ADVANCED MICROECONOMIC THEORY (GEOFFREY A. JEHLE) GENERAL EQUILIBRIUM - EQUILIBRIUM IN COMPETITIVE MARKET SYSTEMS THEOREM 5.1 Basic Properties of Demand If u' satisfies Assumption 5.1 then for each p > 0, the consumer's problem (5.2) has a unique solution, x'(p, p . e'). In addition, x'(p, p . e') is continuous in p on R* . THEOREM 5.2 Properties of Aggregate Excess Demand Functions If for each consumer i, u' satisfies Assumption 5.1, then for all p >> 0, 1. Continuity: z(.) is continuous at p. 2. Homogeneity: z(p) = z(p) for all > > 0. 3. Walras' law: p . z(p) = 0. THEOREM 5.3 Aggregate Excess Demand and Walrasian Equilibrium Suppose z: R" -> R" satisfies the following three conditions: 1. z(.) is continuous on RH; 2. p . z(p) = 0 for all p >> 0; 3. If (p") is a sequence of price vectors in R" converging to p # 0, and pk = 0 for some good k, then for some good k' with PK = 0, the associated sequence of excess demands in the market for good k', (zk (p")), is unbounded above. Then there is a price vector p* >> 0 such that z(p*) = 0.THEOREM 5.4 Utility and Aggregate Excess Demand If each consumer's utility function satisfies Assumption 5.1, and if the aggregate endow- ment of each good is strictly positive (i.e., _ _1 e' > 0), then aggregate excess demand satisfies conditions 1 through 3 of Theorem 5.3. THEOREM 5.5 Existence of Walrasian Equilibrium If each consumer's utility function satisfies Assumption 5.1, and __1 e' >> 0, then there exists at least one price vector, p* >> 0, such that z(p*) = 0. THEOREM 5.6 Core and Equilibria in Competitive Economies Consider an exchange economy (u', e')iET. If each consumer's utility function, u', is strictly increasing on ", then every Walrasian equilibrium allocation is in the core. That is, W(e) C C(e). THEOREM 5.7 First Welfare Theorem Under the hypotheses of Theorem 5.6, every Walrasian equilibrium allocation is Pareto efficient. THEOREM 5.8 Second Welfare Theorem Consider an exchange economy (u', e')ieI with aggregate endowment __ e' >> 0, and with each utility function u' satisfying Assumption 5.1. Suppose that & is a Pareto-efficient allocation for (u', e' )iET, and that endowments are redistributed so that the new endow- ment vector is x. Then & is a Walrasian equilibrium allocation of the resulting exchange economy (u', X')ieI