Chapter 33 Welfare

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

3 Aggregating Preferences x, y, z denote different economic states. 3 agents: Anna, Bill and Cindy. Use simple majority voting to decide a state?

4 Aggregating Preferences More preferred Less preferred

5 Aggregating Preferences Majority Vote Results x beats y

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

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

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

9 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!

10 Aggregating Preferences

11 Aggregating Preferences Rank-order vote results (low score wins).

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

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

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

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

16 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).

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

18 Manipulating Preferences These are truthful preferences.

19 Manipulating Preferences These are truthful preferences. Cindy introduces a new alternative

20 Manipulating Preferences These are truthful preferences. Cindy introduces a new alternative

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

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

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

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

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

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

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

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

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

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

31 Social Welfare Functions 1. If all individuals’ preferences are complete, reflexive and transitive, then so should be the social preference created by the voting rule. 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.

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

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

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

35 Social Optima & Efficiency Any social optimal allocation must be Pareto optimal. Why? If not, then somebody’s utility can be increased without reducing anyone else’s utility; i.e. social suboptimality  inefficiency.

36 Utility Possibilities OBOB OAOA 0

37 Utility Possibilities OBOB OAOA 0

38 Utility Possibilities OBOB OAOA 0

39 Utility Possibilities OBOB OAOA 0

40 Utility Possibilities OBOB OAOA 0

41 Utility Possibilities OBOB OAOA 0

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

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

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

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

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

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

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

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

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

51 Fair Allocations If agent A prefers agent B’s allocation to his own, then agent A envies agent B. An allocation is fair if it is  Pareto efficient  envy free (equitable).

52 Fair Allocations Must equal endowments create fair allocations?

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

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

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

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

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

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

59 Fair Allocations This proves: If every agent’s endowment is identical, then trading in competitive markets results in a fair allocation.

60 Fair Allocations OAOA OBOB Equal endowments.

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

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

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

64 Fair Allocations OAOA OBOB Post-trade allocation -- is it fair? Swap A’s and B’s post-trade allocations

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

66 Fair Allocations OAOA OBOB Post-trade allocation -- is it fair? Swap A’s and B’s post-trade allocations Post-trade allocation is Pareto-efficient and envy-free; hence it is fair.