Price-dependent consumption externalities and non-existence of equilibria

Abstract

We construct a concrete example of an exchange economy E which admits no equilibria in the presence of price-dependent consumption externalities. In our example, agents form a finite atomless measure space ( T , T , μ ) . A reference coalition of each agent t at a price system p is the subset C ( t , p ) ⊂ T consisting of those agents whose income falls into a certain income range associated with t. The interdependence of the taste of agent t operates through a reference consumption vector η , which represents the aggregated effect on t’s taste of the various consumption choices made by all agents in C ( t , p ) . We choose η = 1 / ( μ [ C ( t , p ) ] ) ∫ C ( t , p ) x ( s ) d μ ( s ) , the mean consumption bundle of all agents in C ( t , p ) , where x ( s ) denotes the assignment of a commodity bundle to agent s and where the measure ( 1 / ( μ [ C ( t , p ) ] ) ) μ may be interpreted as the relative frequencies of contact that t makes with other members in C ( t , p ) . In light of the law of large numbers, the above η can be viewed as the consumption trend of the agents in C ( t , p ) , which influences the taste of t.