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 crucialExample 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