Dynamically complete markets under Brownian motion

This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

Existence of an equilibrium for pure exchange economy with fuzzy preferences

This paper focuses on a new model to reach the existence of equilibrium in a pure exchange economy with fuzzy preferences (PXE-FP). The proposed model integrates exchange, consumption and the agent’s fuzzy preference in the consumption set. We set up a new fuzzy binary relation on the consumption set to evaluate the fuzzy preferences. Also, we prove that there exists a continuous fuzzy order-preserving function in the consumption set under certain conditions. The existence of a fuzzy competitive equilibrium for the PXE-FP is confirmed through a new result on the existence of fuzzy Nash equilibrium for fuzzy non-cooperative games. The payoffs of all strategy profiles for any agent are fuzzy numbers in fuzzy non-cooperative games. Finally, we show that the fuzzy competitive equilibrium could be characterized as a solution to an associated quasi-variational inequality, giving rise to an equilibrium solution.

Pure exchange and general equilibrium

This chapter focuses on situations of pure exchange, where consumers wish to exchange bundles of goods they hold at the outset for other bundles they will subsequently consume. It uses this setting to introduce the theory of price-mediated market transactions and, more particularly, the theory of general equilibrium, in which all markets in all goods are considered simultaneously. Following in the footsteps of generations of classical microeconomists, the chapter makes the assertion that in many situations of pure exchange, the consumer will wind up at the consumption allocation part of some Walrasian equilibrium for the economy, and insofar as there are markets in these goods, prices will correspond to equilibrium prices. One thing that the concept of a Walrasian equilibrium does not provide is any sense of how market operates. There is no model here of who sets prices, or what gets exchanged for what, when, and where.