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,∑ ,vj
Social 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 endowment 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
Definition 1.5. An allocation ( y), 20, is feasible if n≤∑m1∑y;G()≤0 Definition 1.6.(ar*, y,,, p") is a Walrasian equilibrium if (1)a solves max mila s t 访P (2)yf solves max G(3)≤0 (3(r*, y") is feasible x2≤>m Example 1.19. Add a firm to the economy in Example 1. 13. The firm inputs to produce y by production function y= Ve. 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 a t produce y by production function y=Va. The two consumers share the firm equally. Find the equilibrium price In equilibrium, MRS(i)=MRT (yi) p￥ l, h Definition 1.7. A feasible allocation(a, y) is Pareto optimal if 3 no feasible alloca- tion(a,, y) s.t. iriTi, vi. Proposition 1.31. Suppose that u; is differentiable, quasi-concave and Drui(a)>0, for all i, and G, is differentiable, quasi-convex and D,, (vi)>0. Then, a feasible allocation(a, y) is Pareto optimal iff MRS出(x1)=…=MRSn(xn)=MRnh()=…=MBmh(ym),V1,h.■(4.7)
Definition 1.5. An allocation (x, y), x ≥ 0, is feasible if [n i=1 xi ≤ [n i=1 wi +[m j=1 yj ; Gj (yj ) ≤ 0, ∀ j. Definition 1.6. (x∗, y∗, p∗) is a Walrasian equilibrium if (1) x∗ i solves max ui(xi) s.t. p∗ · xi ≤ p∗ · wi + Sm j=1 αij p∗ · y∗ j (2) y∗ j solves max p∗ · yj s.t. Gj (yj ) ≤ 0 (3) (x∗, y∗) is feasible: [n i=1 x∗ i ≤ [n i=1 wi +[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, MRSlh i (xi) = MRTlh j (yi) = pl ph , ∀ l, h, i, j. Definition 1.7. A feasible allocation (x, y) is Pareto optimal if ∃ no feasible allocation (x0 , y0 ) s.t. x0 i "i xi, ∀ i. Proposition 1.31. Suppose that ui is differentiable, quasi-concave and Dxui(x) > 0, for all i, and Gj is differentiable, quasi-convex and DyGj (yj ) > 0. Then, a feasible allocation (x, y) is Pareto optimal iff MRSlh 1 (x1) = ··· = MRSlh n (xn) = MRTlh 1 (y1) = ··· = MRTlh m (ym), ∀ l, h. (4.7) 1 — 22
Theorem 1.6.(The First Welfare Theorem). If (a*, y,, p*) is a Walrasian equi- librium,(az, y)is PO.I Theorem 1.7.(The Second Welfare Theorem). Suppose preferences are contin- uous, strictly monotonic, and strictly convex. Then, any PO allocation (a', yr)is a Walrasian equilibrium allocation with proper distributions of profit shares and endow- The two results about social optimality also hold 4.4. General Equilibrium with Uncertainty Let T=all possible states=(1, 2, .. T. State t occurs with probability Tt Individuals i=1, 2, .., n, with endowment w; E Ri for t. Given t, i's consumption bundle is2∈R+, In a plan, i's consumption bundle is a;≡(x2,x2,…,x)∈R*with 丌(2 t=1 m firms j=1, 2,., m, with production possibilities G;: RKT-R, j=1, 2,., m, whee≡(,,…,)∈ R is the net output vector and p=(p3,p2,…,p)∈R杆 is the price vector. The firms are owned by individuals, where aii is the share of firm j owned by agent 0≤0;≤1,∑:a;=1,V Contingent contracts economy: contracts on buying and selling are contingent on all the states of the economy Definition 1.8. A allocation(a, y), a 20, is feasible if ∑≤∑v+∑功,;G()≤0 Definition 1.9.(a, y* p*) is a Walrasian equilibrium if (1)∈ RRT solves t.∑p”·x≤∑p” t=1j=1
Theorem 1.6. (The First Welfare Theorem). If (x∗, y∗, p∗) is a Walrasian equilibrium, (x∗, y∗) is PO. Theorem 1.7. (The Second Welfare Theorem). Suppose preferences are continuous, strictly monotonic, and strictly convex. Then, any PO allocation (x∗, y∗) is a Walrasian equilibrium allocation with proper distributions of profit shares and endowments. 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 πt . Individuals i = 1, 2, . . . , n, with endowment wt i ∈ Rk + for t. Given t, i’s consumption bundle is xt i ∈ Rk +. In a plan, i’s consumption bundle is xi ≡ (x1 i , x2 i ,...,xτ i ) ∈ Rkτ + with ui(xi) = [τ t=1 πt iui(xt i). m firms j = 1, 2, . . . , m, with production possibilities Gj : Rkτ → R, j = 1, 2, . . . , m, where yj ≡ (y1 j , y2 j ,...,yτ j ) ∈ Rkτ is the net output vector and p = (p1, p2,...,pτ ) ∈ Rkτ + is the price vector. The firms are owned by individuals, where αij is the share of firm j owned by agent i, 0 ≤ αij ≤ 1, S i αij = 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 xt i ≤ [n i=1 wt i +[m j=1 yt j , ∀ t; Gj (yj ) ≤ 0, ∀ j. Definition 1.9. (x∗, y∗, p∗) is a Walrasian equilibrium if (1) x∗ i ∈ Rkτ + solves max ui(xi) s.t. Sτ t=1 p∗t · xt i ≤ Sτ t=1 p∗t · wt i + Sτ t=1 Sm j=1 αij p∗t · y∗t j 1 — 23
(2)∈卫灯 solves ∑p2·v stG(v,2,…,)≤0 (3)(z*, y") is feasible Definition 1.10. A feasible allocation (, y) is Pareto optimal if 3 no feasible allo- cation (,y) s.t. Ti>i i, V i Since this economy is a special case of the previous one, the four welfare theorems still hold This economy requires k x T markets- informationally inefficient 24
(2) y∗ j ∈ Rkτ solves max Sτ t=1 p∗t · yt j s.t. Gj (y1 j , y2 j ,...,yτ j ) ≤ 0 (3) (x∗, y∗) is feasible: [n i=1 x∗t i ≤ [n i=1 wt 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 allocation (x0 , y0 ) s.t. x0 i "i xi, ∀ 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 inefficient. 1 — 24
