# SFB 303 Discussion Paper No. A - 184

**Author**: Werner, Jan

**Title**: Equilibrium with Incomplete Markets without Ordered Preferences

**Abstract**: The purpose of this note is to prove the existence of Radner equilibrium of plans, prices and price
expectations (Radner (1972)) in an exchange economy with incomplete financial markets under weak conditions
on consumers' preference relations. All the previous existence results in the case of markets for financial assets
(i.e., assets with exogenously specified money returns) require monotonicity of consumers' preferences (see
Werner (1985), Cass (1984), Duffie (1987)). Strict monotonicity was used for proving generic existence of
equilibrium with markets for commodity futures (see Duffie and Shafer (1985), Magill and Shafer (1985)), since
the results rely on the regular economy arguments. Evidently, monotonicity is a restrictive assumption in the
context of uncertainty. The result of this note allows preferences to be non-monotonic and even non-complete
and non-transitive. Assumptions are essentially the same as in the case of an economy with complete markets
(see Shafer (1975) which is also the main reference for the idea of the proof). Only the non-satiation assumption
is strengthened to non-satiation in every state of nature. We also prove (Remark 2) a variant of the existence
result which allows for "zero-probability" states, i.e., states such that consumer's satisfaction does not depend on
consumption conditional on this state.

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**Creation-Date**: July 1988

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