Competitive General Equilibrium with network externalities

Highlights

• Indivisibilities are frequent in markets with Network Externalities.

• Externalities coupled with indivisibilities make existence of equilibrium non-trivial.

• We provide an existence theorem in a model with a measure space of consumers.

• Key assumptions are anonymity of network effects and a dispersed income distribution.

Abstract

We identify sufficient conditions for existence of competitive equilibrium with network externalities and indivisibilities. Such combination of externalities and indivisibilities is present in many goods and services with network effects, and it makes existence of equilibrium non-trivial. We provide an existence theorem in a model with a measure space of consumers. Key conditions for existence are anonymity of network effects and dispersion in the economy’s income distribution.

Introduction

Network externalities (NE in what follows) arise when the satisfaction that a consumer obtains from the consumption of a given good or service depends, usually positively, on the number of consumers that consume the same good or service.¹ Typical examples are telephone, e-mail and social media: the fact that a new individual uses the service enhances its usefulness for existing users, who can then interact with an additional member. Most of the theoretical literature on markets with NE is based on partial equilibrium models, and it studies questions of competition among networks and compatibility decisions of network goods’ providers.² The present paper studies the problem of existence of general equilibrium with external network effects.

On the General Equilibrium side, Starr (1999) proposes a model in which network goods are produced by firms using a technology characterized by set-up costs, hence displaying economies of scale. Starr (1999) proves existence of an Average Cost Pricing Equilibrium in that economy. In the language of the literature on external effects, ibid. studies equilibrium existence in the presence of “pecuniary externalities”. In contrast, we consider the problem of equilibrium existence with direct NE, that is, when the consumption of network goods by an individual enters other individuals’ utility functions.

A usual feature of markets with NE is tipping behavior: the tendency for one of the competing goods or protocols to pull away from its competitors and win a substantial share of the market. Indeed, in virtually all papers in which there is tipping in equilibrium it is implicitly or explicitly assumed that there is an indivisibility intrinsic to the network goods. We do not formally address the question of whether indivisibilities are necessary for tipping behavior to arise. However, we informally argue that the occurrence of tipping is due not only to NE, which make it advantageous for consumers to join popular networks, but also to an indivisibility in consumption of network goods. Indivisibilities encourage consumers to coordinate, to avoid having to join multiple networks of the same type. Motivated by this informal argument, in this paper we study the problem of equilibrium existence with NE and indivisibilities combined.

The general problem of externalities has been previously studied in the General Equilibrium literature. Arrow and Hahn (1971) prove existence of equilibrium with externalities assuming a finite number of agents with convex preferences and no indivisibilities. More recently, Balder (2004) obtains existence results in a model of an exchange economy with a measure space of consumers and externalities, but also without indivisibilities. Balder (2004) is the first to use an “externality mapping” to model external effects by means of a continuous mapping of aggregate consumption, which enters consumers’ utility functions. Such mapping is necessary to avoid the potential technical problems for existence of equilibrium with extenalities and a measure space of consumers first pointed out by Balder (2000).³ In the current paper we use an externality mapping which is a special case of the class proposed by Balder (2004), and using that device we show existence of equilibrium in an economy with indivisibilities in consumption, NE, and in which network goods are produced by a set of firms with convex technologies.

Indivisibilities have also been studied in the General Equilibrium literature before. For example, Mas-Colell (1977) and Yamazaki (1978) obtain existence of equilibrium in economies with a non-atomic measure space of consumers and indivisibilities/non-convex preferences. Their key assumption to ensure existence of equilibrium is dispersion in the economy’s income distribution. These papers do not, however, allow for externalities.

Externalities combined with indivisibilities and/or non-convex preferences can pose problems for existence of competitive equilibrium. These problems were illustrated by Noguchi and Zame (2006), which by means of an example shows that an equilibrium may not exist with a pollution externality and non-convex preferences. A key feature of the example in ibid. is that households are impacted by consumptions of different consumers differentially.⁴⁵ In our paper we rule out such differential external effects by assuming that households only care about the configuration of network sizes in the economy. That assumption is key to our results. Indeed, we show that together with the assumption on dispersion of the income distribution mentioned above, this anonymity assumption is sufficient to avoid the existence problems identified by Noguchi and Zame (2006).

The rest of the paper is organized as follows. In Section 2 we introduce the model and state our main result, the existence theorem. In Section 3 we present an example to illustrate the type of economy discussed in the paper, and construct an equilibrium in this economy. Section 4 contains the proof of the existence theorem. Section 5 concludes.