Welfare Social Welfare Functions uix is individual is

Welfare

Social Welfare Functions ui(x) is individual i’s utility from overall allocation x. n Utilitarian: n n Weighted-sum: n Minimax: 2

Social Welfare Functions Suppose social welfare depends only on individuals’ own allocations, instead of overall allocations. n I. e. individual utility is ui(xi), rather than ui(x). n Then social welfare is n where is an increasing function. 3

Social Optima & Efficiency Any social optimal allocation must be Pareto optimal. n Why? n If not, then somebody’s utility can be increased without reducing anyone else’s utility; i. e. social suboptimality inefficiency. n 4

Utility Possibilities OB 0 0 OA 5

Utility Possibilities OB 0 0 OA 6

Utility Possibilities Utility possibility frontier (UPF) OB 0 OA 0 Utility possibility set 7

Social Optima & Efficiency UPF is the set of efficient utility pairs. Social optimum is efficient. Higher social welfare Social indifference curves 8

Fair Allocations Some Pareto efficient allocations are “unfair”. n E. g. one consumer eats everything is efficient, but “unfair”. n Can competitive markets guarantee that a “fair” allocation can be achieved? n 9

Fair Allocations If agent A prefers agent B’s allocation to his own, then agent A envies agent B. n An allocation is fair if it is n ¨ Pareto efficient ¨ envy free (equitable). 10

Fair Allocations Must equal endowments create fair allocations? n No. Why not? n 11

Fair Allocations 3 agents, same endowments. n Agents A and B have the same preferences. Agent C does not. n Agents B and C trade agent B achieves a more preferred bundle. n Therefore agent A must envy agent B unfair allocation. n 12

Fair Allocations 2 agents, same endowments. n Now trade is conducted in competitive markets. n Must the post-trade allocation be fair? n Yes. Why? n 13

Fair Allocations Endowment of each is n Post-trade bundles are and n Then n and 14

Fair Allocations Suppose agent A envies agent B. n I. e. n n Then, for agent A, n Contradiction. is not affordable for agent A. 15

Fair Allocations n This proves: If every agent’s endowment is identical, then trading in competitive markets results in a fair allocation. 16

Fair Allocations OB OA Equal endowments. 17

Fair Allocations OB Slope = -p 1/p 2 Given prices p 1 and p 2. OA 18

Fair Allocations OB Post-trade allocation -is it fair? OA 19

Fair Allocations Swap A’s and B’s post-trade allocations OB Post-trade allocation -is it fair? OA A does not envy B’s post-trade allocation. B does not envy A’s post-trade allocation. 20

Fair Allocations n Post-trade allocation is ¨ Pareto-efficient and ¨ Envy-free; n Hence it is fair. 21