Price-dependent consumption externalities and non-existence of equilibria

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)dμ(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.