OLIGOPOLY AND COST SHARING IN ECONOMIES WITH PUBLIC GOODS

• Published date: 01 May 2016
• DOI: http://doi.org/10.1111/iere.12165
• Date: 01 May 2016
• Author: Maria Gabriella Graziano,Achille Basile,Marialaura Pesce

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 2, May 2016

OLIGOPOLY AND COST SHARING IN ECONOMIES WITH PUBLIC GOODS∗

BYACHILLE BASILE,MARIA GABRIELLA GRAZIANO,AND MARIALAURA PESCE1

Universit`

a degli Studi di Napoli Federico II, Italy; Universit`

a degli Studi di Napoli Federico II,

Italy and CSEF; Universit`

a degli Studi di Napoli Federico II, Italy and CSEF

We study economies that involve both small and large traders as well as the choice of a public project. Within this

framework, we establish two sufficient conditions under which the set of competitive allocations coincides with the core.

Our first core equivalence result holds under the assumption that there is a countably infinite set of large traders similar

to each other. The second result, independent of the number of large traders, requires the existence of a coalition of

small traders with the same characteristics of the large traders. Finally, we show how the generalized Aubin approach

to cooperation may dispense with both conditions.

1. INTRODUCTION

In a large economy, the core of the market coincides with the set of competitive allocations.

Although the core acknowledges that coalitions of traders may cooperate to improve their own

welfare, a multitude of negligible agents ends up generating the same outcome they would

get by acting as price-takers. Edgeworth conjectured such equivalence of cooperative versus

competitive behavior, later proved in Aumann’s seminal contribution (Aumann, 1964) where

the space of agents is postulated to be a continuum. The ensuing literature sought more re-

alistic assumptions, introducing two major features: (i) nonnegligible market participants and

(ii) public goods.

The first issue looms when traders may concentrate in their hands an initial ownership of

commodities that is sufficiently large with respect to the total market endowment. This is

typical in monopolistic or, more generally, oligopolistic markets. Another case arises when,

while the initial endowment is spread over a continuum of negligible traders, some of them may

join forces and decide to act as a single player on the market. Under any agreement that prevents

further splits in subcoalitions, nonnegligible market participants may organize themselves in

the form of cartels, syndicates, or similar institutions. Since this reduces the number of active

coalitions, the core becomes larger and the equivalence theorem may fail.

In economies with public goods, core equivalence may fail even when all traders are negligible.

Roughly speaking, the intuition is the following: When perfect competition requires a large

number of agents and the per capita cost for a public good is decreasing, this weakens the

influence of small coalitions and makes the core larger. Indeed, if a blocking coalition is expected

to finance the public project by itself, too small coalitions might not be able to cover the

whole cost and, consequently, blocking becomes harder. This happens in the typical setup that

combines the assumption of a uniform distribution among agents for the cost of providing the

public good (as in the Lindahl competitive approach) with a Foley veto mechanism for modeling

the objections underlying the construction of the core (see Foley, 1970).

∗Manuscript received March 2014; revised July 2014.

1The authors thank the editor and two referees for helpful comments. The financial support from research grants

PRIN 20103S5RN3 “Robust Decision Making in Markets and Organization”and Programma STAR Napoli-call2013-89

“Equilibrium with Ambiguity” is gratefully acknowledged. Please address correspondence to: Achille Basile, Diparti-

mento di Scienze Economiche e Statistiche. Universit`

a degli Studi di Napoli Federico II, Complesso Monte S. Angelo,

Via Cintia, 80126 Napoli, Italy. Tel.: +39 081675104; fax: +39 081675009. E-mail: basile@unina.it.

487

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

488 BASILE,GRAZIANO,AND PESCE

Since either feature may lead to a failure of core equivalence, the literature has provided

several conditions under which the equivalence holds true when one of the two features is

present; see Shitovitz (1973); Mas-Colell (1980); Diamantaras and Gilles (1996); Gilles and

Diamantaras (1998); Weber and Wiesmeth (1991). This article answers the same question when

both features are present; that is, we provide sufficient conditions for core equivalence under

the simultaneous presence of negligible/nonnegligible traders and of collective goods.

This enriched model fits well problems of international cooperation for the provision of a

global public goods (e.g., climate control) where few large players, corresponding to rich coun-

tries, interact with many small traders, corresponding to poor countries; see Buchholz et al.

(2006) and Shitovitz and Spiegel (1998) among others. These situations have attracted much

interest toward the noncooperative provision of public goods (when countries are assumed to

be independent players) as well as toward cooperative solutions coordinated through interna-

tional agreements to overcome the inefficiency of noncooperative policies. To the best of our

knowledge, however, only very little attention has been devoted to a comparison of the two

approaches.

In markets with public goods, the entities that benefit from them (consumers, firms, munic-

ipalities, regions, countries, etc.) are usually different both in the size of their endowments as

well as in their contributive capacity. An additional contribution of our model is that it helps

to manage the dichotomy between small and large traders along with the distinction between

small and large tax-payers (or contributors), clarifying the relations among them.

The main elements in our setup of mixed economy (Shitovitz, 1973) are an atomless sector

of consumers representing the ocean of negligible traders, a set of atoms representing the

influential agents, and a contribution scheme that specifies the cost for the provision of

the public good for individual agents as well as for coalitions. Negligible and influential agents

are defined with respect to the size measure in the agents’ space. Small and large contributors

are similarly defined with respect to the measure underlying the contribution scheme.

We do not assume any mathematical structure for the set of public projects. This choice

does not preclude the standard Euclidean structure, but carries the important advantage of

naturally embedding the case where traders have misaligned preferences over public goods

and a common ordering cannot be assumed. Moreover, if public projects are interpreted as

environments (i.e., collections of variables common to all the agents but determined outside the

market mechanism), this general framework encompasses many different economic problems;

see Diamantaras and Gilles (1996); Gilles and Diamantaras (1998); Diamantaras et al. (1996);

Gilles and Scotchmer (1997); Basile et al. (2005); Graziano (2007); Hammond and Villar (1998),

Hammond and Villar (1999).

Our notion of competitive equilibrium originates in Mas-Colell (1980) and in the subsequent

papers Diamantaras and Gilles (1996) and Gilles and Diamantaras (1998). Compared to Gilles

and Diamantaras (1998), we do not assume that the individual contribution is the same for

each public project. Consequently, our cost share equilibrium allows individual payments to

vary according to individual benefits. This situation naturally arises when a level of provision

is interpreted as a whole configuration of public policies or when cost share functions are

interpreted as voluntary contributions (see Mas-Colell, 1980) instead of predetermined tax

systems. From a technical viewpoint, the cost share equilibrium generalizes the linear cost

share equilibrium introduced in Gilles and Diamantaras (1998), by combining the notion of

cost share function (see Mas-Colell, 1980, and Gilles and Diamantaras, 1998, Definition 2) with

the idea of cost distribution introduced in Gilles and Diamantaras (1998). On the cooperative

side, we borrow the veto mechanism from Gilles and Diamantaras (1998) and Graziano and

Romaniello (2012). This involves a (contribution)measure, defined on the set of all coalitions,

that pinpoints the level of contribution that each blocking coalition is expected to cover. Due

to the motivations above, our model allows that the contribution measure varies across public

projects. Then, we conveniently remold the σ-core of Gilles and Diamantaras (1998) into a

definition of the core, here dubbed weak σ-core, that naturally fits our characterization of cost

share equilibria.