Abstract
Abstract In this paper, we consider a production economy with an unbounded attainable set where the consumers may have non-complete non-transitive preferences. To get the existence of an equilibrium, we provide an asymptotic property on preferences for the attainable consumptions and we use a combination of the nonlinear optimization and fixed point theorems on truncated economies together with an asymptotic argument. We show that this condition holds true if the set of attainable allocations is compact or, when the preferences are representable by utility functions, if the set of attainable individually rational utility levels is compact. This assumption generalizes the CPP condition of [N. Allouch, An equilibrium existence result with short selling, J. Math. Econom. 37 2002, 2, 81–94] and covers the example of [F. H. Page, Jr., M. H. Wooders and P. K. Monteiro, Inconsequential arbitrage, J. Math. Econom. 34 2000, 4, 439–469] when the attainable utility levels set is not compact. So we extend the previous existence results with non-compact attainable sets in two ways by adding a production sector and considering general preferences.
Highlights
With the exception of the seminal paper of Mas-Colell [14] and a first paper of Shafer-Sonnenschein [18], equilibrium for a finite dimensional standard economy is commonly proved using explicitly or implicitly equilibrium existence for the associated abstract economy in which agents are the consumers, the producers and an hypothetical additional agent, the Walrasian auctioneer. In exchange economies, it is well-known that the existence of equilibrium with consumption sets that are not bounded from below requires some nonarbitrage conditions
In [7], it is shown that these conditions imply the compactness of the individually rational utility level set, which is clearly weaker than assuming the compactness of the attainable allocation, and the authors prove an existence result of an equilibrium under this last condition
The purpose of our paper is to extend this result to finite dimensional production economies with non-complete, non-transitive preferences, which may not be representable by a utility function
Summary
With the exception of the seminal paper of Mas-Colell [14] and a first paper of Shafer-Sonnenschein [18], equilibrium for a finite dimensional standard economy is commonly proved using explicitly or implicitly equilibrium existence for the associated abstract economy (see [3], [9], [8], [12], [16], [17]) in which agents are the consumers, the producers and an hypothetical additional agent, the Walrasian auctioneer. In [7], it is shown that these conditions imply the compactness of the individually rational utility level set, which is clearly weaker than assuming the compactness of the attainable allocation, and the authors prove an existence result of an equilibrium under this last condition. The purpose of our paper is to extend this result to finite dimensional production economies with non-complete, non-transitive preferences, which may not be representable by a utility function. To compare our work with the contribution of Won and Yannelis [21], we provide an asymmetric assumption (EWH3) for exchange economies which is less demanding for one particular consumer. Our condition can be identically stated for an exchange economy or for a production economy This means that even, if there exists non compact feasible productions, an equilibrium still exists if the attainable consumption set remains compact.
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