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Basic assumption of welfarism

- Distribution of individual’s well-being is the only information that is required to compare social state on the basis of general interest
- This position has been challenged by many philosophers and economists
- It ignores other important issues (rights, freedom, discrimination, ressources inequalities) when these have no impact on individual’s well-being

Examples of situations where other information than well-being may be used?

- Assume Bob has initially a well-being of 4 while Mary has a well-being of 10 and that these well-being levels are comparable
- Assume that Bob wants to have sex with Mary but Mary does not want to have sex with Bob
- Specifically, assume that the situation in which Mary and Bob have sex together (rape) generates a distribution of utilities of 8 and 8 (Bob benefits more from having sex than Mary suffers)
- Hammond’s equity principle would consider rape to be better than non rape
- So will maxi-min, so will utilitarianism
- Plausible ?

Examples of situations where other information than well-being may be used?

- Another example: Equal budget Walrassian Equilibrium.
- Suppose a standard economy in which you give to everyone an equal share of all ressources of the economy
- Define a general competitive (Walrassian) equilibrium for this economy
- For many philsophers and economists, the resulting situation is « fair »
- It is Pareto-efficient (1st Welfare Theorem)
- It is Envy-free (no one envies any other)

Illustration: An Exchange economy

xA2

1/2

B

xB1

Equal share of

the endowment

2

2/2

Walrassian

Equilibrium

xA1

A

1

xB2

Complement/substitutes to Welfare

- An important one: Freedom
- The well-being achieved by an individual is not society’s problem. Society’s problem is to allocate efficiently individuals’ opportunities
- Freedom is the important distributandum
- What is individual freedom ?

Important notion: an opportunity set

- Opportunity set: set of all options that are available for choice to an individual.
- Obvious example: Budget set: set of all consumption bundles that an individual can afford given prices and wealth
- Other examples: set of all candidates available at some election (political freedom), set of all available means of transportation for commuting between two cities, set of all religious rituals that are allowed (including none)
- Policies affect opportunity sets: How can the impact of policies on an individual opportunity set be appraised ?

Important notion: an opportunity set

- Opportunity set: encompass in a single concept several notions of freedom discussed by philosophers.
- Famous distinction: Positive vs negative freedoom
- Negative freedom: Freedom from constraints imposed by the behavior of other humans (e.g. I’m not allowed to sunbath topless)
- Positive freedom: Freedom from technological constraint (An individual living in Rome in 2007 has the freedom to spend a week end to Beijing; An individual living in Rome in 50 b.c did not have this freedom)
- We incorportate all constraints in the definition of opportunity sets.

Indirect utility ranking of opportunity sets

- X set of all conceivable options that the individual can choose
- a family of subsets of X. Typical elements A, B, C, etc. of are interpreted as alternative opportunity sets (menus)
- : a ranking of opportunity sets (A B means « opportunity set A is weakly better than opportunity set B)
- Assume that the individual has a preference ordering R over the option in X (perhaps based on his/her well-being
- Obvious ranking: A B for all b B, there exists some a in A such that aR b. (indirect utility ranking)
- This ranking does not attach intrinsic importance to freedom

Intrinsic importance of freedom ?

Money available to other use than alcohol

I

opportunity set if alcohol

is legal

-p

alcohol

I/p

Intrinsic importance of freedom ?

Money available to other use than alcohol

Optimal choice if

Alcohol is legal

I

alcohol

I/p

Intrinsic importance of freedom ?

Money available to other use than alcohol

I

Opportunity set

If alcohol is illegal

alcohol

I/p

Intrinsic importance of freedom ?

Money available to other use than alcohol

I

The opportunity set if

alcohol is illegal…

alcohol

I/p

Intrinsic importance of freedom ?

Money available to other use than alcohol

I

The opportunity set if alcohol

is illegal…is equivalent

to the opportunity set when

alcohol is legal for a person

who does not drink alcohol when

it is legal.

alcohol

I/p

Intrinsic importance of freedom ?

Money available to other use than alcohol

I

Is choosing freely not to drink

equivalent to being forced to

abstinence ?

alcohol

I/p

Two broad approaches to freedom

- Defining freedom of choice without resorting to choice motivation (evaluating opportunities)
- Defining freedom of choice by appealing to preferences (example: indirect utility ranking, with the preference R interpreted as refleting the motivation of the individual. This motivation need not be entirely connected to well-being.

Freedom without reference to choice motivations: set inclusion

- Elementary principle: Weak Monotonicity with Respect to Set Inclusion. Adding an option to a set does not reduce freedom
- Formally: A B A B
- Weak principle, agreed upon by Indirect utility ranking (someone who chooses optimally from a set can not loose from its enlargement).
- Stronger notion: Strong Monotonicity with respect to set Inclusion. Adding an option to a set strictly increases freedom
- Formally: A B A B
- Some economists and philosophers don’t like this second version (does adding the option of « being beheaded at dawn » to my opportunity set really increase my freedom ?)
- Limitation of Monotonicity to set inclusion principles: Highly incomplete!.
- N.B. not a problem for budget sets evaluated at a given prices configuration p (B(p,y) B(p,y’) y y’

Freedom without reference to choice motivations: cardinality rule

- Assume that consists of all non-empty finite subsets of X.
- A possible (naïve ?) freedom-based ranking of sets: counting the number of options
- AcardB #A #B
- Pattanaik and Xu (1990), Rech. Econ. Louv. have proposed an elegant characterization of this ranking based on 3 axioms.
- 1) Indifference between no-choice situations. For all x, y X, {x} {y}
- 2) Preference for choice over no choice.For all distinct x, y X, {x,y} {y}
- 3) Independance: A, B, C such that A C = B C = A B A C B C
- Theorem: A reflexive and transitive binary relation on satisfies axioms 1-3 if and only if = card

Freedom without reference to choice motivations: cardinality rule

- Critique 1 of Pattanaik and Xu results: Indifference between no choice situations: Extreme. If x = « going to a dental surgery with anaesthesia » and y = « going to the same dental surgery without anesthesia », is {x} {y} plausible ?
- A way out of this criticism: Additive generalization of the cardinality rule AaddB aAv(a) bBv(b) for some (freedom weight) function v: X++
- card is a particular case of this class of rules for which v: X++ is defined by v(x)=v(y) for all x, y X
- AaddB satisfies preference for choice over no-choice and independance but not indifference between no-choice situations (except if v(x)=v(y) for all x, y X)
- But there are other rankings that satisfy axioms 2 and 3 and that are not additive generalization of the cardinality rule
- Gravel, Laslier and Trannoy (1998) provides a characterization of the additive generalization (see below)

Freedom without reference to choice motivations: cardinality rule

- Critique 2 of Pattanaik and Xu results: Independence. Suppose x = « commuting between Aix and Eguilles on a red bicycle » and y = « commuting between Aix and Eguilles in a red car ». A plausible ranking of these two (zero-choice) situations would be {x} {y}.
- Suppose now that the option z = « commuting between Aix and Eguilles on a blue bicycle » is added to both opportunity set. Is the ranking {x,z} {y,z} plausible ?
- Thomas d’Aquinae (XIIth century) ‘An angel is more valuable than a stone; it does not follow, however, that two angels are more valuable than one angel and a stone’.
- Independance says that the contribution of adding any option to freedom is independant from the set to which it is added. Neglects the diversity of options.

What is diversity?

- Important in these days (biodiversity, UN convention on cultural diversity, product diversity, etc.)
- Framework for measuring diversity: the same than that for measuring freedom (A B « A offers weakly more diversity than B »)
- In biology, the elements of the sets (living individuals) are grouped into collections (species)
- Assume there are n species
- Every set A can be represented by an n-dimensional vector sA=(sA1,…,sAn) where sAi 0 is the number of individuals belonging to species i (for i = 1,…,n) in set A (the possibility of sAi = 0 is of course not ruled out)
- Biologists compare sets (ecosystems) on the basis of their generalized entropy (GE)

Generalized entropy ?

Contains several indices

Generalized entropy ?

Measure of the disorder (unpredictability) of a system

in information theory (Entropy is maximized in ecosystems

in which all species have the same number of members)

Generalized entropy ?

Species counting

Problem with general entropy

- No justifications in terms of diversity appraisal
- Treat all species symmetrically (an ecosystem in which all living individuals are split equally between two different species of fly is just as diverse as an ecosystem in which they are all split between sea horse and chimpanzee
- Violates monotonicity with respect to set inclusion

Other approach: aggregate dissimilarity

- There exists an underlying notion of (pairwise) dissimilarity between objects (« a blue bike is more similar to a red bike than to a blue car », etc.)
- Diversity of a set is then seen as aggregate dissimilarity
- Pioneer of the approach Weizman (QJE 1992; 1993; Econometrica 1998)
- Dissimilarity is assumed to be available in the form of a distance function d: XX d(w,z) d(x,y) means « w is weakly less similar to z than x is to y

Other approach: aggregate dissimilarity (2)

- Distance function satisfies: d(x,y) = d(y,x) d(x,x) for all x and y distinct.
- Satisfies also the so-called « triangle » inequality (i.e. d(x,z) ≤ d(x,y) + d(y,z) for all x, y and z.
- Weizman procedure: take any finite set A.
- For each element a A, calculate the vector of distances between a and all elements of A (including a itself) and write this vector in an increasing ordered fashion
- Compare these vectors by the lexi-min criterion
- Remove from A the element whose ordered vector of distances is the lowest for the lexi-min criterion (they may be several such elements in which case any one of them will do).
- Record, for the element removed, the value of the smallest non-zero component of the ordered vector of distances
- Redo the procedure on the remaining set thus created.
- At the end one is left with one element and one has obtained a list of smallest non-zero distances
- Weizman: compare sets on the basis of the sum of these non-zero distances

Weizman procedure:

- Ingenious, but complex
- Has been axiomatized by Bossert, Pattanaik and Xu (2002). Axioms are not appealing.
- Rides crucially on a cardinally meaningful notion of distance which can be added
- Realistic ? Are our notions of dissimilarity sufficiently precise to say things like « the dissimilarity between a blue car and a red bicycle is twice that between a blue and a red car ».

Alternative approach to diversity as aggregate dissimilarity

- Using a qualitative (ordinal) notion of dissimilarity.
- (w,z) Q (x,y) = « w is weakly more dissimilar from z than x is from y »
- Q is a quaternary relation
- QA (strictly more dissimilar), QS (equally dissimilar)
- Q satisfies (x,y) Q (y,x) Q (x,x) for all x and y.
- Let us also assume that it satisfies (x,y) QA (x,x) for all distinct x and y.
- Can we define diversity by using only ordinal information on dissimilarity

Alternative approach to diversity as aggregate dissimilarity (2)

- Attempts in Bervoets & Gravel (2007).
- 3 axioms
- 1: For all w, x, y, z, (w,z) Q (x,y) {w,z} {x,y}
- 2: A B AB for all A and B.
- 3: For all sets A, B, C and D such that B C = C D = B D = , AB C, AB D and AC D imply that AB C D and AB D, A B C and AC D imply that AB C D
- Theorem: A reflexive and transitive ranking of all finite subsets of some finite set X satisfy axioms 1-3 if and only if it is the maxi max ranking (compare sets on the basis of the dissimiliarity of the most dissimilar pair of objects)
- Extension: Lexi-max ranking of sets

Other approach: diversity as the value of the realized attributes

- Nehring and Puppe (Econometrica, 2002).
- Objects in X have attributes (being a mamal, being a car, etc.)
- An attribute is modelled a subset of all the objects having the considered attributes.
- , the family of relevant attributes
- Nehring and Puppe propose and axiomatically characterize the following rule for comparing sets based on their diversity

Other approach: diversity as the value of the realized attributes (2)

- v is a function that values the attributes.
- An object contributes to the diversity of set only through the value of its attributes
- v has cardinal significance in Nehring and Puppe’s approach
- Difficulty: identifying and weighting the relevant attributes

Diversity and freedom ?

- Complex relation
- Why is diversity an important element of freedom ?
- It seems difficult to answer this question without thinking more about why freedom is important

Freedom as indeterminacy with respect to preferences

- Freedom is important only through the use made of it by the individual
- Yet, the individual is not always completely determined with respect to the criterion he or she will use for making a choice in an opportunity set
- This suggests that we should appraise freedom of choice by resorting to a family of possible preferences that the individual could use
- 2 interpretations of this family:
- 1: the preferences that the individual could possibly use for choosing and
- 2, the preference that any reasonable person could use

Freedom as indeterminacy with respect to preferences (1)

- Example: Kreps (1979) preference for flexibility
- Opportunity sets are compared on the basis of their expected maximal utility, with the expectation taken over all possible utility functions that represent the preferences in

where URis a numerical representation of

the preference R.

Freedom as indeterminacy with respect to preferences (1)

- Such a ranking of sets satisfies two axioms
- 1: weak monotonicity with respect to set inclusion
- 2: Contraction consistency: For all sets A and B, if A B and if A {x} A, then B {x} B
- Theorem (Kreps (1979): An ordering of all finite subsets of a finite set X satisfies weak monotonicity with respect to set inclusion and contraction consistency if and only if there exists a finite set of preference orderings on X and a probability distribution p on such that :

holds for some numerical representation UR of the

preferences R in

Freedom as indeterminacy with respect to preferences (1)

- Another approach: Foster (1993)
- A B for all R , and for all b B, there exists some a in A such that aR b
- If = {R}, this ranking is the indirect utility ranking
- If is the set of all logically conceivable orderings of X, then this ranking is nothing else than set inclusion.
- In between these two extreme cases, the ranking is incomplete but transitive and satisfies monotonicity with respect to set inclusion (prove it!).

Freedom as indeterminacy with respect to preferences (2)

- contains preferences that any reasonnable person (but not necessarily the individual to which we are interested) could have
- Example: Allowing women to enroll in the army
- For any set A, let E(A) denote the set of all (essential) options in A that could be choosen by some preference in R (that is E(A) = {a A: R such that a R a’ a’ A}
- Pattanaik and Xu (1998): rank sets on the basis of their number of essential alternatives for the preferences in

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