2. Chapter Thirty-One Welfare

3. 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?

4. 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?

5. Aggregating Preferences More preferred Less preferred

6. Aggregating Preferences Majority Vote Results x beats y

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

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

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

10. 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!

11. Aggregating Preferences

12. Rank-order vote results (low score wins).

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

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

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

17. 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).

18. 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.

19. Manipulating Preferences These are truthful preferences.

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

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

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

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

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

25. 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

26. 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!!

27. 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.

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

29. 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.

30. 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.

31. 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?

32. 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?

33. 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.

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

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

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

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

38. 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.

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

40. 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.

41. Utility Possibilities OBOB OAOA 0 0

42. OBOB OAOA 0 0

43. OBOB OAOA 0 0

44. OBOB OAOA 0 0

45. OBOB OAOA 0 0

46. OBOB OAOA 0 0

47. OBOB OAOA 0 0 Utility possibility frontier (upf)

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

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

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

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

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

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

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

55. 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?

56. 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).

57. Fair Allocations u Must equal endowments create fair allocations?

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

59. 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.

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

61. 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?

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

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

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

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

66. 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.

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

68. Fair Allocations OAOA OBOB Equal endowments.

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

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

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

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

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

74. 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.

75. 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.