Discussion Paper No. 252
Freedom and achievement of well-being in theadaptive dynamics of capabilities
Jun Matsuyama and Kenji Mori

January 2010TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY

KAWAUCHI, AOBA-KU, SENDAI,

980-8576 JAPANFreedom and achievement of well-being in theadaptive dynamics of capabilities
Jun Matsuyama∗and Kenji Mori†
January 2010

Abstract
Sen’s capability approach has been studied by many researchers including not only economists but also political philosophers. Almost all
researchers consider capabilities as a static concept. However, D’Agata(2007) points out a need for consideration within a dynamic concept in
the capability approach. D’Agata (2007) develops a model with utilization functions determined endogenously through the adaptive dynamics. Motivated by his work, we also consider the adaptive dynamics
of capabilities. Our model formalized below focuses on a relationshipbetween goods and capability for simpliﬁcation. In this respect, ourmodel seems to diﬀer from D’Agata’s one which focuses an evolution ofutilization functions themselves. However, we assume implicitly thatthe evolution of capabilities depends on that of utilization functions.
By applying Rosenbaum’s formula (Rosenbaum, 2000; D’Agata, 2009a)to the dynamic context, to deﬁne well-being freedom, we show the existence of a distribution of goods equalizing well-being freedom of eachindividual, total amount of goods being ﬁxed. Furthermore, the valueof freedom is non-decreasing as the total amount of goods increases.
On the other hand, by means of counter examples, we exemplify thatone’s well-being achievement, deﬁned as in the dynamic context, candecrease even if he/she is given more (or equal) goods. In terms of thetwo-way evaluation of well-being à la Sen, the fact that a distributionof goods can negatively aﬀect some individuals characterizes a normative quality of the adaptive process. Our model shows that economic
distributive measures may have a limitation in improving individuals’
well-being.
∗

The Doctoral Program, Graduate School of Economics and Mnagement,
Tohoku University, 27-1 Kawauchi, Aoba-ku, Sendai 980-8576, Japan.
E-mail: jun-matsu@mwd.biglobe.ne.jp†
Graduate School of Economics and Mnagement, Tohoku University, 27-1 Kawauchi,
Aoba-ku, Sendai 980-8576, Japan. E-mail: mori@econ.tohoku.ac.jp1

1

Introduction

Since Sen’s article of the title of “Equality of what ?,” the capability approach, which is one of the evaluating systems of individual’s well-being,
has been studied by many researchers including not only economists but
also political philosophers1 . In economics, usually, individual’s well-being isrepresented by a numerical value such as utility or income. However, Sencriticizes the utility-based or income-based approach in order to propose
the concept of functionings. Furthermore, the capability approach focuseson not only his/her achievement of functionings but also freedom to choose
alternatives.
Sen (1985b) evaluates an individual’s advantage in terms of the four
aspects which are summarized by the following table.

achievement
freedomwell-being

agencywell-being achievement
well-being freedom

agency achievement
agency freedom

Table 1: The four aspects of an individual’s advantage.
The ﬁrst and second rows of table 1 represent achievement and freedom
aspect of an individual, respectively. Achievement is the concept which
reﬂects “what he/she is.” Freedom is the concept which represents “whathe/she can do or be.” The ﬁrst and second columns of table 1 are well-being
and agency aspect of an individual, respectively. The agency aspect of anindividual reﬂects an ethical point of view in a sense that an individual asagency does not always act as a well-being or utility maximizer. According
to our interpretation, the agency is an advantage of the individual necessary
to implement his well-being as the overall advantage. In this paper, we
conﬁne our argument to the well-being without explicitly considering theprocess of implementation.
Next, we refer to Sen’s capability approach where an individual wellbeing is measured by his capability. In Sen’s original formula, capability isdeﬁned as a set of functioning vectors which can be chosen by an individual. The functioning is not a mono-dimensional concept of well-being such1

Debates on distributive justice have been discussed since Rawls’ work “A Theory ofJustice.” (See e.g., Arneson, 1989; Cohen, 1989; Dworkin, 1981a, 1981b; Nozick, 1974;
Sen, 1980, 1985a, 1992; Rawls, 1971, and Roemer, 1996).

2

as utility or income but a multi-dimensional one in the form of a functioning vector. As an example of functioning, one might think of the Human
Development Index (HDI) which consists of three dimensions such as life
expectancy, the literacy rate and GDP per capita.
The capability is deﬁned by the following three stages. The ﬁrst stage
is the commodity space. Consider an individual endowed with a set ofgoods. When he chooses a bundle of goods, he goes to the second stage,that is, the characteristic space by transforming the bundle of goods into avector of characteristics by means of a characteristic function which maps
goods into their desirable features (Lancaster, 1966). For example, a bicycle
has transportation as its characteristic. An important point is that thecharacteristics of goods are determined in an objective manner. Whether
the owner of a bicycle is a disable or not, it has transportation as a feature in
itself. Then, he goes to the third stage, which is called the functioning spacein Sen’s term, by transforming a vector of characteristics into a vector offunctionings by means of a utilization function which is diﬀerent from person
to person. According to his life plan, he chooses a utilization function andthen obtains corresponding functionings. Lastly, the capability can be seen
as a set of possible vectors of functionings to be chosen by an individual.
Following Sen (1996), we interpret the value of chosen functionings as wellbeing achievement and that of the capability as well-being freedom, which
are depicted in table 1.
Almost all researchers consider the capability as a static concept. Their
models do not take the time aspect into consideration. As the ﬁrst attempt to consider the capability approach in a dynamic context, D’Agata(2007) develops a model with utilization functions determined endogenously
through the adaptive dynamics. In the adaptive dynamics, the capability
at time t + 1 depends on the chosen functionings at time t. Our model
formalized below builds on D’Agata’s pioneering model, where we pay more
attention to a relationship between goods and capability.
The aim of this paper is to assess critically the concept of distributive
justice and a normative quality of the adaptive process. Before entering into
a main argument, in section 2, we review brieﬂy D’Agata (2007) in order tocontrast with our model and add some remarks to justify adopting Rosembaum’s measure. In section 3, we deﬁne the well-being freedom which is theconcept proposed by Sen (1985b), by applying Rosenbaum’s formula to the
dynamic context. We show the existence of a distribution of goods equalizing well-being freedom of each individual with the total amount of goods
being ﬁxed. Furthermore, the value of freedom is non-decreasing as the total
amount of goods increases. In section 4, we deﬁne well-being achievement,
3

contrary to section 3, by means of counter examples, we exemplify that one’s
well-being achievement can decrease even if he/she is given more (or equal)
goods under the similar conditions. Section 5 is concluding remarks.

2

D’Agata’s adaptive dynamic model of capability

Let us start by brieﬂy reviewing D’Agata’s article in 2007, where he provides
a formal treatment of the capability approach within a dynamic framework.
According to D’Agata (2007), this attempt is justiﬁed by Sen himself. He
points out that, in Sen’s original formulation of capabilities (Sen, 1985a),
utilization functions are function of time-dependent variables of functionings such as age, health (see D’Agata, 2007, p.181, footnote 4). Thus, it issaid to be natural to assume that not only utilization functions but also acharacteristic function change over time. In D’Agata’s model, he assumes
that, under a given bundle of commodity, utilization functions are determined endogenously through the adaptive dynamics, but the characteristic
function does not change over time. The adaptive dynamics is formalized asfollows. At the beginning of time 0, one chooses a utilization function which
maximizes his valuation of each functioning over a given initial capability
set. During time 0, by employing the utilization function, he develops his
ability (i.e., new utilization functions) “around” the used utilization function. At the beginning of time 1, he obtains a new capability set and thenchooses a utilization function in the same manner. And so on. As time goesby, we obtain a sequence of utilization functions chosen each time. D’Agata
investigates mathematical properties of the sequence of utilization functions;
he shows the existence of limit points of the sequence of utilization functions
under certain conditions and provides conditions ensuring their uniqueness.
Furthermore, he critically investigates Sen’s following view: if two individuals are exactly the same, their equality in one focal variable space coincides
with their equality in other focal variable spaces. By means of numericalexamples, he argues that Sen’s view is not always valid. That is, even if two
individuals are exactly the same, their functioning may be diﬀerent according to their own choices; an important insight we are starting from (apartfrom his negative view of Sen).
Since D’Agata is devoted to an evolution of utilization functions, he does
not explicitly focus on the concept of distributive justice (i.e., equality ofcapabilities). In order to consider distributive justice properly, our model
pays attention to a relationship between goods and capability, as opposed
to his model which focuses only on an evolution of utilization functions

4themselves. Although our model seems to diﬀer from D’Agata’s one at
ﬁrst sight, we assume implicitly as well that the evolution of capabilities
depends on that of utilization functions. Our model can been seen rather
as an extension of D’Agata’s one in the following respects. Firstly, in orderto focus on distributive justice, we consider n individuals in an economy.
Secondly, while the amount of goods is ﬁxed in his model, we view it asvariable (see below corollary 1 in section 3, and section 4), to clarify arelationship between the well-being freedom and goods, and a relationshipbetween the well-being achievement and goods. Thirdly, we focus on boththe aspect of the well-being achievement and the aspect of the well-being
freedom, while D’Agata considers only the former. Thus, our model provides
a broader framework compared to D’Agata’s one.
We would like to make some additional remarks here to justify adopting
Rosenbaum’s measure in our paper. As we mentioned above, the capability approach has been scrutinized by many researchers. While in empiricalstudies, capabilities have been measured by using statistical data (Kuklys,
2005; Anand and van Hees, 2006), in theoretical studies, especially the social
choice theory, a strand of researches focus on a ranking rule of opportunity
sets, in particular of capability sets, which is each deduced by certain plausible axioms (Pattanaik and Xu, 1990; Sen, 1991; Klemisch-Ahlert, 1993;
Bossert, Pattanaik and Xu, 1994; Puppe, 1995; Pattanaik and Xu, 1998;
Sugden, 1998; van Hees, 1998; Pattanaik and Xu, 2000; Barberà, Bossert
and Pattanaik, 2004, and Xu, 2008). For example, we might introduce thenumber of elements of a set as such a measure. The measure is called the
cardinality criterion, which is characterized axiomatically by Pattanaik andXu (1990). It is very simple and easy to understand intuitively, but diﬃcult
to apply to an inﬁnite set. D’Agata (2009) investigates four plausible measures including the cardinality criterion and then examines critically whether
they lead to counter-intuitive results or not. The other three measures are
the social freedom measure (Steiner, 1994; Carter, 1999), the social-cardinal
freedom measure (Kramer, 2003) and Rosenbaum’s measure (Rosenbaum,2000). He interprets the ﬁrst two measures of the three as corollaries of the
cardinal criterion by pointing out that the unit of social freedom measure
and that of social-cardinal freedom measure can be reduced to that of the
cardinal criterion. This would imply that the three measures except Rosenbaum’s measure are only applicable to a ﬁnite set. In this sense, we consider
Rosenbaum’s measure as a ﬁrst candidate of the speciﬁc measure of freedom
which can be deﬁned on inﬁnite sets.
D’Agata, on the other hand, points out that Rosenbaum’s measure leads
to a counter-intuitive result. Rosenbaum’s measure is deﬁned as the ratio
5between maximal distance between two elements belonging to the set of socially technical feasible actions and maximal distance between two elements
elements belonging to the set of free actions. Consider any square set andanother set which consists only of its diagonal. In this case, the value of thesquare set measured à la Rosenbaum is the same as the value of the set of its
diagonal. This fact yields a counter-intuitive result because of violation of
so-called Kramer’s rejection rule, that is, the rule which rejects any measurethat gives the same degree of freedoms to any two sets such that the one
is a proper subset of the other. However, as we see below, in our model,such a counter-intuitive result does not occur due to assumptions of no sudden shrinkage and comprehensiveness of capabilities (see below assumption
1 (ii) and (vi)). Thus, the rejection rule does not hinder us from adopting
Rosenbaum’s formula, as ﬁrst approach, to measure capability sets.

3

Freedom of well-being

To begin with, we introduce some preliminary notations and deﬁnitions.
Let N = {1, . . . , n} be the ﬁnite set of individuals in an economy and the
cardinality of N , |N | be the number of individuals. The economy is endowedwith a certain amount of goods. We denote the total amount of goods inthe economy as non-negative real number X ∈ R+ . In most (though notall) parts of this paper, we assume a real number X to be ﬁxed. Let xi be
an amount of goods distributed to individual i. Then,sum of each xi∑thenis the total amount of goods in the economy, that is, i=1 xi = X. Let mbe the number of functionings. A functioning space stated above is deﬁnedmon the set of non-negative m-dimensional real space Rm+ . Let y ∈ R+m
be a vector of functionings and Y ⊂ R+ be a set of technically possiblefunctioning vectors. Let Qi ⊂ Rm
+ be a capability of individual i. In thispaper, we introduce time into a capability. Let us put a superscript t on Qi .
Qti represents a capability of individual i at time t. Following Sen (1985a),
we deﬁne a valuation function of individual i as a function vi : Rm
+ → R.
We consider the well-being freedom, in other words, a individual’s capability, in the dynamic context. For this purpose, we provide a model whichlinks the capability approach with the adaptive model (see e.g., Atkinson
and Stiglitz, 1969; Day and Kennedy, 1970; Cherene, 1978; D’Agata, 2005,2009b). Then, we investigate the existence of an initial distribution of goods
equalizing each capability in the economy. To begin with, we formally deﬁne
an initial capability as follows.

6

Deﬁnition 1. (Initial capability)
Firstly, we deﬁne a correspondence qi : R+ → Y, xi 7→ qi (xi ). The imageof xi , qi (xi ), is interpreted as the initial capability of individual i with anamount of goods xi , that is, Q0i = qi (xi ).
We assume the following properties of the initial capability.
Assumption 1.
The correspondence qi has the following six properties:(i) compact valued,(ii) continuous,(iii) qi (0) = {0},′′(iv) qi (xi ) ⊂ qi (xi ) for xi < xi ,(v) ∃xi ∈ R+ , qi (xi ) = Y,(vi) ∀xi , qi (xi ) is comprehensive.
Property (i) means that the initial capability is bounded and closed.
Property (ii) means that the initial capability does not expand or shrink
suddenly as the amount of goods changes. Property (iii) means that theinitial capability contains nothing but null vector whenever an individual
has no goods. Property (iv) means the monotonicity of the initial capability with respect to the amount of goods. Property (v) means that there is
the amount of goods at which the individual can reach the technically feasible set. Property (vi) means that the initial capability contains segments
between any point of boundary and the origin.
Next assumptions concern a set of technically possible functioning vector.
Assumtion 2.
Y is compact, 0 ∈ Y and Y \ {0} ̸= ∅.
The boundedness of Y means that no one grows limitlessly throughout
one’s life. The closedness of Y means that every goal of growth is feasible.
The literature on the adaptive dynamic model listed above emphasize
bounded rationality of individuals rather than perfect foresight. Individuals
behave locally with knowledge based on the past behavior. The following
deﬁnition means such individuals.
Deﬁnition 2. (Evolution of capability)
For all i, the capability at time t + 1 (t = 0, 1, · · · ), Qt+1
i , depends on the7chosen functioning vector at time t, yit , in the manner of a correspondence= ψ(yit ).
ψ : Y → Y, y 7→ ψ(y), that is, Qt+1
i
Assumption 3.
The correspondence ψ has the following four properties:(i) compact valued,(ii) continuous,(iii) ψ(0) = {0},(iv) y ∈ ψ(y) for all y ∈ Y.
Properties (i), (ii), and (iii) of assumption 3 correspond to those of assumption 1, respectively. Property (iv) means that, whichever we chose any
functioning vector at time t of Y , the corresponding image at time t + 1
necessarily contains the chosen functioning vector at time t.
Next deﬁnition concerns the principle of individuals’ behavior. Eachtime, individuals with limited knowledge act as locally maximizer. Then,we get a path of the chosen functionings generated by locally maximizing
behavior of each time. We call the path adaptive process.
Deﬁnition 3. (Adaptive process)
Adaptive process {ȳit }t∈N0 is deﬁned by ȳit = argmaxy∈Qti vi (y) and Qt+1=itψ(ȳi ) for t ∈ N0 .
Deﬁnition 4. (Feasible set of functionings)
Let ψ t (Q0i ) = ψ t (qi (xi )) := ∪y∈ψt−1 (qi (xi )) ψ(y) (t = 1, 2, · · · ) with ψ 0 (qi (xi )) :=tqi (xi ), then we deﬁne a correspondence, Yit : R+ → Rm
+ , xi 7→ Yi (xi ), in thefollowing manner:
Yit (xi ) := ψ t (Q0i ) = ψ t (qi (xi )),where ψ t (Q0i ) means a closure of ψ t (Q0i ). Yit (xi ) is called the feasible set offunctionings of individual i at time t.
Remark 1.
Readers may be a little bit diﬃcult to understand deﬁnition 4 intuitively. To
understand deﬁnition 4 clearer, recall the principle of individuals’ behavior
in our model, locally maximizing behavior. The concept of a feasible set offunctionings is not limited to this maximizing behavior. We suppose any
possible principles of individuals’ behavior whatever we think of, which is8

indeed relevant to the concept of freedom. Given an initial capability anda principle of individuals’ behavior, individuals would not need to choose
a functioning vector maximizing their own valuations. Hence, we could
get paths diﬀerent from the adaptive process. Then, we would also get thesequence of capabilities associated with each of them at each time. By doing
so, we could take a union of capabilities over all possible paths at each time.
Thus, deﬁnition 4 means that the union generated by the above procedure
is deﬁned as a feasible set of functionings at each time. Therefore, Qti is asubset of Yit for each t.
Deﬁnition 5. (Temporal measure of freedom)
Applying Rosenbaum’s formula to the dynamic context, we deﬁne the temporal measure of freedom of individual i at time t as:
φti = φti (xi ) :=maxy1 ,y2 ∈Yit (xi ) |y1 − y2 |maxy1 ,y2 ∈Y |y1 − y2 |.

Deﬁnition 6. (Measure of freedom)
The measure of freedom of individual i is deﬁned as:
φi = φi (xi ) := lim φti (xi ) =limt→∞ maxy1 ,y2 ∈Yit (xi ) |y1 − y2 |maxy1 ,y2 ∈Y |y1 − y2 |t→∞.

Remark 2.
The measure of freedom of individual i, φi (xi ), is well deﬁned. Clearly, thedenominator in the above formula has a maximum value and is not zero(see assumption 2). For all t ∈ N0 , i and xi , Yit (xi ) is compact. Especially,
given i and xi , a feasible set of functionings at time t + 1 contains that attime t, that is, Yit+1 (xi ) ⊃ Yit (xi ) (see assumption 3 (iv) and deﬁnition 4).
Thus, maxy1 ,y2 ∈Yit (xi ) |y1 − y2 | is monotonously non-decreasing with respect
to t. Also, it is bounded because of the compactness of Y . Therefore,maxy1 ,y2 ∈Yit (xi ) |y1 − y2 | is convergent as t → ∞.
Before proving proposition 1, we prove the following lemmata.
Assumption 4
For all i, function φti is uniformly convergent to φi .
Lemma 1.
Function φi is continuous.

9

proof.
As is known, if f : A → B, g : B → C are two continuous correspondences,then the composite correspondence g ◦ f : A → C, a 7→ g ◦ f (a), g ◦ f (a) :=∪b∈f (a) g(b) is also continuous. By deﬁnitions, correspondences qi and ψ aswell as the function | · | : Rm × Rm → R+ , (y1 , y2 ) 7→ |y1 − y2 | are eachcontinuous, therefore for all i and t, {|y1 −y2 ||y1 , y2 ∈ Yit (xi )} is continuouswith respect to xi , which implies that φti (xi ) is a continuous function of xi .
Since φti is uniformly convergent to φi , φi is continuous. ■Lemma 2.
Yit (xi ) and φi (xi ) is monotonously non-decreasing with respect to xi .
proof.
Obvious. ■Proposition 1.
There exists an initial distribution of goods(x1 , x2 , . . . , xn ) so that φ1 (x1 ) =∑φ2 (x2 ) = · · · = φn (xn ) =: φ(X) with ni=1 xi = X.
proof.
According to lemma 1, for all i, φi (xi ) is continuous with φi (0) = 0 and
φi (xi ) = 1 for suﬃciently large xi because of Assumption 1 (v). Therefore, for every φ ∈ [0, 1], there exists at least one value of xi fulﬁlling
φi (xi ) = φ because of the intermediate value theorem. Deﬁnex+i (φ) :=∑n++ (φ) :=
max{xi |φi (xi ) ∑= φ}, x−
(φ)
:=
min{x
|φ
(x
)
=
φ},
X
i
i
i
i=1 xi (φ)
i
n
−
−
+
−
and X (φ) := i=1 xi (φ). Note that X (φ) and X (φ) are monotonously
increasing with X − (0) = 0 and X + (1) = ∞. Then, (i) for any X̄ ∈[0, X − (1)), there exist a unique φ̄ ∈ [0, 1] and a unique λ̄ ∈ [0, 1] sothat λ̄X − (φ̄) + (1 − λ̄)X + (φ̄) = X̄ is valid. Choose x̄1 , x̄2 , . . . , x̄n so that+x̄i = λ̄x−1, 2, . . . , n. Then, we obtain φ1 (x̄1 ) =i (φ̄) + (1 − λ̄)xi (φ̄) for i = ∑n>· · · = φn (x̄n ) = φ̄ = φ(X̄) with
i=1 x̄i = X̄. (ii) In case of X̄ =−−>X (1), there exists a unique µ̄ = 1 so that µ̄X (1) = X̄ is valid. Choosex̄1 , x̄2 , . . . , x̄n so that x̄i = µ̄x−i (1) for i∑= 1, 2, . . . , n. Then, we obtain
φ1 (x̄1 ) = · · · = φn (x̄n ) = 1 = φ(X̄) with ni=1 x̄i = X̄. ■+−+Note that x−i (φ̄) = xi (φ̄) implies x̄i = xi (φ̄) = xi (φ̄).

Corollary 1.
φ(X) is monotonously non-decreasing.

10

proof.
Since X + (φ) and X − (φ) are monotonously increasing, φ(X) is monotonously
non-decreasing. ■4Achievement of well-being

In this section, we provide two examples of the well-being achievement. As
we proved above, the well-being freedom is monotonously non-decreasing
with respect to the amount of goods. Likewise, we are likely to speculate
on the well-being achievement in the manner that. That is to say, the wellbeing achievement of each individual cannot decrease as the quantity ofgoods distributed to them increases. However, under the same conditions as
section 3, our examples suggest that there can exist an individual whose wellbeing achievement decreases suddenly at one point of distribution. Somepeople may take our examples as arbitrary ones. We do not have to consider
them to be the expression of a reality, but a logically counter-example tothe intuition that one’s well-being achievement increases if he/she is given
more goods.
Assumption 5.
Valuation function vi (y) is continuous for all i.
Deﬁnition 7. (Global maximum)
Global maximum of individual i (GMi ) is deﬁned as GMi := maxy∈Y vi (y).
Deﬁnition 7 says that, given Y and vi , the global maximum of individual
i is deﬁned as an achievable maximum value of vi on Y .
Deﬁnition 8. (Well-being achievement)
The well-being achievement of individual i is measured by the adaptive
process {ȳit }t∈N0 (see deﬁnition 3) as follows:
αi = αi (xi ) := lim vi (ȳit ).
t→∞

Given vi , we can obtain a sequence of valuation of individual i associatedwith his/her adaptive process. We deﬁne the well-being achievement as alimit of this sequence.
The following examples are the simplest cases that satisfy the assumptions we stated above. The diﬀerence between two examples we intend toemphasize is the number of a point of contact between a valuation function
11and an initial capability; just one point in example 1 and an inﬁnite pointin example 2.
Example 1.
We consider a representative individual i and deﬁne Y, vi , qi and ψ as
Y := {(y1 , y2 )| 0 <= y1 <= 12, 0 <= y2 <= 12},vi (y1 , y2 ) := 2y1 + y2 ,3
3 2
qi (xi ) := {(y1 , y2 )| 0 <= min{ 2 xi , 12}, 0 <= min{ 4 xi , 12}} and= y1 <= y2 <
ψ(ỹ1 , ỹ2 ) :={{(y1 , y2 )| 0 <= y1 <= min{ỹ1 + |ỹ1 − ỹ2 |, 12}, 0 <= y2 <= min {ỹ2 , 12}}<<<<{(y1 , y2 )| 0 = y1 = min {ỹ1 , 12}, 0 = y2 = min{ỹ2 + |ỹ1 − ỹ2 |, 12}}for ỹ1 >= ỹ2 ,>for ỹ2 = ỹ1 ,which are depicted in ﬁgure 1. The initial capability of individual i is given asa rectangle. As the amount of goods increases, the initial capability expands
while it’s northeast apex moving along y2 = 34 y12 up to the point (6,12) andthen moving along the ceiling from (6,12) to (12,12). Clearly, for each xi , vi
touches the northeast apex of the initial capabilities at one point. In case ofa point where the ﬁrst component is greater than the second, the northeast
apex of the initial capabilities evolves to the right side until achieving theeast side of Y . In case of the opposite, the point of apex evolves upward untilachieving the ceiling of Y . So, the set of every achievable point by startingfrom the initial capability consists of the thick parts of the ceiling and theright side of Y , the origin and the point (3,3) which do not move. We cansummarize these observations as ﬁgure 2. According to this ﬁgure, the wellbeing achievement of individual i increases as the amount of goods increases
up to the neighborhood of 2. Yet, her well-being achievement decreases
suddenly when she gets 2. Furthermore, even though she gets more on theinterval (2,5), her well-being achievement is less than 27 at 2 − ε (see table
2).
xi
αi

0
0

···
↗

2−ε
27

2
9

2+ε
18

···
↗

5
27

···
↗

8
36

···
→

Table 2: The relationship between xi and αi .
Insert ﬁgure 1 and 2 here.
Example 2.
This is another extreme case. Deﬁnitions of Y and ψ(ỹ1 , ỹ2 ) are the same12

as those of example 1. We redeﬁne vi and qi asvi (y1 , y2 ) := y1 + y2 ,qi (xi ) = Y ∩ {(y1 , y2 )| y1 + y2 <= xi },
which are depicted in ﬁgure 3. The initial capability of individual i is given
as a southwest area of a dotted line for each xi . Since a slope of vi corresponds that of each dotted line, there exists an inﬁnite number of maximum
valuation points. Consider qi (16) and then choose any point on the corresponding dotted line. If the ﬁrst component of the point is strictly greaterthan the second, the chosen point evolves to the right side until arriving at
the east wall of Y according to ψ. In case of the opposite, the point evolves
up to the ceiling of Y . If the ﬁrst of the point equals the second, the chosen point stays here. Every achievable point from the dotted line of qi (16)consist of the ﬁne dotted parts on the edge of Y and point (8,8). We canalso consider cases of qi (12) and qi (4) in the same manner. Then, we obtaina mountain-shaped symmetrical curve for each initial capability depicted
in ﬁgure 4. If we look at the highest mountain-shaped symmetrical curve,
which corresponds to qi (16) where αi is deﬁned on the interval [4,12], we canﬁnd that it decreases suddenly at 8. We can also ﬁnd out the same featurein the second highest mountain-shaped curve which corresponds to qi (12)and the lowest mountain-shaped curve which corresponds to qi (4). A point
we would like to emphasize is that the well-being achievement with moreamount of goods can be lower than that with less amount of goods. If one
chooses yi = (8, 8) under qi (16), then αi = 16. However, according to ﬁgure
4, under qi (12) which is associated with less amounts of goods compared toqi (16), she can obtain more well-being achievement by choosing any pointin {(y1 , y2 )|y1 + y2 = 12, y1 ∈ (4, 8) \ {6}}. Furthermore, this fact provides
us with a remarkable matter. In our model, each individual is characterized
by a triple (xi , qi , vi ). So, example 2 also suggests that, even if individuals
are completely identical with respect to a triple (xi , qi , vi ), each well-being
achievement can vary largely according to each functioning to be chosen by
them; moreover, even the same individual can increase or decrease her wellbeing achievement depending on the chosen functioning even if she gets thesame amount of goods.
Insert ﬁgure 3 and 4 here.

5

Concluding remarks

Freedom as a main constituent of the individual well-being has been widely
acknowledged. Thus, when thinking about the individual well-being, we fo13

cus on both achievement aspect and freedom one. In this paper, on the basisof the two-way evaluation of well-being à la Sen, diﬀerent aspects of the wellbeing are distinguished. Under these aspects, we constructed a model of thecapability approach through the adaptive dynamics, as opposed to the conventional static treatment of the capability approach. Then, following Sen’s
original idea on distributive justice, we could show the existence of a distribution of goods equalizing well-being freedom (i.e., capabilities) in termsof Rosenbaum’s measure. Furthermore, by dealing with the distributed
amount of goods as variable, which is diﬀerent from D’Agata’s model, we
presented the monotonicity of the well-being freedom with respect to theamount of goods. This fact is a very plausible consequence because it is difﬁcult to understand that one’s well-being freedom would not increase eventhough she gets more. What about the well-being achievement? Our examples lead to a counter-intuitive result. In other words, one’s well-being
achievement can decrease even if he/she gets more under the same conditionsof the well-being freedom. Furthermore, even if individuals are completely
identical with respect to a triple (xi , qi , vi ), each well-being achievement canchange a lot depending on the course of functioning to be chosen.
What does it mean? The fact that an additional distribution of goods cannegatively aﬀect some individuals can characterize a normative quality of
the adaptive process, that is, in the adaptive process, economic distributive
measures may have a limitation in improving the individuals’ well-being. It
is necessary for the capability approach to be put into a broader perspective
considering not only economic measure but also other supplementations suchas communication, cooperation and so on. In this respect, our view is sharedwith Sen. “I shall consider the following types of instrumental freedoms:(1) political freedom, (2) economic facilities, (3) social opportunities, (4)transparency guarantees and (5) protective sequlity.”(Sen, 1999, p.38, italic
in original.)

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17

y2

(3, 12)(6, 12)(12, 12)qi (4)qi (2)(12, 3)(3, 3)y1O

qi (0)
Figure 1: Dynamics of the well-being achievement in example 1.

αi
36
27
18
9
xi
O

2

5

8

Figure 2: The relationship between xi and αi .

18

y2

(4, 12)(12, 12)qi (16)(8, 8)qi (12)(12, 4)qi (4)y1O

qi (0)
Figure 3: Dynamics of the well-being achievement in example 2.

αiqi (16)

20
18
16
14
12

qi (12)qi (4)
4

y1
O

2

4

6

8

12

Figure 4: The relationship between y1 and αi .

19