Freedom, Economic Resources and Wellbeing: A Stochastic
Dominance Approach∗
PRELIMINARY DRAFT
Please do not cite or circulate without authorization
Paul Makdissi†
University of Ottawa
University of Queensland
Myra Yazbeck‡
University of Queensland
Daouda Sylla§
University of Ottawa
July 2013
Abstract
In this paper we reconcile research on wellbeing measurement with Sen’s freedom to achieve
approach. In doing so we develop procedural ethical principles and the associated stochastic
dominance conditions as well as capability ethical principles and the associated stochastic dominance conditions. To provide an empirical illustration, we use information on women’s’ individual
freedom available in Demographic Health Surveys (DHS).
Keywords: Freedom, Wellbeing, Stochastic Dominance
JEL Codes: D63, I31
∗
Part of this paper was written while the first author was visiting the School of Economics of the University of
Queensland.
†
CIRPÉE and Department of Economics, University of Ottawa, 55 Laurier E. (10125), Ottawa, Ontario, Canada,
K1N 6N5; Email: [email protected]
‡
School of Economics, 612- Colin Clark (39) Building, University of Queensland, Brisbane, QLD 4072, Australia;
Email: [email protected]
§
Department of Economics, University of Ottawa, 55 Laurier E. (10125), Ottawa, Ontario, Canada, K1N 6N5;
Email: [email protected]
1
Introduction
Freedom is a a key concept in Sen’s capability approach in which the evaluation of well-being focuses
on the space of capabilities (available achievement) rather of the space of functionings (actual
achievement) (Sen, 1985).1 When an individual has access to the most extensive set of freedom,
then the actual achievement observed is a good indicator of available achievement.2 However,
when access to the set of freedom is limited, the actual achievement does not reflect the available
achievement. While the notion of freedom is ambiguous and plural, Sen argues that it is concerned
with the ”opportunity” to achieve those things that a person values (i.e., economic freedom), and
that another aspect of freedom (i.e., the process aspect) is concerned with process by which you
achieve (i.e., “whether the person was free to choose herself, whether others intruded or obstructed,
and so on”, Sen, 2002, p. 10).
Most of the wellbeing measurement literature focuses is based on the comparison of actual
achievement. In this paper we focus on both aspects of wellbeing: the actual achievement and
freedom to achieve. The capabilities approach to assess freedom can be used at various levels of
sophistication yet, Sen (1999) admits that practicality and data limitations often force the researcher
to confine her analysis to the achieved bundle only. This being said, one must always be aware of this
substantial drawback and should differentiate between what is acceptable because of feasibility and
what is the right procedure. This paper is motivated by the importance of freedom in development
and is concerned with the problems encountered when applying the capabilities approach to freedom.
Its objective is to provide a first attempt at measuring freedom using Sen’s capabilities approach
by exploiting available information on individual freedom available in DHS data. While much of
the theoretical work starts from the theory then goes to the data, this paper’s research question
is motivated by the availability of data, previously considered unavailable, and seeks a formal
1
2
Capabilities are defined as the set of all social functionings available to an individual.
This assumes that the individual has chosen her most preferred functionings in her capabilities set.
1
axiomatic framework that allows researchers to exploit this information. In this perspective, the
contribution of this paper is twofold. The first contribution is at the general conceptual level,
it resides in its focus on the ”freedom to achieve” (i.e., process freedom) based on ”functionings”.
This contribution reconciles research on freedom based on actual achievement with Sen’s perspective
regarding how such research should be conducted (i.e., taking into account freedom to achieve). Thus
by using information available in DHS questionnaires, this work addresses the following argument
made by Sen: ‘information concerning these functionings has to be sought from both non-market
direct observations and from questionnaires” (Sen, 1985, p.47). The second contribution is at the
measurement level, it relates to the measurement of multidimensional wellbeing. The measurement
of multidimensional wellbeing can be divided in two broad strands. The first is concerned with
the development of multidimensional indices following Bourguignon and Chakravarty (1999, 2003),
Atkinson (2003), Alkire and Foster (2011). The second focusses on the development of stochastic
dominance tests following Duclos, Sahn and Younger (2006, 2011). This paper contributes to this
second strand of the literature since we adopt a stochastic dominance approach to use available
information on freedom and assess the extent of individuals’ freedom and its impact on wellbeing.
Given that the available information on freedom is discrete in nature, the technicalities of this
paper is related to the sequential stochastic dominance literature that developed tests for household
of heterogenous sizes [See, Atkinson and Bourguignon (1982, 1987), Atkinson (1992), Jenkins and
Lambert (1993), Chambaz and Maurin (1998) and Duclos and Makdissi (2005)]. Moyes (2012)
reinterprets this literature and replaces “household size by “ability” and points that some level of
cardinality may be imposed on ability. This allows him to introduce inequality aversion in the
dimension of ability. This paper builds on Moyes (2012) and considers sets of freedoms instead
of “ability”. It also develops higher order ethical principles in the dimension of freedoms. We
introduce the concept of procedural dominance in the space of freedoms and develop the associated
stochastic dominance tests. As a result we reinterpret standard unidimensional income (economic
2
resources in our paper) stochastic dominance as “consequential” dominance since it focusses on
actual achievements. Finally, we consider a capability dominance where we use the information
on both economic resources (consequence) and freedoms (process) to assess the wellbeing of the
individuals. We proposes some desirable ethical principles and develop the tests that identify robust
partial orderings. This paper is organized as follows. Section 2 provides the theoretical framework.
Section 3 presents consequential ethical principle and dominance. Section 4 presents procedural
ethical principles and dominance. Section 5 presents capability ethical principle and dominance.
Section 6 provides an empirical illustration on the evolution of women wellbeing in Sub Sharan
Africa.
2
Theoretical Framework
Consider a population P := {1, 2, . . . , n} of n individuals. Each individual is described by two
attributes: economic resources (income, wealth, ...) and a set of freedoms. We assume that there
exist a finite number of sets of freedoms K, where 2 ≤ K ≤ n. We denote the collection of all
possible sets of freedoms by L := {L1 , L2 , . . . , LK }, where Li represents a set of freedoms. We
assume that the sets of freedom are ordered such as L1 ⊆ L2 ⊆ . . . ⊆ LK 3 . In this framework, LK
may be interpreted as the most extensive set of freedoms compatible with a similar set for others4 .
Individual i’s economic resources are represented by xi ∈ D := [x, x] ⊂ <. Individual i’s set of
freedoms is represented by li ∈ L .
The standard practice in the literature on the measurement of wellbeing is to evaluate social
wellbeing of two different populations by comparing their distribution of economic resources5 . This
approach is purely consequential since it focusses only on individual socioeconomic achievements.
We assume that each individual uses her economic resources within the limit of her freedoms to
3
We will discuss in Section 5 how this assumption can be relaxed.
Note that Rawls’ (1971) first principle of justice requires that all individuals have access to LK
5
An equivalence scale and regional deflators may be applied in order to account for differences in household sizes,
regional price differentials, handicap, etc.
4
3
achieve some social functionings. Sen (1992) defines the capability of an individual as the various
functionings that this person can achieve. The capability of a person reflects her freedom to lead
one type of life or another. More precisely, Sen (1992) argues that “a person’s position in a social
arrangement can be judged in two different perspectives, viz. (1) the actual achievement6 , and (2)
the freedom to achieve” (Sen, 1992, p.31). We take this last perspective and make the assumption
that individual i’s freedom to achieve is determined jointly by her economic resources and her set of
freedoms. Formally, we assume that the distribution of capabilities in population P is represented
by the joint distribution of economic resources and freedoms:


x1 l1
 ..
.. 
 .
. 



c ≡ [x||l] :=  xi li 
.
 ..
.. 
 .
. 
xn ln
(1)
Each row c represents the capability of an individual i that is described by the value of her economic
resources, xi , and her set of freedoms, li . We denote by
C n := {c ≡ (x;l)|x ∈ D n ∧ l ∈ L n , }
(2)
the set of joint distributions of economic resources and freedoms for a population of size n. Since
we may have to compare populations of different sizes, we make the assumption that c is drawn
from:
C :=
∞
[
C n.
(3)
n=1
Taking into account freedom to achieve in wellbeing assessment is crucial. In our framework,
agents differ in their economic resources and in their sets of freedoms. They are homogenous
otherwise7 . Assume first that there is no differences in freedoms. If individual i’s actual achievement
6
Actual achievement can be in measured in the space of utility, primary goods (Rawls, 1971), resources (Dworkin,
1981), social functionings, etc.
7
We can assume that an equivalent scale has been applied on the value of economic resources to take into account
differences in family sizes, handicap, etc.
4
is a vector of her actual social functionings, fi , it makes sense to assume that her capability set was
the set of all other social functionings f that can be achieved using the same economic resources.
Formally, assume that choosing a vector of social functionings fi implies a cost v(fi ) ≤ xi . The
capability set S (ci ) of this individual is then define as:
S (ci ) := {f |v(f ) ≤ xi }.
(4)
In this context, using economic resources xi as an indicator of wellbeing makes sense since it directly
identifies the capability set of individual i. In our context, this will describe the situation of
a population P whose all individual members have access to the freedom set LK . Let us now
introduce heterogeneity in freedoms. Let T (Lj ) represents the sets of social functionings that are
not allowed or impossible to achieve under the set of freedoms Lj . The capability set of an individual
endowed with ci = (xi , li ) is now defined as
S (ci ) := {f |v(f ) ≤ xi ∧ f 6∈ T (li )}.
(5)
From equation (5), it appears that identifying the capability set of individual i requires both xi and
li .
We assume that the evaluation of social wellbeing is made by a a sympathetic ethical observer
who wishes to evaluate achievement in the social functionings space. This sympathetic ethical
observer relies on a social wellbeing measure to assess and compare the wellbeing of two different
populations P 0 and P 1 which may be of different sizes.
Definition 1 A social wellbeing measure is a function W (c) : C → < whose value indicates de
level of social wellbeing of a population.
In this context, for a given social wellbeing measure, W (c), each distribution of capabilities is
assigned a real number that is called a social wellbeing index. Any choice of social wellbeing
measure embodies a set of value judgements on the “quality” of a distribution of capabilities. These
5
value judgements are described by a set of axioms that are imposed on the wellbeing measure.
Since we do not axiomatize a particular class of indices and to avoid confusion, we will called
these axioms “ethical principles” hereafter. At this point, we shall introduce two widely recognized
ethical principles that should be imposed on any social wellbeing measure. Assume that we have
two distributions of capabilities c0 ∈ C n0 and c1 ∈ C n1 .
Definition 2 c1 ∈ C n1 is obtained from c0 ∈ C n0 by a (k-) replication if, for some positive integer
k, n1 = k · n0 and



c1 = 

Ethical Principle 1 Replication Invariance:
c0
c0
..
.
c0



.

(6)
W (c1 ) = W (c0 ) whenever c1 is obtained from
c0 by a (k-) replication.
This ethical principle was first introduced by Chakravarty (1983) and Thon (1983) in the income
inequality literature. This ethical principle allows for the comparison of distributions of different
sizes since any two different sized distributions can be replicated to the same size.
Definition 3 c1 ∈ C n is obtained from c0 ∈ C n by a permutation if c1 = Πn×n · c0 , where Πn×n
is a permutation matrix.8
Ethical Principle 2 Symmetry: W (c1 ) = W (c0 ) whenever c1 is obtained from c0 by a permutation.
This ethical principle implies that the name of the recipient of a capability vector ci = (xi , li ) does
not matter in the measurement of wellbeing. In empirical applications, this ethical principle does
not impose any real restriction since the identity of a recipient is not disclosed in cross sectional
data sets.
8
A permutation matrix has elements of 0 and 1 only and each row and column sums up to one.
6
As in Atkinson (1970), we assume that the sympathetic ethical observer relies on the average level
of wellbeing to compare the distribution of wellbeing between different societies. This assumption
implies that Ethical Principle 1 is obeyed. It is important to point that our approach differs
from Atkinson’s approach in one crucial aspect. Atkinson (1970) average utilitarian social welfare
function is purely consequential. It evaluates wellbeing of individual i as being solely determined by
her achievement in economic resources, xi . In this paper, we adopt Sen’s (1992) capability approach
to assess social wellbeing. Formally, we assume that the sympathetic ethical observer assess the
wellbeing of individual i using a individual wellbeing function ω(xi , li ) that may depend on both the
individual’s economic resources and set of freedoms. Using the same individual wellbeing measure
for each observation insures that the social wellbeing measure obeys Ethical Principle 2. Since
this sympathetic ethical observer relies on average wellbeing to assess social wellbeing W (c; ω) in
population P, we get
n
1X
W (c; ω) :=
ω(xi , li )
n
(7)
i=0
Once the sympathetic ethical observer has chosen her individual wellbeing function ω, a complete
ordering of the different distributions is obtained. In other words:
∀c0 , c1 ∈ C : c0 ω c1 ⇔ W (c0 ; ω) ≥ W (c1 ; ω)
(8)
In this paper we take a somewhat different approach. We assume that in order to avoid arbitrariness
in the choice of the individual welfare function, the sympathetic ethical observer tries to identify
partial orderings that are unanimously approved by all individual welfare function ω belonging to
a predefine set Ω. In other words:
∀c0 , c1 ∈ C : c0 Ω c1 ⇔ W (c0 ; ω) ≥ W (c1 ; ω) ∀ω ∈ Ω.
(9)
Identifying partial orderings by computing the values of all indices belonging to Ω is an impossible
task since some definitions of Ω may imply an infinite number of comparisons to be performed.
7
The solution to this problem is to develop the stochastic dominance tests that identify these partial
orderings. Stochastic dominance tests compare the structure of F (x, Lj ), the joint cumulative
distribution function associated with c ≡ (x;l) ∈ C without relying on the use of a specific social
wellbeing measure. This joint cumulative distribution indicates the proportion of individuals whose
freedoms set is a subset of Lj and whose income does not exceed x. Formally, F (x, Lj ) is drawn
from:
F := {F : D × L → [0, 1] |F is nondecreasing, right continuous and onto} .
(10)
In this context, social wellbeing in equation (7) may be rewritten as:
Z
ω(x, Lj )dF (x, Lj ).
W (F ; ω) =
(11)
D×L
At this point, it is worth to mention that public policy may alter both economic resources (economic policy) and freedoms (legal framework and public education on social, gender and religious
norms). In this context, it is useful to define four concepts:
1. the marginal distribution of freedoms, F (x, Lj ), indicates the proportion of the population
with a freedom set included in Lj ,
2. the proportion of the population having freedom set Lj is then given by:
F (x, L1 )
if Lj = L1
P (Lj ; F ) =
,
F (x, Lj ) − F (x, Lj−1 ) if Lj ∈ L \ L1
(12)
3. the marginal distribution of economic resources, F (x, LK ), indicates the proportion of the
population with economic resources valued less than x,
4. the conditional distribution of economic resources, F (x|Lj ), indicates the proportion of individuals with economic resources valued less than x among those having a freedom set Lj .
Using these definitions, equation (11) can be usefully rewritten as:
W (F ; ω) =
K
X
Z
x
j=1
8
x
ω(x, Lj )dF (x|Lj ).
P (Lj ; F )
(13)
The stochastic dominance results presented in the next section will also be based on the expression
depicted in equation (13). For analytical tractability, we follow Fishburn (1976) and assume that
the marginal contribution of economic resources to wellbeing vanishes at x. Formally, this implies
that if the level of economic resources is at the upper bound of the domain, adding additional
resources would not increase wellbeing. However, the level of wellbeing at x may still be different
for different sets of freedoms.
Assumption 1 ω1 (x, l) = 0
Essentially, this assumption leads to simpler stochastic dominance tests for order 3. Although it
eases the stochastic dominance tests, equivalent results may be obtained by assuming that the
domain of economic resources D is the set of real numbers <.9
3
Consequential dominance
In this section, we present an overview of the well known standard unidimensional stochastic dominance tests and their associate ethical principles. Since these tests have been developed to assess
orderings of wellbeing in the dimension of economic resources, they focuss on achievements instead
of on freedom to achieve. For this reason, we will term these tests consequential dominance tests.
Since a consequential observer does not value differences in freedoms, the individual wellbeing function, ω(x, l), will be such that ω(x, L1 ) = ω(x, L2 ) = . . . = ω(x, LK ) for all x ∈ D. Let denote by
Ξ the set of all consequential individual wellbeing functions. In this context, equation (11) can be
rewritten as:
Z
W (F ; ω) =
x
ω(x, LK )dF (x, LK ),
∀ω ∈ Ξ.
(14)
x
In addition to Ethical Principles 1 and 2, a consequential analyst may impose more ethical
judgements by imposing additional ethical principles on the social wellbeing measure. An ethical
9
The reader can refer to Fishburn (1980) and Duclos and Makdissi (2004).
9
principle that is widely accepted among consequential analysts is that an increase in the economic
resources of one individual, other things held constant, should be deemed good. Formally,
Definition 4 c1 ∈ C n is obtained from c0 ∈ C n by an increment in economic resources if there
exists an individual i such that x1i > x0i , x1j = x0j ∀j ∈ P \ {i} and l1 ∈ L n .
Ethical Principle 3 Resources Monotonicity: W (c1 , ω) ≥ W (c0 , ω) whenever c1 is obtained
from c0 by an increment in economic resources.
The set of consequential individual wellbeing measures that obey Ethical Principles 1, 2 and 3 will
be denoted by Ω1,0 . Formally,
Ω1,0 := ω ω(·) ∈ C11 ∪ Ξ
∧
ω1 (x, l) ≥ 0
∧
ω1 (x, l) = 0 ,
(15)
where C1s is the set of all continuous function that are s−times differentiable in their first argument
and ωs (x, l) =
∂ s ω(x,l)
∂xs ,
and ω0 (x, l) = ω(x, l).
It is standard practice to assume that the sympathetic ethical observer display inequality aversion. In other words, a transfer of economic resource from a richer to a poorer individual will
be deemed desirable. This ethical principle is known in the income distribution literature as the
Pigou-Dalton Transfer Principle. In the context of this paper, we will term it Resources Transfer
Principle. Formally,
Definition 5 c1 ∈ C n is obtained from c0 ∈ C n by a progressive transfer of economic resources if
there exist two individuals i, j ∈ P such that x0j −x0i > x1i −x0i = x0j −x1j > 0, x1k = x0k ∀k ∈ P \{i, j}
and l1 ∈ L n . Alternatively, we say that c0 is obtained from c1 by a regressive transfer of economic
resources.
Ethical Principle 4 Resources Transfer Principle:
Wc (c1 , ω) ≥ Wc (c0 , ω) whenever c1 is
obtained from c0 by a progressive transfer of economic resources.
10
The set of consequential individual wellbeing function ω(·) that obey Ethical Principles 1, 2, 3 and
4 will be denoted by Ω2,0 ⊂ Ω1,0 . Formally,
Ω2,0 := ω ω(·) ∈ C12 ∪ Ω1,0 ∧
ω2 (x, l) ≤ 0 .
(16)
Kolm (1976) argues that inequality aversion may be more important in the bottom than in the
upper part of the income distribution. To formalize this idea, he introduces an ethical principle
that is based on composite transfers.
Definition 6 c1 ∈ C n is obtained from c0 ∈ C n by a progressive composite transfer of economic
resources if there exist four individuals i, j, k, m ∈ P such that x0j − x0i = x0m − x0k > x1i − x0i =
x1m −x0m = x0j −x1j = x0k −x1k > 0, x0k ≥ x0j , x1p = x0p ∀p ∈ P \{i, j, k, m} and l1 ∈ L n . Alternatively,
we say that c0 is obtained from c1 by a regressive composite transfer of economic resources.
Ethical Principle 5 Resources Transfer Sensitivity:
W (c1 , ω) ≥ W (c0 , ω) whenever c1 is
obtained from c0 by a progressive composite transfer of economic resources.
Ethical Principle 5 implies that the ethical observer prefers progressive transfers that happen in the
bottom of the distribution. This implies that, although deemed negative by the ethical observer, a
regressive transfer may be more than compensated by an equivalent progressive transfer happening
in lower down in the distribution of economic resources10 .
The set of consequential individual wellbeing functions ω(·) that obey Ethical Principles 1, 2, 3,
4 and 5 will be denoted by Ω3,0 ⊂ Ω2,0 ⊂ Ω1,0 . Formally,
Ω3,0 := ω ω(·) ∈ C13 ∪ Ω2,0 ∧
ω3 (x, l) ≥ 0 .
(17)
As argued in the previous section, identifying partial orderings as defined in equation (9) for
all function ω ∈ Ωs,0 is an impossible task since it implies an infinite number of comparisons to
10
Fishburn and Willig (1984) have introduced a class of ethical transfer principles called generalized transfer sensitivity. We do not address these more general ethical transfer principles in this paper. The extension is relatively
straightforward
11
be performed. The literature on income distribution have developed stochastic dominance tests
that identify these orderings. In this paper, we will refer to these tests as consequential stochastic
dominance tests since they focus solely on the distribution of economic resources. In order to present
these tests, it is useful to define following stochastic dominance functions Ds,0 (x, LK ) as
Ds,0 (x, Lj ) :=
LK )
if s = 1
RFx(x, s−1,0
D
(x,
L
)dx
if s ∈ {2, 3}
K
x
(18)
Using these functions, partial orderings can be identified using the following result:
Theorem 1 F1 (x, l) Ωs,0 F0 (x, l), s ∈ {1, 2, 3} if and only if:
D0s,0 (x, LK ) − D1s,0 (x, LK ) ≥ 0
4
∀x ∈ D
(19)
Procedural dominance
In this section, we present the ethical principles and develop the associated stochastic dominance
tests that we term procedural dominance tests. A procedural ethical observer does not consider the
distribution of economic resources at all. Instead, she focusses only on the distribution of freedom
to chose social functionings. Freedom to chose is different from freedom to achieve in the sense that
in order to be free to achieve, an individual needs economic resources in addition to her freedom to
chose. This case will be considered in the next section.
We do not consider freedoms to chose as defined only by the legal institutions of the country.
In addition, we assume that these freedoms can be influenced by the other dimensions of the
institutional context like the social, religious or family norms that the individual has to obey. In
this context, the individual wellbeing function, ω(x, l) will be such that ω(x, Lj ) = ω(x, Lj ) for all
x ∈ D and all Lj ∈ L . Let denote by Π the set of all consequential individual wellbeing functions.
In this context, equation (11) can be rewritten as:
W (F ; ω) =
K
X
P (Lj ; F )ω(x, Lj ),
j=1
12
∀ω ∈ Π.
(20)
Following the lines of the literature on income inequality, it is useful to define procedural ethical
principles. In order to present these ethical principles and their associated stochastic dominance
tests, it is useful to define the following collections of sets:
L (Lk ) :=
L (Lk ) :=
L \ {L1 }
if k = 1
L (Lk−1 ) \ {Lk } if k ∈ {2, 3, . . . , K}
(21)
L \ {LK }
if k = K
L (Lk+1 ) \ {Lk } if k ∈ {1, 2, 3, . . . , K − 1}
(22)
Let us also define ∆t ωs (x, Lj ), an order t change in freedoms sets as
t
∆ ωs (x, Lj ) :=
ωs (x, Lj+1 ) − ωs (x, Lj ) ∀j ∈ {1, . . . , K − 1}
if t = 1
∆t−1 ωs (x, Lj+1 ) − ∆t−1 ωs (x, Lj ) ∀j ∈ {1, . . . , K − 1} if t ∈ {2, 3}
(23)
Let first start with a purely ordinal view on freedoms sets. In this context, the only information
that is deemed valuable is the ranking of wellbeing of different individuals according to their freedoms
set. If an ethical observer takes this standpoint, an ethical principle that is useful to add to Ethical
Principles 1 and 2 is a monotonicity principle. Formally,
Definition 7 c1 ∈ C n is obtained from c0 ∈ C n by an increment in freedoms if there exists an
individual i such that li1 ∈ L (li0 ), lj1 = lj0 ∀j ∈ P \ {i} and x1 ∈ D n .
Ethical Principle 6 Freedoms Monotonicity:
W (c1 , ω) ≥ W (c0 , ω) whenever c1 is obtained
from c0 by an increment in freedoms.
The set of procedural individual wellbeing measures that obey Ethical Principles 1, 2 and 6 will be
denoted by Ω0,1 . Formally,
Ω0,1 := ω ω(·) ∈ Π
∧
∆1 ω(x, Lj ) ≥ 0
∀j ∈ {1, . . . , K − 1} ,
(24)
Building on the literature on income distribution, it is possible to develop higher order procedural
ethical principle similar to the consequential ones. However, it is important to make a comment
before introducing these other ethical principles. If an ethical observer consider that the ranking
13
of freedom sets is purely ordinal, then Ethical Principle 6 is the only ethical principle that will
deemed acceptable by this ethical observer. When introducing higher order procedural ethical
principles, one makes the implicit assumption that the information on the impact of freedoms on
wellbeing is cardinal in nature since in involves making judgments on at least two different changes
in freedoms sets. In addition, improving every individual’s freedom is not subject to the same kind
of constraint than increases in economic resources. Economic resources are limited and increasing
some individual economic resources may imply a reduction of another person economic resources.
Individual freedoms are defined by the institutional framework. It is virtually possible to have a
distribution in which every individual has a freedoms set LK . Human rights activist and many
political philosopher will point to this situation as the only acceptable alternative. In this context,
introducing ethical principles based on transfers of freedoms may be a little hazardous. However,
one have to bear in mind that the objective here is to compare two existing distributions, one of
which may be theoretically seen as been obtained by a series of transfers. If we take this position
and we assume some level of cardinality, making higher order procedural judgements makes some
sense.
A first ethical principle based on transfers of freedoms sets is linked with freedom inequality
aversion. Formally,
Definition 8 c1 ∈ C n is obtained from c0 ∈ C n by a progressive transfer of freedoms if there exist
1 = l0
two individuals i, j ∈ P such that li0 = Lki , li1 = Lki +1 , lj0 = Lkj ∈ L (Lki +1 ), lj1 = Lkj −1 , lm
m
∀m ∈ P \ {i, j} and x1 ∈ D n . Alternatively, we say that c0 is obtained from c1 by a regressive
transfer of freedoms.
Ethical Principle 7 Freedoms Transfer Principle:
W (c1 , ω) ≥ W (c0 , ω) whenever c1 is
obtained from c0 by a progressive transfer of freedoms.
The set of procedural individual wellbeing function ω(·) that obey Ethical Principles 1, 2, 3 and 7
14
will be denoted by Ω0,2 ⊂ Ω0,1 . Formally,
Ω0,2 := ω ω(·) ∈ Ω0,1
∧
∆2 ω(x, Lj ) ≤ 0
∀j ∈ {1, . . . , K − 2} .
(25)
Once the concept of cardinality accepted, we can also increase the order of ethical principles in a
similar way to economic resources and introduce freedoms transfer sensitivity.
Definition 9 c1 ∈ C n is obtained from c0 ∈ C n by a progressive composite transfer of freedoms
if there exist four individuals i, j, m, p ∈ P such that li0 = Lki , li1 = Lki +1 , lj0 = Lkj ∈ L (Lki +1 ),
0 = L
1
0
1
1
0
lj1 = Lkj −1 , lm
km ∈ L (Lki +1 ), lm = Lkm −1 , lp = Lkm +kj −ki , lp = Lkm +kj −ki +1 , lq = lq
∀q ∈ P \ {i, j, m, p} and x1 ∈ D n . Alternatively, we say that c0 is obtained from c1 by a regressive
composite transfer of freedoms.
Ethical Principle 8 Freedoms Transfer Sensitivity:
W (c1 , ω) ≥ W (c0 , ω) whenever c1 is
obtained from c0 by a progressive composite transfer of freedoms.
The set of procedural individual wellbeing function ω(·) that obey Ethical Principles 1, 2, 6, 7 and
8 will be denoted by Ω0,3 ⊂ Ω0,2 . Formally,
Ω0,3 := ω ω(·) ∈ Ω0,2
∧
∆3 ω(x, Lj ) ≥ 0
∀j ∈ {1, . . . , K − 2} .
(26)
An important remark should be made at this point. Since the levels of freedoms (defined by the
freedom sets) are finite, there is a limit on the order of procedural ethical principles even if we ought
to accept some cardinality in the concept. To illustrate this, imagine that K = 2. In this case,
the only ethical principle that can be applied is Ethical Principle 6. In this context, performing
a progressive transfer will imply moving one person from L1 to L2 and the other from L2 to L1 .
Since Ethical Principle 2 implies anonymity in the social evaluation, such a transfer implies that the
index should remain constant. For the same kind of reason, the highest order of procedural ethical
principle applicable is Ethical Principle 7 when K = 3. At least four different sets of freedoms are
needed to introduce Ethical Principle 8.
15
As in the previous section, identifying partial orderings as defined in equation (9) for all function
ω ∈ Ω0,t is an impossible task since it implies an infinite number of comparisons to be performed.
As for the distribution of economic resources, we can develop procedural stochastic dominance tests
that identify these orderings. In order to present these tests, it is useful to define the following
stochastic dominance functions D0,t (x, Lk ) as
0,t
D (x, Lk ) :=
F (x, L )
if t = 1
Pk k 0,t−1
(x, Lj ) if t ∈ {2, 3}
j=1 D
(27)
Using these functions, partial orderings can be identified using the following result:
Theorem 2 F1 (x, l) Ω0,t F0 (x, l), for t ∈ {1, 2} if and only if:
D00,t (x, l) − D10,t (x, l) ≥ 0
∀l ∈ L (LK )
(28)
Theorem 3 F1 (x, l) Ω0,3 F0 (x, l) if and only if:
D00,3 (x, l) − D10,3 (x, l) ≥ 0 ∀l ∈ L (LK−2 )
and,
0,2
D0 (x, LK−1 ) − D10,2 (x, LK−1 ) ≥ 0.
5
(29)
Capability dominance
In this section, the ethical observer evaluates wellbeing in the space of individual capabilities. To
assess an individual’s capability, information is needed on both economic resources and sets of
freedoms. In order to be free to achieve a given social functioning, an individual must have the
freedom to adopt this functioning and the economic resources to finance its cost.
In this section, we adapt the ethical principles of Sections 3 and 4 to this more general framework of capability dominance and add some other ethical principles that describe the interrelations
between the freedoms and economic resources. The widest set of indices may be defined by imposing
conditional monotonicity on both freedoms and economic resources together with an adaptation of
Sen’s Weak Equity Axiom which goes: “Let person i have a lower level of welfare than person j for
16
each level of individual income. Then in distributing a given total of income among n individuals
including i and j, the optimal solution must give i a higher level of income than j.” (see Sen, 1992,
p. 18). In our framework, this would translate in an increase of the welfare index if we perform
a marginal transfer of income from an individual to another one with the same level of economic
resources and a smaller set of freedoms. Let us now define formally this set of indices.
Definition 10 c1 ∈ C n is obtained from c0 ∈ C n by a freedom neutral increment in economic
resources if there exists an individual i ∈ P such that x1i > x0i , x1j = x0j ∀j ∈ P \ {i} and l1 = l0 .
Ethical Principle 9 Conditional Resources Monotonicity: W (c1 , ω) ≥ W (c0 , ω) whenever
c1 is obtained from c0 by a freedom neutral increment in economic resources.
Definition 11 c1 ∈ C n is obtained from c0 ∈ C n by a resource neutral increment in freedoms if
there exists an individual i ∈ P such that li1 ∈ L (li0 ), lj1 = lj0 ∀j ∈ P \ {i} and x1 = x0 .
Ethical Principle 10 Conditional Freedoms Monotonicity: W (c1 , ω) ≥ W (c0 , ω) whenever
c1 is obtained from c0 by a resource neutral increment in freedoms.
Definition 12 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive transfer of resources if there
exist two individuals i, j ∈ P such that x0i = x0j , li0 < lj0 , x1i = x0i + limδ→0 δ, x1j = x0j − limδ→0 δ,
x1k = x0k ∀k ∈ P \ {i, j} and l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak
regressive transfer of resources.
Ethical Principle 11 Weak Equity: W (c1 , ω) ≥ W (c0 , ω) whenever c1 is obtained from c0 by
a weak progressive transfer of resources.
At this point, it is important to note that although Defintion 12 and the associated Ethical Principle
11 are formulated differently from the favorable permutation in Moyes (2012) they are in fact
17
equivalent. The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10 and 11
will be denoted by Ω1,1 . Formally,
Ω1,1 := ω ω(·) ∈ C11
∧
ω1 (x, l) ≥ 0
∧
ω1 (x, l) = 0
∧
∆1 ω(x, Lj ) ≥ 0
∧
∆1 ω1 (x, Lj ) ≤ 0 .
(30)
Inspection of equation (32) indicates that Ω1,0 ⊂ Ω1,1 since ω1 (x, l) ≥ 0, ω1 (x, l) = 0, ∆1 ω(x, Lj ) = 0
and ∆1 ω1 (x, Lj ) = 0 for all ω(·) ∈ Ω1,0 . Similarly, Ω0,1 ⊂ Ω1,1 since ω1 (x, l) = 0, ∆1 ω(x, Lj ) ≥ 0
and ∆1 ω1 (x, Lj ) = 0 for all ω(·) ∈ Ω1,0 . This implies that consequential and procedural individual
wellbeing measures are two extreme particular cases of the more general set of individual wellbeing
measures based on a capability approach.
If the ordering between two distributions is not unanimous for all indices belonging to Ω1,1 , we
may wish to restrict the set of admissible indices. This can be done by imposing additional ethical
principles in the dimension of economic resources, in the dimension of freedoms or in both dimension
simultaneously. When increasing the order of ethical principles in one or both dimension, we must
add at the same time a generalized version of Ethical Principle 11. Each particular case of this
generalization will be explained below. In this setting, Ethical Principle 11 will be referred as the
Generalized Weak Equity Principle of order (1,1).
Let us first restrict the set of admissible indices by adding ethical principles in the dimension
of economic resources. In a first step, we impose inequality aversion in the dimension of economic
resources and add a weak version of transfer sensitivity that compares the impact of progressive
transfers at similar income levels but for different sets of freedoms. Formally,
Definition 13 c1 ∈ C n is obtained from c0 ∈ C n by a freedom neutral progressive transfer of
economic resources if there exist two individuals i, j ∈ P such that x0j − x0i > x1i − x0i = x0j − x1j > 0,
x1k = x0k ∀k ∈ P \ {i, j}, li0 = lj0 and l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a
freedom neutral regressive transfer of economic resources.
18
Ethical Principle 12 Conditional Resources Transfer Principle:
Wc (c1 , ω) ≥ Wc (c0 , ω)
whenever c1 is obtained from c0 by a freedom neutral progressive transfer of economic resources.
Definition 14 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive composite transfer of order
(2,1) of economic resources if there exist four individuals i, j, k, m ∈ P such that x0j = x0m > x0i =
0 ∈ L (l0 ), x0 − x0 > x1 − x0 = x1 − x0 = x0 − x1 = x0 − x1 > 0, x1 = x0
x0k , li0 = lj0 , lk0 = lm
m
m
n
n
i
j
i
i
i
j
j
k
k
∀n ∈ P \ {i, j, k, m} and l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak
regressive composite transfer of order (2,1) of economic resources.
Ethical Principle 13 Generalized Weak Equity of order (2,1):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive composite transfer of order (2,1) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 12 and 13 will
be denoted by Ω2,1 . Formally,
Ω2,1 := ω ω(·) ∈ Ω1,1
∧
ω2 (x, l) ≤ 0
∧
∆1 ω2 (x, Lj ) ≥ 0 .
(31)
We then add transfer sensitivity in the dimension of economic resources together with a weak version
of Fishburn and Willig (1984) fourth order generalized transfer principle that compares the impact
of composite transfers at similar levels of economic resources but for different sets of freedoms.
Formally,
Definition 15 c1 ∈ C n is obtained from c0 ∈ C n by a freedom neutral progressive composite
transfer of economic resources if there exist four individuals i, j, k, m ∈ P such that x0j − x0i =
x0m − x0k > x1i − x0i = x1m − x0m = x0j − x1j = x0k − x1k > 0, x0k ≥ x0j , x1p = x0p ∀p ∈ P \ {i, j, k, m},
0 and l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a freedom neutral
li0 = lj0 = lk0 = lm
regressive composite transfer of economic resources.
19
Ethical Principle 14 Conditional Resources Transfer Sensitivity:
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a freedom neutral progressive composite transfer of economic
resources.
Definition 16 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive generalized transfer of
order (3,1) of economic resources if there exist eight individuals i, j, k, m, p, q, r, u ∈ P such that
0 ,
x0i = x0p , x0j = x0q , x0k = x0r , x0m = x0u , x0m = x0u > x0k = x0r > xj = xq > xi = xp , li0 = lj0 = lk0 = lm
lp0 = lq0 = lr0 = lu0 ∈ L (li0 ), x0j − x0i = x0m − x0k = x0q − x0n = x0u − x0r > x1i − x0i = x1m − x0m = x1q − x0q =
x1r − x0r = x0j − x1j = x0k − x1k = x0p − x1p = x0u − x1u > 0, x1a = x0a ∀a ∈ P \ {i, j, k, m, p, q, r, u} and
l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak regressive generalized transfer
of order (3,1) of economic resources.
Ethical Principle 15 Generalized Weak Equity of order (3,1):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive generalized transfer of order (3,1) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 12, 13, 14 and
15 will be denoted by Ω3,1 . Formally,
Ω3,1 := ω ω(·) ∈ Ω2,1
∧
ω3 (x, l) ≥ 0
∧
∆1 ω3 (x, Lj ) ≤ 0 .
(32)
Let us now restrict the set of indices by adding ethical principles in the dimension of freedoms
instead of economic resources. We proceed in a similar way by introducing inequality aversion
in freedoms together with a different weak version of transfer sensitivity where we compare two
marginal transfers of economic resources between 4 individuals having all the same level of economic
resources but different sets of freedoms. The first transfer is a progressive transfer (from the person
with more freedoms to the person with less freedoms) that occurs between the two individuals with
more restricted sets of freedoms and the second transfer is a regressive transfer (from the person
20
with less freedoms to the person with more freedoms) occurring between the two individuals with
less restricted sets of freedoms. Formally,
Definition 17 c1 ∈ C n is obtained from c0 ∈ C n by a resource neutral progressive transfer of
freedoms if there exist two individuals i, j ∈ P such that li0 = Lki , li1 = Lki +1 , lj0 = Lkj ∈ L (Lki +1 ),
1 = l0 ∀m ∈ P \ {i, j}, x0 = x0 and x1 = x0 . Alternatively, we say that c0 is obtained
lj1 = Lkj −1 , lm
m
i
j
from c1 by a resource neutral regressive transfer of freedoms.
Ethical Principle 16 Conditional Freedoms Transfer Principle:
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a resource neutral progressive transfer of freedoms.
Definition 18 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive composite transfer of
order (1,2) of resources if there exist four individuals i, j, k, m ∈ P such that x0i = x0j = x0k = x0m ,
0
1
0
0 , x1 = x1 = x0 + lim
1
1
li0 < lj0 ≤ lk0 < lm
δ→0 δ, xj = xk = xj − limδ→0 δ, xp = xp ∀p ∈ P \ {i, j, k, m}
m
i
i
and l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak regressive composite transfer
of order (1,2) of resources.
Ethical Principle 17 Generalized Weak Equity of order (1,2):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive composite transfer of order (1,2) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 16 and 17 will
be denoted by Ω1,2 . Formally,
Ω1,2 := ω ω(·) ∈ Ω1,1
∧
∆2 ω(x, Lj ) ≤ 0
∧
∆2 ω1 (x, Lj ) ≥ 0 .
(33)
We then add transfer sensitivity in the dimension of freedoms together with a weak version of
Fishburn and Willig (1984) fourth order generalized transfer principle that compares the impact
of two groups of composite transfers between individuals that all have the same level of economic
resources but different sets of freedoms. Formally,
21
Definition 19 c1 ∈ C n is obtained from c0 ∈ C n by a resource neutral progressive composite
transfer of freedoms if there exist four individuals i, j, m, p ∈ P such that li0 = Lki , li1 = Lki +1 ,
0 = L
1
0
0
lj0 = Lkj ∈ L (Lki +1 ), lj1 = Lkj −1 , lm
km ∈ L (Lki +1 ), lm = Lkm −1 , lp = Lkm +kj −ki , lp =
Lkm +kj −ki +1 , lq1 = lq0 ∀q ∈ P \ {i, j, m, p}, x0i = x0j = x0m = x0p and x1 = x0 . Alternatively, we say
that c0 is obtained from c1 by a resource neutral regressive transfer of freedoms.
Ethical Principle 18 Conditional Freedoms Transfer Sensitivity:
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a resource neutral progressive composite transfer of freedoms.
Definition 20 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive generalized transfer of
order (1,3) of economic resources if there exist eight individuals i, j, k, m, p, q, r, u ∈ P such that
0 ≤ l 0 < l 0 ≤ l 0 < l 0 , x1 = x1 = x1 =
x0i = x0j = x0k = x0m = x0p = x0q = x0r = x0u , li0 < lj0 ≤ lk0 < lm
p
q
r
u
m
q
i
x1r = x0i + limδ→0 δ, x1j = x1k = x1p = x1u = x0j − limδ→0 δ, x1y = x0y ∀n ∈ P \ {i, j, k, m, p, q, r, u} and
l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak regressive generalized transfer
of order (1,3) of economic resources.
Ethical Principle 19 Generalized Weak Equity of order (1,3):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive generalized transfer of order (1,3) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 16, 17, 18 and
19 will be denoted by Ω1,3 . Formally,
Ω1,3 := ω ω(·) ∈ Ω1,2
∧
∆3 ω(x, Lj ) ≥ 0
∧
∆3 ω1 (x, Lj ) ≤ 0 .
(34)
Let us now restrict the set of indices Ω1,1 by adding ethical principles in both dimension simultaneously. A first step will be to impose inequality aversion in the two dimensions together
with their associated generalized weak equity principles (i.e. or order (2,1) and (1,2)). We add to
this another weak version of Fishburn and Willig (1984) fourth order generalized transfer principle
22
that compares four transfers of income between four pairs of individuals. For each pair, we have
an individual with a lower level of income than the other but both have the same sets of freedom.
The two income levels are the same for all pairs of individuals. The first transfer is a progressive
transfer between the two individuals having the most restricted sets of freedoms in this group of
eight individuals. The second and the third transfer are regressive transfers between individuals
having the intermediate sets of freedoms in the group. Finally, the last transfer is a progressive
transfer between the two individuals with the less restricted set of freedoms in the group. Formally,
Definition 21 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive generalized transfer of
order (2,2) of economic resources if there exist eight individuals i, j, k, m, p, q, r, u ∈ P such that
0 ∈ L (l0 ), l0 = l0 ∈ L (l0 ) ∪ {l0 },
x0j = x0m = x0q = x0u > x0i = x0k = x0p = x0r , li0 = lj0 , lk0 = lm
n
q
i
k
k
lr0 = lu0 ∈ L (ln0 ), x0j − x0i > x1i − x0i = x1m − x0m = x1q − x0q = x1r − x0r = x0j − x1j = x0k − x1k = x0p − x1p =
x0u − x1u > 0, x1a = x0a ∀a ∈ P \ {i, j, k, m, p, q, r, u} and l1 = l0 . Alternatively, we say that c0 is
obtained from c1 by a weak regressive generalized transfer of order (2,2) of economic resources.
Ethical Principle 20 Generalized Weak Equity of order (2,2):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive generalized transfer of order (2,2) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 12, 13, 16, 17
and 20 will be denoted by Ω2,2 . Formally,
Ω2,2 := ω ω(·) ∈ Ω2,1 ∩ Ω1,2
∧
∆2 ω2 (x, Lj ) ≤ 0 .
(35)
Let us now restrict the set of indices Ω2,2 by adding transfer sensitivity in the dimension of
economic resources with its associated generalized weak equity principle (order 3,1). We add to
these two principles a weak version of Fishburn and Willig (1984) fifth order generalized transfer
principle that compares eight transfers of economic resources between four groups of four individuals.
23
In each group we have four individuals with four different levels of economic resources. However,
these four levels of economic resources are the same four all four groups. All individuals in a group
have the same sets of freedoms, however, two individuals from two different groups have different
sets of freedoms. We consider that a composite transfer within each group. For the groups with the
most and the least restricted sets of freedoms, the composite transfers will be progressive. For the
two groups with intermediate sets of freedoms, the composite transfers will be regressive. Formally,
Definition 22 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive generalized transfer of order
(3,2) of economic resources if there exist sixteen individuals i, j, k, m, p, q, r, u, y, z, a, b, d, e, g, h ∈ P
such that x0m = x0u = x0b = x0h > x0k = x0r = x0a = x0g ≥ x0j = x0q = x0z = x0e > x0i = x0p = x0y = x0d ,
x0j −x0i = x0m −x0k > x1i −x0i = x1m −x0m = x1q −x0q = x1r −x0r = x1z −x0z = x1a −x0a = x1d −x0d = x1h −x0h =
x0j − x1j = x0k − x1k = x0p − x1p = x0u − x1u = x0y − x1y = x0b − x1b = x0e − x1e = x0g − x1g > 0, x1α = x0α
0 , l0 = l0 = l0 = l0 ∈ L (l0 ),
∀α ∈ P \ {i, j, k, m, p, q, r, u, y, z, a, b, d, e, g, h} li0 = lj0 = lk0 = lm
p
q
r
u
i
ly0 = lz0 = la0 = lb0 ∈ L (lp0 ) ∪ {lp0 }, ld0 = le0 = lg0 = lh0 ∈ L (ly0 ), and l1 = l0 . Alternatively, we say that
c0 is obtained from c1 by a weak regressive generalized transfer of order (3,2) of economic resources.
Ethical Principle 21 Generalized Weak Equity of order (3,2):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive generalized transfer of order (3,2) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 12, 13, 14, 15,
16, 17, 20 and 21 will be denoted by Ω3,2 . Formally,
Ω3,2 := ω ω(·) ∈ Ω2,2
∧
ω3 (x, l) ≥ 0
∧
ω1 (x, l) = 0
∧
∆2 ω3 (x, Lj ) ≥ 0 .
(36)
Alternatively, we can restrict Ω3,2 by imposing transfer sensitivity in the dimension of freedoms
with its associated generalized weak equity principle (order 1,3). We add to these two principles a
weak version of Fishburn and Willig (1984) fifth order generalized transfer principle that compares
24
eight transfers of economic resources between eight pairs of individuals. In each group we have two
individuals with two different levels of economic resources. However, these two levels of economic
resources are the same four all eight groups. All individuals in a group have the same sets of freedoms, however, two individuals from two different groups have different sets of freedoms. Transfers
go like this. Assume that we rank the pairs of individuals in increasing order of freedoms. In the
first pair, the transfer of economic resources is progressive. It is regressive for the second pair,
regressive for the third pair, progressive for the fourth pair, regressive for the fifth pair, progressive
for the sixth pair, progressive for the seventh pair and regressive for the eight pair. Formally,
Definition 23 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive generalized transfer of order
(2,3) of economic resources if there exist sixteen individuals i, j, k, m, p, q, r, u, y, z, a, b, d, e, g, h ∈ P
such that x0j = x0m = x0q = x0u = x0z = x0b = x0e = x0h > x0i = x0k = x0p = x0r = x0y = x0a = x0d = x0g ,
x0j − x0i > x1i − x0i = x1m − x0m = x1q − x0q = x1r − x0r = x1z − x0z = x1a − x0a = x1d − x0d = x1h − x0h =
x0j − x1j = x0k − x1k = x0p − x1p = x0u − x1u = x0y − x1y = x0b − x1b = x0e − x1e = x0g − x1g > 0, x1α = x0α
0 ∈ L (l0 ), l0 = l0 ∈ L (l0 ) ∪ {l0 },
∀α ∈ P \ {i, j, k, m, p, q, r, u, y, z, a, b, d, e, g, h} li0 = lj0 , lk0 = lm
p
q
i
k
k
lr0 = lu0 ∈ L (lp0 ), ly0 = lz0 ∈ L (lp0 )∪{lp0 }, la0 = lb0 ∈ L (ly0 ), ld0 = le0 ∈ L (ly0 )∪{ly0 }, lg0 = lh0 ∈ L (ld0 ), and
l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak regressive generalized transfer
of order (2,3) of economic resources.
W (c1 , ω) ≥ W (c0 , ω)
Ethical Principle 22 Generalized Weak Equity of order (2,3):
whenever c1 is obtained from c0 by a weak progressive generalized transfer of order (2,3) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 12, 13, 16, 17,
18, 15, 20 and 22 will be denoted by Ω2,3 . Formally,
Ω2,3 := ω ω(·) ∈ Ω2,2
∧
∆3 ω(x, Lj ) ≥ 0
∧
∆3 ω1 (x, Lj ) ≤ 0
∧
∆3 ω2 (x, Lj ) ≥ 0 .
(37)
25
Let us now restrict the set of indices Ω2,2 by adding ethical principles in both dimension simultaneously. We impose transfer sensitivity in the two dimensions together with their associated
generalized weak equity principles (i.e. or order (3,2) and (2,3)). If we add to this a weak version of Fishburn and Willig (1984) sixth order generalized transfer principle that compares eight
composite transfers of economic resources between eight groups of individuals. In each group we
have four individuals with four different levels of economic resources. However, these four levels of
economic resources are the same four all eight groups. All individuals in a group have the same
sets of freedoms, however, two individuals from two different groups have different sets of freedoms.
We consider that a composite transfer within each group. Transfers go like this. Assume that we
rank the eight groups in increasing order of freedoms. In the first group, the composite transfer of
economic resources is progressive. It is regressive for the second group, regressive for the third, progressive for the fourth, regressive for the fifth, progressive for the sixth, progressive for the seventh
and regressive for the eighth. Formally,
Definition 24 c1 ∈ C n is obtained from c0 ∈ C n by a weak progressive generalized transfer of order
(3,3) of economic resources if there exist thirty individuals i, j, k, m, p, q, r, u, y, z, a, b, d, e, g, h, α, β,
γ, δ, , ζ, η, θ, ι, λ, µ, ν, ξ, π, ρ, σ ∈ P such that x0m = x0u = x0b = x0h = x0δ = x0θ = x0ν = x0σ > x0k =
x0r = x0a = x0g = x0γ = x0η = x0µ = x0ρ ≥ x0j = x0q = x0z = x0e = x0β = x0ζ = x0λ = x0π > x0i = x0p = x0y =
x0d = x0α = x0 = x0ι = x0ξ , x0j − x0i = x0m − x0k > x1i − x0i = x1m − x0m = x1q − x0q = x1r − x0r = x1z − x0z =
x1a − x0a = x1d − x0d = x1h − x0h = x1β − x0β = x1γ − x0γ = x1 − x0 = x1θ − x0θ = x1ι − x0ι = x1ν − x0ν =
x1π − x0π = x1ρ − x0ρ = x0j − x1j = x0k − x1k = x0p − x1p = x0u − x1u = x0y − x1y = x0b − x1b = x0e − x1e =
x0g − x1g = x0α − x1α = x0δ − x1δ = x0ζ − x1ζ = x0η − x1η = x0λ − x1λ x0µ − x1µ = x0ξ − x1ξ = x0σ − x1σ > 0,
x1τ = x0τ ∀τ ∈ P \ {i, j, k, m, p, q, r, u, y, z, a, b, d, e, g, h, α, β, γ, δ, , ζ, η, θ, ι, λ, µ, ν, ξ, π, ρ, σ}, li0 =
0 , l0 = l0 = l0 = l0 ∈ L (l0 ), l0 = l0 = l0 = l0 ∈ L (l0 ) ∪ {l0 }, l0 = l0 = l0 = l0 ∈ L (l0 ),
lj0 = lk0 = lm
p
q
r
u
y
z
a
p
p
e
g
y
i
b
d
h
lα0 = lβ0 = lγ0 = lδ0 ∈ L (ly0 ) ∪ {ly0 }, l0 = lζ0 = lη0 = lθ0 ∈ L (ld0 ), lι0 = lλ0 = lµ0 = lν0 ∈ L (ld0 ) ∪ {ld0 },
lξ0 = lπ0 = lρ0 = lσ0 ∈ L (lι0 ), and l1 = l0 . Alternatively, we say that c0 is obtained from c1 by a weak
26
regressive generalized transfer of order (3,3) of economic resources
Ethical Principle 23 Generalized Weak Equity of order (3,3):
W (c1 , ω) ≥ W (c0 , ω)
whenever c1 is obtained from c0 by a weak progressive generalized transfer of order (3,3) of economic
resources.
The set of individual wellbeing measures that obey Ethical Principles 1, 2, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 21 and 23 will be denoted by Ω3,3 . Formally,
Ω3,3 := ω ω(·) ∈ Ω3,2 ∩ Ω2,3
∧
∆3 ω3 (x, Lj ) ≤ 0 .
(38)
It is theoretically possible to extend the logic and develop higher order generalized transfer principles
following Fishburn and Willig (1984). We do not follow this path since the complexity of the ethical
principles for each order increases quickly since we deal with two dimensions simultaneously.
We follow an approach similar to the two previous sections to identify partial orderings as defined
in equation (9) for all function ω ∈ Ωs,t . We first need to define the following stochastic dominance
functions Ds,t (y, Lj ):

if s = 1, t = 1
 RF (x, Lj )
x s−1,1
s,t
(x, Lj )dy if s ∈ {2, 3, . . .}, t = 1
D (x, Lj ) :=
x D
 P
j
s,t−1
(x, Li ) if s ∧ t ∈ {2, 3, . . .}
i=1 D
(39)
Using these functions, partial orderings can be identified using the following result:
Theorem 4 F1 (x, l) Ωs,t F0 (x, l), for s ∈ {1, 2, 3} and t ∈ {1, 2}, if and only if:
1) F1 (x, l) Ω0,t F0 (x, l)
2) F1 (x, l) Ωs,0 F0 (x, l)
3) D0s,t (x, l) − D1s,t (x, l) ≥ 0 ∀x ∈ D ∧ ∀l ∈ L (LK )
(40)
Theorem 5 F1 (x, l) Ωs,3 F0 (x, l), for s ∈ {1, 2, 3}, if and only if:
1)
2)
3)
4)
F1 (x, l) Ω0,3 F0 (x, l)
F1 (x, l) Ωs,0 F0 (x, l)
D0s,3 (x, l) − D1s,3 (x, l) ≥ 0 ∀x ∈ D ∧ ∀l ∈ L (LK−1 )
D0s,2 (x, LK−1 ) − D1s,2 (x, LK−1 ) ≥ 0 ∀x ∈ D
27
(41)
Inspection of conditions (1) and (2) in Theorems 4 and 5 indicates that in order to have capability
dominance for all indices belonging to Ωs,t , we must have consequential dominance of order s and
procedural dominance of order t. This result relates to Moyes’ (2012) observation that the marginal
distributions of the continuous and the discrete variables both play an important role in assessing
the dominance of the joint distributions of the two attributes.
An important assumption that was made in our theoretical framework is that there is a complete
orderings of the sets of freedoms that takes the form of L1 ⊆ L2 ⊆ · · · ⊆ LK . What happens if
we have a situation where for instance we have L1 ⊆ L2 , L3 ⊆ · · · ⊆ LK . In this case L2 6⊂ L3
and L3 6⊂ L2 . In a situation like this it would still be possible to establish dominance by applying
Theorem 4 or 5 twice. The first time we apply it as if L2 ⊆ L3 and the second time as if L3 ⊆ L2 .
However, the complexity of the test increases quickly with the number of incomplete orderings of
sets of freedoms.
6
The evolution of women wellbeing in Sub Saharan Africa in the
2000s
Demographic and Health Surveys (DHS) are nationally-representative household surveys that provide data for a wide range of monitoring and impact evaluation indicators in the areas of population,
health, and nutrition.
The wealth index:
The wealth index is a composite measure of a household’s cumulative
living standard. The wealth index is calculated using easy-to-collect data on a household’s ownership
of selected assets, such as televisions and bicycles; materials used for housing construction; and types
of water access and sanitation facilities. The wealth index is generated by applying a principal
components analysis to the information on asset ownership. We can interpret the wealth index as
an indicator of the capacity of a woman to achieve a certain social functioning, contingent to her
freedom.
28
Using a wealth index based on asset ownership implies that it will be more difficult to identify
robust changes in economic resources (consequential dominance), implying also that it is more
difficult to identify robust changes in capabilities.
Information regarding freedom: In this paper we approach individual freedom from a very
fundamental perspective by using women’s freedom of choice within the household. We believe that
this form of freedom is particularly interesting since it is the cornerstone of individuals freedom
since the absence of women’s individuals freedom within the household is often a good reflection of
the absence of women’s individual freedom at the country level.
Information regarding women’s individual freedom is provided through the following question:
In your opinion, is a husband justified in hitting or beating his wife in the following situations:
1. If she goes out without telling her husband?
2. If she neglects the children?
3. If she argues with him?
4. If she refuses to have sex with him?
5. If she burns the food?
As one can see, these question covers many dimensions of individual freedoms starting from the
freedom in the personal sphere to freedom in the social sphere. List of Countries:
The empirical application will use the DHS for the following countries and years:
1. Burkina Faso 2003 (12477 observations) - 2010 (17087 observations)
2. Ethiopia 2005 (14070 observations) - 2011 (16515 observations)
3. Ghana 2003 (5691 observations) - 2008 (4916 observations)
4. Lesotho 2004 (7095 observations) - 2009 (7624 observations)
29
5. Kenya 2003 (8195 observations) - 2009 (8444 observations)
6. Malawi 2004 (11698 observations) - 2010 (23020 observations)
7. Madagascar 2004 (7949 observations) - 2009 (17375 observations)
8. Nigeria 2003 (7620 observations) -2008 (33385 observations)
9. Senegal 2005 (14602 observations) - 2010 (15688 observations)
10. Rwanda 2005 (11321 observations) - 2010 (13671 observations)
11. Zimbabwe 2006 (8907 observations) - 2011 (9171 observations)
12. Tanzania 2005 (10329 observations) - 2010 (10139 observations)
6.1
Consequential dominance
When we apply the consequential dominance tests from Theorem 1, we have the following results:
1. Lesotho ∆W ≥ 0 for all indices belonging to Ω3,0 (third order dominance)
2. Madagascar ∆W ≥ 0 for all indices belonging to Ω1,0 (first order dominance)
3. Tanzania ∆W ≤ 0 for all indices belonging to Ω2,0 (second order dominance)
We also have on case of restricted dominance in the Burkina Faso. If we restrict our attention
to the population between 0 and the 90th percentile of the wealth index, we get ∆W ≥ 0 for all
indices belonging to Ω2,0 (second order restricted dominance). This result is valid for all indices for
which the marginal impact of an increase in wealth vanishes at 90th percentile of the wealth index
instead of vanishing at x.
30
6.2
Procedural dominance
To assess the incidence of the threat of domestic violence, we count the number of yes to the
questions presented earlier in this section. In doing so, we consider four different sets of freedoms:
L1 is the worst level of punishment with X to X threat of domestic violence; L2 is the middle level
of punishment with X to X threat of domestic violence; L3 is a light level of punishment with X to
X threat of domestic violence and L4 corresponds to the most extensive set of freedom without any
threat of domestic violence.
When we apply the consequential dominance tests from Theorems 2 and 3, we have the following
results:
1. Burkina Faso ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
2. Ethiopia ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
3. Gnana ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
4. Kenya ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
5. Lesotho ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
6. Madagascar ∆W ≤ 0 for all indices belonging to Ω0,1 (first order dominance)
7. Malawi ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
8. Nigeria ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
9. Rwanda ∆W ≤ 0 for all indices belonging to Ω0,1 (first order dominance)
10. Senegal ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
11. Tanzania ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
12. Zimbabwe ∆W ≥ 0 for all indices belonging to Ω0,1 (first order dominance)
31
We observe an Increase in wellbeing everywhere except in Rwanda and Madagascar where there
is a decrease in wellbeing. It is important to note that in this part wellbeing is assessed only on the
basis of the sets of freedom (i.e., rights).
6.3
Capability dominance
At this point it is important to note that Theorems 4 and 5 state that both consequential and
procedural dominance are necessary conditions for capability dominance. Looking at the results
obtained in the previous sections, we notice that only 3 countries have a consequential dominance
result, namely Lesotho, Madagascar and Tanzania. In the case of Madagascar, wellbeing decreases
when we adopt procedural approach. This leaves us with two countries eligible for capability dominance: Tanzania and Lesotho. The capability dominance tests for Tanzania indicate no dominance
for any Ωs,t , s ∈ {1, 2, 3} and t ∈ {1, 2, 3}. As for Lesotho, we have ∆W ≥ 0 for all ω ∈ Ω3,2 ⊂ Ω3,3 .
We have metionned earlier that if we restrict the definition of dominance for the subset of the
population between 0 and the 90th percentile, the Burkina Faso has a restricted dominance result.
Thus if we restrict our attention on this population, we have ∆W ≥ 0 for all ω ∈ Ω2,1 . This is a
very strong result. We consider all indices that display inequality aversion in the space of economic
resources, and only monotonicity in the space of freedoms (even if we have a purely ordinal view of
freedom) in addition to generalized weak equity of order 1,1 and 2,1.
32
References
[1] Alkire, S. and J. Foster (2011), Counting and multidimensional poverty measurement, Journal
of Public Economics, 95, 476-487.
[2] Atkinson, A. B. (1970), On the Measurement of Inequality, Journal of Economic Theory, 21,
24417263.
[3] Atkinson, A.B. (1992), Measuring Poverty and Differences in Family Composition, Economica,
59, 11716.
[4] Atkinson, A.B. (2003), Multidimensional Deprivation: Contrasting Social Welfare and Counting Approaches, Journal of Economic Inequality, 1, 51-65.
[5] Atkinson, A.B. and F. Bourguignon (1982), The comparison of multidimensioned distributions
of economic status, Review of Economic Studies, 49, 18317201.
[6] Atkinson, A.B. and F. Bourguignon (1987), Income distributions and differences in needs, in
G. Feiwel (Ed.), Arrow and the Foundations of the Theory of Economic Policy, MacMillan,
New York, 35017370.
[7] Bourguignon, F. and S. Chakravarty (1999), A family in multidimensional poverty measures, in
D.J. Slottje (Ed.) Advances in Econometrics, Income Distribution and Scientific Methodology:
Essay in Honor of C. Dagum, Physica-Verlas, New York.
[8] Bourguignon, F. and S. Chakravarty (2003), The measurement of multidimensional poverty,
Journal of Economic Inequality, 1, 25-49.
[9] Chakravarty, S. R. (1983), A New Index of Poverty, Mathematical Social Science, 6, 307-313.
33
[10] Chambaz, C. and E. Maurin (1998), Atkinson and Bourguignons dominance criteria: Extended
and applied to the measurement of poverty in France, Review of Income and Wealth, 44,
49717513.
[11] Duclos, J.-Y. and P. Makdissi (2004), Restricted et Unrestricted Dominance for Welfare,
Inequality and Poverty Orderings, Journal of Public Economic Theory, 6, 145-164.
[12] Duclos, J.-Y. and P. Makdissi (2005), Sequential Stochastic Dominance and the Robustness
of Poverty Orderings, Review of Income et Wealth, 51, 63-87.
[13] Duclos, J.-Y., D.E. Sahn and S.D. Younger (2006), Robust Multidimensional Poverty Comparisons, Economic Journal, 116, 943-968.
[14] Duclos, J.-Y., D.E. Sahn and S.D. Younger (2011), Partial multidimensional inequality orderings, Journal of Public Economics, 95, 225-238.
[15] Dworkin, R. (1981), What is Equality? Part 2: Equality of Resources, Philosophy and Public
Affairs, 10, 283-345.
[16] Fishburn, P.C. (1976), Continua of stochastic dominance relations for bounded probability
distributions, Journal of Mathematical Economics, 3, 29517311.
[17] Fishburn, P.C. (1980), Stochastic Dominance and Moments of distributions, Mathematics of
Operations Research, 5, 94-100.
[18] Fishburn, P.C., and R.D. Willig (1984), Transfer Principles in Income Redistribution, Journal
of Public Economics, 25, 3231728.
[19] Jenkins, S.P. and P.J. Lambert (1993), Ranking income distributions when needs differ, Review
of Income and Wealth, 39, 33717356.
[20] Kolm, S.C. (1976), Unequal Inequality: I, Journal of Economic Theory, 12, 41617442.
34
[21] Moyes, Patrick (2012), Comparisons of heterogeneous distributions and dominance criteria,
Journal of Economic Theory, 147, 1351-1383.
[22] Rawls, J. (1971), A Theory of Justice, Harvard University Press, Cambridge, MA.
[23] Sen, A.K. (1985), Commodities and Capabilities, North Holland, Amsterdam.
[24] Sen, A.K. (1992), On Economic Inequality, expanded edition, Norton, New York.
[25] Sen, A.K. (2002), Rationality and Freedom, Belknap Press, Cambridge, MA.
[26] Thon, D. (1983), A Note on A Troublesome Axiom for Poverty Indices, Economic Journal,
93, 199-200.
[27] Whitmore, G.A. (1970), Third Degree Stochastic Dominance, American Economic Review,
60, 457-459.
35
A
Appendix
A.1
Proof of Theorem 1
The result of this theorem is not new. It is already well known in the financial economics and
inequality measurement literature. Whitmore (1970) had already introduced the concept of thirdorder dominance. The result is however used in the proof of Theorems 4 and 5, this is why we are
presenting it. In order to prove Theorem 1, we first need to integrate equation (14) by parts:
Z
x
ω(x, LK )dFi (x, LK ) =
x
x
ω(x, LK )Di1,0 (x, LK )
x
Z
−
x
x
ω1 (x, LK )Di1,0 (x, LK ) dx.
(42)
Since Di1,0 (x, LK ) = 0 by definition, equation (42) can be rewritten as:
Z
x
ω(x, LK )dFi (x, LK ) =
x
ω(x, LK )Di1,0 (x, LK )
Z
−
x
x
ω1 (x, LK )Di1,0 (x, LK ) dx.
(43)
Using equation (43) and keeping in mind that Di1,0 (x, LK ) = 1, ∆W (F1 , F0 ; ω) = W (F1 ; ω) −
W (F0 ; ω) can be rewritten as:
Z
∆W (F1 , F0 ; ω) = −
x
x
h
i
ω1 (x, LK ) D11,0 (x, LK ) − D01,0 (x, LK ) dx
(44)
Integrating by parts twice equation (43) using Assumption 1, we obtain:
s
Z
∆W (F1 , F0 ; ω) = (−1)
x
x
h
i
ω1 (x, LK ) D1s,0 (x, LK ) − D0s,0 (x, LK ) dx for s ∈ {1, 2, 3}.
(45)
To prove sufficiency we only need to use equation (45) and the definition of the set Ωs,0 . We
can then conclude that, if we have D0s,0 (x, LK ) − D1s,0 (x, LK ) ≥ 0 for all x ∈ D, we must have
∆W (F1 , F0 ; ω) ≥ 0.
In order to establish necessity, we use a function ω(x, l)), the (s − 1)th derivative of which
(s ∈ {1, 2, 3}), for all l ∈ L is

if x ≤ x0 ,
 (−1)s−2 ε
(−1)s−2 (x0 + ε − x) if x0 ≤ x ≤ x0 + ε,
ωs−1 (x, l) =

0
if x ≥ x0 ,
36
(46)
where x0 ∈ \{x, x}. This yields

if x ≤ x0 ,
 0
(−1)s−1 if x0 ≤ x ≤ x0 + ε,
ωs (x, l) =

0
if x ≥ x0 .
(47)
Imagine now that D0s,0 (x, LK ) − D1s,0 (x, LK ) < 0 on an interval [x0 , x0 + ε] with x0 ∈ \{x, x} and ε
arbitrarily close to 0. For ω(x, l) defined as in (46), ∆W (F1 , F0 ; ω) < 0. Wellbeing functions ω(x, l)
having the above-given form for ωs−1 (x, l) do not belong to the class Ωs,0 (s ∈ {1, 2, 3}). However,
there is a sequence {ω n (x, l)} of real-valued measurable functions belonging to Ωs,0 (s ∈ {1, 2, 3})
such that limn→∞ ω n (x, l) = ω(x, l). Applying Lebesgue’s dominated convergence theorem we
know that limn→∞ ∆W (F1 , F0 ; ω n ) = ∆W (F1 , F0 ; ω). Hence it cannot be that D0s,0 (x, LK ) −
D1s,0 (x, LK ) < 0 for some [x0 , x0 + ε] when x0 ∈ \{x, x}. This proves the necessity of the condition.
A.2
Proof of Theorem 2
In order to prove Theorem 2, we first need to apply Abel’s decomposition to equation (20) yielding
W (Fi ; ω) = ω(x, LK )Di0,1 (x, LK ) −
K−1
X
∆1 ω(x, Lj )Di0,1 (x, Lj ).
(48)
j=1
Using equation (48) and keeping in mind that Di0,1 (x, LK ) = 1, ∆W (F1 , F0 ; ω) can be rewritten as:
∆W (F1 , F0 ; ω) = −
K−1
X
h
i
∆1 ω(x, Lj ) D10,1 (x, Lj ) − D00,1 (x, Lj ) .
(49)
j=1
Applying again Abel’s decomposition to equation (49) yields
h
i
∆W (F1 , F0 ; ω) = − ∆1 ω(x, LK−1 ) D10,2 (x, LK−1 ) − D00,2 (x, LK−1 )
+
K−2
X
h
i
∆2 ω(x, Lj ) D10,2 (x, Lj ) − D00,2 (x, Lj ) .
(50)
j=1
To prove sufficiency we only need to use equation (49) or equation (50) and the definition of the set
Ω0,t (t ∈ {1, 2}). We can then conclude that, if we have D00,t (x, l)−D10,t (x, l) ≥ 0 for all l ∈ L (LK ),
we must have ∆W (F1 , F0 ; ω) ≥ 0.
37
Proving necessity is quite simple. For t = 1, consider the wellbeing function such that ∆1 (x, l0 ) >
0 for l0 ∈ L (LK ) and ∆1 (x, Lj ) = 0 for all Lj 6= l0 . This wellbeing function belongs to Ω0,1 . Imagine
now that D00,1 (x, l0 ) − D10,1 (x, l0 ) < 0. For this particular case of ω(x, l), ∆W (F1 , F0 ; ω) < 0. Hence
it cannot be that D00,1 (x, l0 ) − D10,1 (x, l0 ) < 0. This proves the necessity of the condition for t = 1.
For t = 2, we proceed in two steps. First consider the wellbeing function such that ∆1 (x, LK−1 ) > 0
and ∆1 (x, Lj ) = 0 for all Lj 6= LK−1 . This wellbeing function belongs to Ω0,2 . Imagine now that
D00,2 (x, LK−1 ) − D10,2 (x, LK−1 ) < 0. For this particular case of ω(x, l), ∆W (F1 , F0 ; ω) < 0. Now,
consider a wellbeing function such that ∆2 (x, l0 ) < 0 for l0 ∈ L (LK−1 ) and ∆2 (x, Lj ) = 0 for all
Lj 6= l0 . This wellbeing function belongs to Ω0,2 . Imagine now that D00,2 (x, l0 ) − D10,2 (x, l0 ) < 0.
For this particular case of ω(x, l), ∆W (F1 , F0 ; ω) < 0. This proves the necessity of the condition
for t = 2.
A.3
Proof of Theorem 3
In order to prove Theorem 3, we apply again Abel’s decomposition to equation (50) yielding
h
i
∆W (F1 , F0 ; ω) = − ∆1 ω(x, LK−1 ) D10,2 (x, LK−1 ) − D00,2 (x, LK−1 )
h
i
+ ∆2 ω(x, Lj ) D10,3 (x, LK−2 ) − D00,3 (x, LK−2 )
−
K−3
X
h
i
∆3 ω(x, Lj ) D10,3 (x, Lj ) − D00,3 (x, Lj ) .
(51)
j=1
To prove sufficiency we only need to use equation (51) and the definition of the set Ω0,3 . We can
then conclude that, if we have D00,3 (x, l) − D10,3 (x, l) ≥ 0 for all l ∈ L (LK−1 ) and D00,2 (x, LK−1 ) −
D10,2 (x, LK−1 ) ≥ 0, we must have ∆W (F1 , F0 ; ω) ≥ 0.
To prove necessity, we proceed in three steps. First consider the wellbeing function such that
∆1 (x, LK−1 ) > 0 and ∆1 (x, Lj ) = 0 for all Lj 6= lK−1 . This wellbeing function belongs to
Ω0,3 . Imagine now that D00,2 (x, LK−1 ) − D10,2 (x, LK−1 ) < 0. For this particular case of ω(x, l),
∆W (F1 , F0 ; ω) < 0. Second, consider the wellbeing function such that ∆2 (x, LK−2 ) < 0 and
∆2 (x, Lj ) = 0 for all Lj 6= lK−2 . This wellbeing function belongs to Ω0,3 . Imagine now that
38
D00,3 (x, LK−2 ) − D10,3 (x, LK−2 ) < 0. For this particular case of ω(x, l), ∆W (F1 , F0 ; ω) < 0. Now,
consider a wellbeing function such that ∆3 (x, l0 ) > 0 for l0 ∈ L (LK−2 ) and ∆3 (x, Lj ) = 0 for all
Lj 6= l0 . This wellbeing function belongs to Ω0,3 . Imagine now that D00,3 (x, l0 ) − D10,3 (x, l0 ) < 0.
For this particular case of ω(x, l), ∆W (F1 , F0 ; ω) < 0. This proves the necessity of the condition.
A.4
Proof of Theorem 4
In order to prove Theorem 4, we first integrate equation (13 by parts, yielding to:
K n
X
W (Fi ; ω) =
ω(x, Lj )P (Lj , Fi )Fi (x|Lj )|x0
o
j=1
Z
−
K
xX
ω1 (x, Lj )P (Lj , Fi )Fi (x|Lj )dx
(52)
x j=1
Applying Abel’s decomposition to both summation in equation (52) and remembering that Fi (x|Lj ) =
0 yields to:
ω(x, LK )Di1,1 (x, LK )
W (Fi ; ω) =
−
K−1
X
j=1
x
Z
−
x
∆1 ω(x, Lj )Di1,1 (x, Lj )
ω1 (x, Lj )Di1,1 (x, Lj )dx
x K−1
X
Z
+
x
∆1 ω(x, Lj )Di1,1 (x, Lj )dx
(53)
j=1
Using equation (53) and keeping in mind that D11,1 (x, LK ) = D01,1 (x, LK ) = 1, ∆W (F1 , F0 , ω)can
be written as:
∆W (F1 , F0 , ω) = −
K−1
X
j=1
x
Z
−
x
Z
+
x
h
i
∆1 ω(x, Lj ) D11,1 (x, Lj ) − D01,1 (x, Lj )
h
i
ω1 (x, Lj ) D11,1 (x, Lj )dx − D01,1 (x, Lj ) dx
x K−1
X
h
i
∆1 ω1 (x, Lj ) D11,1 (x, Lj ) − D01,1 (x, Lj ) dx
j=1
39
(54)
Integrating by parts twice the two integrals in the r.h.s. of equation (54) and using Assumption 1,
we can write:
∆W (F1 , F0 , ω) = −
K−1
X
h
i
∆1 ω(x, Lj ) D11,1 (x, Lj ) − D01,1 (x, Lj )
j=1
x
Z
s
h
i
ωs (x, Lj ) D1s,1 (x, Lj )dx − D0s,1 (x, Lj ) dx
+ (−1)
x
s+1
x K−1
X
Z
+ (−1)
x
h
i
∆1 ωs (x, Lj ) D1s,1 (x, Lj ) − D0s,1 (x, Lj ) dx
j=1
for s ∈ {1, 2, 3}.
(55)
Applying Abel’s decomposition to the two summations on the r.h.s. of equation (55) yields:
h
i
∆W (F1 , F0 , ω) = − ∆1 ω(x, LK−1 ) D11,2 (x, LK−1 ) − D01,2 (x, LK−1 )
+
K−2
X
h
i
∆2 ω(x, Lj ) D11,2 (x, Lj ) − D01,2 (x, Lj )
j=1
s
x
Z
i
h
ωs (x, Lj ) D1s,1 (x, Lj )dx − D0s,1 (x, Lj ) dx
+ (−1)
x
s+1
x
Z
+ (−1)
x
s
Z
+ (−1)
x
i
h
∆1 ωs (x, LK−1 ) D1s,2 (x, LK−1 ) − D0s,2 (x, LK−1 ) dx
x K−2
X
i
h
∆2 ωs (x, Lj ) D1s,2 (x, Lj ) − D0s,2 (x, Lj ) dx
j=1
for s ∈ {1, 2, 3}.
(56)
Equation (55) and (56) prove the sufficiency of the condition.
Necessity of conditions (1) and (2) is easy to prove. Just recall that Ωs,0 ⊂ Ωs,t and Ω0,t ⊂ Ωs,t
and using the necessity part of Theorems 1 and 2 establishes necessity of conditions (1) and (2).
In order to prove for necessity of condition (3) for t = 1, consider a wellbeing function such that
ω(x, L1 ) = ω(x, L2 ) = · · · = ω(x, LK ) and such that ∆1 ωs (x, l) = 0 for all l ∈ L \ {l0 }, l0 ∈ L (LK ),
and (−1)s+1 ∆1 ωs (x, l0 ) ≤ 0 takes the following form:

if x ≤ x0 ,
 0
(−1)s if x0 ≤ x ≤ x0 + ε,
∆1 ωs (x, l) =

0
if x ≥ x0 .
40
(57)
Imagine now that D0s,1 (x, l0 ) − D1s,1 (x, l0 ) < 0 on an interval [x0 , x0 + ε] with x0 ∈ \{x, x} and
ε arbitrarily close to 0. For any ω(x, l) such that ∆1 ωs (x, l0 ) takes the form defined as in (57),
∆W (F1 , F0 ; ω) < 0. Hence it cannot be that D0s,1 (x, l0 ) − D1s,1 (x, l0 ) < 0 for some [x0 , x0 + ε] when
x0 ∈ \{x, x}. This proves the necessity of condition (3) for t = 1.
To prove necessity for t = 2, we proceed in two steps. First, consider a wellbeing function such
that ω(x, L1 ) = ω(x, L2 ) = · · · = ω(x, LK ) and such that ∆1 ωs (x, l) = 0 for all l ∈ L (LK−1 ),
and (−1)s ∆1 ωs (x, LK−1 ) ≤ 0 takes the form in equation (57). Imagine now that D0s,2 (x, LK−1 ) −
D1s,2 (x, LK−1 ) < 0 on an interval [x0 , x0 + ε] with x0 ∈ \{x, x} and ε arbitrarily close to 0. For
any ω(x, l) such that ∆1 ωs (x, LK−1 ) takes the form defined as in (57), ∆W (F1 , F0 ; ω) < 0. Hence
it cannot be that D0s,2 (x, LK−1 ) − D1s,2 (x, LK−1 ) < 0 for some [x0 , x0 + ε] when x0 ∈ \{x, x}.
Now consider a wellbeing function such that ω(x, L1 ) = ω(x, L2 ) = · · · = ω(x, LK ) and such that
∆2 ωs (x, l) = 0 for all l ∈ L \ {l0 }, l0 ∈ L (LK−1 ), and (−1)s ∆2 ωs (x, l0 ) ≤ 0 takes the following
form:

if x ≤ x0 ,
 0
(−1)s+1 if x0 ≤ x ≤ x0 + ε,
∆2 ωs (x, l) =

0
if x ≥ x0 .
(58)
Imagine now that D0s,2 (x, l0 ) − D1s,2 (x, l0 ) < 0 on an interval [x0 , x0 + ε] with x0 ∈ \{x, x} and
ε arbitrarily close to 0. For any ω(x, l) such that ∆2 ωs (x, l0 ) takes the form defined as in (57),
∆W (F1 , F0 ; ω) < 0. Hence it cannot be that D0s,2 (x, l0 ) − D1s,2 (x, l0 ) < 0 for some [x0 , x0 + ε] when
x0 ∈ \{x, x}. This proves the necessity of condition (3) for t = 2.
41
A.5
Proof of Theorem 5
First applying Abel’s decomposition to the two summations on the r.h.s. of equation (56) yields:
h
i
∆W (F1 , F0 , ω) = − ∆1 ω(x, LK−1 ) D11,2 (x, LK−1 ) − D01,2 (x, LK−1 )
h
i
+ ∆2 ω(x, LK−2 ) D11,3 (x, LK−1 ) − D01,3 (x, LK−1 )
+
K−3
X
h
i
∆3 ω(x, Lj ) D11,3 (x, Lj ) − D01,3 (x, Lj )
j=1
s
x
Z
h
i
ωs (x, Lj ) D1s,1 (x, Lj )dx − D0s,1 (x, Lj ) dx
+ (−1)
x
s+1
x
Z
+ (−1)
x
+ (−1)s
x
Z
h
i
∆2 ωs (x, LK−2 ) D1s,3 (x, LK−2 ) − D0s,3 (x, LK−2 ) dx
x
s+1
h
i
∆1 ωs (x, LK−1 ) D1s,2 (x, LK−1 ) − D0s,2 (x, LK−1 ) dx
Z
+ (−1)
x
x K−3
X
h
i
∆3 ωs (x, Lj ) D1s,3 (x, Lj ) − D0s,3 (x, Lj ) dx
j=1
for s ∈ {1, 2, 3}.
(59)
Equation (55) and (56) prove the sufficiency of the condition.
Necessity of conditions (1) and (2) are proved by the fact that Ωs,0 ⊂ Ωs,t and Ω0,t ⊂ Ωs,t . Using
the necessity part of Theorems 1 and 3 establishes necessity of conditions (1) and (2).
In order to prove for necessity of condition (4), consider a wellbeing function such that ω(x, L1 ) =
ω(x, L2 ) = · · · = ω(x, LK ) and such that ∆1 ωs (x, l) = 0 for all l ∈ L (LK−1 ), and (−1)s ∆1 ωs (x, LK−1 ) ≤
0 takes the form in equation (57). Imagine now that D0s,2 (x, LK−1 ) − D1s,2 (x, LK−1 ) < 0 on an
interval [x0 , x0 + ε] with x0 ∈ \{x, x} and ε arbitrarily close to 0. For any ω(x, l) such that
∆1 ωs (x, LK−1 ) takes the form defined as in (57), ∆W (F1 , F0 ; ω) < 0. Hence it cannot be that
D0s,2 (x, LK−1 ) − D1s,2 (x, LK−1 ) < 0 for some [x0 , x0 + ε] when x0 ∈ \{x, x}.
In order to prove for necessity of condition (3), we proceed in two steps. First, consider a
wellbeing function such that ω(x, L1 ) = ω(x, L2 ) = · · · = ω(x, LK ) and such that ∆1 ωs (x, l) = 0
for all l ∈ L \ {LK−2 }, and (−1)s+1 ∆2 ωs (x, LK−2 ) ≤ 0 takes the form in equation (58). Imagine
42
now that D0s,3 (x, LK−2 ) − D1s,3 (x, LK−2 ) < 0 on an interval [x0 , x0 + ε] with x0 ∈ \{x, x} and ε
arbitrarily close to 0. For any ω(x, l) such that ∆2 ωs (x, LK−2 ) takes the form defined as in (58),
∆W (F1 , F0 ; ω) < 0. Hence it cannot be that D0s,3 (x, LK−2 )−D1s,2 (x, LK−2 ) < 0 for some [x0 , x0 +ε]
when x0 ∈ \{x, x}. Now consider a wellbeing function such that ω(x, L1 ) = ω(x, L2 ) = · · · =
ω(x, LK ) and such that ∆3 ωs (x, l) = 0 for all l ∈ L \{l0 }, l0 ∈ L (LK−2 ), and (−1)s+1 ∆3 ωs (x, l0 ) ≤
0 takes the following form:

if x ≤ x0 ,
 0
(−1)s if x0 ≤ x ≤ x0 + ε,
∆3 ωs (x, l) =

0
if x ≥ x0 .
(60)
Imagine now that D0s,3 (x, l0 ) − D1s,3 (x, l0 ) < 0 on an interval [x0 , x0 + ε] with x0 ∈ \{x, x} and
ε arbitrarily close to 0. For any ω(x, l) such that ∆3 ωs (x, l0 ) takes the form defined as in (60),
∆W (F1 , F0 ; ω) < 0. Hence it cannot be that D0s,3 (x, l0 ) − D1s,3 (x, l0 ) < 0 for some [x0 , x0 + ε] when
x0 ∈ \{x, x}. This proves the necessity of condition (3).
43