On the Bertrand core and equilibrium of a market

A striking result in economic theory is that price competition between a small number of sellers producing a homogeneous good may result in the perfectly competitive market outcome. We analyze the formation of price-making contracts when there is the possibility of coalitional deviations from the market. We consider a market with a finite number of buyers and sellers and standard market primitives. In this context we introduce a new core notion which we term the Bertrand core. A trading price is said to be in the Bertrand core if all sellers quoting this price constitutes an equilibrium and no subset of traders, buyers and sellers, can leave the market and improve their outcomes by trading by themselves. Under standard assumptions we show that the Bertrand core is non-empty. Moreover, we are able to obtain a partial equilibrium analogue of the well-known Debreu-Scarf (1963) result by showing that as the set of market traders is replicated then any price other than the competitive equilibrium can be blocked by some subset of traders provided that the market is replicated sufficiently many times.