Social welfare function W : R n → R gives social utility W(u 1 ,u 2 ,...,u n ).Wis strictly increasing. x ? is socially optimal if it solves: max W[u 1 (x 1 ),u 2 (x 2 ),...,u n (x n )] s.t. n [ i=1 x i ≤ n [ i=1 w i 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 a i > 0,i=1,...,n, s.t. x ? solves max W ≡ [ a i u i (x i ) s.t. [ x i ≤ [ x ? i , where the weights a i 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: G j ,j=1,2,...,m. The firms are owned by n individuals i =1,2,...,n, with endowments w i ,i= 1,2,...,n. Let α ij be the share of firm j owned by agent i, 0 ≤ α ij ≤ 1, S i α ij = 1, ? j. 1—21 Definition 1.5. An allocation (x, y),x≥ 0, is feasible if n [ i=1 x i ≤ n [ i=1 w i + m [ j=1 y j ; G j (y j ) ≤ 0, ? j.  Definition 1.6. (x ? ,y ? ,p ? ) is a Walrasian equilibrium if (1) x ? i solves max u i (x i ) s.t. p ? ·x i ≤ p ? · w i + m S j=1 α ij p ? · y ? j (2) y ? j solves max p ? ·y j s.t. G j (y j ) ≤ 0 (3) (x ? ,y ? ) is feasible: n [ i=1 x ? i ≤ n [ i=1 w i + m [ j=1 y ? j .  Example 1.19. Add a firm to the economy in Example 1.13. The firm inputs x to produce y by production function y = √ x. The two consumers share the firm equally. Find the equilibrium price. Example 1.20. Add a firm to the economy in Example 1.14. The firm inputs x to produce y by production function y = √ x. The two consumers share the firm equally. Find the equilibrium price. In equilibrium, MRS lh i (x i )=MRT lh j (y i )= p l p h , ? l, h, i, j. Definition 1.7. A feasible allocation (x, y) is Pareto optimal if ? no feasible alloca- tion (x 0 ,y 0 ) s.t. x 0 i null i x i , ? i.  Proposition 1.31. Suppose that u i is di?erentiable, quasi-concave and D x u i (x) > 0, for all i, and G j is di?erentiable, quasi-convex and D y G j (y j ) > 0. Then, a feasible allocation (x,y) is Pareto optimal i? MRS lh 1 (x 1 )=···= MRS lh n (x n )=MRT lh 1 (y 1 )=···= MRT lh m (y m ), ? l, h.  (4.7) 1—22 Theorem 1.6. (The First Welfare Theorem). If (x ? ,y ? ,p ? ) is a Walrasian equi- librium, (x ? ,y ? ) is PO.  Theorem 1.7. (The Second Welfare Theorem). Suppose preferences are contin- uous, strictly monotonic, and strictly convex. Then, any PO allocation (x ? ,y ? ) is a Walrasian equilibrium allocation with proper distributions of profit shares and endow- ments.  The two results about social optimality also hold. 4.4. General Equilibrium with Uncertainty Let T ≡ {all possible states} = {1,2,...,τ}. State t occurs with probability π t . Individuals i =1,2,...,n, with endowment w t i ∈R k + for t. Given t, i’s consumption bundle is x t i ∈R k + . In a plan, i’s consumption bundle is x i ≡ (x 1 i ,x 2 i ,...,x τ i ) ∈R kτ + with u i (x i )= τ [ t=1 π t i u i (x t i ). m firms j =1,2,...,m, with production possibilities G j : R kτ → R,j=1,2,...,m, where y j ≡ (y 1 j ,y 2 j ,...,y τ j ) ∈R kτ is the net output vector and p =(p 1 ,p 2 ,...,p τ ) ∈R kτ + is the price vector. The firms are owned by individuals, where α ij is the share of firm j owned by agent i, 0 ≤ α ij ≤ 1, S i α ij =1, ? j. Contingent contracts economy: contracts on buying and selling are contingent on all the states of the economy. Definition 1.8. A allocation (x,y),x≥ 0, is feasible if n [ i=1 x t i ≤ n [ i=1 w t i + m [ j=1 y t j , ? t; G j (y j ) ≤ 0, ? j.  Definition 1.9. (x ? ,y ? ,p ? ) is a Walrasian equilibrium if (1) x ? i ∈R kτ + solves max u i (x i ) s.t. τ S t=1 p ?t · x t i ≤ τ S t=1 p ?t · w t i + τ S t=1 m S j=1 α ij p ?t · y ?t j 1—23 (2) y ? j ∈R kτ solves max τ S t=1 p ?t · y t j s.t. G j (y 1 j ,y 2 j ,...,y τ j ) ≤ 0 (3) (x ? ,y ? ) is feasible: n [ i=1 x ?t i ≤ n [ i=1 w t i + m [ j=1 y ?t j ,t=1,2,...,τ.  Definition 1.10. A feasible allocation (x,y) is Pareto optimal if ? no feasible allo- cation (x 0 ,y 0 ) s.t. x 0 i null i x i , ? i.  Since this economy is a special case of the previous one, the four welfare theorems still hold. This economy requires k ×τ markets – informationally ine?cient. 1—24