Two realizations of this idea were proposed so far

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Two realizations of this idea were proposed so far. Gay (1978), Danilov and Sotskov (1990), Mertens (1996), and Florig (1998, 2001) developed an approach based on a notion of a hierarchic price. At equilibrium, all commodities (or commodity bundles treated as separate goods) are divided into several disjoint classes and traded against commodities of the same class according to prices which are an element of some set called a hierarchic price. Moreover, the set of such classes is ordered, superior class commodities cost infinitely much compared to the inferior class ones. Marakulin (1990) uses non-standard prices in the sense of Robinson’s infinites- imal analysis (Robinson (1966)). A similar hierarchic structure of submarkets arises. An idea to use non-standard numbers to measure prices may look odd at first sight. But a second thought shows that it is not a much bigger abstraction than the use of real numbers for this purpose. Hardly anyone ever paid to anyone else a price of 2 . Besides, non-standard prices are even natural, since they reflect the fact that costs and values (which are no more than mere numbers) are usually “more divisible” than quantities of consumption goods such as cars, houses, pieces of clothing, etc. The only disadvantage of this approach seems to be that there are still relatively few working economists trained in non-standard analysis. On the other hand, it clearly exceeds standard ways in elegance of proofs and generality of results. This paper is a continuation of the study of generalized equilibria in a model without the survival assumption. Its first contribution is the reconciliation of standard and non-standard approaches. Starting with non-standard equilib- rium, we derive a unique representation of non-standard prices by a hierarchic price, which allows us to characterize non-standard budget sets in pure standard terms. The equivalence between non-standard equilibria and Florig’s hierarchic 2
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equilibria follows. By use of these results, we prove that it is the possible to represent the set of all equilibrium hierarchic prices as a union of manifolds of dimension less than the number of goods in an economy. This fact allows us to show generic finiteness of hierarchic equilibria for any specified system of dividends. We also provide an existence theorem for this class of equilibria that generalizes the existence result of Marakulin (1990). Furthermore, we show that the set of such equilibria is generically a subset of the union of competitive equilibria and Dr` eze-M¨uller coupons equilibria. Finally, we give an easy proof of the equivalence between non-standard (and, therefore, hierarchic) equilibria and the fuzzy rejective core of an economy, a refinement of the weak core introduced in Konovalov (1998). Section 2 provides the reader with the definition of an equilibrium with non- standard prices and an example which motivates the use of this concept. Sec- tion 3 contains a number of auxiliary results that allow us to describe the set of non-standard equilibria in pure standard terms and establish the equivalence between non-standard and hierarchic equilibria. In Section 4 the structure of the
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  • Spring '16
  • Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

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