Robert M. Anderson

“Equilibrium in Measure-Space Economies with Bads”

by Robert M. Anderson, Haosui Duanmu, M. Ali Khan, and Metin Uyanik

Existence of free-disposal and non-free-disposal Walrasian Equilibrium in the presence of bads and other externalities, and price-dependent preferences, has been completely solved in the case of a finite number of agents.  We seek a similar result for measure-space economies in which allocations are L1 functions and each agent’s preference depends on the allocation, price as well as on that agent’s private consumption.  Using nonstandard analysis, we embed a measure space economy in a hyperfinite economy, i.e. a nonstandard economy satisfying all the formal properties of a finite economy, in which existence of equilibrium follows from existing theorems in the finite-agent case.  We then show that, assuming the hyperfinite equilibrium satisfies a nonstandard version of uniform integrability, it generates an equilibrium on a measure space economy called the Loeb economy.  If agents’ preferences are continuous in the weak topology on L1, the equilibrium on the Loeb economy generates an equilibrium on the original measure space economy.  If preferences are merely continuous on L1 in the norm topology, the equilibrium lives on a measure space that is larger than the original measure space, to which the preferences of the agents must be extended; we show this can be done in a natural way that preserves the essential characteristics of the original economy.