THREE ESSAYS ON THE CRITERIA TO BE USED IN WELFARE ECONOMICS by NICOLAS GRAVEL B.Sc., Universite de Montreal, 1986 M.Sc, Universite de Montreal, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF ECONOMICS We accept this thesis as conforming to the required standard, THE UNIVERSITY OF BRITISH COLUMBIA 1993 © Nicolas Gravel 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of ECONOK CS The University of British Columbia Vancouver, Canada Date DE-6 (2/88) 0 f' 07 — 1713 Abstract This dissertation consists of three essays devoted to the general problem of combining various criteria for evaluating the collective desirability of social states. The first essay discusses the problem of combining a quasi-ordering of some set of alternatives (interpreted as a criterion for an increase in actual social welfare) with an extension of this quasi-ordering to the power set of this set (interpreted as a criterion for an increase in potential social welfare) to obtain a quasi-ordering of some subset of the cartesian product of this set and its power set lexicographically based on the criterion for an increase in actual welfare. The main result of this essay is that, in order for such a quasi-ordering to exist, it is necessary and sufficient that the subset to which it is applied is such that the extension subsumes the original criterion. When applied to the standard Pareto quasi-ordering and its extension defined by the Chipman-Moore (1971) - Samuelson (1950) quasi-ordering, and under standard assumptions on the economic domain, this result is shown to imply Gorman's (1955) conjecture for the transitivity of the Compensation criterion a la KaldorHicks-Scitovsky. The second essay examines Sen's (1991) suggestion that preference information be used to supplement the criterion of freedom of choice for ranking opportunity sets. This paper shows, with some generality, that, in order for this supplementation to produce a transitive ranking of the opportunity sets, it is necessary and sufficient to assume that the domain ranked is such that the individual preference ordering encompasses the criterion of freedom of choice. However, it is also shown that the quasi-transitivity of such a ranking can be obtained without further assumption. The lesson of this paper is thus that there is little room for constructing a ranking of opportunity sets that attaches value to their freedom of choice ii while giving some weight to individual preferences. If freedom of choice is to have any value in the ranking, then in order for the ranking to be transitive, this value will have to be instrumental rather than intrinsic (using Sen's (1988) terminology). Finally, the third essay tries to make sense of the notion of exploitation set forth by Marxists and others and to relate it to that of bargaining power. For this task, a definition of exploitation is proposed which, it is contended, captures the intuitive meaning of the word as the act of taking unfair advantage of someone. More precisely, the definition considers a relationship between two agents to be exploitative if one agent (the exploiter) obtains an advantage from this relationship which can be shown to depend upon the initial deprivation of the other (the exploited) with respect to some poverty threshold. To assess whether the advantage of the exploiter is indeed due to the deprivation of the exploited, the definition considers a counterfactual experiment in which the state of deprivation of the exploited is eliminated and examines the welfare consequences of this experiment for the presumed exploiter. If the latter becomes worse off from this elimination of the other's deprivation, then it is asserted that the presumed exploiter is indeed taking an advantage of the other's deprivation. The problem of specifying an "adequate" poverty threshold is also examined by appealing to bargaining theory. This examination is based upon the somewhat intuitive idea that exploitation is related to an "excessive" bargaining power on the part of the exploiter. The definition of the poverty threshold should therefore be made in such a way as to make exploitation a good measure of the bargaining power of the exploiter in the bargaining game representation of the relation between the exploited and the exploiter. iii Table of Contents Abstract ^ Table of Contents ii ^ iv List of Figures^ vi Acknowledgement General Introduction^ 1 Chapter 1:On the Difficulty of Combining Actual and Potential Criteria for an Increase Social Welfare^ 8 1.1: Preliminaries^ 8 1.2: The Formal Setting^ 17 1.3: General Results^ 28 1.4: Interpretation of the Results in a Standard Economic Environment ^35 1.5: Conclusion^ 41 Chapter 2: Ranking Opportunity Sets on the Basis of their Freedom of Choice and their Ability to Satisfy Preferences: A fundamental difficulty 2.1: Preliminaries 2.2: The Model ^ ^ 2.3: The Results 2.4: Conclusion ^ 42 42 47 ^ 56 ^ 60 iv Chapter 3: Exploitation and Bargaining Theory: A Suggested Interpretation^61 3.1: Preliminaries^ 61 3.2: A New Definition of Exploitation^ 67 3.2 - (i) Formal Statement and Intuition of the Definition ^67 3.2 - (ii) A General Equilibrium example ^ 71 3.2 - (iii) Further Discussion^ 75 3.3 - Exploitation and Bargaining Theory ^ 77 3.3-i) Exploitative relations as bargaining sessions^77 3.3-ii) A definition of Bargaining Power^ 80 3.3-iii) A reinterpretation of the Nash Solution Concept ^82 3.3-iv) Exploitation as a Measure of Bargaining Power ^87 3.4 - Conclusion^ General Conclusion References Appendix 91 ^ 93 ^ 97 ^ 104 v List of Figures Figure 1.1 Figure 1.2 Figure 1.3 Figure 2.1 Figure 3.1 vi Acknowledgements My greatest debt goes to Chuck Blackorby for his very competent supervision. My work greatly benefited from his ready availability, his broad-mindedness, his acute knowledge of the field as well as his kindness. Since the very beginning of my research, he showed confidence in my work which, more than everything else, enhanced my self-assurance. My hope is that he will find in the finished product nothing to undermine this confidence. I also greatly benefited from the comments of John Weymark and Michele Piccione who read very closely various versions of my work in progress. Their rigour has exerted a counterweight to my own tendency for sloppiness. I received valuable comments from Bill Schworm for the first chapter and from David Donaldson for the third one. My debt to David Donaldson is especially great as he provided me with the original impetus to work in this field. Passionate discussions on various aspects of the topics studied herein with colleagues and friends, especially Shasikanta Nandeibaum, Stephan Vigeant, Ivan Alves and Keir Armstrong, have also enhanced my grasp of the field as a whole. Keir Armstrong must be further credited for his help in editing and improving the english of the text. Financial support from SHRCC and University of British Columbia is also gratefully acknowledged. Last but not least, Nathalie has been behind all the work leading to the final outcome and has had to bear all of its costs since the very beginning. Cette these lui est dediee, avec tout mon amour. vii General Introduction Welfare economics is concerned with the evaluation of the consequences of various economic phenomena on the well-being (or "welfare") 1 of the individuals. To perform such a task, it is clearly important to have some information about what this well-being consists of. Yet this information is not easy to obtain. In the conduct of their individual and/or collective lives, human beings manifest concerns for issues as diverse as their freedom, the discrimination they feel being the object of, their ability to better satisfy their preferences, the poverty of some of their peers, their power, the feeling they are being exploited, and so on and so forth. All of these issues are, or can be, affected by phenomena belonging to the realm of economics (whatever that is). Furthermore, a given economic policy is likely to affect these different issues differently for different people. How can economists weight these various consequences, borne differently by different individuals, in their assessment of the effect of various policies on overall social well-being? For quite a long time, economists have held the believe that they could handle this question by resting on nothing but what they view as the most minimal criterion: that of Pareto. The vulgate of this profession of faith is well known: the only situation where an economist, as such, can conclude that a policy has improved the overall welfare of society is when none of the affected individuals suffer as a consequence and at least one individual is 1 Although the terms "welfare" and "well-being" are often used synonymously, this is not so in economics. For an economist, the word "welfare" refers to the position achieved by an individual on an a priori defined preference scale (usually represented by a utility function). The term "well-being" is more general. The two are identical only if one accepts, as many economists do, the credo that the only source of an individual well-being is her ability to fulfil her preferences. For more on this important semantic issue see Sen 11987; 1991). 1 a strict beneficiary. This criterion serves as a basis for what economists call efficiency and, as mentioned by Dupuy (1992; p.53), was supposed to serve as the foundation for the division of labour among social scientists, as viewed by economists: "we [economists] take care of the issues related to efficiency; the rest belong to 'ethics', 'philosophy', 'sociology' or to the sphere of the political debates. To us: the 'objective', the 'undeniable'. To others (i.e. philosophers, politicians, etc.) the realm of infinite and indecisive quibbles." In the last two decades or so, an increasing number of economists have realised that things are not quite so simple. First, it seems clear that the Pareto criterion alone can not take one very far in the direction of making sensible policy recommendations. After all, unanimously-approved policies have been rather rare animals in the history of human kind And when they occur, it is very unlikely that the services of an economist (or any other social scientist) would be needed. This is not to say that the Pareto criterion is useless. As an analytical tool to characterise various allocation devices (like that of walrassian general equilibrium), it has been proved to be of paramount importance. But it is not a sufficient criterion to assess the impact, on human well being, of concrete economic phenomena. Second, even what appears to be the main asset of the Pareto criterion, its incontestable ethical reasonableness, has been the object of some questioning. In particular, Sen's (1970) "liberal paradox" illustrates how the Pareto criterion could crudely contradict very minimal individual rights in the absence of certain restrictions on the pattern of possible preferences individuals can have.' Thus, if economists are interested in assessing the impact of economic phenomena on overall well-being, they can not avoid going beyond the prescriptions of the 2 _proac.h_to_the. -problem-see-Gibbarcl-(-197 41. For a critique ofSen!&ap - 2 Pareto principle alone and considering other criteria of social well-being. This thesis is a very modest attempt to explore some ways by which such a consideration could be made. Of course, the following objection comes immediately to mind. Should economists be interested in assessing the impact of economic phenomena on well-being? Is the job of an economist not more concerned with the positive analysis of the phenomena under consideration, with making theoretical and/or empirical predictions about what will happen to prices, quantities, etc. if such and such changes occur while remaining silent about the effect of these changes on the somewhat blurred notion of "well-being"? In other words, does the normative branch of economics really have a raison d'être? Concerning this fundamental epistemological question, which challenges the relevance of the whole exercise I present herein (as well as a significant part of the intellectual effort pursued under the label of "economics"), I have three remarks. The first two are concerned with the insuperable difficulty of separating the normative and positive aspects of any phenomenon having to do with the conduct of human affairs, as economic phenomena are. The third remark is more prosaic. First, a large part, if not all, of the research agenda of positive economics originates with normative considerations. It is because of the belief that these issues have an impact on overall well-being that the measurement of income inequality, inflation, unemployment, the effect of trade barriers on economic growth, etc. are seen as belonging to the sphere of positive economics. Or, to take an example that shall be examined more closely below, if a consensus arose in Canada to the effect that the Mandan notion of exploitation is an important object of collective concern, then the study of the impact of, say, the Canada-US free trade agreement on the rate of exploitation of workers by firms belongs automatically to the sphere of positive economics. 3 Second, even if one is interested in purely "positive" issues (such as the prediction of the likely reaction of some agent to some change in the parameters of the situation in which she is involved) there is very little hope that a model that is entirely silent about normative issues will succeed in accurately predicting human behaviour. Since human beings care about ethical issues, they are likely to react in a particular way to phenomena that affect these ethical issues. Ignoring this aspect of human nature in economic modelling is very likely to be misleading. 3 Third, and last, expending effort to define a narrow division of labour among researchers, such as that between "normative" and "positive" economics or between what belongs to the realm of economics and what belongs to the realm of philosophy, sociology, political science, etc., is unlikely to be a very efficient strategy for enhancing our overall understanding of human affairs. This is especially true when such a division does not emerge naturally, through the daily work of researchers, but is the result of some ex ante conception of what "should" be, ideally, the task of an economist, a philosopher, a sociologist, etc. Like it or not, economists have always manifested an interest in policy recommendations. The important question is not whether this interest is "legitimate" but, rather, how these recommendations can be made correctly. As mentioned above, the present dissertation tries to shed some light on this issue. It comprises three essays, each analyzing how certain criteria that have appeared historically to be a source of concern to many could serve as a basis for designing policy evaluations. Before giving the flavour of the content of these essays, a few semantical remarks are in order. An economic phenomenon (e.g. a government policy, a price change, a 3 The discussion about focal equilibria in game theory (see for instance Schelling (1980)) provides II I^I. II" a further illustrati^ s. 4 technological innovation) is anything that moves a community of individuals from some initial state' to a new one. Evaluating a phenomenon consists, therefore, in ranking the various possible states. A criterion is simply a particular rule for ranking these states. A criterion can be complete (i.e. it can produce a pairwise ranking of all conceivable states) or incomplete (e.g. it is unable, for some states, to say anything about how they rank relative to one another). A criterion can also be consistent (e.g. there is no contradiction implied by the ranking) or inconsistent (e.g. it may rank state a above b and b above c while, at the same time, ranking c above a). Throughout this dissertation, a strong emphasis is put on consistency; completeness receives very little attention. This choice is no accident as it reflects my strong preference for being right about something rather than wrong about everything. The first essay examines the solution, proposed by economists, to the aforementioned problem of the frequent indecisiveness of the Pareto principle as a criterion for ranking social states: the so-called compensation criteria by which the mere possibility (rather than the actuality) of unanimous improvement becomes a sufficient condition for recommending a particular policy. It provides a general framework in which one can make sense of these compensation criteria and examines, in that context, the two best known members of this family. The Kaldor (1939)-Hicks (1939)-Scitovsky (1941) compensation criterion and the Chipman-Moore (1971)-Samuelson (1950) criterion. It also shows how these two criteria are related and provides a general condition that is necessary and sufficient for the transitivity of the former as well as for the transitivity of the combination of latter with the more traditional Pareto criterion. This general condition appears to be very stringent since, when interpreted in the context of a standard economic environment (with the states being 4 A "state" is a complete description of all aspects of the community that are relevant to the problem at hand. In economics, a state is usually described by an allocation of goods, services, and shares of firms among individuals, together with specifications of the preferences of these individuals and the technologies used by firms. 5 utility vectors), it is equivalent to requiring all individuals to have Engel's curve which are parallel straight lines as in Gorman (1953). The second essay looks at the problem of ranking opportunity sets on the basis of both the freedom of choice they provide and their ability to satisfy preferences. More precisely, it examines Sen's (1991) suggestion that freedom of choice should be the first criterion by which opportunity sets are ranked and, when it does not lead to a clear-cut ranking of the sets, should be supplemented by some information about the preferences of the individuals over the elements of these sets. The conclusion is that for this (particular) combination of the two criteria (freedom and ability to fulfil preferences) to be transitive, it is necessary and sufficient that the individual preferences encompass the freedom criterion. Thus the margin for valuing freedom of choice in the ranking of social states while respecting individual preferences appears to be rather thin. If freedom of choice is to have any value in the ranking then, in order for the ranking to be transitive, this value will have to be instrumental rather than intrinsic (using Sen's (1988) terminology). However, it is also shown in the second essay that, if one is willing to weaken the requirement of transitivity to quasitransitivity (transitivity of the asymmetric factor of the criterion), then the combination of the two criteria can be obtained without much difficulty. The third essay is more exploratory and constitutes an attempt to make sense of the notion of exploitation set forth by Marxists and others and to relate it to that of bargaining power. For this task, a definition of exploitation is proposed which, it is contended, captures the intuitive meaning of the word as the act of taking unfair advantage of someone. More precisely, the definition considers a relationship between two agents to be exploitative if one agent (the exploiter) has an advantage which can be shown to depend upon the initial deprivation of the other agent (the exploited) with respect to some poverty threshold. To assess 6 whether the advantage of the exploiter is indeed due to the deprivation of the exploited, the definition resorts to a counterfactual experiment in which the state of deprivation of the exploited is eliminated and examines the welfare consequences of this experiment for the presumed exploiter. If the latter becomes worse off, then it is asserted he is indeed taking advantage of the former's deprivation. The problem of specifying an "adequate" poverty threshold is also examined by appealing to bargaining theory. This examination is based upon the somewhat intuitive idea that exploitation induces a somewhat "excessive" bargaining power from the part of the exploiter. The definition of the poverty threshold should therefore be made in such a way as to make exploitation an important source of the bargaining power of the exploiter in the bargaining game representation of the relation between the exploited and the exploiter. This essay is more exploratory than the other two since it does not attempt to formally integrate a precise criterion into a formal ranking of states. Instead, the essay tries to make sense of an ethical issue about which many individuals have expressed a vivid concern. 7 Chapter 1 On The Difficulty of Combining Actual and Potential Criteria For an Increase in Social Welfare 1.1) Preliminaries The Pareto principle asserts that a sufficient condition for ranking one economic state above another is when no one in society strongly prefers the latter and at least one person strongly prefers the former.' Economists like to think that, with this principle, they have at their disposal a minimal and (relatively) non-controversial criterion to assess whether or not economic transformations are worth doing. A major problem with this criterion is that the domain of cases to which it can be applied conclusively is rather narrow. To overcome this difficulty in part, without invoking other ethical (and presumably more controversial) principles, it could be suggested that this domain be extended from actual cases to potential or hypothetical ones. In effect, it can be argued that the states ranked by the Pareto principle are, in fact, made of two formally distinct objects: an actual state and a set of potential states considered as feasible given some notion of feasibility. For example, the current income distribution in Canada is only one of the many feasible income distributions which could be reached under different tax regimes; the particular allocation of two goods among two individuals obtained as a Walrassian equilibrium is only one of the many points of the Edgeworth box. At a formal level, this view of social states as pairs of objects (an 5 This statement is, in fact, what Sen (1970) calls the strong version of the Pareto principle. The weak version requires that everyone in society strictly prefer one state over another as a suffcient condition for ranking the former above the latter. The difference between the two versions has no consequence for the_problemat hand 8 actual state and a set of potential states) is all the more acceptable as it does not rule out the particular notion of feasibility for which only the actual state is really feasible and where, therefore, the set of potential states is nothing but the singleton containing the current state as its unique element. To avoid terminological confusion, I refer to the pair composed of an actual state and a set of potential ones as a position of the economy, and to the set of potential states as a situation. The term "state" is to be used, in the discussion that follows, to designate the first coordinate of a position. In ranking two or more positions, the Pareto principle only uses information gleaned from their respective first "coordinates" (i.e. their actual states); it ignores completely all information contained in their situations. A tempting way to extend the usefulness of the Pareto principle in this setting would be to use, in a somewhat "Paretian way", some information about the situations to supplement, when necessary, the largely incomplete ranking of positions provided by the pairwise comparison of their actual states on the basis of the Pareto criterion. The famous Kaldor (1939) - Hicks (1939) - Scitovsky (1941) compensation (KHSC) criterion, which, even today, remains at the basis of many applied cost-benefit analyses, is undoubtedly the result of such a temptation. Recall that an economic position is ranked above another by the KHSC criterion if and only if there exists, in the situation set of the former, a state that Pareto dominates the actual state of the latter. It is worth noticing that, although this criterion extends that of Pareto by ranking conclusively far more positions than the latter, it nonetheless agrees with the Pareto principle in the sense that it never rejects a move that leads to an actual Pareto improvement. This supports the traditional view that actual improvements (no matter how they are defined) should bear an ethical priority over potential improvements: "a bird in hand is worth two in the bush." As is well known, the KHSC criterion can easily give rise to severe 9 Figure 1.1 individual 2's utility Individual l's utility Figure 1.1: This figure displays a situation where a systematic application of the compensation principle leads to inconsistent decisions in a case where the states are utility allocations and the situations are utility possibility sets in a two-individual economy. Clearly,(u",U") would be considered to be better than (u"',U"') by the compensation criterion since ft" in U' is Pareto better than u"'. Similarly, one can see that (u',U) would be considered to be better than (u",U") by the compensation criterion. However (u',U') is not considered to be better than (u'",U'") by the compensation criterion since there is no element in the set U which Pareto dominates u". The compensation criterion leads to an intransitive ranking in this case. Furthermore, if one proceeds one step further and moves to (u,U) from (u',U) as would be recommended by an adherence to the compensation criterion, then one is led to a state that is worse than the original state u"' even according to the Pareto criterion. 10 inconsistencies that are illustrated in figure 1.1 above. Less well known is the exact set of conditions that one needs to impose on the domain of positions in order to avoid these inconsistencies. Given the widespread use of this criterion in applied cost benefit analysis, such a lack of knowledge is a bit surprising. Some time ago, Gorman (1955) made the somewhat imprecise conjecture that preference cycles of the type illustrated in figure 1.1 can occur every time two utility possibility frontiers intersect. Such a conjecture is difficult to interpret since Gorman did not make precise whether he meant that the non-crossing of utility frontiers is a necessary condition for eliminating intransitive uses of the compensation criterion or that it is a sufficient condition for this elimination or both. Neither was he clear about how rich the the domain of positions need to be in order for his conjecture to hold. If one interprets Gorman's conjecture as saying that the requirement of non-crossing of utility possibility frontiers is equivalent to that of transitivity of the ranking generated by the KHSC criterion, and assuming of course that the conjecture is correct, it can safely be said that the assumption that the compensation criterion can be used consistently is a rather stringent one. This is especially the case if one interprets the utility possibility sets as being the image, under the vector of utility functions, of the set of distributions, among members of the society, of all the possible production plans contained in the aggregate production set. For, there is very little reason to assume that all of the possible aggregate production sets could never overlap each other to start with. But even in the most favourable setting where one abstracts from the difficulties generated by production, it has been shown by Gorman (1953) that, under standard assumptions on preferences (convexity, continuity and monotonicity), the requirement of noncrossing of the utility possibility frontiers is in turn equivalent to the assumption that all individuals' Engel curves are parallel straight lines. Here again, there is very little evidence that real households, say in Canada, behave as if they were maximising almost the same 11 homothetic utility functions. 6 The truth and the exact meaning of Gorman's (1955) conjecture notwithstanding, the relative ease with which one can obtain inconsistent applications of the compensation criterion has motivated several economists (see for example Samuelson (1950), Chipman and Moore (1971; 1973; 1976; 1980)) to develop another approach for extending the Pareto criterion to potential states; this is, the Chipman-Moore-Samuelson (CMS) criterion.' This criterion ranks the various positions solely by comparing their situation sets and, more precisely, it considers a situation as being "better" than another if and only if any particular state belonging to the latter is Pareto dominated by some state belonging to the former. It has the merit of being a quasi-ordering s of the power set of the universal set of actual states (see Chipman and Moore (1971; theorem 1)) and is therefore free from the inconsistencies illustrated in figure 1.1. 9 . Since this quasi-ordering is a formal extension' of the standard 'If one makes the extra assumption that all consumer consumption sets have the same origin, then the requirement of parallel straight line Engel curves is clearly equivalent to identical preferences across households. 7 Also known as the Kaldor-Hicks-Samuelson or the Samuelson criterion. For a thorough survey of the history of this criterion and of its relationship to the compensation criterion discussed above, see Chipman and Moore (1978). A quasi-ordering is a reflexive and transitive but not necessarily complete binary relation. 9 This simple fact has been the source of a great excitement for those who believe that potential or hypothetical states should be relevant in evaluating actual ones as the following quote from RuizCastillo (1987) illustrates: "Whatever the shortcoming of this relation [the CMS quasi-ordering] as a basis for passing social judgement, it is at least free from inconsistency and has been shown to be reflexive and transitive" (p.35) 19 Formally, given a set X quasi-ordered by Q, a quasi-ordering Q of the Power set P(X) ofX is said to be an extension of Q if for all y, y' belonging to X, y Q y' implies {y} Q In addition to this basic requirement, an extension can satisfy many other properties. A large portion of the social choice literature is devoted to the study of various extension methods. Representative pieces of this literature • !I•^- •:- • e .^• 12 Pareto criterion (which is defined on the universal set of alternatives and not on its power set), it has been perceived by many (e.g. Chipman and Moore (1973; 1975; 1980), Chipman (1987) and Ruiz-Castillo (1987)) as the natural analogue (for situations sets) to the Pareto criterion (for actual states). The only problem with the CMS criterion is that, as defined above, it has very little to say about actual Pareto improvements and may well consider as non-comparable two positions in which one actual state Pareto dominates the other for the sole reason that the two sets to which the actual states belong are not ranked by the CMS criterion (see figure 1.2). Given the wide acceptance of the Pareto criterion and, more fundamentally, the fact that the whole purpose of constructing the CMS criterion (or the KHSC one) was to extend the usefulness of the Pareto criterion, this state of affairs is clearly undesirable. A natural thing to do in this context would be, it seems, to combine the Pareto criterion with the CMS one in a way that respects, as does the compensation criterion, the ethical priority of the former. In comparing any two positions, such a combination would first check whether one actual state Pareto dominates another and, in the case of Pareto noncomparability (or perhaps Pareto indifference), would compare their respective situations on the basis of the CMS quasi-ordering. Which conditions, if any, are necessary and sufficient for this combination (to be called, for obvious reasons, lexicographic) of the Pareto and the CMS criterion to be a quasi-ordering of the set of positions? How does this problem relate to the one of finding the conditions necessary and sufficient for the transitivity of the KHSC criterion discussed above? How can one interpret Gorman's (1955) conjecture in this context? These are the questions addressed, at a somewhat general level, by the present paper. 13 figure 1.2 individual 2's utility individual l's utility figure 1.2: Assume, as in figure 1.1, that the states consist in utility allocations in a two- individual economy. Clearly, u Pareto dominates u'. However none of the sets of utility allocations to which each state belongs dominates the other as per the CMS criterion. It is first shown that, although related, the problem of finding necessary and sufficient conditions for the transitivity of the compensation criterion and that of finding necessary and sufficient conditions for the transitivity of the lexicographic combination are formally different and, moreover, that the graph of the lexicographic combination is a subset of the graph of the compensation _criterion ris-e-tu a theoretical h43e that the - 14 - - necessary and sufficient conditions for the transitivity of the former are less stringent than the ones for the transitivity of the latter. The rest of the paper shows that such a hope is purely theoretical. More precisely, it is shown that requiring the transitivity of the lexicographic combination is equivalent to requiring this combination to coincide exactly with the CMS criterion alone. To say the same thing a bit differently, it appears to be impossible to make the CMS consistent with the Pareto criterion without making the later redundant and ruling out de facto cases such as the one depicted in figure 1.2. It is also shown that requiring the combination to coincide with the CMS one is equivalent to requiring the KHSC criterion to be transitive. In a sense then, the failure of the compensation criterion to be transitive is entirely due to the failure of the lexicographic combination to be transitive. These results are demonstrated in a very general setting where no metric or topological properties need to be assumed and where the positions ranked by the various criteria can be virtually anything. It is simply supposed that, for each position the actual state is Pareto optimal in the situation set and the set of all conceivable positions is minimally comprehensive. The advantage of working in such a general setting lies in the proof it provides that the difficulty of obtaining a consistent combination of actual and a potential pareto criteria is entirely logical and does not depend upon the somewhat special structure imposed by economic theory. The results do have, however, an immediate implication when interpreted in such a standard economic environment where the situation sets are thought to be compact and disposable subsets of Rn + (interpreted as utility possibility sets) satisfying the property that their frontier in R a+ coincides with their set of Pareto optima. Namely, that the transitivity of this lexicographic combination (and of the compensation criterion) is formally equivalent to the requirement that no utility frontier cross another in_ the Gorman (1953)-sense,Gorm-anls15 (1955) conjecture is thus entirely confirmed and the present paper reemphasises the strong case that already exists against the use of the compensation criterion in cost-benefit analysis. But also, and perhaps more importantly, the result of this paper questions the relevance of using the CMS criterion as well. In effect, even if it is true that this criterion is free from the inconsistencies that characterises the KHS compensation criterion and, as such, seems to give some new merit to the idea that potential welfare considerations are helpful for choosing among alternative social states'', it seems difficult to deny that, if such a consideration is to be made, it should at least be in accordance with the Pareto principle. By establishing that a combination of the Pareto and the CMS criteria is as likely to be consistent as the compensation criterion is, this paper raises also some doubt about the usefulness of the CMS quasi-ordering as an ingredient of normative economics'. It appears to be clear, therefore, that actual and potential welfare do not go hand and hand The organisation of this chapter is as follows. The first section sets out the notation and establishes the correspondence between the lexicographic combination and the compensation criterion. The second section presents the assumptions used in the statement and proof of two general theorems concerning the equivalence between the transitivity of both 11 This idea has been phrased by Chipman (1987) as follows: "There is a sense in which one person might be said to be basically healthier than another even though, at the present moment such a person might have a cold and the other one not. The compensation is usually used to make comparisons in this sense: One state of the economy is sounder, healthier, more robust or has a greater productive potential than another." 12 It is to be hoped that these results will be of some interest to those trying to overcome the so called Boadway (1974) paradox by constructing exact monetary indicators of improvement in the KHSC and/or the CMS sense (see for instance Schweizer (1983), Dierker and Leninghaus (1986) or Keenan and. Snow (199-14 - 16 the KHSC criterion and the lexicographic combination and the equality of the lexicographic combination and the CMS criterion. The third section applies these results to an economic environment and shows that, in this context, requiring the lexicographic combination and the CMS criterion to coincide is equivalent to imposing the non-crossing property of the utilitypossibility sets as defined by Gorman (1953). 1.2: The Formal Setting Let X be a set quasi-ordered by Q with asymmetric, symmetric and non- comparable factors Q, Qs and Q a respectively. Let P(X) denote the power set ofX, and define the extension Q of Q to P(X) as follows. Definition 1.1: V Y, reP(X),YQY' E-> V y' E Y', 3 y E Y such that y Given Y E P(X) Q y'. and an arbitrary quasi-ordering Q, let MQ (Y) denote the set of Q-maximal elements of Y defined by: Definition 1.2: MQ (Y) = {y e Y: y' Q a y for no y' in Y} Let ..9 c - P(X) be some family of subsets of X and let assumed to satisfy the following. Assumption A1.1: V (y,Y) E S, y E MQ(Y). 17 S be a subset of X x The interpretation of this framework is as follows. X is the universal set of economic states. Q (to be though of some version of the Pareto quasi-ordering) is a minimal criterion aggreed upon by members of society to rank, when possible, economic states. Q is the extension of Q to the power set proposed originally by Samuelson (1950) and formalised by Chipman and Moore (1971). It is straightforward to show that Q is reflexive and transitive (see Chipman and Moore (1971; theorem 1). At any single instant, the society is assumed to be located on some position consisting of an actual state and a set of potential ones drawed from some family of subsets of X. Following Samuelson (1950), I shall refer to any such a set as to a situation. The set S is the set of all conceivable positions that the economic system can reach. One can think of S as being generated from some binary relation B which associates, to any actual state y, a set Y of potential states that are assumed to be achievable from y via appropriate and feasible compensation devices." To further enhance one's intuition it can be enlightening to consider, as an example of a position, the current allocation of goods among Canadian households (as an actual state) with the set of allocations of goods that are considered as achievable through the implementation of acceptable tax policies (as a situation). Another position, which could be compared to the previous one, is the actual allocation that would result from the currently discussed enlargement of the North-American Free Trade Aggreement (NAFTA) to Mexico as well as the (possibly different) set of allocations that would then be considered as feasible. 13 my knowledge, the only other attempt to integrate the compensation principle into such an abstract logical framework is Arrow (1963; Chapter 4; in particular pages 40-45). He however assumed that, for each actual state, its situation set was an equivalence class (see Tarski (1965)) induced by the binary relation B. Arrow thus assumed that B was reflexive, symmetric and transitive. The above setting is more general in that it may dispense with the assumption that the relation B is symmetric and transitive. For example, one could well have y' E Y for a position (y,Y) E S without this implying the existence of a position (y',Y') a S with the property that y E Y'. Arrow's assumption is captured in the gompensatien-unrestri-et-edriess vile used in tlappendix (see discussion below). - — 18 Assumption ALI requires simply that, in each position, the actual state is Q-maximal in its situation. For obvious reasons, any weakening of this assumption increases the ease with which one can obtain intransitive applications of either the KHSC criterion or the lexicographic combination I am about to define. Despite the appearance, such an assumption does not imply a restriction to the scope of the framework to the so called "first best cases". One has to recall that the notion of feasibility used to construct the situation corresponding to each actual state need not coincide with that usually assumed in economic analysis. In particular, it may well include political, ethical and informationnal feasibility. The meaning of this notion can also be extended to account for "second best" cases (such as the ones studied by Foley (1970), Hahn (1971) and Arrow (1981)) where lump-sum taxes and transfers are not assumed to be possible. As mentioned in the introduction, even the point of view that no hypothetical states should be used in the comparison of two positions is consistent with assumption A1.1 since the the situation set can well be a singleton. The overall rationale behind assumption A1.1 is that, given a particular notion of feasibility, if a situation is really thought to be made of feasible states and if Q is really seen as being the minimal criterion for ranking the set X, then it is very difficult to understand why the actual state would not be Q- maximal in the situation. If there exists a truly feasible state that is Q-better than the actual one, why not move to it in the first place? The problem of the social planner is to rank the set S in a way which, above everything, respects the minimal criterion Q. That is to say, in a more formal manner. Axiom A1.2: V (y,Y), (y',Y') E 5, y Q y' implies that the position (y',Y') is not ranked strictly above the position (y,Y). 19 This axiom, usually known as the Pareto principle, undoubtedly constitutes the ethical pillar of modern normative economics. The relative scarcity of cases where the Pareto principle can be of some help in choosing among alternative positions, together with a deep reluctance to resort to other (presumably more controversial) criterion, has motivated several generations of economists to extend the usefulness of Q by resorting to its extension Q. The KHSC criterion discussed in the introduction was the first attempt in its direction. This ranking of S, to be denoted K, can be formally defined as follows: Definition 1.3 : V (y,Y),(y',Y') e S, (y,Y) K (y',Y') <--) Y Q (y'}. Strictly speaking, the compensation criterion defined in 1.3 is slightly different from the one originally proposed by Kaldor (1939) since it uses the quasi-ordering Q instead of its asymmetric factor Qa in the definition of the extension Q. This difference bears absolutely no significance for the results of this paper. It is important to notice, after Blackorby and Donaldson (1990; footnote 19), that Kaldor's original definition of the compensation criterion is not the asymmetric factor of K as per definition 1.3. The asymmetric factor of K (or of the original Kaldor's compensation criterion) is what is sometimes called the Scitovsky (1941) criterion. A large part of the debate around the compensation criterion in the forties and the fifties arose from a misunderstanding of this fact. Clearly, since y belongs to Y for each (y,Y) in S, the binary relation K subsumes the Q-ranking of the first coordinates of any two positions and satisfies therefore axiom A1.2. It is well-known that K (or its asymmetric factor Ka ) need not, in general, be transitive since, for an arbitrary sequence (y,Y),(y',Y') and (y",Y"), the fact of having Y Q {y'} and Y Q {y"} imposes nothing whatsoever on the relation between Y and {y"} in terms of Q. For K to rank 20 transitivaly S, some further restrictions must be imposed on S. The exact and rigorous statement of these restrictions remains yet to be provided. However, an informal conjecture made some years ago by Gorman (1955) suggests that these restrictions could be quite severe since, in a model where the states are utility allocations and the situations are utility possibility sets, they could go as far as requiring that no utility possibility sets frontiers cross. In view of the possible intransitivity of K, Samuelson (1950) and Chipman and Moore (1971) have proposed to rank S by comparing the various positions solely on the basis of their situations as per the quasi-ordering Q. That is to say, they have proposed the following ranking C of S. Definition 1.4: V (y,Y),(y',Y') a S, (y,Y) C (y',Y') E— Y Q Y The obvious pitfall with C is that, transitive as it be, it does not necessarily satisfy axiom A1.2. Another approach for ranking S in a way that respects A1.2 while using information contained in the situations as per the extension Q would be to combine Q and Q in the following binary relation R of S: Definition 1.5: V (y,Y),(y',Y') a IS, (y,Y) R (y',Y') <-4 y Q y' v (--.(y' Q y) A Y Q Y') This combination is called lexicographic by analogy with the usual lexicographic ordering of the n-dimensional Euclidian space. It ranks two arbitrary positions by giving a priority to the Q ranking of their actual allocations and, in the case where this ranking is not 21 ^ conclusive, by comparing the two situation sets on the basis of the extension Q. In the aforementionned example of the extension of NAFTA to Mexico, the binary relation R would (weakly) recommand the extension of the aggreement if either of the two following possibilities hold (or both): 1) no Canadians lose in the move from the current allocation of goods to that brought about by the enlargement of NAFTA to Mexico or, 2) some canadians lose and some others gain in the move but, given any allocation that is feasible before the admission of Mexico, one can think of an allocation, that would be feasible after the admission of Mexico in the aggreement, that all canadians weakly prefer. Note that the asymmetric, symmetric and non-comparable factors of R can be written, respectively, as: Lemma 1.1a: V (y,Y),(y',Y) E S, (y,Y) Ra (Y', 1 ") H y Qa y v^Q y) n Y Q. Y) lemma .1.1b: V (y,Y),(y',Y) E S, (y,Y) Rs (y',Y') H y Q s y v (y Q ,, y' n Y Q8 F) Lemma 1..lc: V (y,Y),(y',Y') E S, (y,Y) R a (y',Y) <--> y Q„ y' A Y Q„ Y' Proof (lemma 1.1a): By definition of the asymmetric factor, (y,Y) R a (y',Y) H (y,Y) R (y',Y) ^R (y,Y)). That is to say, from definition 1.5, (y,Y) R. (y',Y') <—> Q v^Q y) Y Q Y)1 A^Q y v (- , (y Q y') A^Q Y)}. After distributing the logical operator "--." (see Margaris (1990)), one obtains the equivalent statement: (y,Y) R. (y',Y) H [y Q y' v (-'(y' Q y) Y Q Y')] n [(-(y' Q y) n y Q y') v^Q y) n -,(Y' Q17))]. Distributing the logical connector 22 A and rearranging the terms yields, using the definition of Q. and Qa : V (y,Y),(y',Y) E S, (y,Y) (y',Y')^y Qa y'^Q y) Y^Y'). ^ The proofs of parts b and c of lemma 1.1 are similar to that of part a and are therefore omitted. A fact that is noteworthy about the asymmetric factor of R is that it recognises a strict domination of one position by another if, and only if, there exists either a strict Qdomination between their respective actual states or a Q-non-comparability between these actual states and a strict Q-domination between their situations. Two positions for which the actual states are Q-indifferent but whose situations are in relation of strict (asymmetric) Qdomination are considered to be indifferent by R. In this sense, R can be said to be lexicographically based on Q in a strong sense. One may not see such a feature as appealing and may think that, in cases where two states are Q-indifferent but are such that their situations are in relation of strict Qdomination, one should observe a relation of strict domination between the two positions. But R is not the only method for generalising the usual lexicographic combination of orderings on the N-dimensional Euclidian space into more abstract spaces. One can also contemplate the following alternative lexicographic combination R' of Q and Q which, perhaps, bears more resemblance with the standard lexicographic ordering of R n : Definition 1.6: V (y,Y), (y',Y') E S, (y,Y) R' (y',Y) H y Q a y v^Q. y) n Y Q Y') " Indeed, there is no straightforward and unique generalisation of the lexicographic ordering of Rn in the present context where S is a subset of the Cartesian product of two sets that are only partially ordered 23 which has asymmetric, symmetric and non comparable factors: Lemma 1.2a: V (y,Y), (y',Y) E S, (y,Y) R.' (y',Y) 4-> y Q. y' v^Qa y) n Y Q a Y' Lemma 1.2b: V (y,Y), (y',Y) E S, (y,Y) Rs ' (y',Y') ÷-> (y Q. y' v y Qa y') Y Q. 1" Lemma 1.2c: V (y,Y),(y',Y) E S, (y,Y)^(y Q. y v y Q„ y') n Y Q. Y' The proof of these lemmas is as straighforward as that of lemma 1.1a and is therefore omitted. As can be noted, this combination considers a position to be strictly better than another even if there is a Q-indifference between the two actual states of these positions (provided that a strict Q domination exists). In this sense, R' can be considered as giving a weak priority to the quasi-ordering Q as it disagrees with Q on the symmetric factor. This paper is centred around the strong relation R. I believe that analogous analysis could be carried out using the weak relation R' instead. The analysis starts with the following lemma. Lemma 1.3: V (y,Y), (y',Y) E S, (y,Y) R (y',Y) H y Q y' v^Q. y) n Y Q Y') Proof Assume (y,Y) R (y',Y). From definition 1.5, this implies y Q y' v ^Q y) n Y Q Y). From the very definition of the asymmetric and symmetric factor of Q, this is equivalent to y 24 Q y' v ( -i(y' Qs y v y' Q. y) A Y Q Y'). Equivalently, using the distributive law of the logical operator "-.", y Q y' v ( '(y' Qs y) A -(y( Q. y) A - Y Q Y'). But this implies (see Margaris (1990; p.2, expression (4)) y Q y' v (-- (y' Q. y) A Y Q Y'). , Suppose now that y Q y' v (-- (y' Qa y) A Y Q Y') holds. Clearly, -'(y' Q. y) implies either of the , following possibilities: y QN y', y Qs y' or Y Q. y'. That is to say y Q y' v ( (Y' Q. y) A Y Q Y') -, implies y Q y' v ((y QN y' v y Q s y' v y Q. y') A Y Q Y). Written differently, this latter expression is y Q.)/ v ((y QAT Y' v .Y Q. Y') A Y Q ir ) v (y Qs)/ A Y Q Y. ). Since (y Qs Y' A Y Q Y) implies y Q y', the whole expression implies y Q.)/ v ((y QN y' v Y Q. Y') A Y Q Y). By definition of QN and Q., this implies y Q y' v (-- (y' Q y) A Y Q I') which, from definition 1.5, implies in , turn (y,Y) R (y',Y). ^ Most of the results of the paper depend upon the following lemma and its implications. Lemma 1.4: Under assumption A1.1, V (y,Y), (y',Y') Proof Suppose .1/ Qa y. Since y E E S, Y Q 177 -3 V Qa .0. —I MQ (Y) from assumption A1.1, it follows from the transitivity of Q that there can not be any St E Y such that , Q y'. It follows from definition 1.1 that Y Q Y' does not hold. ^ The two following results can be immediately derived from lemmas 1 3-1.4: Proposition 1.1: V (y,Y), (y, Y) E S, (y,Y) R (y',Y') 4-- -> y Q y' v Y Q 1" Proposition 1.2: V (y,Y), (y',Y) E S, (y,Y) R s (y',Y) H y Q s y' v Y Q. IT' 25 The basic consequences of lemmas 1 3 and 1.4 is that, given the particular nature of the two quasi-orderings Q and Q and the definition of S, the strong lexicographic combination of Q and Q constructed in definition 1.5 can be written as the union of the two quasi-orderings (proposition 1.1). Thus, the mere fact that Q is related to Q by definition 1.1 and that y is Q-maximal in Y renders redundant the priority given to Q in the combination R. Proposition 1.2 just states that this consequence affects only the symmetric factor of R. It is clear that R need not, in general, be transitive (as it is formally possible, for an arbitrary sequence (y,Y), (y',Y) and (y",Y") in S to have, say, y Q y' and Y' Q Y" without having y Q y" nor Y Q Y"). Is there some reasonable additional structure that one can impose on S in order to preclude such cycles? That is the question to which the second section of this paper provides an answer. But before attempting such an answer, it is of some interest to relate this question to that asked (and roughly answered) almost forty years ago by Gorman (1955) for the KHSC criterion. More precisely, what is the exact relationship between the binary relation R and the binary relation K? The following proposition provides the answer to this question. Proposition 1.3: Let G(R,S) and G(K,S) denote the graph, on S, of the binary relations R and K respectively.' Then, under assumption Al, G(R,S) c G(K,S). 15 The graph of any binary relation L.- is the set of all ordered pairs (y,y') such that y y' is true. For example G( RA ) = {((y,Y),(y',Y')) E S x S: (y,Y) R (y',Y')}. I adopt throughout the paper the convention that any binary relation L- is entirely characterised by its graph. This convention is well accepted in modern formalised treatments of set theory (see for instance Halmos (1974; p.26) and Suppes (1972; p.5'7)) although some (e.g. Kelley (1975; p.260) or Royden (1988; p. 23, footnote 4) prefers to distinguish between a relation_ancLits_graph. 26 figure 1.3 figure 1.3: In the situation illustrated above with Q being the usual quasi ordering of R 2 +, it is clear that neither y __y' or Y. Y hold so that ((y,Y),(y',Y)) does not belong to G(R,S). However, there exists an 9 E Y such that 9 ?..y' so that ((y,Y),(y',Y)) belongs to G(0). Proof Consider any ((y,Y),(y',Y))€ G(R,S). Then, from definition 1.5 and proposition 1.1, one has either y Q y' or Y Q Y'. But since y E Y, it follows that y Q y' implies Y Q {y'}. If, on the other hand, one has Y Q Y it is clear that, since y' E Y, Y Q {y'} holds. Thus, ((y Q y') v (Y Q Y)) —* Y Q {y'} and, from definition 1.3, ((y,Y),(y',Y)) E GM S) . ^ Remark 1.1: On most domains of interest, G(R,S) will in fact be a proper subset of GOO) (that is G(R,S) c G(K,S) but not G(K,S) c G(R,S)) as illustrated in figure 1.3. 27 Proposition 1.3 shows, in general, that the compensation criterion K ranks pairwise at least as many positions as does the lexicographic relation R. One may therefore hope the requirement of transitivity for R to be less demanding than that of transitivity of K since the former involves less pairwise comparisons than the latter. It will now be shown that such a hope is misplaced. 1.3: The general results The main object of this section is the statement and the proof of two theorems establishing, under fairly mild assumptions concerning the richness of S, that the requirement of transitivity for the relation R (theorem 1.1) and for the relation K (theorem 1.2) is equivalent to requiring the binary relations R and C to coincide on S. If this is the case, the CMS criterion as defined by the relation C is a fully appropriate method for ranking S since it aggrees totally with the Pareto criterion and cases such as the one mentioned in the introduction (figure 1.2) are ruled out de facto. The fact that the equality of R and C is sufficient for the transitivity of R is a triviality given the transitivity of C. More interesting and surprising, at least to my view, is the fact that, under weak assumptions concerning the richness of S, the equality of R and C is also necessary for R as well as necessary and sufficient for K to be transitive. The assumptions that are sufficient to endow S with the appropriate richness to establish this latter result, are introduced by the mean of the following definitions. Definition 1.7: A subset Y of X is said to be Q-compact if and only if y a Y --> 3 9 such that 9 Q y. 28 E Mg (Y) Definition 1.8: An actual state y e Y is said to be plausible in a set of positions S c X x if and only if there exists Y E ...9r such that (y,Y) E S. The following three assumptions are then imposed on the domain S. Assumption A1.3: ..9r is a collection of Q-compact subsets of X closed under intersection. Assumption A1.4: Let (y,Y) E S. Then s , E MQ (Y) implies that 9 is plausible in S. Assumption A1.5: Let (y,Y) E S. Then Y^c Y and y E MQ (Y') implies that (y,Y') E S. Q-compactness of a situation Y is a weaker analogue (see section 1.3; lemma 1.6) to the requirement of compactness for spaces for which a topological or a metric structure is assumed.' It is therefore a natural assumption to make if one is interested in using this result in an economic environment where all sets are typically assumed to be compact. The further requirement of closedness under set intersection is solely technical. The Q-compactness assumption allows one to rely on the following interesting result: 16 In fact, it is sometimes the case in the economic literature (see for example Arrow (1951; lemma 1) or Von Neumann and Morgenstern (1947; p.593)), that the property of Q-compactness is sometimes derived from that of compactness (in spaces that have the adequate topological structure) instead of being assumed in the beginning._ 29 Lemma 1.5: Let .9 be a family of Q-compact subsets of X. Then, for all Y, Y' E ,9, Y Q - Y' H MQ (Y) Q MQ (Y) Proof 1) Y Q Y' ---> MQ (Y) Q MQ (Y). Let Y Q Y'. Then V y' E Y, 3 y e Y such that y Q y'. In particular, this is true for S.' E MQ(Y ) ' which, under Q-compactness, is non-empty. Thus 3 S E Y such that .S Q S '. Since Y is Q-compact, there is no loss of generality in assuming 9 a , , MQ (Y). Thus one has MQ (Y) Q MQ (Y'). 2) To prove the sufficiency part of the lemma, suppose that M Q (Y) Q MQ (Y) and consider any y' E Y'. From Q-compactness, 3 S.' E MQ (Y) such that S.' Q y'. Since MQ (Y) Q MQ (Y'), 3 y a MQ (Y) such that y Q 9'. From the transitivity of Q, one has y Q y'. Since y' was arbitrary, one has, from definition 1.5, Y Q Y. ^ Loosely speaking, lemma 1.5 establishes that if the family of sets to which the extension Q is to be applied consists in Q-compact sets, then one needs only to apply the extension to the maximal sets of this family in order to rank it. In other words, the only alternatives that are relevant to assess whether or not a situation potentially dominates another are the ones that are Q-maximal in these situations. This should not be very surprising if one keeps in mind that the whole purpose of using the extension Q is to complete the basic quasi-ordering Q considered as lexicographically prior. Why should the availability of Q-dominated alternatives bears any importance if these alternatives will never be chosen anyway? Assumption A1.3 requires simply that, if a position (y,Y) is known to belong to S, then any Q-maximal element of the situation Y can be the actual state of some position in S. Notice that this assumption is perfectly compatible with cases, sometimes assumed in applied 30 cost- benefit analysis, where a move from y to some other Pareto optimum 9 in the same set has such an irrevocable effect that it permanently alters the whole set Y. For instance, if one thinks that a particular redistribution of the current national income in Canada is feasible and recommends, for exogenous ethical reason, to realise it, it may well be the case that, after the realisation of the redistribution, the former distribution becomes unattainable. By allowing for the possibility of all states not to be mutually accessible, assumption A1.4 is weaker than the compensation unrestrictedness one that was implicitly assumed in Arrow (1963)'s contribution mentioned above (p.15; footnote 14) and that is used in the Appendix. Compensation unrestrictedness requires indeed all feasible states of a situation to be mutually accessible. Assumption A1.5 requires that, if one thinks of Y as being some set of alternatives thought to be achievable from y, then discarding from Y alternatives different than y should not change the fact that the remaining alternatives are still thought as achievable from y. Such an assumption is, once again, hardly objectionable. Before to prove the theorems, it is worth recalling that, reasonable as they are, assumptions A1.3-A1.5 play only the role of providing S with a richness sufficient to obtain the results. That is, their function is simply to allow one to think of certain positions as theoretically possible in way that has some intuitive appeal. But they clearly impose much more richness than what is needed to obtain the announced results. It is with the aim of further emphasizing the non necessity of assumption A1.3-A1.5, that identical results are proved in appendix A with assumptions A1.4 and A1.5 replaced with the different compensation unrestrictedness assumption. 31 Before proceeding, one can note the following (whose proof is obvious) 17 . Claim 1.1: Let Y and Y' be two sets. Then Y' c Y -4 Y Q Y'. The following theorem can now be established: Theorem 1.1: Let g- be a family of subsets ofX satisfying assumption A1.3 and let S be any subset of X x .satisfying assumptions A1.1, A1.4 and A1.5. Then R is a quasi ordering of S if and only if R = C. Proofi Given the definition of Rs and the reflexivity of both Q and Q, it is obvious that R is reflexive. The proof needs therefore only to be concerned with transitivity. The sufficiency part of the result is an immediate consequence of the fact that C is transitive. The proof of the necessity part is by contraposition. Thus, suppose that C and R disagree as to how to rank two positions (y,Y), (y',Y) a S. From proposition 1.1, it is clear that C is a subrelation of R so that the non coincidence of R and C on (y,Y) and (y',Y) must necessarily come from the fact that (y,Y) R (y',Y) holds but (y,Y) C (y',Y) does not. That is to say, it must be the case that y Q y' Q Y'). From definition 1.1, it follows that 3 E 1r such that 9Q fr' for no j) E Y. In particular, this implies that y Q S.' does not hold. It is also clear that y' Q can not hold as assuming otherwise would contradict, given the transitivity of Q, the previous claim that y Q .r does not hold. On the other hand, one can not have S.' Qa y' as such a case would contradict assumption A1.1 that y' E MQ(Y). It follows from the two last sentences that y' QN lemma 1.5, there is no loss of generality in assuming that assumption A1.4 that 3 Y" a .53" such that (S, ',Y") 17.. ^-lemma- I -of ehipmm -arid Proore -(19 /1). 32 E E Mci ( r). Y. From It then follows from S. Now clearly Y" n Y # 4:0 since E Y" n Y'. Moreover, it is true that (Y" nY') c Y", that (Y" nY') c Y' and that S' E M9 (Y" n Y'). It follows at once from A1.5 and A1.1 that (Y,Y"n Y') a S. From definition 1.5 and claim 1.1, (y',Y) R (9',Y' n Y') (in fact (y',Y') R a (9',Y" n Y')). However since S. Q 9' for no 9 E Y, it is clear that Y Q r'n Y' does not hold and, from what has been established above, that y Q does not hold either. This, together with definition 1.5, gives the desired intransitive chain. El Remark 1.2: If assumptions A1.4-A1.5 as well as the closedness under finite intersection part of assumption A1.3 are dropped and replaced by the requirement that, whenever a position (y,Y) belongs to S, so does (y,{y}), the theorem remains true. This fact, together with yet another version of theorem 1.1 proved in appendix with A1.4-A1.5 replaced by compensation unrestrictedness, emphasises further the non-necessity of assumptions A1.3 A1.5. What about the compensation criterion? The following theorem establishes an analogous result for K. Theorem 1.2: Let .9T be a family of subsets of X satisfying assumption A1.3 and let S be any subset of X x .57- satisfying assumptions A1.1, A1.4 and A1.5. Then K is a quasi ordering of S if and only if R = C. Proof The reflexivity of K is an immediate consequence of the reflexivity of both Q and Q and of definition 1.3. As in theorem 1.1, the proof is therefore only concerned with transitivity. The proof of the sufficiency part is by contraposition. Assume, in effect, that the transitivity of K is violated over at least one chain (y,Y), (y',Y') and (y",Y") of positions in S. That is, assume that (y,Y) K (y',Y') and (y',Y') K (y",Y") hold but that (y,Y) K (y",Y") does not. The objective is 33 to establish that, for any such an intransitive chain, the equality of R and C is violated for at least one pair of positions in S. To see this, one must note first that (y,Y) K (y",Y") being false implies, necessarily, that Y Q Y" is false. Now, consider the statement (y,Y) K (y',Y'). From definition 1.3, there exists assuming E s. E Y such that 9 Q y'. From A1.3, there is no loss of generality in MQ (Y). From A1.3, 3 Z E .such that (S ,Z) E S. Under A1.4, Sris closed under , intersection so that (Z n Y) E Sr and, from A1.5, (9,Z n Y) E S. Now, we know that S. Q y'. If (Z n Y) Q Y' does not hold, we find the desired violation of the equality of R and C. If, on the other hand, (Z n Y) Q Y' does hold, then, by virtue of claim 1 1 and the transitivity of Q, Y Q Y' must hold as well. Now an exactly analogous reasoning performed with the statement (y',Y') K (y",Y") lead to exactly the same conclusion. Namely, that either the equality of R and C is violated for (9',Z' n Y') and (y",Y") or that Y' Q Y". Clearly Y Q Y' and Y' Q Y" can not be both true as, otherwise, the transitivity of Q, will imply Y Q Y" (contrary to the assumption that there exists no y E Y such that y Q y'). Therefore, the equality of R and C must be violated at least once for the K-intransitive chain (y,Y), (y',Y') and (y",Y") of positions in S to exist. The proof of the necessity part is exactly like that of theorem 1.1 (with the replacement of R by K) until the step of comparing the positions (y,Y) and (Y ,(Y'n Y') in terms of K is reached. This convenient state of affairs is an immediate consequence of proposition 1.3. Now, as one recalls from theorem 1.1, is such that 9 Q for no j) E Y. That is, from definition 1.3, (y,Y) K (r,(Ynn Y') does not hold. Li If one replaces assumptions A1.4-A1.5 by compensation unrestrictedness as done in the appendix, one can show that the equality of C and R is necessary and sufficient for the compensation criterion to be transitive (theorem A'). 34 1.4: Interpretation of the Results in a Standard Economic Environment In this section, it is shown that in a standard economic environment, the condition of equality of C and R is formally equivalent to the requirement that no utility possibility sets cross. For this task, assume an economy with n consumers indexed by a set N c N+, each of whom is represented by some interval /L i g R+ (i c N) thought to be the image of some set under some continuous utility function f (i E N). Given any 2 points u, v e Rn+, let {u,v) denote the closed segment connecting u to v (that is, [u,v] = {u' E Rn+ I 3 t E [0,1] such that u'= (1-t)u + tvp. Given any w E Rn +, a neighbourhood of radius e around w is denoted Ne(w) and is defined by Ne (w) = {w' E W'+ (V i E N), j w'i - w t I < a}. In this setting any u E Rn+ is interpreted as a utility allocation and any U c Rn+ as a utility possibility set. Let U be some family of subsets of Rn + and let S' be some subset of Rn+ x U. Let F(U) denotes the frontier of the utility possibility set U in Rn+ (that is F(U) = {u E Ulu is adherent to + \ U}) 18 , let U- denote the closure of U and let U ° denote the interior of U. Occasionnaly, when there is no risk of ambiguity concerning the sets U, U',etc, I shall use F, F', etc. to denote their frontiers in Rn+. The notation for the quasi-ordering of vectors in Rn+ is > and >>. The standard practice in economics is to impose a definite structure on the inverse image of any set U (inverse image that is assumed to be a set of allocations of M consumption goods among the N consumers which, in turn, can be seen as being determined by some aggregate production set) and to derive the structure of the sets U from the structure ILSeeDebreu -(4959). 35 imposed on the inverse image, given the assumed properties of the utility functions and the production set. In this section, I proceed instead by imposing the structure directly on the family U and on the set S'.' For this task, consider the following definitions and assumptions: Definition 1.9: A utility possibility set U E W1+ is disposable if and only if (u E U, U' E Rn+ and u' < u) ---> u' E U and is said to be strictly disposable if and only if it is disposable and u E U, u' E Rn+ and u' < u --> 3 ft E U such that & >> u'. Definition 1.10: Two utility possibility frontiers F and F' are said to cross each other in the sense of Gorman (1953) at point u E R n+ if F # F' and u E Fn F'. Assumption A1.6: U is a family of compact and strictly disposable subsets of Rn+ closed under finite intersections. Assumption A1.7: S' = G(M). Now, it is easy to verify that this standard economic framework fits well into the more abstract setting of the last section by substituting R n+, U, ., 111,(U), S' and G(MM) for X, ST, Q, Q, MQ (Y), S and G(MQ) respectively. Indeed, under A1.6 and A1.7, 5' satisfies assumptions A1.3-A1.5 as well as compensation unrestrictedness and, of course, under A1.6, U is closed under finite intersections. For .-compactness, it suffices to show the following 19 The relation between the two spaces (the domain and the image) is well known. An excellent lid Bixby (1973). 36 result. Lemma 1.6: Let U c Rn+ be compact. Then U is .-compact. Proof let u e U with U c Rn compact. Consider 4 = sup G(u) where G(u) = fu' 1 u' .. u and u' E (../). G(u) is clearly compact since it is a closed subset of a compact set. Therefore 4 E G(u) c U. The fact that 4 E MAD is an immediate consequence of the definition of a supremum.^ This lemma concludes the demonstration that the present setting is a formal interpretation of that of the last section. The only thing that remains to be shown is that, in such an economic environment, the equality of R and C is exactly equivalent to Gorman's (1953) condition of non-crossing of any two utility possibility frontiers. Recall that the interest of this latter equivalence lies in the fact that non-crossing of utility possibility frontier is in turn equivalent, when the utility possibility sets are assumed to be generated from an exchange economy, to the requirement of parrallel straight lines Engel curves for all individuals, as shown in Gorman (1953). But this result rests on the assumption that, for each utility possibility set, its frontier in lan+ coincides with its set of Pareto optima. It is in order to obtain, in the present context, such a coincidence that assumption A1.6 of strict disposability has been imposed. Given strict disposability, one obtains the following result. Lemma 1.7: For any strictly disposable U c Rn+, M,(U) = F(U). 37 Proof I first show that M,(U) c F(U). Let u E M,(U). Then, consider any A e Rn + such that ft > u. By definition of M,(U), /I 0 U. Define the sequence {u,}, =7 by u, = u(1-1/i) +^for By construction, one has u, > u and thus u, U for all^But clearly, {u,},:1 converges to u. u is therefore adherent to the complement of U in )18n 1 . - - To show that F(U) c M,(U). Let u' e F(U) but, contrary to the claim, assume that u' M,(U). Then, 3 u such that u > u'. Since U is strictly disposable, 3 ll e U such that t2 » u'. Let 5 = min (it - it:). Since » u', 5 > 0. Consider now any v E N 1.(u'). By definition of a ieN neighbourhood one has I v, - u,' < 5/2 < 5 5. (A, - u,') for all i E N. If u,' > v i , it is clear that fL i > > v,. If v, > u,', one has v, - u,' < h i - u,' which clearly imply v, < A,. Thus v « It and, from disposability, v E U. But this imply that N(u') c U, a contradiction of the original claim that u' e F(U). ^ The present context is thus entirely equivalent to that of Gorman (1953) and one can thus establish the last result of this paper. For this task, some use is made of the following lemmas. Lemma 1.8: Let U be any non-empty disposable subset of IV+. Then U is connected. Proof Since U is a non-empty disposable subset of 1V+, 0 e U. Associate to any u E U the set [0,u]. [0,u] is obviously connected. The rest of the proof is to show that u [0,u] = U. It is UE U straightforward to establish that U c u [0,u]. To show that u [0,u] c U, let u' E u [0,u] . Then, uEU^ for some u e U, E ueU^ uEU [0,u]. By definition of [0,u], u' S u. By disposability u' E U. The fact that U is connected is an immediate consequence (see Binmore (1981; theorem 17.5)) of the fact that U can be written as the union of a collection of connected sets all of which containing 0 as a common element. ^ 38 Lemma 1.9: Let U be any closed subset of Rn . Then, VueU and V v a R n/U, F(U) n [u,v] # 0. Proof Notice first that [u,v] is connected. The proof is then by contradiction. Suppose that for some u E U and for some v E W /U, one has F(U) n f u,v] = 0. In particular, this imply ' trivially that u a U ° . Therefore, it must be the case that [u,v] c (U° u Rn/U) and, by definition, [u,v] = ([u,v] n tr) u ([u,v] n Rn/U). Now clearly, ([u,v] n U° ) # 0 since u E ([u,v] n tr) and ([u,v] n Rn/U) # 0 since v E ([u,v] n Rn/U). Moreover, ([u,v] nir) c U ° so that ([u,v] n tr)c U°- = U (since U is closed). Now the obvious emptyness of U n Rn/U implies the emptyness of ([u,v] n U°)- n ([u,v] n R n/U). By an analogous reasonning, one can establish that ([u,v] n R" /U)- n ([u,v] n U ° ) is also empty. But this implies that [u,v] can be written as the union of two non empty separated sets (see Kelley (1975; p.52), a contradiction of [u,v] being connected. One can now prove the following theorem: Theorem 1.3: let U be a family of subsets of Rn + satisfying A1.6 and let S' c Rn+ x U be a domain of positions satisfying A1.7. Then R and C coincide on S' if and only if no utility frontiers cross. Proof. The proof of the necessity part is by contraposition. Suppose that two frontiers F and F' cross at some point u. Since F # F' (definition 1.10), one has either (3 F') or (3 a E F such that a e a' E F' such that a' 0 F). The two cases are symmetric so that there is no loss of generality in considering only one of the two. Suppose therefore that 3 a a F such that a 0 39 F'. a E F implies, by assumption A1.6 and lemma 1.7, that Q is not comparable with v as per the quasi-ordering From lemma 1.5 and lemma 1.6, it suffices to consider the two positions (v ,F) and (v ,F'). From the reflexivity of the quasi-ordering one has v v. However, either F > F' or F' F is false, so that the equality of R and C is violated. To see this last claim, assume that F' F holds and, therefore, that 3 E F' such that a' 6. Since z F', z # and, therefore, 6' > 12. But if such a exists, there can not be any u E F such that u et' for, assuming the existence of such a u would, given the transitivity of contradict the assumption that fL E F c 111,(U). Thus F' F being true implies necessarily F F' being false so that the equality of R and C is inevitably violated. The proof of the sufficiency part is also by contraposition. Suppose that for two positions (u,U), (u',U') E S , ' one has u u' but not U U'. Recall from the proof of theorem 1.1 that this is the only way by which the equality of R and C can be violated. From definition 1.5, lemma 1.5, and assumption A1.6, 3 v' E F' such that v v' for no v E F. In particular, this is true for u so that u v' does not hold and, by transitivity of u' v' does not hold either. The reflexivity and transitivity of >_ insures also that (v' E F') --> u'). By reflexivity of and definition 1.5, v' U. Define W(u) and W(v') by W(u) = E Rn + u} and W(v') = {O' E R'+ 0' v'}. Let a be the greatest element of W(u) n W (v') . It is not difficult to show that such a unique greatest element exists, provided that W(u) n W(v') is non-empty. W(u) n W(v') is certainly non-empty since it contains 0 as an element. Since -'((u v') v (v' ?_ u)), one has that a < u and a < v'. Since U is disposable, a has from lemmas 1.5 and 1.7 that 3 u* E E U. Since U is compact (closed and bounded), one [a, v'] n F. By construction, u* < v'. V U E U, define the mapping pu : U -4 R+ by p u (u) = 6 f 1 where f a F(U) is such that u E [0,f]. Lemmas 1.6, 1.7 and 1.9 insure that the mapping p u is a continuous function on U. Define now the function U u U' -3 R by qi(x) = p u (x) - p u-(x). It is a continuous function which, by lemma 1.8, is defined on a connected set U u U'. Now, 41 (u*) = H u* p u,(u*) < 0 since u* v', v' E F(U') 40 and u* E U'. However, u _ u' together with the fact that u E F(U) and u' e F(U') imply that liqu') = II u i- II u1 > 0. By the intermediate value theorem, 3ft e U n U' such that ‘1 1(it) = 0. By construction, ft is a point where the two frontiers F(U) and F(U') cross. ^ 1.5: Conclusion Assuming that utility possibility sets can never cross is not an innocuous assumption. In an exchange economy, Gorman (1953) has shown that requiring this condition to hold is equivalent to requiring all individual Engel curves to be parrallel straight lines at identical prices. In the presence of production, the condition of non-crossing of utility possibility sets may be even more stringent since it requires additionnal restrictions on the shape of the admissible aggregate production sets. Stringent as it be, the condition of non crossing of utility possibility sets is, nonetheless, necessary and sufficient for, either the Kaldor-Hicks-Scitovsky compensation criterion to be used without intransitivity, or for the CMS quasi-ordering to be applied in accordance with the Pareto criterion. This is the main lesson of this paper. 41 Chapter 2: Ranking Opportunity Sets on the Basis of their Freedom of Choice and their Ability to Satisfy Preferences: A Fundamental Difficulty 2.1 - Preliminaries. How can we say that one opportunity set faced by an individual is "better" than another? To this question, which is of central importance for policy evaluation, standard economic theory provides the following answer: an opportunity set is better when, and in so far as, it allows the individual to reach a higher level of maximal utility (or a more preferred chosen element). 2° Despite the appeal of its definiteness, this answer has been the object of several criticisms. Among other things, it has been reproached as being insensitive to the intrinsic value of the freedom of choice allowed by the various sets, irrespective of the particular choice that the individual will end up making in each of them. 21 This line of argument rests on the presumption that, even for, say, someone who dislikes alcohol, being given the choice to drink it (although choosing not to) is intrinsically better than living under some prohibitive law. How could one construct a ranking of opportunity sets that respects the intrinsic value of such a freedom of choice? An immediate welfarist answer to this question would be to let the utility function used to represent individual preferences (see Debreu (1959; 20 The philosophical view that situations (like opportunity sets) should be evaluated only in terms of their final consequences is called consequentialism and has been strongly advocated, in economics, by Hammond (1986). The additional postulate that these final consequences should be assessed in terms of individual welfare, as defined by preference fulfilment, is known as welfarism and has been extensively discussed by Sen (1974; 1979; 1991). 21 rya-Sen-- (4985-; 1990; 1991)- iie Vl 42 attae x. 1.4-k)) depend upon whatever measure of freedom of choice is used. After all, a broad enough definition of "preferences" can always encompass everything that is deemed relevant in evaluating individual welfare. 22 One may of course "be deeply sceptical of such allencompassing utility functions" as Suppes (1987; p.243) put it. But even if one is not, there seems to be some specific difficulty in extending the domain of the utility function to include some measure of freedom of choice. In effect, the very notion of freedom of choice seems closely related to the idea of the "size" of the opportunity set from which the individual is choosing. Given an opportunity set, it is intuitively clear that the addition of any new alternative increases the freedom of choice attached to it. In other words, the measure of freedom of choice should agree with the set theoretic notion of inclusion.' Individual preferences, on the other hand, as they are usually understood in economics, are defined as the binary relation that rationalises the individual choice within each of these sets.' Therefore, it seems very difficult, at least a priori, to let the utility function used to represent individual preferences also depend upon the size of the opportunity set. 22 For an argument along this line, see Hammond (1986). 23 In fact, to the best of my knowledge, all attempts to axiomatise the notion of "freedom of choice" (see Kreps (1979), Suppes (1987), Pattanaik and Xu (1990) and Sen (1990; 1991)) have either included this requirement as an axiom or as a consequence of other axioms. See Sen (1970, ch 1; 1971) and, for more technical aspects of the relation between the choice function and its rationalisation by means of a binary relation, Herzberger (1973). As Sen (1991) pointed out, the term "preferences" is not free of ambiguities. A common use of the word "preferences" (sometimes associated with the one just described) is as an index of individual well-being or welfare. Formally speaking though, the two meanings are distinct. If one defines preferences as an index of individual well-being, then identifying these preferences with the binary relation underlying the individual choice (with a possible abstraction from strategic considerations) requires one to take the extra step of assuming that the individual always chooses what is best for her in terms of this wellbeing. As mentioned above (see footnote 1) the doctrine known as welfarism does take this step. The present paper does not. Therefore by "individual preferences", I shall mean the binary relation underlying individual hypothetical choices (abstracting from strategic considerations) without bothering with the issue of whether or not these choices bear any relation to individual "well-being" -er-2 welfarc". 24 43 To get a flavour of this difficulty, consider the framework used in economic theory to study consumer choices in a competitive environment. In this setting, alternatives are assumed to belong to the n-dimensional Euclidian space, preferences are restricted to be continuous and convex, and opportunity sets (budget sets) are assumed to be simplexes. These opportunity sets are ordered by the indirect utility function (the maximal utility achievable in each budget set as a function of the income/price parameters). For the ranking of the budget sets induced by the pairwise comparison of the value of the indirect utility function to include set theoretic inclusion as a subrelation (see Sen (1970; 1.4)), one needs to add strict monotonicity of the preferences and strict convexity of the budget sets.' These assumptions may not always be natural, even in the context specific to economic models. These potential difficulties, together with a more fundamental rejection of a preference-based ranking of opportunity sets, 26 have motivated some authors (see Suppes (1987) and Pattanaik and Xu (1990)) to propose various axioms for ranking opportunity sets solely on the basis of the freedom of choice they offer. Their axioms, which do not assume the existence of individual preferences, lead to a unique ranking of sets: the one induced by the pairwise comparison of their cardinal numbers. 27 In Pattanaik and Xu (1990) this ranking 25 Suppose the budget set is not convex (perhaps because of the fact that the price of some commodity is not really "parametric" but depends upon the quantity purchased as in the case of "quantity discounts"). Then, even if the preference ordering is strictly monotonic and convex, the indirect utility function will not necessarily increase with an enlargement of the budget set. This rejection is obvious in Suppes (1987). Pattanaik and Xu (1990) have also rejected a preference-based comparison of the opportunity sets by condemning what they see as the welfarist bias of this approach. However, given what has been said above (in footnote 5), it is worth remembering that a preference-based approach to the ranking of opportunity sets need not be welfarist. As Sen (1991;p.19-20) puts it, it can easily be argued that the "goodness" of opportunity sets is to be judged with some "regard to what people would themselves like to choose - no matter what motivates them toward that choice" (Sen's emphasis) - without this implying "ground[s] for a 'welfarist' bias in interpreting and doctoring individual choices in the direction of individual welfare." 26 The cardinal number of a finite set is defined as the number of its elements. The cardinal "number" of an infinite but countable set is denoted by N. The cardinal number of any set that is -homeomorphic intei v [0,1] i5 denoted by c (standing for continuum). Finally, the cardinal 27 44 is the only ranking of opportunity sets that satisfies their three axioms. In Suppes (1987), the only examples of rankings that satisfy his five axioms are also rankings obtained by comparing cardinal numbers. Despite informal arguments to the contrary given by Hayek (1960; p.32-35), it seems clear that "cardinal rankings" are of very limited interest, even for the most ardent lover of liberty. First, ranking by cardinality results in a very coarse partition of the family of opportunity sets under consideration, placing all singleton sets in the same equivalence class, all uncountable sets in another, etc.' With such a ranking, "the choice among 'a nasty life', a 'terrible life' and 'an unspeakable life' gives us just as much freedom as the choice among 'a fine life', 'an excellent life' and 'a wonderful life"' as Sen (1991; p.26) puts it. Second, it is not difficult to see that the cardinality rankings violate what might appear to be a very minimal principle of freedom - namely, if one set is a proper subset of another, then the freedom of choice offered by the latter must be strictly larger than that offered by the former. For opportunity sets with finite number of alternatives, rankings in terms of cardinality satisfy this principle. However, when opportunity sets are allowed to be infinite (as is usually the case for budget sets), this principle is violated. For example, according to this cardinality criterion, two individuals facing the same prices but having different incomes possess the same freedom of choice because their budget sets both have the cardinality of the continuum. To overcome this impasse, Sen (1991; p.25) has suggested (following Arrow (1987; p.730)) the valuation of preference choice as a supplement to freedom of choice rather "number" of any set that homeomorphic to the set of all numerical functions defined on the interval [0,11 is f. The ranking of these "numbers" is as follows: (V n, m E N such that n < m) n <m <N<c < f. Readers not familiar with cardinal arithmetic may find Halmos (1974; ch.24) enlightening. 28 Ii A V :^SOO:. 45 • I^• • • ass. than as a substitute for it. Roughly speaking, Sen's idea is to rank sets primarily on the basis of their freedom of choice and, in cases where such a basis does not yield clear-cut conclusions, to complete this ranking by using information on individual preferences. The aim of this chapter is to examine Sen's suggestion more closely. More specifically, the following question is posed: Is it possible to construct a consistent transitive ranking of opportunity sets that respects, intrinsically and above all, the aforementionned minimal principle of freedom of choice and, in cases where this principle is not conclusive, respects individual preferences over the alternatives contained in the opportunity sets? The answer that obtains is no. As shown below, provided that the domain of opportunity sets is minimally rich, a necessary and sufficient condition for such a transitive ranking to exist is to require freedom of choice, as captured by the set theoretic notion of inclusion, to be included as an argument of the preference ordering. Thus, obtaining a transitive ranking of opportunity sets that respects, primarily, a minimal principle of freedom of choice as well as individual preferences over the alternatives contained in the opportunity sets (for cases where no comparisons of these sets solely on the basis of their freedom can be obtained) appears to be equivalent to giving only an instrumental value to freedom of choice instead of an intrinsic one (to use Sen's (1988) terminology). However, the nihilism of this result is tempered by showing that, if one is willing to weaken the requirement of consistency from transitivity to quasi-transitivity, the construction of such a ranking can be achieved without any difficulty. The plan of this chapter is as follows. Section 2.2 sets out the notation and discusses the formal similarities between the present problem and the one that I have addressed in chapter 1. In effect, the lexicographic combination of the Pareto criterion and its 46 extension to feasible sets of social states studied in the first chapter is somewhat analogous to the lexicographic combination of the freedom criterion and the preference criterion that constitutes the object of the present chapter. Section 2.3 states and proves my results and section 2.4 offers some conclusions. 2.2 - The Model Let X be an arbitrary universe of alternatives and let R be a complete, reflexive and transitive binary relation on X (with asymmetric and symmetric factors P and and I respectively). Let irbe a family of subsets of X and let S c ..9rx X be a set satisfying the following two properties. Assumption A2.1: V (A,a) E S, a E A and a R a' V a' c A. Assumption A2.2: For any (A,a), (B,b) E S such that A # X, B # X and a I b, 3 (C,c) a S such that A st C, C ct A, B zC,Cst.B andaIbIc. The intuitive interpretation of this framework in terms of the informal discussion of the preceding section is as follows. X is the universal set of alternatives over which an individual has preferences that are represented by the ordering R. the class of opportunity sets that the individual expects to face. For each opportunity set, the individual chooses (either actually or hypothetically) an alternative that she considers to be at least as good as any other in the set. A typical element (A,a) of S is therefore thought of as a choice situation formed by the opportunity set A and a most prefered (chosen). element a in A 47 Assumption A2.1 merely says that S is a collection of choice situations. 29 The set S is to be ranked in a manner which respect both freedom of choice and preferences. Assumption A2.2 endows this collection with a fairly minimal richness. In words, Assumption A2.2 requires that the domain of choice situations under consideration be such that, if there are two feasible choice situations in which the individual is indifferent between her optimal choices in each of the opportunity sets (which need not be distinct) and neither opportunity set is the universal set, then there must exist at least one other choice situation with the following properties: (i) its opportunity set does not include (nor is included in) either of the two original opportunity sets, and (ii) its most-preferred alternative makes the individual just as well off, according to her preferences, as she was in either of the two original choice situations. The possibility that this new chosen alternative is equal to either (or both) of the originally-chosen elements is not excluded by the assumption. To get a better intuition of the meaning of Assumption A2.2 (as well as its reasonableness) consider the two following examples. In the first example, illustrated by Figure 2.1, an individual is assumed to have a continuous and monotonic preference ordering over all possible bundles containing two goods (i.e. X = R2 +). In such a familiar context, Assumption A2.2 requires simply that, if the two choice situations where the individual chooses a and b given budgets A and B (respectively) belong to the collection of choice situations under consideration, then at least one choice situation, such a (c,C) in Figure 2.1, must be in the collection as well. Clearly, in cases like this where all budget-choice pairs in W'+ are though of as possible choice situations, 29 It is important to recall that, in the present context, preferences need not have meanings other than that of being the binary relation that rationalises individual choices. One may, of course, consider the implicit assumption that individual choices are sufficiently consistent to be rationalised by an ordering as being too strong (see Sen (1971)). However, since I am concerned with the difficulty of consistently combining two principles for ranking S, one of which being the preference relation R, I doubt very- much that this- cliffieulty-eoukt-be-initigated--by making R less consistent 48 good 1 Assumption A2.2 is fairly innocuous. In the second example, the universe X consists of all activities in which an individual called Vanessa can partake in and the family .?of opportunity sets consists of those activities that are available to her in a certain subset of the world's cities within a given daily budget 49 As one might expect, none of the cities offers all conceivable activities. Furthermore, for any pair of opportunity sets, it is not the case that one is the subset of the other. e.g. having a picnic on the beach while enjoying a bottle of Sancerre, and sunbathing topless on the patio of an apartment building are activities that do not belong to Vancouver's opportunity set but do belong to Nice's opportunity set. The converse is true with respect to the activities of watching a baseball game on television or drinking O'keefe Old Stock beer. Assumption A2.2 would require the sample to be rich enough so that, whenever Vanessa is indifferent between her most preferred package of activities in any pair of cities (which could be the same), one can find a third (a second if the previous ones are the same) city which offers a package of options that she likes just as well (but not more). One case that is ruled out by assumption A2.2 is when, say, Vanessa strongly prefers sunbathing topless and picnicing on the beach with a fresh Sancerre to watching baseball and drinking O'keefe Old Stock and where the only cities in the considered subset are Vancouver and Nice. The argument against such an impoverished domain of choice situations, ruled out by A2.2, is that it does not provide a very interesting basis on which the notion of freedom of choices can be discussed. Assumption A2.2 does not apply if either of the choice situations has the universal set X as opportunity set. In contrast with the approach taken in standard economic theory (where the universe is some closed and unbounded subset of R il ordered by a non- satiable preference relation), such a possibility is not precluded in the present framework. Although this state of affairs may seem to be nothing more than an insignificant technical detail, it turns out to have some importance for one of the two main results herein. For this reason, I shall be explicit in denoting by choice situations for which T the (possibly empty) subset of S consisting of all X is the opportunity set. 50 I now turn to the main problem examined in this paper: that of ranking, pairwise, the various choice situations in S. As mentioned in the introduction, economists tend to base such comparisons exclusively upon the individual preferences over the chosen alternatives in the various choice situations of S. That is to say, economist's favourite ranking of S is the one induced by the binary relation R* defined as follows. Definition 2.1: V (A,a), (B,b) E S, (A,a) R* (B,b) E—> a R b. A priori, R* does not seem sensitive to differences in the freedom of choice offered by the various opportunity sets encountered in S. For instance, if Vanessa lives in Vancouver and does not enjoy sunbathing topless, from her point of view, R* will not consider a decision taken by the Vancouver authorities to allow women to go topless as worth doing, even if such a decision undoubtedly enhance her freedom of choice. For many writers (Pattanaik and Xu (1990), Suppes (1987) and Sen (1985; 1988; 1990; 1991), among others), such a state of affairs is unsatisfactory. These authors have argued, in effect, that the mere fact of choosing from a set offering more choice opportunities is, in itself, a desirable thing, irrespective of the preferences over the new opportunities offered by the larger set. If one agrees with this line of argument, it seems desirable to require the ranking of the various choice situations of S to satisfy some notion of freedom of choice. In the present paper, this requirement is modeled via the two following axioms (discussed in different contexts by Kreps (1979), Suppes (1987), Pattanaik and Xu (1990) and Sen (1990; 1991). Axiom A2.3: V (A,a), (B,b) E S, A D B implies that (A,a) should be considered at least as good as (B,b). 51 Axiom A2.4: V (A,a), (B,b) E S, A D B implies that (A,a) should be ranked strictly above (B,b). A priori, if one abstracts from the preference-choice theoretic framework which underlies Assumption A2.1, the two axioms are completely independent since one can satisfy A2.4 without satisfying A2.3 and vice-versa. However, this independence does not hold under A2.1 (Proposition 2.1 below) which makes A2.3 redundant. Both seem to be very natural axioms of freedom since they capture the intuitive idea that, when new options are added to an opportunity set, the freedom of choice it offers is enhanced. As previously mentionned, the preference-based ranking of S of definition 2.1 does not satisfy A2.4 although it does, under A2.1, satisfy A2.3 as the following proposition establishes. Proposition 2.1: Let S satisfy A2.1. Then R* satisfies A2.3. Proof. Let (A,a), (B,b) E S be such that A D B. Hence, b E B -4 b E A. By A2.I, a R a' for all a' E A and, therefore, for a' = b. ^ 30 For R* to satisfy A2.4, one would have to make sure that the following assumption holds on the domain S. Assumption A2.5: V (A,a), (B,b) E 5, A D B- a P b. 30 ^is nothing more in this proposition than a restatement of the well known fact (see for instance Sen (1970; ch. 1*)) that a choice function induced by the maximisation of an ordering satisfies -6^• 52 This assumption could be imposed by endowing X and Jrwith some structure, or by imposing some restrictions on the individual preference ordering R, or by a combination of the two methods as is done in standard consumer theory. However, it may not always be natural to impose such an assumption which, in Sen's (1988) terminology, forces one to view the value of freedom as instrumental (freedom of choice is important only because the individual prefer more freedom to less according to the preference relation R) rather as intrinsic (freedom of choice is important for its own sake). At the other extreme, some have argued in favour of a purely libertarian ranking of S which would completely ignore the preferences of the individual over the chosen alternatives in the choice situations. Such an approach is attempted by Pattanaik and Xu (1990). They assume a finite universe of alternatives X and use the domain of choice situations generated by all possible subsets of the universe as opportunity sets. Because they completely ignore individual preferences, their analysis is entirely centred around the first coordinate of choices situations. They impose three axioms on their ranking of S which, given the finiteness of their universe, satisfy both A2.3 and A2.4, and they show that the only ranking that satisfies their three axioms is the ranking of sets induced by the comparison of the number of their elements. As discussed in the preceding section, it goes without saying that such a ranking is of very limited interest. What is patently clear, at least for a domain satisfying Assumption A2.2, is that, unless the only choice situations in S are those for which the entire universe X is the opportunity set, the freedom axioms A2.3 and A2.4 do not, by themselves, provide the basis 53 for a complete ranking of S. 3 ' Noticing this, as well as recognising the importance of attaching some weight to individual preferences, has led Sen (1991) to argue in favour of an intermediate position between a ranking based entirely on freedom, such as that of Pattanaik and Xu (1990), and a ranking based entirely on individual preferences such as that of definition 2.1. Roughly speaking, Sen's suggestion is to value intrinsically freedom of choice as much as one can but, for cases where Axioms A2.3 and A2.4 lead to non-comparability, to supplement them by comparing the choice situation with respect to their chosen elements as evaluated by the preference relation R. Although Sen's discussion is of an informal nature, the binary relation defined on S as follows, seems to be an appropriate formalisation of this idea. Definition 2.2. V(A,a), (B,b) S, (A,a) (B,b) HADBv(AaBnaRb). This binary relation ranks various opportunity sets primarily on the basis of their freedom of choice (as defined by the quasi-ordering c) and, for cases where this principle is not conclusive, supplements it by ranking sets on the basis of their chosen elements as compared by the preference ordering R. Given the reflexivity of both c and R and the completeness of R, it is clear that is both reflexive and complete. However, at least at this level of generality, need not be transitive. The asymmetric and symmetric factors of (denoted by >- and — respectively) can be shown to be given by the following lemmas. Lemma 2.1: V (A,a), (B,b) E S, (A,a) >- (B,b)^ADBv (A gt BnaP b). Under Assumption A2.2, and provided (once again) that S does not consist only of choices situations for which the entire universe X is the opportunity set, the binary relation c is an incomplete quasi-ordering of he various opportunity sets. 31 54 Lemma 2.2: V (A,a), (B,b) E S, (A,a) — (B,b) E--> A = B v (A cz B A B a A A a / b). As these statements make clear, L satisfies both Axioms A2.3 and A2.4. To simplify the notation, I shall write, to represent any two sets A and B for which A B a B and a A, A E B. Before getting to the main results, a few words on the stuctural similarity between the present problem and the one studied in Chapter 1 are in order. The binary relation .>.: presents indeed some similarity with respect to the lexicographic combination of the Pareto and the Chipman-Moore-Samuelson quasi-orderings of the preceding chapter. Both relations combine lexicographically a binary relation over alternatives (the Pareto criterion in Chapter 1 and the individual preference relation R here) and a binary relation over sets of alternatives (the Chipman-Moore-Samuelson criterion in Chapter 1 and the inclusion relation c here). There are, however, three important differences between the two combinations. First, in chapter 1, the lexicographic combination is based on two incomplete quasi-orderings whereas L is based on one incomplete quasi-ordering (the set relation c) and the complete preference ordering R. Second, in Chapter 1, lexicographic priority is given to the binary relation over alternatives (the Pareto quasi-ordering) whereas in the present paper, priority is given to the binary relation over sets (the set-theoretic relation of inclusion). Third, although c is a subrelation of the set relation used in the preceding chapter (Definition 1.1 and Claim 1.1), the two relations are distinct nonetheless. 32 This limits the extent to which the result of the first paper can be used to handle the problem addressed here. 32 For equality, assuming X = R N , one must further restrict ..r -- to belong to the class of disposable and compact scts (sec Chipman and Moore (1971)). 55 2.3: The Results For the purpose of policy evaluation, it appears important to require the binary relation L- to generate a transitive ranking of the choice situations belonging to S. However, given A2.1 and A2.2, it turns out that a necessary and sufficient condition for L- to provide such a transitive ranking of the choice situations in S is that the subset of these choice situations containing only those which do not involve the entire universe X as the opportunity set satisfies A2.5. That is to say, requiring the relation to be transitive over all choice situations in S is formally equivalent to requiring the preference relation R to include the set relation c as a subrelation (with the exception of the choice situations involving X as the opportunity set). This result is the content of the following theorem. Theorem 2.1: Assume S satisfies A2.1 and A2.2. Then of definition 2.2 is transitive on S if and only if S \T satisfies A2.5. Proof To prove the sufficiency part of the result, suppose S/T satisfies A2.5. That is, V (Y,y), (Z,z) E S such that Y D Z and Y # X, one has y P z. Consider now any (A,a), (B,b) and (C,c) E S such that (A,a) (B,b) and (B,b) (C,c). From Definition 2.2, this implies that [A D B v (A c4 BnaR b)] and [BDCv(BctCAbRc)] are both true. One must consider each of the four logical possibilities generated by the distribution of the logical connector "and" between the two statements. Case (i) A D B and B C. In this case A D C and, from definition 2.2, (A,a) (C,c) follows directly from the transitivity of D. Case (ii) A D B and (BctCAbRc). For this case, one can first notice, as an immediate 56 consequence of the transitivity of D, that A a C holds. Second, by virtue of Proposition 2.1, a R b holds and, from the transitivity of R, a R c holds as well. One thus has A a C and a R c which, from Definition 2.2, implies (A,a) (C,c). Case (iii) (A a B n a R b) and B D C. This case is analogous to (ii) and the implication (A,a) (C,c) by a similar reasoning. case (iv): (A aBnaR b) and (B cCAbRc). Consider first the case where none of the sets A, B and C is equal to the universe. In that case, by assumption, A2.5 holds. Since a R b and b R c, it follows from the transitivity of R that a R c holds. Since R is complete, one has, from the contrapositive of A2.5, that A a C. From Definition 2.2, a R c and A a C imply (A,a) (C,c). One must now consider the possibility of one (or more) of the three sets (A, B or C) being the universal set X. Consider first the expression (A aBnaR b) and assume that A = X. But then A D Z for any Z c X and, in particular, A D C. The implication (A,a) (C,c) follows then immediately from Definition 2.2. Suppose now that B = X. But since A a B, it follows that A = X which brings us to the preceding situation. Finally, suppose that X = C. Since B a C, it must also be the case that B = X and, therefore, that A = X. Thus, the implication (A,a) (C,c) can also be derived for this case. To establish the necessity part, assume that A2.5 is violated on S/T. That is, there exist two choice situations (A,a) and (C,c) in S/T such that C D A and a R c. From Proposition 2.1, c R a so that a I c. Since (C,c) E^C # X so that, from A2.2, there exists (B,b) E S such that B E C, B EA and b I c. Since b R c and B E C, it follows from Definition 2.2 that (B,b) (C,c) holds. Also, since a I b and A E B holds from A2.2 it follows from Definition 2.2 (lemma 2.2) that (A,a) (B,b) holds. If L- were transitive, one would thus have (A,a) (C,c). But because C D A, it follows from Lemma 2.1 that (C,c) ›- (A,a), a contradiction of transitivity. 57 ^ Remark: Although A2.2 is compatible with very sparse domains of choice situations, it is not necessary for Theorem 2.1 to hold. Moreover, A2.2 is not needed for the sufficiency part of the proof. In proving the necessity part of Theorem 2.1, the intransitive chain was obtained from the symmetric factor of the relation L-. This suggests a possible avenue for escaping from the nihilism of the result of theorem 2.1 with respect to Sen's (1991) suggestion. It is based on the view that the full transitivity of a binary relation is, perhaps, an overly strong consistency requirement. Two arguments support this view. First, from the vantage point of an external observer who can only observe the choices generated by the maximisation of a preference ordering (and not the ordering itself), it is very difficult to obtain much information about the symmetric factor of the ordering. Indeed, for the choices made by an individual to reveal information about her "indifference curves", it is necessary for the image of her choice correspondence (see Sen (1970; ch. 1*) not to be single valued (i.e. not to be a choice function). That is to say, the only way by which one can infer, from an observation of individual choices, that a is indifferent to b is by seeing her choosing both a and b from an opportunity set in which both were available. But in practice, it is only in very rare circumstances that one can observe such multiple choices - when confronted with a given opportunity set, individuals make a unique choice (perhaps by breaking ties due to indifference by some randomizing device) so that their indifference is not observable. Second, from a policy evaluation point of view, it can be argued that the symmetric factor of the relation that underlies the social planning criterion of choice among alternative policies is not of a considerable interest. Usually, a policy is recommended (and 58 subsequently adopted) only if it consists of a strict improvement in terms of the relevant criterion of evaluation of the various social states. If a policy is proposed that is judged as being just equivalent to the current state of affairs, the status quo is likely to be recommended. Why recommend change for change's sake? If one follows one of these lines of argument, it is tempting to ask whether the result of Theorem 2.1 holds when the requirement of full transitivity of the binary relation L is replaced by the weaker requirement of quasi-transitivity, that is, transitivity of the asymmetric factor ›. alone. It turns out that the answer to this question is negative, as the following theorem illustrates. Theorem 2.2: Let S satisfy A2.1. Then L of definition 2.2 is quasi-transitive. Proof Consider any (A,a), (B,b) and (C,c) in S such that (A,a) >- (B,b) and (B,c) >- (C,c). By Definition 2.2 and Lemma 2.1, we have: [A D B v (A ct BAaP b)] A [B D C v (B st C A b P c)]. By the transitivity of c (and of c), this implies: AD C v (A zBABDCAaP b) v (A D BABstCAbPc)v(AstBABstCAaPc). Let us first consider the logical statement (A stBABDCAaPb). Since c is transitive, it is clear that AzBABDC implies A st C. By virtue of Assumption A2.1, B D C implies b R c. As an immediate consequence of the transitivity of R (see Sen (1970; lemma 1*a), we have a P c. That is to say, (A sz B and B D C and a P b) implies A z C and a P c Similar reasoning can also establish that the logical statement (ADBABstCAbPc) implies A st C and a P c. Let us now consider the statement (A sz/3 AB stCAaPc). Clearly, a P c --> -'(c R a) so that, from A2.1, A z C holds. We thus have ADCv (A zCAaPc) and, by Definition 2.2 (Lemma 2.1) (A,a) ›- (C,c). ^ 59 2.4 Conclusion The central message of this chapter can be summarized in one sentence. Unless one is willing to live with intransitivities, there is very little trade-off between a ranking of social states based entirely on the freedom of choice they offer and completely ignorant about the individual preferences over these states, and a ranking in which freedom of choice is nothing but an instrument in the pursuit of a (more fundamental) objective of preferences maximisation. The relatively disappointing tone of this conclusion is, however, tempered by the result of theorem 2.2 which shows that such a ranking can be obtained if one is ready to sacrifice transitivity in favour of the weaker quasi-transitivity. This conclusion should also be qualified by the following two observations: (i) In this chapter, freedom of choice has been identified with the set theoretic notion of inclusion. However, the latter concept can by no means exhausts the meaning of a notion as rich as that of freedom. (ii) The lesson has been drawn from the examination of a somewhat special method for ranking social alternatives. Namely, the binary relation L. There are perhaps other methods for ordering consistently the set S in a way that respects both individual preferences over X and the freedom of choice contained in the various opportunity sets that do not imply the satisfaction of Axiom A2.5. 60 Chapter 3 Exploitation and Bargaining Theory: a Suggested Interpretation 3.1- Preliminaries There is some evidence that many people, all over the world, are concerned about exploitation. The relatively high degree of consensus that exists, in western societies, around the necessity of maintaining a legislation on minimum wage and on the length of the period of work provides an immediate evidence of such a concern. 33 So does, to some degree, the historical success of Marxism, based on a virulent denunciation of capitalist exploitation, to mobilise the revolutionary ardour of millions of men and women and to give an ideological caution to a wide variety of totalitarian regimes. Yet, this concern is not reflected in the analytical framework developed by modern welfare economists to evaluate the desirability of various policies. Given the place occupied, in this framework, by ethical issues as various as equality of choice 36 34 , equity 35 , freedom as well as the eternal Pareto efficiency, such an absence is rather surprising. A possible explanation for this state of affairs is that no convincing and rigorous definition of exploitation, as a relevant ethical issue in itself distinct from the aforementioned ones, has 33 In effect, this legislation is usually justified as being a safeguard against possible exploitation of workers by firms. Beside, this legislation appears very difficult to justify on the basis of the two ethical pillars of our western societies that are the respect of individual freedom and some minimal aversion toward inequality. 34 See for example Atkinson (1970), Blackorby and Donaldson (1978), Kolm (1976; 1977), Le Breton and Trannoy (1987) and Sen (1973; 1992). 35 In the sense of Foley (1967), Kolm (1972) or Varian (1974). 36 Bossert, Pattanaik and Xu (1992), Pattanaik and Xu (1990), Sen (1988); 1990; 1991) or Suppes 61 ever been produced. And there is undoubtedly some truth in this statement. The famous definition of exploitation as an unequal exchange of labour set forth by Marx (1867) and his followers has patently failed in its intended objective of characterising a certain class of ethically indefensible situations, as Roemer (1982a; 1982b; 1986; 1988a) convincingly argued. In view of this failure, Roemer (1982a; 1986) has proposed to eliminate exploitation from the list of preoccupation of Marxists (and other social scientists) and to concentrate instead on what is, for him, the "real" ethical issue at stake: the original inequality in the ownership of productive assets. As he (1986) puts it himself: "The central ethical question, which exploitation theory is imperfectly equipped to answer is: what distribution of assets is morally all right? Exploitation, I have claimed, is a useful measure when, and in so far as, it correlates with the ownership of alienable productive assets." (p.281-282) If Roemer is right in his contention that, behind people's aversion to what is called 'exploitation', one finds only an abhorrence of the original inequality in assets ownership, then, clearly, exploitation is not a distinct issue. Accordingly, it does not deserve any particular attention from the part of the economist involved in the making of policy recommendations. The present paper is the result of the somewhat deep conviction that this is not the case. More precisely, it is argued in this paper that: 1) There exists, in reality, an ethical issue distinct from the usual inequality in ownership and the other aforementioned ones behind the widespread aversion to what is called "exploitation", 62 2) such an ethical issue is a natural object of concern for an economist involved in the making of policy evaluations since it is related to inequality in bargaining power, and 3) such an issue is nonetheless difficult to discern in reality and to define formally because it depends upon an adequate solution concept for bargaining games and an acceptable definition of a poverty threshold, two missing objects in the conceptual apparatus developed by economists The starting point of my investigation is to realise, after Elster (1982), that exploitation is a general notion (more so then the standard Marxian definition of it) and, as any dictionary will suggest, is related to the idea of "taking an unjust advantage of someone". Thus, behind one's aversion to exploitation, we find more a dislike of the 'advantage' obtained from someone than an abhorrence of the underlying 'injustice' which has given rise to this 'advantage'. The difference between the 'advantage' and the 'injustice' is that the former necessarily implies the latter but not vice-versa. This last point can perhaps be made clearer by considering the following example. Suppose Stephane is walking in the desert and is very thirsty as a consequence of having no water for 2 days. He then meets a nice guy, named Keir, who has in his possession a couple of canteens of fresh water. He charges Stephane $2000 per canteen at which price Stephane expresses a "desire" to purchase 1 unit. There are of course various grounds on which one could challenge, like Roemer, the morality of the initial situation (prior to the deal proposed to Stephane by Keir) which exhibit an arbitrary inequality between opulent Keir and thirsty Stephane. I believe however that what most people are likely to view as exploitative in the situation portrayed by this example is less this original inequality in the possession of water than the advantage that Keir extracts out of it. As I believe that the historical aversion, which has made Marxism so famous, that many people have developed against pure capitalism is less related to the original 63 inequality in ownership in productive assets per se than to the benefit, which Marx thought could be measured in labour terms, that the capitalists obtain out of this inequality The object of this paper is to pursue further Elster's (1982) idea in proposing, somewhat rigorously, a new definition of exploitation entirely based on this idea of taking an unfair advantage of someone. To obtain such a definition is simply to make precise, from the view point of modern economic theory, the statement: 'an agent obtains an unjust advantage from an other agent'. This requires one to make precise: 1) The type of injustice considered and 2) The "numeraire" in which the advantage is to be measured Point (2) is easy to clarify since whatever profit, utility or, more generally, payoff function of the presumed exploiter can be used to measure the 'advantage'. As far as point (1) goes, this paper adopts the view point that "unjust" means "arising from someone's deprivation in terms of original endowments"?' The deprivation shall be defined in terms of some pre-specified level of certain assets. This level shall be referred to as the poverty threshold. Now, the only thing that remains to be made precise is the meaning given to the expression "arising from". For this purpose, I follow David Hume in resorting to a counterfactual experiment in which the deprivation of the agent from which some advantage is assumed to be obtained is eliminated with everything else remaining the same. 38 More 37 It is not the object of this paper to provide an elaborate philosophical justification for this particular definition of injustice. 38 According to Hume, a phenomenon A is said to cause a phenomenon B if the occurrence of B would not have been possible without the occurrence of A or, as he puts it "We may define a cause to be an object followed by another (...) where, if the first object had not been, the second never had existed" (An inquiry concerning Human Understanding, section VII). In the case above, the advantage is said to be caused by the deprivation (i.e. to be unjust) if the advantage would not have been obtained without deprivation. For an cndorcemcnt of Hume'c definition of causality see Lewis (1973). 64 precisely, to assess whether or not an agent i exploits an agent j, the definition considers a counterfactual situation in which j is just given the amount of relevant assets prescribed by the poverty threshold and examines the direction of the resulting change in the value of i's objective function. If i turns out to be worse off from such an elimination of fs deprivation, then there is really a sense in which it can be said that i was obtaining an advantage out of j's deprivation. As it seems reasonably safe to say that, in the example above, Keir is exploiting Stephane because he would suffer in the counterfactual situation where Stêphane would be given some decent quantity of water at the beginning. Yet it is not entirely clear why exploitation, as I just defined it, is ethically bothering. Quite clearly, the ethical distaste that one can have for the phenomenon captured by my definition is to be strongly influenced by the exact specification of the poverty threshold adopted as well as the determination of the items included in it. For example the advantage obtained by Keir would seem, I believe, fairly unjust to many. However, another free trade situation involving, say, Diana and Charles, the latter starting endowed with one box of biscuits and one tea bag, the former disposing of no cookies at all and two tea bags, trying to exchange their respective endowments in order to satisfy better their convex preferences for the two items may not bother ethically a lot of people, even if it can be characterised as exploitative for some specification of the poverty threshold. Thus, the very reason which underlies one's ethical feeling with respect to these examples is not entirely transparent from the definition of exploitation proposed. To provide some help in rationalising the common dislike for exploitative situations, dislike which, as stated, depends to some extent upon the definition of the poverty threshold, the present chapter appeals to bargaining theory. Representing any bilateral For a critics sec Bcrofsky (1973) or Elstcr (1982). 65 exchange relation like the one between Stephane and Keir in the example above as a bargaining game, it seems intuitively clear that one's dislike for exploitation is intimately related to the somewhat excessive bargaining power it entails on the part of the exploiter. This intuitive idea is to be examined, in this chapter, more closely. In particular, the class of bargaining situations originally studied by Nash (1950) and reexamined recently, in closer connection with economics, by Roemer (1988b; 1990), Binmore, Rubinstein and Wolinsky (1986) and Rubinstein, Safra and Thompson (1992) among others, is used as an idealisation of the various types of relationships that could prevail between exploiters and exploited. It is suggested that, in order for exploitation, as defined in this chapter, to be an adequate measure of the bargaining power of the exploiter in the bargaining representation of her relation with the exploited, very specific conditions need to be imposed on the poverty threshold. The analysis is purely exploratory and depends highly upon one's acceptance of Nash's bargaining framework (and his solution concept) as well as on the definition of a good measure of bargaining power. It is my hope that, despite its incompleteness, the analysis proposed in this chapter helps in shedding some light on the deep connection existing between exploitation and bargaining power. The Chapter is divided as follow. The first part sets the notation, states formally the definition of exploitation presented above and discusses some of its implication for welfare economics. The second section explore the connection with bargaining theory. 66 3.2: A new definition of exploitation. 3.2 i) Formal statement and intuition of the defintion. In the background, we have a set N c N + with card N = n < 00 of individuals and a set M c N + with card M = m < 00 of commodities.' Each individual i E N is identified by a continuous and monotonic preference ordering L i40 defined over some set Xi c^. In addition to her preferences for fixed consumption bundles in X i , each individual i is assumed to have preferences for finite lotteries over X i that satisfy the Von Neuman-Morgenstern axioms of expected utility. Formally: Assumption A3.1: V i E N, 3 a,: Rm --> R+ such that V x, y E^p E [0,1], u i (x) ui (y) x L i y and us (px(131(1-p)y) = pu i (x) + (1-p)u i (y). 41 Clearly, if u i ( ) has the Von Neuman-Morgenstern properties stated in A3.1, so does any positive affine transform of it. ss Commodities are here to be conceived at a more general level then they are usually in economics and may include items like health status, individual rights, criteria against which some discrimination is exerted and so on. 40 A preference ordering L i on Xi is a complete, reflexive and transitive binary relation on Xi . The asymmetric and symmetric factors of L i are denoted by >- i and respectively. The ordering is continuous if, for every x a Xi , the sets B(x) = {± E X i I t L i x} and W(x) = {x' E Xi I x L i x'} are both closed in Xi . Using > and » for the ranking of vectors, the ordering is monotonic if x x' implies x L i x' and x > x' implies x ›- i x'. 41 (px (p( p)y) denotes the lottery in which x is assigned with probability p and y, with probability 1-p. Strictly speaking, assumption A3.1 is not needed in what follows. What is needed is the assumption that each individual preference ordering has a meaningful cardinal utility representation and, for section 2, satisfies some minimal rationality properties (like the general ones proposed by Rubinstein, Safra and Thompson (1992)) under uncertainty. The expected utility hypothesis satisfies these two requirements without too much arm. 67 Given any set X let 6(X) denote the set of all monotonic continuous preference orderings on X satisfying A3.1. Furthermore, each individual i vector v), E E N is endowed with some Rm of various assets. A list (o)„...,w„) of individual endowments is denoted by <a)>. An analogous signification is given to <L i >. Given a list <6.),>, the situation where some individual h E N gets the endowment co and every other agent j (j EN\ (h)) is endowed with the jth component of the vector <o) i > is denoted by <v),co. h >. Also, since each co, is itself a mdimensional vector, I shall denote by 0, (when necessary) the quantity of productive asset k initially owned by individual i. Now let Tc: x (6 ) (XdxRm ) —> Drun be a process (perhaps a market with or without iE N production activity) which assigns, to any list of preferences-endowments pairs (one for each individual) a unique allocation of the m commodities to the n individuals'. Consider some subset K c M with Card K = k of relevant assets and let ti E individual i is considered as deprived if col,` te for all k for at least one j E E R be the poverty threshold. An K with the inequality being strict K. One is now equipped to define exploitation. Definition 3.1: An individual i is said to exploit an individual j in the economy defined by (<co i >,<L i >,70.)) if j is deprived and if u i (70<co,>)) > u i (7c(<td,w_i >)). Note that definition 3.1 says absolutely nothing about the feasibility of giving (somehow from an outside source) individual j the bundle *CS which eliminates her deprivation. But this should by no means constitute a problem. Such a theoretical experiment is concerned solely with assessing whether or not an exploitative relationship exists between two 42 This uniqueness is colely a matter of convenience. The process could well be a correspondence 68 individuals, not with the object of actual implementation. One can also see the importance of having a cardinal utility representation for the preferences of individual i. Without such a representation, definition 3.1 would have only been suitable for giving a crude "yes or no"-type of answer to the question of whether one individual is exploiting an other. That is, it would have been absolutely silent about the degree or the intensity of this exploitation. And as an ethical issue, exploitation appears clearly to be a matter of degree. What can be seen as morally shocking in the situation portrayed in the Keir-Stephan example is the 2000 dollars extracted by Keir. But what would have been one's reaction if 2000 had been replaced by 5? or by 1? 43 In what follows the degree at which individual i exploits individual j in the economy (<o),>,<L i >,7C(.)) is denoted by 8 ij ko),>,<?._- i >,7c(.)) and is defined as follows: Sii (<wi >,<L i >, n(.)) 0 if cok > M k for some k E K = MAX (0, u t (70<u),>)) - u,(70<ta,o)_,>)) otherwise I shall refer to N(.) as the exploitation function of j by i. As defined above, exploitation is the result of the conjunction of two "variables": A particular level of deprivation and a specific allocation process. Given an allocative process TC(.), exploitation becomes entirely determined by deprivation (although not vice-versa). One may therefore wonder whether a broad enough objective of "poverty alleviation" will not at once eliminate exploitation. In other words, given the process, is not exploitation a secondary ethical issue with respect to the more common one of poverty alleviation? Note that for a sufficiently low price charged by Keir for his water, it is not at all clear whether exploitation as in definition 3.1 will occur. Stephane might well decide to buy some canteens from Keir even in the counterfactual were he is given some initial provision of water if the price originally charged by Keir was reasonable. 69 The answer to this question is undoubtedly yes in theory. Given the specification of a poverty threshold til, a complete elimination of deprivation will obviously imply the elimination of exploitation. In practice however, it is doubtful that one can ever achieve, one day, such an ambitious program. The objective of a complete elimination of deprivation appears particularly utopian when, as should be done, one includes in the items entering into the composition of las inalienable personal assets such as the health status, innate abilities, etc. In waiting for the complete and final alleviation of whatever concept of deprivation is used, it may be a worthwhile objective to try to limit the extent to which some individuals can benefit from the deprivation of the others. From a policy oriented point of view, both the allocative process and the initial level of deprivation can be used to reduce exploitation while, obviously, only the latter of the two instruments will be considered in the reduction of deprivation only. In that sense, the objective of reducing exploitation and that of reducing poverty may have very different implications for policy recommendation. It is also worth mentioning that definition 3.1 does not betray the spirit of Marx's (1867) thought when it is applied to a pure capitalist system, that is, for him, a system where... "...the owner of money must meet in the market with the free labourer, free in the double sense that, as a free man, he can dispose of his labour power as his own commodity, and that on the other hands he has no other commodity for sale, is short of everything necessary for the realisation of his labour power." (p.187-188; My emphasis) To see this, as well as to get a foretaste of how definition 3.1 can be applied in a somewhat common economic setting, the following simple Cobb-Douglas general equilibrium example is proposed. 70 3.2 ii) A general equilibrium example: A particular economy consists of N capitalists and M 4N+1 workers. A unique constant return to scale technology is used to produce a single consumption good c (to serve as numeraire) from two factors: labour L and capital K. The frontier of the production set described by the technology is given by the following production function: c K1/2L 1/2 All agents have identical preferences over consumption and leisure represented by the following Cobb-Douglas utility function: Ui(ci,/,) = ci 1/2 ,1 /2 for i=1,...,N+M. Each agent is initially endowed with one unit of time (to be allocated between labour effort L and leisure 1). In addition, each capitalist is endowed with one unit of capital (which is not an argument of her utility function). More over, the capitalists own the technology (a fact which does not make any difference since the only equilibrium prices at which profit maximisation is well defined are those which yields zero-profit). Acknowledging this fact allows one to write a typical capitalist i optimisation problem as: MAX ci ll2/i v2 subject to ci 5_ wL + rk i , L i +l i =1 and k i 5 1. {c o l i ,L i ) where w and r are the wage rate and the renting price of capital respectively and where k, (not to be formally confused with K) is the capital service supply by capitalist i. Capitalist i's labour supply L i *, obtained as a solution for this problem, is (assuming strictly positive w): Li * = w-r 2w ^ if w > r L i * = 0^otherwise. Analogously, a typical worker j's optimisation problem is: 71 1121 1/2 MAX^J subjectJ to cJ < W/J and LJ +1=1 {Cp ipLi } While her labour supply Li* is: Li * = 1/2 Profit maximisation together with the zero-profit condition implies that the following relationship must hold between the equilibrium wage w* and the equilibrium rental price of capital r* r* = 1/4w* Now since capital is not by itself subjectively valued by its owner, it follows that each capitalist is interested in supplying all her capital stock at any positive price r. Together with profit maximisation, this implies that labour demand L d* is: Ld* = N/4w2 To find the prices (w* ,r*) which clear the market, it suffices to solve the following equation M + N - rN = N 2 2w 4w 2 if w > r. Otherwise the market clearing equation is simply: M/2 = N/4w 2 . It is not difficult to see that w* =(N/2M) 1/2 = (N/(8N+2)) 1/2 and, correspondingly, r* = 1/2 ((4N-F1)/22012 is the unique price equilibrium for this economy. At such a price equilibrium, no capitalist works and each enjoys 1/2 ((4N+1)/2N) 1/2 units of the consumption good. Each 72 worker spends half of her time working and consumes 1/2 (N/(2(4N+1)))' units of c. Since 1/2 ((4N+1)/2N) 1/2 > 1/2 (N/(2(4N+1))) 112 it is obvious that the value of capitalist consumption is larger than the value of the worker's consumption for any numeraire used and, therefore, that Mandan exploitation (positive difference, in labour value, between capitalists and workers consumption) is taking place. Is such an economy also exhibiting exploitation as per definition 1? To answer this question, one needs to compute the general equilibrium prices and quantities for a slightly different economy in which everything is just as before except that one worker, say the first one, now owns some amount 0 < k 1 units of capital prescribed by the poverty threshold. Just like a typical capitalist, worker 1 is interested in supplying inelastically all her capital stock k at any positive price r. Acknowledging this fact allows one to write worker l's problem as: 1 subject to c 1 W/ 1 + kr, L 1+1 1 =1 MAX^cl y2,1/2 t {c l 1 , , L , The labour supply of worker 1 becomes: L 1 * = w-kr 2w ^ if w > kr L 1 * = 0^otherwise. Profit maximising labour demand Ld* becomes: Ld * = N + k 4w 2 The zero-profit condition remaining, r*=1/4w* must still hold in equilibrium. The labour market clearing condition now takes the following form: M+N - Nr - kr = N + k^if w* > r*, 2^2w^4w2 73 (M-1)w - kr = N + k^if r* > w* > kr* and 2w^4w2 M-1 = N + k ^ 2^4w2 otherwise. The assumption that M 4N+1, together with the fact that k 1, eliminates the first of the three market clearing conditions. If the specification of the poverty threshold requires endowing worker 1 with k < N/(2N-1), then the market clearing wage is 1 (2N+30/2 and the 2 (4N+1) 1/2 corresponding equilibrium rental price of capital is 1 (4N+1) u2 . In this equilibrium, no 2(2N+3k) 112 capitalist works and each enjoys 1 (4N+1) 1/2 units of the consumption good. Exploitation as 2 (2N+3k) 1/2 per definition 3.1 does take place since the exploitation of worker 1 by a typical capitalist i in this case is: ^ 811 (.) = 1 (4N+1)"4 - 1 (4N+1) 1/4 4 (2N) 14^4 (2N+3k) 114 >0. If, on the other hand, k N/(2N-1), the new equilibrium wage is 1/2 [(N+k)/(2N)1 1/2 and, correspondingly, the new rental price of capital is [N/(2(N+k)P 2 . In this equilibrium, no capitalist works (unless k > (2N+1)/3). However, 1/2 ((4N+1)/2N) v2 (the initial capitalist consumption) is strictly greater than the consumption enjoyed in the new equilibrium ([N/(2(N+k)] 1/2 ) so that, here again, the capitalist exploits worker 1. Moreover, the degree by which a capitalist i exploits worker 1 is: 811 (.) = ((4N+1)18N) 114 - N 1(2(N +k)1 114 > 0. This example illustrates some of the difficulties that could arise from the use of definition 3.1 to measure the degree of exploitation prevailing in general equilibrium models. In effect, the reckoning of the value of the exploitation function for each pair of 74 individuals involves the comparison of two different general equilibria: the observed one and the counterfactual one where the deprivation of the exploited is eliminated. As is known from Arrow and Hahn (1971;ch.10), no general conclusions can be derived from these comparisons so that each particular model, such as the highly simplistic one above, needs to be worked out completely. Notice also that such a reckoning may have to become quite involved since the comparison may have to be done for as many as n!/2(n-2)! pairs of individuals. Moreover, as this example illustrates, the exploitation function 5, (.) is not a continuous function of the j poverty threshold. Thus, the evaluation of the degree of exploitation prevailing in an economy approximated by a general equilibrium model is likely to be a rather tedious exercise. Beside, it is not clear that a general equilibrium setting where no interaction between agents (except through an anonymous price system) is taken into consideration, is fully appropriate to study such a phenomenon as exploitation which, by its very nature, involves some sort of an interaction between the exploiter and the exploited. This provides another argument in favour of the bargaining approach proposed in section 3.2. 3.2 iii) Further discussion Another important issue related to the measurement of exploitation is that of aggregation. Exploitation, as it is defined in this chapter, is a measure of some advantage obtained by i from her relation with j through the process 7c(.). But it does not say anything about how these measures are to be aggregated over all possible relations of exploitation that can simultaneously take place in an economy. The problem is not worse here than in standard welfare economics and social choice since it involves the definition of a method for making interpersonal utility comparisons (comparing the various function i E NO between the various N) as well as a method for aggregating these functions (sum, product, etc.). But it is nonetheless a real one and shall not be addressed further in this paper. 75 Moreover, the consideration of exploitation, as captured by definition 1.1, as a possible criterion on which to base certain collective decisions has specific and delicate implications for the shape of the social welfare function when, as it is likely, other criteria are to be used. Although a full account of the issue is beyond the scope of this chapter, a few illustrations of the difficulties are worth mentioning. A problem is raised when the social welfare function is assumed to be sensitive to both the Pareto criterion and whatever aggregate measure of exploitation is used. In the Keir-Stephane example, the exploitative deal offered by Keir to Stêphane constitutes a Paretoimprovement over the situation that was prevailing prior to their meeting. How should one rank two situations such as these in which a Pareto improvement is accompanied by an increase in the degree of exploitation? More precisely, what should be the trade-off between a Pareto improvement and a increase in exploitation? Another example of the possible difficulties is given by the historical phenomenon of slavery, for which the existence of exploitation has never been the object of much controversy. Assessing the existence of exploitation in such a context requires the inclusion, as a component of the threshold ti of some indicator of self ownership (say a dummy variable of value 1 if the individual is free and 0 if she is a slave)." This being done, it does not necessarily follow from definition 3.1 that a master exploits her slaves. For example, if the master treats her slaves well enough, providing them with refined food, sophisticated shelter, etc. it is not clear that the master would end up worse off in the counterfactual situation where the slaves would be given self ownership. And, looking in the American history, it has " Strictly speaking, definition 3.1 does not apply to an economy with slavery since the counterfactual elimination of the property right of the master over the slave does not only affect the "endowment" of the slave (by giving her freedom) but also reduces the endowment of the master. I am indebted to Michele Piccione for pointing me out this state of affairs. 76 been argued that when slavery was abolished in 1861, many southern cotton producers became in fact better off since it was now cheaper to hire at low wage the newly emancipated blacks as workers than to feed and dwell them as slaves." How should the two situations be ranked? Slavery without exploitation or "free" labour market with exploitation? The two examples illustrate some of the difficulties that the construction of an exploitation averse social welfare function would encounter when other independent criteria, such as the Pareto criterion or the respect of certain basic rights, are deemed relevant to evaluate social welfare. I shall not address these questions further in this chapter. 3.3: Exploitation and Bargaining Theory 3.3 i) Exploitative relations as bargaining sessions Intuitively, what appears ethically disturbing in the fact that someone obtains an advantage out of the state of deprivation of someone else is the somewhat extreme bargaining power that this advantage seems to give to the exploiter. Our aversion toward exploitation seems therefore closely related to our dislike of an excessive inequality in bargaining power. For instance, the argument, sometimes heard, that the relation between a capitalist and a worker is not per se fundamentally different than that prevailing between a master and her slave" rests on some idea of the two agents (exploiters and exploited) facing some sort of bargaining problem a la Nash (1950). In both cases, the exploiter can obtain a 45 Kolm (1985; p. 302-303) attributes to Adam Smith this conclusion that cotton growers of southern United States would benefit from slave emancipation. 46 See, for in stance„Cohen_(.1983)^ 77 very high level of utility on the frontier of the bargaining set by threatening the exploited with some "punishment" if she fails to cooperate. In the capitalist case, the "punishment" comes from the deprivation of the worker which, if a solution to the bargaining problem is not achieved (i.e. if she does not agree with the condition imposed by the capitalist), will drive her to the wall of starvation or of severe destitution.' In the other case, the punishment feared by the slave is more direct (torture, whip, etc.). But in both cases, a bargaining agreement very favourable to the exploiter will be achieved by whatever threat of "punishment" is made credible by the institutional setting in which the bargaining game takes place (slavery, free market with private ownership, etc.). The object of this section is to provide a partial foundation to these well known ideas. Once again, the following discussion is exploratory and far from being definite. For this task, it is important to reinterpret the formal setting of the previous section somewhat differently. In particular, one shall now think of the process 10.) as being the result of a certain number of bilateral bargaining sessions taking place between the agents. In any of these possible bargaining sessions, two agents, say i and j with their preferences satisfying A3.1, meet carrying with them their stock o and co, of assets. To each of these meetings, nature associates a set Xi; of possible physical agreements. The exact process by which these sets are actually generated and assigned to each meeting will be ignored but it may enhance one's intuition to think of them as being either the Edgeworth box of a two person-exchange economy determined by all possible allocations of the sum of their endowments or the split, between i and j, of all the additional output brought about by the work effort of j in the firm partially or completely owned by i. If the two agents agree on some 47 Of course, this will not be the case if full employment prevails and workers can move from one job to another costlessly. It is on the assessment of whether or not these conditions hold in reality that marxists and neo-classical economists disagree about the actuality of the exploitation of workers under capitalism. For more on this issue, see Cohen (1988). 78 element x E Xti , they sign a contract and obtain this agreement. Otherwise, they leave the bargaining session and are stuck with the best option o i (co) and oi (o.),) allowed by their endowments co, and (Or " The quintuplet (L i ,Lr oj (co,),oi (co,),X,i ) constitutes a standard bargaining problem described in physical terms. From now on, the subscripts i and j are replaced by 1 and 2. Formally, our interest lies in the family E of bargaining sessions defined as follow. Definition 3.2: E {(} 1 L.- 2 ,o i (co,),o,(co2 ),X„) I (i=1,2) , E P(X 12 ), E Rm+, X„ c Rm+x Rm+) The following assumption, standard in bargaining theory, is imposed on E. Assumption A3.2: V (L i , L2,0 1 ((.0 1 ),0 2(0)2 ),X 12 ) (i) X1 2 (ii) 3 x E E, is compact in Rm+x Rm+ E X 12 , such that x i ›- i o,(o),) for i=1,2 (iii) V x,y c {(o 1 (co 1 ),o 2 (o32 ))) v X12 , V p E 49 [0,1], 3 z E X12 such that prEE4(1 -p)y^z (i=1,2). 5° 48 ^again, I shall not have anything to say about what this option is Think of it as being the consumption of some component of the vectors co, and co, or the future gains that an agent can expect to make in subsequent bargaining sessions with these endowments. As mentioned in Binmore, Rubinstein and Wolinsky (1986;p.185), the exact specification of these outside options is of a paramount importance for the determination of the solution of the game. 49 As so a matter of notation, x denotes the bundle obtained by individual i in the allocation x E i X12. Under this assumption, the utility possibility sets of the bargaining problem are convex and cnniprebensivp (SPP Rossprt (1992) nr Thompson (1987)) 79 What is needed now is a solution concept for the class E of bargaining sessions as well as a satisfactory definition of the bargaining power of an agent in any bargaining session (L- 1 , L- 2 ,o i (w i ),o 2 ((o2 ),X„) she may be involved in. I shall start by the latter since it can be defined without the former. 3.3 ii) A Definition of Bargaining Power Intuitively, the bargaining power of agent 1 in the bargaining session a = (L 1 ,L 2 ,o 1 (co l ),o 2 (co 2 ),X 12 ) is related to her ability to end up close to the largest gain she can expect to make after agreeing with 2 upon some x in X12 . Formally, for i E {1,2}, i*j, let /i = Arg Max u i (x) subject to ui (x) ui (o,(6)) be individual i's ideal point in the bargaining session xelf.12 a. That is, given the bargaining opportunities allowed by the set X„, I, represents the best outcome she can expect, given that individual j must have at least an incentive to participate to the bargaining session. Quite intuitively, the closest i will end up being, at the solution, to her ideal point, the largest will have been her bargaining power in the session. This suggests the following approach to defining bargaining power. Given any solution s(o) to the bargaining situation a, define, for i E {1,2}, the function 71 : [-1,1] by (u(s0) u(o((.4) -(4.1)-4s 0)) - y,(0 )- 41)-44 (4 1^1 1 (1) To understand what y,(.) does, consider the case where individual i gets the utility of her ideal point at the solution of the bargaining session and where, therefore, her bargaining power is maximal. Then y,(a) = 1. On the other hand, if i merely gets her reservation level of utility at the end of the session, her bargaining power is clearly absent and yi(a) = -1. Between these two bounds, y,(.) increases continuously with respect to the utility achieved by i at the solution s(a) 80 of the bargaining session a. As it can be seen easily, the function y,(.) is invariant to any positive affine transform of u t (.) and does not depend upon the particular preference scale used to measure individual i's cardinal preferences. Two other properties of y,(a), as an index of i's bargaining power, are also worth noticing. First, y,(.) is a measure of i's bargaining power that is valid for a given preference ordering y,C) is patently inadequate to compare the bargaining power exerted by i in different bargaining sessions if i's preferences are allowed to change from one session to another. Second, it is interesting to note that y,(.) is an ex post or "empirical" measure of i's bargaining power since it does not depend in any way upon the definition of a particular solution concept.' In what follows, the bargaining power of agent i in bargaining session a, to be denoted (3,(a) is defined by the following convenient re-scaling of the function V.): 1310 - 1 +y (o) 2 (2) The interest of this function lies not only in the fact that it maps monotonically [-1,1] to [0,1] (a more natural scale) but also, that it provides another useful interpretation of the type of bargaining power captured by it. Indeed, from (1), it is easy to see that 13, E [0,1] is the number that satisfies: u i(s(a)) =^+ (1-13du i (o,(03,))^ (3) 51 In that sense, it differs from the commonly used notion of bargaining power specifically derived from the asymmetric version of the Nash (1950) solution concept (see below) developed by Roth (1979). 81 3.3 iii) A Reinterpretation of the Nash Solution Concept Given this notion of bargaining power, we can now proceed and define an appropriate solution concept for the bargaining sessions ( L i , _>.-. 2 ,o,(oh),o 2 (o.)2 ),X„) in E. By a solution concept, it is meant a function s: I —>^le+ that associates a unique agreement L 2 ,0 1 (0) 1 ),0 2 (0) 2 ),X 12 )) E^{(01(0)1),02(0)2))) v Xid to each bargaining problem ( L- 2 ,o,(o),),o 2 (co2 ),X, 2 ) in E. Before proposing a solution concept that seems suitable for the problem at hand, a few remarks on the relation between classical bargaining theory and the approach adopted here are in order. In classical bargaining theory, a typical bargaining problem like ( L2,°1(co1),02(0)2),X12) is presented in its condensed version (d,U) where U is the image, under the vector of utility functions (u,(.),u 2 (.)) of assumption A3.1, of X 12 and d, usually called the disagreement point, is the vector of utilities the two individuals get from their outside options (that is, d = u i (o i (coi )),u 2 (o2 (o)2 ))). Let denote the class of condensed bargaining problems corresponding to E. 52 A solution concept for the class of condensed bargaining problems is a function II: Eu —> R2 + that associates the unique vector of utilities ii(d,U) to each condensed bargaining problem (d,U). A famous solution concept for the class of condensed bargaining problems, proposed by Nash (1950), is the following. Definition 3.3: Ii(d,U) = MAX (u 1 -d i )(u 2-d2 ) subject to u i d1 for i=1,2 (u,,u 2 ) 52 x That is, E u = {(d,U) I 3 ( L i , k-. 2 ,o,(o),),o 2 (co2 ),X, 2 ) with u=(u,(x,),u 2 (x2 ))). E E s.t. E X 12 82 d, = u,(o i (w i )) (i=1,2) and (u E^—) (3 That is, the solution to any condensed bargaining situation (d,U) is obtained by maximising the product of the two individual Von Neuman-Morgenstern utility gains with respect to their disagreement position. The justification usually given to this particular solution concept is that, under A3.1 and A3.2, it is the only solution that satisfies the following well known axioms. Invariance to person-by-person positive affine transforms (IAT): If (d,U) and (d',U') E Eu are such that, for some a,^0 (i=1,2) and b i E R (i=1,2), ((u 1 ,u2 ) E U) ÷-> (a 1 u 1 +b 1 ,a 2 u2 +b 2 ) E U' and d' = (a i d i +b ,,,a 2d2 +b 2 ), then v i = pi(d,U) E--> i +b = pi(d' ,U'). Symmetry (SYM): if U is symmetric with respect to the main diagonal and if d 1 = d2 , then (v 1 ,v 2 ) = p(d,U) -3 v, = v 2 . Pareto Optimality (PO): V (d,U) E Yu, (u E U)^> p(d,U)). Independence of Irrelevant Alternatives (IIA): (U' c U n p(d,U) E U') -4 (p(d,U) = p(d,U')53 This practice of concentrating "real" bargaining problems like ( L- 2 ,o 1(o) 1 ),o 2 (w2 ),X 12 ) into their utility representations (d,U) has been the object of increasing criticisms from various parts (see for example Rubinstein (1982), Binmore, Rubinstein and Wolinsky (1986), Roemer (1988b; 1990) and Rubinstein, Safra and Thompson (1992)). The backbone of these criticisms is that the condensed bargaining problem (d,U) abstracts away 53 Other solutions concepts for the condensed problems, as well as their axiomatic justifications have been proposed by Kalai and Smorodinsky (1975) and Kalai (1977) among others. 83 many things of paramount importance for the prediction of the outcome that is likely to emerge from a typical bargaining process.' Among these things are the particular specification of the bargaining procedure (that is, the extensive form of the bargaining game) (Rubinstein (1982) and Binmore, Rubinstein and Wolinsky (1986)) and the nature of the physical objects on which the two agents bargain (Roemer (1988b; 1990)). In particular, the acceptability of the axioms invoked to justify a particular solution concept is crucially dependant upon the type of physical bargaining problem the modeller has in mind Another problem with classical bargaining theory, related to the previous one, is that the distinction between the normative and the positive aim of the theory is not always clearly made. Although both aims are important, they have to be clearly distinguished. A "just" solution to a bargaining problem that could be proposed, say, by some benevolent arbitrator is unlikely to be the same as the solution that two self-motivated individuals would arrive at in a particular institutional setting. In what follows, my objective is "positive" since I want to predict the likely outcome of a typical bargaining session of the type described above. Although I shall abstract from the exact specification of the extensive form of the bargaining game, it appears clearly An other way to say this in a social choice theoretic language is that classical bargaining theory is "welfarist" in the sense that the only information it uses to predict the outcome of a bargaining game is utility. Two real bargaining problems that generate the same utility possibility sets are considered identical. To give an example, suppose that two condensed problems (d,U) and (d',U') in E u are such that one is the result of increasing the other by agent-specific affine transforms but that the two "real" bargaining problems (L i , L 2 ,o 1 (o.)1 ,,) o2,(co X ) and ( L2',01(C)ir,02(CO2Y,X/2) that have generated (d,U) 2, ,) 12, and (d',U') are entirely different situations with different preference orderings and different bargaining sets. Then requiring the solutions for these two different bargaining problems to be related by the same affine transforms as above seems completely unnatural. For other examples in those lines concerning IIA as well, see Roemer (1988b; p.28-29) and Rubinstein, Safra and Thompson (1992; p.1173-1174) 84 desirable, given my definition of exploitation, to present the analysis in terms of the "real" bargaining problems instead of their condensed version. With this in mind, the solution concept I shall use to discuss the relation between exploitation and bargaining power is the following (Rubinstein, Safra and Thompson (1992)): Definition 3.4: V (L- 1 ,L- 2 ,o,(co,),o 2 (co2 ),X„) E E s(.) is such that, for i=1,2, i^for all x E X12 , {(o 1 (0) 1 ),0 2 (0)2 ))) and all p E [0,1], (px,ED(1-p)o i (coi ) >- i si •)) -4 (ps3(.)®(1-p)oj (coi )^x). 56 In words, s is the solution of a bargaining problem if, given any other possible agreement x demanded by either of the two agents and any credible treat, made by this agent, of leaving the bargaining session with probability (1-p) if her demand is rejected, it is in the best interest of the other agent to oppose to this demand, even when accounting for the risk (1-p) of a breakdown in the negotiation. Beside its intuitive appeal, this solution concept can be shown to be the approximation of the unique perfect equilibrium of the Rubinstein (1982) alternating offer model (see Binmore, Rubinstein and Wolinsky (1986) and Rubinstein, Safra and Thompson (1992). Moreover this solution concept has the following relation with the Nash solution concept for Fes,. 56 SP and x, refer to individual i's bundle in the solution s(.) and the aggreement x respectively (recall that both the aggreement x and the image of s() are in r i x1r). 85 Proposition 3.1: (Rubinstein, Srafa and Thompson (1992)): Under A3.1, s(.) as per definition 3.4 is the solution of the bargaining problem (L i , L- 2 ,o 1 (co i ),o 2 (o)2 ),X12 ) if and only if p(.) as per definition 3.3 is the solution for the condensed version (d,U) of ( L 2 ,43 1(c 1 ),o 2(co2 ),X 12). - Proof Let (u 1 *,u2 *) be the Nash solution for (d,U). Thus, u l *u 2 *. Max u 1 (x 1 )u 2 (x2 ) subject to seX12 ZIA) u,(o,(co,)) (i=1,2) after normalising the Von Neuman Morgenstern utilities ut (.) in such a way as to set u i (oi (o),) = 0 for i=1,2. Since u t (.) is unique up to an positive affine transform, such a normalisation is clearly legitimate. In view of this normalisation, and given the constraint that u i (x i ) u i (oi(coi )) for i=1,2 and assumption A2.3-ii), one needs only to consider utility pairs (u 1 ,u2 ) E U such that u i > 0. Now, for any such (u 1 ,u2 ) E U, ij u i *uj * /L i u, and hence, u i */u, upti *. Suppose, contrary to the claim, that s satisfy definition 3.3. Then, 3 x p)oi (o),) for some ij E E X 12 . p E E E {1,2} and i#j, u,*(.)-1 does not [0,1] such that (px0(1-p)o i (co,) >- i s and x i (psEB(1- {1,2}, i#j. That is, pu t (x) > u t *(s) and uf(x) > pu,*(s). But then, uf(x)/uNs) > p > u,*(s)lu i (x), a contradiction. An analogous reasoning starting with the solution s(.) for the "real" bargaining problem establishes the reverse implication of the proposition. ^ Moreover, under A3.1 and A3.2, the solution s(.) exists for all bargaining sessions contained in E (Rubinstein, Srafa and Thompson (1992; proposition 2*)). However s(.) need not be unique if Arg Max u 1 (x)u 2 (x) is not single valued. But clearly, the utility numbers xe X12 u,* and u 2 * defined by u i *u2 *. Max u 1 (x)u 2 (x) are unique. Of course, given proposition 3.1, the xe X12 solution s(.) satisfies all the Nashaxioms (IAT, SYM, PO and HA). Given this solution concept, we can explore the link between exploitation and bargaining power. 86 3.3-iv) Exploitation as a Measure of Bargaining Power Keeping the same notation as in section 1, one can reinterpret definition 3.1 in the light of the present setting as follows. Definition 3.5: Agent 1 exploits agent 2 in the bargaining session a =(?_-_,,L-. 2 ,0,(0),),0 2 (0),),x,2 ) E Z if u,(s i (a)) - u 1 (g 1 ) > 0 with g = s(.?_-. 1 ,L 2 ,o 1 (o 1 ),o 2 (ti),X 12 ). In this definition (which is nothing else than a reinterpretation of definition 3.1 in the present context), the counterfactual situation with respect to which the exploitative nature of the relationship between 1 and 2 in the bargaining session a is assessed consists in exactly the same session as a with the only difference that 2 original endowment co 2 is replaced by CU. What is the link between the exploitation of 2 by 1 and the bargaining power of 1 as captured by V.)? The following proposition provides a partial answer to this question. Proposition 3.2:Let individual 2 start the bargaining session a = (L- 1 , L- 2 ,o,(co i ),o 2 (o32 ),X 12 ) deprived. Then using the solution concept of definition 3.4, and under A3.1 and A3.2, 1 exploits 2 as per definition 3.4. Moreover, the more deprived is 2, the more she is exploited. Proof Let s(a) be the solution to the bargaining session a. By definition 3.4, for all agreements x E X12 and for all p E [0,1] such that px2 ED (1-p)o 2(o32 ) > 2 S 2 (0), one has ps,(a) ED - (1-p)o l (w 1 ) x 1 . The first step of the proof is to show that, given any such x, one can find some q E [0,1] such that qx 2 ® (1-q)o 2 (co2 ) L 2 s2 (u) and qs,(a) (1-q)o 1 (w 1 ) — 1 x 1 . For this purpose, - define m(x) = MIN {q E [0,1] I qx2 ® (1-q)o2((o2) Y2 s2 (a)}. Since L 2 is continuous and - monotonic, m(.) is continuous and is such that m(x)x 2 ® (1-m(x))0 2 (0)2 ) - 2 S2 (6). One needs to 87 establish that m(x)s 1 (a) ED (1-m(x))o 1 (o) 1 ) — 1 x 1 . By contradiction, suppose first that x 1 >- 1 m(x)s 1 (a) O (1-m(x))o 1 (co 1 ). Since L i is continuous, one can find a sufficiently small E such that x 1 >- 1 (m(x)+E)s,(45) (1-m(x)-E)o 1 (o) 1 ). By definition of m(x), (m(x)+E)x 2 (1-m(x)-E)o 2 (o)2 ) >- 2 S2 (6). But this contradicts s(a) being the solution as per definition 3.4. Suppose instead that m(x)s 1 (a) (1 - m(x))o 1 (co i ) >- 1 x i . Define then q by qs,(a) (1-q)o 1 (co 1 ) — 1 x,. From monotonicity of L i , q < m(x). From the monotonicity of L 2 and the definition of m(.), one has s2 (u) > 2 (qx2 0 (1 q)o,(co z ). Clearly, s2 (u) >- 2 ps2(a) O (1-p)o 2 (o)2 ) for all p E [0,1[. But then, by continuity of the orderings L i (i=1,2), one can find a Pareto optimal z close enough to s(a) and a small 8 such that z2 >2 (1-8)s 2 (a) ED 8(31 2 (0)2 ) and ( 1-5 )z1^8o 1 (oh) >- 1 s i (a). But this again contradicts the statement that s(o) is the solution as per definition 3.4. Therefore, for x E X 12 such that, for some p E [0,1], px2^(1-p)o 2 (co2 ) >- 2 92 (6) and ps i (a) 0 (1-p)o 1 (co 1 ) L i x 1 , there is no loss of , generality in assuming ps i (a) ® (1-p)o 1 (o 1 ) — 1 x 1 . Let us consider now the solution g to the "counterfactual" bargaining session (?.-. 1 ,L 2 ,o 1 (co 1 ),o 2 (ti),X 12 ). Since the preferences of individual 2 are monotonic, px2 O (1 -p)o2 (ti) > 2 px2 (1-p)o 2 (co2 ) it is clear that by demanding some new agreement w - 2 s 2 (a). E X12 Since ps i (a) ED (1-p)o 1 (co 1 ) — 1 x1, while threatening 1 to leave the bargaining session with probability (1+E p), individual 2 will force 1 to give up her demand since (p - Os,(a) 0 (1+e-p)o 1 (o 1 ) >- 1 x1 >i w / . 1 will end up worse off in the counterfactual , situation while 2 will end up better off. It is straightforward from the above argument to establish the second sentence of the proposition. ^ It should be noted that proposition 3.2 is a mere reinterpretation, in the present context, of a well-known result in classical bargaining theory, which relates to the adverse effect, on a player's final position in the game, of an increase in her risk aversion (Kihlstrom, Roth and Schmeidler (1981), Roth (1985)). An alternative proof of this result is 88 given in Rubinstein, Srafa and Thompson (1992; proposition 3). Although proposition 3.2 is consistent with our original intuition that the counterfactual improvement in individual 2's endowment makes her bold enough to insist on her demands for better agreements because she does not fear as much as before the adverse consequences of a negotiation breakdown, it does not quite say that exploitation as per definition 3.4 measures adequately the bargaining power of the exploiter. The only thing that proposition 3 2 affirms is that if in a bargaining session, one increase the endowment of one agent while leaving intact the other parameters of the problem, then the agent whose endowment is increased will benefit while the other will suffer. This increase in one's agent original situation may well have nothing to do with the reduction of her "deprivation". In effect, the definition of deprivation is entirely redundant for the result of proposition 3.2 to hold. Moreover, the bargaining model explored in this section does not make clear what is the exact relevance, for assessing the bargaining power of individual 1 in a particular bargaining session a, of invoking some counterfactual bargaining session (r 1 , L- 2 ,o 1 (o) 1 ),o 2 (ts),X 12 ). Since our objective is to explain individual l's bargaining power in the actual session a, why not stick with the utility reached by individual 1 in a without bothering about some metaphysical counterfactual session? In the absence of a more precise definition of the poverty threshold as well as of the influence this threshold can have on the parameters of the bargaining games, there is absolutely nothing that can answer this question. It seems therefore to be an important objective to explore the type of properties a "good" poverty threshold should satisfy in order to make exploitation a significant source of the bargaining power of individual 1. Quite clearly, the deprivation would have to be defined 89 an example with respect to a threshold that is of such a vital importance for individual 2 that she would be very afraid, in her bargaining with individual 1, to make any threat that could give rise to a small risk of negotiation breakdown. In other words_the threshold would have 90 to be such that the individual has much to loose if the bargaining session fails to produce an agreement. This is illustrated in Figure 3.1 which represents the utility possibility set of a typical bargaining session a normalised by setting u t (o,(to,) = 0 for i=1,2 and ug)=1. This figure exhibits where the threshold is (and its corresponding outside option for individual 2) is such that individual 2 is so badly off at this threshold that she is willing to do anything to get an agreement. The figure illustrates that, at the Nash solution (identified by the rectangle contained in U with the largest area), individual 1 obtains a lot from 2's deprivation ((3 1 is close to 1). The fact that this gain comes, for a large part, from the deprivation of 2 is illustrated in that example by considering what is the Nash equilibrium of the counterfactual session where 2's deprivation is eliminated. As indicated on the picture, exploitation as per definition 3.4, explains a significant portion of 13 1 in this particular example. The reason for this lies in the fact that the poverty threshold with respect to which individual 2's deprivation is assessed is such that the marginal utility of a small increase in her endowment (below the threshold) is "large" (the slope of the frontier of the utility possibility set is steep). For this reason, individual 2 is not very "bold" in asking for a better aggreement (with some treat of breaking the bargaining session in case of non aggreement from individual 1) since she fears very much the adverse consequences of a negotiation breakdown. Defining a good poverty threshold consists, therefore, in finding an individual endowment below which any individual's marginal utility is sufficiently large. 3.4: Conclusion Instead of attempting to summarise the discussion presented in this paper, I conclude with the four objectives that further investigations into the problem of making sense of the notion of exploitation should have in mind. 91 1) The specification of a poverty threshold that makes exploitation a significant source of the bargaining power achieved by individuals in the various bargaining situations in which they are involved appears to be of paramount importance and call for further investigation. 2) Bargaining theory provides a useful framework for discussing the issue of exploitation but it needs to be interpreted in physical and economic terms and not in pure utility terms. 3) It is certainly worth continuing to look for other solution concepts than that of Nash (as reexamined by Rubinstein, Srafa and Thompson (1992)) considered in this paper. The verdict as to whether exploitation is a good measure of bargaining power depends crucially on how the bargaining sessions are conducted. 4) More ambitiously, some gap must be filled between bargaining theory and general equilibrium theory. A theory explaining the functioning of the markets as if they were bargaining games played by sellers and buyers is needed if one wants to aggregate the issue of exploitation or bargaining power over all bargaining sessions and, therefore, to provide some meaning to the objective of an aggregate reduction in exploitation. 92 General conclusion At a rather general level, the whole exercise of this dissertation can be described as an attempt to highlight, somewhat rigorously, various criteria used to assess the desirability of (economic) states of affairs. These criteria are formalised either as binary relations over alternative states of affairs (in the first two essays) or as a numerical index attached to them (in the third one). In the first two essays, states of affairs consist of pairs, each element of which being ranked by a particular binary relation. In the first essay, the first element of a typical pair is interpreted as the actual state of the economic system (for instance the current distribution of incomes prevailing in Canada) and the second element, called a situation, is viewed as the set of all states (including the actual one) that are potentially feasible given whatever technical, economic, informational, political etc. constraints one is willing to consider. The first essay looks at the problem of combining transitively a quasi-ordering of states (as a criterion for an actual improvement) with an extension of it to situations (as a criterion for potential improvement). The combination considered is lexicographically based on the criterion for actual improvements to reflect the usual ethical priority given to actual improvements over potential ones. The domain of pairs ranked by the combination is such that each actual state is maximal, with respect to the quasi-ordering of states, in its situation. This domain is also endowed with a minimal richness. The main result of this essay is that, in order for this combination to generate a transitive ranking of the pairs, it is necessary and sufficient to assume that the domain of pairs is such that, given any two pairs, the domination of an actual state by a second one in terms of the quasi-ordering of states imply necessarily the domination of the former's situation by that of the latter. That is to say, loosely, the only way to consistently combine a criterion for actual improvement with a criterion for potential 93 improvement while giving priority to the former is to assume a world where, in fact, the criterion for actual improvements is redundant. If this result is interpreted in the light of a more standard economic environment, with the quasi-ordering of actual states being the standard Pareto quasi-ordering, the quasi-ordering of situations being the Chipman-Moore (1971)-Samuelson (1950) (CMS) quasi-ordering, the states being utility allocations and the situations being utility possibility sets, this result is shown as saying that the only way by which the CMS and the Pareto criteria can be combined together transitively is if no utility possibility frontiers cross together in the sense of Gorman (1953). As shown by this author, the non-crossing of utility frontiers is in turn equivalent, when a typical utility possibility set is generated by all possible allocations of an aggregate endowment of k goods among n individuals with continuous, convex and monotonic preference, to assuming that all individuals have preferences that generate parallel straight lines Engel curves, an obviously implausible restriction. In short, this paper shows that, unless one is willing to use it in a way which does not aggree with the Pareto principle (i.e which does not always recommand Pareto improvements), the CMS criterion is not a better criterion for ranking alternative pairs than the original Kaldor-Hicks-Scitovsky compensation criterion was. In the second essay, the pairs ranked are choice situations, each of which consists of an opportunity set (that a given individual could possibly face at some stage of her life time) and a chosen element in it (element that is assumed to maximise the individual's complete, reflexive and transitive preference ordering). The essay considers the problem of ranking these choice situations primarily on the basis of the freedom of choice offered by their opportunity sets and, in the cases where the choice situations are not comparable on the sole basis of their freedom of choice, on the basis of the individual preferences over the most preferred elements in each of them. The criterion used to assess the freedom of choice offered by the opportunity set is that of set inclusion. An opportunity set is thus seen as offering at 94 least as much freedom as a second one if any option available in the latter is also available in the former. The relation will be strict if the two sets are not equal and weak otherwise. The main result of this essay is that, in order for the combination of the freedom and the preference criteria to transitively rank the various choice situations, it is necessary and sufficient to assume a domain of choice situations in which the criterion of freedom of choice is redundant. However, if one is willing to weaken the requirement of consistency from transitivity to quasi-transitivity, then a combination of the criterion of freedom of choice and that of preferences which gives priority to the former can be obtained without much difficulties. In the third essay, the objective pursued is more to transform an intuitive ethical issue, that of exploitation, into a well defined criterion for evaluating economic states than to combine two already well defined criteria. This is accomplished by providing and motivating a somewhat new and general definition of exploitation viewed as an extraction of an unfair advantage out of someone else's situation. The unfairness in someone's situation is defined as the initial deprivation of this person, the exploited, with respect to some prespecified poverty threshold of some relevant assets. An advantage obtained by another individual is said to be unfair (exploitative) if the value of the objective function of this individual is reduced in the counterfactual situation where the exploited is given the poverty threshold as her initial endowment. Obviously, the acceptability of this definition depends upon a good definition of a poverty threshold. In order to help in providing such a definition, I resort to bargaining theory. Using the Nash solution concept, as reexamined recently by Rubinstein, Safra and Thompson (1992), as a stylised description of the outcome of bargaining situations, I try to establish what kind of property the poverty threshold must satisfy in order for exploitation to be an adequate measure of the bargaining power of the exploiter in the (implicit) bargaining game played with the exploited. The informal finding of this essay is that the poverty 95 threshold must be such that any individual endowed with less of the relevant assets than what is prescribed by the threshold will be very reluctant to exchange a small improvement over her original endowment for a bet involving a larger gain and a return to her original endowment as outcomes. With such a poverty threshold, a deprived individual will not be bold enough to ask for a more favourable aggreement in her bargaining with another individual who will then obtain most of the mutual benefits arising as the result of the bargaining session. exploiter. This is why exploitation can take place. Further research shall soon give more definiteness to this result. The overall lesson that I draw from the exercise presented in this dissertation is that clarifying rigorously seemingly well defined principles for evaluating social states is a difficult task which must be accomplished with care. This should be contrasted with the arrogant assurance sometimes exhibited by economists in their comments about various issues such as that of free trade between US, Canada and Mexico. Hearing many economists expressing their view on this particular issue, the free trade aggreement will definitely constitute an improvement for all three countries. What do they mean by that? 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Von Neumann, J and Morgenstern, 0 (1947) "Theory of Games and Economic Behavior", 2nd edition, Princeton, Princeton University Press. 103 Appendix This appendix proves other versions of theorems 1.1 and 1.2 by replacing assumption A1.3 and A1.4 by the following compensation unrestrictedness assumption. A set S is said to be compensation unrestricted (CU) if and only if (y,Y) implies that (y*,Y) E E S and y* E MQ(Y) S. This assumption is often made in applied cost benefit analysis and asserts, roughly, that all states of the economic system are mutually accessible. Theorem A: Let Jambe a family of Q-compact subsets of X and let S be any subset of X x satisfying A1.1 and CU. Then R is a quasi ordering of S if and only if R = C on S. Proof As in theorem 1.1, the proof of the sufficiency part is trivial. To prove the necessity part, I follow the same steps than in theorem 1.1 (starting with (y,Y) and (y',Y') such that y Q y' and -'(Y Q Y')) until I am led to the existence of an S.' Y . Since S is compensation unrestricted,^E S. E Y' such that S Q SY for no Si e Now, y' Q S' is clearly false since assuming otherwise would contradict, given the transitivity of Q, the claim that y Q is false. On the other hand, Q. y' is precluded as otherwise, a contradiction of y' E MQ (Y') would occur. One thus has y' QN . From the reflexivity of Q, Y' Q Y' and, from definition 1.4, (y',Y') R (S',Y'). However (y,Y) R ,Y') does not hold since neither y Q S' or Y Q Y' are true. One thus has (y,Y) R (y',Y') and (y',Y') R ,Y') but not (y,Y) R^an intransitive chain. 104 ^ It is also easy to show the following analogue to theorem 2.2 stating that, under compensation unrestrictedness, the equality of R and C is a necessary and sufficient condition for the compensation criterion to be transitive. Theorem A': Let .5rbe a family of Q-compacts subsets of X and let Sc X x irbe a domain of positions satisfying A1.1 and compensation unrestrictedness. Then, R = C on S if and only if K is transitive on S. Proof Consider any sequence (y,Y), (y',Y') and (y",Y") E S such that (y,Y) K (y',Y') and (y',Y') K (y",Y"). From definition 1.3, 3 9 E Y such that 9 Q y' and 3 9' E Y' such that I' Q y". Compensation unrestrictedness implies that (5, ,Y) E S and (r ,Y') E S. If R = C on S, it must be the case that Y Q Y' and Y' Q Y". The transitivity of Q and proposition 1.1 imply that (y,Y) K (y",Y"). Suppose now that R and C disagree as to how to rank two positions (y,Y) and (y',Y') in S. As in theorem 1.1, it must be the case that y y', it follows from definition 1.5 that (y,Y) Q y' holds but Y Q Y' does not. Since y Q R (y',Y') and, from proposition 1.3, that (y,Y) K (y',Y). Now the fact that Y Q Y does not hold implies, from definition 1.1, that there exists 9' E Y such that 9 Q 9' for no 9 in Y. Since Y is Q-compact, there is no loss of generality in assuming 9' e M Q (Y). Since S satisfies compensation unrestrictedness, definition 1.5 and proposition 1.1, one has (y',Y') (r ,Y) E S. From R (r,Y) (in fact (y',Y') Rs (S ',Y) from proposition 1.2). From proposition 1.3, it follows that (y',Y') K , (9',Y') holds. Thus, one has (y,Y) K (y',Y') and (y',Y')K (9',Y'). However, since 9 Q 9' for no 9 in Y, it follows from definition 1.3 that (y,Y) K (9',Y'), a violation of transitivity. ^ 105
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Three essays on the criteria to be used in welfare economics Gravel, Nicolas 1993
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Title | Three essays on the criteria to be used in welfare economics |
Creator |
Gravel, Nicolas |
Date Issued | 1993 |
Description | This dissertation consists of three essays devoted to the general problem of combining various criteria for evaluating the collective desirability of social states. The first essay discusses the problem of combining a quasi-ordering of some set of alternatives (interpreted as a criterion for an increase in actual social welfare) with an extension of this quasi-ordering to the power set of this set (interpreted as a criterion for an increase in potential social welfare) to obtain a quasi-ordering of some subset of the Cartesian product of this set and its power set lexicographically based on the criterion for an increase in actual welfare. The main result of this essay is that, in order for such a quasi-ordering to exist, it is necessary and sufficient that the subset to which it is applied is such that the extension subsumes the original criterion. When applied to the standard Pareto quasi-ordering and its extension defined by the Chipman-Moore (1971) - Samuelson (1950) quasi-ordering, and under standard assumptions on the economic domain, this result is shown to imply Gorman’s (1955) conjecture for the transitivity of the Compensation criterion a la Kaldor-Hicks-Scitovsky. The second essay examines Sen's (1991) suggestion that preference information be used to supplement the criterion of freedom of choice for ranking opportunity sets. This paper shows, with some generality, that, in order for this supplementation to produce a transitive ranking of the opportunity sets, it is necessary and sufficient to assume that the domain ranked is such that the individual preference ordering encompasses the criterion of freedom of choice. However, it is also shown that the quasi-transitivity of such a ranking can be obtained without further assumption. The lesson of this paper is thus that there is little room for constructing a ranking of opportunity sets that attaches value to their freedom of choice while giving some weight to individual preferences. If freedom of choice is to have any value in the ranking, then in order for the ranking to be transitive, this value will have to be instrumental rather than intrinsic (using Sen's (1988) terminology). Finally, the third essay tries to make sense of the notion of exploitation set forth by Marxists and others and to relate it to that of bargaining power. For this task, a definition of exploitation is proposed which, it is contended, captures the intuitive meaning of the word as the act of taking unfair advantage of someone. More precisely, the definition considers a relationship between two agents to be exploitative if one agent (the exploiter) obtains an advantage from this relationship which can be shown to depend upon the initial deprivation of the other (the exploited) with respect to some poverty threshold. To assess whether the advantage of the exploiter is indeed due to the deprivation of the exploited, the definition considers a counterfactual experiment in which the state of deprivation of the exploited is eliminated and examines the welfare consequences of this experiment for the presumed exploiter. If the latter becomes worse off from this elimination of the other's deprivation, then it is asserted that the presumed exploiter is indeed taking an advantage of the other’s deprivation. The problem of specifying an "adequate" poverty threshold is also examined by appealing to bargaining theory. This examination is based upon the somewhat intuitive idea that exploitation is related to an "excessive" bargaining power on the part of the exploiter. The definition of the poverty threshold should therefore be made in such a way as to make exploitation a good measure of the bargaining power of the exploiter in the bargaining game representation of the relation between the exploited and the exploiter. |
Extent | 4583631 bytes |
Subject |
Welfare economics. |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-15 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0098862 |
URI | http://hdl.handle.net/2429/2012 |
Degree |
Doctor of Philosophy - PhD |
Program |
Economics |
Affiliation |
Arts, Faculty of Vancouver School of Economics |
Degree Grantor | University of British Columbia |
Graduation Date | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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