Social ChoiceSession 4

Carmen Pasca and John Hey

Preamble• In sessions 2 and 3 we asked the question: is there some

way we can aggregate individual preferences into social preferences.

• The answer was a resounding ‘NO’.• So getting the State to work by aggregating preferences is

not going to work.• Let us try something else: can we specify general principles

that the state should use for deciding things?• Rather than talk about ‘things’ over which people might

have preferences...• ...we talk here about distributions of income over which

the state must decide.

Plan for this session 1• In session 2 we started with preferences and asked if we

could aggregate them into social preferences. Answer? NO.• This session we are going to ask a different question: can we

start with principles for choosing?• We start with a very simple one-dimensional context:

choosing income distributions, where each individual’s income is a random drawing from the distribution.

• We report on an experiment to see if there is agreement on principles determining distribution and redistribution.

• Then we make the problem two-dimensional – income is partly chosen and partly random. Which (re-)distributions should society chose ... using which principles?

Plan for this session 2• We start with Amiel and Cowell’s book, Thinking about

Inequality , Cambridge University Press, 1999.• And report on an experiment seeing if there is agreement about

principles of choosing between distributions (and redistributions) in one dimension. (45 minutes)

• We then take a break. (15 minutes)• We then return and look at Fleurbaey’s book, Fairness,

Responsibility and Welfare – where there are two dimensions.• We propose some reasonable principles and show that they work

in one context. (45 minutes)• In a second break we ask you to apply them in another context. • We then return and discuss what we find for the last 30 minutes.

The first part of this session is actually a published paper “On Choosing a Constitution

(at least the part relating to the distribution of income)”

by

John Hey and Carmen Pasca

Applied Economics Letters 2011.The paper can be found here.

Outline• This is a part of the published paper.• For the purpose of the course of Social Choice it is an

exercise in seeing which of various axioms used in the literature for choosing between distributions of income are acceptable.

• Each axiom has an implication also for re-distribution – taxation if you like. This is one of the main concerns of social choice.

• We will not look at all the paper but just the bits relevant to the course.

Introduction and Motivation

• Here we ask directly which principles people prefer for choosing between income distributions.

• Previous experiments have asked about preferences over distributions.

• Well-summarised in Amiel and Cowell, Thinking about Inequality, Cambridge University Press, 1999.

• In this experiment we ask subjects with which principles/axioms they are in agreement, and their responses are appropriately incentivated.

The Axioms• 5 fundamental axioms and their “opposites”:

• Rawls.• Dispersion.• Transfer.• Transfer in groups.• Lorenz.• In the following slides we are not going to detail the ‘opposites’ of the principles.

Full details can be found in the published paper. Moreover we are not going to detail all the axioms.

Rawls

• “If the poorest in distribution 1 has a higher income than the poorest in distribution 2, then I prefer distribution 1.”• This axiom is implementated

lexicographically.

Dispersion

• “I prefer the distribution in which the dispersion of income is the smallest.”

• We measure dispersion with standard deviation.

The Transfer Principle.

• “I think that the distribution improves if we take away 1 euro from one individual and we give it to someone poorer.”• The subjects were advised that this

axiom could be invoked several times.

The Lorenz principle.

• “If the n poorest individuals in distribution 1 have a larger proportion of the total income than the n poorest in distribution 2, for all values of n between 1 and N then I prefer distribution 1.”• (In the experiment N was 10.)

The Experiment (1)• Two treatments (5 sessions of each):• (1) 10 subjects, one of them chosen at random to be the

Social Planner (who will be part of the society) and then the position of the 10 in society randomised.

• (2) 11 subjects, one of them chosen at random to be the Social Planner (who will NOT be part of the society) and then the position of the remaining 10 in society randomised.

• Three repetitions.• Forced to spend at least 60 (30,20) seconds on each reply on

Repetition 1 (2,3).

The Experiment (2)

• At the end of the experiment, one subject chosen at random to be the Social Planner and then the order of the subjects in society randomly chosen.

• The preferred distribution of the Social Planner, using his or her preferences to chose one from a randomly generated set of 100 distributions (all with the same mean), distributed to the 10 members of society.

Partecipante 1 2 3 4 5 6 7 8 9 10

Distribuzione 1 2 5 6 8 9 14 17 29 30 30

Distribuzione 2 0 3 8 14 14 17 20 21 23 30

Distribuzione 3 0 1 7 11 13 19 21 23 27 28

Distribuzione 4 0 3 12 14 17 18 20 21 22 23

Distribuzione 5 3 3 9 10 10 20 21 21 24 29

Distribuzione 6 2 2 5 7 15 16 16 28 29 30

Distribuzione 7 3 4 11 14 15 16 17 19 25 26

Distribuzione 8 0 6 8 12 16 16 17 21 26 28

Distribuzione 9 2 2 7 9 14 14 19 26 27 30

Distribuzione 10 0 2 4 13 18 18 19 24 26 26

Distribuzione 11 0 5 11 12 15 17 18 20 25 27

Distribuzione 12 0 4 8 10 11 18 21 24 27 27

Distribuzione 13 0 4 5 9 12 17 24 26 26 27

Distribuzione 14 1 2 2 11 16 20 20 21 27 30

Distribuzione 15 0 2 9 11 14 14 22 24 26 28

Distribuzione 16 2 5 9 11 12 15 18 22 26 30

Distribuzione 17 1 1 8 11 11 18 21 23 26 30

Distribuzione 18 2 3 7 7 15 16 19 25 26 30

Distribuzione 19 4 9 10 12 14 14 19 21 22 25

Distribuzione 20 7 9 10 10 11 13 14 19 28 29

Distribuzione 21 0 5 11 12 13 17 19 20 26 27

Distribuzione 22 3 4 4 14 17 18 18 21 22 29

Distribuzione 23 1 3 11 15 16 18 19 21 22 24

Distribuzione 24 2 4 4 4 18 21 23 23 24 27

Distribuzione 25 0 4 6 6 13 15 23 25 28 30

Esempi delle distribuzioni possibili

The Experiment (3)• On each repetition, subjects were first asked to state which

axioms they agreed with...• ... and then in which order they wanted them

implemented.• Conflicts were pointed out.• If a unique distribution was not implied by the axioms they

wanted implemented, they were asked to implement more.• Subjects were told that if they had no axioms left to

implement, the distribution would be chosen at random.

Results• There was not a big difference between the two

treatments.• We note that there were subjects who agreed with several

principles but did not try to implement them simultaneously.

• Many subjects agreed with a principle and its opposite (which is OK if implemented sequentially but not if implemented simultaneously).

• Perhaps subjects were confused about the implications of the principles?

• Or they deliberately wanted to implement them sequentially?

Agreement and Implementation (in percentage terms)

Axiom Treatment 1 Treatment 2

Agreed-to Implemented Agreed-to Implemented

R 14.6 18.3 15.8 20.2 RR 11.8 9.6 11.2 7.9

D 9.5 19.1 12.1 26.4 RD 9.2 15.7 7.0 10.3

T 11.1 5.7 11.4 6.6 RT 2.4 0.9 3.1 1.7

GT 11.4 10.0 12.5 9.9 RGT 3.0 3.0 2.5 2.1

L 14.6 8.7 14.7 7.9RL 12.4 9.1 9.7 7.0

Conclusion and questions for the break

• We have reported on an experiment designed to see which principles relating to the distribution and redistribution of income are agreed with.

• It is the axioms that are important for this course.• Suppose society could choose any possible redistribution of a

given total income, which would it choose? Obviously the answer depends on the principles it agrees with. Answer this question supposing that society agrees with

• (1) Rawls (2) Transfer • (3) Reverse Rawls (4) Reverse Transfer• (at home you might like to think about the other axioms and

their reverses).

Fairness, Responsibility and Welfare

• Marc Fleurbaey, Oxford UP, 2008.• We focus on social issues concerned with distribution

and redistribution of income.• We continue with the theme of whether we can agree

on general principles determining distribution and redistribution.

• We will focus on two key dimensions – those of effort and luck.

• So income is determined by luck and effort.• Following Fleurbaey’s initial treatment we will assume

that these two are independent of each other.

Responsibility and Fairness• So a person’s income is determined partly by his work but partly

by his luck. This is a generalisation of what we did before.• We assume that the first is determined solely by the individual

and the latter solely by fate.• To keep our discussion simple we assume that each of these

takes just two values: the individual either works or does not; the individual has bad or good luck.

• We consider what the state ‘should’ do in terms of redistribution to ‘appropriately’ take into account these two features.

• We might agree that the state ‘should’ compensate for bad luck but should it do so equally for those who work and those who do not?

Income• We suppose that individuals get income which depends on

whether they work or not and also whether they have bad or good luck. Here is an example:

• In this context, Social Choice is about redistribution of the earnings. Should society redistribute and if so how?

No Work WorkBad luck 1 4

Good luck 3 6

Transfers

• We suppose that Social Choice here is about deciding some transfers (you can think of these as being taxes and subsidies). Further we suppose that people’s happiness depends upon their income after transfers.

• Let us suppose for convenience that the total transfers must be zero – so it is simple redistribution.

• What principles should be used in deciding on these transfers?

• One obvious principle in this context is no envy.• In the next slide we define it and explore its

implications.

No Envy• We have assumed that people decide whether to work or

not but then fate decides their income.• NO ENVY here means that no-one envies anyone else

given their decision whether to work or not.• This implies the following:• After-tax income is the same for all people who work.• After-tax income is the same for all people who don’t work.Fleurbaey calls this Equal Well-Being for Equal Responsibility.• The tax/subsidy is the same for all people with bad luck.• The tax/subsidy is the same for all people with good luck.Fleurbaey calls this Equal Treatment for Equal Circumstances.

No envy here?

Before-tax incomes No Work WorkBad luck 1 4

Good luck 3 6

Transfers No Work WorkBad luck +1 +1

Good luck -1 -1

After-tax incomes No Work WorkBad luck 2 5

Good luck 2 5

Yippee!• We seem to be able to have no envy here.• Two questions:• Is this the only way to do it with these before-tax incomes?• Is it always possible? Consider the following:

• Why is this different?• These are questions for the break.

Before-tax incomes No Work WorkBad luck 1 4

Good luck 3 8

No envy in this second case?

Before-tax incomes No Work WorkBad luck 1 4

Good luck 3 8

Transfers No Work WorkBad luck +1 +1

Good luck -1 -1

After-tax incomes No Work WorkBad luck 2 5

Good luck 2 7

That does not work – does this?

Before-tax incomes No Work WorkBad luck 1 4

Good luck 3 8

Transfers No Work WorkBad luck +2 +2

Good luck -2 -2

After-tax incomes No Work WorkBad luck 3 6

Good luck 1 6

What does?

Before-tax incomes No Work WorkBad luck 1 4

Good luck 3 8

Transfers No Work WorkBad luck

Good luck

After-tax incomes No Work WorkBad luck

Good luck

No Envy does not always work• The reason is simply that luck has a different effect on

income in the work and no-work states.• The implication is that we have to weaken one of the two

implications of No Envy, either Equal Well-Being for Equal Responsibility or Equal Treatment for Equal Circumstances.

• Which?• This is a real pity as it seems we might have a principle to

which everybody might agree.• It gives more work to Social Choice theorists...• ... their work is never (can never be?) done.

Conclusions

• Once again we seem to have an impossibility result.

• Even if we agree on the general principle of No Envy, it is impossible to implement it everywhere...

• ...there is bound to be envy somewhere.• Is this surprising?• What do we do about it?