1. Chapter 34
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; Bill, Bertha and Bob.
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 No
socially
best
alternative!
x beats y
y beats z
z beats x

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

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

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

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

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

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.
Bob introduces a
new alternative

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

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

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

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

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
z-score = 6

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
-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.

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.
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
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.

33. Social Welfare Functions
ui(x) is individual i’s utility from overall allocation x.

34. Social Welfare Functions
ui(x) is individual i’s utility from overall allocation x.
Utilitarian:

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

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

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

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

39. 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.

40. Utility PossibilitiesOB OA

41. Utility PossibilitiesOB OA

42. Utility PossibilitiesOB OA

43. Utility PossibilitiesOB OA

44. Utility PossibilitiesOB OA

45. Utility PossibilitiesOB OA

46. Utility Possibilities
Utility possibility
frontier (upf) OB OA

47. Utility Possibilities
Utility possibility
frontier (upf) OB
Utility possibility set OA

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

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

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

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

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

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

54. 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?

55. 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).

56. Fair Allocations
Must equal endowments create fair allocations?

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

58. 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.

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

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

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

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

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

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

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

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

67. Fair Allocations OB OA
Equal endowments.

68. Fair Allocations OB
Given prices
p1 and p2.
Slope
= -p1/p2 OA

69. Fair Allocations OB
Given prices
p1 and p2.
Slope
= -p1/p2 OA

70. Fair Allocations OB
Given prices
p1 and p2.
Slope
= -p1/p2 OA

71. Fair Allocations OB
Post-trade
allocation --
is it fair? OA

72. Fair Allocations OB Swap A’s and B’s post-trade allocations Post-trade
is it fair? OA

73. Fair Allocations OB Swap A’s and B’s post-trade allocations Post-trade
is it fair? OA
A does not envy B’s post-trade allocation.
B does not envy A’s post-trade allocation.

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