Normative foundations of
public Intervention
General normative evaluation
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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
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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 ?
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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
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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 1
Marine
Nicolas
François
Individual 2
Individual 3
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
François
Marine
Nicolas
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
François
Marine
Nicolas
A majority (1 and 3) prefers Marine to Nicolas
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
François
Marine
Nicolas
A majority (1 and 3) prefers Marine to Nicolas
A majority (1 and 2) prefers Nicolas to François
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
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
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
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………….
the Condorcet paradox
Individual 1
Individual 2
Marine
Nicolas
François
Nicolas
François
Marine
Individual 3
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
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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 1
Individual 2
Marine
Nicolas
Jean-Luc
François
Nicolas
François
Jean-Luc
Marine
Individual 3
François
Marine
Nicolas
Jean-Luc
Borda rule
Individual 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
François
Jean-Luc
Marine
Individual 3
4
3
2
1
François
Marine
Nicolas
Jean-Luc
4
3
2
1
Borda rule
Individual 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
François
Jean-Luc
Marine
Sum of scores Marine = 8
Individual 3
4
3
2
1
François 4
Marine
3
Nicolas
2
Jean-Luc 1
Borda rule
Individual 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
François
Jean-Luc
Marine
Sum of scores Marine = 8
Sum of scores Nicolas = 9
Individual 3
4
3
2
1
François
Marine
Nicolas
Jean-Luc
4
3
2
1
Borda rule
Individual 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François 1
Nicolas
François
Jean-Luc
Marine
Sum of scores Marine = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Individual 3
4
3
2
1
François
Marine
Nicolas
Jean-Luc
4
3
2
1
Borda rule
Individual 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
4
François 3
Jean-Luc 2
Marine
1
Sum of scores Marine = 8
Sum of scores Nicolas = 9
Sum of scores François = 8
Sum of scores Jean-Luc = 5
Individual 3
François
Marine
Nicolas
Jean-Luc
4
3
2
1
Borda rule
Individual 1
Individual 2
Marne
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
4
François 3
Jean-Luc 2
Marine 1
Individual 3
François 4
Marine
3
Nicolas
2
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
Borda rule
Individual 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
4
François 3
Jean-Luc 2
Marine 1
Individual 3
François
Marine
Nicolas
Jean-Luc
4
3
2
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 1
Individual 2
Marine 4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
4
François 3
Jean-Luc 2
Marine 1
Individual 3
François
Marine
Nicolas
Jean-Luc
4
3
2
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 1
Individual 2
Marine
4
Nicolas
3
Jean-Luc 2
François
1
Nicolas
François
Jean-Luc
Marine
Individual 3
4
3
2
1
François
Marine
Nicolas
Jean-Luc
4
3
2
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 1
Individual 2
Marine
4
Jean-Luc 3
Nicolas
2
François
1
Nicolas
4
François 3
Marine
2
Jean-Luc 1
Individual 3
François
Marine
Nicolas
Jean-Luc
4
3
2
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 1
Individual 2
Marine 4
Jean-Luc 3
Nicolas
2
François
1
Nicolas
4
François 3
Marine
2
Jean-Luc 1
Individual 3
François
4
Marine 3
Nicolas
2
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
Borda rule
Individual 1
Individual 2
Marine 4
Jean-Luc 3
Nicolas
2
François
1
Nicolas
4
François 3
Marine
2
Jean-Luc 1
Individual 3
François
4
Marine 3
Nicolas
2
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
Borda rule
Individual 1
Individual 2
Marine
4
Jean-Luc 3
Nicolas
2
François
1
Nicolas
4
François 3
Marine 2
Jean-Luc 1
Individual 3
François
Marine
Nicolas
Jean-Luc
4
3
2
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 ?
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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
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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
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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
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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
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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)
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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
z
x
xA1
A
1
xB 2
Illustration: An Edgeworth Box
xA2
B
xB1
x is efficient
z is not efficient
y
x
z
xA1
A
xB 2
Illustration: An Edgeworth Box
xA2
B
xB1
y
x
z
x is efficient
z is not efficient
x is not socially
better than z as
per the Pareto
principle
xA1
A
xB 2
Illustration: An Edgeworth Box
xA2
B
xB1
y is better than
z as per the
Pareto principle
y
x
z
xA1
A
xB 2
Should we relax the Pareto principle ? (2)
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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)
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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)
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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
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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
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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)
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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)
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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 ?
Single-peaked
left
Jean-Luc
François
Nicolas
right
Single peaked preference ?
Single-peaked
left
Jean-Luc
François
Nicolas
right
Single peaked preference ?
Single-peaked
left
Jean-Luc
François
Nicolas
right
Single peaked preference ?
Single-peaked
left
Jean-Luc
François
Nicolas
right
Single peaked preference ?
Not Single-peaked
left
Jean-Luc
François
Nicolas
right
Single peaked preference ?
Not Single-peaked
left
Jean-Luc
François
Nicolas
right
Comments on Black theorem
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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
Jean-Luc
François
Nicolas
Individual 2
Individual 3
Individual 4
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
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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 »
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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 ?
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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 ?
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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
Aggregate consumer’s surplus ?
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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)  min
x1 ,... xl
l
p x
j 1
j
j
subject to U ( x1 ,... xl )  u
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 ?
Direct money metric:
 ( p, x )  E ( p, u( x ))
Gives the amount of money needed at prices p to be as
well-off as with bundle x
Indirect money metric:
 ( p, q, w )  E ( p,V (q, w ))
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 ?
These money metric utilities are connected to observable
demand behavior
( x1M ( p, w ),..., xlM ( p, w ))  arg max u( x )
xB ( p , w )
Marshallian (ordinary) demand functions
( x1H ( p, u),..., x1H ( p, u))  arg min
( x1 ,..., xl )
l
p x
j 1
j
j
s.t . U ( x1 ,..., xl )  u
Hicksian (compensated) demand functions (depends upon
unobservable utility level)
Aggregate consumer’s surplus ?
Six important identities (valid for every p  +l, w  + and u  ):
E ( p,V ( p, w ))  w
(1)
V ( p, E ( p, u))  u
(2)
H
xM
(
p
,
E
(
p
,
u
))

x
j
j ( p, u)
(3)
x Hj ( p,V ( p, w))  x M
j ( p, w )
(4)

V ( p, w ) / p j
 xM
j ( p, w )
(5)
Roy’s identity
E ( p , u )
 xH
j ( p, u )
p j
(6)
Sheppard’s Lemma
V ( p, w ) / w
Aggregate consumer’s surplus ?
( p, w1 ,..., w n ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
   i ( p,Vi ( p, w i ))    i ( p,Vi ( p' , w'i ))
n
  (  i ( p,Vi ( p, w i ))   i ( p,Vi ( p' , w'i )))  0
i 1
n
  (  i ( p,Vi ( p, w i ))   i ( p' ,Vi ( p' , w'i ))   i ( p' ,Vi ( p' , w'i ))   i ( p,Vi ( p' , w'i )))  0
i 1
n
  (w i  w'i   i ( p' ,Vi ( p' , w'i ))   i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (w i  w'i    x Hi
j ( p1 ,..., p j 1 , q j , p' j  1 ,..., p'l ,u'i )dq j )  0
i 1
j 1 p '
j
surplus Hicksien du consommate ur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
identity (1)
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
Recurrent application of Sheppard’s lemma
  (wi  w'i   x ( p1 ,..., p j 1 , q j , p' j 1 ,..., p'l ,u 'i )dq j )  0
i 1
j 1 p '
Hi
j
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
Aggregate consumer’s surplus ?
( p, w1 ,..., wn ) R (q, w'1 ,..., w'n )
n
n
i 1
i 1
  i ( p,Vi ( p, wi ))   i ( p,Vi ( p' , w'i ))
n
  ( i ( p, Vi ( p, wi ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  ( i ( p, Vi ( p, wi ))  i ( p ' ,Vi ( p' , w'i ))  i ( p ' ,Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
  (wi  w'i  i ( p' , Vi ( p' , w'i ))  i ( p,Vi ( p' , w'i )))  0
i 1
n
l
pj
  (wi  w'i   x Hi
j ( p1 ,..., p j 1 , q j , p ' j 1 ,..., p 'l ,u 'i ) dq j )  0
i 1
j 1 p '
j


surplus Hicksien du consommateur
A one good, one price illustration
price
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’)
Aggregate consumer’s surplus ?



Usually done with Marshallian demand
(rather than Hicksian demand)
Marshallian surplus is not a correct measure
of welfare change for one consumer but is
an approximation of two correct measures of
welfare change: Hicksian surplus at prices p
and Hicskian surplus at prices p’ (Willig
(1976), AER, « 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
Can we measure happiness ? (2)







Ordinal measure of happiness: defined up to an increasing
transform.
Definition: g: A  (where A  ) is an increasing
function if, for all a, b  A, a > b  g(a) > g(b)
If Ui is a numerical representation of Ri, and if g:   is
an increasing function, then the function h: X   defined
by: h(x) = g(U(x)) is also a numerical representation of 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
R
Can we measure happiness ? (3)




The three functions of the previous example are
ordinally equivalent.
Definition: Function U is said to be ordinally
equivalent to function G (both functions having X
as domain) if, for some increasing function
g:  , one has U(x) = g(G(x)) for every x  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)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers.
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2.
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved
by a monotonic transformation.
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved
by a monotonic transformation. U(b) > [U(c)+U(a)]/2 being
true does not imply that g(U(b)) > [g(U(c))+g(U(a))]/2 is true
for every increasing function g:  .
Can we measure happiness ? (4)



Ordinal notion of well-being does not enable one to talk
about changes in well-being.
For example a statement like « I get more extra happiness
from my first beer than from my second » is meaningless
with ordinal measurement of well-being.
proof: let a, b and c be the alternatives in which I drink,
respectively, no beer, one beer and two beers. If U is a
function that measures ordinally my happiness, the
statement « I get more extra happiness from the first beer
than from the second » writes: U(b)-U(a) > U(c) – U(b) 
U(b) > [U(c)+U(a)]/2. Yet this last statement is not preserved
by a monotonic transformation. U(b) > [U(c)+U(a)]/2 being
true does not imply that g(U(b)) > [g(U(c))+g(U(a))]/2 is true
for every increasing function g:  . For example,
having 3 > (4+1)/2 does not imply having 33 > (43+13)/2
Can we measure happiness ? (5)





Stronger measurement of well-being: cardinal.
Suppose U: X  and G: X  are two measures of wellbeing. 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)



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)



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)




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)




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
u2
utility possibility set
u1 = u2
u1
Contrasting utilitarianism and max-min
u2
u’
-1
Utilitarian optimum
u1 = u2
u
u
u’
u1
Contrasting utilitarianism and max-min
u2
u’
-1
u1 = u2
Rawlsian
optimum
u
u
u’
u1
Contrasting utilitarianism and max-min
u2
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
u2
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r =-
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r =-
r=0
u1 = u2
r=1
u1
Mean-of-order r functional
u2
r =-
r=0
u1 = u2
r=1
r=+
u1
Mean-of-order r functional
u2
u1 = u2
r=+
Max-max indifference
curve
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 wellbeing 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 wellbeing 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 wellbeing 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 wellbeing iff for all (U1,…Un) and (G1,…,Gn)  DU such that Ui
= aGi for some strictly positive real number a (for i =
1,…n), one has W (U1,…Un) = W(G1,…,Gn)
Information used by a social
welfare functional



There are some connections between these various
informational invariance requirements
Specifically, ONC  CNC  CUC  CFC  RSFC and,
similarly, OFC  CFC and CUC  CFC. On the other
hand, it is important to notice that CUC does not imply nor
is implied by OFC.
What information on individual’s well-being are the
examples of welfare functional given above using ?
Information used by a social
welfare functional




Max-min, Max-max, lexi-min, lexi-max are all using
OFC information.
Utilitarianism: uses CUC 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
Worse than
u by Pareto
D
u1
Illustration
u2
By NC, all points
in C are ranked
in the
same way
vis-à-vis u
A
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
By NC, all points
in C are ranked
in the
same way
vis-à-vis u
A
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
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
Worse than
u by Pareto
D
u1
Illustration
u2
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
all points here
are also ranked
in the same way
vis-à-vis u
u1
Illustration
by Pareto, a and b
can not be
u2 indifferent to u
(and to themselves)
by transitivity)
a
A
b
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
all points here
are also ranked
in the same way
vis-à-vis u
u1
Illustration
u2
A
C
u
u
B
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
D
u1
Illustration
u2
A
C
u
u
B
D
u1
Illustration
u2
A
c
uu
B
D
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
C
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
A
Worse
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
Better
u1
Illustration
u2
A
Worse
u
u
B
Better
u1
Illustration
u2
A
Worse
Individual 1
is the dictator
u
u
B
Better
u1
Illustration
u2
A
C
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
D
u1
Illustration
u2
A
Better
Better than
u by Pareto
u
u
B
Worse than
u by Pareto
Worse
u1
Illustration
u2
A
Better
u
u
B
Worse
u1
Illustration
u2
A
Better
Individual 2
Is the dictator
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 25, 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 ?
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
We go continuously
from the better…
u’
worse
u’2
u’2 = u’(1)
u’1
better
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse
u’2 = u’(1)
u’1
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse to the
worse
u’2 = u’(1)
u’1
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse to the
worse
u’2 = u’(1)
u’1
u1
u2
better
Continuity ?
u2 = u1
u’(.)
u’1
= u’(2)
worse
u’
u’2
better
worse Without encountering
indifference
u’2 = u’(1)
u’1
u1
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 wellbeing 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 VonNeuman 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
Generalized utilitarianism



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


Theorem: An anonymous social welfare
functional W: n   satisfies Paretoindifference, 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 wellbeing
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): PigouDalton principle of equity
A social welfare functional W satisfies the PigouDalton 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)
g(u)
g(u+(1-)v)
IX
g(u) +
a
(1-)g(v)
b
g(v)
v
u+(1-)v
u
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 Paretoindifference, 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 ?
Some issues with variable population





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 ?
Some issues with variable population


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
Some issues with variable population



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.
Example of a social choice
function (2)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas Marine Marine
François François François
Marine
Nicolas Nicolas
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)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas Marine Marine
François François François
Marine
Nicolas 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)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas Marine Marine
François François François
Marine
Nicolas 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.
A difficulty with a social choice
function (or correspondence) (2)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas Marine Marine
François François François
Marine
Nicolas Nicolas
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.
A difficulty with a social choice
function (or correspondence) (2)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas François Marine
François Marine François
Marine
Nicolas Nicolas
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.
A difficulty with a social choice
function (or correspondence) (2)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas François Marine
François Marine François
Marine
Nicolas Nicolas
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.
A difficulty with a social choice
function (or correspondence) (2)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas François Marine
François Marine François
Marine
Nicolas Nicolas
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.
A difficulty with a social choice
function (or correspondence) (2)
R1
François
Nicolas
Marine
R2
Nicolas
Marine
François
R3
R4
R5
Nicolas François Marine
François Marine François
Marine
Nicolas Nicolas
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 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
80000
70000
60000
Australia
50000
France
Germany
disposable income 40000
Italy
Spain
30000
sweden
UK
USA
20000
India
10000
0
1
2
3
4
5
6
individual rank
7
8
9
10
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 ?
70.00
65.00
60.00
55.00
50.00
45.00
40.00
Number of deads per
35.00
1000
30.00
25.00
20.00
15.00
10.00
5.00
0.00
Australia
France
Germany
Italy
Spain
Sweden
UK
US
India
1
2
3
infant mortality groups
4
5
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 happinessinequality
Extreme form of aversion toward happinessinequality (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 PigouDalton (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 ?
80000
70000
60000
50000
40000
UK
USA
30000
20000
10000
0
1
2
3
4
5
6
7
8
9
10
Mana ?
80000
70000
60000
50000
US
40000
UK
30000
20000
10000
0
1
2
3
4
5
6
7
8
9
10
Robin Hood and Mana ?
60000
50000
40000
30000
Australi
Canada
20000
10000
Robin Hood and Mana ?
12000
10000
8000
Australia
6000
Canada
4000
2000
0
1
2
3
Robin Hood and Mana ?
12000
10000
8000
Australia
6000
Canada
4000
2000
0
1
2
3
Robin Hood and Mana ?
12000
10000
8000
Australia
6000
Canada
4000
2000
0
1
2
3
Robin Hood and Mana ?
12000
10000
8000
Australia
6000
Canada
4000
2000
0
1
2
3
Robin Hood and Mana ?
60000
50000
40000
Australia
30000
Canada
20000
10000
0
1
2
3
4
5
6
7
8
9
10
Robin Hood and Mana ?
60000
50000
40000
Australia
30000
Canada
20000
10000
0
1
2
3
4
5
6
7
8
9
10
Robin Hood and Mana ?
60000
50000
40000
Australia
30000
Canada
20000
10000
0
1
2
3
4
5
6
7
8
9
10
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
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
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
13337
18142
1309
17377
16588
16839
15092
15854
21581
1462
20456
18665
19494
16966
18579
25206
1649
24203
20921
22382
19169
21574
29387
1859
28467
24042
25955
22382
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
There9438are 27549poor 8646
in France
11982
19806
and 1 poor in germany
14857
10933
8666
10113
13371
22044
but poverty gap in Germany
16614
12629
10028
is 3745
while
it is11656only14723 24554
18376
11415
13639
16147
27696
346514769
in France
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.
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.
250000
200000
cumulated total income
Australia
France
150000
Germany
Italy
Spain
Sweden
100000
UK
US
India
50000
0
1
2
3
4
5
6
individual rank
7
8
9
10
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
Lorenz dominance chart
Switzerland
US
Austria
UK
France
Germany
Australia
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)
Aversion to correlation ?
a red society
Literacy rate (%)
and a white society
70
60
50
40
400 500 600 700
Income (rupees/month)
Aversion to correlation ?
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
Australia
Canada
UK
France
Sweden
US
Austria
Spain
Portugal
India
Italy