Ichiro Obara UCLA December 3, 2008 Obara (UCLA) Core December 3, 2008 1 / 22
in Edgeworth Box Core in Edgeworth Box Obara (UCLA) Core December 3, 2008 2 / 22
in Edgeworth Box Motivation How should we interpret the price in Walrasian equilibrium? The notion of core provides a different way to understand competitive equilibria. Obara (UCLA) Core December 3, 2008 3 / 22
in Edgeworth Box Example Consider a pure exchange economy with two consumers and two goods. Contract curve is the set of Pareto efficient allocations which are better than the initial endowments for both consumers. Obara (UCLA) Core December 3, 2008 4 / 22
in Edgeworth Box Example Contract Curve 2 e 1 Obara (UCLA) Core December 3, 2008 5 / 22
in Edgeworth Box Example Consider a pure exchange economy with four consumers and two goods where two consumers are type 1 and two consumers are type 2 (type i: ( i, e i )). This economy is called a (2-)replica economy of the previous pure exchange economy with two consumers. Consider a feasible allocation x A (the same type consumes the same amount). x can be blocked by a coalition of one type 1 consumer and two type 2 consumers. Type 1 consumes at y 1 and type 2 consumes at y 2 where y 1 + 2y 2 = e 1 + 2e 2. Obara (UCLA) Core December 3, 2008 6 / 22
in Edgeworth Box Example 2 consumers for each type 2 y x y e 1 Obara (UCLA) Core December 3, 2008 7 / 22
in Edgeworth Box Example Consider only symmetric allocations (witin each type). Then we can use the same Edgeworth box to represent allocations in r-replica economies for r = 2, 3,... Every inefficient allocation can be blocked by the grand coalition. Pareto efficient allocations which are not on the contract curve are blocked by just one consumer. More and more (symmetric) allocations on the contract curve are blocked for r-replica economies as r increases. In the limit, every alocation other than the CE allocation in the two-consumer economy is blocked. Obara (UCLA) Core December 3, 2008 8 / 22
in Edgeworth Box Example With more consumers. 2 y x y e 1 Obara (UCLA) Core December 3, 2008 9 / 22
in Edgeworth Box Example in the limit 2 e 1 Obara (UCLA) Core December 3, 2008 10 / 22
Core Obara (UCLA) Core December 3, 2008 11 / 22
Assumptions We consider a pure exchange economy E pure. Assume that, for every i I, X i = R L +, i is locally nonsatiated, continuous and strictly convex, ei 0. Obara (UCLA) Core December 3, 2008 12 / 22
Definitions A coalition S I blocks x A if (x i ) i S R S L + such that x i x i for every i S and i S x i i S e i. Core x A has the core property if there is no coalition that can block it. The set of feasible allocations that have the core property is the core of the economy. Obara (UCLA) Core December 3, 2008 13 / 22
Remark. We know that every Walrasian equilibrium allocation has the core property. Since there exists an equilibrium in this economy, the core is not empty. Obara (UCLA) Core December 3, 2008 14 / 22
Replica Economy Fix one pure exchange economy E pure = ( {X i, i, e i } i I ). An r-replica economy of E pure, denoted by Er pure, is a pure exchange economy where, for each i I, there are r consumers whose preference is i and endowment is e i. A consumption vector of the q th consumer of type i is denoted by x i,q. The set of feasible allocations in E pure r is denoted by A r. Obara (UCLA) Core December 3, 2008 15 / 22
Equal Treatment Property First, it is without loss of generality to focus on symmetric allocations. Equal Treatment Property Suppose that x A r has the core property for E pure r. Then x i,q = x i,q for all q, q and i I. Obara (UCLA) Core December 3, 2008 16 / 22
Proof Step 1. Pick the worst guy for each type. Suppose not. Without loss of generality, assume that x i,q i x i,1 i I and xi,q i xi,1 for some (i, q). for q = 2,..., r for every Step 2. The coalition of the worst guys can block x. Let x i = 1 r r q=1 x i,q be the average consumption of type i consumers given x. Then x i ( )x i,1 for every (some) i I and i I x i i I e i. With a slight redistribution from (x i ) i I, the coalition of consumer 1s can block x A. Obara (UCLA) Core December 3, 2008 17 / 22
Remark For any x A of E pure, let x r A r be (x 1,..., x }{{} 1,..., x I,..., x }{{} I ). Since r times r times a competitive equilibrium is in the core, the equal treatment property holds for CE as well, i.e. if (x, p) is a CE for Er pure, then there exists x A such that x = x r. Also note that (x, p) is a competitive equilibrium for E pure. A corollary of ETP: the core of E pure r+1 Hence the core is shrinking as r is increasing. is smaller than the core of Epure r. It turns out that the core converges to the set of competitive equilibrium allcations as r. Obara (UCLA) Core December 3, 2008 18 / 22
Core Convergence Core Convergence Theorem Suppose that, for some x A, x r A r is in the core of E pure r for r = 1, 2,... Then there exists p R L + such that (x, p ) is a competitive equilibrium for E pure. Remark. (x r, p ) is a competitive equilibtium for E pure r for every r. Obara (UCLA) Core December 3, 2008 19 / 22
Proof Step 1. Z = the (average) resources required to block x Suppose that x r x A. Define Z i is in the core of E pure r for r = 1, 2,... for some { z i R L : z i + e i R L + and z i + e i i x i } Define a convex (check!) set Z by { Z z R L : a I & z i Z i, i I s.t. z i I a i z i } (Interpretation: if rz is at your disposal for some z Z, you can find a coalition which consists of ra i consumers of type i and blocks x in E pure r ). Obara (UCLA) Core December 3, 2008 20 / 22
Proof Step 2. 0 Z Suppose 0 Z (This immediately leads to a contradiction if a i, i I are all rational numbers). Then there exist a I and z i Z i, i i such that z i + e i i x i and 0 i I a iz i = 0. For each i I, let a r i be the smallest integer such that ra i ai r. Define zi r = ra i ai r z i. Then z r i + e i R L + for r = 1, 2,..., z r i z i (thus z r i + e i i x i 0 i I ar i zr i for r = 1, 2,... for large r), and this contradicts to x r being in the core for every r. Obara (UCLA) Core December 3, 2008 21 / 22
Proof Step 3. Apply SHT and the standard trick. By SHT, there exists p ( 0) R L such that p z 0 for all z Z. Clearly p > 0. By local nonsatiation, you can show that p x i p e i for any x i i x i (set a j = 0 for any j i). Since i I x i r, we have p xi = p e i. Since p xi = p e i > 0, cost minimization implies utility maximization, i.e., xi x i for any x i such that p x i p xi = p e i. Obara (UCLA) Core December 3, 2008 22 / 22