Welfare

1
Welfare
2
Social Welfare Functions

• ui(x) is individual is utility from overall
  allocation x.
• Utilitarian
• Weighted-sum
• Minimax

3
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 iswhere is an
  increasing function.

4
Social Optima Efficiency

• Any social optimal allocation must be Pareto
  optimal.
• Why?
• If not, then somebodys utility can be increased
  without reducing anyone elses utility i.e.
  social suboptimality ? inefficiency.

5
Utility Possibilities
OB
0
0
OA
6
Utility Possibilities
OB
0
0
OA
7
Utility Possibilities
Utility possibility frontier (UPF)
OB
0
0
Utility possibility set
OA
8
Social Optima Efficiency
UPF is the set of efficient
utility pairs.
Social optimum is efficient.
Higher social welfare
Social indifference curves
9
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?

10
Fair Allocations

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

11
Fair Allocations

• Must equal endowments create fair allocations?
• No. Why not?

12
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.

13
Fair Allocations

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

14
Fair Allocations

• Endowment of each is
• Post-trade bundles are
  and
• Thenand

15
Fair Allocations

• Suppose agent A envies agent B.
• I.e.
• Then, for agent A,
• Contradiction. is not affordable
  for agent A.

16
Fair Allocations

• This proves If every agents endowment is
  identical, then trading in competitive markets
  results in a fair allocation.

17
Fair Allocations
OB
OA
Equal endowments.
18
Fair Allocations
OB
Given prices p1 and p2.
Slope -p1/p2
OA
19
Fair Allocations
OB
Post-trade allocation -- is it fair?
OA
20
Fair Allocations
OB
Swap As and Bs post-trade allocations
Post-trade allocation -- is it fair?
OA
A does not envy Bs post-trade allocation. B does
not envy As post-trade allocation.
21
Fair Allocations

• Post-trade allocation is
• Pareto-efficient and
• Envy-free
• Hence it is fair.