A static microeconomic model of pure competition (Q1210796)

Pure competition entails that actions of a single agent do not affect the market. Two approaches to the modelling of microeconomic exchange markets with pure competition, are used widely. One is the replica model: Large markets are obtained from a small one by replications; the asymptotic behavior depicts pure competition. The second approach employs a continuum of agents, and individual traders are interpreted as infinitesimals, in a standard or a nonstandard analysis. The celebrated Edgeworth conjecture, namely the equality of the core and the competitive equilibria, was verified in the two models, by Debreu and Scarf in the replica model and by Aumann, who introduced the continuum model. Applications the two approaches go far beyond the Edgeworth conjecture; plenty of results are available, that enhance the understanding and insight about pure competition. The treatise under review offers and analyzes another approach to the modelling of pure competition in exchange markets: The number of agents is countable, and actions of groups of traders are determined by finitely additive measures. The latter allow, in particular, that the influence of a single agent is nil, while large coalitions can affect the market. Within this model the author verifies the Edgeworth conjecture. Further applications are not offered here. Also described is the interpretation of the replica model in the framework of the paper. The mathematics involved is that of finitely additive measures, Stone spaces and Boolean algebra, the Darboux integration, etc. The text is self-contained, and all the mathematical notions are explained well. Elaborate discussions connect the mathematics to the economic model. In particular, a statistics interpretation is stressed, which relates actions of coalitions to observable ensembles. At present the model seems (to the reviewer) to be on the abstract side, and lacks the concreteness that the continuum or the replica models offer when handling economic agents. This may be changed when more applications are developed.