Alternative Proposals to Measure the Standard of Living when its Development over Time Is Uncertain∗ Wulf Gaertner Department of Economics University of Osnabrück D–49069 Osnabrück, Germany E-mail: WGaertner@oec.uni-osnabrueck.de Yongsheng Xu Department of Economics Andrew Young School of Policy Studies Georgia State University Atlanta, GA 30303, U.S.A. Email: prcyyx@langate.gsu.edu January 2005 ∗ Helpful comments by members of the Human Development Report Office at the UN are gratefully acknowledged. We also wish to thank Eugenio Peluso and Ernesto Savaglio for allowing us to study some of their unpublished work. Finally, we are grateful to Christian Aumann for preparing the figures of this paper. Abstract. Human development is about expanding the choices human beings have to lead lives that they value and is captured by its capability sets which consist of various functioning vectors. The standard of living is then reflected in capability sets. This paper proposes some particular ways of measuring the standard of living available to an agent, be it an individual or a whole country, when the direction of the development of society represented by a reference functioning vector or a reference cone is uncertain. We provide axiomatic characterizations of the various measures proposed. Keywords: Functioning, Capability, Uncertainty, Social Progress, Standard of Living JEL Classification Numbers: C78, D63, D71, D74, O12 1 Introduction Income and wealth are important factors in order to provide and secure a decent standard of living. Economic growth may help to improve this situation. But human development is about much more. As the human development report 2001 asserts, development is about expanding the choices human beings have to lead lives that they value. Fundamental is “building human capabilities – the range of things that people can do or be in life” (Human Development Report (HDR), p. 9). And the report spells out the most basic capabilities for human development: To lead a long and healthy life, to be knowledgeable, and to have access to the resources needed for a decent standard of living. The human development index aggregates these three basic dimensions of human development into a summary measure. As an indicator for a long and healthy life, life expectancy at birth is used from which a life expectancy index is constructed. An education index represents knowledge. The latter concept considers the adult literacy rate and a so–called gross enrolment ratio. From each of these an index is formed and then, the two indices are multiplied by factors of two–thirds and one–third respectively and additively combined to yield the education index. The logarithm of income is used to construct a GDP index that is supposed to capture the notion of a decent standard of living. All three indices thus formed are weighted by one–third and this weighted sum yields the human development index HDI. The human poverty index for developing countries (HPI-1) and the poverty index for selected OECD countries (HPI-2) are constructed in a similar way. HPI-1, for example, is composed of three indices, each is weighted by the same exponent α and in their symmetric linear representation, each index is multiplied by one third (actually both HPI-1 and HPI-2 have the structure of a CES production function). In the HDR of 2001, a value of 3 is attached to the exponent α in order “to give additional but not overwhelming weight to areas of more acute deprivation” (HDR, p. 241). There are various other indices of development which were developed recently. Their structure is roughly the same: they are multidimensional in character but eventually reduced to one numerical index. This reduction procedure involves an exercise in weighting as spelt out in our examples above. Clearly, a change of weights means affecting the aggregate outcome. In relation to the α–exponent, Anand and Sen (1997) admit that “there is an inescapable arbitrariness ” (p. 16) in its choice. Earlier on in their paper, they are more explicit on this issue. “Since any choice of weights should be open to questioning and debating in public discussions, it is crucial that the judgments that are implicit in such weighting be made as clear and comprehensible as possible, and thus be open to public scrutiny” (p. 6). The human development index is a handy tool without any doubt but as Sen, one of the originators of this index, emphasizes the choice of weights is 1 a sensible issue and ultimately a matter for social choice based on valuational arguments (Sen, 2002, p. 7). Sen goes one step further and stresses the time dimension. “When the ingredients of a judgment are diverse, any aggregate index with constant weights (the emphasis is by the author) over its diverse constituent elements would tend to oversimplify the evaluative exercise” (p. 12). One has to be interested in the present situation of countries but sometimes, changes over time are of particular interest. The spread of diseases as well as a more restricted access to clean water resources are important for life expectancy in developing countries. So a higher weight for these aspects would signal particular attention. In more developed countries where death at an early age is no longer a pressing issue, social exclusion measured by long–term unemployment may justify a higher weight in future investigations. Therefore, departures from the current structure and usage of the various indices may seem legitimate. In this paper, we propose a particular way of measuring the standard of living available to an agent as well as to a whole country. The agent or country will be characterized by a capability set consisting of various vectors of functionings possible at any given time. The basis for our theoretical analysis is Lancaster’s (1966) characteristics approach to consumer theory. In this approach consumer goods generate characteristics, and this is done according to a linear “input– output” relationship. The higher the income of a consumer or country, the higher are the maximally possible purchases of a particular good. However, in general, the consumer can choose among different consumer goods and, moreover, the consumer can spend part of his income on commodity a, let’s say, another part on good b, a third part on commodity c, etc. In other words, combinations of different commodities are possible and income–wise feasible. In the space of characteristics, we obtain, due to the linear “production technology”, star–shaped convex spaces. In our context, we assume linear input–output relationships in a twofold way. Consumer goods (but also investment goods, like capital investments in land irrigation or education) generate characteristics and these characteristics lead to different functionings or functioning vectors. These represent health, longevity, literacy and other basic qualities. Given a particular income (for an individual) or a particular budget (for a country), the individual (or country, respectively) can acquire various consumer goods (a country would, additionally, run different investment projects). These yield various functioning vectors and combinations of these generate convex spaces of functionings. These spaces span the agent’s as well as a country’s capability set. Due to the underlying linearity, they are star–shaped. The human development index as well as the other indicators mentioned above produce a real number for each country under investigation. By doing so, a complete ordering over all countries concerned is generated. Both the ordering as well as measured differences in the HDI, for example, between two countries a and b reveal deficiencies. Among the countries with high human development, 2 an HDI value in 1999 of 0.939 for Norway and a value of 0.831 for Slovakia show quite a large gap between the two countries, whereas Slovakia and Hungary seem to be at a very similar stage of human development, the latter’s HDI index being 0.829 for the same year. In this paper, we do not consider indices or real numbers as indicators or benchmarks for comparisons. As has become clear above, we shall focus on vectors of functionings. In order to be judged living a satisfactory life, an agent or a country must have a given functioning vector in her capability set. We readily admit that determining such a reference functioning vector is, conceptually speaking, not easy. For the moment, we wish to assume that this problem has been solved (we just refer to the development and refinement of the HDI and other indices over the last twelve years). To improve her standard of living in terms of functionings, given the uncertainty associated with the development of society (and the world economy), it is not immediately clear along which direction the agent’s or the country’s functioning vector will grow as time progresses. Furthermore – and now we come back to Sen’s remarks on constant weights and the aspect of changes over time, the reference functioning vector may, and perhaps should, change over time in its composition, paying, perhaps, more attention to the access to clean water resources and adult illiteracy in developing countries, and, perhaps, paying more attention to long–term unemployment and youth unemployment in more developed countries. We investigate how the agent’s or country’s standard of living may be measured, given these uncertainties within and among societies. We shall also examine the case where the reference functioning vector lies outside the capability set of the agent or the country considered. The structure of the paper is as follows. Section 2 introduces the basic notation and definitions. Section 3 presents the axioms that we need for our first characterization result. Section 4 states this theorem and provides a proof. Section 5 introduces a deprivation–gap ordering and discusses a second result. Section 6 generalizes our previous approach by introducing a cone with a reference surface. The final section 7 is devoted to some concluding remarks. 2 Basic Notation and Definitions Let IRn+ be the non-negative orthant of the n-dimensional real space. The vectors in IRn+ will be denoted by x, y, z, a, b, · · · , and are interpreted as functioning vectors (Sen (1985, 1987)). For all x = (x1 , · · · , xn ), y = (y1 , · · · , yn ) ∈ IRn+ , define x ≥ y as xi ≥ yi for all i = 1, . . . , n, x > y when xi ≥ yi for all i = 1, · · · , n and xj > yj for some j ∈ {1, · · · , n}, and x >> y when xi > yi for all i = 1, · · · , n.pFor all x, y ∈ IRn+ , we define the distance between them as Pn 2 follows: ||x − y|| = i=1 (xi − yi ) . At any given point of time, the set of all vectors that may be available to the individual is a subset of IRn+ . Such a set will be called the individual’s capability 3 set. We will use A, B, C, etc. to denote the capability sets. Our concern in this paper is to rank different capability sets in terms of the standard of living that they offer to the individual. In particular, we confine our attention to opportunity sets that are (2.1) compact: a capability set A ⊆ IRn+ is compact iff A is closed and bounded, (2.2) convex: a capability set A ⊆ IRn+ is convex iff, for all x, y ∈ IRn+ and all α ∈ [0, 1], if x, y ∈ A, then αx + (1 − α)y ∈ A, (2.3) star-shaped: a capability set A ⊆ IRn+ is star-shaped iff, for all x ∈ IRn+ and all t ∈ [0, 1], if x ∈ A, then tx ∈ A. Let K be the set of all capability sets that are compact, convex and star-shaped. For all A, B ∈ K, we write A ⊆ B for “A being a subset of B” and A ⊂ B for “A being a proper subset of B”. For all A, B ∈ K and all x∗ ∈ IRn+ , let A >x∗ B denote: [whenever x∗ ∈ B, there is a neighborhood, N (x∗ , ǫ) = {x ∈ IRn+ : x ≥ x∗ , ||x−x∗ || ≤ ǫ} where ǫ > 0, of x∗ such that N (x∗ , ǫ) ⊆ A] and [for all b ∈ B with b > x∗ , there exists a ∈ A such that a >> b]. Let x0 ∈ IRn+ be the deprivation vector of functionings below which the individual’s standard of living is judged to be “poor”. Throughout this paper, we assume that x0 is fixed. For all t > 0, let X(x0 , t) = {x ∈ IRn+ : x ≥ x0 , ||x − x0 || ≤ t}. Scalar t measures the Euclidean distance between two vectors in the functioning space. This, of course, presupposes that we can quantify each of the functionings appropriately so that there is a measurement scale common to all functionings considered. An analogy, but only an analogy, is a world where the utility functions of all individuals are fully cardinally comparable. Transformations would have to be positive affine with the same scale factor and the same shift of level for each functioning. Of course, other measures of distance could be considered as well, but we shall not do this in this paper. For all A ∈ K, let  −1 if x0 6∈ A r(A) = maxt {t ∈ IR+ : {x ∈ IRn+ : x ≥ x0 , ||x − x0 || ≤ t} ⊆ A} if x0 ∈ A Figure 1 depicts the maximal t ∈ IR+ for two capability sets A and B when x0 ∈ A ∩ B. 4 f2 A x° B f1 Figure 1: comparison of two capability sets A and B Let  be a binary relation over K that satisfies reflexivity: [for all A ∈ K, A  A], transitivity: [for all A, B, C ∈ K, if A  B and B  C then A  C], and completeness: [for all A, B ∈ K with A 6= B, A  B or B  A]. Thus,  is an ordering. The intended interpretation of  is the following: for all A, B ∈ K, [A  B] will be interpreted as “the degree of the standard of living offered by A is at least as great as the degree of the standard of living offered by B”. ≻ and ∼, respectively, are the asymmetric and symmetric part of . 3 Axiomatic Properties In the following two sections, we present an axiomatic characterization of the standard of living ranking defined below: For all A, B ∈ K, A r B ⇔ r(A) ≥ r(B). We begin by listing a set of axioms. Definition 3.1.  over K satisfies (3.1.1) Monotonicity iff, for all A, B ∈ K, if B ⊆ A then A  B. (3.1.2) Betweenness iff, for all A, B ∈ K, if A ≻ B with x0 ∈ A ∩ B, then there exists C ∈ K such that A >x0 B and A ≻ C ≻ B. 5 (3.1.3) Dominance iff, for all A, B ∈ K, if x0 6∈ B, then A  B, and furthermore, if x0 ∈ A, then A ≻ B. (3.1.4) Domination in Terms of Uncertain Development iff, for all A, B ∈ K, if there exists t > 0 such that X(x0 , t)∩A = X(x0 , t), and B ∩X(x0 , t) ⊂ X(x0 , t), then A ≻ B. The intuition behind Monotonicity is simple and easy to explain. It requires that whenever B is a subset of A, then A is ranked at least as high as B concerning standards of living offered. Betweenness requires that when A is judged to offer a higher standard of living than B relative to the deprivation vector x0 , there must exist a set C such that C >x0 B and A offers a higher standard of living than C, which in turn offers a higher standard of living than B. Dominance requires that whenever the deprivation vector x0 is not achievable in B, the standard of living offered by B cannot be higher than that offered by any other capability set A, and furthermore, if the deprivation vector x0 is achievable under A, then A offers a higher standard of living than B. Domination in Terms of Uncertain Development requires that, for two capability sets A and B, whenever A results from progress made in all dimensions of functioning vectors, while B does not offer this particular kind of progress, the standard of living under A is judged to be higher than that offered by B. 4 A First Characterization Result Theorem 4.1. Suppose  over K is an ordering. Then,  satisfies Monotonicity, Betweenness, Dominance, and Domination in Terms of Uncertain Development if and only if =r . Proof. It can be checked that r is an ordering and satisfies Monotonicity, Betweenness, Dominance and Domination in Terms of Uncertain Development. We now show that if  over K satisfies Monotonicity, Betweenness, Dominance and Domination in Terms of Uncertain Development, then =r . (i) We first show that, for all t > 0 and all A, B ∈ K, if r(A) = t = r(B) and B ∩ X(x0 , t) = X(x0 , t) = A ∩ X(x0 , t), then A ∼ B. Suppose A ≻ B. Then, by Betweenness, there exists C ∈ K such that B lies entirely in C relative to x0 , and A ≻ C ≻ B. Since r(B) = t > 0, C >x0 B, B and C are compact and star-shaped, and for some positive t′ > t and some set C ′ ∈ K, {x ∈ IRn+ : x ≥ x0 , x ∈ C ′ } = X(x0 , t′ ), C ′ ⊆ C, and B ∩ X(x0 , t′ ) ⊂ X(x0 , t′ ). By Monotonicity, C  C ′ and by Domination in Terms of Uncertain Development (henceforth, Domination for short), C ′ ≻ B. Hence, A ≻ C ′ ≻ B. Noting that r(A) = t < t′ = r(C ′ ), we must have X(x0 , t) ∩ C ′ is a proper subset of C ′ ∩ X(x0 , t′ ) = X(x0 , t′ ). By Domination, since A ∩ X(x0 , t′ ) ⊂ X(x0 , t′ ), we obtain C ′ ≻ A which is in 6 contradiction to A ≻ C ′ . Therefore, it is not true that A ≻ B. Similarly, it can be shown that it is not true B ≻ A. Therefore, A ∼ B. (ii) Second, we show that for all A, B ∈ K, if r(A) > r(B) > 0, then A ≻ B. Let A, B ∈ K be such that r(A) > r(B) > 0. Consider A′ , B ′ ∈ K such that A′ ∩ X(x0 , r(A)) = X(x0 , r(A)), B ′ ∩ X(x0 , r(B)) = X(x0 , r(B)) and B ′ ⊂ A′ . Since r(A) > r(B) > 0, such A′ and B ′ exist. From (i), A′ ∼ A and B ′ ∼ B. By Domination, A′ ≻ B ′ . Then, A ≻ B follows from transitivity of . (iii) Third, we show that for all A, B ∈ K, if r(A) > 0 = r(B), then A ≻ B. Note that, since r(A) > 0 = r(B), it must be true that B ∩ X(x0 , r(A)) ⊂ A ∩ X(x0 , r(A)) = X(x0 , r(A)). By Domination, A ≻ B follows easily. (iv) We next show that, for all A, B ∈ K, if x0 6∈ A and x0 6∈ B, then A ∼ B. Since x0 6∈ A, by Dominance, B  A. Similarly, by Dominance, from x0 6∈ B, it follows that A  B. Therefore, A ∼ B. (v) We now show that for all A, B ∈ K, if B = {x ∈ IRn+ : x = tx0 , ∀t ∈ [0, 1]}, x0 ∈ A and r(A) = 0, then A ≻ B for t < 1 and A ∼ B for t = 1. For the first case, since x0 ∈ A and x0 ∈ / B, we obtain, by Dominance, that A ≻ B. Consider now the case that t = 1. Then x0 ∈ B and x0 ∈ A ∩ B. First, since A ∈ K, clearly, B ⊆ A. By Monotonicity, A % B. Suppose that A ≻ B. Then, by Betweenness, there exists C ∈ K such that C >x0 B and A ≻ C ≻ B. Note that, since C >x0 B and x0 ∈ B, r(C) > 0. From (iii) above and r(A) = 0, C ≻ A follows immediately, which is a contradiction to A ≻ C obtained earlier. Therefore, A ∼ B. (vi) From (v), for all A, B ∈ K, if x0 ∈ A ∩ B and r(A) = r(B) = 0, then A ∼ B follows immediately. (vii) To complete the proof, we note that, for all A, B ∈ K, if x0 ∈ A and x0 6∈ B and r(A) = 0, by Dominance, A ≻ B. Therefore, (i) – (vii), together with the transitivity of  complete the proof of Theorem 4.1. 5 Further Properties and A Deprivation-Gap Ordering The ordering r defined in Section 3 and characterized in Section 4 ranks all capability sets A, B ∈ K with x0 6∈ (A ∪ B) equally in terms of the standard of living. This is rather unsatisfactory. In this section, we first propose a ranking 7 rule that avoids this undesirable feature. We then propose several properties to characterize this new ranking rule. For all A, B ∈ K, A > B is to denote that, [for all b ∈ B with b >> 0, there exists a neighborhood N (b, ǫ) = {x ∈ IRn+ : x ≥ b, ||x − b|| ≤ ǫ} where ǫ > 0 of b such that N (b, ǫ) ⊆ A] and [for all b ∈ B, there exists a ∈ A such that a >> b]. Note that whenever A > B, then A >x0 B. For all t > 0 and all x0 ∈ IRn+ , let O(x0 , t) = {x ∈ IRn+ : ||x − x0 || ≤ t}. For all A ∈ K, let  − mint {t ∈ IR+ : {x ∈ IRn+ : ||x − x0 || ≤ t} ∩ A 6= ∅} if x0 6∈ A ∗ r (A) = n 0 0 maxt {t ∈ IR+ : {x ∈ IR+ : x ≥ x , ||x − x || ≤ t} ⊆ A} if x0 ∈ A f2 A x° B f1 Figure 2: comparison of two capability sets A and B in terms of a deprivation–gap ordering Figure 2 depicts the minimal t ∈ IR+ for two capability sets A and B when x0 ∈ / A ∪ B. Define the following deprivation-gap ordering: ∗ For all A, B ∈ K, A r B ⇔ r∗ (A) ≥ r ∗ (B). Consider the following axioms: Definition 5.1.  over K satisfies 8 (5.1.1) Strong Betweenness iff, for all A, B ∈ K, if A ≻ B, then there exists C ∈ K such that C > B and A ≻ C ≻ B. (5.1.2) Regressive Domination iff, for all A, B ∈ K, if there exists t > 0 such that O(x0 , t) ∩ A 6= ∅, and B ∩ O(x0 , t) = ∅, then A ≻ B. Strong Betweenness requires that if the standard of living offered by a capability set A is higher than the standard of living offered by a capability set B, then there always exists a capability set C which has C > B and which offers a standard of living between A and B. Given that A > B implies that A >x0 B, it is straightforward to check that Strong Betweenness implies Betweenness. Regressive Domination is the counterpart of Domination in terms of Uncertain Development, and deals with the situation in which there is a possibility of “regressive development”: if the capability set A dominates the capability set B in the fashion of “regressive development”, then A offers a higher standard of living than B. It can be checked that Regressive Domination implies that, whenever x0 ∈ A while x0 6∈ B, we must have A ≻ B. Thus, Regressive Domination is a stronger requirement than Dominance proposed in Section 3. Theorem 5.2. Suppose  over K is an ordering. Then,  satisfies Monotonicity, Strong Betweenness, Domination in Terms of Uncertain Development and ∗ Regressive Domination if and only if =r . ∗ Proof. We first note that r satisfies Monotonicity, Strong Betweenness, Domination in terms of Uncertain Development and Regressive Domination. Therefore, we need to show if  satisfies Monotonicity, Strong Betweenness, Domination in Terms of Uncertain Development and Regressive Domination, then ∗ =r . Let  be an ordering that satisfies the four properties specified in Theorem 5.2. Note that, since Strong Betweenness implies Betweenness, and Regressive Domination implies Dominance, from the proof of Theorem 4.1, the following must be true: (*) for all A, B ∈ K, if x0 ∈ A and x0 6∈ B, then A ≻ B, and if x0 ∈ A ∩ B, then r ∗ (A) ≥ r ∗ (B) ⇔ A  B. Therefore, it remains to be shown that, if x0 6∈ A ∪ B, then r ∗ (A) ≥ r ∗ (B) ⇔ A  B. Let A, B ∈ K be such that x0 6∈ A ∪ B and r ∗ (A) = r ∗ (B). Clearly, r ∗ (A) < 0 since A is closed, compact, star-shaped, and x0 6∈ A. For such A and B, we need to show that A ∼ B. Suppose to the contrary that A ≻ B or B ≻ A. If A ≻ B, by Strong Betweenness, there exists C ∈ K such that C > B and A ≻ C ≻ B. Note that, since C > B, there exists a positive number t < −r ∗ (A) such that O(x0 , t) ∩ C 6= ∅. Since −r ∗ (A) > t, it must be the case that A ∩ O(x0 , t) = ∅. By Regressive Domination, C ≻ A, a contradiction. Similarly, B ≻ A leads to a similar contradiction. Therefore, A ∼ B. 9 Next, for all A, B ∈ K, if A, B are such that x0 6∈ A ∪ B, r ∗ (A) > r ∗ (B), then A ≻ B follows directly from Regressive Domination. Therefore, for all A, B ∈ K, if x0 6∈ A ∪ B, then A  B ⇔ r ∗ (A) ≥ r ∗ (B). This, together with (*), proves Theorem 5.2. 6 Generalization: The ǫ–Cone Up to this point, it was assumed that the reference vector of functionings is a point on a straight line from the origin of the functioning space. This concept is easy to define and relatively easy to “handle”. But is it realistic? Should the “point” of reference perhaps be a set of points so that different combinations of functionings would be equally suitable as points of reference? Once this idea is introduced, it would be possible to consider “trade–off” relationships between functionings so that we would have various combinations of functionings available which would be equivalent in terms of serving as appropriate reference points. To be somewhat more explicit, let us take health, education and being nourished as basic functionings. A better education will teach people to avoid certain illnesses, also a higher quality of food intakes will reduce health outlays so that in terms of functionings there seem to be “efficiency–frontiers” in the functioning space. And was it justified to treat the path to reference vector x0 as a straight line from the origin or would it be more appealing to conceive of this path as “somewhat meandering” in the north–east direction? Both points can be taken care of by introducing, what we shall call, an ǫ– cone stretching north–east with its vertex at the origin. A cone with angle ǫ and length λ has the property that its north–east boundary with all points having length λ from the origin constitutes a (reference) surface. This surface with concave curvature looks like an efficiency frontier. Its north–east boundary will extend in length, the larger λ which means that trade–offs among functioning vectors are possible over a wider range. This seems plausible since countries with higher levels of functionings apparently have more trade–off possibilities than poorer countries (with lower levels of functionings). Geometrically, λ can also be interpreted as the radius of a virtual circle from the origin, where only a small segment in direction north–east is considered. The larger the radius, the greater the absolute values of the components of the functioning vectors can get that belong to the ǫ–cone. Turned around and viewed from the aspect of deprivation level, the larger λ, the higher are the demands for minimally required vectors of functionings. The latter is a feature that was, of course, already true in the case of a straight line. Note that when ǫ converges to zero, we are back to our original approach. We want to become somewhat more formal now. Let ǫ > 0 and λ > 0. Consider a cone with angle ǫ, vertex O and length λ. We shall call such a cone an ǫ-cone. From now on, an ǫ-cone with length of λ 10 will be denoted by ǫ(λ). In the following discussion, ǫ(λ) is taken as given. Let ∂ǫ(λ) := {x ∈ ǫ(λ) : there exists no y ∈ ǫ(λ) such that y > x}. Let xi ∈ ∂ǫ(λ). For any t ≥ 0, define X(ǫ(λ), xi , t) := {x ∈ IRn+ : x ≥ xi , ||x − xi || ≤ t}. For any capability set A and any xi ∈ ∂ǫ(λ), let t(A, xi ) = max{t : X(ǫ(λ), xi , t) ⊆ A}. For any capability set A and any xi ∈ ∂ǫ(λ), let A(xi , t(A, xi )) = X(ǫ(λ), xi , t(A, xi )). For any ǫ(λ) and any j = 1, . . . , n, let ǫ∗j (λ) = min{xji |xi ∈ IRn+ : xi ∈ ∂ǫ(λ)} and let ǫ∗ (λ) = (ǫ∗1 (λ), . . . , ǫ∗j (λ), . . . , ǫ∗n (λ)). For any t ≥ 0 and ǫ(λ), we define X(ǫ∗ (λ), t) = {x ∈ IRn+ : ||x − ǫ∗ (λ)|| ≤ t, x ≥ ǫ∗ (λ)}. In this section, we propose several new methods of ranking capability sets, based on the ǫ–cone. Consider the first one: (1) For any capability set A, let ŝ(A) = max{t ∈ IR+ : X(ǫ∗ (λ), t) ⊆ A}. Define r̂(A) = ŝ(A) if ŝ(A) ≥ 0 and r̂(A) = −1 if otherwise. Then, for all capability sets A and B, A ≥ǫ∗ B ↔ ŝ(A) ≥ ŝ(B) . Let us explain in more detail what we have done in the last two paragraphs. Given all xi ∈ ∂ǫ(λ), we have constructed a point that consists of minimal components in all n dimensions. Due to the construction of the ǫ–cone, the point lies inside the cone. This point is then taken as the point of reference for finding the largest circle such that its segment in north–east direction still lies in capability set A. In many respects we are now back to sections 2–4, since our reference vector has again become a single point. Instead of looking for the minimal coordinates of all points in ∂ǫ(λ), we could have also determined the maximal coordinates for all dimensions of functionings. This point would lie outside the ǫ–cone (see Figure 3 below). 11 f2 A ε*(λ) ε-cone Figure 3: minimal and maximal coordinates of ∂ε (λ) f1 We just asserted that for this particular proposal we are, to a great extent, back to our first approach. A characterization of this proposal would largely follow our first characterization result in section 4. Therefore, we omit further details. (2) The second proposal concerns a dominance relationship. The situation is depicted in Figure 4. 12 f2 A B ε-cone Figure 4: set inclusion with ε-cone (B ⊂ A) f1 First, we introduce set X(xi , t). For all t ≥ 0 and xi ∈ IRn+ , let X(xi , t) = {x ∈ IRn+ : x ≥ xi , ||x − xi || ≤ t}. For any A, B ∈ K, we say that B is a proper subset of A relative to ǫ(λ), to be denoted by B ⊂ǫ(λ) A, if B is a subset of A and for all t ≥ 0 and all xi ∈ ∂ǫ(λ), X(xi , t) ⊆ B → X(xi , t) ⊆ A, and for some t′ ≥ 0 and some x′i ∈ ∂ǫ(λ), X(x′i , t′ ) ⊆ A and X(x′i , t′ ) is not a subset of B. Define dominance over K as follows: for all A, B ∈ K, if ∂ǫ(λ) ∩ B = ∂ǫ(λ) ∩ A = ∅, then A ∼dominance B, if ∂ǫ(λ) ∩ B = ∅, ∂ǫ(λ) ∩ A 6= ∅, then A ≻dominance B, if ∂ǫ(λ) ∩ B 6= ∅, ∂ǫ(λ) ∩ A 6= ∅, then A dominance B ⇔ [B(xi , t(B, xi )) ⊆ A(xi , t(A, xi )) for all xi ∈ ∂ǫ(λ)]. It is clear from this definition that the dominance relationship can only yield a partial ordering. Definition 6.1.  over K satisfies Cone Dominance iff, for all A, B ∈ K, if ∂ǫ(λ) ∩ B = ∅, then A  B, if ∂ǫ(λ) ∩ B = ∅ and ∂ǫ(λ) ∩ A 6= ∅, then A ≻ B. 13 Strong Domination in Terms of Uncertain Development iff, for all A, B ∈ K, if for all t ≥ 0 and all xi ∈ ∂ǫ(λ), X(xi , t) ⊂ B → X(xi , t) ⊆ A, then A  B, if for some t ≥ 0 and some xi ∈ ∂ǫ(λ), X(xi , t) is a subset of A but X(xi , t) is not a subset of B, then A  B → A ≻ B. Strict Betweenness iff, for all A, B ∈ K, if A ≻ B, then there exists C ∈ K such that C ⊂ǫ(λ) A, B ⊂ǫ(λ) C and A ≻ C ≻ B. Theorem 6.1. Suppose that  is reflexive and transitive. Then,  satisfies Cone Dominance, Strong Domination in Terms of Uncertain Development, and Strict Betweenness if and only if =dominance . Proof. Suppose that  is reflexive and transitive and satisfies Cone Dominance, Strong Domination in Terms of Uncertain Development, and Strict Betweenness. Let A, B ∈ K. We consider the following cases. (i) ∂ǫ(λ) ∩ A = ∂ǫ(λ) ∩ B = ∅. Since ∂ǫ(λ) ∩ A = ∅, by Cone Dominance, B  A. Similarly, since ∂ǫ(λ) ∩ B = ∅, by Cone Dominance, A  B. Therefore, A ∼ B follows immediately. (ii) ∂ǫ(λ)∩A 6= ∅ and ∂ǫ(λ)∩B = ∅. By Cone Dominance, from ∂ǫ(λ)∩A 6= ∅ and ∂ǫ(λ) ∩ B = ∅, it follows that A ≻ B. (iii) ∂ǫ(λ) ∩ A = ∅ and ∂ǫ(λ) ∩ B 6= ∅. In this case, B ≻ A follows from a similar argument as in (ii). (iv) ∂ǫ(λ) ∩ A 6= ∅ and ∂ǫ(λ) ∩ B 6= ∅. Consider first that [B(xi , t(B, xi )) ⊆ A(xi , t(A, xi )) for all xi ∈ ∂ǫ(λ)] and [A(xi , t(A, xi )) ⊆ B(xi , t(B, xi )) for all xi ∈ ∂ǫ(λ)]. By Strong Domination in Terms of Uncertain Development, we obtain A  B and B  A, that is, A ∼ B. Next, consider [B(xi , t(B, xi )) ⊆ A(xi , t(A, xi )) for all xi ∈ ∂ǫ(λ)] and [for some t ≥ 0 and x0 ∈ ∂ǫ(λ), X(x0 , t) is a subset of A but X(x0 , t) is not a subset of B]. Note that, in this case, by the first part of Strong Domination in Terms of Uncertain Development, A  B. Then, by the second part of Strong Domination in Terms of Uncertain Development, A ≻ B follows easily. Similarly, when [A(xi , t(A, xi )) ⊆ B(xi , t(B, xi )) for all xi ∈ ∂ǫ(λ)] and [for some t ≥ 0 and x0 ∈ ∂ǫ(λ), X(x0 , t) is a subset of B but X(x0 , t) is not a subset of A], B ≻ A follows immediately. Finally, consider [B(x1 , t(B, x1 )) is a proper subset of A(x1 , t(A, x1 )) for some x1 ∈ ∂ǫ(λ)], and, [A(x2 , t(A, x2 )) is a proper subset 14 of B(x2 , t(B, x2 )) for some x2 ∈ ∂ǫ(λ)]. We need to show that not(A  B) and not(B  A). If A ∼ B, by Strong Domination in Terms of Uncertain Development, from [B(x1 , t(B, x1 )) is a proper subset of A(x1 , t(A, x1 )) for some x1 ∈ ∂ǫ(λ)] implying X(x1 , t(A, x1 )) is a subset of A and X(x1 , t(A, x1 )) is not a subset of B, we must have A ≻ B, a contradiction. From A ∼ B, by Strong Domination we arrive at B ≻ A via an analogous argument, again a contradiction. If A ≻ B, by Strict Betweenness, there exists C ∈ K such that B ⊂ǫ(λ) C, C ⊂ǫ(λ) A, and A ≻ C ≻ B. Note that [B(x1 , t(B, x1 )) is a proper subset of A(x1 , t(A, x1 )) for some x1 ∈ ∂ǫ(λ)], and, [A(x2 , t(A, x2 )) is a proper subset of B(x2 , t(B, x2 )) for some x2 ∈ ∂ǫ(λ)]. Therefore, A is not a subset of B and B is not a subset of A in this case. Note, however, that C is proper subset of A and B is a proper subset of C. Hence, B must be a proper subset of A, a contradiction. Following a similar argument, B ≻ A leads to another contradiction. Therefore, it must be the case that not(A  B) and not(B  A). (3) The third case which still has to be worked out in detail concerns the situation that B(x1 , t(B, x1 )) is a proper subset of A(x1 , t(A, x1 )) for some x1 ∈ ∂ǫ(λ), and A(x2 , t(A, x2 )) is a proper subset of B(x2 , t(B, x2 )) for some x2 ∈ ∂ǫ(λ). This case is depicted in Figure 5. A simpler situation is shown in Figure 6. 15 f2 A xi ε-cone xj B Figure 5: comparison of sets A and B (A ∩ B ≠ ∅) f2 f1 t*(A) A t*(B) B ε-cone Figure 6: comparison of sets A and B (A ∩ B ≠ ∅) a simpler case f1 Let us consider the following criterion: For any capability set A, let t∗ (A) = min{t(A, xi ) : for all xi ∈ ∂ǫ(λ)}. Define r(A) = t∗ (A) if ∂ǫ(λ) ⊆ A and r(A) = −1 if otherwise. Then, for all capability sets A and B, A maximin B ⇔ r(A) ≥ r(B). 16 We have argued at length in this paper that the development of the reference vector over time is, to a certain degree, uncertain. This means that in case of a reference surface ∂ǫ(λ), there is no a priori reason to discriminate among the points on the surface. The dominance relation is only applicable for cases where B(xi , t(B, xi )) ⊆ A(xi , t(A, xi )) for all xi ∈ ∂ǫ(λ). Let us define A(xi , t∗ (A)) as the smallest “north–east circle” still contained in A with xi ∈ ∂ǫ(λ) as its “origin” or vertex. We also define B(xj , t∗ (B)) with xj ∈ ∂ǫ(λ) in an analogous way. The simplest case would be a situation in which B(xj , t∗ (B)) ⊆ A(xi , t∗ (A)) with xi = xj . For this case, for example, the newly introduced criterion would yield that A maximin B. We now exploit the idea that movements along the north–east reference surface should, under uncertainty, be considered as equivalent. We take B(xj , t∗ (B)) with xj ∈ ∂ǫ(λ) and transfer this north–east ball along the reference surface to xi , where the smallest north–east circle with respect to A is located. Once this ball has settled at xi ∈ ∂ǫ(λ), we shall define it as B̃(xj → xi , t∗ (B)). And we now claim that if B̃(xj → xi , t∗ (B)) ⊆ A(xi , t∗ (A)), then A  B. The axiom which would allow for these movements along ∂ǫ(λ) should be called an axiom of equivalent movements along the reference surface under uncertainty. Again, the simplest case would be a situation where B(xj , t∗ (B)) ⊆ A(xi , t∗ (A)) with xj = xi , but the new axiom of equivalent movements would allow us to check for set inclusion of any two north–east balls with origins on the reference surface. 7 Concluding Remarks In this paper we have proposed several new ways to measure the standard of living and to compare the standards of living of two persons or, more importantly, two countries. The basis for our approach is Sen’s proposal to consider functioning vectors and capability sets. The human development index constructs a real number for each country under investigation. Comparisons among different countries are done by calculating numerical differences of their respective development index. The issue of determining the appropriate weights for those components that enter a development index is central for constructing this index. These weights can and will change over time. Consequently, the aggregate real number will vary under different weighting schemes. In this paper, comparisons among different countries are based on a reference functioning vector that may change as time progresses or on a reference surface. The different functionings that constitute the reference vector undergo a re–evaluation over time, also in relation to each other. Functionings are the focus of attention in many investigations on human development these days. Therefore, we think that the approach formulated here can be used for real–world applications. It should be interesting to see how the 17 rankings according to the currently used HDI would fare in comparison with our new measures. To conclude this paper, two remarks are in order. First, due to our assumption of the linear “production technology” in producing functioning vectors, we have focused on capability sets that are compact, convex and star-shaped. We realize that the linear “production technology” is a restrictive assumption. It would, therefore, be interesting to relax this assumption and examine the problem of ranking capability sets thus obtained in terms of standards of living offered. Secondly, it is implicitly assumed, given the uncertainty associated with the development of society, that directions along which the agent’s or the country’s functioning vector will grow as time progresses are “equally likely”. One may argue that, even though it is not possible to know the precise direction along which the agent’s functioning vector will grow as time goes by, a range of possible directions can be identified and this range is much narrower than the range of all possible directions implicitly assumed in our framework. It would then be interesting to explore ways of measuring the standard of living offered by capability sets when the information about the range of possible growth directions becomes available. But we leave this point for another occasion. 18 References Anand, S. and A. Sen (1997): Concepts of Human Development and Poverty: A Multidimensional Perspective. In: Poverty and Human Development. Human Development Report Office, The United Nations Development Programme, New York. Lancaster, K.J. (1966): A New Approach to Consumer Theory. Journal of Political Economy 74, 132–157. Sen, A. (1985): Commodities and Capabilities. Amsterdam: North Holland. Sen, A. (1987): The Standard of Living. Cambridge: Cambridge University Press. Sen, A. (2002): Foreword to “Human Development: Concepts and Measures”, ed. by S. Fukuda–Parr and A. K. Shiva Kumar. Forthcoming. United Nations Development Programm (UNDP) (2001): Human Development Report 2001. New York: Oxford University Press. 19