Definition 1.2.(a*, P,)is a Walrasian equilibrium if (1) For any i, a solves max ui(i) s.t.p·n;≤ (2)I*is feasible r≤ Equilibrium price p*. Equilibrium allocation: 2),H Note: A p* for any A>0 is also an equilibrium price Ofer curve: Pi(p)=Ti(p, p wi. The equilibrium is the intersection point of the ffer curves Excess demand function P* is determined by 2(p*)so Proposition 1. 25.( Walras Law). If preferences are strictly monotonic, then p 2(p) Lemma1.1.Let△k-1={p∈R|∑P=1}.Iff:△k-1→ r is continuous and satisfies p·f(p)=0,Vp∈△-l,then3p∈△k-1st.f(p)≤0 Theorem 1.2.(Existence of Equilibrium). If preferences are strictly convex, strictly monotonic, and continuous, then an equilibrium exists. L Good j is desirable if p;=0=2(p)>0 Proposition 1. 26.(Market Clearing). Suppose the preferences are strictly If good j is desirable, then 2(p)=0 and P>0 in equilibrium
Definition 1.2. (x∗, p∗) is a Walrasian equilibrium if (1) For any i, x∗ i solves ⎧ ⎪⎨ ⎪⎩ max ui(xi) s.t. p∗ · xi ≤ p∗ · wi (2) x∗ is feasible: [n i=1 x∗ i ≤ [n i=1 wi. Equilibrium price p∗. Equilibrium allocation: x∗ i = xi(p∗ , p∗ · wi), ∀ i. Note: λ p∗ for any λ > 0 is also an equilibrium price. Offer curve: ϕi(p) ≡ xi(p, p · wi). The equilibrium is the intersection point of the offer curves. Excess demand function: z(p) ≡ [n i=1 xi(p, p · wi) −[n i=1 wi. 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 7 k−1 ≡ {p ∈ Rk + | Spi = 1}. If f : 7k−1 → Rk is continuous and satisfies p · f(p)=0, ∀ p ∈ 7k−1, then ∃ p∗ ∈ 7k−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 pj = 0 ⇒ zj (p) > 0. Proposition 1.26. (Market Clearing). Suppose the preferences are strictly monotonic. If good j is desirable, then zj (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 differentiable, and all goods are gross substitutes and desirable, then the equilibrium price p" is unique up to a positive multiplier, i.e (i/pk,……,r-1/p) Is unique Example 1. 13. Find the equilibrium for u1(a, g)=ry, u1=(10,20); u2(, 9)=a y 2=(20,5) Example 1. 14. Find the equilibrium for u1(, y)=mina, yF, 1=(40,0) u2(a, y)=mina, y1, 4.2. Optimality of Equilibria Definition 1.3. A feasible allocation is Pareto optimal or weakly Pareto optimal if there is no feasible allocation a' s.t. iri i, v i. That is, one can no longer make everyone better off. Definition 1. 4. A feasible allocation is strongly Pareto optimal if there is no feasible allocation a'st.(1)ai i,V i, and(2)3 io s.t. i io Tio. That is, one can no longer make anyone better off without hurting others Example 1.15. Suppose that there is only one good and two agents. Individual 1's con a =(a1, r2)E R2. The feasible set of allocations is the shaded area in Figure 4/Lector sumption is 1 E R and individual 2s consumption is 2 E R. The allocation is a Feasible setc Figure 4.5. One Good and Two agents 19
Theorem 1.3. (Uniqueness of Equilibrium). If the preferences are strictly monotonic, all demand functions are differentiable, 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 u1(x, y) = xy, w1 = (10, 20); u2(x, y) = x2y, w2 = (20, 5). Example 1.14. Find the equilibrium for u1(x, y) = min{x, y}, w1 = (40, 0); u2(x, y) = min{x, y}, w2 = (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 x0 s.t. x0 i "i xi, ∀ i. That is, one can no longer make everyone better off. Definition 1.4. A feasible allocation x is strongly Pareto optimal if there is no feasible allocation x0 s.t. (1) x0 i i xi, ∀ i, and (2) ∃ i0 s.t. x0 i0 "i0 xi0 . That is, one can no longer make anyone better off without hurting others. Example 1.15. Suppose that there is only one good and two agents. Individual 1’s consumption is x1 ∈ R and individual 2’s consumption is x2 ∈ R. The allocation is a vector x = (x1, x2) ∈ R2. The feasible set of allocations is the shaded area in Figure 4.5. 1 x 2 x Feasible set A B C D Figure 4.5. One Good and Two Agents 1 — 19
Example 1.16. Consider (x,y) 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 is the slope of the indifference curve, measuring the substitutability of the two goods, defined by dui(ai)/aui(ai) MRSI(i)=a axh pay in good h for one more unit of good Z Proposition 1. 28. Suppose ui is differentiable, quasi-concave and Dru; (a)>0, V Then, a feasible allocation a is po iff MRSI(a1)=MRSI(a2) MRS(an) The contract curve is the set of all po allocations Cxample 1.17. For the agents in Example 1. 13, find the contract curve. L Example 1. 18. For the agents in Example 1. 14, find the PO allocations Theorem 1. 4.(First Welfare Theorem). If (a, p*) is a Walrasian equilibrium,I* 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 is a Walrasian equilibrium allocation with a proper redistribution of endowments Note: convexity of preferences is crucial
Example 1.16. Consider u1(x, y) = xy, u2(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. MRSlh i is the slope of the indifference curve, measuring the substitutability of the two goods, defined by MRSlh i (xi) ≡ ∂ui(xi) ∂xl i ! ∂ui(xi) ∂xh i ≡ pay in good h for one more unit of good l. Proposition 1.28. Suppose ui is differentiable, quasi-concave and Dxui(x) > 0, ∀ i. Then, a feasible allocation x is PO iff MRSlh 1 (x1) = MRSlh 2 (x2) = ··· = MRSlh n (xn), ∀ 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
