MSc Economics Extended Essay Candidate Number: 65794 Option: Public Economics Social Welfare, Individual Well-being and Opportunity Sets 5900 words approx.
1. Social Welfare and Social Choice Theory The problem of ranking economic states of affairs in terms of some notion of goodness or desirability is of central importance for public economics. It is with respect to such an ordering 1 that, for example, a tax schedule is deemed optimal or an income distribution considered more desirable than another. The traditional approach for making normative judgements of this kind is to employ a Bergson- Samuelson social welfare function (henceforth swf) W ( ), representing some notion of aggregate well-being : state x is considered better for society than state y if and only if W (x) > W (y). Value judgements as to what constitutes the good of society are embodied in the specification of the swf W ( ). For example, a widely held view is that social welfare should depend only on the welfare of the individuals in the society, so that W (x) = W (w 1 (x),..., w n (x)), where w i (x) is the well-being of individual i in state x. Moreover, the utilitarian outlook that pervades economic thought manifests itself through the common specification letting the swf be the sum of its arguments 2, that is W (x) = i w i(x). On the other hand, it is possible to argue, as Rawls (1971) did, that the good of society should consists of the wellbeing of its most disadvantaged individual, so that W (x) = min i w i (x). In view of this multiplicity of views, it is natural to ask whether it is possible to provide some justification for choosing, say, a utilitarian versus a Rawlsian functional form for our swf. Since different specifications will usually give rise to different rankings of social states, this is an issue of fundamental importance for the normative use of the swf approach. 1.1. Arrow s Impossibility Theorem The first step for answering this question was taken by Arrow (1951) who singlehandedly gave birth to the field of social choice theory. In his classic Social Choice and Individual Values, the object of study is an aggregation process or rule 3 that takes profiles of individual orderings of social states to produce corresponding social rankings. The question he asked is whether it is possible to construct such an aggregation rule that satisfies the following mild-looking conditions: 4 (u) Universal domain: all possible profiles of individual rankings are allowed as inputs to the aggregation rule; (p) Pareto principle: if a profile is such that every individual ranks x higher than y then the corresponding social ranking should also rank x higher than y; (i) Independence of irrelevant alternatives: if two profiles coincide in their ranking of x with respect to y, then the corresponding social rankings should also coincide in the way they rank x against y. 1 The terms ranking and ordering will be used interchangeably throughout to mean a complete and transitive binary relation, unless otherwise stated. 2 This additive specification is often motivated by analytical convenience rather than a belief in the tenets of utilitarianism 3 Confusingly, these aggregation rules are also called social welfare functions by Arrow (1951), but they are to be distinguished from the aforementioned Bergson-Samuelson social welfare functions (swf). 4 Strictly speaking, these are not the conditions in Arrow s (1951) original contribution, but those used by Sen s (1970b) in his equivalent formulation. 1
His celebrated result, known as Arrow s Impossibility Theorem, states that the only aggregation rules that satisfy these three conditions are dictatorial, that is, they are such that there is one individual whose own ranking always coincides with the social one, regardless of those of the others. Therefore, if we add the requirement that the aggregation rule be non-dictatorial, then Arrow s result seems to effectively declare the impossibility of sensible aggregation procedures. 5 Despite its great analytical beauty, the relevance of this result for our original question is not immediately obvious. Given the way it was originally formulated, the theorem seems to apply mainly to the problem of social decision making (hence the name social choice theory), and in this guise it could be regarded as a generalisation of the so-called Condorcet paradox. Indeed, this was the dominant interpretation until Sen (1970b) embedded Arrow s result in an informationally richer setting. 1.2. Sen s swfl approach and welfarism In his seminal Collective Choice and Social Welfare, Sen considers social welfare functionals (swfl), namely, rules for assigning social rankings to profiles of individual welfare functions w i ( ), where w i (x) represents the well-being of individual i in state x, precisely as mentioned previously. The significance of swfls for our problem is clear: the objective is to find conditions on a swfl which ensure that it will always produce a social ordering that ranks states in terms of a swf (or any positive monotonic transformation thereof) with a particular functional form. For example a utilitarian swfl would produce social orderings that rank states in terms of the sum of the individual welfare functions, whereas a Rawlsian (leximin) swfl would give rankings based on the minimum 6 level of individual welfare. In this setup, the universal domain (u), Pareto (p) and independence (i) conditions appropriately restated can be shown to imply that the social ranking depends only on the profile of individual welfare functions 7. In terms of the swf approach, acceptance of the above three conditions implies that W ( ) must be a function of the w i ( ) only. But more importantly, the exclusive role of welfare information in the determination of the social ranking, known in the literature as welfarism, rules out deontological procedures that take into account the concept of rights and liberty 8 or even the way in which states of affairs are brought about, such as Nozick s (1974) theory of entitlements. 5 Following its publication, controversy raged over the relevance of Arrow s theorem for the derivation of a social ranking from one particular profile of individual orderings. In particular, it was argued that since Arrow used multi-profile conditions such as independence of irrelevant alternatives (i), his result need not apply to the single-profile case. In fact, it turned out that exactly analogous impossibilities arise when the above conditions are appropriately modified to the single-profile case, as shown by Kemp and Ng (1976) and Parks (1976), among others. 6 In case of ties, the ranking would be decided lexicographically by comparing the next lowest welfare levels. 7 Strict speaking, welfarism requires the Pareto indifference (or strong Pareto) condition rather than the weak Pareto condition given above, that is, we need the social ranking to respect unanimous indifference between two states. 8 Sen s (1970a) famous Impossibility of a Paretian liberal examined the conflict between the Pareto principle and a particular interpretation of libertarian rights, sparking off the study of rights in social choice theory. See Suzumura (1983, ch. 7) for a partial survey of the resulting literature. 2
1.3. A taxonomy of informational assumptions The richness of Sen s extended framework comes from the possibility of imposing informational invariance conditions of the following form: if a profile of individual welfare functions is transformed in a way that preserves all the information contained in it, then the corresponding social ranking should also be preserved 9. By specifying what transformations leave welfare information unchanged, it is possible to express measurability and interpersonal comparability properties of the individual welfare functions. For example, if the social ranking corresponding to the profile (w 1,..., w n ) has to stay the same after every w i ( ) is transformed by a different positive monotonic function φ i ( ), then we are saying that individual welfare functions are ordinal and non-comparable. Similarly, if the allowed transformations are those where the φ i ( ) are different positive affine functions, i.e. φ i (w) = a i + b i w with b i > 0, then the w i ( ) are said to be cardinal and non-comparable. If instead, information-preserving transformations are those with φ i = φ for all i for some positive monotonic function φ, then the w i ( ) are ordinal and levelcomparable. This is because statements such as w i (x) > w j (x), meaning that in state x the level of well-being of individual i is higher than that of individual j, can now be used in the construction of the social ranking, since they are preserved by the allowed invariance transformations. However, if the φ i ( ) are required to be of the form φ i (w) = a i + b w, then the w i ( ) are said to be cardinal and unit-comparable; now statements about welfare differences such as w i (x) w i (y) > w j (y) w j (x), meaning that in moving from state y to state x the welfare gain of individual i is greater than the welfare loss of individual j, can enter the aggregation process. Putting the two together, so that the only information-preserving transformations are those with φ i ( ) being the same positive affine function φ(w) = a + b w for all i, the individual welfare functions are cardinal and fully comparable, so that both levels and differences of welfare can be compared across individuals. 1.4. (Im)possibilities with(out) interpersonal comparability How does this help us understand the implications of Arrow s Impossibility Theorem? It turns out that the linchpin of his result is the assumption stated explicitly but not in formal terms that interpersonal comparison of utilities has no meaning and, in fact, that there is no meaning relevant to welfare comparisons in the measurability of individual utility (Arrow, 1951, p. 9). In terms of the taxonomy given above, this translates to the assumption that the individual welfare functions are ordinal and non-comparable. His theorem can then be reformulated as saying that there exists no swfl that satisfies the universal domain (u), Pareto (p), independence (i), non-dictatorship and ordinal non-comparability conditions. Allowing cardinal welfare information without interpersonal comparability does not in any overcome this impasse, as the result holds even if ordinal non-comparability is replaced by cardinal non-comparability (Sen, 1970b, Theorem 8*2). Intuitively, the poverty of welfare information inherent in the inability to make any comparisons of levels or differences of well-being, in conjuction with the requirement that this be the only allowed information as implied by the welfarist trio (u), (p), 9 This is the formulation of invariance requirements in Roberts (1980). Sen (1970b) used a different, but equivalent formulation in terms of comparability sets. 3
(i) allows only very crude, namely dictatorial, aggregation procedures, and is thus responsible for precipitating Arrow s impossibility. On the other hand, this gloom is completely lifted as soon as we introduce interpersonal comparability of some kind. If we allow level- but not unit-comparability, so that equity considerations come into play, then Rawlsian swfls are acceptable methods of aggregation. Allowing unit- but not level-comparability, so that welfare difference comparisons between gains and losses can be made, gives rise to utilitarian swfls. Moreover, with full comparability, where both utilitarian and leximin swfls are possible, one can discriminate between the two by imposing (roughly speaking) equity and continuity axioms that respectively rule out the former and the latter 10. Thus social choice theory gives us axiomatic characterisations for utilitarian and Rawlsian methods of social evaluation, and provides us with a precise, formal language in which to frame the age-old debate of how the good of society should be determined. The crucial message that comes out of research on swfls is that informational issues play a fundamental role in the construction of social rankings, and in particular, interpersonal comparisons of well-being are absolutely indispensable for the traditional welfarist swf-based approach to social evaluation. 2. Individual Welfare and Utility Given the fundamental and usually exclusive role played by the information contained in individual welfare functions, it is natural to ask what really constitutes a person s well-being. The approach adopted by most economists is to use utility as the relevant concept of welfare. Unfortunately, this practice can give rise to considerable ambiguity and confusion due to the multiplicity of meanings and interpretations attributable to this versatile term. It is therefore useful to spell out in some detail the most important ways in which the word utility is used in the economic literature. 2.1. Utility as a subjective ranking In its most abstract and formal incarnation, a utility function is nothing more than a real-valued representation of an individual s binary relation representing her preferences over alternatives. Thus u(x) > u(y) only means that alternative x is preferred to or considered better than alternative y. In the context of welfare analysis, the individual s preference relation embodies subjective judgements that rank states of affairs in terms of her own well-being; her utility function is only a mathematically convenient way of expressing these judgements, rather than a measure of any property of these states. Then u i (x) > u i (y) is simply interpreted as saying that individual i considers state x to be better than state y in terms of her well-being. The problem with this interpretation is that, as it stands, it is difficult to make interpersonal comparisons of utility. The assertion u i (x) > u j (x) is meaningless since it is always possible to reverse the inequality by scaling up u j ( ) without altering anyone s individual ranking of states. This is precisely the reason that induced Arrow (1951) to affirm that interpersonal comparison of utilities has no 10 See Sen (1977) and Blackorby, Donaldson, and Weymark (1984) for accessible overviews of these results. 4
meaning, leading to his famous impossibility. Moreover, as mentioned above, cardinalising people s utility functions by considering, say, their rankings of lotteries over states does not in any way resolve this difficulty. 11 2.2. Utility as objective valuation One way of dealing with this problem is to impose more structure on the states of affairs and to introduce some notion of objective valuation. Concretely, a state x can assumed to be characterised by some vector (x 1,..., x n ), where the component x i measures individual i s quantity of some good (which could be multidimensional) in that state; this good is what determines each person s well-being in a state. As an illustration, in the problem of income inequality measurement, x would be an income distribution and x i would be the level of income of individual i. Furthermore, it is assumed that there is a common welfare ranking over quantities of this good (say, income), represented by a utility function u( ); individual i s utility function over states is then given by u i (x) = u(x i ). This representative individual formulation is still purely formal, since as it stands, the common ranking 12 represented by u( ) is not constructed using any particular notion of well-being; all that matters is that it be the same for all individuals. This unanimity then enables interpersonal comparisons of utilities: u i (x) > u j (x), now equivalent to u(x i ) > u(x j ), means that individual i s deal in state x is better than that of individual j, according to the common valuation embodied in u( ). But how do we arrive at such a common ranking? One way is to consider a feature of the good under consideration which enables it to be judged by some objective criteria: one could try to justify a common increasing concave utility function over money by arguing that more money is always better than less, and that people are naturally (and perhaps even similarly ) risk averse. 13 The validity and feasibility of this kind of approach clearly depends on the problem at hand, which will determine both what the relevant good is, and the extent to which it is possible to agree on some objective valuation of it. For some problems this might be quite a sensible way to proceed: for example, Sen (1985b, 1987, 1992) argues that in conditions of deprivation, a person s well-being is given by the vector of functionings that she achieves, that is, objective features of one s life such as being nourished, sheltered and free of disease 14, and that even if different people might rank these functionings in different ways, the differences in this case 11 Nonetheless, as Hammond (1991) mentions in his survey of interpersonal comparisons of utility, there are have been proposals for making von Neumann-Morgernstern utility functions interpersonally comparable by assuming that they are bounded above and below, and normalising them by affine transformations to make their upper and lower bounds equal, say 1 and 0, respectively. Unsuprisingly, such proposals have not gained much acceptance due to the lack of any ethical or substantive justification. 12 At the risk of belabouring the point, it should be stressed that this is not a common ranking of states but of the good that characterises them. Using the example above, it s a ranking of incomes, not of income distributions; the same utility function over money can be consistent with many different rankings of income distributions depending on the social welfare function used. 13 One could even go as far as arguing, as Harsanyi (1955) did, that there exist fundamental preferences on lotteries over states and personal characteristics; these are the preferences that an impartial observer uses to makes judgements about states of affairs behind a veil of ignorance to use Rawls (1971) famous expression as to his personal characteristics. See Weymark (1991) for an overview of the debate on this controversial claim. 14 Sen envisages other more complex functionings, such as having self-respect or taking part in the life of the community, as being constitutive of a person s well-being. 5
are, by their very nature, more limited; the intersections of the rankings typically quite large; and the practical problems of sensible aggregation less exacting. (Sen, 1985b, p. 56) 2.3. Utility as consumer preferences On the other hand, other problems are not so amenable to this common-valuation approach, as in, for example, multi-person analysis of commodity taxation. In this context, a state is a vector of competitive budget sets, one for each consumer, where these are determined by a combination of prices, commodity taxes and income. Each consumer is assumed to have preferences over commodity bundles representable by a utility function u i ( ). Faced with a budget set, he chooses the (not necessarily unique) bundle that maximises u i ( ) and consumes it. The value of u i ( ) associated with this maximal bundle is called his indirect utility, and is the notion of welfare used to rank budget sets. If everyone had the same utility function, this case wouldn t be any different from that of common objective valuation presented in section 2.2. However, in our particular example, if we wish to study distributional features of optimal taxation that arise from heterogeneity of preferences, than it is useless to assume the existence of a common utility ranking of bundles. But as long as consumers have different preferences and hence different utility functions, we run into exactly the same problems of interpersonal comparability that we saw in section 2.1: the utility functions of different consumers could be transformed by separate positive monotonic transformations, where these transformations are directly reflected in the indirect utility functions. It is therefore impossible to make interpersonal comparisons of any kind using this abstract preference-based interpretation of welfare. 2.4. Utility as a mental state However, if we add substantive independent content to the notion of utility, on top of its purely formal role as a device for modelling consumer choice, interpersonal comparisons are apt to become more tractable. A possibility is to let u i (x) represent the happiness or desire-fulfilment achieved by individual i through consumption of bundle x. Thus a person s consumption choice is guided by utility in the sense of something he derives or enjoys from a bundle of goods, in line with the non-technical meaning of the term 15. If utility is considered some mental state or sensation, it is not altogether impossible to make rough comparisons across people, something that is done quite often in informal terms. On the other hand, such comparisons are the exception rather than the norm and there is (yet) no scientific basis for making them in a systematic way. 16 15 Despite declarations that the only meaning the concepts of utility can be said to have is their indications of actual behavior (Arrow, 1951, p. 9) and the ostensible preponderance of its purely formalistic interpretation, many economists find some intuitive appeal in the idea of utility as a measure of happiness or desire-fulfilment. 16 A separate criticism is that even if happiness and desire-fulfilment were accurately comparable, the problem of expensive tastes and adaption to deprivation would still remain: someone hooked on plovers eggs and pre-phylloxera claret is unhappy unless he gets these extraordinary provisions (Sen, 1985a, p. 197), whilst a person who is ill-fed, undernourished, unsheltered and ill can still be high up in the scale of happiness or desire-fulfilment if he or she has learned to have realistic desires and to take pleasure in small mercies (Sen, 1985b, p. 21). This indeed a serious problem, but it is a veritable ethical quagmire and the issue cannot be given adequate treatment here. 6
3. Well-being and Opportunities Suppose that somehow we manage to come up with some notion of utility that overcomes all the difficulties mentioned above by taking into account objective circumstances, personal preferences and mental states such as happiness or desirefulfilment. It is then possible to believe that utility, thus defined, is a sound and ethically appealing measure of individual well-being. But even then, it can be argued that such a concept still does not provide a wholly satisfactory basis for the evaluation of social states. As in the commodity taxation example given in Section 2.3, an economic state of affairs can often be characterised as a profile of budget sets, one for each individual in the society. More generally one could also consider opportunity sets, that is, sets of alternatives from which individuals can freely choose. In this context, the hypothetical perfect utility function (which could be different across individuals) would measure the well-being that an individual achieves through the choice and consumption of a particular alternative (or commodity bundle). The most natural way i.e., the prevailing one in the economic literature to evaluate the goodness of particular opportunity sets, and hence of states of affairs, is to use the individual s indirect utility function, that is, the utility from the best available alternative. But this would rank opportunity sets, and hence states, solely on the basis of achieved well-being, completely ignoring any concept of opportunities for achieving such well-being. One argument against the indirect utility approach asserts the importance of the act of choice for an individual s well-being: [c]hoosing may itself be a valuable part of living, and a life of genuine choice with serious options may be seen to be for that reason richer. (Sen, 1992, p. 41) Thus freedom of choice, the possibility of choosing meaningfully among alternative, has an instrumental role in increasing a person s well-being. But perhaps, even more importantly, the indirect utility approach can be rejected as the basis of social evaluation if we believe that a person s advantage and condition in a social state is best captured by the opportunities he has to attain well-being. We might believe that a good and just society is one that offers rich and plentiful opportunities for all its members, and that to improve their lot it is necessary to increase their freedom to achieve welfare, in short, their well-being freedom is what matters to society. 17 In fact, this concept of freedom for welfare is a fundamental component in Sen s vision for welfare evaluation. In numerous writings (Sen, 1985a,b, 1987, 1992), he proposes the notion of capabilities, namely the set of functionings (as described above) that a person can choose from, as being the notion that best describes individual advantages and conditions. But the notion of well-being freedom is more general than Sen s capabilities, and indeed, after Sen s (1988, 1991) calls for the inclusion of the concept of freedom in economic analysis, there exists now a whole literature dealing with freedom of choice 18. What will be of interest to us are contributions from this literature that suggest ways to make the concept of well-being freedom operational as the basis of social evaluation. 19 17 To use Berlin s (1969) terminology, this is a positive notion of freedom, different from the negative one expounded in the writings of philosophers of the libertarian tradition, from Mill to Nozick. 18 The concept is quite an elusive one, and the object of much of the literature is to provide axiomatic characterisations for it, which we will not dwell upon. 19 The papers reviewed in the next section interpret their rankings as capturing either wellbeing or freedom of choice, but we will instead evaluate their contributions as rankings in 7
4. Rankings of Opportunity Sets The problem at hand is to find ways for ranking opportunity sets in terms of the opportunities they offer for achieving welfare. We will deal with this question in rather formal terms: the alternatives among which the agent chooses can be interpreted as commodity bundles, functionings or just any kind of resources. What is important is that each alternative offers the individual some well-being, which, for ease of exposition, we ll simply call utility. The following notation will be used: Ω is the finite universal set of alternatives a, b, c,... ; u( ) is the individual s utility function over alternatives; 2 Ω is the set of subsets of Ω and its elements A, B, C,... are opportunity sets; is a transitive binary relation on 2 Ω ranking opportunity sets, with and being the asymmetric and symmetric parts of, respectively. Indirect Utility The simplest ranking is the traditional indirect utility ranking mentioned above: Ranking A B max x A u(x) max y B u(y). According to this ranking, removing all the alternatives except the best one(s) will result in no change in opportunities, so that freedom considerations play no role whatsoever. This unsatisfactory neglect of freedom led Pattanaik and Xu (1990) to seek a way of directly incorporating such considerations into the ranking of opportunity sets. They propose the following three axioms: (ins) Indifference between No-choice Situations: {x} {y} for all x, y Ω. (ind) Independence: A B A {x} B {x} for all x X such that x / A and x / B. (sm) Strict Monotonicity: {x, y} {x} for all x, y Ω with x y. The interpretation of (ins) is that if two opportunity sets only offer one alternative then they both offer no freedom and hence should be ranked equally. Note that this implies that welfare or utility plays no role whatsoever in ranking (at least) singleton opportunity sets. Independence (ind) requires that the ranking between two opportunity sets should not be altered by the addition of a common alternative and can thus be seen as a separability condition: the effect of the additional element is assumed to be independent of the opportunity sets it is being added to, ruling out menudependence of the alternatives. Strict monotonicity (sm) is justified by Pattanaik and Xu by saying that some choice is strictly better than no choice, but its implications are rather stronger. In conjunction with (ind), it implies that A {x} A for all A 2 Ω and x Ω. From this it follows (by transitivity) that if B A, then A B, i.e., expansion of opportunities is always strictly preferred to the status quo, so (sm) is quite a strong assumption. Cardinality Using these axioms, Pattanaik and Xu (1990) prove that a ranking satisfies ranking (ins), (sm) and (ind) if and only if it ranks opportunity sets on the basis of their cardinality, i.e., A B A B, where A is the number of elements in A. terms of freedom to achieve well-being. Also, the notation has been changed from the original papers in order to provide a coherent presentation. 8
This cardinality rule is a very simplistic one for assessing opportunities and the authors themselves reject it as being trivial. In a reply to their paper, Sen (1990) points the blame at axiom (ins), which, as mentioned earlier, dissociates freedom from well-being: [f]reedom is not just a matter of having a larger number of alternatives; it depends on what kind of alternatives they are. (Sen, 1990, p. 470) 4.1. Utility-based rankings Taking into account such objections, Klemisch-Ahlert (1993) proposes the following axiom as a replacement for Pattanaik and Xu s (ins): (ex) Extension 20 : u(x) u(y) {x} {y}. Klemisch-Ahlert describes u(x) as a strictly positive weight given to alternative x, but in our context it is appropriate to interpret it as its utility. In fact, given an everywhere positive utility function u( ), it is natural to ask whether (ex), together with (ind) and (sm), characterises particular rankings of opportunity sets in the same way that (ins) gives rise to the cardinality rule. Klemisch-Ahlert notes that the ranking defined by A B u(x) u(y), (1) x A y B satisfies (ex), (ind) and (sm), but it is not the only one that does. In fact, any monotonically increasing transformation of u( ) will give rise to a different ranking that satisfies these three axioms. But more importantly, these axioms do not even guarantee the above additive form. Lexicographic Indeed, Bossert, Pattanaik, and Xu (1994) propose a ranking of opportunity ranking sets that satisfies (ex), (ind)and (sm), in terms of a lexicographic ordering on the utilities of the alternatives, i.e., A B (u(a (1) ),..., u(a (r) ), 0,..., 0) l (u(b (1) ),..., u(b (s) ), 0,..., 0) where (a (1),..., a (r) ), (b (1),..., b (r) ) are the elements of A and B in descending order of their (strictly positive) utility and l is the lexicographic ordering on R n, with n = Ω. Although this ranking differs from the indirect utility one in considering an expansion of an opportunity set as an improvement, it still places too much importance on the (ordinal) utility of the best alternative. As long as the best element x of A is ranked higher in terms utility than the best one y in B, A will be judged to have more well-being freedom than B. One might instead think if x is slightly better than y, it could be possible to add enough alternatives to B to make it rank higher in terms of opportunities. To make such statements it is clear that utility that is more than ordinal is required. This is the route taken by Bossert (1997), who starts with numerically significant 21 utility and examines the problem of characterising rules that assign opportunity set rankings u to utility functions u( ), rather than obtaining one 20 Thus called because is an extension of the ranking over Ω embodied in u( ) to a ranking of 2 Ω. 21 That is, the numbers themselves has meaning so that no transformation will leave all the significant information intact. 9
ranking for a fixed utility function 22. The technical details will not concern us here, but the essence of the main result is that imposing (ex), (ind) and a continuity condition 23 gives rise to an additive ranking rule such that A u B g(u(x)) g(u(y)) x A y B for some increasing function g : R R. The continuity condition effectively rules out lexicographic rankings like the one studied by Bossert et al. (1994), and is thus fundamental for obtaining the additive form 24. Note that the function g( ) is the same for all possible utility functions, so that an opportunity set ranking rule provides a consistent mechanism for making such rankings. Furthermore, if instead of being numerically significant, utility functions u( ) are assumed to be ratio-scale measurable 25 then the function g( ) is restricted to be of the form g(u) = u r for some positive constant r, so that the rankings take the form A u B u(x) r u(y) r x A y B The parameter r determines how the relative importance of the best alternative with respect to the others: as r gets arbitrarily large the ranking approaches the indirect utility one, whereas as r tends to 0, it reduces to the cardinality one. On the other hand, if utility functions are only cardinally measurable, then the additive form above is not possible. 26 At this point, it is natural to interpret an opportunity set ranking rule as an objective procedure for deriving rankings of opportunity sets whilst taking into account individual circumstances embodied in the utility functions u( ). That is, once we agree on some reasonable value for the parameter r, we could tell every member of society how he or she should rank opportunity sets in terms of wellbeing freedom, on the basis of his or her utility function. Moreover, the above additive form embodies tradeoffs between quality and quantity of opportunities in a rather natural and intuitively plausible way. But unfortunately, this appealing formulation comes at a very high informational cost: in order to have the additive form we need very detailed utility information, with at least ratio-scale measurability. 22 The difference is analogous to the aforementioned distinction between the multi- and singleprofile approaches to social choice 23 Its statement is rather cumbersome and not particularly illuminating and is therefore omitted. It suffices to mention that it is similar in spirit to standard continuity conditions for preference relations requiring at-least-as-good-as and at-least-as-bad-as sets to be closed. 24 Note that Bossert (1997) does not use the monotonicity assumption (sm), and allows the addition of alternatives to make the opportunity set worse, if these alternatives have large enough negative utility. This does not play a central role in the discussion, and the requirement of positive utility would not alter the result. 25 In this case two utility functions that are positive multiples of each other represent exactly the same information and should hence be assigned exactly equal opportunity set rankings. 26 This is because a translation of the origin cannot guarantee invariance of the ranking: given two opportunity sets with one having more elements than the other, it is always possible to make the larger one rank higher by shifting the origin of the utility function by a sufficiently large number. Of course translations could also have the effect of making positive utilities negative and viceversa and so change the ranking, but the other problem just mentioned here applies even if the utility values are always positive and the translations are only allowed to be positive. 10
4.2. Subjective rankings An alternative way of looking at well-being freedom, is to take rankings of opportunity sets as its primitive notion. That is, instead of trying to derive opportunity set rankings from utility functions, we could instead take the former as given and analyse their structure. As a first step, we could impose the previously used independence axiom (ind) Independence: A B A {x} B {x} for x not in A or B, and a weaker version of the monotonicity axiom (sm) (wm) Weak Monotonicity: if B A then A B, and we could regard these as minimal rationality requirements. In particular, we could argue that one s well-being freedom cannot possibly decrease after an expansion of alternatives, since whatever could be chosen beforehand can still be chosen afterwards. Furthermore, in light of the discussion of the papers by Klemisch-Ahlert (1993) Additive and Bossert (1997), we could seek an additive representation of the form representation A B p(x) p(y). ( ) x A y B The change in notation from u( ) to p( ), which we ll call an opportunity valuation, is to emphasize the fact that whereas previously we were taking utility information as given and constructing the opportunity set ranking, now we are doing the reverse by constructing the function p( ) to represent. Here, p(x) can be interpreted as the additional contribution of alternative x to the individual s opportunities for welfare; it is a quantification of the value the individual places on x in terms of its potential for achieving well-being. The natural question is, under what circumstances does an opportunity valuation p( ) exist? Note that if the above representation is possible, then axioms (ind) and (wm) are automatically satisfied, so they are necessary conditions for the existence of p( ). But are they sufficient? The answer lies in noticing the similarity to the problem of finding a probability measure that represents a subjective probability relation. 4.3. Opportunity sets and subjective probability 27 Given a universal set Ω, a subjective probability is a complete binary relation on 2 Ω, where A B is interpreted as event A is at least as likely as event B. A probability measure P ( ) is a function P : 2 Ω R such that (i) P (Ω) = 1, (ii) P (A) 0 for all A 2 Ω, (iii) P (A B) = P (A) + P (B) for all disjoint A, B 2 Ω, and corresponds to the usual notion of quantitative probability. A probability measure P ( ) is said to agree with the subjective probability relation if A B P (A) P (B). Note that for finite Ω, the existence of such a measure is equivalent to that of the additive representation in ( ) if we define p( ) as p(x) = P ({x}). Hence the problem of finding an agreeing measure boils down to finding a solution to the system of inequalities that arises from the additive 27 The following overview of subjective probability theory is necessarily very brief. See Fishburn (1986) and Krantz, Luce, Suppes, and Tversky (1971) for more detailed expositions. 11
representation. For example, if {x 1, x 2 } {x 3, x 4 } then we need to have p( ) such that p(x 1 ) + p(x 2 ) p(x 3 ) + p(x 4 ), and similarly for all the other set comparisons embodied in. The problem is that might be such that the corresponding system of inequalities for p( ) is inconsistent, so that the additive representation cannot be achieved. The aim is then to find conditions that ensure that there exists a solution p( ). Note the exact analogy to our initial problem of representing opportunity set rankings: the same conditions that guarantee an agreeing measure would also be sufficient for the existence of an opportunity valuation p( ) in ( ). The issue of sufficiency was first investigated by de Finetti (1931), who noted that the following three conditions are necessary for agreement, that is, they follow from the existence of an agreeing measure P ( ): (nt) Nontriviality: Ω ; (nn) Nonnegativity: A for all A 2 Ω ; (add) Additivity: A B A C B C for C disjoint from A and B. Again, note the equivalence of axioms (nn) and (add) with our axioms (wm) and (ind). Moreover, (nt) is really a minimal requirement that will be satisfied by any reasonable opportunity set ranking. It was conjectured that the above conditions were not only necessary but also sufficient for the existence of an agreeing measure. The question was finally settled in the negative through a counterexample 28 due to Kraft, Pratt, and Seidenberg (1959), who consider the following condition as a replacement for (add): (add + ) Strong Additivity: if (A 1,..., A n ) and (B 1..., B n ) are collection of sets such that, if every x Ω is contained in exactly as many A j s as B j s, and A j B j for all j < n, then B n A n. Note that this is quite a strong condition since it implies both (add) and transitivity of. The sufficiency result they prove is the following: Theorem 1 (Kraft et al., 1959). There exists a probability measure on 2 Ω that agrees with a complete subjective probability relation on 2 Ω, if and only if the relation satisfies (nt), (nn) and (add + ). The first (and apparently the only) application of this result to ranking opportunity sets is provided by Gravel, Laslier, and Trannoy (1997), who apply it to the case where is a ranking of well-being freedom common to all individuals. Their chief contribution is the following logically equivalent reformulation of (add + ): (iugr) Impossibility of Unanimous Gains from Redistribution: Given any number n > 1 of individuals, and any two profiles of opportunity sets (A 1,..., A n ) and (B 1..., B n ), where A j and B j are the opportunity sets of individual j, such that (A 1,..., A n ) is obtained from (B 1..., B n ) by transfers of alternatives amongst individuals, if A i B i for some i then there is at least one individual j for whom B j A j. The interpretation is that a mere redistribution of alternatives amongst a fixed number of people cannot strictly increase the freedom of choice of an individual 28 If we have Ω = {x 1, x 2, x 3, x 4.x 4 } and a qualitative probability such that {x 4 } {x 1, x 3 }, {x 2, x 3 } {x 1, x 4 }, {x 1, x 5 } {x 3, x 4 } and {x 1, x 3, x 4 } {x 2, x 5 }, then to have an agreeing p( ), we would need to solve the system of inequalities p(x 4 ) > p(x 1 ) + p(x 3 ), p(x 2 ) + p(x 3 ) > p(x 1 ) + p(x 4 ), p(x 1 ) + p(x 5 ) > p(x 3 ) + p(x 4 ), p(x 1 ) + p(x 3 ) + p(x 4 ) > p(x 2 ) + p(x 5 ). These inequalities turn out to be inconsistent so no such agreeing measure exists. 12
without strictly reducing that of someone else. Gravel et al. (1997) then show 29 that (iugr) implies (add) and that (iugr) together with (wm) gives a representation of of the form ( ), with non-negative p( ). The problem with this approach is that it requires that be the same for all individuals, that is, that well-being freedom be evaluated in exactly the same way for everyone. For this to be the case one would need some objective method of ranking opportunity sets, and if this relies on utility information then there has to be agreement on that too, thus limiting the appeal of this formulation. 4.4. Unique agreement Theorem 1 (Kraft et al., 1959) gives us sufficient conditions for the existence of a p( ) representing, but it does not guarantee its uniqueness: one could have multiple such functions p( ) all representing the same opportunity set ranking. It is therefore interesting to investigate conditions that ensure the existence of a unique opportunity valuation p( ). Note that if p( ) is unique up to multiplication by a positive constant, then we can add the normalisation requirement customary for probability measures that P (Ω) = x Ωp(x) = 1, to make it truly unique. Hence we will say that a p( ) uniquely agrees with if it is unique up to positive scalar multiplication. The first to provide such a result was Suppes (1987) who, in addition to nontriviality (nt), nonnegativity (nn), and additivity (add), considers the following condition: (u 1 ) If A B then there exists a C 2 Ω such that A B C. This axiom does seem at first glance quite appealing: it says that it always possible to expand an opportunity set so as to match the well-being freedom offered by another better set. Suppes uses this condition to prove Theorem 2 (Suppes, 1987). If a complete and transitive ranking on 2 Ω satisfies (nt), (nn), (add) and (u 1 ), then there exists a p( ) that uniquely agrees with. Unfortunately, this result is marred by fact that axiom (u 1 ) is quite a strong one, and has the rather undesirable consequence that for any x, either p(x) = 0 or p(x) = p for some positive constant p, so that the same opportunity valuation is given to all valuable alternatives (those with p(x) > 0). Given the resemblance to the (ins) axiom, this implies that the ranking is the same as Pattanaik and Xu s (1990) cardinality ordering with the difference that now the addition of alternatives does not always increase freedom, since only the number of valuable alternatives is counted. On the other hand, Fishburn and Roberts (1989) consider the following more appealing condition for unique agreement (for convenience, we denote {x} as x): (u 2 ) For every x Ω such that x y for some y Ω, there exists an A 2 Ω such that x A and x y for every y A. Let a unit be an x Ω for which there exists no y Ω such that x y 0. Then what (u 2 ) is saying is that for every x that is not a unit, one can always find enough units y 1, y 2,..., y k such that x y 1 y 2 y k. Using this we have 29 Given the formal equivalence of (iugr) and (add + ), it seems somewhat bizarre that Gravel et al. (1997) give their own proofs which are more involved than the existing ones. 13
Theorem 3 (Fishburn and Roberts, 1989). If a complete and transitive ranking on 2 Ω satisfies (nt), (nn), (add) and (u 2 ), then there exists a p( ) that uniquely agrees with. The essence of this result is that one can measure the freedom in an opportunity set by counting the number of units to which it is equivalent. Of course the alternatives that are designated as units will depend on the particular ranking that is being considered. It is interesting to note that this representation has considerable informational richness. For example, the equality p(x) = 2p(y) now has an intuitive meaning: the set containing only alternative x is considered as offering as much well-being freedom as that containing y and another alternative equivalent to it. 30 Note that axiom (u 2 ) is weaker than axiom (u 1 ), so there are more rankings satisfying (u 2 ) than (u 1 ). Nonetheless one could argue that (u 2 ) is still too strong a requirement: intuitively, what is required is that given any opportunity set one can find enough small alternatives the units to form an equivalent opportunity set. Probably, the larger the universal set of alternatives Ω, the more plausible the existence of such units becomes. Unfortunately the only weaker axioms for unique agreement are very technical and nowhere near as amenable to intuitive interpretation. 4.5. Interpersonal comparability In terms of interpersonal comparability, Pattanaik and Xu s (1990) cardinality ranking is the most straightforward: since all it does is count the number of alternatives, it is the same for all individuals and hence fully comparable. On the other hand, the comparability of the utility-based rankings discussed in Section 4.1 depends on that of the individual utility functions themselves. The more interesting case is provided by the subjective rankings analysed in the previous two sections. These rankings are, by their very nature, subjective ones and might seem to be prone to precisely the same problems as those of subjective utility rankings discussed in Section 2.1. However, in the case of opportunity sets, the states of affairs have a lot more structure. In particular, since we are dealing with a finite set of alternatives, we could impose a normalisation requiring that P (Ω) = x Ωp(x) = 1. Then given an opportunity set A, we could then interpret P (A) as the proportion of total opportunity available in set A. If P i ( ) and P j ( ) are the well-being freedom measures of individuals i and j, and A and B are their respective opportunity sets in a particular state, then we could interpret P i (A) > P j (B) as saying that in that state individual i has a higher proportion of total opportunities for welfare than individual j, as measured by their own standards of well-being freedom. The ethical appeal of such comparisons is not altogether clear, but they do seem to possess some intuitive plausibility. 5. Conclusion In this essay we have discussed the connection between social welfare, individual well-being and opportunities. In particular we noted the shortcomings of the traditional utility-based approach to welfare evaluation and looked at the notion 30 Perhaps one could adopt this approach to the problem of cardinalizing utilities. Instead of considering preferences over lotteries of alternatives, one could consider preferences over opportunity sets. 14
of well-being freedom, or opportunities for welfare, as an alternative. Having analysed ways to make these ideas operational, we should briefly mention some of the limitations of our analysis. A major drawback is that we made extensive use of the independence and additivity axioms (ind) and (add). As mentioned previously, this rules out menudependence, that is, the effect of adding an alternative does not depend on the set to which it is being added. One can imagine many circumstances in which menu-dependence plays an important role. To give a simple example, the act of not eating can be valued differently according to the circumstances: it is fasting when food is available and starving when it is not. Nonetheless, independence assumptions are very useful since they greatly simplify the analysis: any attempt to do without them would run into formidable technical and conceptual problems. Another limitation is that we have only been dealing with finite opportunity sets, even though in many economic contexts, it is more natural to think of infinite ones, such as consumer budget sets. If these sets are uncountably infinite then the additive representation ( ) is useless and alternative formulations must be sought. Perhaps it is possible to recast it in terms of integrals, but the level of technical sophistication required is quite beyond the scope of this essay. On the other hand, having finite opportunity sets is probably not such a great limitation, especially considering that a continuous setting is often used for analytical convenience than for realism. But in any case, the extension of the idea of measurement of freedom and opportunity to such settings is undoubtedly worth studying. References Arrow K.J. (1951). Social choice and individual values. Wiley, New York. 2nd edition 1963. Berlin I. (1969). Four Essays on Liberty. Oxford University Press, 2nd edition. Blackorby C., Donaldson D., and Weymark J.A. (1984). Social Choice with Interpersonal Utility Comparisons: A Diagrammatic Approach. International Economic Review, 25(2):327 356. Bossert W. (1997). Opportunity sets and well-being. Social Choice and Welfare, 18(1):97 112. Bossert W., Pattanaik P.K., and Xu Y. (1994). Ranking Opportunity Sets: An Axiomatic Approach. Journal of Economic Theory, 63(2):326 345. de Finetti B. (1931). Sul significato soggettivo della probabilià. Fundamenta Mathematicae, 17:298 329. Fishburn P.C. (1986). The Axioms of Subjective Probability. Statistical Science, 1(3):335 358. Fishburn P.C. and Roberts F.S. (1989). Axioms for Unique Subjective Probability on Finite Sets. Journal of Mathematical Psychology, 33(2):117 130. Gravel N., Laslier J.F., and Trannoy A. (1997). Individual Freedom of Choice in a Social Setting. In J.F. Laslier, M. Fleurbaey, N. Gravel, and A. Trannoy (editors) Freedom in Economics: New Perspectives in Normative Analysis. Routledge, London. Hammond P.J. (1991). Interpersonal Comparisons of Utility: Why and how they are and should be made. In J. Elster and J.E. Roemer (editors) Interpersonal Comparisons of Well-being. Cambridge University Press. Harsanyi J.C. (1955). Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility. Journal of Political Economy, 63(4):309 321. Kemp M.C. and Ng Y.K. (1976). On the Existence of Social Welfare Functions, Social Orderings and Social Decision Functions. Economica, 43(169):59 66. 15