Bilateral Oligopoly with a Competitive Fringe

The model of strategic market games due to Shubik (1973), Shapley (1976) and Shapley and Shubik (1977) is based on the assumption of strategic behavior on the part of buyers and sellers. Unlike Cournot who assumed that buyers are price takers, in strategic market games all the agents are assumed to behave strategically. A particular case of the more general strategic market games is the case of a bilateral oligopoly. In this paper we are concerned with the version of bilateral oligopoly due to Gabszewicz and Michel (1997). However, we assume that on the fringe of this bilateral oligopoly is a market in which the buyers act as price takers. Thus in our model there are two goods X and Y. X is the numeraire good and also plays the role of money in our model. The other good is Y which is a consumption good. The sellers of Y are initially endowed with Y and no X; the buyers of Y are initially endowed with X and no Y. Ordinarily, with price-taking behavior on the part of buyers, and all agents caring for both X and Y, our model would be no different from the one proposed by Codognato and Gabszewicz (1991) and reproduced in Gabszewicz (2002, 2006). In this paper we assume that while buyers care for both X and Y, sellers care only for X and hence are profit maximizers. The sellers are assumed to behave strategically and there are two types of buyers - those who behave strategically and those who are price takers. In the bilateral oligopoly, each seller offers a portion of his initial endowment of Y to the buyers who submit bids in units of X. If the total bids and offers in this market are positive, then the price of Y is determined solely by the ratio of bids to offers. This also determines the price of Y in the market where buyers are price takers. In fact if the price of Y differed on the two markets there would always be scope for arbitrage - someone could buy Y on the market where it is cheaper and sell it for a profit on the market where it is more expensive. The price-taking or competitive buyers express the quantity of Y that they demand at this price. Since the sellers have no use for Y, they offer to the competitive buyers whatever of Y that remains after they have made offers to the strategic buyers. The allocation that is determined after the bids and offers are submitted is as follows. Each seller recovers the value of his offer in the bilateral oligopoly from the strategic buyers. Each strategic buyer gets the quantity of Y that he can purchase with the bid that he has placed in the market for Y. The amount that the sellers offer on the competitive market is distributed among the buyers by using a proportional rule: each buyer obtains an amount of Y that is proportional to the quantity of Y that he demands. Each competitive buyer pays for the Y that he has purchased its value in units of X at the price determined by the bilateral oligopoly. Each seller sells an amount of Y that is proportional to the quantity of Y that he offered on the market and recovers from the competitive market its value in units of X. There are two possibilities in the competitive market: (a) there is excess demand for Y so that the buyers are rationed; or (b) there is excess supply so that each buyer gets whatever of Y he demanded but the sellers sell only a portion of what they offered in the competitive market. We look for an equilibrium in this model where each seller is satisfied with the quantity he offers on the bilateral oligopoly, given the bids and offers of all other strategic players and each strategic buyer is satisfied with his bid, given the bids and offers of all other strategic players. In other words the equilibrium is self enforcing. It turns out that in this model there is a trivial equilibrium: one in which no bids or offers are submitted. Hence we narrow our scope to a particular case of non-trivial equilibrium, i.e. an active equilibrium one in which all strategic players submit either a positive bid or a positive offer. In this class we further narrow down our interest to only those equilibria where no one is rationed in the competitive market. We call such equilibria, exact active. Our main result says that if the economy is replicated giving rise to a convergent sequence of (type) symmetric exact active equilibria (i.e. exact active equilibria where all replica of an agent in the original economy choose the same strategy) then the corresponding sequence of price-allocation pairs converge to a competitive equilibrium for the original economy. This result is analogous to the asymptotic convergence of Cournot equilibria that is discussed in Lahiri (2010) or Lahmandi-Ayed (2001). In other words as the number of agents become large, there is at least one sequence of equilibrium price-allocation pairs that approximates a competitive equilibrium, provided there exists a convergent sequence of symmetric exact active equilibria. In a final section we discuss an example of an economy where all buyers have Cobb-Douglas utility functions and show that the concepts introduced in this paper (as also the convergence result) are non-vacuous. Similar analysis for oligopoly in the context of pure exchange economy can be found in Lahiri (2011). Ordinarily the justification for competitive price taking behavior is the presence of a large number of agents on the same side of the market. However, here we see that even with a small number of agents there is the distinct possibility of price-taking behavior being sustainable. We do not need an auctioneer to call out the prices on the competitive market. The price is determined by strategic interaction that takes place in a bilateral oligopoly on whose fringe the competitive market is located. Hence this is one case where competitive price formation without an auctioneer or the assumption of a large number of buyers, is possible.