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Normative foundations of public Intervention

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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.
- Ria preference ordering of individual i on X (with asymmetric and symmetric factors Piand 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 xRiy 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: xRiy} #{i N:yRix}. Widely used, but does not always lead to a transitive ranking of social states (Condorcet paradox).

the Condorcet paradox

Individual 3

Individual2

Individual1

Nicolas

François

Marine

Marine

Nicolas

François

the Condorcet paradox

Individual 3

Individual2

Individual1

François

Marine

Nicolas

Nicolas

François

Marine

Marine

Nicolas

François

the Condorcet paradox

Individual 3

Individual2

Individual1

François

Marine

Nicolas

Nicolas

François

Marine

Marine

Nicolas

François

A majority (1 and 3) prefers Marine to Nicolas

the Condorcet paradox

Individual 3

Individual2

Individual1

François

Marine

Nicolas

Nicolas

François

Marine

Marine

Nicolas

François

A majority (1 and 3) prefers Marine to Nicolas

A majority (1 and 2) prefers Nicolas to François

the Condorcet paradox

Individual 3

Individual2

Individual1

François

Marine

Nicolas

Nicolas

François

Marine

Marine

Nicolas

François

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

the Condorcet paradox

Individual 3

Individual2

Individual1

François

Marine

Nicolas

Nicolas

François

Marine

Marine

Nicolas

François

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………….

the Condorcet paradox

Individual 3

Individual2

Individual1

François

Marine

Nicolas

Nicolas

François

Marine

Marine

Nicolas

François

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 3

Individual2

Individual1

François

Marine

Nicolas

Jean-Luc

Nicolas

François

Jean-Luc

Marine

Marine

Nicolas

Jean-Luc

François

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

Sum of scores Marine = 8

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

Sum of scores Marine = 8

Sum of scores Nicolas = 9

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

Sum of scores Marine = 8

Sum of scores Nicolas = 9

Sum of scores François = 8

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

Sum of scores Marine = 8

Sum of scores Nicolas = 9

Sum of scores François = 8

Sum of scores Jean-Luc = 5

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marne 4

Nicolas 3

Jean-Luc 2

François 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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 1

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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Jean-Luc 2

Marine 1

Marine 4

Nicolas 3

Jean-Luc 2

François 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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Marine 2

Jean-Luc 1

Marine 4

Jean-Luc 3

Nicolas 2

François 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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Marine 2

Jean-Luc 1

Marine 4

Jean-Luc 3

Nicolas 2

François 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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Marine 2

Jean-Luc 1

Marine 4

Jean-Luc 3

Nicolas 2

François 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

Borda rule

Individual 3

Individual2

Individual1

François 4

Marine 3

Nicolas 2

Jean-Luc 1

Nicolas 4

François 3

Marine 2

Jean-Luc 1

Marine 4

Jean-Luc 3

Nicolas 2

François 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 Nshould 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 xRiy x R’i y for all i, one must have xR 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

x is efficient

z is not efficient

y

x is not socially

better than z as

per the Pareto

principle

z

x

xA1

A

xB2

Illustration: An Edgeworth Box

xA2

B

xB1

y is better than

z as per the

Pareto principle

y

z

x

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 ifor 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 (andon 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.

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

Individual1

Individual2

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 ? alternatives »

- 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

Aggregate consumer’s surplus ? alternatives »

- Individuals live in a perfectly competitive environment
- Individual i faces prices p =(p1,….,pl) and wealth wi.
- B(p,wi)={x +lp.x wi} (Budget set)
- Individual ordering Ri on +l induces the dual (indirect) ordering RDiof 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

Aggregate consumer’s surplus ? alternatives »

- 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.

Aggregate consumer’s surplus ? alternatives »

Direct money metric:

Gives the amount of money needed at prices p to be as well-off as with bundle x

Indirectmoney 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

Aggregate consumer’s surplus ? alternatives »

These money metric utilities are connected to observable

demand behavior

Marshallian (ordinary) demand functions

Hicksian (compensated) demand functions (depends upon

unobservable utility level)

Aggregate consumer’s surplus ? alternatives »

Six important identities (valid for everyp +l, w +and u ):

(1)

(2)

(3)

(4)

Roy’s identity

(5)

Sheppard’s Lemma

(6)

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

identity (1)

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Recurrent application of Sheppard’s lemma

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

Aggregate consumer’s surplus ? alternatives »

A one good, one price illustration alternatives »

price

a

pj’

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’)

Aggregate consumer’s surplus ? alternatives »

- 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 metric a collective decision rule?

- 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 ? metric a collective decision rule?

- 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) metric a collective decision rule?

- 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
- Uiis such that, for every x and y in X, Ui(x) Ui(y) x Ri y
- Ordinal measure of happiness

Can we measure happiness ? (2) metric a collective decision rule?

- 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) = eU(z1,z2) = elnz1elnz2 = z1z2
- H(z1,z2) = -1/G(z1,z2) = -1/(z1z2) all represent numerically Ri

Can we measure happiness ? (3) metric a collective decision rule?

- 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.

Can we measure happiness ? (4) metric a collective decision rule?

- 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) metric a collective decision rule?

- 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) metric a collective decision rule?

- 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) metric a collective decision rule?

- Ordinal notion of well-being does not enable one to talk about changes in 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) metric a collective decision rule?

- Ordinal notion of well-being does not enable one to talk about changes in 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) metric a collective decision rule?

- 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))

Can we measure happiness ? (6) metric a collective decision rule?

- 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 Kelvin
- With cardinal measurement, units and zero are meaningless but a difference in values is meaningful.

Can we measure happiness ? (7) metric a collective decision rule?

- 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).

Can we measure happiness ? (8) metric a collective decision rule?

- 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.

Can we measure happiness ? (9) metric a collective decision rule?

- 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 nthe 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 individuals’ well-being ?

- Utilitarianism: x R yiUi(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 individuals’ well-being ?

u2

utility possibility set

u1 = u2

u1

Contrasting utilitarianism and max-min individuals’ well-being ?

u2

u’

-1

u1 = u2

Utilitarian optimum

u

u1

u

u’

Contrasting utilitarianism and max-min individuals’ well-being ?

u2

u’

-1

u1 = u2

Rawlsian

optimum

u

u1

u

u’

Contrasting utilitarianism and max-min individuals’ well-being ?

u2

Utilitarian optimum

u1 = u2

Rawlsian

optimum

Best feasible

egalitarian

outcome

u1

Contrasting utilitarianism and Max-min individuals’ well-being ?

- 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 individuals’ well-being ?

- 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 yilnUi(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.

Extension of Max-min individuals’ well-being ?

- 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 individuals’ well-being ?

- 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) individuals’ well-being ?

- 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) individuals’ well-being ?

- 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 (ratio-scale)

- 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 ?

Information used by a social welfare functional (ratio-scale)

- 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 (ratio-scale)

- 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 (ratio-scale)

- 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 Nimplies 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) DUand 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 xR y x R’ y where R = W(U1,…,Un)) and R’ = W(U’1,…,U’n))

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

Welfarist lemma (ratio-scale)

- 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 interpersonnaly comparable

u2

A

C

Better than

u by Pareto

u

u

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Worse than

u by Pareto

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Illustration interpersonnaly comparable

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By NC, all points

in C are ranked

in the

same way

vis-à-vis u

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Better than

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in C are ranked

in the

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Better than

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Illustration interpersonnaly comparable

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Illustration interpersonnaly comparable

- 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 interpersonnaly comparable

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Illustration interpersonnaly comparable

by Pareto, a and b

can not be

indifferent to u

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by NC, the

(strict) ranking

of region C

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be the opposite

of the (strict)

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vis-à-vis u

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Illustration interpersonnaly comparable

- 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 interpersonnaly comparable

u2

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Better than

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Moral of this story interpersonnaly comparable

- 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 interpersonnaly comparable

- 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 interpersonnaly comparable

- 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: interpersonnaly comparable

- 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 interpersonnaly comparable

- 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, andUj(y) > Uj(x) > Ui(x) > Ui(y), one has xPy where R = W(U1,…,Un))

The Lexi-min theorem: interpersonnaly comparable

- 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 interpersonnaly comparable

- 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 ? interpersonnaly comparable

u interpersonnaly comparable 2

better

Continuity ?

u2 = u1

u’(.)

u’1

= u’(2)

worse

We go continuously

from the better…

better

u’

u’2

worse

u1

= u’(1)

u’2

u’1

u interpersonnaly comparable 2

better

Continuity ?

u2 = u1

u’(.)

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= u’(2)

worse

better

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u’2

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u interpersonnaly comparable 2

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Continuity ?

u2 = u1

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= u’(2)

worse

better

u’

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to the

worse

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= u’(1)

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u interpersonnaly comparable 2

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Continuity ?

u2 = u1

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= u’(2)

worse

better

u’

u’2

worse

to the

worse

u1

= u’(1)

u’2

u’1

u interpersonnaly comparable 2

better

Continuity ?

u2 = u1

u’(.)

u’1

= u’(2)

worse

better

u’

u’2

worse

Without encountering

indifference

u1

= u’(1)

u’2

u’1

Continuity interpersonnaly comparable

- 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: uR* u’} are both closed in n

Bad news ? interpersonnaly comparable

- 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 interpersonnaly comparable

- 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 interpersonnaly comparable

- No need of continuity
- If 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 utilitarianism interpersonnaly comparable

- 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 interpersonnaly comparable

- 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/rif r > 0, g(u) = ln(u) if r = 0 and g(u) = -u1/rif r < 0

Generalized utilitarianism interpersonnaly comparable

- 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))

Generalized utilitarianism interpersonnaly comparable

- 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) interpersonnaly comparable

- 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, andUj(x) = Uj(y) - Ui(x) = Ui(y) + , one has xPy where R = W(U1,…,Un))

Remarks on this theorem (2) interpersonnaly comparable

- 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? interpersonnaly comparable

g(x)

g(u)

g(u+(1-)v)

IX

a

g(u) +

(1-)g(v)

b

g(v)

u+(1-)v

v

u

Equity respectful Generalized utilitarianism interpersonnaly comparable

- 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 xR y ig(Ui(x)) ig(Ui(y)) where R = W(U1,…,Un) for some increasing and concave function g: n

Ratio-scale comparability interpersonnaly comparable

- 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 interpersonnaly comparable

- 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 interpersonnaly comparable

- 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 ?

Some issues with variable population interpersonnaly comparable

- 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 = nn
- for all u X, n(u) is the dimension of u (number of people)
- How should we compare these vectors ?

Some issues with variable population interpersonnaly comparable

- Classical utilitarianism u RCU v n(u)i=1ui n(v)i=1vi
- 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=1ui)/n(u) (n(v)i=1vi)/n(v)
- Note: AU (c(u) =(n-1)(u)) and CU (c=0) are particular cases of CL

Some issues with variable population interpersonnaly comparable

- 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 interpersonnaly comparable

- 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 interpersonnaly comparable

- 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= {xnl+ : x1j+…+xnjj 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)= {xA : zPix 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) function

- 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.

R function5

R3

R4

R1

R2

François

Nicolas

Marine

Nicolas

Marine

François

Nicolas

François

Marine

Marine

François

Nicolas

Marine

François

Nicolas

Example of a social choice function (2)Then C(R1, R2, R3, R4 ,R5) = Nicolas

Indeed, in the first round, Nicolas and Marine

are the options which receive the most vote

Example of a social choice function (2) function

R5

R3

R4

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

Example of a social choice function (2) function

R5

R3

R4

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.

R correspondence) (1)5

R3

R4

R1

R2

François

Nicolas

Marine

Nicolas

Marine

François

Nicolas

François

Marine

Marine

François

Nicolas

Marine

François

Nicolas

A difficulty with a social choice function (or correspondence) (2)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.

R correspondence) (1)5

R3

R4

R1

R2

François

Nicolas

Marine

Nicolas

Marine

François

Nicolas

François

Marine

Marine

François

Nicolas

François

Marine

Nicolas

A difficulty with a social choice function (or correspondence) (2)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.

R correspondence) (1)5

R3

R4

R1

R2

François

Nicolas

Marine

Nicolas

Marine

François

Nicolas

François

Marine

Marine

François

Nicolas

François

Marine

Nicolas

A difficulty with a social choice function (or correspondence) (2)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.

R correspondence) (1)5

R3

R4

R1

R2

François

Nicolas

Marine

Nicolas

Marine

François

Nicolas

François

Marine

Marine

François

Nicolas

François

Marine

Nicolas

A difficulty with a social choice function (or correspondence) (2)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.

R correspondence) (1)5

R3

R4

R1

R2

François

Nicolas

Marine

Nicolas

Marine

François

Nicolas

François

Marine

Marine

François

Nicolas

François

Marine

Nicolas

A difficulty with a social choice function (or correspondence) (2)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’isuch that (R1,… R’i,…, Rn)D and C(R1,… R’i,…,Rn) PiC(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 electoral system is manipulable

- Definition 1: A social choice function C:D A is dictatorial if there exists some individual hNsuch 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:nA is manipulable on at least one profile in n.

Illustration: robust measurement of inequalities electoral system is manipulable

- 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 electoral system is manipulable

- 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 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 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 ? electoral system is manipulable

- 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 electoral system is manipulable

- 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) comparisons

- 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) comparisons

- 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: comparisons

- 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 ? comparisons

- 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 comparisons

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

- nindividuals 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 societies where individuals differer in one attribute

- 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 ? societies where individuals differer in one attribute

Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Robin Hood and Mana ? societies where individuals differer in one attribute

Answer no 2: Poverty dominance societies where individuals differer in one attribute

- 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

2 measures of poverty societies where individuals differer in one attribute

- 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

Contrasting headcount and poverty gap societies where individuals differer in one attribute

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

Poverty dominance societies where individuals differer in one attribute

- 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 societies where individuals differer in one attribute

- 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.

Answer no 3: Lorenz dominance societies where individuals differer in one attribute

- 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 dominance answer

- 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

Lorenz dominance chart dominance answer

Switzerland

US

Austria

Australia

UK

France

Germany

Canada

Sweden

Italy

Spain

Portugal

India

Important challenge: to extend to many attributes dominance answer

- Same welfarist ethics
- Suitable generalization of poverty notions (poverty in several dimensions)
- No Lorenz curves
- New issue: Correlation between attributes

Aversion to correlation ? dominance answer

a red society

Literacy rate (%)

70

60

50

40

700

400

600

500

Income (rupees/month)

Aversion to correlation ? dominance answer

a red society

Literacy rate (%)

and a white society

70

60

50

40

700

400

600

500

Income (rupees/month)

Aversion to correlation ? dominance answer

a red society

Literacy rate (%)

and a white society

white society is more

just

70

60

50

40

700

400

600

500

Income (rupees/month)

Bidimensional dominance chart dominance answer

Germany

Sweden

France

Switzerland

US

UK

Austria

Australia

Canada

Spain

Italy

Portugal

India

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