22 therefore b i p b i p for every i which implies

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closedness (see, for instance, Anderson (1991), Proposition 2.2.2). Therefore, B i p ) = ¯ B i p ) for every i, which implies that ¯ x is a Walrasian equilibrium. 2 Reversely, an equilibrium with non-standard prices p is a Walrasian equilibrium if the survival condition is satisfied for each agent i at prices ¯ p = ( p/ k p k ) . Recall that a for a * IR l denotes a standard part of a, that is an element of IR l such that a a. Proposition 2.4 Suppose that ¯ x is a non-standard equilibrium with non-standard prices p * Q. Let ¯ p = ( p/ k p k ) . If inf ¯ pX i < ¯ pw i , i N, then ¯ B i ( p ) = B i p ) for every i N, and ¯ x is a Walrasian equilibrium sustained by the price system ¯ p. The proof of this fact is relegated to an appendix since it uses Proposition 3.1 that appears later in the paper. Replacing ( iii ) in the Definition 2 . 2 by X i N ¯ x i X i N w i , gives a definition of a semi-equilibrium with non-standard prices . A satiation effect and dividends Unfortunately, non-standard equilibria typically do not exist due to a satiation effect caused by measuring prices in non-standard numbers. Recall that an agent is said to be satiated if his demand does not belong to the boundary of his budget set. If there is at least one such an agent in an economy, Walras’ law is violated and no competitive equilibrium exists. In our model, even if agent i ’s preferences are locally non-satiated, his demand belongs to the boundary of the standard set ¯ B i ( p ) but not necessarily to the boundary of the set * B i ( p ) , which almost entirely consists of non-standard points. In other words, infinitesimal budget excess may be created, which in its turn may result in non-standard equilibrium existence failure. The problem is alike the one caused by indivisibilities. Suppose that a smallest available quantity of the good that I need is ε and its price is p 1 . If the amount δ of free value at my disposal is less than p 1 ε, I can not use it to increase my 5
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utility. All the more, if δ is infinitesimal while p 1 is not, no standard quantity of such a good is achievable. In any case, the consequence of this infinitely small value being unused is an inequality i N p ¯ x i < i N pw i , where ¯ x i are individual demands, which is inconsistent with the equilibrium market clearing condition 1 . Semi-equilibria with non-standard prices exist if all consumption sets are pos- itive orthants (Marakulin (1990)). In a more general case, where X i are con- vex closed bounded from below sets, there exist dividend equilibria with non- standard prices. A notion of dividend equilibrium was proposed by several authors (Makarov (1981), Aumann and Dr` eze (1986), Mas-Colell (1992)) in or- der to analyse economies which allowed for possibly satiated preferences. In a dividend equilibrium, each agent i ’s budget constraint is relaxed by some slack variable d i in order to allow for redistribution of a budget excess created by sa- tiated agents among non-satiated ones. Such a slack variable can be interpreted as an agent’s endowment of coupons (as in Dr` eze and M¨uller (1980)) or paper money (as in Kajii (1996)).
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  • Spring '16
  • Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

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