Social welfare function W: Rn-R gives social utility W(u1, u2,. un ). W is strictly increasing is socially optimal if it solves max Wu(a1), u2(a2),., un(n) st>Tis>w Proposition 1.29. If is SO, it is PO. I Proposition 1. 30. Suppose that preferences are continuous, strictly monotonic, and strictly convex. Then, for any PO allocation x* with >>0,v i, there exist ai> 0.t=1 n. s.t. a' solves maxW≡∑a11(xz) s t where the weights ai are the reciprocals of the marginal utilities of income. I Note: competitive market favors individuals with large incomes Summary (1)Competitive equilibria and SO allocations are PO (2)PO allocations are competitive equilibria under convexity of preferences and endow- ment redistribution (3)PO allocations are SO under convexity of preferences and a linear social welfare function with special weights 4.3. General Equilibrium with Production m firms: i=1, 2, .. m. Production possibilities: Gi,j= 1, 2,...,m The firms are owned by n individuals i= 1, 2, .. n, with endowments wi, i 1,2,…,n. Let aii be the share of firm j owned by agent i,0≤a≤1,∑ ,vjSocial welfare function W : Rn → R gives social utility W(u1, u2,...,un). W is strictly increasing. x∗ is socially optimal if it solves: max W[u1(x1), u2(x2),...,un(xn)] s.t. [n i=1 xi ≤ [n i=1 wi Proposition 1.29. If x∗ is SO, it is PO.  Proposition 1.30. Suppose that preferences are continuous, strictly monotonic, and strictly convex. Then, for any PO allocation x∗ with x∗ i >> 0, ∀ i, there exist ai > 0, i = 1, . . . , n, s.t. x∗ solves max W ≡ [aiui(xi) s.t. [xi ≤ [x∗ i , where the weights ai are the reciprocals of the marginal utilities of income.  Note: competitive market favors individuals with large incomes. Summary: (1) Competitive equilibria and SO allocations are PO. (2) PO allocations are competitive equilibria under convexity of preferences and endow￾ment redistribution. (3) PO allocations are SO under convexity of preferences and a linear social welfare function with special weights. 4.3. General Equilibrium with Production m firms: j = 1, 2, . . . , m. Production possibilities: Gj , j = 1, 2, . . . , m. The firms are owned by n individuals i = 1, 2, . . . , n, with endowments wi, i = 1, 2, . . . , n. Let αij be the share of firm j owned by agent i, 0 ≤ αij ≤ 1, S i αij = 1, ∀ j. 1 — 21