that propose some 
improvements of the HDI on those respects and are closely related to our aim.  
Hicks (1997) and Foster, López-Calva & Székely (2005) introduce 
distributive considerations into the HDI, following a constructive approach 
(incorporating the Gini index on the additive structure, in the first case, and in 
terms of a generalized mean in the second one). Chakravarty (2003) provides a 
simple characterization of the additive aggregation formula when average  
 
 
 
 
http://www.upo.es/econ 
 
values (rather than individual values) are taken as the inputs of the social 
evaluation index. Finally, Herrero, Martínez & Villar (2009) characterize a 
multiplicative version of the HDI that is flexible enough to admit incorporating 
distributive considerations. 
We provide here a unified treatment of all those indices following an 
axiomatic approach. Extending the notion of additive separability, we 
characterize a family of development indices that corresponds to the 
generalized mean of order α  of the egalitarian equivalent values of the different 
dimensions considered. We call this property separability of degree  α (see 
below). This result can be regarded as giving support to the proposal in Foster, 
López-Calva & Székely (2005). Yet it is more flexible concerning distributive 
aspects. It includes as particular cases the standard HDI (the arithmetic mean) 
or the multiplicative index in Herrero, Martínez & Villar (2009) (the geometric 
mean), among others.    
  
 
2.  The model 
 
Let N = {1,2,…,n} denote a society consisting of n individuals and  K = 
{1,2,…,k}  a set of characteristics. Each characteristic corresponds to a variable 
that approximates one relevant dimension of social development. A social 
state is a matrix Y with n rows (one for each individual) and k columns (one for 
each characteristic). The element   [, 1 ] ij y c ∈  of matrix Y describes the value of 
the variable j for individual i, where c > 0 is an arbitrarily small scalar. That is, 
we assume that the values of each of those characteristics are already 
normalized and bounded above from zero.
1 Therefore,  [, 1 ]
nk c Ω=   is the space 
of admissible social state matrices. We denote by Y* the matrix all whose 
                                                 
1 We can think that the original variable is 
min max [, ] ij j j zz z ++ ∈⊂ R  and then simply define 
max
ij
ij
j
z
y
z
=  , with 
min
max min
j
j
j
z
c
z
⎧⎫ ⎪⎪ = ⎨⎬
⎪⎪ ⎩⎭
.   
 
 
 
 
http://www.upo.es/econ 
 
elements are equal to 1. A bold letter   j y   indicates the jth column of matrix Y. It 
describes the distribution of the jth characteristic in the population. The vector 
() n j 1  corresponds to the jth column of matrix Y*. We denote by  j Y−  the 
(1 ) nk ×−   matrix obtained from Y  by deleting its j-th column. We can therefore 
write  (,) j j YY − = y , in the understanding that  j y  actually occupies the j-th 
position in the array of columns. 
 
A  Social Evaluation Index is a continuous single-valued mapping   
     I : Ω→R  that provides a numerical evaluation of social states.
2 
For a given evaluation index I and a given social state matrix Y, we 
define the egalitarian equivalent value of the jth characteristic as the number 
(,) ( 0 , 1 ] jj Y ξ − ∈ y  such that  () () , ()( , ) jn j j IY IY j Y ξ −− = 1 y . When  (,) j j Y ξ − y  does not 
depend on   j Y−  we shall simply write  () j ξ y . 
We first introduce two basic requirements on the social evaluation index: 
neutrality and normalization. Neutrality makes it explicit that all characteristics 
enter the evaluation function on an equal foot. That can be formalized by 
requiring that a permutation of the characteristics does not affect the social 
evaluation (recall that all variables vary in the interval [c, 1], so that their mean 
differences have already been neutralized). Normalization fixes the scale of the 
index. It requires that when the matrix is uniform (i.e. all entries are identical), 
the index takes on the very same value. Formally:   
•  Neutrality. For each Y ∈Ω , if   () C Y π  denotes a permutation of the 
columns of Y, then:  (( ) ) ( ) C I YI Y π = .  
•  Normalization. Let  [ ] ( ) (1),..., ( ) , nn Yq q k = 11  for some  [, 1 ] qc ∈ . Then,   
(() ) I Yq q = .  
                                                 
2 Note that we introduce the requirement of continuity in the very definition of the index. That is, 
we focus on those mappings for which small changes in the variables imply small changes in 
the index.  
 
 
 
 
http://www.upo.es/econ 
 
 
We now establish conditions on the behaviour of the index when the 
social state matrix changes. Suppose that the jth column of matrix Y changes 
from  j y  to  'jj =+ yy a, for some 
n ∈ a R . We say that index I is additively 
separable if there exists 
     
sj :R
2n → R such that:  
     
I(Y− j,y j +a) = I(Y)+ sj(y j,a) 
This property says that when the change in the jth column of social state matrix 
Y is the result of an additive composition, then its social evaluation index also 
corresponds to an additive composition of the original index and a real-valued 
function that depends on the change experienced in  j y  (more specifically, it 
depends on the particular characteristic –hence the subindex in the function, 
the original value   j y , and the perturbation term 
n ∈ a R ).
3
 
The notion of separability of degree α extends this idea as follows:  
•  Separability of degree α. Let  ,' YY∈Ω be such that  '( , ) jj YY − =+ y a , for 
some  admissible 
n ∈ a R ,  and let α∈R be given. Then,  
[] [ ] (' ) () ( ,) jj IY IY s
α αα ⎡ ⎤ =+ ⎣ ⎦ y a . 
 
According to this notion, the index associated with a matrix that results from 
an additive change in its jth column is an additive transformation of order α of 
the original index and a function that depends on the original value and the 
perturbation term a . The coefficient α parameterizes impact of the change on 
index and is related to the elasticity of substitution of the different characteristics 
(see below for a discussion).  
 
                                                 
3 This notion is related to that of “consistency in aggregation” introduced in Chakravarty (2003). 
Note that the very definition of a social evaluation index implies that all tose functions sj are to  
be continuous.  
 
 
 
 
http://www.upo.es/econ 
 
 
The following result is obtained: 
 
 
 Theorem 1: A social evaluation index I satisfies neutrality, normalization and 
separablity of degree α, if an only if it corresponds to the generalized mean of 
order α ∈R of the egalitarian equivalent values. That is: 
1
1/
1
() , 0
()
() , 0
j
jK
k
j jK
k IY
α α
ξα
ξα
∈
∈
⎧
⎡⎤ ⎪ ⎡⎤ ≠ ⎢⎥ ⎪ ⎣⎦ = ⎨⎣⎦
⎪
⎡⎤ = ⎪ ⎣⎦ ⎩
∑
∏
y
y
              [1] 
Morever, those properties are independent. 
 
Proof.- 
Let Y ∈Ω and   0 α ≠  be given. By separability of degree α we can write:  
() ()
**
11 1 1 (,) () ( 1 ) , ( 1 ) nn IY IY s
α αα
− ⎡ ⎤ ⎡⎤ ⎡ ⎤ =+ − ⎣⎦ ⎣ ⎦ ⎣ ⎦ y1 y 1  
Define now:  () () 1 () : ( ) , ( ) jj n j n ts j j =− y1 y 1 . Then,  
[]
[] [ ] [ ]
[]
**
11 11
*
12 22 11
*
(,) () )
( , , (3),..., ( )) ( ) ( ) )
....
() ( ) ( )
nn
jj jK
IY IY t
Ik t I Y t
IY IY t
αα α
α αα α
α α α
−
∈
⎡⎤ ⎡ ⎤ =+ ⎣⎦ ⎣ ⎦
⎡⎤ =++ ⎣⎦
⎡⎤ ⎡⎤ =+ ⎣⎦ ⎣⎦ ∑
y( y
yy1 1 y ( y
y
 
 
By neutrality,  (.) (.) j tt =  for all j. That is, separability of degree α and 
neutrality imply: 
     
I(Y) = I(Y
*) ⎡ ⎣ ⎤ ⎦
α
+ t(y j) ⎡ ⎣ ⎤ ⎦
α
j∈K ∑
⎛
⎝
⎞
⎠
1/α
. 
For the special case in which   jn q y =1   for all j, by normalization we get: 
[] () 1 () n I qY k t q q
α α α ∗ ⎡⎤ =+ = ⎣⎦ 1  
which implies  []
1
() n
q
tq
k
α
α −
= 1    
 
 
 
 
http://www.upo.es/econ 
 
Now observe that, for each  [, 1 ]
n
j c ∈ y , there exists a scalar  () j ξ y such 
that 
     
t(y j) = t(1nξ(y j)) (by normalization this is true in particular for 
     
ξ(y j) = t(y j)). 
Therefore, we conclude,  
1
1
() 1 1
() 1 ( )
j
j
jK jK
IY
kk
α α
α α ξ
ξ
∈∈
⎡⎤ ⎛⎞ ⎡⎤ − ⎡ ⎤ ⎣⎦ ⎢⎥ ⎜⎟ ⎡⎤ =+ = ⎢ ⎥ ⎣⎦ ⎢⎥ ⎜⎟ ⎣ ⎦ ⎝⎠ ⎣⎦
∑∑
y
y
 
 
For the case 0 α =  we substitute the value of the function, which is not 
defined, by the value of its limit as  0 α → . The standard procedure (L’Hôpital 
rule) gives us the desired result. 
 
Consider now the following indices:  
(1) 
1/
()
nk
ij
iNjK
IY y
∈∈
⎡⎤
= ⎢⎥
⎣⎦ ∏∏ .  
(2)
1
() ( ) j
jK
IY
α α
ξ
∈
⎡⎤
⎡⎤ = ⎢⎥ ⎣⎦
⎣⎦ ∑ y .  
(3) 
1
1
() ( ) jj
jK
IY
k
α α
βξ
∈
⎡⎤
⎡⎤ = ⎢⎥ ⎣⎦
⎣⎦ ∑ y  with   1 j jKβ
∈ = ∑  and  1/ j k β ≠ , for some j.  
Index (1) satisfies neutrality and normalization but not α-consistency. 
Index (2) satisfies neutrality and α-consistency but not normalization. Index (3) 
satisfies α-consistency and normalization but not neutrality. 
                                   Q.e.d. 
Remark 1.- The restriction  0 ij y >  is introduced in order to avoid the 
indeterminacy of the formula for negative values of the parameter α.  
 
 
  
 
 
 
 
http://www.upo.es/econ 
 
 
 
3.  Discussion  
 
Theorem 1 above identifies a family of indices that permits measuring 
development in a multidimensional context, in terms of the generalized mean of 
order α of the corresponding egalitarian equivalent values. The properties of the 
generalized means are well known: they are homogeneous of degree one, 
monotone in ξ, increasing in α, concave for  1 α ≤  and convex for  1 α ≥ .  
There are some specific values of the parameter α worth considering. 
For  1 α =  we get additive separability and the formula yields the arithmetic 
mean (the type of the HDI). By taking limits, as α → 0, we obtain the geometric 
mean. For  1 α =−  we get the harmonic mean (used for the analysis of gender 
differences in the Human Development Reports). Finally, taking limits when 
α →−∞ we get the leximin criterion:  { } () m i n jK j IY ξ ∈ = .    
The parameter α may thus be regarded as controlling the degree of 
substitutability between the different dimensions (equation 1 is actually a 
symmetric CES function). Or, put differently, the parameter α measures our 
concern for equality across dimensions, with lower values of α corresponding to 
higher concern and viceversa. Additive separability implies constant rates of 
substitution (linear indifference curves) whereas separability of degree zero 
implies decreasing rates of substitution (a standard Cobb-Douglas symmetric 
function). In the limit, separability of degree α →− ∞ implies full 
complementarity (Leontief indifference curves).  
Indeed, we can singularize the leximin criterion as the only member of 
this family that satisfies “minimal lower boundedness” (Bossert and Peters, 
(2000), Herrero, Martínez & Villar (2009)). That property says that there is no 
trade-off between dimensions when all members of the society are at their worst 
level in some of them. Formally: for all Y ∈Ω, all  j K ∈ ,  I(Y− j,c1n( j)) ≥ I(Y).   
 
 
 
 
http://www.upo.es/econ 
 
The following result is obtained: 
 
Corollary:  A social evaluation index satisfies neutrality, normalization α-
consistency, and minimal lower boundedness if an only if it corresponds to the 
leximin criterion. That is: 
{ } () m i n jK j IY ξ ∈ =                
 
(The proof is trivial and thus omitted)  
 
Following the theory of income inequality measurement [see for instance 
Cowell (1995), Sen & Foster (1997), Goerlich & Villar (2009)] we can identify 
the egalitarian equivalent value of a characteristic,  () j ξ y , with its mean value 
deflated by some inequality measure, that is:  
() () 1 () jj j f ξμ ⎡ ⎤ =− ⎣ ⎦ yy y                       [2] 
where  () j f y  is an inequality index (e.g. the Gini index or a member of the 
entropy  or the Atkinson family). For that we need to assume that the inequality 
index satisfies anonymity (a permutation of the rows does not change the value 
of the index) and quasi-concavity (redistribution improves the value of the 
index).
4  
Note that a generalized mean of order α corresponds to Atkinson’s 
egalitarian equivalent value for  1 αε =−. Therefore, one could choose precisely 
this notion of egalitarian equivalent and obtain a formula that gives us the 
geometric mean of order α of the geometric mean of order α of each dimension. 
This is precisely the proposal in Foster, López-Calva & Székely (2005). Yet one 
may consider that inequality among people with respect to some dimension 
(e.g. income) may be treated differently that inequality across dimensions. In 
that respect our formulation is flexible enough to allow for a different treatment 
of people and dimensions. Be as it may, observe that the neutrality property 
                                                 
4  If we also impose monotonicity (larger values of the variables imply larger values of the 
function), then the inequality index will have to vary into the interval [0,1].  
  
 
 
 
 
http://www.upo.es/econ 
 
prevents us from using different definitions of the egalitarian equivalent values 
for different characteristics.  
  
 
References 
 
Chakravarty, S.R. (2003), A Generalized Human Development Index, Review 
of Development Economics, 7 : 99-114. 
Cowell, F. A, (1995), Measuring Inequality, 2
nd Ed., Harvester Wheatsheaf, 
Hemel Hempstead. 
Foster, J.E., López-Calva, L.F. and Székely, M. (2005), Measuring the 
Distribution of Human Development: Methodology and an Application to 
Mexico, Journal of Human Development and Capabilities, 6 : 5-29. 
Goerlich, F.J. & Villar, A. (2009), Desigualdad y Bienestar Social. De la 
Teoría a la Práctica. Fundación BBVA, Madrid.  
Herrero, C., Martínez, R. & Villar, A. (2009), Multidimensional Social Evalutaion. 
An Application to the Measurement of Human Development, Review of 
Income and Wealth, forthcoming. 
Hicks, D.A. (1997), The Inequality-Adjusted Human Development Index: A 
Constructive Proposal, World Development, 25 : 1283-1298. 
Sagar, A.D. & Najam, A. (1999), The Human Development Index: A Critical 
Review, Ecological Economics, 25 : 249-264. 
Sen, A.K. & Foster, J.E. (1997), On Economic Inequality, 2
nd Ed., Oxford 
University Press, Oxford. 
Stiglitz, J.E., Sen, A. & Fitoussi, J.P. (2009), Report by the Commission on the 
Measurement of Economic Performance and Social Progress, European 
Commission, Beyond GDP. 
United Nations Development Program (2009), Human Development Report 
2009, Palgrave Macmillan, New York. 

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Human Development Report

Measuring Inequality, 2 nd Ed., Harvester Wheatsheaf,

Measuring the Distribution of Human Development: Methodology and an Application to Mexico,

Multidimensional Social Evalutaion. An Application to the Measurement of Human Development, Review of Income and Wealth,

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The Inequality-Adjusted Human Development Index: A Constructive Proposal,

Distribution Sensitive Multidimensional Development Indices

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Distribution Sensitive Multidimensional Development Indices

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