aa r X i v : . [ ec on . T H ] M a r J. S. MILL’S LIBERAL PRINCIPLE AND UNANIMITY
EDWARD J. GREEN
Abstract.
The broad concept of an individual’s welfare is actually a clusterof related specific concepts that bear a “family resemblance” to one another.One might care about how a policy will affect people both in terms of their sub-jective preferences and also in terms of some notion of their objective interests.This paper provides a framework for evaluation of policies in terms of welfarecriteria that combine these two considerations. Sufficient conditions are pro-vided for such a criterion to imply the same ranking of social states as doesPareto’s unanimity criterion. Sufficiency is proved via study of a communityof agents with interdependent ordinal preferences. Introduction
The broad concept of a person’s welfare is actually a cluster of related, specificconcepts. Economists have tended to focus attention almost exclusively on just oneof these specific concepts, which interprets questions regarding people’s welfare asbeing about their subjective preferences. One might care also about some notionof people’s objective interests. I use the term ‘objective interest’ to refer to anyspecific notion of a person’s well-being that might be quantified by an index ofmeasurements of various aspects of the person’s situation.Obviously there can be many distinct notions of a person’s objective interest. Forinstance, some notions are based exclusively on the adequacy of a person’s access tobasic physical necessities, while other notions encompass social and psychologicalaspects of the person’s situation as well. (Notably, Rawls’s (1971) characterizationof a person’s objective interest in terms of “primary goods” includes these broaderaspects.) It should be possible to design a satisfactory framework for evaluation ofpolicies in terms of welfare criteria based on notions of people’s objective interests,and also in terms of criteria that combine considerations of objective interests andpreferences, just as neoclassical economists have succeeded in doing for preference-based notions.Moreover, once it is granted that there may several alternative concepts of welfarethat are all worthy of consideration in policy making, then some interesting generalquestions arise concerning the relationships between them. Are there salient fea-tures common to various specific notions of a person’s objective interest—analogousto the ordinal features by which economists characterize preferences in an abstract,general way—that might serve as a basis for economic analysis? If so, can anygeneral comparison be made between the conclusions of welfare analyses basedon various notions of objective interest possessing these features, or between theconclusions of such an analysis and a neoclassical analysis based exclusively onpreferences? In particular, are there any interesting conditions under which all of
Date : 2019 (first draft, 1994). these various analyses would reach identical conclusions? This paper studies thesequestions.The main goal of the paper is to compare the neoclassical welfare criterion ofunanimous preference (formulated by Pareto (1909) and emphasized in welfare anal-ysis by Wicksell (1935)) to an alternative criterion that is somewhat in the samespirit, but that relies in an essential way on a notion of each person’s objectiveinterest as well as on a notion of persons’ preferences. The formulation of thisalternative criterion is suggested by John Stuart Mill’s discussion of liberalism. Mill emphasizes two connected ideas: that there is a notion of each person’s in-terest that is distinct from that person’s preferences, and that a person ought tobe permitted to do as he pleases so long as he does not damage the interests (inthis non-preferential sense) of others (Mill 1859, Book I and Book IV). He extendsthis idea of liberal permissibility to coalitions. That is, he proposes that a groupof people ought to be permitted to act unanimously as long as they do not damagethe interest of anyone outside the group. This proposal would license actions ofwhich the entire community approves, because the group that includes everyonewould act unanimously and there would be no outsider whose interest could bedamaged. Thus Mill’s criterion of liberally permissible action is at least as broadas the Pareto-improvement criterion. Furthermore Mill states explicitly that heinterprets the notion of people’s interests narrowly enough so that some actionswould also be permissible although they were not unanimously approved. Thatis, Mill takes the position that a person’s mere preference that an action shouldnot be taken does not make the action damaging to the person’s objective interest.He proposes that someone should be permitted to take an action of which no oneelse approves, as long as those others’ disapproval reflects such mere preferencesunconnected to their interests. On this interpretation, Mill’s proposed criterion ofliberal permissibility is strictly broader than the Pareto-improvement criterion.The preceding discussion makes it clear that Mill’s views on liberally permissibleaction are based on two separate considerations: a specific notion of what objectiveinterests people possess, and a general characterization of the relationship betweenpeople’s objective interests and the permissibility of actions that other people mightwish to take. Thus someone might endorse Mill’s general characterization, but differwith him about concrete cases on the basis of holding a different view of what arepeople’s objective interests. (Indeed, someone might suppose contrary to Mill that amere preference does create a corresponding objective interest. When Mill’s generalcharacterization is interpreted according to that view, the resulting criterion issimply a reformulation of the Pareto-improvement criterion.) The formal theory tobe presented here will emphasize the separation between Mill’s two considerations.The main result to be proved is that, even in the context of an assumption aboutthe relationship between preferences and interests that is significantly weaker thanto posit an identification between them, Mill’s general characterization of liberalpermissibility coincides exactly with the Pareto-improvement criterion. This resultprobably would have surprised both Mill and his philosophical critics. It showsthat the conflicts between some alternative welfare criteria are much less pervasive Mill’s thinking has also been influential in stimulating the formal investigation (beginningwith Sen 1970) of the problem of constraining Paretian welfare analysis to respect persons’ rights.Schick (1980) and Riley (1988) have employed formal social-choice theory to interpret Mill’spolitical philosophy as a whole. than might have been believed. However it does identify some particular situations(namely, those in which the relationship between preferences and welfare assumedin the theorem are implausible) in which the liberal and Paretian criteria are likelyto lead to different conclusions.2.
The setting of the problem
In order to examine carefully the logic behind Mill’s suggestion that his liberalprinciple permits a wider class of actions than just those that accomplish unani-mously preferred changes, I am going to restate the suggestion in language thatis parallel to that of neoclassical welfare economics. There a person’s preferencesare represented formally by a binary relation that may hold between social states ,which in turn represent possible situations of the community. A person’s objectiveinterest can also be also be represented formally by a binary relation between so-cial states. As in the case of preferences, a distinct relation is identified with eachperson or agent in the community.In welfare economics, the relation of unanimous preference (often called the“Pareto-improvement relation”) is defined from the preference relations of individ-uals. Specifically, one social state is the unanimous successor of another, status-quosocial state if every agent prefers the change to it from the status quo, with at leastone agent’s preference being strict. Analogously, define a social state to be the liberal successor of a status-quo social state if all members of some group of agentsprefer the change to it from the status quo, with at least one member of the grouphaving a strict preference, and if also the change does not reduce the welfare of anyagent outside the group. (In both of these definitions, I refer to one social state asthe status quo only to distinguish it from the other social state in the pair. I donot mean to imply that it is historically determined or special in any other way.)Using these definitions, Mill’s suggestion can be restated as the thesis that onesocial state may be a liberal successor of another without necessarily being itsunanimous successor. Now I specify a formal theory in which this thesis can beexpressed. The aim of this theory is to characterize the logic of the thesis in a waythat is completely explicit, and that is also general in the sense of being independentof specific proposals regarding what might constitute a person’s interest.Consider a finite set I of agents, and a set X of social states. ( i, j, and k willrange over I, and w, x, y, and z will range over X. ) To each agent j are associatedtwo binary relations on X, W j and R j . When xW j y, this means that state x provides for the objective interest of agent j as well as state y does. I will referto W j as the interest relation, or simply the interest, of agent j. When xR j y, thismeans that j weakly prefers x to y. Define connectedness of a relation to mean that each pair of states is related inat least one direction, and for each agent j, assume that(1) W j is transitive and reflexive, and R j is transitive and connected. V j and P j will denote the asymmetric (or strict) parts of W j and R j respectively. E j and I j will denote the symmetric parts of W j and R j respectively.Within this simple formal language, it is possible to state three substantiveconditions that may be placed on agents’ preferences. The first condition actuallyguarantees three things: that each agent cares only about his own interest andpossibly the interests of others, that each agent prefers a social state that betterprovides for his own interest if the interests of others are held constant, and that each interest relation is connected on any set of states on which all other agents’interests are constant. Formally, for any agent j and for any states x and y, (2) If ∀ i = j xE i y, then [ xW j y ⇐⇒ xR j y ] . This will be referred to as the condition that preferences are based on interests.
Second, agent j will be called nonpaternalistic if his preferences are consonantwith his own interest and the preferences of others. Formally, j is nonpaternalisticif, for all x and y, (3) If xW j y and ∀ i = j xR i y, then xR j y. The third condition that can be stated regarding an agent’s preferences is that hispreferences about trade-offs between the interests of agents within any one group inthe population are independent of the situation regarding interests of other agentsoutside the group. Formally, agent i has preferences that are separable in interests if for any partition of I into groups J and K and for any states w, x, y, and z, (4) If ∀ j ∈ J [ wE j y and xE j z ] and ∀ k ∈ K [ wE k x and yE k z ] , then wR i x ⇐⇒ yR i z. Separability of preferences in interests will play an important technical role inthe arguments below. It is a necessary condition for an agent’s preferences to berepresentable by a utility function that is a sum of functions that in turn are closelyrelated to his and other agents’ interest relations. Given the other assumptions thathave been made here, and a few others that will be introduced shortly, it is alsosufficient for the existence of such a representation. It should be noticed, though,that separability is an assumption with a lot of substantive content. For example,if each agent’s interest is taken to be his wealth, then it rules out the possibilitythat an agent prefers small increases in wealth for others whenever they are poorerthan he is, but that (perhaps because of envy) he prefers that they should suffersmall decreases in wealth when they are richer than he is. This paper will concludewith an example that shows that the liberal and unanimity relations may coincideeven when agents’ preferences are not separable in interests.Three technical assumptions about the topological structure of the set of socialstates, and of agents’ interest and preference relations, will also be needed. Incontrast to the separability assumption, these assumptions do not seem to raisesignificant issues of interpretation. The meanings of the first two assumptionsare immediately obvious. The third assumption states that if E j is viewed as acorrespondence, then it is lower hemicontinuous:(5) X is a connected, separable topological space.(6) For every agent j, W j and R j have closed graph in X . (7) For every agent j, pair of states y and z, and neighborhood Z of z, if yE j z, then there exists a neighborhood Y of y such that ∀ y ′ ∈ Y ∃ z ′ ∈ Z y ′ E j z ′ . The use of ‘nonpaternalism’ for this condition is well entrenched in the literature on theinterdependent preferences. It is easy to think of examples in which the condition applies inways that intuitively have nothing to do with nonpaternalism, though. For example, Sen (1970),considers a prude who would prefer to be the one to read
Lady Chatterley’s Lover if anyone hasto read it at all,and someone else who would like to read the book but who maliciously wouldeven more enjoy forcing the prude to read it. Regarding someone who prefers that his ownwelfare should be maximized certeris paribus, the condition formulated here is necessary, but notsufficient, for the person’s preferences to be genuinely non-paternalistic.
The unanimity and liberal relations now have to be defined. State x is a unani-mous successor to y (and is also described as being Pareto superior to y or a Paretoimprovement of y ) if(8) ∀ i xR i y and ∃ j xP j y. Define x to be a liberal successor of y if, for some coalition J ⊆ I, (9) ∀ i ∈ J xR i y and ∃ j ∈ J xP j y and ∀ k J xW k y. Mill and Pareto contrasted
At the close of section 1, I provided an informal summary of Mill’s liberal prin-ciple. I pointed out the principle always countenances Pareto improvements. Theformal theory just set forth affords an explicit proof of this. In particular, taking J = I in (9) yields (8), showing that the unanimous-succession relation is a subre-lation of the liberal-succession relation. This is exactly the formal statement of theassertion.I also mentioned Mill’s thesis that his principle countenances some changes ofsocial state that are not Pareto improvements. I emphasized that the thesis followsfrom his view that there is at most a very loose relationship between a person’spreferences and his objective interest or even people’s objective interests in gen-eral. However, in section 2 I have assumed that all agents have nonpaternalisticpreferences that are based on interests and separable in interests. These conditionsall relate preferences to interests. One might wonder whether these relationshipsrestrict the scope of the liberal principle so tightly that it must coincide exactlywith the Pareto principle. Because the conditions in section 2 have been specifiedin an explicit and formal way, it is possible to construct an example that showsdefinitely that the conditions do not imply that Mill’s principle and the Paretoprinciple coincide. I will provide one such example next. This example should notnecessarily be interpreted to show that Mill’s thesis is incorrect, though. Rather itshows that the conditions introduced in section 2 are weaker in some circumstancesthan they might seem to be. After presenting the example. I will introduce somefurther assumptions that rule out such degenerate circumstances. Once these as-sumptions have been set forth, the stage will be set to examine the relationshipbetween Mill’s liberal principle and the Pareto principle. Now, here is the exampleof a community in which the liberal-succession relation is much more inclusive thanthe unanimous-succession relation.Let I = { , , , } , and let X be an interval of the real line. Suppose that xR y and xW y if x ≥ y, and that xW y and xR y if x ≤ y. Let agent 3 be This relation of liberal succession is one of two possible ways (within the present theory) toformalize Mill’s criterion for when a change of social state should be countenanced. The otherpossibility would be to stipulate that a coalition may make any change of state that does not givenon-members strictly lower levels of welfare than they previously enjoyed. That is, x could bedefined to be a liberal successor of y if, for some coalition J ⊆ I, (9 ′ ) ∀ i ∈ J xR i y and ∃ j ∈ J xP j y and not ∃ k J yV k x. In general, (9 ′ ) defines a more permissive (i.e., set-theoretically larger) relation than does (9).However, (9) and (9 ′ ) define the same relation if all welfare relations W j are connected. Whileconnectedness is not assumed directly, it is implied by other assumptions that will be used here(cf. lemma 6 below). identical to agent 1, and agent 4 to agent 2. It is easily verified that all of theassumptions (1)–(7) are satisfied. However, no pair of social states is related by theunanimous-succession relation, but every pair of distinct social states is related bythe liberal-succession relation.In this example, the assumptions (2) and (3) of interest-based preference andnonpaternalism have been trivialized. That is, each assumption is an implicationwith a hypothesis that is never satisfied. Now I will define four conditions whichare intended to identify environments that are rich enough for the two assumptionsto have nontrivial content. Note that the example above does not satisfy any ofthese conditions.First, agents’ interests will be said to have product structure if (10) ∀{ x i } i ∈ I ∃ y ∀ i yE i x i . Second, agent i will be said to have idiosyncratic interest if(11) ∃ x ∃ y [ xV i y and ∀ j = i xE j y ] . Third, analogously, agent i will be said to have idiosyncratic preferences if(12) ∃ x ∃ y [ xP i y and ∀ j = i xI j y ] . Fourth, it will be said that an unambiguous improvement is possible if(13) ∃ x ∃ y ∀ i [ xV i y and xP i y ] . Agents’ judgments and interpersonalagreement regarding interests
Taken together, assumptions (2) that agents’ preferences are based on interestsand (4) that agents’ preferences are separable in interests have an interpetationthat each agent’s preferences are based on considerations of trade-offs between hisown interest and other agents’ interests. (There might well be other possible in-terpretations of agents’ preferences as well.) The idea of making trade-offs amonginterests implies that the difference between the degree to which an agent’s inter-est is satisfied in two distinct states is conceived as being a cardinal magnitude.Moreover, agents may either agree or disagree with one another about comparisonsbetween these magnitudes. This is a subtle point, and it is also an important onefor the welfare analysis which is to follow. Therefore let us consider it carefullynow.Consider two agents i and j , each of whose interest is completely determined bywhat kind of house he lives in. Suppose that there are four types of house—hovels, It would be desirable to make only an assumption to the effect that there is some latitude foragents’ interests to vary independently. The present, inordinately strong, assumption is made inorder to invoke a theorem of conjoint measurement theory below. Krantz, et. al. ((1971), pp. 275-6) discuss the general need to have a theory of measurement based on a weaker assumption. Thusthe objectionable feature of the present assumption seems to be a convenient technical fictionthat need not be viewed as limiting the validity of the results to be derived here. Of course,social states must be viewed as conceptually possible profiles of interests that are not necessarilytechnically feasible to satisfy all at once. cottages, mansions, and palaces—in increasing order of satisfaction of an agent’sinterest. Suppose that a social state is simply a specification of a type of house foreach agent.The ordinal ranking of types of house determines the answers to some questionsregarding cardinal comparisons, such as “Would it make more difference for thesatisfaction of agent j ’s interest if he were to move from a cottage to a palace thanif he were to move from a cottage to a mansion?” Clearly the former differenceis larger, since the two changes begin at the same point but the latter changeaccomplishes only part (in terms of the ranking) of what the former accomplishes.However, the ranking does not determine the answers to other questions such as“Would it make a bigger difference for the satisfaction of agent j ’s interest if hewere to move from a mansion to a palace, than if he were to move from a hovel toa cottage?” That is because neither of the ordinal intervals described by these twochanges includes the other. If one were to represent the magnitudes of differencesin satisfaction of an agent’s interest as distances between points on a line, thenpictures like either of the following would be possible.[hovel] [cottage] [mansion] [palace][hovel] [cottage] [mansion] [palace]Information about agents’ preferences can be interpreted to represent the agentsas implicitly answering such questions, though. For example, consider the followingtwo pairs of states: x and y , and x ′ and y ′ .State x : Agent i lives in a mansion; Agent j lives in a mansion.State y : Agent i lives in a cottage; Agent j lives in a palace.State x ′ : Agent i lives in a mansion; Agent j lives in a hovel.State y ′ : Agent i lives in a cottage; Agent j lives in a cottage.Suppose that agent i strictly prefers state y to state x and also strictly prefersstate x ′ to state y ′ . Then I will interpret agent i as judging that agent j ’s interestis affected more significantly by a move from a mansion to a palace than by amove from a hovel to a cottage, in the following sense. Agent i would be willingto bear the sacrifice of having to move from a mansion to a cottage in order toenable j to move from a mansion to a palace, but he would be unwilling to bearthe same sacrifice in order to enable j to move from a hovel to a cottage. This pairof preferences can be interpreted as reflecting a judgment on the part of i that amove from a mansion to a palace would make a larger difference for the satisfactionof j ’s interest than would a move from a hovel to a cottage.In an environment with many social states, this judgments-about-interests in-terpretation of agents’ preferences will have restrictive implications. For example,consider another two pairs of social states.State w : Agent i lives in a cottage; Agent j lives in a mansion.State z : Agent i lives in a hovel; Agent j lives in a palace. State w ′ : Agent i lives in a cottage; Agent j lives in a hovel.State z ′ : Agent i lives in a hovel; Agent j lives in a cottage.Suppose that i were strictly to prefer state w to state z and state z ′ to state w ′ .The interpretation of preferences as reflecting judgments about cardinal differencesin satisfaction of interests would suggest that i considers a move from a hovel to acottage to matter more for the satisfaction of j ’s interest than would a move froma mansion to a palace. The argument for this implication is the same as before,except now i ’s contemplated sacrifice is a move from a cottage to a hovel instead offrom a palace to a mansion. If i were to hold all of the preferences that have beendiscussed in this paragraph and in previous one, then the judgments-about-interestsinterpretation of preferences would impute two inconsistent judgments to him. Iwill demonstrate later in this paper (in lemma 10) that the assumptions made in thepreceding two sections are sufficient to rule out such imputations of inconsistency.Technically, in addition to the assumptions that have been introduced so far, lemma10 also requires an additional double-cancellation condition in the case of two-agentcommunities and of some larger communitites. This condition will be set forthwhen the lemma is stated formally.Agent j (or any other agent in the community) might either agree or disagreewith agent i ’s assessments of how much difference various changes of social statewould make to the satisfaction of j ’s interest. The main result to be proved inthis paper will be that, in the presence of the assumptions that have already beenmade (along with the double-cancellation condition), the additional assumptionthat agents always agree with one another about such assessments is sufficient forthe liberal-succession relation and the unanimous-succession relation to coincide.5. Equivalence of the welfare criteria: a preliminary result
The model of a community set forth in sections 2 and 3 should seem familiarin many respects, especially in relation to work of Debreu. As in Debreu (1960),preferences of an individual agent are described in terms of ordinal and topologicalassumptions along with an assumption regarding separability. As in Debreu (1959),a welfare proposition regarding a community of agents described in such terms isto be investigated. This investigation divides logically into two parts. First arepresentation theorem should be proved, asserting the existence of well behavedutility functions which represent agents’ preferences and also of analogous func-tions which represent agents’ interests. Then, taking advantage of these numericalrepresentations, the proposition about welfare should be proved.Since welfare analysis provides the motivation for the representation theorem,and also since the particular set of assumptions studied here in this paper not bethe only ones which yield the sort of representation that is derived, I will begin byformulating the representation that is needed and by proving that it implies thatthe liberal-succession and unanimous-succession relations coincide.Throughout this section and the remainder of the paper (except where explicitlystated to the contrary), I will assume that conditions (1)–(6) and (10)–(13) hold.The representation theorem that I will prove later from these assumptions (andthe two further ones discussed at the end of the preceding section) contains the Debreu (1959) did not impose separability, however. following three assertions (14)–(16) regarding the possibility of representing agents’preferences and interests by numerical functions. (14) v j : X → R is continuous and ∀ x ∀ y [ xW j y ⇐⇒ v j ( x ) ≥ v j ( y )] . (15) p j : X → R is continuous and ∀ x ∀ y [ xR j y ⇐⇒ p j ( x ) ≥ p j ( y )] . (16) There exist scalars α ij and κ i such that ∀ i ∀ x p i ( x ) = Σ j ≤ n α ij v j ( x ) + κ i . The first two of these assertions are just the sort of ordinal representations thatone would expect. (Note, however, that (14) implies the connectedness of interestrelations, which has not been assumed directly in (1).) The real force of the repre-sentation is provided by assertion (16). What this assertion obviously accomplishesis to reflect numerically the assumptions (2) and (4) that agents’ preferences arebased on interests and separable in interests. The assertion states significantly morethan that, though. Each agent’s preferences might individually be based on inter-ests and separable in interests, and yet the numerical representation of someone’sinterest (agent i , say) that entered one agent’s (agent j ) utility function might be anarbitrary continuous, strictly increasing transformation of the numerical represen-tation of i ’s interests that entered another agent k ’s utility function. Assertion (16)states that the representation of i ’s interest that enters j ’s utility function mustbe a strictly increasing affine transformation of the representation of i ’s interestthat enters k ’s utility function. This means that when the preferences of agents j and k reflect cardinal comparisons of differences in satisfaction of i ’s interest (suchas that it would make a greater difference for i to move from a hovel to a cottagethan it would make for him to move from the cottage to a mansion), then j and k must agree with one another. As has already been explained in the precedingsection, the existence of such agreement throughout the community regarding car-dinal intrapersonal comparisons across social states of each agent’s interest is a keyassumption of the proof that liberal succession and unanimous succession coincide.This assumption does not follow from the assumptions about the community thathave been made so far. An ordinal assumption that does imply assumption (16)(in the presence of the other assumptions) will be introduced below in section 8.Now all of the assumptions have been introduced that are needed to prove thecoincidence of the liberal-succession and unanimous-succession relations. In theproof, it will be convenient to write (16) in matrix form. To do so, define v : X → R n and p : X → R n , where n is the number of agents in i, by(17) ∀ x ∀ j [ v ( x )] j = v j ( x )and(18) ∀ x ∀ j [ p ( x )] j = p j ( x ) . Let A be the matrix with coefficients α ij . Then (16) can be written in matrixnotation as:(19) ∀ x p ( x ) = Av ( x ) + κ. The scalar κ i in (16) could be eliminated if p i were adjusted by subtracting it. However, thepresent formulation simplifies the proof of lemma 5 below. Now the proof of the welfare theorem will be presented in a series of lemmas.
Lemma 1. v ( X ) contains a non-empty open subset of R n . Proof.
Let x j V j y j for each j, as is guaranteed by (11). By continuity of v j andconnectedness of X, the interval [ v j ( y j ) , v j ( x j )] is a subset of v j ( X ) . This intervalhas nonempty interior, since v j represents W j . Since agents’ interests have productstructure, then, ( v ( y ) , v ( x )) × . . . × ( v n ( y n ) , v n ( x n )) ⊆ v ( X ) . (cid:3) Lemma 2. A is nonsingular.Proof. By idiosyncracy of preferences, for each i there exist x and y ; such that xP i y and ∀ j = i xI j y. Thus p ( x ) − p ( y ) = A [ v ( x ) − v ( y )] is a scalar multiple of the usualbasis vector in dimension i . (cid:3) Let B = ( β ij ) = A − . Then, for every i there exists a scalar λ i such that(20) ∀ x p i ( x ) = α ii v i ( v ) + X j = i α ij [ X k ≤ n β jk p k ( x )] + λ i . Equation (20) provides an implicit characterization of the preferences of agent i in terms of i ’s interest and other agents’ preferences. However, p i occurs on bothsides of the equation. In order to have an explicit characterization, p i should notoccur on the right. That is, the term P j = i α ij β ji p i ( x ) should be subtracted fromboth sides of (20), and then both sides of the resulting equation should be dividedby 1 − P j = i α ij β ji . The result of these operations, defining γ ik = P j = i α ij β jk ,δ ii = α ii / (1 − γ ii ) , µ i = λ i / (1 − γ ii ) , and δ ik = γ ik / (1 − γ ii ) for k = i, is(21) ∀ x p i ( x ) = δ ii v i ( x ) + X k = i δ ik p k ( x ) + µ i . There is one thing that has to be verified for this derivation of (21) to be sound.That is that γ ii = 1 . Otherwise, the final step of the derivation would have beendivision by zero. This verification is now provided.Recall once again, that by idiosyncracy of i ’s preference, there exist states x and y such that xP i y and ∀ k = i xI k y. By nonpaternalism, i ’s strict preference requiresthat xV i y. Equation (15) implies that p k ( x ) = p k ( y ) for all k = i, so subtraction of(20) from the corresponding equation for state y yields [ p i ( y ) − p i ( x )] = α ii [ v i ( y ) − v i ( x )] + γ ii [ p i ( y ) − p i ( x )] . By (14), [ v i ( y ) − v i ( x )] is strictly positive.Therefore γ ii = 1 if α ii = 0 . However, if α ii = 0 , then it would follow from (11),(14), (15) and (16) that y j R j x j (where x j and y j are the pair of states that (11)guarantees to exist for j ) , contrary to (2).Since γ ii = 1 , then, preferences representable by (16) are also representable by(21). This latter representation is now studied further. Lemma 3.
In (21), for every i , δ ii > . Proof.
By idiosyncracy of preferences, there exist x and y such that xP i y and ∀ j = i xI j y. Thus, by nonpaternalism of i, xV i y. Therefore δ ii > (cid:3) Lemma 4.
In (21), for every i and k, δ ik ≥ . Proof.
By lemma 3, it is sufficient to prove this inequality for i = k. By lemma 1,there exist a subset Y ⊆ X and a nonempty open set U ⊆ R n such that v maps Y onto U. By lemma 2, p ( Y ) = A ( U ) is open in R n . For in define q i : Y → R n by[ q i ( x )] j = δ ii v i ( x ) + µ i if i = j, and [ q i ( x )] j = p j ( x ) if i = j. Note that, by (21), q i = Q i p, where Q i is a matrix that will also be denoted by ( χ jk ) . This matrix is thesame as the identity matrix except for its i th row, and χ ii = 1 and χ ik = − δ ik for k = i. It is obvious (using row i to expand the determinant) that Q i is nonsingular,so q i ( Y ) = Q i A ( U ) , which is open in R n . Thus there exist states x and y such that q i ( x ) − q i ( y ) is a positive scalar multiple of the usual basis vector in the k dimension.That is, xE i y, xP k y, and xI j y for all other agents j. By nonpaternalism of i, xR i y, so δ ik ≥ (cid:3) These results have a useful geometric interpretation. Consider the vector spacethat is obtained from the space of all continuous, real-valued functions on X whenfunctions that differ by a constant are identified with one another. Let V be thefinite-dimensional subspace generated by { v i } i ∈ I . Lemma 1 asserts that V is iso-morphic to R n . For a set F of vectors in V, let K [ F ] be the convex cone generated by F. The possibility of an unambiguous improvement (i.e., assumption (13)) impliesthat K [ { v i } i ∈ I ∪ { p i } i ∈ I ] does not contain any linear subspace (except for {∅} )of V. Lemma 2 asserts that { p i } i ∈ I is a basis of V. Lemma 4 asserts that, for each i, p i ∈ K [ { v i } ∪ { p j } j = i ] . These facts are now used to establish the coincidence ofthe two social-choice relations.
Theorem 1.
The additively-separable representation of preferences discussed hereis sufficient for the Mill’s principle to be equivalent to the Pareto principle. Specif-ically, suppose that the environment is as described in (1)–(6) and (10)–(13). Ifagents’ interests and preferences can be represented as in (14)—(16), then Paretosuperiority and liberal succession coincide.Proof.
Since the Pareto relation is contained in the relation of liberal succession,it is sufficient to prove that x is Pareto superior to y if x is a liberal succession of y. A contradiction will be obtained from the contrary assumption. In particular,assume that J ⊆ I satisfies (9) but that yP k x. Then p k ∈ K [ { v i } i J ∪ { p j } j ∈ J ] . Thus K [ { v i } i J ∪ { p j } j ∈ I ] is not a subset of K [ { v i } i J ∪ { p j } j ∈ J ] . By (13), K [ { v i } i J ∪ { p j } j ∈ I ] contains no linear subspace (except {∅} ) of V. Therefore,by (12) and Corollary 18.5.2 of Rockafellar (1970), there is an agent k such that p k K [ { v i } i J ∪ { p j } j = k ] . (Note that (12) rules out that p j could be a scalarmultiple of p k for any j = k. ) This contradicts lemma 4, since k J . (cid:3) Nonmalevolence
Theorem 1 has antecedents in the work of Bergstrom ((1971), (1989), (1988)),Green ((1979), (1982)), and Pearce (1983), all of which assume some version of(16). In the remaining part of this paper, it will be shown that the assumptionson ordinal preference and interest relations that have been introduced above (alongwith the double-cancellation condition and an ordinal assumption that will implyagents’ agreement about intrapersonal interest comparisons) can replace (16) inthe theorem. Before showing this, though, I will now briefly discuss one differencebetween theorem 1 and it antecedents. The remainder of the paper is independentof this discussion. In the antecedent theorems just mentioned, the possibility of an unambiguousimprovement (13) has not been assumed but an additional assumption of nonmalev-olence has been made. Formally, agent j is nonmalevolent if(13 ′ ) ∀ x ∀ y [if ∀ i xW i y, then xR i y ] . Neither of assumptions (13) and (13 ′ ) implies the other. In particular, note thatthe example in section 3 satisfies (13 ′ ) but not (13). To see that (13) does notimply (13 ′ ) , consider two agents and let X be the Euclidean plane, with v ( x ) = x and v ( x ) = x . Define p = 2 v − v and p = 2 v − v . Obviously neitherpreference relation defined by these utility functions satisfies (13 ′ ) , but (1 ,
1) is anunambiguous improvement over (0 , . Conditional on the other assumptions that have been made here, though, theassumptions (13) and (13 ′ ) are equivalent. It is easy to see that the assumptionsof this paper (specifically (1), (4), (10), (11) and (13 ′ )) imply (13). The converseimplication is proved analogously to theorem 1. Clearly (13 ′ ) implies that all of thecoefficients α ij in (16) are nonnegative. Thus, if some agent j does not satisfy (13 ′ ) , then p j K [ { v i } i ∈ I ] . Thus K [ { v i } i ∈ I ∪ { p j } j ∈ I ] is not a subset of K [ { v i } i ∈ I ] . By (13), K [ { v i } i ∈ I ∪ { p j } j ∈ I ] contains only the trivial linear subspace {∅} of V, sothere is an agent k such that p k K [ { v i } i ∈ I ∪ { p j } j = k ] . This contradicts lemma4, though, so the failure of (13 ′ ) for j implies the failure of (13). That is, if (13)holds, then all agents must satisfy (13 ′ ) . Additive conjoint measurement of preference
The proof of theorem 1 has relied heavily on two assumptions that are notobvious consequences of what had previously been assumed. The first of these isassumption (14), that for each agent j there is a continuous function v j : X → R such that ∀ x ∀ y [ xW j y ⇐⇒ v j ( x ) ≥ v j ( y )] . This assumption implies in particularthat agents’ interest relations are connected on X, which is a stronger connectednessclaim than the limited one that follows from assumptions (1) and (2). The otherassumption is (16), that the functions v j can be specified in such a way that thereexist scalars α ij and κ i such that ∀ i ∀ x p i ( x ) = P j ≤ n α ij v j ( x ) + κ i . These assumptions will be derived from ordinal and topological assumptions.The first step to accomplish this is to formulate three conditions (i.e., (21) - (23)below) that jointly imply (14) and (16). The first of these conditions will be estab-lished in this section, and the second and third in the next.Call a function f : X → R compatible with an equivalence relation E on X if,for all x and y, xEy implies that f ( x ) = f ( y ) . Say that f is strictly increasing with respect to a binary relation V on X if, for all x and y, xV y implies that f ( x ) > f ( y ) . Note that (14) is equivalent to the statement that, for each j, thereis a continuous function v j that is both compatible with E j and strictly increasingwith respect to V j . The first of the three conditions incorporates this informationfrom (14) directly in a way that closely resembles (16). For all agents i and j, let N ( i ) be a set of agents, let ν i be a scalar, and let p i and u ij be functions from X to R . Condition (16) will be paraphrased by taking N ( i ) to be the set of j such that α ij = 0 and u ij to be α ij v j . Now the first two conditions can be stated: (14) resembles (13) and (18) in its form, but it follows routinely from previous assumptions,using a result of Debreu (1959). The crucial difference is that R i is assumed to be connected,while the connectedness of W i must be proved. (22) ∀ i ∀ x p i ( x ) = P j ∈ N ( i ) u ij ( x ) + ν i , where each p i satisfies (15) and each u ij for j ∈ N ( i ) is continuous, non-constant, and compatible with E j , and(23) Each u ij for j ∈ N ( i ) is strictly increasing with respect to V j . Conditions (22) and (23) do not quite imply (16), because (16) entails (accordingto the paraphrase that has just been described) the third condition that (24) If j ∈ N ( h ) ∩ N ( i ) , then there exist scalars σ hij and τ hij such that u hj = σ hij u ij + τ hij . Lemma 5.
Conditions (22) and (24) imply condition (16).Proof.
Consider a well ordering of I. For every j, define ι ( j ) to be the first i suchthat j ∈ N ( i ) . Note that ι ( j ) ≤ j (because j ∈ N ( j ) by (2), (11), and (22)). Foreach h, define M ( h ) to be the union of the sets N ( i ) for i preceding h. For each j, let v j = u ι ( j ) j . Define α hj = σ hι ( j ) j if j ∈ N ( h ) and ι ( j ) < h, α hj = 1 if ι ( j ) = h, and α hj = 0 otherwise. Define κ h = ν h + P j ∈ M ( h ) ∩ N ( h ) τ hι ( j ) j . It is routinelyverified that these definitions satisfy (16). (cid:3)
The remainder of this section is devoted to deriving (22) as a consequence ofordinal and topological assumptions. Besides the assumptions that have alreadybeen stated, only one further assumption is needed. This assumption, called the double-cancellation condition , strengthens the separability condition (4) in the caseof an agent whose preferences depend only on the interests of himself and one otheragent:(25) If ∃ i = j ∀ x ∀ y [ xW i y and xW j y jointly imply that xR j y ] , then the follow-ing condition holds for all states r, s, t, x, y, and z : if rR j x and sR j y andif rW i t, xW i s, yW i z, rW j y, sW j t, and xW j z, then tR j z. Like separability, this condition is necessary (even without any other assumptions)for the existence of an additive conjoint representation. Note that the condition issatisfied trivially if N ( j ) is not a two-element set.Now it will be shown that the ordinal and topological assumptions that have beenmade so far are sufficient to establish (22). The proof that (22) holds for agent i depends on the cardinality of N ( i ) . If N ( i ) has more than one element, then (22)will follow from a representation theorem for additive conjoint measurement due toDebreu (1959). If N ( i ) is a singleton (i.e., if i ’s preferences depend only on his owninterest), then it is sufficient to take p i = u ii = v i and ν i = 0 , where v i satisfies(14). The existence of such a function v i is now established. Lemma 6.
Assumptions (1), (2), (5), (6), and (10) imply assumption (14), i.e.,that for each agent j there is a continuous function v j : X → R such that ∀ x ∀ y [ xW j y ⇐⇒ v j ( x ) ≥ v j ( y )] . Proof.
By a result of Debreu (1959), it is sufficient under these assumptions to showthat each W j is connected on X. Consider any agent j and any pair of states x and y. By (10), there exists a state z such that xE j z and ∀ i = j yE i z. Either yR j z or zR j y by (1), so either yW j z or zW j y respectively by (2). Then by (1), either yW j x or xW j y respectively because xE j z . (cid:3) σ hij = α hj / α ij and τ hij = 0 . In the case general that R i and W i may not coincide, the derivation of (22)depends on Debreu’s representation theorem, which is now stated as lemma 7. Fora proof, see Debreu (1959), or Krantz, et al. (1971, § u ij asserted in the lemma is not explicitly mentioned in those sources, but it isclear from a careful inspection of Debreu’s proof, in which he has noted on pp.22-24 that u ij is obtained from a continuous function and that the several steps ofthe construction of u ij preserve continuity. Lemma 7 (Representation theorem for additive conjoint measurement) . Supposethat R i = W i and that assumptions (1) - (6), (11) and (25) are satisfied. Supposethat X = Π j ∈ I X j , where each X j is a topological space and X has the producttopology, and that, for each j , ∀ x ∀ y [ x j = y j implies xE j y ] . Then ∀ x p i ( x ) = P j ∈ N ( i ) u ij ( x ) + ν i , where i ∈ N ( i ) ⊆ I, p i satisfies (15) and each u ij for j ∈ N ( i ) is continuous and non-constant and factors through X j (i.e., U ij ( x ) depends onlyon x j ) . Lemma 7 falls short of the objective of this section in two respects. First, evengiven the hypotheses of the theorem, the conclusion that each function u ij factorsthrough X j is weaker than the desired conclusion that each u ij is compatible with W j . Second, the hypothesis that X is a cartesian product of spaces that individuallydetermine agents’ interest levels is stronger than the product-structure assumption(10). The cartesian-product representation restricts the economic applicability ofthe lemma to private-goods environments, since the level of provision of a publicgood would be a coordinate of the social state that would affect the interests of allagents. The conclusion of lemma 7 can actually be strengthened to include compat-ibility of each u ij with E j , and the hypothesis of a cartesian-product representationcan be weakened to assumption (10) if (7) is also assumed. These amendments arenow made in lemma 8 and lemma 10. Lemma 8.
Suppose that assumptions (1) - (6), (11) and (25) are satisfied. Supposethat X = Π j ∈ I X j , where each X j is a topological space and X has the producttopology, and that, for each j, ∀ x ∀ y [ x j = y j implies xE j y ] . Then ∀ x p i ( x ) = P j ∈ N ( i ) u ij ( x ) + ν i , where i ∈ N ( i ) ⊆ I, p i satisfies (15) and each u ij for j ∈ N ( i ) is continuous, non-constant, and compatible with E j . Proof.
Lemma 7 establishes everything except for the compatibility of u ij with E j , and it establishes that each u ij factors through X j . A contradiction will bederived from the assumption that, for some j ∈ N ( i ) , u ij is not compatible with E j . Specifically, suppose that xE j y but that u ij ( x ) < u ij ( y ) . Let z j = x j but let z h = y h for all h = j. Then ∀ h zE h y, so zR i y by (2). However, p i ( z ) < p i ( y ) , contradicting (15). (cid:3) A technical result is needed in order to relax the cartesian-product hypothesisto (10). Define a set Y of states to be saturated with respect to an equivalencerelation E if x ∈ Y whenever, for some y, y ∈ Y and xEy. Note that a functionis compatible with E if and only if its level sets are saturated with respect to E. Unlike Debreu’s proof, the proof given by Krantz, et al. does not assume the separability of X, and it does not assert the continuity of the functions u ij . I do not know whether the functionsconstructed in that proof actually are continuous. Since continuity of the functions representingagents’ interest relations was used in lemma 1 and will be used again in the next section, I haveasserted the separability of X in (6). It will be required that the property of saturation with respect to the relations E j should be preserved when topological closures are taken. Lemma 9 asserts that thisrequirement can be met. Lemma 9.
If (1), (6), and (7) hold and Y is a set of states that is saturated withrespect to E j , then cℓ ( Y ) is saturated with respect to E j . Proof.
Equation (1) entails that E j is an equivalence relation. By (6), the graph of E j is closed. Suppose that cℓ ( Y ) is not saturated with respect to E j . Then thereexist states y ∈ cℓ ( Y ) and x cℓ ( Y ) such that xE j y. Since x cℓ ( Y ) , there isan open set U containing x and disjoint from Y. By (7), there is an open set V containing y and satisfying ∀ v ∈ V ∃ u ∈ U uE j v. Thus Y cannot be saturated withrespect to E j , because Y ∩ V = ∅ . (cid:3) Recall that if E is an equivalence relation on X, then the quotient space X / E is the set of E -equivalence classes of elements of X, with the finest topology thatmakes the mapping from elements to their equivalence classes continuous. Lemma 10.
Suppose that assumptions (1)–(7), (10), (11) and (25) are satisfied.Then ∀ y p i ( x ) = P j ∈ N ( i ) u ij ( x ) + ν i , where i ∈ N ( i ) ⊆ I, p i satisfies (15) andeach u ij for j ∈ N ( i ) is continuous, non-constant, and compatible with E j . Thatis, (22) is satisfied.Proof.
Begin by defining xE I y if and only if ∀ i xE i y. Obviously E i is an equivalencerelation and each W i induces a relation on X / E I . By (2), each R i also inducesa relation on X / E I . (‘ W i ’ and ‘ R i ’ will denote the relations on X / E I , as wellas the relations on X ) . There are functions v i and p i satisfying (14) and (15) bylemma 6 and (Debreu (1959)), respectively, and these induce functions from X / E I to R that are compatible with E i (resp. I i ) and strictly increasing with respectto V i (resp. P i ) . The induced functions are continuous by Bourbaki (1965, I.3.4,Prop. 6). Thus the induced functions satisfy (14) and (15) relative to the inducedrelations on
X / E I . Therefore the relations R i and W i on X / E i have closed graph.Now, for each i, define X i = X / E i . There is a canonical bijection between
X / E I and Π j ∈ I X j . By lemma 9 and Bourbaki (1965, I.5.4, Prop 6 and corollary to Prop.7), this bijection is a homeomorphism. Thus R i and W i induce relations withclosed graph on Π j ∈ I X j . These induced relations also satisfy the other hypothesesof lemma 8. The conclusion of this lemma is derived by composing the canonicalsurjection of X onto Π j ∈ I X j with the functions from Π j ∈ I X j to R that lemma 8asserts to exist. (cid:3) Publicity of cardinal intrapersonal interest comparisons
Conditions (23) and (24) still have to derived. In combination with (22), condi-tion (23) states that, when one agent cares about the interest of another at all, thenhe strictly prefers an increase in the interest of that agent if the interests of othersare held constant. Condition (24) states that agents’ preferences reflect agreementabout the cardinal magnitudes of intrapersonal interest differences between states.(Note that this as an assertion of coincidence between various agents’ subjectivepreferences regarding relevant sets of social states. It is not being asserted that thereis any objective basis for making interpersonal comparisons of either preferences orinterests.) Thus, both of these conditions are substantive ones. The assumptions that have been made so far are insufficient to guarantee eitherof these conditions. However, both conditions can be derived if the assumption (27)that will be stated below is added to those already made. The assumption states(in the context of (4)) that agents’ preference relations reflect agreement about thedirections and relative magnitudes of pairs of interest differences, so evidently it isnecessary for (22), (23), and (24) to hold. The statement of (27) will be simplifiedby the following lemma.
Lemma 11.
If assumptions (2), (10), (11), and (22) hold, then ∀ i i ∈ N ( i ) and (26) j ∈ N ( i ) ⇐⇒ ∃ x ∃ y [ xP i y and ∀ h = j xE h y ] . Proof. If j N ( i ) and ∀ h = j xE h y, then (22) implies that xI i y. Thus, if xP i y and ∀ h = j xE h y, then j ∈ N ( i ) . By (2) and (11), this implication entails that i ∈ N ( i ) . The converse implication follows from (15) and the nonconstancy andcompatibility assertions of (22). If j ∈ N ( i ) , then there exist states w and z suchthat u ij ( w ) > u ij ( z ) . By (10), there are states x and y such that xE j w, yE j z, and ∀ h = j xE h y. These states therefore satisfy xP i y, as well. (cid:3) Now the ordinal assumption regarding publicity of cardinal intrapersonal interestcomparisons can be stated. For brevity, this assumption will be called interestcardinality. (27) Let N ( i ) be defined by (26). Suppose that j ∈ N ( h ) ∩ N ( i ) . Then thefollowing implication holds for all states w h , w i , x h , x i , y h , y i , z h , and z i satisfying w h E j w i , x j E j x i , y h E j y i , z h E j z i , and ∀ g = j [ w h E g x h , w i E g x i ,y h E g z h and y i E g z i ]: if w h R h y h , z h R h x h , and y i R i w i , then z i R i x i . To understand this assumption, think of agents h and i as forming their prefer-ences by weighing interest gains to agent j against interest losses to other agents.Agent h judges that j ’s gain moving from w h to y h would not outweigh others’losses but that j ’s gain moving from x h to z h would at least balance others’ losses.Since others’ interest is the same in w h as in x h and also in y j as in z h , h mustjudge that j gains at least as much by moving from x h to z h as by moving from w h to y h . If i shares this judgment, then he should also judge that j gains at leastas much by moving from x i to z i as by moving from w i to y i , since j has the sameinterest in each of the ‘ i ’ states as in the corresponding ‘ h ’ state. If i also judgesthat j ’s gain moving from w i to y i is sufficient to balance others’ losses, then heshould judge as well that j ’s gain in moving from x i to z i is sufficient to balanceothers’ losses, as the conclusion of (27) reflects. Lemma 12.
If (2), (10), (11), (22) and (27) hold and j ∈ N ( i ) , then u ij is strictlyincreasing in V j . That is, under these hypotheses (23) holds and v j = u ij satisfies(14) for j. Proof.
First, in (27) substitute w for w h and w i , x for x h and x i , y for y h and y i , and z for z h and z i . Next substitute j for i, i for h, and x for w and y. Theresulting statement is the implication that, if zR i x and ∀ g = j xE g z, then zR j x. Now, consider any states x and w such that xV j w. By (10) there exists z such that wE j z and ∀ g = j xE g z. Note that xV j z, so that xP j z by (2) and therefore xP i z by (27) (using the substitutions that have just been made). Therefore by (22), u ij ( x ) > u ij ( z ) = u ij ( w ). (cid:3) Now the affine-relation condition (24) can be studied. The first step is to derivefrom (24) a condition that resembles (27) more closely.
Lemma 13.
If (2), (10), (11), (22) and (27) hold and j ∈ N ( h ) ∩ N ( i ) , but (24)does not hold, then there exist states x, y, and z such that xV j y, yV j z, and (28) [ u hj ( y ) − y hj ( z )] / [ u hj ( x ) − u hj ( z )] = [ u ij ( y ) − u ij ( z )] / [ u ij ( x ) − u ij ( z )] . Proof.
Consider any states r and s such that rV j s. Define a function f : X → R by f ( t ) = u ij ( s ) + { [ u ij ( r ) − u ij ( s )] / [ u hj ( r ) − u hj ( s )] } [ u hj ( t ) − u hj ( s )] . Note that the denominator of the fraction in the expression is nonzero by lemma12, and that f ( t ) = u ij ( t ) for t = r, s. If (24) does not hold, then there must besome other state t such that f ( t ) = u ij ( t ) , and it is impossible that tE j r or tE j s. Let x, y, and z be r, s, and t arranged in order of decreasing interest afforded to j. The conclusion of the lemma is obtained by noting that[ u hj ( y ) − u hj ( z )] / [ u hj ( x ) − u hj ( z )] = [ f ( y ) − f ( z )] / [ f ( x ) − f ( z )] , that u ij ( t ) = f ( t ) for exactly one of the values t = x, y, z, and that [ a − c ] / [ b − c ]is strictly monotone in each of its variables on { ( a, b, c ) | a > b > c } . (cid:3) Lemma 14.
Suppose that (2), (6), (10), (11) and (27) hold that j ∈ N ( h ) ∩ N ( i ) , where h and i are distinct agents. Suppose that each of N ( h ) and N ( i ) contains anelement distinct from j. Then (24) holds with respect to j. Proof.
Consider any three states x, y, and z such that xV j y and yV j z. To simplifynotation, it will be assumed without loss of generality that j is neither h nor i. Let r h V h s h and r i V i s i , as guaranteed by (11). Choose any natural number n sufficientlylarge so that(29) u hj ( x ) − u hj ( z ) < n [ u hh ( r h ) − u hh ( s h )] and u ij ( x ) − u ij ( z ) < n [ u ii ( r i ) − u ii ( s i )] . Because the functions u hh and u ii are continuous and X is connected, there exist t h and t i such that u hj ( x ) − u hj ( z ) = n [ u hh ( t h ) − u hh ( s h )] and u ij ( x ) − u ij ( z ) = n [ u ii ( t i ) − u ii ( s i )] . Now, because u hj is continuous and X is connected, there existsa function η n : { , , . . . , n } → X such that, for m ≤ n, u hj ( η n ( m )) = u hj ( z ) + m [ u hj ( t h ) − u hj ( s h )] . (In particular, this implies that η n (0) E j z and η n ( n ) E j x. )Assumption (27) will now be used to show that, for all m ≤ n, u ij ( η n ( m )) = u ij ( z ) + m [ u ij ( t i ) − u ij ( s i )] . Since η n (0) E j z and η n ( n ) E j x, this is equivalent to ∀ m < n u ij ( η n ( m + 1)) − u ij ( η n ( m )) = u ij ( t i ) − u ij ( s i ) . Suppose, to the contrary,that u ij ( η n ( m +1)) − u ij ( η n ( m )) = u ij ( t i ) − u ij ( s i ) for some m < n. Without loss ofgenerality, suppose that u ij ( η n ( m + 1)) − u ij ( η n ( m )) < u ij ( t i ) − u ij ( s i ) . Then, forsome p, u ij ( η n ( p + 1)) − u ij ( η n ( p )) > u ij ( t i ) − u ij ( s i ) . (Otherwise u ij ( z ) − u ij ( x ) = P k < n [ u ij ( η n ( k + 1)) − u ij ( η n ( k ))] < n [ u ij ( t i − u ij ( s i )] . ) Now, using (10), states w h , w i , x h , x i , y h , y i , z h , and z i will be specified that satisfy the hypotheses of(27). Here, when a state is subscripted by ‘ g ’, the subscript takes both values h and i. The subscript ‘ f ’ takes all values except j and the value taken by g in the This entails that h and i are elements of N ( h ) and N ( i ) , respectively, that are distinct from j. In general, agents h ′ ∈ N ( h ) and i ′ ∈ N ( i ) will have to be specified that are distinct from j. same expression. (That is, ‘ f ’ can always take any value except h, i, and j, and itcan take whichever of { h, i } is not taken by g. ) Let q be a fixed, arbitrary state.Suppose that w g E j η n ( p ) , x g E j η n ( m ) , y g E j η n ( p + 1) , and z g E j η n ( m + 1); that w g E g t g , x g E g t g , y g E g s g , and z g E g s g ; and that w g E f q, x g E f q, y g E f q, and z g E f q. With respect to the states so specified, (27), fails to hold if (22) holds. Sincethis contradicts the hypothesis of the lemma, ∀ m ≤ n u ij ( η n ( m )) = u ij ( z ) +( m / n ) [ u ij ( x ) − u ij ( z )] . For any n large enough to satisfy (29), there is a natural number µ n < n satisfy-ing η n ( µ n + 1) V j y and yW j η n ( µ n ) . Thus both [ u hj ( y ) − u hj ( z )] / [ u hj ( x ) − u hj ( z )]and [ u ij ( y ) − u ij ( z )] / [ u ij ( x ) − u ij ( z )] are in the interval [ µ n / n, ( u n + 1) / n ) . Since n can be arbitrarily large, [ u hj ( y ) − u hj ( z )] / [ u hj ( x ) − u hj ( z )] = [ u ij ( y ) − u ij ( z )] / [ u ij ( x ) − u ij ( z )] . Therefore (24) holds, by lemma 13. (cid:3)
Lemma 15.
If conditions (2), (6), (10), (11), (22) and (27) hold, then condition(24) holds for all agents h, i, and j. Proof.
In view of lemma 14, (24) only has to be established now in the case that h = j, N ( j ) = { j } and j ∈ N ( i ) for some i = j. In that case, simply take u hj = u ij for some i satisfying j ∈ N ( i ). (cid:3) The foregoing lemmas immediately establish the following theorem.
Theorem 2.
Suppose that assumptions (1)–(6), (10)–(13), (25) and (27) are satis-fied, and that either (7) is satisfied or else X possesses the cartesian-product struc-ture described in lemma 7. Then Pareto superiority and liberal succession coincide. An alternative cardinality condition on preferences
Within many communities there is wide, if approximate agreement, regardingcardinal intrapersonal comparisons of interest. It is uncontroversial that someone’sinterest would be more significantly advanced if he were to move from a hovel toa decent house than if he were to move from the house to a mansion, for example.The prevalence of this kind of shared intuition about welfare contributes to theplausibility of condition (27).However, some people may find it less plausible that there is public agreementabout intrapersonal cardinal interest comparisons, than that there is public agree-ment about intrapersonal cardinal preference comparisons. For one thing, it isoften more obvious what people prefer than what is objectively good for them. Foranother, people who endorse expected-utility theory are already committed to ac-cept one form of the cardinality of persons’ preferences, even if they do not acceptthe cardinality of persons’ interests. People who hold these views may prefer toassume the publicity of intrapersonal cardinal preference comparisons rather thanto assume (27). It will be shown here that such an assumption can be formulatedin a way similar to (27), and that the formal assumption (together with the otheraxioms) implies the existence of common additive interest factors.Consider how to express the agreement of two agents, j and k, about intrap-ersonal cardinal preference comparisons concerning agent i. Suppose that ∀ h i, j } [ wI h y and xI h z ] , and that wE j y and xE j z. Make an analogous assumption I am indebted to Arthur Robson for stating the two considerations that I discuss now. with k substituted for j, but with respect to a different set of social states. Specif-ically, suppose that ∀ h
6∈ { i, k } [ w ′ I h y ′ and x ′ I h z ′ ] , and that w ′ E j y ′ and x ′ E j z ′ . Let i be indifferent between the corresponding social states mentioned in these twoassumptions. That is, let wI i w ′ , xI i x ′ , yI i y ′ , and zI i z ′ . Now suppose that xR j w, yR j z, and w ′ R k x ′ . By reasoning analogous to thatregarding (27), the two preferences of j imply that j must regard i ’s interest gainfrom being in x rather than w as being at least as great as i ’s interest gain frombeing in z rather than y would be. The preference of k implies that k must regard i ’s interest gain from being in x ′ rather than w ′ as being insufficient to outweighstrictly the net disadvantages of x ′ relative to w ′ for everyone else. (Note that thesewill be in terms of preference for everyone except k, and in terms of interest for k. ) Thus, if k agrees with j about intrapersonal cardinal preference comparisonsregarding i, then k should also regard i ’s interest gain from being in z ′ rather than y ′ as being insufficient to outweigh strictly the net disadvantages of z ′ relative to y ′ for everyone else. That is, it should be the case that y ′ R k z ′ . The implication from these various assumptions to their conclusion jointly consti-tute the assumption that intrapersonal cardinal preference comparisons are public: (30) Define j ∈ S ( i ) if and only if, for some states x and y, xE i y, ∀ h
6∈ { i, j } xI h y, and xP h y. If j ∈ S ( h ) ∩ S ( i ) , then following implications holds for allstates w h , w i , x h , x i , y h , y i , z h , and z i satisfying w h I j w i , x h I j x i , y h I j y i ,z h I j z i , ∀ g
6∈ { h, j } [ w h I g x h and y h I g z h ] , ∀ g
6∈ { i, j } [ w i I g x i and y i I g z i ] ,w h E h x h , y h E h z h , w i E i x i , and y i E i z i : if w h R h y h , z h R h x h , and y i R i w i , then z i R i x i . It seems very plausible that this condition can play an analogous role to (27)in guaranteeing the coincidence of Pareto superiority and liberal succession on thebasis of ordinal and topological assumptions.10.
Coincidence without separability or publicity
Theorem 2 and the alternative just suggested each hypothesize (a) product struc-ture of interests and separability of preferences in interests, and (b) public agree-ment regarding some form of intrapersonal cardinal comparison. The standardconstruction of nonpaternalistic preferences, by beginning with cardinal interest-representation functions as in theorem 1, yields a family of preferences that satisfyall of these qualitative hypotheses. Although the example provided in section 3 hasshown that the hypotheses of interest-determination and nonpaternalism alone aretoo broad to characterize the class of nonunanimous preference profiles on whichthe liberal principle and unanimity coincide, it might well be suspected that theadditional qualitative hypotheses would prove to be necessary as well as sufficientfor this coincidence. A counterexample that disproves this conjecture is now pro-vided. Let there be three agents, I = { , , } , and let X = R . Define(31) v i ( x ) = x i , This example will involve preferences that are represented by utility functions defined bytaking the minimum of several numbers. Such utility functions are pointwise limits of sequencesof CES utility functions. Every CES utility function represents a preference relation satisfying theseparability assumption. Thus, despite the example there may still be a close connection betweenseparability and the coincidence of the two welfare relations. and define(32) p i ( x ) = min { x i / , x j , x k } , where I = { i, j, k } . Then define W i and R i as in (14) and (15).It is evident from (31) and (32) that agents’ preferences are determined by in-terests, and that the agents are not unanimous. It will now be shown that theirpreferences are also nonpaternalistic and that the relation of liberal succession co-incides with that of Pareto improvement. Since the latter claim implies the former,only the coincidence of the two welfare orderings needs to be shown. Recall thatevery Pareto improvement is automatically a liberal successor, so it is sufficient toestablish the converse inclusion.Suppose, then, that y is a liberal successor of x. Specifically, suppose that(33) ∀ i ∈ C p i ( y ) ≥ p i ( x ) , that(34) ∀ j C y ≥ x, and that(35) p k ( x ) > p k ( y ) . By (35), y is not a Pareto improvement over x. This will be shown to be a contra-diction.By (32) and (35),(36) min { y i , y j , y k / } < min { x i , x j , x k / } . By (32) and (35), k C, so y k ≥ x k by (34). Therefore (36) implies that, for one ofthe other agents h ∈ { i, j } , (37) y h = min { y i , y j , y k / } < min { x i , x j , x k / } ≤ x h . Therefore h ∈ C by (34). But, by (32) and (37),(38) p h ( y ) ≤ y h / < min { x h / , x i , x j , x k } = p h ( x ) . Therefore h C by (33), so a contradiction has been reached. This establishesthat liberal succession coincides with Pareto improvement, and consequently thatall agents have nonpaternalistic preferences.However, agents’ preferences are not separable in interests, and cardinal inter-personal comparisons of neither preferences nor interests are public. To see thatthe preferences of agent 1 are inseparable, let J = { } and K = { , } , and de-fine w = (1 , , , x = (2 , , , y = (1 , , , and z = (2 , , . Then (4) fails tohold because, although its hypothesis holds, xP w while yR z. By the symmetry ofthe example, the preferences of the other two agents are not separable in interestseither. To show that intrapersonal cardinal interest comparisons are not public, let i = 1 and j = 2 in (27). Define w = z = (3 , ,
2) and x = y = (2 , , . Then thehypothesis of (27) holds, but wR x and not yR z, so the conclusion of (27) fails tohold. Therefore (27) (which is an implication) fails to hold.Intrapersonal cardinal preference comparisons can similarly be shown not to bepublic. To do so, let i = 1 , j = 2 , and k = 3 in (30), and define: w = z = (3 , , x = y = (2 , , w ′ = z ′ = (3 , , x ′ = y ′ = (2 , , { w, x, y, z } but hastotally “altruistic” preferences for the interest of agent 1 among { w ′ , x ′ , y ′ , z ′ } . From these considerations, it can easily be seen that (30) fails to hold.11.
Examples and conclusion
Strictly speaking, Mill’s principle concerns only the permissibility of private ac-tivities. Contemporary libertarians advocate that public authorities should nothave the power to take any action that Mill’s principle would prohibit to the coali-tion of persons who prefer the action. Such a position obviously has strong dis-tributive implications. An alternative view would hold that public authorities canlegitimately take some actions that would be prohibited to private persons andcoalitions, but would also recognize that public actions should be structured in away that will minimize “transactions costs.” On this view, the public authoritiesshould not take an action if some person or coalition can propose an alternativeaction that the liberal principle would endorse as a replacement for the initiallycontemplated action. Such a view implicitly defines a notion of “liberal efficiency”that is analogous to the familiar concept of Pareto efficiency.On this latter normative view about government action, there is a broad scopefor welfare analysis but the familiar Paretian analysis is foundationally inadequate.There are several issues in public finance, with respect to which such a view seemsto be widely held. One of these issues is whether redistributive policy ought tobe implemented by cash transfers or by transfers in kind (e.g., provision of publichousing). One of the common arguments in favor of cash transfers is that it isdemeaning to the recipients of transfers not to be granted autonomy over the useof the resources that society is willing to transfer. This argument makes animplicit appeal to the liberal principle. It is clear that some of the proponentsof the argument regard it as being conceptually distinct from the argument thattransfers in kind are inefficient, although those proponents (especially when theyare economists) typically produce the latter argument as a corollary. It might bethought that there could be transfer programs with respect to which the argumentabout recipients’ autonomy could be raised although cash transfers would not bePareto superior to transfers in kind. The results of this paper indicate the specialfeatures that such a program would have to possess. In cases where such featuresare absent, an efficient transfer never violates the preferences of recipients abouthow the resources dedicated to them should be used.It would be widely agreed that the assumptions about interests and preferencesthat have been discussed here are reasonable ones to make in the context of an Conversely, one of the arguments sometimes made in favor of transfers in kind is that someclasses of recipients (e.g., addicts) are incapable of exercising such autonomy. evaluation of transfers to competent adults. The one assumption from which therewould possibly be substantial dissent is the additive separability of preferences ininterests. As was shown in the preceding section, this assumption will not be satis-fied if people’s preferences reflect maximin considerations regarding the satisfactionof interests. However, the example studied in that section makes it plausible thatthe results proved here are robust to this specific kind of failure of the separabilityassumption. This is a matter for further research. If these results are indeed robust,then it will be fair to conclude that the welfare evaluation of transfers to competentadults can safely focus on efficiency questions to the exclusion of questions aboutliberal choice. Another issue concerns the regulation of markets for services such as educationor medical care. Because these services are differentiated products, it is conceiv-ably efficient to limit the number of product varieties in order to take advantage ofeconomies of scale. Suppose, for simplicity, that any feasible allocation involvingproduction of more than one product variety is Pareto dominated by some feasibleallocation in which only a single variety is produced. Thus, according to either theefficiency criterion or the liberal criterion, only an allocation with a single varietycan be optimal. The question is, which varieties may be produced in optimal allo-cations? Parallel to the case of transfers just discussed, the answer to this questionis the same for both criteria if the assumptions studied in this paper hold. Alsoparallel to that case, the appropriateness of separability and cardinality assump-tions is open to question on account of features of the situation to which the resultsof this paper plausibly are robust. However, in the case of differentiated-productallocation, there are also intrinsic difficulties about the publicity of intrapersonalcardinal interest comparisons. That is, the failure of the cardinality assumption tohold may not be simply a by-product of a violation of the separability assumption.To understand the nature of these possible intrinsic difficulties, consider some ofthe troublesome and divisive questions about resource allocation in education andmedicine. How much emphasis should schools give to the development of students’skills in reading and arithmetic, versus the development of critical thinking andappreciation of the arts? How much of medical research funding should be spenton finding cures for rare diseases that strike people early in life, versus commondiseases of older people? Debates about these questions have to do partly withdistributional issues, but they also reflect pronounced differences in the participants’views about the relative importance of disparate elements of the good life. That is,they reflect the fact that participants are (at least implicitly) expressing cardinaljudgments about the components of persons’ interests, and that they disagree aboutthose judgments although their ordinal judgments may be unanimous. It is not atall clear whether the results of this paper can be extended to cover situations thatpossess this complication.These examples illuminate the philosophical significance of the technical resultsthat have been derived here. These results constitute a limited defense of the study It might still be argued that there is something ethically preferable about letting recipientsmake their own choices, rather than making choices for them that are consistent with their pref-erences. Nevertheless, this argument does not seem as compelling as an argument that people’spreferences about matters that are their own business are actually being thwarted. For example,most people would judge that rather small reductions of administrative cost constitute sufficientgrounds to justify centralized allocation if the result is consistent with recipients’ preferences. of the unanimity relation in welfare analysis to address concerns that may funda-mentally be about respect for persons’ objective interests. This defense acknowl-edges the legitimacy of these concerns, but argues that they actually are addressed,even though implicitly, by the technically simpler efficiency analysis. FollowingMill, a theoretical concept of a person’s non-preferential interest has been intro-duced in order to relate the specification of rights and liberties to the specificationof preferences. While there may be many possible ways to give a substantive def-inition of this concept, only a few qualitative assumptions about the concept areneeded in order to make the defense. For the most part, these assumptions seemto be appropriate for the discussion of distributional issues and of other issues withwhich applied welfare analysis typically deals. A formal example has shown thatthe assumption that is likely to be regarded as the most restrictive one, separability,may be stronger than is actually needed. The two economic examples that havejust been discussed show that, even if this conjecture is true, the limited defenseof the unanimity criterion falls short of showing that explicit consideration of theliberal criterion would be completely dispensable for applied welfare analysis. References
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