Normative foundations of public Intervention. General normative evaluation X, a set of mutually exclusive social states (complete descriptions of all.

Normative foundations of public Intervention

General normative evaluationX, a set of mutually exclusive social states (complete descriptions of all relevant aspects of a society)N a set of individuals N = {1,..,n} indexed by iExample 1: X= +n (the set of all income distributions)Example 2: X = +nm (the set of all allocations of m goods (public and private) between the n-individuals. Ri a preference ordering of individual i on X (with asymmetric and symmetric factors Pi and Ii).Ordering: a reflexive, complete and transitive binary relation.x Ri y means individual i weakly prefers state x to state yPi = strict preference, Ii = indifference Basic question (Arrow (1950): how can we compare the various elements of X on the basis of their social goodness ?

General normative evaluationArrows formulation of the problem. = (R1 ,, Rn) a profile of preferences. the set of all binary relations on X , the set of all orderings on XD n, the set of all admissible profilesGeneral problem (K. Arrow 1950): to find a collective decision rule C: D that associates to every profile of individual preferences a binary relation R = C()x R y means x is at least as good as y when individuals preferences are ()

Examples of normative criteria ?1: Dictatorship of individual h: x R y if and only x Rh y (not very attractive)2: ranking social states according to an exogenous code (say the Charia). Assume that the exogenous code ranks any pair of social alternatives as per the ordering (x y means that x (women can not drive a car) is weakly preferable to y (women drive a car). Then C()= for all profiles (). Notice that even if everybody in the society thinks that y is strictly preferred to x, the social ranking states that x is better than y.

Examples of collective decision rules3: Unanimity rule (Pareto criterion): x R y if and only if x Ri y for all i. Interesting but deeply incomplete (does not rank alternatives for which individuals preferences conflict) 4: Majority rule. x R y if and only if #{i N: x Ri y} #{i N :y Ri x}. Widely used, but does not always lead to a transitive ranking of social states (Condorcet paradox).

the Condorcet paradox

the Condorcet paradoxIndividual 1Individual 2Individual 3

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranois

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarine

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarineFranoisMarineNicolas

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarineFranoisMarineNicolasA majority (1 and 3) prefers Marine to Nicolas

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarineFranoisMarineNicolasA majority (1 and 3) prefers Marine to NicolasA majority (1 and 2) prefers Nicolas to Franois

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarineFranoisMarineNicolasA majority (1 and 3) prefers Marine to NicolasA majority (1 and 2) prefers Nicolas to FranoisTransitivity would require that Marine be socially preferred to Franois

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarineFranoisMarineNicolasA majority (1 and 3) prefers Marine to NicolasA majority (1 and 2) prefers Nicolas to FranoisTransitivity would require that Marinene be socially preferred to Franois but.

the Condorcet paradoxIndividual 1Individual 2Individual 3MarineNicolasFranoisNicolasFranoisMarineFranoisMarineNicolasA majority (1 and 3) prefers Marine to NicolasA majority (1 and 2) prefers Nicolas to FranoisTransitivity would require that Marine be socially preferred to Franois but.A majority (2 and 3) prefers strictly Franois to Marine

Example 5: Positional BordaWorks if X is finite. For every individual i and social state x, define the Borda score of x for i as the number of social states that i considers (weakly) worse than x. Borda rule ranks social states on the basis of the sum, over all individuals, of their Borda scoresLet us illustrate this rule through an example

Borda ruleIndividual 1Individual 2Individual 3MarineNicolasJean-LucFranoisNicolasFranoisJean-LucMarineFranoisMarineNicolasJean-Luc

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8Sum of scores Jean-Luc = 5

Borda ruleIndividual 1Individual 2Individual 3Marne 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8Sum of scores Jean-Luc = 5Nicolas is the best alternative, followed closely by Marineand Franois. Jean-Luc is the worst alternative

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8Sum of scores Jean-Luc = 5Problem: The social ranking of Franois, Nicolas and Marinedepends upon the position of the (irrelevant) Jean-Luc

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8Sum of scores Jean-Marie = 5Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc belowMarine for 2 changes the social ranking of Marine and Nicolas

Borda ruleIndividual 1Individual 2Individual 3Marine 4Nicolas 3Jean-Luc 2Franois 1Nicolas 4Franois 3Jean-Luc 2Marine 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8Sum of scores Jean-Luc = 5Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc belowMarine for 2 changes the social ranking of Marine and Nicolas

Borda ruleIndividual 1Individual 2Individual 3Marine 4Jean-Luc 3Nicolas 2Franois 1Nicolas 4Franois 3Marine 2Jean-Luc 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 8Sum of scores Nicolas = 9Sum of scores Franois = 8Sum of scores Jean-Luc = 5Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc belowMarine for 2 changes the social ranking of Marine and Nicolas

Borda ruleIndividual 1Individual 2Individual 3Marine 4Jean-Luc 3Nicolas 2Franois 1Nicolas 4Franois 3Marine 2Jean-Luc 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 9Sum of scores Nicolas = 8Sum of scores Franois = 8Sum of scores Jean-Luc = 5Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc belowMarine for 2 changes the social ranking of Marine and Nicolas

Borda ruleIndividual 1Individual 2Individual 3Marine 4Jean-Luc 3Nicolas 2Franois 1Nicolas 4Franois 3Marine 2Jean-Luc 1Franois 4Marine 3Nicolas 2Jean-Luc 1Sum of scores Marine = 9Sum of scores Nicolas = 8Sum of scores Franois = 8Sum of scores Jean-Luc = 5The social ranking of Marine and Nicolas depends upon the individual ranking of Nicolas vs Jean-Luc or Marine vs Jean-Luc

Are there other collective decision rules ?Arrow (1951) proposes an axiomatic approach to this problemHe proposes five axioms that, he thought, should be satisfied by any collective decison ruleHe shows that there is no rule satisfying all these propertiesFamous impossibility theorem, that throw a lot of pessimism on the prospect of obtaining a good definition of general interest as a function of the individual interest

Five desirable properties on the collective decision rule1) Non-dictatorship. There exists no individual h in N such that, for all social states x and y, for all profiles , x Ph y implies x P y (where R = C() 2) Collective rationality. The social ranking should always be an ordering (that is, the image of C should be ) (violated by the unanimity (completeness) and the majority rule (transitivity)3) Unrestricted domain. D = n (all logically conceivable preferences are a priori possible)

Five desirable properties on the collective decision rule4) Weak Pareto principle. For all social states x and y, for all profiles D , x Pi y for all i N should imply x P y (where R = C() (violated by the collective decision rule coming from an exogenous tradition code)5) Binary independance from irrelevant alternatives. For every two profiles and D and every two social states x and y such that x Ri y x Ri y for all i, one must have x R y x R y where R = C() and R = C(). The social ranking of x and y should only depend upon the individual rankings of x and y.

Arrows theorem: There does not exist any collective decision function C: D that satisfies axioms 1-5

All Arrows axioms are independentDictatorship of individual h satisfies Pareto, collective rationality, binary independence of irrelevant alternatives and unrestricted domain but violates non-dictatorshipThe Tradition ordering satisfies non-dictatorship, collective rationality, binary independance of irrelevant alternative and unrestricted domain, but violates ParetoThe majority rule satisfies non-dictatorship, Pareto, binary independence of irrelevant alternative and unrestricted domain but violates collective rationality (as does the unanimity rule)The Borda rule satisfies non-dictatorship, Pareto, unrestricted domain and collective rationality, but violates binary independence of irrelevant alternativesWell see later that there are collective decisions functions that violate unrestricted domain but that satisfies all other axioms

Escape out of Arrows theoremNatural strategy: relaxing the axioms It is difficult to quarel with non-dictatorship We can relax the assumption that the social ranking of social states is an ordering (in particular we may accept that it be incomplete)We can relax unrestricted domainWe can relax binary independance of irrelevant alternativesShould we relax Pareto ?

Should we relax the Pareto principle ? (1)Most economists, who use the Pareto principle as the main criterion for efficiency, would say no! Many economists abuse of the Pareto principle Given a set A in X, say that state a is efficient in A if there are no other state in A that everybody weakly prefers to a and at least somebody strictly prefers to a.Common abuse: if a is efficient in A and b is not efficient in A, then a is socially better than b Other abuse (potential Pareto) a is socially better than b if it is possible, being at a, to compensate the loosers in the move from b to a while keeping the gainers gainers!Only one use is admissible: if everybody believes that x is weakly better than y, then x is socially weakly better than y.

Illustration: An Edgeworth BoxABxB2xA1xA2xB1xyz21

Illustration: An Edgeworth BoxABxB2xA1xA2xB1xyzx is efficientz is not efficient

Illustration: An Edgeworth BoxABxB2xA1xA2xB1xyzx is efficientz is not efficientx is not socially better than z asper the Paretoprinciple

Illustration: An Edgeworth BoxABxB2xA1xA2xB1xyzy is better thanz as per the Pareto principle

Should we relax the Pareto principle ? (2)Three variants of the Pareto principle Weak Pareto: if x Pi y for all i N, then x P yPareto indifference: if x Ii y for all i N, then x I yStrong Pareto: if x Ri y for all i for all i N and x Ph y for at least one individual h, then x P y A famous critique of the Pareto-principle: When combined with unrestricted domain, it may hurt widely accepted liberal values (Sen (1970) liberal paradox).

Sen (1970) liberal paradox (1)Minimal liberalism: respect for an individual personal sphere (John Stuart Mills)For example, x is a social state in which Mary sleeps on her belly and y is a social state that is identical to x in every respect other than the fact that, in y, Mary sleeps on her backMinimal liberalism would impose, it seems, that Mary be decisive (dictator) on the ranking of x and y.

Sen (1970) liberal paradox (2)Minimal liberalism: There exists two individuals h and i N, and four social states w, x,, y and z such that h is decisive over x and y and i is decisive over w and z Sen impossibility theorem: There does not exist any collective decision function C: D satisfying unrestricted domain, weak pareto and minimal liberalism.

Proof of Sens impossibility resultOne novel: Lady Chatterleys lover2 individuals (Prude and Libertin)4 social states: Everybody reads the book (w), nobody reads the book (x), Prude only reads it (y), Libertin only reads it (z), By liberalism, Prude is decisive on x and y (and on w and z) and Libertin is decisive on x and z (and on w and y)By unrestricted domain, the profile where Prude prefers x to y and y to z and where Libertin prefers y to z and z to x is possibleBy minimal liberalism (decisiveness of Prude on x and y), x is socially better than y and, by Pareto, y is socially better than z. It follows by transitivity that x is socially better than z even thought the liberal respect of the decisiveness of Libertin over z and x would have required z to be socially better than x

Sen liberal paradoxShows a problem between liberalism and respect of preferences when the domain is unrestrictedWhen people are allowed to have any preference (even for things that are not of their business), it is impossible to respect these preferences (in the Pareto sense) and the individuals sovereignty over their personal sphereSen Liberal paradox: attacks the combination of the Pareto principle and unrestricted domainSuggests that unrestricted domain may be a strong assumption.

Relaxing unrestricted domain for Arrows theorem (1)One possibility: imposing additional structural assumptions on the set XFor example X could be the set of all allocations of l goods (l > 1) accross the n individuals (that is X = nl)In this framework, it would be natural to impose additional assumptions on individual preferences.For instance, individuals could be selfish (they care only about what they get). They could also have preferences that are convex, continuous, and monotonic (more of each good is better)Unfortunately, most domain restrictions of this kind (economic domains) do not provide escape out of the nihilism of Arrows theorem.

Relaxing unrestricted domain for Arrows theorem (2)A classical restriction: single peakednessSuppose there is a universally recognized ordering of the set X of alternatives (e.g. the position of policies on a left-right spectrum)An individual preference ordering Ri is single-peaked for if, for all three states x, y and z such that x y z , x Pi z y Pi z and z Pi x y Pi x A profile is single peaked if there exists an ordering for which all individual preferences are single-peaked. Dsp n the set of all single peaked profilesTheorem (Black 1947) If the number of individuals is odd, and D = Dsp then there exists a non-dictatorial collective decision function C: D satisfying Pareto and binary independence of irrelevant alternatives. The majority rule is one such collective decision function.

Single peaked preference ?leftrightJean-LucFranoisNicolasSingle-peaked

Single peaked preference ?leftrightJean-LucFranoisNicolasNot Single-peaked

Comments on Black theorem Widely used in public economicsIn any set of social states where each individual has a most preferred state, the social state that beats any other by a majority of vote (Condorcet winner) is the most preferred alternative of the individual whose peak is the median of all individuals peaks (median voter theorem)Notice the odd restriction on the number of individuals

Even with single-peaked preferences, the majority rule is not transitive if the number of individuals is evenIndividual 1Individual 2Individual 3Jean-LucFranoisNicolasFranoisJean-LucNicolasNicolasFranoisJean-LucIndividual 4NicolasFranoisJean-LucJean-Luc is weakly preferred, socially, to Nicolas Nicolas is weakly preferred, socially, to Franois but Jean-Luc is not weakly preferred, socially, to FranoisPreferences are single peaked (on the left-right axe)

Domain restrictions that garantees transitivity of majority votingSen and Pattanaik (1969) Extremal Restriction conditionA profile of preferences satisfies the Extremal Restriction condition if and only if, for all social states x, y and z, the existence of an individual i for which x Pi y Pi z must imply, for all individuals h for which z Ph x, that z Ph y Ph x. Theorem (Sen and Pattanaik (1969). A profile of preferences satisfies the extremal restriction condition if and only if the majority rule defined on this profile is transitive. See W. Gaertner Domain Conditions in Social Choice Theory, Cambridge University Press, 2001.

Relaxing Binary independence of irrelevant alternativesJustification of this axiom: information parcimoniousnessDe Borda rule violates itIn economic domains, there are various social orderings who violate this axiom but satisfy all the other Arrows axiomsAn example: Aggregate consumers surplus

Aggregate consumers surplus ?X = +nl (set of all allocations of consumption bundles)xi +l individual is bundle in xRi, a continuous, convex, monotonic and selfish ordering on +nl Selfishness means that for all i N, w, x, y and z in +nl such that wi = xi and yi = zi, x Ri y w Ri zSelfishness means that we can view individual preferences as being only defined on +l

Aggregate consumers surplus ?Individuals live in a perfectly competitive environmentIndividual i faces prices p =(p1,.,pl) and wealth wi. B(p,wi)={x +l p.x wi } (Budget set)Individual ordering Ri on +l induces the dual (indirect) ordering RDi of all prices/wealth configurations (p,w) +l+1 as follows: (p,w) RDi (p,w) for all xB(p,w), there exists x B(p,w) for which x Ri x.Ui: +l , a numerical representation of Ri (Ui(x) Ui(y) x Ri y) (such a numerical representation exists by Debreu (1954) theorem; it is unique up to a monotonic transform)Vi: +l+1 a numerical representation of RDi Vi(p,wi) = the maximal utility achieved by i when facing prices p +l and having a wealth wiProblem of applied cost-benefit analysis: ranking various prices and wealth configurations

Aggregate consumers surplus ?A money-metric representation of individual preferencesFor every prices configuration p +l and utility level u, define E(p,u) by:

E(p,u) associates, to every utility level u, the minimal amountof money required at prices p, to achieve that utility level. This (expenditure) function is increasing in utility (given prices).It provides therefore a numerical representation (in money units) of individual preferences.

Aggregate consumers surplus ?Gives the amount of money needed at prices p to achieve the level of satisfaction associated to prices q and wealth w . Direct money metric:Gives the amount of money needed at prices p to be as well-off as with bundle x Indirect money metric:money metric utility functions depend upon reference prices

Aggregate consumers surplus ?Hicksian (compensated) demand functions (depends uponunobservable utility level)These money metric utilities are connected to observabledemand behaviorMarshallian (ordinary) demand functions

Aggregate consumers surplus ?Six important identities (valid for every p +l, w + and u ):(1)(2)(3)(4)(5)Roys identity(6)Sheppards Lemma

Aggregate consumers surplus ?

Aggregate consumers surplus ?identity (1)

Aggregate consumers surplus ?Recurrent application of Sheppards lemma

A one good, one price illustrationpricequantityHicksian demandpjpjSurplus= area pjabpjni=1xHij(p1,,pj-1,pj,pj+1,,pl,ui)ni=1xHij(p1,,pj-1,pj,pj+1,,pl,ui)ab

Aggregate consumers surplus ?Usually done with Marshallian demand (rather than Hicksian demand)Marshallian surplus is not a correct measure of welfare change for one consumer but is an approximation of two correct measures of welfare change: Hicksian surplus at prices p and Hicskian surplus at prices p (Willig (1976), AER, consumers surplus without apology).Widely used in applied welfare economics

Is the ranking of social states based on the sum of money metric a collective decision rule?It violates slightly the unrestricted domain condition (because it is defined on all selfish, convex, monotonic and continuous profile of individual orderings on +nl but not on all profiles of orderings (unimportant violation)).It satisfies non-dictatorship and ParetoIt obviously satisfies collective rationality if the reference prices used to evaluate money metric do not changeIt violates binary independence of irrelevant alternatives (prove it).Ethical justification for Aggregate consumers surplus is unclear

Normative evaluation with individual utility functionsWhat does it mean to say that Bob prefers social state x to social state y ? Economic theory is not very precise in its interpretation of preferences A preference is usually considered to be an ordering of social states that reflects the individuals objective or interest and which rationalizes individuals choiceMore precise definition: preferences reflects the individuals well-being (happiness, joy, satisfaction, welfare, etc.)What happens if one views the problem of defining general interest as a function of individual well-being rather than individual preferences ?Philosophical tradition: Utilitarianism (Beccaria, Hume, Bentham): The best social objective is to achieve the maximal aggregate happiness.

What is happiness ?Objective approach: happiness is an objective mental state Subjective approach: happiness is the extent to which desires are satisfiedSee James Griffin Well being: Its meaning, measurement and moral importance, London, Clarendon 1988Can happiness be measured ?Can happiness be compared accross individuals ?If the answers given to these two questions are positive, how should we aggregate individuals happinesses ?

Can we measure happiness ? (1)Suppose Ri is an ordering of social states according to is well-being. Can we get a measure of this happiness ?In a weak ordinal sense, the answer is yes (provided that the set X is finite or, if X is some closed and convex subset of +nl , if Ri is continuous (Debreu (1954))Let Ui: X be a numerical representation of Ri Ui is such that, for every x and y in X, Ui(x) Ui(y) x Ri yOrdinal measure of happiness

Can we measure happiness ? (2)Ordinal measure of happiness: defined up to an increasing transform. Definition: g: A (where A ) is an increasing function if, for all a, b A, a > b g(a) > g(b)If Ui is a numerical representation of Ri, and if g: is an increasing function, then the function h: X defined by: h(x) = g(U(x)) is also a numerical representation of RiExample : if Ri is the ordering on +2 defined by: (x1,x2) Ri (y1,y2) lnx1 + lnx2 lny1 + lny2 , then the functions defined, for every (z1,z2), by:U(z1,z2) = lnz1 + lnz2G(z1,z2) = e U(z1,z2) = elnz1elnz2 = z1z2H(z1,z2) = -1/G(z1,z2) = -1/(z1z2) all represent numerically Ri

Can we measure happiness ? (3)The three functions of the previous example are ordinally equivalent. Definition: Function U is said to be ordinally equivalent to function G (both functions having X as domain) if, for some increasing function g: , one has U(x) = g(G(x)) for every x XRemark: ordinal equivalence is a symmetric relation, because if g : is increasing, then its inverse is also increasing.Ordinal measurement of well-being is weak because all ordinally equivalent functions provide the same information about this well-being.

Can we measure happiness ? (4)Ordinal notion of well-being does not enable one to talk about changes in well-being.For example a statement like I get more extra happiness from my first beer than from my second is meaningless with ordinal measurement of well-being. proof: let a, b and c be the alternatives in which I drink, respectively, no beer, one beer and two beers.

Can we measure happiness ? (4)Ordinal notion of well-being does not enable one to talk about changes in well-being.For example a statement like I get more extra happiness from my first beer than from my second is meaningless with ordinal measurement of well-being. proof: let a, b and c be the alternatives in which I drink, respectively, no beer, one beer and two beers. If U is a function that measures ordinally my happiness, the statement I get more extra happiness from the first beer than from the second writes: U(b)-U(a) > U(c) U(b) U(b) > [U(c)+U(a)]/2.

Can we measure happiness ? (4)Ordinal notion of well-being does not enable one to talk about changes in well-being.For example a statement like I get more extra happiness from my first beer than from my second is meaningless with ordinal measurement of well-being. proof: let a, b and c be the alternatives in which I drink, respectively, no beer, one beer and two beers. If U is a function that measures ordinally my happiness, the statement I get more extra happiness from the first beer than from the second writes: U(b)-U(a) > U(c) U(b) U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved by a monotonic transformation.

Can we measure happiness ? (4)Ordinal notion of well-being does not enable one to talk about changes in well-being.For example a statement like I get more extra happiness from my first beer than from my second is meaningless with ordinal measurement of well-being. proof: let a, b and c be the alternatives in which I drink, respectively, no beer, one beer and two beers. If U is a function that measures ordinally my happiness, the statement I get more extra happiness from the first beer than from the second writes: U(b)-U(a) > U(c) U(b) U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved by a monotonic transformation. U(b) > [U(c)+U(a)]/2 being true does not imply that g(U(b)) > [g(U(c))+g(U(a))]/2 is true for every increasing function g: .

Can we measure happiness ? (4)Ordinal notion of well-being does not enable one to talk about changes in well-being.For example a statement like I get more extra happiness from my first beer than from my second is meaningless with ordinal measurement of well-being. proof: let a, b and c be the alternatives in which I drink, respectively, no beer, one beer and two beers. If U is a function that measures ordinally my happiness, the statement I get more extra happiness from the first beer than from the second writes: U(b)-U(a) > U(c) U(b) U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved by a monotonic transformation. U(b) > [U(c)+U(a)]/2 being true does not imply that g(U(b)) > [g(U(c))+g(U(a))]/2 is true for every increasing function g: . For example, having 3 > (4+1)/2 does not imply having 33 > (43+13)/2

Can we measure happiness ? (5)Stronger measurement of well-being: cardinal.Suppose U: X and G: X are two measures of well-being. We say that they are cardinally equivalent if and only if there exists a real number a and a strictly positive real number b such that, for every x X, U(x) = a + bG(x). We say that a cardinal measure of well-being is unique up to an increasing affine transform (g: is affine if, for every c , it writes g(c) = a + bc for some real numbers a and b > 0Statements about welfare changes make sense with cardinal measurementIf U(x)-U(y) > U(w)-U(z), then (a+bU(x)-(a+bU(y)) = b[U(x)-U(y)] > b[U(w)-U(z)] (if b > 0) = (a + bU(w)-(a+bU(z))

Can we measure happiness ? (6)Example of cardinal measurement in sciences: temperature. Various measures of temperature (Kelvin, Celsius, Farenheit) Suppose U(x) is the temperature of x in Celcius. Then G(x) = 32 + 9U(x)/5 is the temperature of x in Farenheit and H(x) = -273 + U(x) is the temperature of x in KelvinWith cardinal measurement, units and zero are meaningless but a difference in values is meaningful.

Can we measure happiness ? (7)Measurement can even more precise than cardinal. An example is age, which is what we call ratio-scale measurable. If U(x) is the age of x in years, then G(x) = 12U(x) is the age of x in months and H(x) = U(x)/100 is the age of x in centuries. Zero matters for age. A ratio scale measure keeps constant the ratio. Statements like my happiness today is one third of what it was yesterday are meaningful if happiness is measured by a ratio-scaleFunctions U: X and G: X are said to be ratio-scale equivalent if and only if there exists a strictly positive real number b such that, for every x X, U(x) = bG(x).

Can we measure happiness ? (8)Notice that the precision of a measurement is a decreasing function of the size of the class of functions that are considered equivalent. Ordinal measurement is not precise because the class of functions that provide the same information on well-being is large. It contains indeed all functions that can be obtained from another by mean of an increasing transform.Cardinal measurement is more precise because the class of functions that convey the same information than a given function is restricted to those functions that can be obtained by applying an affine increasing transformRatio-scale measurement is even more precise because equivalent measures are restricted to those that are related by a increasing linear function.

Can we measure happiness ? (9)What kind of measurement of happiness is available ? Ordinal measurement is easy: you need to observe the individual choosing in various circumstances and to assume that her choices are driven by the pursuit of happiness. If choices are consistent (satisfy revealed preferences axioms), you can obtain from choices an ordering of all objects of choice, which can be represented by a utility functionCardinal measurement seems plausible by introspection. But we havent find yet a device (rod) for measuring differences in well-being (like the difference between the position of a mercury column when water boils and its position when water freezes).Ratio-scale is even more demanding: it assumes the existence of a zero level of happiness (above you are happy, below you are sad). Not implausible, but difficult to find. Level at which an individual is indifferent between dying and living ?

Can we define general interest as a function of individuals well-being ?As before, we assume that there are n individualsUi: X a (utility) function that measures individual is well-being in the various social states(U1 ,, Un): a profile of individual utility functionsthe set of all logically conceivable real valued functions on XDU n the domain of plausible profiles of utility functions A social welfare functional is a mapping W: DU that associates to every profile (U1 ,, Un) of individual utility functions a binary relation R = W(U1,,Un))Problem: how to find a good social welfare functional ?

Examples of social welfare functionalsUtilitarianism: x R y iUi(x) iUi(y) where R = W(U1,,Un) x is no worse than y iff the sum of happiness is no smaller in x than in yVenerable ethical theory: Beccaria, Bentham, Hume, Stuart Mills.Max-min (Rawls): x R y min (U1(x),, Un(x)) min (U1(y),, Un(y)) where R = W(U1,,Un) x is no worse than y if the least happy person in x is at least as well-off as the least happy person in y

Contrasting utilitarianism and max-minu2u1u1 = u2utility possibility set

Contrasting utilitarianism and max-minu2u1u1 = u2uuuu-1Utilitarian optimum

Contrasting utilitarianism and max-minu2u1u1 = u2uuuu-1Rawlsian optimum

Contrasting utilitarianism and max-minu2u1u1 = u2Rawlsian optimumUtilitarian optimumBest feasible egalitarian outcome

Contrasting utilitarianism and Max-minMax-min and utilitarianism satisfy the weak Pareto principle (if everybody (including the least happy) is better off, then things are improving).Max-min is the most egalitarian ranking that satisfies the weak Pareto principleMax-min does not satisfy the strong Pareto principle (Max min does not consider to be good a change that does not hurt anyone and that benefits everybody except the least happy person)Utilitarianism does not exhibit any aversion to happiness-inequality. It is only concerned with the sum, no matter how the sum is distributed

Examples of social welfare functionalsUtilitarianism and Max-min are particular (extreme) cases of a more general family of social welfare functionalsMean of order r family (for a real number r 1) x R y [iUi(x)r]1/r [iUi(y)r]1/r if r 0 and x R y ilnUi(x) ilnUi(y) otherwise (where R = W(U1,,Un)) If r =1, UtilitarianismAs r -, the functional approaches Max-minr 1 if and only if the functional is weakly averse to happiness inequality.

Mean-of-order r functionalu2u1u1 = u2r=1r=0

Mean-of-order r functionalu2u1u1 = u2r=1r=0r =-

Mean-of-order r functionalu2u1u1 = u2r=1r=0r =-r=+

Mean-of-order r functionalu2u1u1 = u2r=+Max-max indifferencecurve

Extension of Max-minMax-min functional does not respect the strong Pareto principleThere is an extension of this functional that does: Lexi-min (due to Kolm (1972)Lexi-min: x R y There exists some j N such that U(j)(x) U(j)(y) and U(j)(x) = U(j)(y) for all j < j where, for every z X, (U(1)(z),,U(n)(z)) is the (ordered) permutation of (U1(z)Un(z)) such that U(j+1)(z) U(j)(z) for every j = 1,,n-1 (R = W(U1,,Un))

Information used by a social welfare functionalWhen defining a social welfare functional, it is important to specify the information on the individuals utility functions used by the functionalIs individual utility ordinally measurable, cardinally measurable, ratio-scale measurable ?Are individuals utilities interpersonally comparable ?

Information used by a social welfare functional (ordinal)A social welfare functional W: DU uses ordinal and non-comparable (ONC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = gi(Gi) for some increasing functions gi: (for i = 1,n), one has W (U1,Un) = W(G1,,Gn) A social welfare functional W: DU uses ordinal and comparable (OC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = g(Gi) for some increasing function g: (for i = 1,n), one has W (U1,Un) = W(G1,,Gn)

Information used by a social welfare functional (cardinal)A social welfare functional W: DU uses cardinal and non-comparable (CNC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = aiGi+bi for some strictly positive real number ai and real number bi (for i = 1,n), one has W (U1,Un) = W(G1,,Gn) A social welfare functional W: DU uses cardinal and unit-comparable (CUC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = aGi+bi for some strictly positive real number a and real number bi (for i = 1,n), one has W (U1,Un) = W(G1,,Gn) A social welfare functional W: DU uses cardinal and fully comparable (CFC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = aGi+b for some strictly positive real number a and real number b (for i = 1,n), one has W (U1,Un) = W(G1,,Gn)

Information used by a social welfare functional (ratio-scale)A social welfare functional W: DU uses ratio-scale and non-comparable (RSNC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = aiGi for some strictly positive real number ai (for i = 1,n), one has W (U1,Un) = W(G1,,Gn) A social welfare functional W: DU uses ratio-scale and comparable (RSC) information on individual well-being iff for all (U1,Un) and (G1,,Gn) DU such that Ui = aGi for some strictly positive real number a (for i = 1,n), one has W (U1,Un) = W(G1,,Gn)

Information used by a social welfare functional There are some connections between these various informational invariance requirements Specifically, ONC CNC CUC CFC RSFC and, similarly, OFC CFC and CUC CFC. On the other hand, it is important to notice that CUC does not imply nor is implied by OFC. What information on individuals well-being are the examples of welfare functional given above using ?

Information used by a social welfare functional Max-min, Max-max, lexi-min, lexi-max are all using OFC information. Utilitarianism: uses CUC informationMean of order r: uses RSC information. Under various informational assumptions, can we obtain sensible welfare functionals ?

Desirable properties on the Social Welfare functional1) Non-dictatorship. There exists no individual h in N such that, for all social states x and y, for all profiles (U1,,Un) DU, Uh(x) > Uh(y) implies x P y (where R = W(U1,,Un)) 2) Collective rationality. The social ranking should always be an ordering (that is, the image of W should be ) 3) Unrestricted domain. DU = n (all logically conceivable combinations of utility functions are a priori possible)

Desirable properties on the Social Welfare Functional4a) Strong Pareto. For all social states x and y, for all profiles (U1,,Un) DU , Ui(x) Ui(y) for all i N and Uh(x) > Uh(y) for some h should imply x P y (where R = W(U1,,Un)) 4b) Pareto Indifference. For all social states x and y, for all profiles (Ui,,Un) DU , Ui(x) = Ui(y) for all i N implies x I y (where R = W(U1,,Un)) 5) Binary independance from irrelevant alternatives. For every two profiles (U1,,Un) and (U1,,Un) DU and every two social states x and y such that Ui(x) = Ui(x) and Ui(y) = Ui(y) for all i, one must have x R y x R y where R = W(U1,,Un)) and R = W(U1,,Un))

Welfarist lemma: If a social welfare functional W satisfies 2, 3, 4b and 5, there exists an ordering R* on n such that, for all profiles (U1,,Un) DU, x R y (U1(x),,Un(x)) R* (U1(y),,Un(y)) (where R = W(U1,,Un))

Welfarist lemmaQuite powerful: The only information that matters for comparing social states is the utility levels achieved in those statesRanking of social states can be represented by a ranking of utility vectors achieved in those states. This lemma can be used to see whether Arrows impossibility result is robust to the replacement of information on preference by information on happinessAs can be guessed, this robustness check will depend upon the precision of the information that is available on individuals happiness.

Arrows theorem remains if happiness is not interpersonnaly comparable Theorem: If a social welfare functional W: DU satisfies conditions 2-5 and uses CNC or ONC information on individuals well-being, then W is dictatorial.Proof: Diagrammatic (using the welfarist theorem, and illustrating for two individuals)

Illustrationu1u2u

Illustrationu1u2uAu

Illustrationu1u2uABu

Illustrationu1u2uABuC

Illustrationu1u2uABuCD

Illustrationu1u2uABuCDBetter thanu by Pareto

Illustrationu1u2uABuCDBetter thanu by ParetoWorse thanu by Pareto

Illustrationu1u2uABuDBetter thanu by ParetoWorse thanu by ParetoBy NC, all pointsin C are ranked in thesame way vis--vis u

Illustrationu1u2uABuDBetter thanu by ParetoWorse thanu by Paretoab

IllustrationThe social ranking of a =(a1,a2) and u=(u1,u2) must be the same than the social ranking of (1a1+1, 2a2+2) and (1u1+1, 2u2+2) for every numbers i > 0 and i (i = 1, 2).Using i = (ui-bi)/(ui-ai) > 0 and i = ui(bi-ai)/(ui-ai), this implies that the social ranking of b = (1a1+1, 2a2+2) and u = (1u1+1, 2u2+2) must be the same than the social ranking of a and u

Illustrationu1u2uABuDBetter thanu by ParetoWorse thanu by Paretoab

Illustrationu1u2uABuBetter thanu by ParetoWorse thanu by Paretoaball points hereare also rankedin the same way vis--vis u

Illustrationu1u2uABuBetter thanu by ParetoWorse thanu by Paretoaball points hereare also rankedin the same way vis--vis u by Pareto, a and bcan not be indifferent to u(and to themselves)by transitivity)

Illustrationu1u2uABuCDby NC, the(strict) ranking of region Cvis--vis u mustbe the oppositeof the (strict) ranking of D vis--vis u

Illustrationu1u2uABuCD

Illustrationu1u2uABuDdc

IllustrationThe social ranking of c =(c1,c2) and u =(u1,u2) must be the same than the social ranking of (1c1+1, 2c2+2) and (1u1+1, 2u2+2) for every numbers i > 0 and i (i = 1, 2).Using i = (di-ui)/(ui-ci) > 0 and i = (u2i-dici)/(ui-ci), this implies that the social ranking of u = (1c1+1, 2c2+2) and d = (1u1+1, 2u2+2) must be the same than the social ranking of c and uIf c is above u, d is below u and if c is below u, d is above u

Illustrationu1u2uABuWorseBetterBetter thanu by ParetoWorse thanu by Pareto

Illustrationu1u2uABuWorseBetter

Illustrationu1u2uABuWorseBetterIndividual 1is the dictator

Illustrationu1u2uABuBetterWorseBetter thanu by ParetoWorse thanu by Pareto

Illustrationu1u2uABuBetterWorse

Illustrationu1u2uABuBetterWorseIndividual 2Is the dictator

Moral of this storyArrows theorem is robust to the replacement of preferences by well-being if well-being can not be compared interpersonally (notice that cardinal measurability does not help if no interpersonal comparison is possible) What if well-being is ratio-scale measurable and interpersonnally non-comparable ?Welfarist theorem gives nice geometric intuition on whats going on, see Blackorby, Donaldson and Weymark (1984), International Economic ReviewGeneralization to n individuals is easy

Allowing ordinal comparabilityA strengthening of non-dictatorship: AnonymityA social welfare functional W is anonymous if for every two profiles (U1,,Un) and (U1,,Un) DU such that (U1,,Un) is a permutation of (U1,,Un), one has R = R where R = W(U1,,Un)) and R = W(U1,,Un))Dictatorship of individual h is clearly not anonymous.Hence, by virtue of the previous theorem, there are no anonymous social welfare functionals that use ON or CN information on individuals well-being and that satisfy axioms 2)-5). We will now show that this impossibility vanishes if we allow for ordinal comparisons of well-being accross individuals. Specifically, we are going to show that if we allow the social welfare functional to use OC information on individual well-being, then the only anonymous social welfare functionals are positional dictatorships

Positional dictatorshipA social welfare functional W is a positional dictatorship if there exists a rank r {1,,n} such that, for every two social states x and y, and every profile (U1,,Un) of utility functions U(r)(x) > U(r)(y) x P y where R = W(U1,,Un)) and, for every z X, (U(1)(z),,U(n)(z)) is the ordered permutation of (U1(z),Un(z)) satisfying U(i)(z) U(i+1)(z) for every i = 1,,n-1Max-min and Lexi-min are positional dictatorships (for r = 1). So is Max-max (r = n). Another one would be the dictatorship of the smallest integer greater than or equal to n/2 (median) Positional dictatorship rules only specify the social ranking that prevails when the positional dictator has a strict preference. They dont impose anything on the social ranking when the positional dictator is indifferent.Hence, positional dictatorship does not enable a distinction between lexi-min and max-min.

A new theorem:Theorem: A social welfare functional W: DU is anonymous, satisfies conditions 2-5 and uses OC information on individuals well-being if and only if W is a positional dictatorship.

Remarks on this theoremIf we drop anonymity, we get other kinds of dictatorships (including non-anonymous ones)Proof of this result is straightforward, but cumbersome (see Gevers, Econometrica (1979) and Roberts R. Eco. Stud. (1980).Max dictatorship is not very appealing. Can we eliminate it ?Yes if we impose an axiom of minimal equity, due to Hammond (Econometrica, 1976)A social welfare functional W satisfies Hammonds minimal equity principle if for every profile (U1,,Un) and every two social states x and y for which there are individuals i and j such that Uh(x) = Uh(y) for all h i, j, and Uj(y) > Uj(x) > Ui(x) > Ui(y), one has x P y where R = W(U1,,Un))

The Lexi-min theorem:Theorem: A social welfare functional W: DU is anonymous, satisfies conditions 2-5, uses OC information on individuals well-being and satisfies Hammonds equity principle if and only if it is the Lexi-min .

Further remarks on Lexi-minIt is not a continuous ranking of alternativesMaxi-min by contrast is continuous (even thought it violates the strong Pareto principle)Suppose we replace in the previous theorem strong Pareto by weak Pareto, and that we add continuity, can we get Maxi-min ?

Continuity ?

u1u2u(.)u1u2u2u1= u(1)= u(2)uu2 = u1Continuity ?betterworsebetterworseWe go continuously from the better

u1u2u(.)u1u2u2u1= u(1)= u(2)uu2 = u1Continuity ?betterworsebetterworse

u1u2u(.)u1u2u2u1= u(1)= u(2)uu2 = u1Continuity ?betterworsebetterworseto theworse

u1u2u(.)u1u2u2u1= u(1)= u(2)uu2 = u1Continuity ?betterworsebetterworseWithout encounteringindifference

ContinuityA social welfare functional W satisfying 2,3, 4a and 5 is continuous if for every profile (U1,,Un), the welfarist ordering R* of n that corresponds to R by the welfarist theorem is continuous where R = W(U1,,Un)) An ordering R* of n is continuous if, for every u n, the sets NWR*(u) = {u n: u R* u} and NBR*(u) = {u n: u R* u} are both closed in n

Bad news ? Theorem 1: There are no anonymous and continuous social welfare functionals W: DU that use OC information on individuals well-being and that satisfy collective rationality, weak Pareto, Pareto-indifference, unrestricted domain, binary independance and Hammonds equity if n > 2Theorem 2: If n = 2, an anonymous and continuous social welfare functional W: DU using OC information on individuals well-being satisfies collective rationality, weak Pareto, Pareto-indifference, unrestricted domain, binary independance and Hammonds equity if and only if it is the max-min Hence, no characterization of max-min in this setting.

Cardinal measurability and unit comparabilityTheorem: An anonymous social welfare functional W: DU satisfies conditions 2-5 and uses CUC information on individuals well-being if and only if it is utilitarian.

Remarks on this utilitarian theoremNo need of continuityIf anonymity is dropped, then asymmetric utilitarianism emerges (social ranking R is defined by: x R y iNiUi(x) iNiUi(y) for some non-negative real numbers i (i = 1,,n) (numbers are strictly positive if strong Pareto is satisfied).Notice that if weak Pareto only is required (some i can be zero), this family of social orderings contains standard dictatorship (which is not surprising)

Other axiomatic justifications of utilitarianismMaskin (1978). Uses CFC along with continuity and a separability condition (independence with respect to unconcerned individuals)Harsanyi (1953) impartial observer theorem. Society is looked at from behind a veil of ignorance. We must choose a social state without knowing in which shoes we are going to be, but by assuming an equal chance of being in anybodys shoesIf the social planner who looks at society from behind this veil of ignorance has Von-Neuman Morgenstern preferences, it should order social state on the basis of the expected utility of being anyoneThis argument is flawed

Generalized utilitarianismUtilitarianism is insensitive to utility inequalityA social ranking that is more general than utilitarianism is, as we have seen, the mean of order rBut one could also consider a more general family of social rankings: symmetric generalized utilitarianismx R y ig(Ui(x)) ig(Ui(y)) where R = W(U1,,Un) for some increasing function g: Mean of order r is a special case of this where g is defined by g(u) = u1/r if r > 0, g(u) = ln(u) if r = 0 and g(u) = -u1/r if r < 0

Generalized utilitarianismA new axiom: Independence with respect to unconcerned individualsA ranking of two states should be independent from the utility function of the individuals who are indifferent (unconcerned ?) between the two statesA social welfare functional satisfies independence with respect to unconcerned individuals if, for all profiles (U1,,Un) of utility functions and all social states w, x, y and z X, the existence of a group G of individuals such that Ug(w) = Ug(x) and Ug(y) = Ug(z) for all g G and Uh(w) = Uh(y) and Uh(x) = Uh(z) for all h N\G implies that w R x y R z where R = W(U1,,Un))

Generalized utilitarianismTheorem: An anonymous social welfare functional W: n satisfies Pareto-indifference, strong Pareto, continuity, independence with respect to unconcerned individuals and binary independence of irrelevant alternative if and only if it is a generalized utilitarian rankingProof: See Blackorby, Bossert and Donaldson, Population Issues in Social Choice Theory, Welfare economics and Ethics, Cambrige University Press, 2005, theorem 4.7

Remarks on this theorem (1)Does not ride on measurability assumption on well-beingDoes not restrict the g function.A way to restrict the g function is to impose utility inequality aversion property on the social rankingAn example of inequality aversion: Hammonds weak equity principleAnother example (weaker than Hammonds): Pigou-Dalton principle of equityA social welfare functional W satisfies the Pigou-Dalton equity principle if for every profile (U1,,Un) and every two social states x and y for which there are individuals i and j and a number > 0 such that Uh(x) = Uh(y) for all h i, j, and Uj(x) = Uj(y) - Ui(x) = Ui(y) + , one has x P y where R = W(U1,,Un))

Remarks on this theorem (2)Both equity principles incorporate implicitly interpersonnal comparability and measurability assumptions on well-beingUtility levels must be compared accross individuals to make sense of Hammonds equity principles.Utility differences of between two individuals must also be meaningful in order for the Pigou-Dalton equity principle of transfer to make senseHammonds equity implies Pigou-Dalton equity but not vice-versaPigou-Dalton equity leads to a significant restriction of the g function: concavity g is concave if, for all numbers u and v and every number [0,1], one has g(u+(1-)v) g(u)+(1-)g(v)

ug(v)vg(u)aIXbg(u) + (1-)g(v)u+(1-)vg(x)g(u+(1-)v)Concavity?

Equity respectful Generalized utilitarianismTheorem: An anonymous social welfare functional W: +n satisfies Pareto-indifference, strong Pareto, continuity, independence with respect to unconcerned individuals, binary independence of irrelevant alternative and Pigou-Dalton equity principle if and only if x R y ig(Ui(x)) ig(Ui(y)) where R = W(U1,,Un) for some increasing and concave function g: n

Ratio-scale comparabilityRequires a meaning to be given to zero levels of happinessA negative happiness is not the same thing then a positive one.Suppose that we restrict the domain DU of admissible profiles of utility functions to n+ where + is the set of all functions U: X +

Ratio scale comparabilityTheorem: An anonymous social welfare functional W: +n satisfies Pareto-indifference, strong Pareto, continuity, independence with respect to unconcerned individuals, binary independence of irrelevant alternative and RSFC if and only if it is the mean of order r rankingProof: See Blackorby and Donaldson, International Economic Review (1982), theorem 2

Some issues with variable populationWe have been so far assuming that the number of individuals is fixed.Yet there are many normative issues that require the comparison of societies with different numbers of membersIs it good to add new people to actual societies (demographic policies) ?With varying numbers of individuals, defining general interest as a function of individual interest becomes trickyHow can someone compares her well-being with the situation in which she does not come to existence ?

Some issues with variable populationProblem (under welfarism and anonymity): Comparing utility vectors of different dimensions.(u1,,um) a vector of utilities in a society with m persons; (v1,,vn) a vector of utilities in a society with n persons (remember that individuals name does not matter under anonymity)X = nnfor all u X, n(u) is the dimension of u (number of people)How should we compare these vectors ?

Some issues with variable populationClassical utilitarianism u RCU v n(u)i=1 ui n(v)i=1 vi Critical level utilitarianism u RCLU v n(u)i=1 (ui c(u)) n(v)i=1 (vi c(v)) where c is a critical utility level which in general depends upon the distribution of utilities)Average Utilitarianism u RAU v (n(u)i=1 ui)/n(u) (n(v)i=1 vi)/n(v)Note: AU (c(u) =(n-1)(u)) and CU (c=0) are particular cases of CL

Some issues with variable populationClassical utilitarianism : Generates the repugnant conclusion (Parfitt, reason and persons, 1984). For any positive level of well-being , however small, it is always possible to improve upon the current state by packing the earth with people even if these people only enjoy level of utilityAverage Utilitarianism: Avoids the repugnant conclusion, because adding people is good only if their well-being is above the average. See Blackorby, Bossert, Donaldson: Population issues in Social Choice Theory, Wefare Economics, and ethics, Cambridge U.Press, 2005.

Collective decision with asymmetric information So far, we have assumed that the information needed to make collective decision (in our setting, on individual preferences or utility functions) is available to the public autority.Yet, one of the main difficulty of public economics is that the public authority does not have the information.What are people preferences for police protection, etc. ?Question: how can the public authority decides when it does not know peoples preference ?

Collective decision making under asymmetric informationX : universe of social statesA, a subset (menu) of XD n, the set of all admissible preference profiles.A social choice correspondence is a mapping C: D A that associates to every preference profile (R1 ,, Rn) D a set C (R1 ,, Rn) of socially optimal alternatives in A. A social choice correspondence is called a social choice function if # C (R1 ,, Rn) =1 for all (R1 ,, Rn) D.

Example of a social choice correspondence that is not a functionX = nl+ (set of all allocations of l goods accross n individuals) A = {x nl+ : x1j++xnj j for j = 1,,l} for some l+ (an Edgeworth box)D: the set of all selfish, continuous, monotonically increasing and convex preference profiles.Pareto correspondence: C: D A defined by: C (R1 ,, Rn) = {x A : z Pi x for some i h N s. t. x Ph z}. The Pareto correspondence selects all allocations in A that are Pareto-efficient for the preference profile (R1 ,, Rn). This set of allocations depends of course upon the preference profile (R1,, Rn).

Example of a social choice function (1)A = {Franois, Marine, Nicolas} A social choice correspondence: two-rounds (or runoff) voting.1st round: select the two alternatives that are the favorite ones of the largest number of individuals.2nd round: select the alternative that beats the other by a majority of votes (in both rounds, unlikely ties are broken by an exogenous device). For example, assume n = 5 and suppose that the profile (R1, R2, R3, R4 ,R5) is as follows.

Example of a social choice function (2)Then C(R1, R2, R3, R4 ,R5) = Nicolas Indeed, in the first round, Nicolas and Marineare the options which receive the most vote

Example of a social choice function (2)R1FranoisNicolas MarineR2NicolasMarineFranois R3NicolasFranoisMarineR4MarineFranoisNicolasR5MarineFranoisNicolasThen C(R1, R2, R3, R4 ,R5) = Nicolas Indeed, in the first round, Nicolas and Marine areare the options which receive the most vote

Example of a social choice function (2)R1FranoisNicolas MarineR2NicolasMarineFranois R3NicolasFranoisMarineR4MarineFranoisNicolasR5MarineFranoisNicolasThen C(R1, R2, R3, R4 ,R5) = Nicolas Indeed, in the first round, Nicolas and Marine areare the options which receive the most vote In the 2nd round, Nicolas beats Marine by 3/5 ofthe votes.

A difficulty with a social choice function (or correspondence) (1)It assumes that the profile of preferences is known. Yet this knowledge is not typically available. Individuals know (presumably) their preferences, but the institution in charge of conducting policies doesntProblem: individuals may have incentive to hide their true preference (and thus to manipulate the social choice function).This possibility is clear in the two-round election given earlier.

A difficulty with a social choice function (or correspondence) (2)The two-round electoral system is easily manipulableIndividuals 4 and 5 dislike heavily Nicolas (given theirtrue preference).Suppose one of them (4 say) lies and claim (by hisVote) that his favorite candidate is Franois.

A difficulty with a social choice function (or correspondence) (2)The two-round electoral system is easily manipulableIndividuals 4 and 5 dislike heavily Nicolas (given theirtrue preference).Suppose one of them (4 say) lies and claims (by hisvote) that his favorite candidate is Franois.

A difficulty with a social choice function (or correspondence) (2)The two-round electoral system is easily manipulableIndividuals 4 and 5 dislike heavily Nicolas (given theirtrue preference).Suppose one of them (4 say) lies and claims (by hisvote) that his favorite candidate is Franois.Then Franois and Nicolas will go to the 2nd round.

A difficulty with a social choice function (or correspondence) (2)The two-round electoral system is easily manipulableIndividuals 4 and 5 dislike heavily Nicolas (given theirtrue preference).Suppose one of them (4 say) lies and claims (by hisvote) that his favorite candidate is Franois.Then Franois and Nicolas will go to the 2nd round. And Franois will win!!

The social choice function underlying the two-round French electoral system is manipulableDefinition: A social choice function C: D A is manipulable at a profile (R1,,Rn) D if their exists some individual i N and a preference Ri such that (R1, Ri,, Rn) D and C(R1, Ri,,Rn) Pi C(R1, Ri,,Rn).In words, a social choice function is manipulable at a profile of preferences if, at this profile, one individual would benefit from announcing another preference than the preference he/she has in this profile. The two-round electoral system discussed above was manipulable at the considered profile.Q: Can we expect social choice function to never be manipulable ?

Gibbard-Satterthwaite theoremDefinition 1: A social choice function C: D A is dictatorial if there exists some individual h N such that, for all profiles (R1,,Rn) D, x Ph y y C(R1,,Rn).Definition 2: A social choice function C: D A is trivial if C(R1,,Rn) = C(R1,,Rn) for all profiles (R1,,Rn) and (R1,,Rn) in D.Theorem: If #A 3, any non-dictatorial and non-trivial social choice function C: n A is manipulable on at least one profile in n.

Illustration: robust measurement of inequalitiesSo far, we have been quite abstract. Public policy evaluation is described by means of social welfare functionals, or collective decesion rules, or social choice functions.Let us illustrate how these abstract tools can be used to evaluate in practice policies.Focus: policies that affect the distribution of individual observable attributes (income, health, education, etc.).

ExampleComparing 12 OECD countries (+ India) based on their distribution of disposable income and some public goods (based on Gravel, Moyes and Tarroux (Economica (2009))Sample of some 20 000 households in each country (1998-2002)Disposable income: income available after all taxes and social security contributions have been paid and all transfers payment have been receivedIncomes are made comparable across households by equivalence scale adjustmentIncomes are made comparable across countries by adjusting for purchasing power differences

Chart1

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Australia

France

Germany

Italy

Spain

sweden

UK

USA

India

individual rank

disposable income

moyenneVingtile

AustraliaAustriaCanadaFranceGermanyItalyPortugalSpainSwedenSwitzerlandUKUSAAustraliaAustriaCanadaFranceGermanyItalyPortugalSpainswedenSwitzerlandUKUSAIndia

1er vingtile219852342513474541922171171716424209559134293173473368154285617058553554254627475808867948985403789

2e vingtile726883956057759575184938337538517408117676368763492371073089779555100126575460254079056146158598110251019

3e vingtile8660100578070894393866162421449508639138227969100101179512850119351179312024805961107045105401733410883146871168

4e vingtile981411404988310167106386987499058639474154089227120391458014725143381344113229943875498646119821980613337181421309

5e vingtile111021238811313112991181877235731660310177167111029513807173771658816839150921485710933866610113133712204415854215811462

6e vingtile1248713313125571228812230839564887487109031795711470155682045618665194941696616614126291002811656147232455418579252061649

7e vingtile1384114219137381315612830906672218261116361924812754172772420320921223821916918376147691141513639161472769621574293871859

8e vingtile1531915230149391372613627981078789031123272036313920190062846724042259552238221221173421393016535181403209525188348192167

9e vingtile16731161071617814601144641056983659764130332143915174206923459228069309582683425201207431811320968210913825430190433732694

10e vingtile180241706817500155831525111296896610463137092264916533224705453738539444574017539217311743204735457308186184949022790304735

11e vingtile19599181591883516471161931213997611118914391238921782224260

12e vingtile213121917120152174601703513120102951212315056252161933626152

13e vingtile232372021021619184891783214246109371304815719267522073628222

14e vingtile251692163323145198501891915292118931423116575286402241330552

15e vingtile272922308824969214952041516632130811562617530308512409533271

16e vingtile296432499626941232682202718052147791744418750333392628036367

17e vingtile327252706129392253522384819712165911958020126364212859840460

18e vingtile364592907632524283162655321774196342235622056400863178246286

19e vingtile422913155537582331563036825301246012864824976460813690756446

20me vingtile6678445523513324719448066370463949442266366607761661136101615

LG

GermanyAustraliaAustriaSpainFranceItalyPortugalSwedenSwitzerlandUKUSACanada

1er vingtile210.4523105818110.0709348906261.774055431182.1893304983241.8955088522109.093471677486.8476440467210.6350409248280.8861576913171.715704607158.6369757152125.6529130492

2e vingtile585.4133705905473.1473751992681.950047159278.4056393982614.6198917297355.1683662183253.6569467569580.8731448747870.2279622136490.0921373758540.4708874101428.737897312

3e vingtile1055.6143179258906.55043431241187.8853439323521.27771443341061.9038391472664.568392163465.00305562411015.7618352371558.5449878619889.29321501421041.0121859342831.9746887143

4e vingtile1585.81853113331396.63198312121758.9054526943814.52930282541571.3388831811014.6912849762722.88312938371486.74058811492331.76244904911350.74521800811642.87714764491326.0747270451

5e vingtile2396.52523029961953.08909251732377.97280676661144.41647974722136.19426801031399.14400337351006.0867133541995.59238977943163.55163439091863.92818077522333.08856900911892.062897341

6e vingtile2780.65024849052576.56215939883044.30994134821518.74527844232749.8509466181818.1507842041328.75682831542540.26021659764068.47445225762438.70578822123111.58473336492519.537820457

7e vingtile3421.95714138713274.13386105673750.50928877271932.56916824133492.24215669072270.89530720941684.92599887943124.36342844625033.58825028273076.7830705533975.41560310473206.6259470423

8e vingtile4104.37423366744034.22881467324521.97054560022388.39692708264089.41465400122761.79668896312079.7971342413742.28324138066047.55223345373772.61966860544926.6337759813953.8408927088

9e vingtile4827.60591839424869.76051460455317.79857226172872.77322954684819.51569699813290.19985390922495.30157662684389.84473462927113.08518327914531.10197791555961.28105585094762.2524518432

10e vingtile5589.75623672815772.06709461876168.2569793813395.28179254345598.8633034033854.50088700312945.57424460285078.72110466148244.3220008595358.33466737887083.66249841085639.3908486251

11e vingtile6400.4465347946751.96204487787075.87485879023952.99206955116423.12907710564461.42894015113436.54745786995797.2596262649443.64025627696254.58133693388296.77903747716578.9214699306

12e vingtile7251.93840078427815.82281275168034.59126408934560.73223164137297.40603710065117.34465119383949.63535773156547.794474137910699.68887198327212.99747126499605.09812396667587.6162499626

13e vingtile8143.03542402888978.84044595179056.27125687795213.29561651878219.71659990995831.70997708244491.8256332667337.08676256112041.41450673688249.846725691811016.09979724758668.1855370854

14e vingtile9089.505909360210237.043869208810128.92715025525924.8916521329212.3403303156595.39454273345087.53137576688163.920977317513468.94451461489370.057237002812543.45326791949825.5351909545

15e vingtile10114.419642709811603.182382280511283.60022827486704.662551609210290.10856811937425.3682616855740.10218187749039.575079149115010.2965852110575.336233923314206.887081251811072.9172785475

16e vingtile11212.776671743713086.247949153712571.23681727827593.263420674711451.55582831518331.13570820916497.53187937499974.936706430616678.331894721611898.47496243116025.551383304512420.4274985199

17e vingtile12403.307621490914729.728213938514243.6475719128553.536818830412716.80653511819316.90947634027307.772436700410983.550228963318502.538352312613321.638565694918048.680394682513894.9773135441

18e vingtile13738.308809566916540.354117353415342.2877423699672.462538449914135.221331020410405.70527761258288.519956010412086.281431923920502.490470221314907.712091815420362.812764832715514.9759510168

19e vingtile15248.66467232718657.419037051516880.207454247911568.059388884415790.025331285211726.4543318739529.384252002813334.426944993822814.098594847516753.571872630423185.37160053117394.4313086762

20me vingtile17646.612643856421991.92289514819156.216470904812994.039503428518148.336577604113490.249551875511484.189607250715163.777653902326680.280894069419806.261136680128265.031582678619960.2755720128

LR

GermanyAustraliaAustriaSpainFranceItalyPortugalSwedenSwitzerlandUKUSACanada

0.050.01192593250.00500506190.01366522750.00632515630.01332879780.00808683870.00756236590.01389067060.01052785610.00866976880.00561248180.0062951492

0.10.03317426310.02151459780.03559941220.02142564210.03386645870.02632778330.02208749210.0383066250.03261689660.02474430350.01912153840.021479558

0.150.05981965710.04122197220.06201043640.04011667920.05851246120.04926286870.04049071560.06698606760.05841561390.04489960060.03683039170.0416815232

0.20.08986532220.06350658780.09181904240.06268484120.08658308030.0752166430.06294594170.09804552810.0873964730.068197890.0581240160.0664356923

0.250.13580653010.08880938250.12413582870.08807241810.11770744160.10371520540.08760624370.13160258840.1185726510.09410802820.08254328540.0947914216

0.30.15757416480.11715947590.15892020980.11688014940.15152082590.1347751780.11570314260.16752159490.15248994070.12312802360.11008601650.126227607

0.350.19391580760.1488789260.19578549320.14872735820.19242767190.16833604880.14671701330.20604123190.18866324050.15534396170.14064783870.1606503846

0.40.23258708720.18344138590.23605760320.18380711610.22533275360.20472539650.18110090530.24679095980.22666748740.19047611470.17430137170.1980854863

0.450.27357125220.22143404820.27760171640.2210839230.26556239340.24389466190.2172814680.28949545650.26660458380.22877119240.21090657690.2385865082

0.50.31676086220.26246304710.32199766530.26129532630.30850559110.28572495060.25648951690.33492453010.3090043180.27053741390.25061576450.282530711

0.550.36270114070.30702008540.36937747440.30421579590.35392384580.33071507850.29924161610.38230972250.35395580330.31578808810.29353510590.3296007335

0.60.41095356640.35539515350.41942474790.35098648350.40209779040.37933654460.3439193790.43180496470.40103359160.36417764170.33982265670.3801358465

0.650.46145034110.40827900720.47275887020.40120669290.4529184570.43229074120.39113126710.48385612940.45132262870.41652721170.38974305640.4342718369

0.70.51508502470.46549107680.52875405570.45596995840.50761348240.48890085520.44300308070.53838305760.50482768780.47308561530.44377991340.4922544859

0.750.5731649380.52761108870.58903073290.51597984980.56700009520.5504248260.499826490.59612949260.56259889630.53393904890.5026312120.5547477157

0.80.63540674340.59504791880.65624842130.58436511750.63099754510.61756720480.56578061680.65781343770.62511830220.60074311250.56697447290.6222573157

0.850.70287186960.66977900410.7435522350.65826618560.70071471730.69064026140.63633331450.72432809820.69349113770.67259734050.63855157360.696131537

0.90.77852384970.75211040870.80090386150.74437687650.77887145580.77135009530.72173311650.79704950230.76845107260.75267674140.72042420.7772926729

0.950.86411284590.84837597540.88118692330.89025890570.87005358670.86925407030.82978290830.87936048980.85509214410.84587251260.82028465220.871452463

1111111111111

What are these data saying on justice ?Except for the 10% poorest, americans in every income group have larger income than French, swedish and German. Does that mean that US is a better society than France, Sweden or Germany? Americans in every income group have larger income than British, Australians, Italians, spanish and Indians. Does that mean that US is a better society than UK, Australia, Italy, Spain or India ?It would seem so if income was the only relevant attribute. But is that so ?

Another attribute: regional infant mortalityInfant mortality (number of children who die before the age of one per thousand births) is a good indicator of the overall working of the medical system of the region where individuals live How do countries compare in terms of the different infant mortality rate that they offer to their citizens on the basis of their place of residence ?

Rev moy quantile

MOYENNE DES VINGTILES

AustraliaAustriaCanadaFranceGermanyItalyPortugalSpainSwedenSwitzerlandUKUSA

0.052193.22491343675234.28923629672512.87914752614710.63270944994185.00618747022163.34742766531705.33570127181641.35710278434208.90817190735574.54871411973425.86378720743172.5973958608

0.17266.72547432538394.82095682136056.72984527737582.85204078347514.59251426474932.4270188663369.65760837173840.40926409267407.846082476311756.00977288896366.6935132097633.9403299423

0.158659.565952593410053.83725636558070.08035399698942.06004290919384.66802755096160.54274972134212.24838650884939.9313974788635.577859344613815.51515361637967.50282007810010.2425778298

0.29813.360850883211401.38464258439883.217557336410165.126009187510636.11326924146982.00234456464980.51034300125862.50842301439471.253152895415404.04894769659223.703575103312039.2159777786

0.2511101.502360589712383.909439631611312.914649750711297.054792078811762.35403914147719.58659511155706.99864076636602.005566071910176.067798731116708.240086835410293.583662615113806.7242940419

0.312484.917585102913306.918922626912557.396077947612285.570061331912139.38048002198393.87337031976472.35951310677486.326725306310902.230269752117951.462977114911470.405865511515567.6527584101

0.3513834.050405787514216.586530950913737.551074037413129.335006089812829.8894961919065.14792838727211.20891350958260.458729085611634.823967000519238.020470100812752.67234195917276.7494948451

0.415311.532638284815222.477410698514938.798176187913692.486812220513626.99676270049809.80849976037875.15926505159025.726156466412323.216274877720356.210977114113918.238796279119006.4186668876

0.4516730.115386888216101.881392835616177.63215141514600.65643735614463.955137100210568.85254921348363.1313446699759.928176357313030.693136055321434.381673868615171.044029574820691.5418976071

0.518023.123170624817068.343355549717500.091865726115582.961093808715250.79547648711295.68280774158963.753329201310462.305528255313707.700723736222647.385782856316530.1716018222469.6761777948

0.5519597.210170877918158.658078122218835.115076891616471.426337598516191.680619433112138.59271128559756.26966598811187.372181254814388.103176543423890.226460522117813.096833729424260.3563039291

0.621310.637631492219171.393353251420151.429839031817459.706038113517034.16781174913119.996972550410290.184072542212121.557217272715054.058815247525214.299406339219328.991321272726151.7630664396

0.6523237.464200210420204.845817903421618.194541220718488.602740553717831.24439819514244.493580720410933.185678391913044.967539731415716.252115649226749.496868855920735.557039657828222.1178281859

0.725168.703642676721624.737033941923144.964767301619848.873528472218919.252658473415289.685093666511892.880451187814228.313500131516571.685129526328637.046454356822413.252869970730551.6273332091

0.7527290.065337580823083.724090836524968.953097205421492.448309963420410.25882081316630.823450119913081.066342194315623.736202868117527.468084806330850.526053524324095.481362774933270.7704199661

0.829638.942426392124971.830969452426941.16551838323263.599430602322021.733135056818049.874458678814762.123310444317424.130549200118748.91790018333339.425207388926271.943581742736367.0542308756

0.8532710.848531579126823.059362543729392.341156993425349.683013812623848.498295721419707.043860489516570.311712500919562.032427695820124.342588741736418.016687176228584.858001789840459.8529604859

0.936446.769380879728776.006043865132523.879052032728313.726838380526543.802371639721767.707782181319631.079430967522352.524551613422054.245860424140081.834076923431774.783853798646285.5545850766

0.9542285.951299253631531.122498626637581.888899907133151.862000829130350.554236013725212.29072061424567.737047057627845.127852787624972.914883758746065.425106262436904.507474019256443.3273027838

166762.044545505745523.154955763451332.053531710947194.196509299448066.024570478236647.379624499939383.473597230838847.338268248936645.022165713977524.75098049661136.1130823335101604.021359068

MOYENNE DES CENTILES

AustraliaAustriaCanadaFranceGermanyItalyPortugalSpainSwedenSwitzerlandUKUSA

0.01-4733.35756018742272.1896521273-311.98462107921634.80647989781517.500355904322.3581497765436.7065267055166.15264064171578.1680894692-5313.15286971021250.3416872804-981.8177564154

0.021527.27711153984512.37143278351770.21553624793865.89291139513296.65742260481230.64952171891140.423926664977.02290491423463.97038484155984.85348013262557.39240918232199.4232893676

0.033645.70911703245732.62779906212863.44388315655300.76783139494597.86612264712480.59669066291861.2303727951889.1835119964570.24119705527924.10556421183690.15849762193915.4809225286

0.044836.20801456456599.75266230653730.92973110426094.96328043735440.20064224693129.67011183042379.91862107832435.76498772535402.06001346889300.11062305384471.6891153424972.347892863

0.055722.5514343577105.60518467354523.80907796816668.99319131266083.25973773723660.86060877132708.98225259372756.41822985356051.970410966810062.53525446955166.28094253225772.0286085167

0.066495.95997901597653.91642294595167.35637724466965.76906355656581.18028931214244.48696867333004.60536510573265.90108835066631.269349218410705.61255105665618.54135261056493.7018395363

0.076893.30073779938041.0233564475658.37178164217319.72517396277099.11228013364576.11404597033301.70132430523630.06303264297090.850174477411321.25216243715978.30537547777120.198668504

0.087319.24322547188432.03174819496062.70477321647622.74630724027473.68817057414961.7156808993430.75909262763925.61974552017473.273658793211816.70428584446347.84068118377665.8915185988

0.097678.5361847038770.24638974066481.29083099377858.42105892767993.03583207065325.57132623493537.34702891074132.82802138977800.553700823112260.92580447296766.15038495548204.6574613659

0.17944.48352139099082.97110761716912.16152344718154.22091076438419.51139474755557.32564324093572.09956231364250.80407874538052.938693544812679.97232401057127.72192397148686.5003953582

0.118166.95785516729436.80954358767313.4254387538435.12117032498711.30283618065859.17883459733788.91788973394465.35391546858292.368364598613082.42233784037422.91751607599147.4087519738

0.128408.39890113419786.01897437677661.97730374728714.42617375749086.17888697516123.08606245014004.34326826144697.85225730098462.209151650213474.50210796367705.85587145619581.9057196508

0.138658.39373834910046.33040788858043.19181585828924.25127810229496.17369213826174.67614615844229.3109203624976.76027235528640.055223091713853.78394793077980.208134284510010.8893368564

0.148920.355580553710364.07414328461.20080534689181.22225748529720.35570673556227.10386565864428.09204077145202.6527135028807.731989747514170.15222172938256.76246358610450.5303340991

0.159147.105205215210639.57864832748867.73525341769454.7538734679913.37843258216429.37306180614615.18406455625353.20365895688978.023064622614501.71590666658473.304001919610862.3567629356

0.169360.757713535710919.08997808769227.29975279239720.048415361610105.88562837126639.14578451184763.15358711785534.1007219019182.413165965814811.49102126838758.698153233611292.0675454166

0.179560.39047929511234.95210758249557.30014267969905.450245123910409.45581256556819.99049341724912.89200891735702.31520779389316.114124130615177.25953474099041.309646474911680.1582292129

0.189796.487326335611418.24897707379890.623085464610137.202479074110663.81979278876983.41655955735011.38760359165893.08457878689481.439579441915423.00744929869227.192112514712069.8582378121

0.1910051.412615747711624.693742153210234.853926970410409.194074384310880.18331978737122.53589431615066.92728707166024.75256341129628.98276613315671.83500101869437.790767528212411.6811242713

0.210298.800690901611794.018054974110507.416879288610652.547252950311123.85918364747342.88145930095152.44210305866157.27693777849752.639528169815936.26943051139656.513830975612741.5194953914

0.2110569.656801712911962.982393942810777.382428413610833.787791043211401.90161273537489.15171301325334.47266155066287.04073386859911.423361269416211.87431070399834.850295803213090.0116020874

0.2210830.287850084612180.319724106811057.202214190311111.783258640811630.73158221297558.22591311095561.85510337126423.44321613710056.665106477816474.303190896210059.743075578113457.3263600917

0.2311091.452539881912395.238745320211317.280595217211350.335790786311809.5826359227692.35323330695743.2005918446577.164177546210160.689076721216705.340727501210262.725193012513806.6199188563

0.2411371.482323934912584.33018533911584.888821631611502.02758233611972.16773354117841.51443026125882.41111190386790.829158095910292.077277591216961.036907041510534.062556520414166.0232652936

0.2511642.241669417412782.549687023911828.628193237511685.503780454211979.19761301648017.41580813266008.15511307826933.35330782810460.64186991417187.780817712410777.085197659914516.7186089783

0.2611915.755748843912978.609455917812059.983379105611874.643576478911979.19761301648122.9198674896155.05853085597092.724418924910617.001407986517424.73123924110972.550201086914865.3045766804

0.2712162.975989103113126.657052878212307.506976775112074.204165046411984.5690027888211.03478779666291.8764309227307.709084236610745.55795318817708.392801702911197.345373722415229.6506915408

0.2812489.174781558213284.753386986112567.935878814812297.20253318812117.62601553888392.7285880086487.40574336137481.825446190410890.54106283917944.559387832611427.893229956915557.520469668

0.2912790.73922189313464.200467570912807.417543437912486.796369787712243.2361824848563.80635637586643.36766565917695.39914685111047.112203737818194.347155024911714.499429286715912.1487979069

0.313064.64973847413675.949336263913045.937006749612695.39145926512359.36668034928681.20744643716780.90261721617856.318557743411210.251723746818482.140500286712035.271001268516274.1142311332

0.3113332.174233637113866.338640522913295.231838402612848.479568647812513.78786289148804.94086406926879.53334461817979.706849572311364.096556882218766.17323083312283.800387203916592.0821935091

0.3213574.801582570213989.609913156313514.70931244312955.39538261312701.14708867968947.6360990227034.62129796838129.058739690611515.310840277419001.571628832712503.953020707516907.7718698599

0.3313805.962480677114228.009328355213728.339643791213104.270955197612843.16820202089072.87665861187209.72435340198259.789027264211634.299949286919258.280581745912759.131444795317271.0997910243

0.3414075.173242886314418.242650018513950.538563782313301.8085823512977.14490795649195.24432895727396.47161258368400.462763864111764.200848567919434.235575307313001.707155251617631.1529032877

0.3514382.90083564714582.703475725314198.364359683713409.438297693613118.83771192079306.24691959097535.06880387268533.156103540811897.816787030419723.496640706613208.526633333717980.214256847

0.3614694.381449306514774.134995916714442.371628390213421.305299994313261.99439924229463.43934120877622.13670529368673.939775538812006.619720850119988.880740976813451.536357588318338.8076498493

0.3715019.056241349415054.669638197614695.979169849213560.654704098813462.82413781059652.36372699267758.82068894088889.449524289112184.917585688720163.007755195513703.671253492818676.5999834519

0.3815301.449957247815252.975934622314947.491935341813649.375707678413627.18530560779806.00798769847874.02218008719042.704825224112343.976999950220357.389209528213960.44886881418996.8995738776

0.3915604.476492822115426.513793601415193.142927305713815.297335900813811.58837730419976.94418487948003.13507900189193.85617033812471.122935210920533.659534627214138.27554

Normative foundations of public Intervention. General normative evaluation X, a set of mutually exclusive social states (complete descriptions of all.

Documents

x ri y

y ri x

ordering x y

x rh y

x women

state x

y women

dictatorship of individual