Extended abstract (March 30, 2007)

Nir Dagan

<nir@nirdagan.com>

Dept. of Economics and Business

The College of Judea and Samaria

P.O.Box 3, Ariel 40700, Israel.

We model an oligopoly as a coalitional game. The economy has a continuum of players, where the oligopolists are positive measure atoms, and the consumers are a non-atomic continuum.

Previous studies, such as Shitovitz (1973) and Gabzewicz and Mertens (1971) assumed that all coalitions can achieve all re-allocations of their joint resources. As a result, the core includes only Pareto efficient outcomes.

For perfectly competitive economies (with no atoms) our model maintains equivalence between core and price equilibrium outcomes. However, for economies with atoms we depart from previous studies, and predict outcomes that are usually not Pareto efficient. Moreover, the predictions of our model differ in many aspects from the strategic analysis of oligopoly, thus providing new insights.

We assume that an agreement with direct mass participation of members from the non-atomic sector is impossible. Instead, atoms present to the members of the non-atomic sector individual menus. The members of the non-atomic sector may choose transactions repeatedly from these menus. We further assume that memebers of the non-atomic sector may form finite coalitions among themselves, and reallocate resources that are obtained from their interaction with the atoms.

For every coalition with a positive measure of members from the non-atomic sector, we define

**Theorem 1** shows that SCS-allocations must be competitive from the point of view of the coalition's non-atomic sector members.

The result makes use of the f-core studied by Hammond, Kaneko and Wooders (1989) and Hammond (1999). It provides a formal support to informal claims found in text books, such as Kreps (1990) and Tirole (1988), trying to provide a rationale for linear pricing of a monopoly.

Next, we define

**Theorem 2** establishes relations between the SCS-core and more familiar equilibrium concepts: Competitive equilibrium allocations are always in the SCS-core; the SCS-core coincides with the classical core both in finite economies and in perfectly competitive economies.

Theorem 3 shows a feature of the SCS-core in mixed economies. Consider a situation were the oligopolists form a cartel and maximize their joint profit by choosing the prices of the commodities as they see fit. When the goods are complements in the consumers' demand, strategic models often predict an equilibrium that is worse than this joint profit maximazation, from the point of view of all traders, oligopolists and small trades alike. **Theorem 3** states that no such inferior than joint profit maximization outcome can ever belong to the SCS-core.

We now turn to study a more specific collection of oligopoly models. We study an economic setting introduced by Shapley and Shubik (1969). In this setting there is one or more firms each of which produces a good with constant marginal cost. The consumers form a continuum and all have an identical quasi-linear quadratic utility function. The goods produced by the firms are not identical, but the utitlity function treats them in a symmetric fashion. This model was extensively studied as a strategic game among the firms with quantity or price competition.

We characterize the equilibria for several situations.

In the case of monopoly, where a single firm produces one good, there are a continuum of equilibrium prices associated with SCS-core allocations. It ranges between the competitive price (marginal cost) and the text-book monopoly price. If the monopoly produces several goods, te result is similar. All goods will have an identical price, and its range is the same.

In a symmetric oligopoly, again all goods with have an identical price, but the price range will be smaller, where the low competitive end stays and the maximum possible price is lower.

we also consider an assymmetric duopoly, where one firm produces two goods and the other firm produces one of the two goods. In this situation the price of the two gooods is identical, but the firm producing one good makes no profit. The price range is smaller compared to the situation where each firm produces a single commodity.

The assymetric duopoly prediction is very different from strategic price competition, in which the good produced by both firms will have a lower price than the other. The coalitional model captures the competition between firms differently, and puts greater emphasize on profits, as it allows to allocate profits among firms by side payments and not only directly from the interaction of each firm with the consumers.

We intend to study further assymmetric situations as we believe that the coalitional theory of oligopoly may provide here results quite different from classical strategic models.

The coalitional theory of oligopoly addresses some issues neglected by the strategic theory. First, the existence of a price system is derrived rather than assumed. Secondly, firms and consumers are treated in a more symmetric fashion. The consumers are no longer passive price takers. Their participation in potentially improving coalitions requires that they'll be better off, hence the firms cannot dictate market terms to them. This is evident from the equilibria obtained in a monopoly situation. Thirdy, the model allows profit allocations to be influenced by side payments, and lends itself to study implicit mergers between firms.

To conclude, the new approach to oligoply theory extends the theory of perfectly competitive economies to economies with some large players, and also provide different predictions than the exsisting theories of oligopoly.

This list includes only references cited in the extended abstract, and not the complete reference list of the paper.

- Gabzewicz, Jean J and Jean-François Mertens (1971)
An equivalence theorem for a core of an economy whose atoms are not "too" big,
*Econometrica*39: 713-721 - Hammond, Peter J (1999)
On f-core equivalence in a continuum economy with general widespread externalities,
*Journal of Mathematical Economics*32:177–184 - Hammond, Peter J, Mamoru Kaneko, and Myrna H Wooders (1989)
Continuum economies with finite coalitions: core, equilibrium and widespread externalities,
*Journal of Economic Theory*49:113–134 - Kreps, David M (1990)
*A Course in Microeconomic Theory,*Harvester Wheatsheaf, England. - Shapley, Lloyd and Martin Shubik (1969)
Price strategy oligopoly with product variation,
*Kyklos*22:30-44 - Shitovitz, Benyamin (1973)
Oligopoly in markets with a continuum of traders,
*Econometrica*41:467-501 - Tirole, Jean (1988)
*The Theory of Industrial Organization,*The MIT Press, Cambridge, Massachusetts.