Pure Strategy Price Equilibrium Research Articles

Information, Bertrand–Edgeworth competition and the law of one price

Homogeneous goods often sell at different prices within the same market. This paper proposes a theoretical foundation for this phenomenon in the context of a capacity-constrained price game. Sellers have asymmetric information about the market demand, modelled by a partition of the state space, and evaluate uncertain profits in a way consistent with ambiguity aversion. We demonstrate that a pure strategy price equilibrium exists if the market demand is uniformly elastic in each state. Interestingly, the sellers may choose different prices, violating the law of one price. Moreover, market demand may be rationed between the sellers, resulting in consumers purchasing at different prices.

Information, ambiguity and price equilibrium

This work analyzes a price choice game with discontinuous payoffs in which sellers may have asymmetric information about the state of the market. The prices which sellers post in the market must be measurable with respect to their private information and sellers are assumed to have ex ante maximin expected utilities. Easily verified conditions upon the primitives which guarantee the existence of a pure strategy price equilibrium in the game, under both equal sharing and winner-takes-all sharing at price ties, are presented. The existence proofs are direct and constructive.

Price competition, free entry, and welfare in congested markets

In this paper we study the problem of price competition and free entry in congested markets. In particular, we consider a network with multiple origins and a common destination node, where each link is owned by a firm that sets prices in order to maximize profits, whereas users want to minimize the total cost they face, which is given by the congestion cost plus the prices set by firms. In this environment, we introduce the notion of Markovian Traffic Equilibrium to establish the existence and uniqueness of a pure strategy price equilibrium, without assuming that the demand functions are concave nor imposing particular functional forms for the latency functions. We derive explicit conditions to guarantee existence and uniqueness of equilibria. Given this existence and uniqueness result, we apply our framework to study entry decisions and welfare, and establish that in congested markets with free entry, the number of firms exceeds the social optimum.

PRODUCT DIFFERENTIATION UNDER CONGESTION: HOTELLING WAS RIGHT

We introduce negative network externalities—“congestion costs”—into H. Hotelling's (1929) model of spatial competition with linear transportation costs. For any firm locations on opposite sides of the midpoint, a pure strategy price equilibrium exists and is unique if congestion costs are strong enough relative to transportation costs. We analyze product differentiation and find that Hotelling's Principle of Minimum Differentiation comes closer to holding in the presence of congestion costs. The greater are congestion costs, the less differentiated products can be in (locationally symmetric) equilibrium. In fact, minimum differentiation comes arbitrarily close to holding depending on the magnitude of these costs relative to transportation costs. Intuitively, greater congestion effects stabilize competition at closer quarters, eliminating aggressive pricing equilibria. Thus, negative network externalities can play a significant role in product differentiation. (JEL D21, L15, R12)

A Hotelling style model of spatial competition for a convenience good

Ordinarily people do not make special purpose trips to acquire goods like gasoline or groceries but instead buy them as the need arises in the course of their daily lives. Such goods are commonly called convenience goods. We modify Hotelling’s model of spatial competition so that we can analyze the price equilibrium of duopolists that retail a convenience good. Certain features of the duopolists’ demand functions suggest that price competition is more severe in the convenience goods model than in the Hotelling model. The same features complicate the analysis because they mean that a pure strategy price equilibrium does not exist for many locational configurations. Although we are not able to find the mixed strategy price equilibrium analytically, we do present some numerical results on equilibrium prices that broadly confirm this suggestion. We also provide a more general product differentiation interpretation of the convenience good model.

Existence Advertising, Price Competition and Asymmetric Market Structure

We examine a two-stage duopoly game in which firms advertise their existence to consumers in stage 1 and compete in prices in stage 2. Whenever the advertising technology generates positive overlap in customer bases, the equilibrium for the stage-1 game is asymmetric in that one firm chooses to remain small in comparison to its competitor. For a specific random advertising technology, we show that one firm will always be half as large as the other. No pure-strategy price equilibrium exists in the stage-2 game, and as long as there is some overlap in customer bases, the mixed-strategy price equilibrium does not converge to the Bertrand equilibrium.

Existence of equilibrium in a differential game of spatial competition with advertising

We investigate a differential duopoly game with horizontal product differentiation and advertising efforts aimed at increasing market demand, to show that the standard approach to spatial competition fails to produce a pure-strategy price equilibrium in a dynamic game framework. This holds independently of the shape of the transportation cost function. Then, we introduce an endogenous cost associated with the choice of location and characterise the feedback equilibrium, identifying the necessary and sufficient condition for the existence of the pure-strategy (stationary) price equilibrium. The same condition is singled out for the static game where consumer population is constant. Finally, we show that the static game cannot be viewed as a special case of the dynamic one.

Aggregation and Imperfect Competition: On the Existence of Equilibrium

We present a new approach to the theory of imperfect competition and apply it to study price competition among differentiated products. The central result provides general conditions under which there exists a pure-strategy price equilibrium for any number of firms producing any set of products. This includes products with multi-dimen- sional attributes. In addition to the proof of existence, we provide conditions for uniqueness. Our analysis covers location models, the characteristics approach, and probabilistic choice together in a unified framework. To prove existence, we employ aggregation theorems due to Prekopa (1971) and Borell (1975). Our companion paper (Caplin and Nalebuff (1991)) introduces these theorems and develops the application to super-majority voting rules. WE PRESENT A NEW APPROACH to the theory of imperfect competition and apply it to study price competition among differentiated products. The central result is that there exists a pure-strategy price equilibrium for any number of firms producing any set of products. In addition to the proof of existence, we provide conditions for uniqueness. Our model both unites diverse strands of the earlier literature and opens up uncharted areas for future analysis. In particular, we expand the traditional one-dimensional framework to allow for multi-dimen- sional product differentiation. Our approach involves twin restrictions on consumer preferences: one on individuals' preferences, the other on the distribution of preferences across society. These are generalizations of the restrictions supporting 64%-majority rule presented in Caplin and Nalebuff (1988). To prove existence, we apply a new technique of aggregation. This technique is valuable in a variety of other problems. In the companion paper, we use the aggregation result to generalize our earlier work on 64%-majority rule and to characterize the relationship between the distribution of human capital and the distribution of income (Caplin and Nalebuff (1991)). There are additional applications in statistics and in search theory. We begin with a brief review of the early literature on imperfect competition, describing in more detail the existence problem and previous solutions. Section 3 presents our twin assumptions, and shows that they cover many standard cases. In Section 4, we introduce the aggregation theorem and use it in the analysis of demand functions. The proof of existence of equilibrium is in Section

MULTI-DIMENSIONAL PRODUCT DIFFERENTIATION AND PRICE COMPETITION

The authors present a framework for analyzi ng the design and pricingof differentiated products. Their approach has two nov el features. Individuals have Cobb-Douglas preferences over multidimensional pro duct characteristics. The distribution of Cobb-Douglas parameters across society is concave: this implies that there is general agreement about which characteri stic mix is most desirable. With these assumptions the authors demonstrate exist ence of a pure strategy price equilibrium for arbitrary pairs of products in two dimensions and also show how raising the dimensionality of the product space pr ovides an advantage to a late entrant in a price- regulated duopoly. Copyright 1986 by Royal Economic Society.

Two Stage (Perfect) Equilibrium in Hotelling's Model

HOTELLING [I 929] suggests that his spatial competition model can be used to describe a wide range of differentiation problems. He points to the selection of product variety within an industry as a potential application. Most of the subsequent work in his tradition has taken the equivalence of the two problems for granted. In this note, we look more closely at this equivalence. We distinguish between two types of differentiated products according to the variety of preferences assumed with respect to those products. Building on this distinction, we recognize that the demand faced by a firm locating on Hotelling's line is analogous to the demand faced by a firm selecting a horizontally differentiated product. It should be emphasized, thus, that the equivalence between the geographical and the product selection problem is by no means general and relies on specific assumptions about consumers' preferences. Next, we formulate a small model, close to Hotelling's one in which firms select a horizontally differentiated product. Within this model, it will be shown that a pure strategy price equilibrium exists for every pair of products. This statement is to be contrasted with the results of d'Aspremont et al. [I979], who have shown that in the original Hotelling's setting an equilibrium fails to exist when firms are too close together. 1 In addition, we will examine the product selection problem faced by firms in a two-stage game: in the first stage, each firm selects a product; in the second stage, the firm defines a price for its product. We will seek the existence of a perfect equilibrium in this game. A perfect equilibrium in this context can be viewed as a Nash equilibrium in a game in which firms select a product, aware of the non-cooperative price equilibrium that will occur for each of their choices. It will be shown that contrary to Hotelling's principle, the two firms will maximise their differentiation (i.e. firms will locate at opposite ends of the market).