handoutve - On Existence of Competitive Equilibria of...

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On Existence of Competitive Equilibria of Sequence Problems In fi nite dimensional problems, under certain conditions we know that we can decentralize a Pareto optimal allocation of a planner’s problem through a competitive price system. In this section we simply establish the conditions under which these theo- rems hold in in fi nite dimensional commodity spaces like that of the Arrow- Debreu complete forward markets environment studied in a previous lec- ture. The equilibrium concept we use to study this issue is called a valuation equilibrium (Debreu 1954). It is called this because the equilibrium is characterized by a linear functional that assigns a value to each commodity bundle. In general , the valuation functional that we are interested in may not be representable as a price system; that is, there may be no price vector such that the value of a commodity bundle is the “inner product” of the price vector and the quantity vector (i.e. p · y = P t =0 p t y t may not be well de fi ned). This section provides su cient conditions for the valuation equilibrium to have a price representation in an in fi nite dimensional commodity space. See S-L Ch. 15 or Prescott and Lucas (1972). Some mathematical essentials Def. A vector space is a set X together with 2 operations: vector ad- dition and scalar multiplication. These operations must satisfy properties such as commutativity (i.e. for any two elements (called vectors) x , y X , x + y = y + x ), associativity (i.e. ( x + y )+ z = x +( y + z )), and distributivity (i.e. α ( x + y ) = αx + αy ). To say how far one element is from another, we need a notion of distance. Def. If X is any vector space, a norm on X is a function k·k : X R such that: (i) k x k 0; (ii) k αx k = | α | k x k ; and (iii) k x + y k k x k + k y k (the triangle inequality). Def. ( X, k·k ) is called a normed vector (or linear) space . Def. A sequence { x 0 , x 1 , ... : x t X for each t = 0 , 1 , ... } in a normed vector space is Cauchy if for any ε > 0 , there exists an integer N such that for all n N and m N , k x n x m k < ε . X is complete if every Cauchy sequence in X converges to a point in X . A complete normed vector space is called a Banach space. Note that in a normed vector space, every convergent sequence is a Cauchy sequence since if x n x , then k x n x m k = k x n x + x x m k k x n x k + 1
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k x x m k 0. In general, however, a Cauchy sequence may not be con- vergent. For example, let
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