Definition 1.2. (x ? ,p ? ) is a Walrasian equilibrium if (1) For any i, x ? i solves ? ? ? ? ? max u i (x i ) s.t. p ? · x i ≤ p ? · w i (2) x ? is feasible: n null i=1 x ? i ≤ n null i=1 w i .  Equilibrium price p ? . Equilibrium allocation: x ? i = x i (p ? ,p ? · w i ), ? i. Note: λ p ? for any λ > 0 is also an equilibrium price. O?er curve: ? i (p) ≡ x i (p, p · w i ). The equilibrium is the intersection point of the o?er curves. Excess demand function: z(p) ≡ n null i=1 x i (p, p · w i )? n null i=1 w i . p ? is determined by z(p ? ) ≤ 0. Proposition 1.25. (Walras Law). If preferences are strictly monotonic, then p·z(p)= 0, ? p.  Lemma 1.1. Let null k?1 ≡ {p ∈ R k + | null p i =1}. If f : null k?1 → R k is continuous and satisfies p · f(p)=0, ? p ∈null k?1 , then ? p ? ∈null k?1 s.t. f(p ? ) ≤ 0.  Theorem 1.2. (Existence of Equilibrium). If preferences are strictly convex, strictly monotonic, and continuous, then an equilibrium exists.  Good j is desirable if p j =0? z j (p) > 0. Proposition 1.26. (Market Clearing). Suppose the preferences are strictly monotonic. If good j is desirable, then z j (p ? )=0 and p ? j > 0 in equilibrium.  1—18 Theorem 1.3. (Uniqueness of Equilibrium). If the preferences are strictly monotonic, all demand functions are di?erentiable, and all goods are gross substitutes and desirable, then the equilibrium price p ? is unique up to a positive multiplier, i.e. (p ? 1 /p ? k , ..., p ? k?1 /p ? k ) is unique.  Example 1.13. Find the equilibrium for u 1 (x,y)=xy, w 1 =(10,20); u 2 (x,y)=x 2 y, w 2 =(20,5).  Example 1.14. Find the equilibrium for u 1 (x,y)=min{x,y},w 1 =(40,0); u 2 (x,y)=min{x,y},w 2 =(0,20).  4.2. Optimality of Equilibria Definition 1.3. A feasible allocation x is Pareto optimal or weakly Pareto optimal if there is no feasible allocation x 0 s.t. x 0 i null i x i , ? i. That is, one can no longer make everyone better o?. Definition 1.4. A feasible allocation x is strongly Pareto optimal if there is no feasible allocation x 0 s.t. (1) x 0 i null i x i , ? i, and (2) ? i 0 s.t. x 0 i 0 null i 0 x i 0 . That is, one can no longer make anyone better o? without hurting others. Example 1.15. Suppose that there is only one good and two agents. Individual 1’s con- sumption is x 1 ∈R and individual 2’s consumption is x 2 ∈R. The allocation is a vector x =(x 1 ,x 2 ) ∈R 2 . The feasible set of allocations is the shaded area in Figure 4.5. 1 x 2 x Feasible set AB C D Figure 4.5. One Good and Two Agents 1—19 Example 1.16. Consider u 1 (x,y)=xy, u 2 (x,y)=1. All the points in the Edgeworth box are weakly PO, but only one point is strongly PO.  Proposition 1.27. A strongly PO allocation is weakly PO. Conversely, if all the utility functions are continuous and strictly monotonic, a weakly PO allocation is strongly PO.  MRS lh i is the slope of the indi?erence curve, measuring the substitutability of the two goods, defined by MRS lh i (x i ) ≡ ?u i (x i ) ?x l i null ?u i (x i ) ?x h i ≡ pay in good h for one more unit of good l. Proposition 1.28. Suppose u i is di?erentiable, quasi-concave and D x u i (x) > 0, ? i. Then, a feasible allocation x is PO i? MRS lh 1 (x 1 )=MRS lh 2 (x 2 )=···= MRS lh n (x n ), ? l, h.  The contract curve is the set of all PO allocations. Example 1.17. For the agents in Example 1.13, find the contract curve.  Example 1.18. For the agents in Example 1.14, find the PO allocations.  Theorem 1.4. (First Welfare Theorem). If (x ? ,p ? ) is a Walrasian equilibrium, x ? is Pareto optimal.  Note: nothing about fairness. Theorem 1.5. (Second Welfare Theorem). Suppose that preferences are continuous, strictly monotonic, and strictly convex. Then, any Pareto optimal allocation x ? is a Walrasian equilibrium allocation with a proper redistribution of endowments.  Note: convexity of preferences is crucial. 1—20