Bertrand-Edgeworth Competition with Substantial Horizontal Product Differentiation

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Bertrand-Edgeworth Competition with Substantial Horizontal Product Diﬀerentiation

Abstract: Since Kreps and Scheinkman’s seminal article (1983) a large number of papers have analyzed capacity constraints’ potential to relax price competition. However, the majority of the ensuing literature has assumed that products are either perfect or very close substitutes. Therefore very little is known about the interaction between capacity constraints and local monopoly power. The aim of the present paper is to shed light on this question using a standard Hotelling setup. The high level of product diﬀerentiation results in a variety of equilibrium firm behavior and it generates at least one pure-strategy equilibrium for any capacity level.

Introduction

The problem of capacity-constrained pricing decision in oligopolies has received considerable attention since Kreps and Scheinkman’s seminal article (1983). Most of the work in the field of Bertrand-Edgeworth oligopolies focused on the case of homogeneous goods and the capacities’ potential impact of relaxing price competition.1 However, a large number of real-world industries characterized by capacity constraints oﬀer diﬀerentiated products. Examples include the airline industry, where capacities clearly play a central role and diﬀerent companies tend to include diﬀerent services in the price of their ticket (checked-in luggage, seat reservation, in-flight meal etc.). In the telecommunication sector, mobile service operators are bound by the size of their 3G and 4G networks, and clearly oﬀer diﬀerentiated products (monthly data cap, speed, network coverage etc.). In the hospitality industry, competing hotels tend to be diﬀerentiated (breakfast, reservation policy, amenities) and constrained by the number of available rooms.
Moreover, taking into account both horizontal product diﬀerentiation and the presence of capacity constraints might lead to novel and surprising theoretical results, as first demonstrated by Wauthy (1996). Despite the prevalence of such industries and the theoretical interest they present, the literature on Bertrand-Edgeworth oligopolies with product diﬀerentiation remains scarce. As Wauthy (2014) points out in a recent survey of this branch of literature:
“ The minimal core of strategic decisions a firm has to make is three-fold: What to produce? At which scale? At what price? A full-fledged theory of oligopolistic competition should be able to embrace these three dimensions jointly. [..] we do not have such a theory at our disposal. [..] it is urgent to devote more eﬀorts to analyze in full depth the class of Bertrand-Edgeworth pricing games with product diﬀerentiation. ”
This paper aims to make a step in this direction. Specifically, it analyzes Bertrand-Edgeworth competition on markets characterized by a substantial level of product diﬀer-entiation. By restricting attention to relatively high levels of product diﬀerentiation in a standard Hotelling setup, it shows that there exists at least one pure-strategy equilibrium for any capacity-pair. This stands in contrast with most models of Bertrand-Edgeworth competition that typically find non-existence for intermediate capacity-levels. The main result of the paper is a complete characterization of the pure-strategy equilibria, which reveals a variety of equilibrium firm behavior in this setting. Note that an even higher level of product diﬀerentiation leads to a trivial pure-strategy equilibrium: non-interacting firms acting as local monopolies.
Most closely related to this paper is Boccard and Wauthy (2011). They investigate the interaction between capacity constraints and Hotelling-type diﬀerentiation and find the absence of an equilibrium in pure strategies for intermediate capacity levels. Their main finding is that the support of equilibrium prices consists of a finite number of atoms, and the number of these atoms is decreasing in the level of product diﬀerentiation. An important assumption their paper makes is that consumers’ valuation for the good is large compared to transportation costs, which results in the market always being covered in equilibrium. While this assumption prevails in the Hotelling literature2, the present paper shows that it hides an interesting setting, namely the case of substantial product diﬀerentiation.
In earlier work, Benassy (1989) and Canoy (1996) also analyze Bertrand-Edgeworth models with horizontal product diﬀerentiation. The main diﬀerence with the present paper is that both of these papers use non-standard specifications of product diﬀerentiation. Specifically, Benassy (1989) captures product diﬀerentiation through demand elasticities in a model of monopolistic competition, whereas Canoy (1996) introduces asymmetries between the firms and allows consumers to buy several units of the good. A common finding of the papers is the existence of pure-strategy equilibrium for suﬃciently high levels of product diﬀerentia-tion. The present paper reformulates this result in the more standard Hotelling framework. Furthermore, contrary to the papers above, the simplicity of the model allows for the com-plete characterization of pure-strategy equilibria for substantial levels of product diﬀerentiation.
The chapter is organized as follows. Section 1.2 describes the model, formulates the profit function and identifies the potential equilibrium strategies. Section 1.3 contains the main result of the paper, the complete characterization of the equilibria. Section 1.4 discusses the results in the light of the existing literature. Section 1.5 examines an asymmetric version of the baseline model. Section 1.6 concludes.

The model

Setting

This paper analyzes a duopoly with firms denoted x and y that produce substitute prod-ucts. They choose a price pi (i 2 fx; y g) for one unit of their product. Assume the firms are located on the two extreme points of a unit-length Hotelling-line (x at = 0, y at = 1) and transportation cost is linear. Moreover, consumers are uniformly distributed along the line but are otherwise identical. They all seek to buy one unit of the product which provides them a gross surplus v. The value of the outside option of not buying the product is normalized to 0. In addition, the firms face rigid capacity constraints kx; ky. For simplicity, assume that marginal costs of production are constant and normalized to zero. The size of the capacities as well as the value of the other parameters of the model are common knowledge. The firms’ objective is to maximize their profit by choosing their price.
A consumer located at point purchasing from firm x has a net surplus of v px t while purchasing from firm y provides her a net surplus of v py t (1 ) where t is the per-unit transportation cost.
Assumption. Assume v=t 1:5, i.e. the products of the firms are substantially diﬀerent from one another. Furthermore, to get rid of some trivial cases I will assume 1 < v=t 1:5 and refer to it as intermediate level of product diﬀerentiation.
Boccard and Wauthy (2011) analyze a similar setting, the key diﬀerence being the level of product diﬀerentiation. They restrict their attention to situations in which products are relatively close substitutes, namely v=t > 2. Below I argue that this simplifying assumption has a surprisingly large impact on the nature of equilibria, hence extending the analysis to the case of intermediate capacity levels provides new insights into the mechanisms of capacity-constrained oligopolies.
Furthermore, it is easy to see that a very high level of product diﬀerentiation, i.e., v=t < 1, leads to the uninteresting case of firms behaving as local monopolists, never interacting.

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The profit function

Assuming rational consumers the following two constraints are straightforward. The participa-tion constraint (PC) ensures that a consumer located at point buys from firm x only if her net surplus derived from this purchase is non-negative: v px + t (PC)
The individual rationality constraint (IR) ensures that a consumer located at point buys from firm x only if this provides her a net surplus higher than buying from the competitor: v px t v py t (1 ) (IR)
Let Tx be the marginal consumer who is indiﬀerent whether to buy from firm x or not. In the absence of capacity constraints it is easy to see that Tx is the minimum of the solutions of the binding constraints (PC) and (IR).
Let T x be the consumer for whom both of the above constraints are binding. Thus this consumer is indiﬀerent among buying form x, buying from y and not buying at all. Formally, py v + t v px tT x = v py t(1 T x) = 0 ) T x = : t
Thus T x plays the role of partitioning the price space according to market coverage. The net surplus being decreasing in the distance from firm x implies that (PC) is binding for Tx T x and (IR) is binding if Tx T x. Symmetric formulas apply to firm y. Therefore, in case capacities are abundant,
8 v t Tx if Tx x,
px = T (1.1)
< py + t 2 t Tx if Tx T x.
Naturally, the existence of capacity constraints means for firm x that it cannot serve more than kx consumers. Assume that after each consumer chooses the firm to buy from (or to abstain from buying), firms have the possibility to select which consumers to serve and they serve those who are the closest to them. In our setting this corresponds to the assumption of eﬃcient rationing rule, which is extensively used in the literature. Therefore the additional constraints caused by the fixed capacity levels can be written as: Tx kx and 1 Ty ky (CC)
It is important to notice that in some cases, when firm y is capacity-constrained, firm x can extract a higher surplus from some consumers by knowing that they cannot purchase from the rival even if they wanted to since firm y does not serve them. Practically, this means that the participation constraint (PC) will always be binding on whenever this interval T x; 1 ky
suﬃciently small: ky 1 T x. Using this is not empty, i.e. whenever the rival’s capacity is observation, one can reformulate (1.1) for any capacity level: px = 8 v t Tx if Tx maxf x; 1 kyg ;
Firm x’s profit can be simply written as x = pxTx. Given the competitor’s capacity and its price choice, determining the unit price px is equivalent to determining the marginal consumer Tx. The observation that prices and quantities can be used interchangeably will simplify the solution of the model.3 Importantly, the firms decide about prices, however, the quantities those prices imply are more directly comparable with the size of capacities.
The profit can thus be rewritten as
x(Tx) = 8 C f g (1.3)
< xLM = (v if Tx ,
t Tx) Tx max T x; 1 ky
x = (py + t 2 t Tx) Tx if Tx > max f ky g
T x; 1
Note that this formula reveals another interpretation of T x: it is the point where the two quadratic curves that constitute the profit function cross (other than their crossing at 0).
The optimization problem of the firm consists of finding the value Tx which maximizes the above expression satisfying the capacity constraint (CC). The superscript LM stands for Local Monopoly because the firm extracts all the consumer surplus from the marginal consumer when (PC) binds. Similarly, the superscript C stands for Competition since the marginal consumer is indiﬀerent between the oﬀer of the two firms whenever (IR) binds.

Potential equilibrium strategies

Define TxLM = arg maxTx xLM and TxC = arg maxTx xC , the values at which the two quadratic curves attain their maxima, hence they are local maxima of the profit function x(Tx).
The relative order of the five variables TxLM ; TxC ; T x; 1 ky and kx
is crucial in solving the maximization problem. The main diﬃculty in the solution of the firms’ maximization program is twofold. On the one hand, the profit function is discontinuous whenever ky 1 T x and kinked otherwise. On the other hand, the values x = py v + t and T C = py + t depend on the choice of the other firm, py. The following lemma simplifies the solution considerably.
Lemma 1.1. TxLM T x implies TxC T x and TxC T x implies TxLM TxC T x.
The proof of the lemma is relegated to the Appendix. The form of firm x’s profit function hinges on the relative order of T x and 1 ky. Therefore in the following discussion Iwill separate two cases: In Case A the capacity of firm y is relatively large, 1 ky < T x. In Case B 1 ky T x which means that firm x may be able to take advantage of the fact that its adversary is relatively capacity-constrained. Case A: 1 ky < T x. When the capacity of firm y is relatively large, (1.1) shows the relation between the price px charged by firm x and its demand (captured by the marginal consumer Tx). Using Lemma 1.1 three diﬀerent subcases can be identified depending on the parameter values of the model and the competitor’s choice.
Lemma 1.2. Assume 1 ky < T x.
(A1) if TxLM T x then the optimal choice of firm x is min(TxLM ; kx),
(A2) if TxC T x then the optimal choice of firm x is min(TxC ; kx),
(A3) if TxC T x TxLM then the optimal choice of firm x is min(T x; kx).

Table of contents :

Introduction
1 Bertrand-Edgeworth Competition with Substantial Horizontal Product Differentiation
1.1 Introduction
1.2 The model
1.2.1 Setting
1.2.2 The profit function
1.2.3 Potential equilibrium strategies
1.3 Equilibria
1.4 Discussion
1.5 An asymmetric model
1.6 Conclusion
1.7 Appendix of Chapter 1
2 Monopoly Pricing with Dual Capacity Constraints
2.1 Introduction
2.1.1 Related literature
2.2 The model
2.2.1 A simple benchmark
2.2.2 The dual capacity model
2.3 Results
2.3.1 Optimal monopoly pricing
2.3.2 Comparative statics
2.4 Welfare
2.5 Endogenous capacity choice
2.6 Incentive compatibility
2.6.1 Incentive compatibility for low-types
2.6.2 Incentive compatibility for high-types
2.7 General distribution of consumers
2.7.1 A simple model of take-out restaurants
2.8 Conclusion
2.9 Appendix of Chapter 2
3 Competition with Dual Capacity Constraints
3.1 Introduction
3.1.1 Related literature
3.2 The model
3.2.1 Rationing rule
3.3 Symmetric pure-strategy equilibria with at most one binding capacity
3.3.1 Excluding low-types
3.3.2 Excluding high-types
3.3.3 Serving some consumers of both types
3.4 Symmetric pure-strategy equilibrium with both capacity constraints binding
3.4.1 Low levels of vH
3.4.2 Very low levels of vH
3.4.3 Medium levels of vH
3.4.4 High levels of vH
3.5 Discussion
3.5.1 A numerical example
3.5.2 Comparative statics
3.5.3 Comparison to monopoly benchmarks
3.5.4 Comparison to the single capacity benchmark
3.6 Conclusion
3.7 Appendix of Chapter 3
Bibliography