Chapter Thirty-One Welfare

Social Choice u Different economic states will be preferred by different individuals. u How can individual preferences be aggregated into a social preference over all possible economic states?

Aggregating Preferences u x, y, z denote different economic states. u 3 agents; Bill, Bertha and Bob. u Use simple majority voting to decide a state?

Aggregating Preferences More preferred Less preferred

Aggregating Preferences Majority Vote Results x beats y

Aggregating Preferences Majority Vote Results x beats y y beats z

Aggregating Preferences Majority Vote Results x beats y y beats z z beats x

Aggregating Preferences Majority Vote Results x beats y y beats z z beats x No socially best alternative!

Aggregating Preferences Majority Vote Results x beats y y beats z z beats x Majority voting does not always aggregate transitive individual preferences into a transitive social preference. No socially best alternative!

Aggregating Preferences

Rank-order vote results (low score wins).

Aggregating Preferences Rank-order vote results (low score wins). x-score = 6

Aggregating Preferences x-score = 6 y-score = 6 Rank-order vote results (low score wins).

Aggregating Preferences x-score = 6 y-score = 6 z-score = 6 Rank-order vote results (low score wins).

Aggregating Preferences x-score = 6 y-score = 6 z-score = 6 No state is selected! Rank-order vote results (low score wins).

Aggregating Preferences x-score = 6 y-score = 6 z-score = 6 No state is selected! Rank-order voting is indecisive in this case. Rank-order vote results (low score wins).

Manipulating Preferences u As well, most voting schemes are manipulable. u I.e. one individual can cast an untruthful vote to improve the social outcome for himself. u Again consider rank-order voting.

Manipulating Preferences These are truthful preferences.

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies.

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7 z-score = 6

Manipulating Preferences These are truthful preferences. Bob introduces a new alternative and then lies. Rank-order vote results. x-score = 8 y-score = 7 z-score = 6 -score = 9 z wins!!

Desirable Voting Rule Properties u 1. If all individuals preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. u 2. If all individuals rank x before y then so should the voting rule. u 3. Social preference between x and y should depend on individuals preferences between x and y only.

Desirable Voting Rule Properties u Kenneth Arrows Impossibility Theorem: The only voting rule with all of properties 1, 2 and 3 is dictatorial.

Desirable Voting Rule Properties u Kenneth Arrows Impossibility Theorem: The only voting rule with all of properties 1, 2 and 3 is dictatorial. u Implication is that a nondictatorial voting rule requires giving up at least one of properties 1, 2 or 3.

Social Welfare Functions u 1. If all individuals preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. u 2. If all individuals rank x before y then so should the voting rule. u 3. Social preference between x and y should depend on individuals preferences between x and y only.

Social Welfare Functions u 1. If all individuals preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. u 2. If all individuals rank x before y then so should the voting rule. u 3. Social preference between x and y should depend on individuals preferences between x and y only. Give up which one of these?

Social Welfare Functions u 1. If all individuals preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. u 2. If all individuals rank x before y then so should the voting rule. u 3. Social preference between x and y should depend on individuals preferences between x and y only. Give up which one of these?

Social Welfare Functions u 1. If all individuals preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. u 2. If all individuals rank x before y then so should the voting rule. There is a variety of voting procedures with both properties 1 and 2.

Social Welfare Functions u u i (x) is individual is utility from overall allocation x.

Social Welfare Functions u u i (x) is individual is utility from overall allocation x. u Utilitarian:

Social Welfare Functions u u i (x) is individual is utility from overall allocation x. u Utilitarian: u Weighted-sum:

Social Welfare Functions u u i (x) is individual is utility from overall allocation x. u Utilitarian: u Weighted-sum: u Minimax:

Social Welfare Functions u Suppose social welfare depends only on individuals own allocations, instead of overall allocations. u I.e. individual utility is u i (x i ), rather than u i (x). u Then social welfare is where is an increasing function.

Social Optima & Efficiency u Any social optimal allocation must be Pareto optimal. u Why?

Social Optima & Efficiency u Any social optimal allocation must be Pareto optimal. u Why? u If not, then somebodys utility can be increased without reducing anyone elses utility; i.e. social suboptimality inefficiency.

Utility Possibilities OBOB OAOA 0 0

OBOB OAOA 0 0

OBOB OAOA 0 0

OBOB OAOA 0 0

OBOB OAOA 0 0

OBOB OAOA 0 0

OBOB OAOA 0 0 Utility possibility frontier (upf)

Utility Possibilities OBOB OAOA 0 0 Utility possibility frontier (upf) Utility possibility set

Social Optima & Efficiency Upf is the set of efficient utility pairs.

Social Optima & Efficiency Upf is the set of efficient utility pairs. Social indifference curves

Social Optima & Efficiency Upf is the set of efficient utility pairs. Social indifference curves Higher social welfare

Social Optima & Efficiency Upf is the set of efficient utility pairs. Social indifference curves Higher social welfare

Social Optima & Efficiency Upf is the set of efficient utility pairs. Social indifference curves Social optimum

Social Optima & Efficiency Upf is the set of efficient utility pairs. Social indifference curves Social optimum is efficient.

Fair Allocations u Some Pareto efficient allocations are unfair. u E.g. one consumer eats everything is efficient, but unfair. u Can competitive markets guarantee that a fair allocation can be achieved?

Fair Allocations u If agent A prefers agent Bs allocation to his own, then agent A envies agent B. u An allocation is fair if it is –Pareto efficient –envy free (equitable).

Fair Allocations u Must equal endowments create fair allocations?

Fair Allocations u Must equal endowments create fair allocations? u No. Why not?

Fair Allocations u 3 agents, same endowments. u Agents A and B have the same preferences. Agent C does not. u Agents B and C trade agent B achieves a more preferred bundle. u Therefore agent A must envy agent B unfair allocation.

Fair Allocations u 2 agents, same endowments. u Now trade is conducted in competitive markets. u Must the post-trade allocation be fair?

Fair Allocations u 2 agents, same endowments. u Now trade is conducted in competitive markets. u Must the post-trade allocation be fair? u Yes. Why?

Fair Allocations u Endowment of each is u Post-trade bundles are and

Fair Allocations u Endowment of each is u Post-trade bundles are and u Then and

Fair Allocations u Suppose agent A envies agent B. u I.e.

Fair Allocations u Suppose agent A envies agent B. u I.e. u Then, for agent A,

Fair Allocations u Suppose agent A envies agent B. u I.e. u Then, for agent A, u Contradiction. is not affordable for agent A.

Fair Allocations u This proves: If every agents endowment is identical, then trading in competitive markets results in a fair allocation.

Fair Allocations OAOA OBOB Equal endowments.

Fair Allocations OAOA OBOB Given prices p 1 and p 2. Slope = -p 1 /p 2

Fair Allocations OAOA OBOB Given prices p 1 and p 2. Slope = -p 1 /p 2

Fair Allocations OAOA OBOB Given prices p 1 and p 2. Slope = -p 1 /p 2

Fair Allocations OAOA OBOB Post-trade allocation -- is it fair?

Fair Allocations OAOA OBOB Post-trade allocation -- is it fair? Swap As and Bs post-trade allocations

Fair Allocations OAOA OBOB Post-trade allocation -- is it fair? Swap As and Bs post-trade allocations A does not envy Bs post-trade allocation. B does not envy As post-trade allocation.

Fair Allocations OAOA OBOB Post-trade allocation -- is it fair? Swap As and Bs post-trade allocations Post-trade allocation is Pareto-efficient and envy-free; hence it is fair.