Normative foundations of public Intervention

Normative foundations of public Intervention

General normative evaluation
X, 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 i Example 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 y » Pi = « 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 evaluation
Arrow’s formulation of the problem. <Ri > = (R1 ,…, Rn) a profile of preferences.  the set of all binary relations on X   , the set of all orderings on X D  n, the set of all admissible profiles General problem (K. Arrow 1950): to find a « collective decision rule » C: D   that associates to every profile <Ri > of individual preferences a binary relation R = C(<Ri >) x R y means « x is at least as good as y when individuals’ preferences are (<Ri >)

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(<Ri >)=  for all profiles (<Ri >). 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 rules
3: 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 paradox Individual 1 Individual 2 Individual 3

the Condorcet paradox Individual 2 Individual 3 Individual 1 Marine
Nicolas François

the Condorcet paradox Individual 2 Individual 3 Individual 1 Nicolas
Marine Nicolas François Nicolas François Marine

Marine Nicolas François Nicolas François Marine François Marine Nicolas

Marine Nicolas François Nicolas François Marine François Marine Nicolas A majority (1 and 3) prefers Marine to Nicolas

Marine Nicolas François Nicolas François Marine François Marine Nicolas A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François

Marine Nicolas François Nicolas François Marine François Marine Nicolas A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François Transitivity would require that Marine be socially preferred to François

Marine Nicolas François Nicolas François Marine François Marine Nicolas A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François Transitivity would require that Marinene be socially preferred to François but………….

Marine Nicolas François Nicolas François Marine François Marine Nicolas A majority (1 and 3) prefers Marine to Nicolas A majority (1 and 2) prefers Nicolas to François Transitivity would require that Marine be socially preferred to François but…………. A majority (2 and 3) prefers strictly François to Marine

Example 5: Positional Borda
Works 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 scores Let us illustrate this rule through an example

Borda rule Individual 2 Individual 3 Individual 1 Nicolas François
Marine Nicolas Jean-Luc François Nicolas François Jean-Luc Marine François Marine Nicolas Jean-Luc

Borda rule Individual 2 Individual 3 Individual 1 Nicolas 4 François 4
Marine Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François Marine Nicolas Jean-Luc

Marine Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François 4 Marine Nicolas Jean-Luc 1 Sum of scores Marine = 8

Marine Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François Marine Nicolas Jean-Luc Sum of scores Marine = 8 Sum of scores Nicolas = 9

Marine Nicolas Jean-Luc 2 François 1 Nicolas François Jean-Luc 2 Marine François Marine Nicolas Jean-Luc Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8

Marine Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François Marine Nicolas Jean-Luc Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5

Marne Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François 4 Marine Nicolas Jean-Luc 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Nicolas is the best alternative, followed closely by Marine and François. Jean-Luc is the worst alternative

Marine Nicolas Jean-Luc 2 François Nicolas François Jean-Luc Marine François Marine Nicolas Jean-Luc Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Problem: The social ranking of François, Nicolas and Marine depends upon the position of the (irrelevant) Jean-Luc

Marine 4 Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François Marine Nicolas Jean-Luc Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Marie = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

Marine Nicolas Jean-Luc 2 François Nicolas François Jean-Luc 2 Marine François Marine Nicolas Jean-Luc Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

Marine Jean-Luc 3 Nicolas François Nicolas François Marine Jean-Luc 1 François Marine Nicolas Jean-Luc 1 Sum of scores Marine = 8 Sum of scores Nicolas = 9 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

Marine 4 Jean-Luc 3 Nicolas François Nicolas François Marine Jean-Luc 1 François Marine Nicolas Jean-Luc 1 Sum of scores Marine = 9 Sum of scores Nicolas = 8 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

Marine 4 Jean-Luc 3 Nicolas François Nicolas François Marine Jean-Luc 1 François Marine Nicolas Jean-Luc Sum of scores Marine = 9 Sum of scores Nicolas = 8 Sum of scores François = 8 Sum of scores Jean-Luc = 5 Raising Jean-Luc above Nicolas for 1 and lowering Jean-Luc below Marine for 2 changes the social ranking of Marine and Nicolas

Marine Jean-Luc 3 Nicolas François Nicolas François Marine 2 Jean-Luc 1 François 4 Marine Nicolas Jean-Luc 1 Sum of scores Marine = 9 Sum of scores Nicolas = 8 Sum of scores François = 8 Sum of scores Jean-Luc = 5 The 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 problem He proposes five axioms that, he thought, should be satisfied by any collective decison rule He shows that there is no rule satisfying all these properties Famous 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 rule
1) Non-dictatorship. There exists no individual h in N such that, for all social states x and y, for all profiles <Ri>, x Ph y implies x P y (where R = C(<Ri>) 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 rule
4) Weak Pareto principle. For all social states x and y, for all profiles <Ri>  D , x Pi y for all i  N should imply x P y (where R = C(<Ri>) (violated by the collective decision rule coming from an exogenous tradition code) 5) Binary independance from irrelevant alternatives. For every two profiles <Ri> and <R’i>  D and every two social states x and y such that x Ri y  x R’i y for all i, one must have x R y  x R’ y where R = C(<Ri>) and R’ = C(<R’i>). The social ranking of x and y should only depend upon the individual rankings of x and y.

Arrow’s theorem: There does not exist any collective decision function C: D   that satisfies axioms 1-5

All Arrow’s axioms are independent
Dictatorship of individual h satisfies Pareto, collective rationality, binary independence of irrelevant alternatives and unrestricted domain but violates non-dictatorship The Tradition ordering satisfies non-dictatorship, collective rationality, binary independance of irrelevant alternative and unrestricted domain, but violates Pareto The 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 alternatives We’ll see later that there are collective decisions functions that violate unrestricted domain but that satisfies all other axioms

Escape out of Arrow’s theorem
Natural 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 domain We can relax binary independance of irrelevant alternatives Should 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 Box
xA2 B xB1 y 2 x z xA1 A 1 xB2

xA2 B xB1 x is efficient z is not efficient y x z xA1 A xB2

xA2 B xB1 x is efficient z is not efficient y x is not socially better than z as per the Pareto principle x z xA1 A xB2

xA2 B xB1 y is better than z as per the Pareto principle y x z xA1 A xB2

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 y Pareto indifference: if x Ii y for all i  N, then x I y Strong 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 back Minimal 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 Sen’s impossibility result
One novel: Lady Chatterley’s lover 2 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 possible By 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 paradox Shows a problem between liberalism and respect of preferences when the domain is unrestricted When 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 individual’s sovereignty over their personal sphere Sen Liberal paradox: attacks the combination of the Pareto principle and unrestricted domain Suggests that unrestricted domain may be a strong assumption.

Relaxing unrestricted domain for Arrow’s theorem (1)
One possibility: imposing additional structural assumptions on the set X For 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 Arrow’s theorem.

Relaxing unrestricted domain for Arrow’s theorem (2)
A classical restriction: single peakedness Suppose 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 <Ri> is single peaked if there exists an ordering  for which all individual preferences are single-peaked. Dsp  n the set of all single peaked profiles Theorem (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 ?
left right Jean-Luc François Nicolas

Single peaked preference ?
Not Single-peaked left right Jean-Luc François Nicolas

Comments on Black theorem
Widely used in public economics In 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 even Individual 1 Individual 2 Individual 3 Individual 4 Jean-Luc François Nicolas François Jean-Luc Nicolas Nicolas François Jean-Luc Nicolas François Jean-Luc Preferences are single peaked (on the left-right axe) Jean-Luc is weakly preferred, socially, to Nicolas Nicolas is weakly preferred, socially, to François but Jean-Luc is not weakly preferred, socially, to François

Domain restrictions that garantees transitivity of majority voting
Sen and Pattanaik (1969) Extremal Restriction condition A profile of preferences <Ri> 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 <Ri> 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 alternatives »
Justification of this axiom: information parcimoniousness De Borda rule violates it In economic domains, there are various social orderings who violate this axiom but satisfy all the other Arrow’s axioms An example: Aggregate consumer’s surplus

Aggregate consumer’s surplus ?
X = +nl (set of all allocations of consumption bundles) xi  +l individual i’s bundle in x Ri, 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 z Selfishness means that we can view individual preferences as being only defined on +l

Individuals live in a perfectly competitive environment Individual 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 x’B(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 wi » Problem of applied cost-benefit analysis: ranking various prices and wealth configurations

A money-metric representation of individual preferences For 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 amount of 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.

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

These money metric utilities are connected to observable demand behavior Marshallian (ordinary) demand functions Hicksian (compensated) demand functions (depends upon unobservable utility level)

Six important identities (valid for every p  +l, w  + and u  ): (1) (2) (3) (4) Roy’s identity (5) (6) Sheppard’s Lemma

identity (1)

Recurrent application of Sheppard’s lemma

A one good, one price illustration
pj’ a Hicksian demand b pj Surplus = area pj’abpj quantity ni=1xHij(p1,…,p’j-1,pj’,pj+1,…,pl,ui’) ni=1xHij(p1,…,pj-1,pj,pj+1,…,pl,ui’)

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, « consumer’s 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 Pareto It obviously satisfies collective rationality if the reference prices used to evaluate money metric do not change It violates binary independence of irrelevant alternatives (prove it). Ethical justification for Aggregate consumer’s surplus is unclear

Normative evaluation with individual utility functions
What 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 individual’s « objective » or « interest » and which rationalizes individual’s choice More precise definition: preferences reflects the individual’s « 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 satisfied See James Griffin « Well being: Its meaning, measurement and moral importance », London, Clarendon 1988 Can 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 i’s 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 y Ordinal measure of happiness

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 Ri Example : 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 + lnz2 G(z1,z2) = e U(z1,z2) = elnz1elnz2 = z1z2 H(z1,z2) = -1/G(z1,z2) = -1/(z1z2) all represent numerically Ri

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  X Remark: 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.

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.

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.

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.

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:  .

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

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 > 0 Statements about welfare changes make sense with cardinal measurement If 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))

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) = U(x)/5 is the temperature of x in Farenheit and H(x) = U(x) is the temperature of x in Kelvin With cardinal measurement, units and zero are meaningless but a difference in values is meaningful.

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-scale Functions 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).

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 transform Ratio-scale measurement is even more precise because equivalent measures are restricted to those that are related by a increasing linear function.

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 function Cardinal measurement seems plausible by introspection. But we haven’t 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 individuals Ui: X   a (utility) function that measures individual i’s well-being in the various social states (U1 ,…, Un): a profile of individual utility functions the set of all logically conceivable real valued functions on X DU  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 functionals
Utilitarianism: 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 y Venerable 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-min
utility possibility set u1 = u2 u1

-1 u1 = u2 Utilitarian optimum u u u’ u1

-1 u1 = u2 Rawlsian optimum u u u’ u1

Utilitarian optimum u1 = u2 Rawlsian optimum Best feasible egalitarian outcome u1

Contrasting utilitarianism and Max-min
Max-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 principle Max-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 functionals
Utilitarianism and Max-min are particular (extreme) cases of a more general family of social welfare functionals Mean 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, Utilitarianism As r  -, the functional approaches Max-min r  1 if and only if the functional is weakly averse to happiness inequality.

Mean-of-order r functional
u1 = u2 r=1 u1

u1 = u2 r=1 r=+ u1

u1 = u2 Max-max indifference curve r=+ u1

Extension of Max-min Max-min functional does not respect the strong Pareto principle There 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 functional
When defining a social welfare functional, it is important to specify the information on the individuals’ utility functions used by the functional Is 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)

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 individual’s well-being are the examples of welfare functional given above using ?

Max-min, Max-max, lexi-min, lexi-max are all using OFC information. Utilitarianism: uses CUC information Mean of order r: uses RSC information. Under various informational assumptions, can we obtain sensible welfare functionals ?

Desirable properties on the Social Welfare functional
1) 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 Functional
4a) 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 (U’1,…,U’n)  DU and every two social states x and y such that Ui(x) = U’i(x) and Ui(y) = U’i(y) for all i, one must have x R y  x R’ y where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n))

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 lemma Quite powerful: The only information that matters for comparing social states is the utility levels achieved in those states Ranking of social states can be represented by a ranking of utility vectors achieved in those states. This lemma can be used to see whether Arrow’s impossibility result is robust to the replacement of information on preference by information on happiness As can be guessed, this robustness check will depend upon the precision of the information that is available on individual’s happiness.

Arrow’s 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)

Illustration u2 u u1

Illustration u2 A u u u1

Illustration u2 A u u B u1

Illustration u2 A C u u B u1

Illustration u2 A C u u B D u1

Illustration u2 A C Better than u by Pareto u u B D u1

Illustration u2 A C Better than u by Pareto u u B D Worse than

Illustration u2 By NC, all points in C are ranked in the A same way
vis-à-vis u A Better than u by Pareto u u B D Worse than u by Pareto u1

Illustration u2 a A b Better than u by Pareto u u B D Worse than

Illustration The 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

Illustration u2 a A b Better than u by Pareto u u B D Worse than

Illustration u2 a A b Better than u by Pareto u u all points here
are also ranked in the same way vis-à-vis u B Worse than u by Pareto u1

Illustration u2 a A b Better than u by Pareto u u all points here
by Pareto, a and b can not be indifferent to u (and to themselves) by transitivity) u2 a A b Better than u by Pareto u u all points here are also ranked in the same way vis-à-vis u B Worse than u by Pareto u1

Illustration u2 by NC, the (strict) ranking of region C
vis-à-vis u must be the opposite of the (strict) ranking of D vis-à-vis u A C u u B D u1

Illustration u2 A C u u B D u1

Illustration u2 A c u u d B D u1

Illustration The 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 u If c is above u, d is below u and if c is below u, d is above u

Illustration u2 A Worse Better than u by Pareto u u B Better
Worse than u by Pareto u1

Illustration u2 A Worse u u B Better u1

Illustration u2 Individual 1 is the dictator A Worse u u B Better u1

Illustration u2 A Better Better than u by Pareto u u B Worse
Worse than u by Pareto u1

Illustration u2 A Better u u B Worse u1

Illustration u2 A Individual 2 Is the dictator Better u u B Worse u1

Moral of this story Arrow’s 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 what’s going on, see Blackorby, Donaldson and Weymark (1984), International Economic Review Generalization to n individuals is easy

Allowing ordinal comparability
A strengthening of non-dictatorship: Anonymity A social welfare functional W is anonymous if for every two profiles (U1,…,Un) and (U’1,…,U’n)  DU such that (U1,…,Un) is a permutation of (U’1,…,U’n), one has R = R’ where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n)) 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 individual’s 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 dictatorship
A 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-1 Max-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 don’t 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 theorem
If 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 Hammond’s 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 Hammond’s equity principle if and only if it is the Lexi-min .

Further remarks on Lexi-min
It is not a continuous ranking of alternatives Maxi-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 ?

Continuity ? u2 better u2 = u1 u’(.) u’1 = u’(2) worse
We go continuously from the better… u’ better u’2 worse u1 u’2 = u’(1) u’1

Continuity ? u2 better u2 = u1 u’(.) u’1 = u’(2) worse u’ better u’2

to the worse u1 u’2 = u’(1) u’1

Without encountering indifference u1 u’2 = u’(1) u’1

Continuity A 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 Hammond’s equity if n > 2 Theorem 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 Hammond’s equity if and only if it is the max-min Hence, no characterization of max-min in this setting.

Cardinal measurability and unit comparability
Theorem: 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 theorem
No need of continuity If anonymity is dropped, then asymmetric utilitarianism emerges (social ranking R is defined by: x R y  iNiUi(x)  iNiUi(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 utilitarianism
Maskin (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 anybody’s shoes If 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 anyone This argument is flawed

Generalized utilitarianism
Utilitarianism is insensitive to utility inequality A social ranking that is more general than utilitarianism is, as we have seen, the mean of order r But one could also consider a more general family of social rankings: symmetric generalized utilitarianism x 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

A new axiom: Independence with respect to unconcerned individuals A ranking of two states should be independent from the utility function of the individuals who are indifferent (unconcerned ?) between the two states A 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))

Theorem: 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 ranking Proof: 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-being Does not restrict the g function. A way to restrict the g function is to impose utility inequality aversion property on the social ranking An example of inequality aversion: Hammond’s weak equity principle Another example (weaker than Hammond’s): Pigou-Dalton principle of equity A 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-being Utility levels must be compared accross individuals to make sense of Hammond’s equity principles. Utility differences of  between two individuals must also be meaningful in order for the Pigou-Dalton equity principle of transfer to make sense Hammond’s equity implies Pigou-Dalton equity but not vice-versa Pigou-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)

Concavity? g(x) IX g(u) + g(u) g(u+(1-)v) a (1-)g(v) b g(v)

Equity respectful Generalized utilitarianism
Theorem: 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 comparability
Requires a meaning to be given to zero levels of happiness A 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 comparability
Theorem: 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 ranking Proof: See Blackorby and Donaldson, International Economic Review (1982), theorem 2

Some issues with variable population
We 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 members Is 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 tricky How can someone compares her well-being with the situation in which she does not come to existence ?

Problem (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 individual’s name does not matter under anonymity) X = nn for all u  X, n(u) is the dimension of u (number of people) How should we compare these vectors ?

Classical 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

Classical 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 utility Average 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 information
X : universe of social states A, a subset (menu) of X D  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 function
X = 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 = {François, 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.

R1 François Nicolas Marine R2 R3 R4 R5 Then C(R1, R2, R3, R4 ,R5) = Nicolas Indeed, in the first round, Nicolas and Marine are the options which receive the most vote

R3 R4 R5 R1 R2 François Nicolas Marine Nicolas Marine François Nicolas François Marine Marine François Nicolas Marine François Nicolas Then C(R1, R2, R3, R4 ,R5) = Nicolas Indeed, in the first round, Nicolas and Marine are are the options which receive the most vote

R3 R4 R5 R1 R2 François Nicolas Marine Nicolas Marine François Nicolas François Marine Marine François Nicolas Marine François Nicolas Then C(R1, R2, R3, R4 ,R5) = Nicolas Indeed, in the first round, Nicolas and Marine are are the options which receive the most vote In the 2nd round, Nicolas beats Marine by 3/5 of the 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 doesn’t Problem: 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.

François Nicolas Marine R2 R3 R4 R5 The two-round electoral system is easily manipulable Individuals 4 and 5 dislike heavily Nicolas (given their true preference). Suppose one of them (4 say) « lies » and claim (by his Vote) that his favorite candidate is François.

François Nicolas Marine R2 R3 R4 R5 The two-round electoral system is easily manipulable Individuals 4 and 5 dislike heavily Nicolas (given their true preference). Suppose one of them (4 say) « lies » and claims (by his vote) that his favorite candidate is François.

François Nicolas Marine R2 R3 R4 R5 The two-round electoral system is easily manipulable Individuals 4 and 5 dislike heavily Nicolas (given their true preference). Suppose one of them (4 say) « lies » and claims (by his vote) that his favorite candidate is François. Then François and Nicolas will go to the 2nd round.

François Nicolas Marine R2 R3 R4 R5 The two-round electoral system is easily manipulable Individuals 4 and 5 dislike heavily Nicolas (given their true preference). Suppose one of them (4 say) « lies » and claims (by his vote) that his favorite candidate is François. Then François and Nicolas will go to the 2nd round. And François will win!!

The social choice function underlying the two-round French electoral system is manipulable
Definition: 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 R’i such that (R1,… R’i,…, Rn)  D and C(R1,… R’i,…,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 theorem
Definition 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(R’1,…,R’n) for all profiles (R1,…,Rn) and (R’1,…,R’n) 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 inequalities
So 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.).

Example Comparing 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 households in each country ( ) Disposable income: income available after all taxes and social security contributions have been paid and all transfers payment have been received Incomes are made comparable across households by equivalence scale adjustment Incomes are made comparable across countries by adjusting for purchasing power differences

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 mortality
Infant 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 ?

General principles that can be derived from these comparisons
Countries differ by the total amount of each attribute they allocate to their citizens :« size of the cake » They also differ by the way they share this cake Less obviously, they also differ by the way they correlate the attribute between people (are individuals who are « rich » in income also those who are « rich » in health, or education? ). Question: how can we use the normative theory seen before to compare these countries.

Remember our welfarist principles (1)
Welfarism: The only thing that matters for evaluating a society is the distribution of welfare - between individuals A just society is a society that maximises an increasing function of individual happiness. Fundamental assumption: individual happiness can be measured and compared (necessary to escape from Arrow’s theorem)

Remember our welfarist principles (2)
We don’t need to know how to measure happiness. But we have to accept the idea that we can measure it in a meaningful way. We have also to make general assumption on the way by which individual welfare depends upon the individual attributes. Here are examples of such assumptions.

Let us assume that: Happiness is increasing with respect to each attribute (more income makes people happier, so does more health, etc.) The extra pleasure brought about by an extra unit of an attribute decreases with the level of the attribute (a rich individual gets less extra pleasure from an extra euro than does an otherwise identical poorer individual) The rate of increase in happiness with respect to a particular attribute is decreasing with respect to every other attribute

Which function of individual happiness should we maximize ?
Classical Utilitarianism (Bentham): the sum Modern view point: a function that exhibits some aversion with respect to happiness-inequality Extreme form of aversion toward happiness-inequality (John Rawls): Maxi-Min, we should focus only on the welfare of the less happy person in the society.

Robust normative dominance
Society A is better than society B if the distribution of happiness in A is considered better than that in B by any function that exhibits aversion to happiness-inequality, under the assumption that the relationship between unobservable individual happiness and obervable individual attributes satisfies the above properties (Welfarist dominance)

Let us apply this notion to the problem of comparing societies where individuals differer in one attribute n individuals identical in every respect other than the considered attribute (income) y = (y1,…,yn) an income distribution y(.) = (y(1),…,y(n)) the ordered permutation of y (considered equivalent to y if the ethics used is anonymous ) Q: When are we « sure » that y is « more just » than z ?

Anwer no 1: Mana and Robin Hood
When y(.) has been obtained from z(.) by giving mana to some, or all, the individuals When y(.) has been obtained from z(.) by a finite sequence of bilateral Pigou-Dalton (Robin Hood) transfers between a donator that is richer than the recipient. When y(.) has been obtained from z(.) by both manas and Robin Hood transfers

Mana ?

Robin Hood and Mana ?

Answer no 2: Poverty dominance
Important issue: poverty How do we define poverty ? Basic principle: You define a (poverty) line that partitions the population into 2 groups: poor and rich Pour commencer, faites ressortir l'intérêt du sujet pour l'assistance. Donnez une brève vue d'ensemble de la présentation. Tenez compte de l'intérêt de l'assistance pour le sujet ainsi que de leurs connaissances en la matière pour choisir votre vocabulaire, des exemples et des illustrations. Insistez sur l'importance du sujet pour capter l'attention des auditeurs.

2 measures of poverty 1) Headcount: Count the number (or the fraction) of people below the line 2) poverty gap: Calculate the minimal amount of money needed to eliminate poverty as defined by the line Pour commencer, faites ressortir l'intérêt du sujet pour l'assistance. Donnez une brève vue d'ensemble de la présentation. Tenez compte de l'intérêt de l'assistance pour le sujet ainsi que de leurs connaissances en la matière pour choisir votre vocabulaire, des exemples et des illustrations. Insistez sur l'importance du sujet pour capter l'attention des auditeurs.

Contrasting headcount and poverty gap
Australia Austria Canada France Germany Italy Portugal Spain sweden Switz. UK USA India 4733 6815 4285 6170 5855 3554 2546 2747 5808 8679 4898 5403 789 9237 10730 8977 9555 10012 6575 4602 5407 9056 14615 8598 11025 1019 11795 12850 11935 11793 12024 8059 6110 7045 10540 17334 10883 14687 1168 14580 14725 14338 13441 13229 9438 7549 8646 11982 19806 13337 18142 1309 17377 16588 16839 15092 14857 10933 8666 10113 13371 22044 15854 21581 1462 20456 18665 19494 16966 16614 12629 10028 11656 14723 24554 18579 25206 1649 24203 20921 22382 19169 18376 14769 11415 13639 16147 27696 21574 29387 1859 28467 24042 25955 21221 17342 13930 16535 18140 32095 25188 34819 2167 34592 28069 30958 26834 25201 20743 18113 20968 21091 38254 30190 43373 2694 54537 38539 44457 40175 39217 31174 32047 35457 30818 61849 49022 79030 4735 Line = 9 600 There are 2 poor in France and 1 poor in germany but poverty gap in Germany is 3745 while it is only 3465 in France Pour commencer, faites ressortir l'intérêt du sujet pour l'assistance. Donnez une brève vue d'ensemble de la présentation. Tenez compte de l'intérêt de l'assistance pour le sujet ainsi que de leurs connaissances en la matière pour choisir votre vocabulaire, des exemples et des illustrations. Insistez sur l'importance du sujet pour capter l'attention des auditeurs.

Poverty dominance Problem with poverty measurement: how do we draw the line ? Criterion: society A is better than society B if, no matter how the line is drawn, poverty is lower in A than in B for the poverty gap (poverty gap dominance)

Answer no 3: Lorenz dominance
Lorenz dominance criterion: Society A is better than society B if the total income held by individuals below a certain rank is higher in A than in B no matter what the rank is. Easy to see with Lorenz curves. Let us draw Lorenz curves with our data.

Cool! the 3 answers are all equivalent to the welfarist dominance answer
It is equivalent to say : society A is more just than society B for any welfarist ethics One can go from B to A by a finite sequence of Robin Hood transfers and/or mana Poverty gap in A is lower than in B for all poverty lines Lorenz curve in A is everywhere above that in B.

This result is a beautiful one
Comes from mathematics: Hardy, Littlewood & Polya (1936), Berge (1959), Adapted to economics by Kolm (1966;1969), Dasgupta, Sen and Starett (1973) and Sen (1973) It provides a solid justification for the use of Lorenz curves Si vous avez plusieurs points, étapes ou idées-clé à traiter, utilisez plusieurs diapositives. Déterminez si l'assistance est censée comprendre un nouveau concept, apprendre une procédure ou approfondir un concept déjà connu. Choisissez l'explication appropriée pour étayer chacun des points traités. Vous pouvez accompagner votre présentation de données techniques supplémentaires sous formes de copies papier, disquettes, messages électroniques ou sites Internet. Développez chaque point de façon à communiquer avec l'assistance.

Lorenz dominance chart
Switzerland US Austria Australia UK France Germany Canada Sweden Italy Spain Portugal India

Important challenge: to extend to many attributes
Same welfarist ethics Suitable generalization of poverty notions (poverty in several dimensions) No Lorenz curves New issue: Correlation between attributes

Aversion to correlation ?
a red society Literacy rate (%) 70 60 50 40 400 500 600 700 Income (rupees/month)

a red society Literacy rate (%) and a white society 70 60 50 40 400 500 600 700 Income (rupees/month)

a red society Literacy rate (%) and a white society white society is more just 70 60 50 40 400 500 600 700 Income (rupees/month)

Bidimensional dominance chart
Germany Switzerland France Sweden US Australia Canada UK Austria Spain Italy Portugal India

Normative foundations of public Intervention

Similar presentations

Presentation on theme: "Normative foundations of public Intervention"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Normative foundations of public Intervention

Similar presentations

Presentation on theme: "Normative foundations of public Intervention"— Presentation transcript:

Similar presentations

About project

Feedback