Social Choice: Lecture 14

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Social Choice: Lecture 14

Ram Singh

Course 001

October 27, 2014

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Social Choice I

Question

How to choose from the feasible set of alternatives?

Do societies have preference relations similar to the ones assumed for individuals?

Is Pareto criterion helpful here?

What are the other approaches possible in a social context?

Ram Singh: (DSE) Social Choice October 27, 2014 2 / 12

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Pareto Criterion I

Let

Nbe the set of individuals.

Sbe the set of feasible alternatives.

uⁱ utility fn fori the individual Ube the set of possible utilities

U={(u¹(x), ...,uⁿ(x))|x∈S}

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Pareto Criterion II

Definition

xis ‘Pareto as goods as’y, i.e.,xRyif (∀i∈N)[xRiy]

Definition

xis Pareto superior toy, i.e.,xPyifxRybut∼yRx. That is, (∀i∈N)[xR_iy]

(∃j ∈N)[xP_jy]

Question

As a preference relation, is ‘Pareto-superior’ a complete relation?

As a preference relation, is ‘Pareto-as good as’ a complete relation?

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Rawls Criterion: Egalitarian World I

Veil of ignorance: Consider distributions of one good, say wealth, acrossn individuals in a society. Let

mini∈N

uⁱ(xⁱ) =minimum{u¹(x¹), ...,uⁿ(xⁿ)}

Definition

Distributionx= (x¹, ...,xⁿ)is Rawls superior to distributiony= (y¹, ...,yⁿ)if mini∈N

uⁱ(xⁱ)>min

i∈N

uⁱ(yⁱ)

When individual preferences are monotonic in the good, this implies:

Distributionx= (x¹, ...,xⁿ)is Rawls superior to distributiony= (y¹, ...,yⁿ)if mini∈N{x¹, ...,xⁿ}>min

i∈N{y¹, ...,yⁿ}

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Rawls Criterion: Egalitarian World II

Proposition LetPn

i=1eⁱ =C. Distributionx= (x¹, ...,xⁿ)is Rawls Best if {x¹, ...,xⁿ}=min{C

n, ...,C n}

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Rawls’ Criterion and Markets

Question

1 Is an Equal division also a Non-envious allocation?

2 Is an Equal division allocation Pareto Efficient?

Question

Suppose we start from a Rawls Best allocation as the endowment. Will competitive equilibrium allocation be egalitarian?

Proposition

When preferences are strongly monotonic and initial allocation is ‘Equal’, the competitive equilibrium is non-envious and Pareto efficient.

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Rawls Criterion: Limitations

In real world

individual welfare has several components;uⁱ(xⁱ), wherexⁱ has several components

Implications of policy interventions are complex

Individuals have different beliefs about the possible outcomes.

For example, consider distribution ofmgoods some of which are legal, economic and social entitlements. Define,

minj∈N

uⁱ(x^j) =minimum{uⁱ(x¹), ...,uⁱ(xⁿ)}

Now, person 1 may feel

minj∈Nu¹(x^j)>min

j∈Nu¹(y^j) But, person 2 may have

minj∈N

u²(x^j)<min

j∈N

u²(y^j)

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Kaldor-Hicks Criterion I

Let

xandybe any two allocations

S(x)be the set of allocations that are accessible fromx.

S(y)be the set of allocations that are accessible fromy.

Definition

xis Kaldor superior toy, i.e.,xKyif there existsz∈S(x)such thatzPy (∀i ∈N)[zR_iy]

(∃j∈N)[zPjy]

However, it is possible that

xKy and yKx.

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Scitovsky Criterion I

Definition

xis Scitovsky superior toy, i.e.,xSyif

xKy but

∼yKx

xKyimplies there existsz∈S(x)such thatzPy. That is, (∀i ∈N)[zRiy]

(∃j∈N)[zP_jy]

But, there is not∈S(y)such that

(∀i ∈N)[tRix]

(∃j ∈N)[tP_jx]

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Scitovsky Criterion II

Definition

All social states/alternatives are accessible from each other if (∀x,z,z)[z∈S(x)⇒z∈S(y)]

Proposition

If all social states/alternatives are accessible from each other thenxSyif and only ifxis P.O butyis not P.O

Proposition

If all social states/alternatives are accessible from each other thenxKyif and only ifyis not P.O.

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Samuelson Criterion

Definition

xis Samuelson superior toy, i.e.,xS¯yif for anyz∈S(y) xKz

That is, for anyz∈S(y), there existsw∈S(x)such thatwPz, i.e.,

(∀i ∈N)[wR_iz]

(∃j ∈N)[wPjz]