We gratefully acknowledge the constructive comments of Professors Ruey-Yih Lin, Yan-Shu Lin, and two anonymous referees. All remaining errors are ours. The financial support by the National Science Council (100-2410-H-031-063) is deeply appreciated.

ARTICLE

# Spatial price discrimination in a symmetric barbell model: Bertrand vs. Cournot^{†}

Article first published online: 27 FEB 2013

DOI: 10.1111/j.1435-5957.2012.00476.x

© 2013 The Author(s). Papers in Regional Science © 2013 RSAI

Additional Information

#### How to Cite

Sun, C.-H. and Lai, F.-C. (2014), Spatial price discrimination in a symmetric barbell model: Bertrand vs. Cournot. Papers in Regional Science, 93: 141–158. doi: 10.1111/j.1435-5957.2012.00476.x

^{†}

#### Publication History

- Issue published online: 24 MAR 2014
- Article first published online: 27 FEB 2013
- Manuscript Accepted: 22 SEP 2012
- Manuscript Received: 4 FEB 2012

#### Funded by

- National Science Council. Grant Number: 100-2410-H-031-063

- Abstract
- Article
- References
- Cited By

### Keywords:

- D43;
- L22;
- R32

- Spatial discrimination;
- Bertrand competition;
- Cournot competition

### Abstract

This paper investigates the theory of spatial discrimination for general demands and general transportation costs in a barbell model, where markets' locations are assumed to be at opposite endpoints of a line. Duopoly firms in the Bertrand model always differentiate maximally, whereas in the Cournot model the market demand structure is not so critical, but the functional form of the transportation cost plays a crucial role in determining equilibrium location. Non-maximal distance appears in the Cournot equilibrium when a convex transportation cost is allowed, bringing about the findings that the equilibrium consumer surplus may be higher and the equilibrium profits may be lower under Cournot competition than under Bertrand competition.

### Resumen

Este artículo investiga la teoría de la discriminación espacial para demandas generales y costos de transporte generales en un modelo de concentración en dos extremos (*barbell*), donde se supone que las ubicaciones de los mercados están en puntos totalmente opuestos de una línea. Las empresas de duopolio en el modelo de Bertrand siempre diferencian al máximo, mientras que en el modelo de Cournot la estructura de la demanda del mercado no es tan importante, sino que la forma funcional del costo del transporte juega un papel crucial a la hora de determinar la ubicación del equilibrio. La distancia no-máxima aparece en el equilibrio de Cournot cuando se permite un costo de transporte convexo, lo que lleva a concluir que el excedente de equilibrio del consumidor puede ser mayor y los beneficios de equilibrio pueden ser inferiores bajo una competencia de Cournot que bajo una competencia de Bertrand.

### Introduction

This paper investigates the theory of spatial discrimination for general demands and general transportation costs in a discrete linear-city model, where markets' locations are assumed to be at opposite endpoints of a line, and discusses two types of competition – Bertrand and Cournot. Our research purpose is two-fold. First, we ask whether or not the functional forms of market demands and transportation costs are determinant in firms' location choices for a model with spatial price discrimination (delivered pricing) under Bertrand competition and Cournot competition. Second, we take a look at whether or not the functional forms of market demands and transportation costs affect consumer surplus, profit, and social welfare rankings between Bertrand and Cournot equilibria.

Spatial competition has spawned a rich and diverse literature ever since the pioneering work by Hotelling (1929), who introduces a Bertrand model with mill pricing. Location models fall into two categories: shipping or delivered pricing models in which firms bear the transportation costs; and shopping or mill pricing models in which consumers pay for transportation. From firms' competition strategy, Cournot competition and Bertrand competition are the two respective standard models. Firms behave as Cournot oligopolists and compete in quantities, whereas firms behave as Bertrand oligopolists and compete in prices (See Gupta et al. 2004; Matsumura and Shimizu 2005 for detailed discussions).

This study is related to the literature on spatial competition with non-linear transportation cost functions and non-linear market demands. Most papers on spatial price discrimination assume market demand functions are linear in price while transportation costs are linear in shipping distance, such as the spatial Cournot models of Hamilton et al. (1989) and Pal (1998), for example.

Examining the location issues associated with a non-linear transportation cost function is an important topic in regional science. Regarding general transportation costs in spatial Cournot models, Anderson and Neven (1991) analyse such costs in a continuous linear-city model, meaning that markets' locations are assumed to be continuous on a line segment, and demonstrate that a unique agglomeration equilibrium exists when the transportation costs are general and convex in distance. Matsumura and Shimizu (2006) discuss general transportation costs in a spatial discriminatory pricing model with a circular city and formulate that the equilibrium outcome under Cournot competition does not depend on whether the transportation cost functions are concave or convex. A relatively limited number of contributions deal instead with general demands in a spatial Cournot model. Weskamp (1985) explores the existence of quantity equilibria, but not location equilibria, for general demands in a model where consumers are located at the nodes in a finite transportation network.

This present study is also related to the literature on comparisons between Bertrand and Cournot equilibria. In their seminal paper, Singh and Vives (1984) show that, in a differentiated duopoly with substitute goods, Cournot competition entails higher prices and profits than Bertrand competition, whereas both consumer surplus and social welfare are higher under Bertrand competition. A wide stream of literature has engaged in challenging and reversing the rankings between Bertrand and Cournot equilibria.

In the case of differentiated products and by considering research and development (R&D) competition among firms, Qiu (1997) presents that Bertrand competition leads to lower social welfare. A firm's profit is shown to be higher under Bertrand competition than under Cournot competition, by allowing for both vertical differentiation and horizontal differentiation in Hackner (2000), and by extending the Singh and Vives (1984) model to a wider range of cost and demand asymmetry between firms in Zanchettin (2006). In the case of homogeneous goods, Delbono and Denicolo (1990) and Boone (2001) discuss the possibility of a reversal in profit ranking between the two versions of competition.

Several papers have already noticed the effects of spatial consideration on the comparisons between Bertrand and Cournot equilibria. Hamilton et al. (1989) evaluate the equilibrium outcomes between the two types of competition in a continuous linear-city model. For larger transportation costs, they find that Bertrand profits are greater than Cournot profits, showing that a spatial barrier may reverse the standard result regarding profit rankings between the two types of competition. Liang et al. (2006) discuss spatial discrimination in a two-market barbell model with asymmetric demands, using linear demand functions and linear transportation cost function.1 They demonstrate that profits and prices are lower and social welfare is higher under Cournot competition than under Bertrand competition if the transportation cost is high and one of the markets is sufficiently large, indicating the demands of the two endpoint markets are sufficiently asymmetric. This means that all the rankings between Bertrand and Cournot equilibria can be reversed owing to both the spatial barrier generated from transportation costs as well as the market size effect of asymmetric demands.

The spirit of the current paper is close to Hamilton et al. (1989) and Liang et al. (2006). We find that in a barbell model with Bertrand competition, duopoly firms always differentiate maximally on locations. Given a linear transportation cost function, the equilibrium location in the Cournot model does not depend on the demand structure. However, the functional form of the transportation cost plays a crucial part in determining the equilibrium location. More specifically, given a linear demand function, the maximal distance between duopoly firms is the unique equilibrium outcome under Cournot competition if the transportation cost function has a general form and is concave in distance, whereas both duopoly firms agglomerated at the centre and separately located inside the linear city may be equilibrium outcomes given strict convexity of the transportation cost.2

This paper contributes first to the study of spatial competition with non-linear market demands and non-linear transportation costs. The results herein show that for a given linear transportation cost incurred by firms, maximum differentiation between duopoly firms' locations under Cournot competition in a symmetric barbell model should be very robust with respect to the demand structure. Our extension leads to new findings for the impact of the functional form of transportation costs on spatial Cournot competition. Anderson and Neven (1991) obtain that in a linear city with a continuous market setting, concavity of the transportation cost function under Cournot competition may yield a different equilibrium location from that under linear transportation costs. Our result from the Cournot model indicates that in a linear city with two endpoint markets, the equilibrium outcome is maximum differentiation, whether or not the transportation cost function is linear or strictly concave in distance. Matsumura and Shimizu (2006) present that the functional form of transportation costs plays no role under a duopoly Cournot model with a circular city. It is worth noting that in a barbell model, strict convexity of the transportation cost yields quite specific results in comparison with the results of a concave transportation cost.

From the viewpoint of comparisons between Bertrand and Cournot equilibria, we find that when maximum differentiation occurs in equilibrium under both the two competitive versions, the total quantities are lower and the total costs are higher under Cournot competition than under Bertrand competition, resulting in the equilibrium price (consumer surplus) being lower (higher) and the equilibrium social welfare being higher under Bertrand equilibrium than under Cournot equilibrium for a concave transportation cost function. On the other hand, when allowing for a strictly convex transportation cost function, the non-maximal distance appears in equilibrium in the Cournot model, bringing about the results that the equilibrium consumer surplus may be higher and the equilibrium profits may be lower under Cournot competition.

Comparing the findings of this model with that of Hamilton et al. (1989) shows that when the transportation costs are strictly convex and the markets are discrete, Cournot competition may lead to higher consumer surplus. This paper is in fact a direct extension/generalization of Liang et al. (2006) and generalizes some of their results by claiming that duopoly firms may locate separately inside a linear city or concentrate at the centre in Cournot competition. It implies that with symmetric demands the introduction of a spatial barrier to competition generated from transportation costs is sufficient to reverse the standard consumer surplus and profit rankings of Bertrand and Cournot equilibria, while the market size effect of asymmetric demands proposed by Liang et al. (2006) is crucial in reversing the standard social welfare ranking between Bertrand and Cournot equilibria. In particular, Liang et al. (2006) analyse a location-price/quantity game with asymmetric demands for which ours is a special case.

### The model

Two firms denoted by 1 and 2 are located on a line with length one between two different markets denoted by *A* and *B*, each of which is located at a point such as Figure 1 shows.

In each market *j*, for *j* = *A*,*B*, the demand function is symmetric and is given by *P _{j}* =

*P*(

*Q*), where

_{j}*Q*=

_{j}*q*

_{1j}+

*q*

_{2j}is the market's industry output (price is a function of quantity

*Q*) and

_{j}*D*=

_{j}*P*

^{−1}(

*p*) (demand is a function of price

_{j}*p*). The demand function

_{j}*P*(

*Q*) is assumed to be continuous, differentiable, and downward sloping (

_{j}*P*′(

*Q*) < 0, for

_{j}*j*=

*A*,

*B*).

The duopoly firms produce homogeneous goods using identical constant-returns-to-scale technologies, and marginal production costs are normalized to zero. It is assumed that each firm *i*, for *i* = 1,2, pays a symmetric unit transportation cost *t*(*x _{i}*) (

*t*(1−

*x*), respectively) to ship a unit of the homogeneous good from its own location

_{i}*x*to market

_{i}*A*(market

*B*, respectively). Without loss of generality, it is assumed that

*x*

_{1}≤

*x*

_{2}. Suppose the unit transportation cost function is strictly increasing in distance such that

*t*′(

*d*) > 0, where

*d*denotes shipping distance, and

*t*(0) = 0. We assume that

*D*=

_{j}*P*

^{−1}(

*t*(1)) > 0, for

*j*=

*A*,

*B*, which ensures that a monopolist can profitably serve the two markets from any location, implying that the unit transportation cost is not too high or the demand of each market is not too low.

The duopoly firms engage in the following competition. In the first stage, each firm *i* simultaneously and independently chooses its location *x _{i}* ∈ [0,1]. In the second stage, each firm

*i*simultaneously and independently chooses its delivered price schedule

*p*∈ [0, ∞) (in the Bertrand model) or quantity schedule

_{ij}*q*∈ [0, ∞) (in the Cournot model), for

_{ij}*i*= 1,2 and

*j*=

*A*,

*B*. We suppose the firms are able to discriminate among consumers, and consumer arbitrage is assumed to be prohibitively costly.

### Analysis

This section discusses the equilibrium in the model formulated above. The equilibrium concept is a subgame perfect Nash equilibrium (SPNE hereafter), which can be derived using backward induction. We first consider the Bertrand model.

#### Bertrand competition

We begin by analysing the second-stage Bertrand competition. It is well known that when the duopoly firms agglomerate (i.e., *x*_{1} = *x*_{2}), the two firms' unit transportation costs and thus unit total costs, which involve transportation costs and production costs, are identical in each market, and each firm's profit is zero from the Bertrand paradox. When the duopoly firms locate separately (i.e., *x*_{1} ≠ *x*_{2}), the high-cost firm in the market is not active, and the low-cost firm obtains the whole market share in Bertrand equilibrium.

Turning to the first stage of this game, if spatial agglomeration occurs (i.e., *x*_{1} = *x*_{2}), then each firm's transportation cost for shipping a unit of the good from its own location to market *j*, for *j* = *A*,*B*, is identical. Bertrand competition leads to marginal cost pricing (*p _{iA}* =

*t*(

*x*) and

_{i}*p*=

_{iB}*t*(1 −

*x*) for

_{i}*i*= 1,2) and zero profits for both firms. Given the rival firm's location, each firm can increase its profit by any relocation. Thus, we have

*x*

_{1}≠

*x*

_{2}.

If *x*_{1} ≠ *x*_{2}, then in the market served by lower-cost firm *i* the price and quantity depend only on the transportation cost of its rival (i.e., firm −*i*) in a limit-pricing equilibrium ( and , where is firm *i*'s monopoly price in market *j*, given its own location *x _{i}*, for

*i*= 1,2 and

*j*=

*A*,

*B*). In a monopoly equilibrium ( and ), the price and quantity in the market served by lower-cost firm

*i*depend only on its own transportation cost. When

*x*<

_{i}*x*

_{−i}(

*x*>

_{i}*x*

_{−i}, respectively), a decrease (an increase, respectively) in

*x*decreases firm

_{i}*i*'s transportation cost and increases firm

*i*'s profit monotonically in each case. In this sense, the best response for the location choices of each firm

*i*must be either

*x*= 0 or

_{i}*x*= 1, and thereby maximum differentiation is the unique equilibrium outcome under Bertrand competition. We yield the simple result of Proposition 1.

_{i}Proposition 1. *For general demand, P*(*Q*_{j})*, and general transportation cost, t*(*d*)*, there exists a unique SPNE outcome showing maximum differentiation,* (*, *) *where superscript “B” signifies “Bertrand” equilibrium, under Bertrand competition. If* *, then* (*p*_{1A} = *t*(1)*, p*_{2A} = *t*(1))* and *(*p*_{1B} = *t*(1), *p*_{2B} = *t*(1))* in equilibrium.*3 *If* , then *and *(*p*_{1B} = *t*(1)*,* )* in equilibrium.*

In a barbell model, consumers concentrating at the two endpoints of the line mitigate the incentive for firms to move toward the centre, as compared with the interior dispersion result, (, ), of a linear city model with a continuous market setting shown by Hamilton et al. (1989). Such a barbell model shows that fierce Bertrand competition leading to maximal location differentiation between two firms is the unique equilibrium outcome for any increasing transportation cost and decreasing demand functions in comparison with the linear model of Liang et al. (2006). The intuition is that a firm wants to locate where it can minimize the cost of serving its customers given its rival's location, and so it will locate at an endpoint if it serves only one market.

#### Cournot competition and comparisons

We next consider the Cournot model. In order to compare earlier findings in the literature, we focus only on a duopoly equilibrium, which means the two endpoint markets are served by both firms in equilibrium.4 Under the assumptions given by Section 2, firm *i*'s total profit is the sum of its profits in markets *A* and *B*, given by *π _{iA}* = [

*P*(

*Q*) −

_{A}*t*(

*x*)]

_{i}*q*and

_{iA}*π*= [

_{iB}*P*(

*Q*) −

_{B}*t*(1 −

*x*)]

_{i}*q*, respectively. We further make the following two assumptions under Cournot competition.

_{iB}Assumption 1. Assumption 1

Assumption 2. Assumption 2

Here, *i* = 1,2, and *j* = *A*,*B*. Assumption 1 ensures a downward sloping Cournot reaction function. Assumption 2 ensures that the reaction functions have a slope less than one in absolute value, and therefore there exists a unique Cournot equilibrium.

We should note that there are two differences between Bertrand and Cournot competition: how price is determined and when production takes place. More specifically, Kreps and Scheinkman (1983) claim that Cournot's model of an oligopoly is used to explain sectors in which firms first determine product quotas and then choose prices to sell the fixed amount of stocks. On the other hand, Bertrand's model of an oligopoly is used to explain sectors in which firms announce prices and then deliver the amount of goods according to orders (i.e., production follows the realization of demand).5

Recall that when unit costs are constant and arbitrage is non-binding, quantities set at different markets by the same firm are strategically independent under Cournot competition. The equilibrium quantity of each firm *i* (for *i* = 1,2) in the game's second stage satisfies the following first-order condition:

We note that the first-order condition can be rewritten as:

- (1)

where for *j* = *A*,*B* denotes the price elasticity of demand in market *j*, and for *i* = 1,2 and *j* = *A*,*B* denotes the price elasticity of firm *i*'s residual demand in market *j*. An intuitive interpretation of the first-order condition in Equation (1) is that each firm acts as a monopolist with respect to its residual demand in each market and the optimal price of a firm is given by the so-called ‘inverse elasticity rule’. In this sense, the less elastic a firm's residual demand is in each market, the more that processing firms can exercise their oligopoly power (i.e., the higher is a firm's Lerner index).

Turning to the first stage of this game, each firm *i* chooses its location *x _{i}* so as to maximize its total profit

*π*=

_{i}*π*+

_{iA}*π*. By symmetry of the model, we concentrate on firm 1 and assume

_{iB}*x*

_{1}≤ 1/2. The total profit of firm 1 for the location subgame can be written as:

With general demand and general transportation cost functions, each firm's location effect can be decomposed into four parts through the implicit-function theorem:

- (2)

Here, for *j* = *A*,*B* denotes the elasticity of the slope of the inverse demand function in each market.

First, a slight increase (decrease, respectively) in *x*_{1} decreases the unit costs of firm 1 in market *B* (market *A*, respectively). Since the two firms' quantity decisions are strategic substitutes, the competitor responds to a reduction in cost by reducing its own quantity, and this then raises firm 1's incentive to move towards the centre (to move away from the centre, respectively). We identify a firm's location choice, which affects the rival firm's quantity decision, as a ‘strategic effect’. The strategic effect of firm 1 in market *B* is positive (i.e., it gives firm 1 an incentive to move toward its opponent), but the strategic effect of firm 1 in market *A* is negative (i.e., it gives firm 1 an incentive to move away from its opponent).

Second, moving towards the centre lowers (raises, respectively) firm 1's unit costs and directly raises (lowers, respectively) its profits in market *B* (market *A*, respectively). Given a firm's quantity supplied, the more the unit transportation cost is reduced (increased, respectively), the more the firm benefits (loses, respectively). The ‘cost effect’ of firm 1 in market *B* is positive, whereas that of firm 1 in market *A* is negative.

The strategic effect and the cost effect of firm 1 in market *A*, which in turn cause a cost differentiation between the firms, are a centrifugal force. On the other hand, the strategic effect and the cost effect of firm 1 in market *B*, which in turn cause cost similarity between the firms, are a centripetal force. These two forces – centrifugal force vs. centripetal force – determine the position of the equilibrium location in this game.

Generally speaking, with symmetric locations (*x*_{2} = 1−*x*_{1} and *x*_{1} < 0.5), , , *E _{A}* =

*E*, and:

_{B}It follows that for a weakly concave inverse demand (*P*″(·) ≤ 0 and thus *E _{j}* ≥ 0), a switch from

*E*= 0 (corresponding to linear demand) to

_{j}*E*> 0 (corresponding to strictly concave demand) increases the centripetal force of a firm.6 The reason is that a slight increase in

_{j}*x*

_{1}raises firm 1's output (and thus total industry output) in market

*B*, which intensely decreases the price in market

*B*for a strictly concave inverse demand function (

*P*″(·) < 0) and

*E*> 0. Since market

_{j}*B*is the major market of firm 2, it responds to an increase in

*x*

_{1}by greatly reducing its own quantity, and this then raises firm 1's incentive to move towards the centre. On the other hand,

*t*′(1−

*x*

_{1}) >

*t*′(

*x*

_{1}) for

*t*″(·) > 0 and

*x*

_{1}< 0.5. It follows that a firm's centripetal force is stronger and a firm's centrifugal force is weaker for a strictly convex transportation cost.

Before examining the equilibrium outcome in the Cournot model, we first discuss the price (consumer surplus) and social welfare rankings between the two competitive versions. Social welfare *W* is defined as total surplus minus total cost and written as:

- (3)

Here, *TC _{A}* =

*q*

_{1A}·

*t*(

*x*

_{1}) +

*q*

_{2A}·

*t*(

*x*

_{2}) and

*TC*=

_{B}*q*

_{1B}·

*t*(1 −

*x*

_{1}) +

*q*

_{2B}·

*t*(1 −

*x*

_{2}) respectively denote the total cost in markets

*A*and

*B*. The following Proposition gives us some useful criterion about the rankings between Bertrand and Cournot equilibria.7

Proposition 2. Proposition 2 *For general demand P*(*Q _{j}*)

*and general transportation cost t*(

*d*)

*, if maximum differentiation occurs under Cournot competition, which indicates SPNE outcomes are the same under the two competitive versions, then: (1) the equilibrium price (consumer surplus) in each market is lower (higher) under Bertrand competition than under Cournot competition,*()

*where superscript “C” signifies “Cournot” equilibrium; (2) social welfare is higher under Bertrand competition than under Cournot competition, W*

^{B}>

*W*

^{C}.

Proof. Proof Under Bertrand competition, the equilibrium price of each market *j* for *j* = *A*,*B* is equal to if . From the above analysis, the equilibrium quantity and price of market *j* under Cournot competition are obtained as:

- (4)

Subtracting from , we get:

- (5)

which is positive for *j* = *A*,*B* since and in duopoly equilibrium. The equilibrium price in each market is lower and thereby the equilibrium quantity in each market is higher under Bertrand competition than under Cournot competition, and for *j* = *A*,*B*, implying that consumer surplus in each market is higher under Bertrand competition, . Combining the fact that the total cost under Bertrand equilibrium is zero, which is unambiguously smaller than the total cost under Cournot equilibrium, we conclude that *W ^{B}* >

*W*.

^{C}If , then the equilibrium price in each market is even lower ( for *j* = *A*,*B*) and thereby the equilibrium quantity of each market is even higher () under Bertrand competition. The same argument for the case of applies to showing that *W ^{B}* >

*W*. Q.E.D.

^{C}From the viewpoint of social welfare, both markets are served by the firms in an efficient manner under Bertrand equilibrium since the transportation cost is zero. On the other hand, each firm works as a local monopolist in each market, while the low-cost firm obtains the whole share of its own market, but bears competitive pressure from its rival in the limit-pricing equilibrium under Bertrand competition, implying the low-cost firm cannot quote an overly high price. Although in the monopoly equilibrium the low-cost firm obtains the whole share of its own market without bearing any competitive pressure from its rival, the monopoly pricing is even lower. Therefore, the equilibrium price (quantity) is lower (higher) under Bertrand competition than under Cournot competition. It is straightforward that Bertrand equilibrium leads to higher social welfare.

We next analyse the equilibrium outcome in the Cournot model. It is quite difficult to analyse a spatial Cournot model with both general demand functions and general transportation cost functions. We first discuss the firms' competition behaviour for linear demand and general transportation cost.

Suppose the market demand function is linear in price, *P _{j}* =

*a*−

*Q*for

_{j}*a*is a positive constant, and the transportation cost function has a general form,

*t*(

*d*). We concentrate on firm 1. A similar reasoning can hold for the second firm. From straightforward derivation, the first-order derivative of firm 1's total profit,

*π*

_{1}=

*π*

_{1A}+

*π*

_{1B}, with respect to

*x*

_{1}in the game's first stage is calculated as:

- (6)

This is obviously equal to zero (∂*π*_{1}(*x*_{1},*x*_{2})/∂*x*_{1} = 0) at central agglomeration (*x*_{1} = 1/2, *x*_{2} = 1/2). The second-order derivative of firm 1's total profit is obtained as:

- (7)

Note that if *t*″(·) ≤ 0 and *t*(1) < 0.5*a* from Equation (7). This means that firm 1's profit function is continuous and convex instead of concave in its own location *x*_{1} if the transportation cost function is concave in distance, *t*″(·) ≤ 0. It follows that, given the opponents' location choices, firm *i* must either set *x _{i}* = 0 or set

*x*= 1. For a concave transportation cost function, only three possible equilibrium outcomes exist: (

_{i}*x*

_{1},

*x*

_{2}) = (0,0), (0,1), and (1,1). Equation (8) indicates that, for a fixed

*x*

_{2}= 1 (

*x*

_{1}= 0, respectively), the best response in the location choices of firm 1 (firm 2, respectively) is given by

*x*

_{1}= 0 (

*x*

_{2}= 1, respectively), and thus duopoly firms have no incentives to deviate from maximum differentiation.

- (8)

This gives the result shown in Proposition 3.8

Proposition 3. Proposition 3 *Suppose the market demand function is linear in price, P _{j}* =

*a*−

*Q*

_{j}*, and the transportation cost function has a general form*

*t*(

*d*)

*and is concave in distance, t*″(·) ≤ 0

*, and t*(1) < 0.5

*a*.

*There exists a unique SPNE outcome showing maximum differentiation,*(, )

*, under Cournot competition.*

The intuition is as follows. From Equation (2), it is easy to see that if *x*_{1} < 0.5, then *t*′(1 − *x*_{1}) ≤ *t*′(*x*_{1}) for a concave transportation cost (*t*″(·) ≤ 0) and *E _{j}* = 0 for linear demand, which indicates that firm 1's direct benefit (i.e., the cost effect in market

*B*) and indirect benefit (i.e., the strategic effect in market

*B*) of moving towards the centre are both offset by its direct loss (i.e., the cost effect in market

*A*) and indirect loss (i.e., the strategic effect in market

*A*), because market

*A*is the major market of firm 1 (). Thus, the centrifugal force dominates the centripetal force between firms and maximum differentiation is the unique equilibrium outcome for linear demand and a concave transportation cost.

As with the reasoning of the weaker centripetal force in a barbell model, the result contrasts sharply to that of a linear city model with a continuous market setting in which duopoly firms agglomerate at the centre with a convex transportation cost as shown by Anderson and Neven (1991). Our result again confirms the earlier finding of Liang et al. (2006) when the demands of the two endpoint markets are symmetric. It reveals that the result of maximum differentiation under Cournot competition holds not only for a linear transportation cost function, but also for a general and concave function. Proposition 3 implies that for a given linear demand function, the non-concavity of transportation cost is a necessary condition for an interior location equilibrium. There is now a direct corollary to the above Propositions 1, 2 and 3.

Corollary 1. Corollary 1 *Suppose the market demand function is linear in price, P _{j}* =

*a*−

*Q*

_{j}*, and the transportation cost function has a general form and is concave in distance, t*″(·) ≤ 0

*, and t*(1) < 0.5

*a*

*, then: (1) the equilibrium price (consumer surplus) in each market is lower (higher) under Bertrand competition than under Cournot competition,*()

*, and (2) social welfare is higher under Bertrand competition than under Cournot competition, W*>

^{B}*W*

^{C}.This means that the introduction of spatial consideration into the models of Bertrand competition and Cournot competition does not change the standard result regarding the two competitive versions' price (consumer surplus) and social welfare rankings.

We next discuss Cournot competition for linear demand (*P _{j}* =

*a*−

*Q*) and quadratic (strictly convex) transportation cost (

_{j}*t*(

*d*) =

*τd*

^{2}, for

*τ*is a positive constant).9 We first briefly discuss the application of a concave and a convex transportation cost technology. One can find real life situations that commonly correspond to a concave transportation cost technology. Intuitively, concavity of transportation costs captures the idea that the marginal transportation costs decrease with respect to distance and can be derived through the existence of fixed costs in the transportation technology. On the other hand, a convex transportation cost technology corresponds to situations where we consider that both distance and time are relevant variables in the calculation of transportation costs. It is also well known that the spatial competition model provides an explanation for product location. In this case, the convexity of the transportation cost function corresponds to a consumer's increasing marginal disutility from not getting the ideal product specification in a mill pricing model and that corresponds to a firm's increasing marginal adjustment cost, which depends on the specification of the standard and the desired products, in a delivered pricing model.10

Let us concentrate on firm 1. For linear demand (*P _{j}* =

*a*−

*Q*) and quadratic transportation cost (

_{j}*t*(

*d*) =

*τd*

^{2}), the total profit of firm 1,

*π*

_{1}=

*π*

_{1A}+

*π*

_{1B}, at the game's first stage is calculated as:

- (9)

We solve the first-order condition in the location choice of firm 1 as:

- (10)

We concentrate only on symmetric equilibrium outcomes to obtain analytical solutions. Substituting *x*_{2} = 1−*x*_{1} into Equation (10) yields three candidate equilibrium locations of firm 1:

- (11)

The third root, , in Equation (11) is unambiguously greater than 1/2. The Appendix shows that both spatial agglomeration, (), and interior dispersion, (, ) for , can be sustained as equilibrium locations for linear demand and quadratic transportation cost.11

Proposition 4. Proposition 4 *Suppose the market demand is linear in price, **P _{j}* =

*a*−

*Q*

_{j}*, and the transportation cost is quadratic in distance, t*(

*d*) =

*τd*

^{2}

*. Under Cournot competition:*

*There exists an SPNE outcome showing spatial agglomeration,*(, )*, for**and a*> 7.072.*There exists an SPNE outcome showing interior dispersion,*(, )*where**is given by Equation*(11)*and**, for**and a*≤ 10*. Moreover, the location differentiation between firms is decreasing in the transportation rate in equilibrium (i.e.,*.*Duopoly firms locating at opposite endpoints of the line,*(*x*_{1}= 0,*x*_{2}= 1)*, are never an SPNE outcome.*

Proof. Proof See the Appendix.

Contrary to the results of Proposition 3, maximum differentiation is never an equilibrium outcome, and both spatial agglomeration and interior dispersion may occur under Cournot competition when the transportation cost is quadratic in distance.12

The intuition behind our results is as follows. From Equation (2), *E _{j}* = 0 for linear demand and

*t*′(1−

*x*

_{1}) >

*t*′(

*x*

_{1}) for

*t*″(·) > 0 and

*x*

_{1}< 0.5, meaning that moving towards the competitor reduces the unit transportation costs of firm 1 in market

*B*by a larger amount than the corresponding increase in the unit transportation costs in market

*A*. Thus, a firm's centripetal force is stronger and a firm's centrifugal force is weaker for a convex transportation cost. To support maximal location differentiation, (, ), as a duopoly equilibrium under Cournot competition for a strictly convex transportation cost, the transportation rate

*τ*cannot be too high (

*τ*< 0.5

*a*) for the reason of guaranteeing that both markets are served by the duopoly firms. Due to the even more trivial centrifugal force of a firm from a lower

*τ*, the duopoly firms indeed have incentives to deviate from maximum differentiation.

For the same reason, the centripetal force due to intensifying quantity competition is no longer dominated by the centrifugal force due to relaxing quantity competition for a convex transportation cost. The duopoly firms' interior locations may then be sustained as equilibrium outcomes.

There are two kinds of interior equilibrium locations: central agglomeration, (, ), and interior dispersion, (, ) for . Given that the rival firm is located at the centre , the weightiness of a firm's centrifugal force is slight, since the maximal distance between the firms is no greater than 1/2. In this case, duopoly firms have no incentives to deviate from spatial agglomeration for a lower transportation rate. Nonetheless, the maximal distance between the duopoly firms is larger and can be greater than 1/2 for an interior dispersion, which increases the weightiness of the centrifugal force.

It follows that interior dispersion can be sustained as an equilibrium outcome only if the transportation rate is high enough (*τ* > 0.558*a*) and then the centripetal force is much stronger. For an interior dispersion, the location differentiation between firms is decreasing in the transportation rate, because the higher the transportation rate is, the stronger a firm's centripetal force will be. Figure 2 illustrates the relationship between the agglomerated equilibrium and the dispersed equilibrium, depending on the value of *a*.13

We conclude that the equilibrium location crucially depends on whether or not the transportation cost is convex in distance, and that convexity of the transportation cost may yield quite specific results, when compared with the results from barbell models such as in Arevalo-Tome and Chamorro-Rivas (2006), Liang et al. (2006) and Sun (2009). It is also worth noting that spatial Cournot competition in a barbell model yields a richer set of spatial configurations, involving central agglomeration, maximum differentiation, and interior dispersion, which may be used to explain a variety of firms' location decisions in reality. To the best of the authors' knowledge, a relatively limited number of works on spatial competition in a linear space achieve all these three kinds of equilibrium locations, such as from minimum differentiation to maximum differentiation.

We next compare consumer surplus and aggregate profits for each market in the two competition versions. For 0.558*a* < *τ* < 0.571*a* and *a* ≤ 10, there exists an interior equilibrium location in the Cournot model. The price difference and the aggregate profit difference between the Cournot and Bertrand equilibria in each market *j*, for *j* = *A*,*B*, are respectively calculated as:

- (12)

- (13)

Thus, consumer surplus is higher and aggregate profits are lower under Cournot competition than under Bertrand competition, and , which are in contrast to the standard results.

For 0 < *τ* < min{2.343*a* − 16.569, 0.8*a*} and *a* > 7.072, there is an agglomeration equilibrium in the Cournot model. When , it is verified that:

- (14)

- (15)

When , it is verified that:

- (16)

- (17)

In this sense, if 0.4*a* < *τ* < min{2.343*a* − 16.569, 0.8*a*} and *a* > 8.528, then consumer surplus is higher under Cournot competition than under Bertrand competition, , which is a reversal of the standard result concerning consumer surplus ranking between the two types of competition. If 0.263*a* < *τ* < min{2.343*a* − 16.569, 0.8*a*} and *a* > 7.966, then Cournot competition leads to lower aggregate profits, , which also reverses the standard result concerning profit ranking.

We next compare social welfare under Bertrand and Cournot competition. For 0.558*a* < *τ* < 0.571*a* and *a* ≤ 10, there exists an interior equilibrium location, (, ), with duopoly firms separately and symmetrically located around the centre under Cournot competition. The welfare difference between the Cournot and Bertrand equilibria is calculated as:

- (18)

which is unambiguously negative. For 0 < *τ* < min{2.343*a* − 16.569, 0.8*a*} and *a* > 7.072, duopoly firms agglomerate at the centre , (*x*_{1} = 1/2,*x*_{2} = 1/2), under Cournot competition. When *τ* ≤ 0.5*a*, it is verified that:

- (19)

When *τ* > 0.5*a*, it is verified that:

- (20)

We summarize the results as follows.

Corollary 2. Corollary 2 *Suppose the market demand is linear in price, **P _{j}* =

*a*−

*Q*(

_{j}, and the transportation cost is quadratic in distance, t*d*) =

*τd*

^{2}.

*Equilibrium price (consumer surplus) in each market is lower (higher) under Cournot competition than under Bertrand competition,*()*, for a higher transportation rate such that*0.4*a*<*τ*< min{2.343*a*− 16.569, 0.8*a*}*and a*> 8.528.*Aggregate profits are lower under Cournot competition than under Bertrand competition,**, when the transportation rate*τ*is relatively high such that*0.263*a*<*τ*< min{2.343*a*− 16.569, 0.8*a*}*and a*> 7.966.*Social welfare is higher under Bertrand competition than under Cournot competition, W*^{B}>*W*^{C}.

Figure 3 illustrates the price, aggregate profit and social welfare rankings between Cournot and Bertrand equilibria for the parameter *a* = 9.282. In this case, central agglomeration can be sustained as equilibrium outcomes in Cournot competition for a lower transportation rate (0 < *τ* < 2.343*a* − 16.569 = 5.179), and interior dispersion can be sustained as an equilibrium outcome for a higher transportation rate (0.558*a* = 5.179 < *τ* < 0.571*a* = 5.3). Moreover, in the Bertrand model, the case of a limit-pricing equilibrium corresponds to a lower transportation rate (*τ* ≤ 0.5*a* = 4.641), and the case of a monopoly equilibrium corresponds to a higher transportation rate (*τ* > 0.5*a* = 4.641).

The intuition behind Corollary 2 is as follows. Duopoly firms differentiate maximally under Bertrand competition, but locate inside the linear city under Cournot competition. Note that each firm obtains the whole share of its own market in the limit-pricing equilibrium (or monopoly equilibrium) under Bertrand competition, and the equilibrium price and quantity depend on the rival firm's transportation cost. For a convex transportation cost, the rival firm's cost is intensely increasing in distance, and thus the equilibrium price is extremely high and the equilibrium quantity is extremely low at the maximum differentiation equilibrium for a higher transportation rate.

Combining the fact that the total cost under Bertrand equilibrium is zero, which is unambiguously smaller than the total cost under Cournot equilibrium, the firms' profits are higher under Bertrand competition than under Cournot competition. The standard social welfare ranking of the two modes of competition is reinforced by the fact that Bertrand competition leads to greater efficiency. The novelty here is that, for a higher transportation rate, a switch from Cournot competition to Bertrand competition increases social welfare by raising firms' profits, which outweigh the decrease in consumer surplus. The standard result is that an increase in consumer surplus outweighs a decrease in firms' profits.

Comparing the results of this model with those of Hamilton et al. (1989), consumer surplus (price) can be higher (lower) under Cournot competition than under Bertrand competition, when the transportation costs are strictly convex and the markets are discrete. Our non-maximal distance result in the Cournot model is also contrary to the result of Liang et al. (2006), bringing about the findings whereby a reversal in price (consumer surplus) and profit rankings does not necessarily depend on a market size effect resulting from asymmetric demands, while the market size effect proposed by Liang et al. (2006) is crucial in reversing the social welfare ranking between Bertrand and Cournot equilibria.

We finally briefly discuss the firms' competition behaviour for general demand (*P*(*Q _{j}*)) and linear transportation cost (

*t*(

*d*) =

*τd*, for

*τ*is a positive constant). After a number of rather complex algebraic manipulations, which are excluded in the paper to save space, we can show that maximum differentiation (, ) is the only possible symmetric equilibrium location that is stable, provided the market demand has a general form and the transportation cost is linear in distance.14 This means that the price (consumer surplus) and social welfare rankings between Bertrand and Cournot equilibria cannot be reversed unless one allows for a strictly convex transportation cost function.15

### Conclusions

In a non-spatial context, it is well known that Bertrand competition entails higher social welfare and consumer surplus, but lower profits than under Cournot competition. In a spatial world, the assumption of a single market must be relaxed insofar as firms choose locations and offer a product to multiple marketplaces. When considering a spatial economy with multiple marketplaces, the analysis incorporates a new element – the transportation cost gives the spatial firm its monopoly power over customers close to it – which creates a spatial barrier and may be a critical factor affecting firms' location choices and comparisons between Cournot and Bertrand equilibria.

This paper investigates Bertrand competition and Cournot competition for general demand functions and general transportation cost functions in spatial discriminatory pricing models, where markets are located at opposite endpoints of a line. Central to our findings is that in the Bertrand model, duopoly firms always differentiate maximally on locations. In the Cournot model, given a linear transportation cost, the market demand structure is not so critical, but the functional form of the transportation cost plays a crucial role in determining equilibrium location. This means that linearity for market demand functions assumed in the literature on spatial discrimination should be innocuous, however, an assumption of linear transportation costs may be non-innocuous. Contrary to previous findings in the literature, with symmetric demands we show that Cournot competition does yield a richer set of spatial configurations under a convex transportation cost function, involving central agglomeration, interior dispersion, and maximum differentiation, which may be used to explain a variety of location decisions by firms in the real world.

For the reason that Bertrand competition entails greater efficiency, the standard social welfare ranking between Cournot and Bertrand competition is reinforced. However, the equilibrium consumer surplus may be higher and the equilibrium profits may be lower under Cournot competition than under Bertrand competition, given the markets in a linear city are discrete and the demands are symmetric. This means that the introduction of a spatial barrier to competition generated from transportation costs is sufficient to reverse the standard consumer surplus and profit rankings of Bertrand and Cournot equilibria.

We are still not able to derive a general property of the equilibrium under Cournot competition for both general transportation costs and general demands. This extension will be a project for future research.

### Appendix

We discuss all three kinds of symmetric location patterns: maximal location differentiation (, ), spatial agglomeration (, ), and interior dispersion (, ) for .

We first present the possibility of maximal location differentiation (, ) in equilibrium. Note that maximal location differentiation is a duopoly equilibrium only if and for *i* = 1,2, which requires 0 < *τ* < 0.5*a*. When 0 < *τ* < 0.5*a*, ∂*π*_{1}(0,1)/∂*x*_{1} = 8*τ*(*a* − 2*τ*)/9 > 0 and ∂*π*_{1}(1,1)/∂*x*_{1} = 8*τ*(−*a* + *τ*)/9 < 0 from Equation (10). The intermediate value theorem then implies that for a given *x*_{2} = 1, firm 1's best response in location choices must be one of the roots satisfying ∂*π*_{1}(*x*_{1},1)/∂*x*_{1} = 0, instead of *x*_{1} = 0. Q.E.D.

We now discuss the possibility of spatial agglomeration (, ) in equilibrium. When *x*_{2} = 1/2, we calculate the second-order derivative of firm 1's profits as:

- (A1)

Substituting *x*_{1} = 1/2 into Equation (A1) yields:

- (A2)

which is negative when 0 < *τ* < 0.8*a*. Thus, given *x*_{2} = 1/2, firm 1's profit function reaches its local maximum at *x*_{1} = 1/2. Solving from Equation (A1) yields two roots.

- (A3)

The first root, *μ*_{1}, in Equation (A3) is unambiguously greater than 1/2, and the second root, *μ*_{2}, is greater than zero (*μ*_{2} > 0) if and only if *τ* > 4*a*/11 ≈ 0.364*a*. This implies function changes its sign at once for *x*_{1} ∈ [0,1/2]. We therefore have the following Claims 1 and 2.

Claim 1. Claim 1 if and only if 0 < *τ* < 0.8*a*.

Claim 2. Claim 2 The sign of function (1) is always negative for 0 < *τ* ≤ 4*a*/11 ≈ 0.364*a*, and (2) it changes once for *τ* > 4*a*/11 ≈ 0.364*a*.

Given *x*_{2} = 1/2, the equilibrium quantity of firm 1 in market *B* when *x*_{1} = 0 is given by . Therefore, we have the next claim.

Claim 3. Claim 3 if and only if *τ* < 4*a*/7 ≈ 0.571*a*.

When 0 < *τ* ≤ 0.364*a*, the sign of is always negative for *x*_{1} ∈ [0,1/2], which means firm 1's profit reaches its global maximum at *x*_{1} = 1/2 given *x*_{2} = 1/2. When 0.364*a* < *τ* < 0.571*a*, the sign of changes once. Since *π*_{1}(1/2, 1/2) − *π*_{1}(0, 1/2) = *τ*(2*a* − 3*τ*)/9 is positive for 0.364*a* < *τ* < 0.571*a*, the best response of firm 1 is *x*_{1} = 1/2. When 0.571*a* ≤ *τ* < 0.8*a*, there is a critical value such that and if . Since the profit function is continuous and decreasing in *x*_{1} for , the optimal choice of firm 1 is *x*_{1} = 0. Subtracting *π*_{1}(0,1/2) from *π*_{1}(1/2,1/2), we get:

- (A4)

which is positive if and *a* > 7.072. Thus, when *τ* < min{2.343*a* − 16.569, 0.8*a*} and *a* > 7.072, the best response in the location choices of firm 1 for 0 ≤ *x*_{1} ≤ 1 is given by *x*_{1} = 1/2. By symmetry, the same argument applies to firm 2. We complete the proof. Q.E.D.

We next discuss the possibility of interior dispersion in equilibrium (, ) for and given by Equation (11). Note first that if and only if 0.5*a* < *τ* < 4*a*/7 ≈ 0.571*a*. When 0.5*a* < *τ* < 0.571*a* and *a* ≤ 10, , for *i* = 1,2 and are all positive:

- (A5)

Since *π*_{1}(*x*_{1}, *x*_{2}) − *π*_{1}(1 − *x*_{1}, *x*_{2}) = −4*τ*^{2}(2*x*_{1} − 1)(2*x*_{2} − 1)/9 > 0 ∀*x*_{1} < 1/2 and *x*_{2} > 1/2, each firm has no incentive to ‘leapfrog’ its rival. Given , the second-order derivative of firm 1's profit is calculated as:

- (A6)

We therefore present the next two claims.

Claim 4. Claim 4 if and only if *τ* > 8*a*/15 ≈ 0.533*a*.

Claim 5. Claim 5 if and only if *τ* > 4*a*/9 ≈ 0.444*a*.

For 0.533*a* < *τ* < 0.571*a*, there is a root , satisfying . Since function changes its sign at most twice, firm 1's profit reaches its local minimum instead of local maximum at . The following Equations (A7) and (A8) guarantee that the best response of firm 1 is given by when 0.558*a* < *τ* < 0.571*a*.16

- (A7)

- (A8)

Furthermore, Equation (A9) indicates that the duopoly firms will locate at a decreasing distance for a higher transportation rate. Q.E.D.

- (A9)

- 1
The literature's analysis on spatial oligopoly with strategic location choice has typically proceeded on the assumption that consumers are uniformly distributed along a line market (i.e., a continuous market setting). Hwang and Mai (1990) counter this with an investigation into firm behaviour when consumers are found in concentration at two endpoint markets on a line (i.e., a discrete two-market setting), which is the so-called barbell model.

- 2
We emphasize that it remains an open question to investigate what could happen under Cournot competition for both general transportation costs and general market demands.

- 3
Without loss of generality, it is assumed that the low-cost firm 1 (firm 2, respectively) obtains the whole share of market

*A*(market*B*, respectively) in equilibrium. - 4
Note that the parameters are restricted to guarantee that the two endpoint markets are served by both firms in equilibrium, rather than for all location combinations of the duopoly firms.

- 5
We thank an anonymous referee for suggesting this recognition.

- 6
Note that the weakly concave inverse demand function is a sufficient condition for the second-order condition of a Cournot equilibrium.

- 7
The two competitive versions' profit ranking cannot be justified unless we incorporate some more restrictions on the functional forms of market demand and transportation cost.

- 8
In this case maximal location differentiation is a duopoly equilibrium only if

*t*(1) < 0.5*a*, which satisfies the condition ensuring*D*=_{j}*P*^{−1}(*t*(1)) > 0 for*j*=*A*,*B*, requiring*t*(1) <*a*. - 9
Note that Proposition 3 gives a sufficient, but not necessary, condition for an equilibrium location.

- 10
There are also some investigations that examine convexity on transportation costs in the spatial competition literature, whatever the market competition type is, such as d'Aspremont et al. (1979), Neven (1985), Anderson (1988), and Economides (1989) who analyze a mill pricing model with convex transportation costs, and Anderson and Neven (1991) who assume convexity on transportation costs in order to investigate spatial Cournot competition.

- 11
In this case spatial agglomeration is a duopoly equilibrium only if

*τ*< 4*a*, interior dispersion is a duopoly equilibrium only if 0.5*a*<*τ*< 4*a*/7 ≈ 0.571*a*, and maximal location differentiation is a duopoly equilibrium only if*τ*< 0.5*a*. Moreover, the condition ensuring*D*=_{j}*P*^{−1}(*t*(1)) > 0 for*j*=*A*,*B*requires*τ*<*a*. - 12
We can also show that for a linear demand function and a linear-quadratic transportation cost function with some parameter restrictions the following three kinds of location patterns can all be sustained as equilibrium outcomes: spatial agglomeration, maximum differentiation, and interior dispersion. We also find that spatial agglomeration occurs for a concave inverse demand function () and a convex transportation cost function (

*t*(*d*) =*τd*^{2}). It follows that linear demands are not necessary conditions for an interior equilibrium. - 13
It can be shown that when multiple equilibria exist, the central agglomeration is a stable equilibrium for 0 <

*τ*< 4*a*/7 ≈ 0.571*a*, and the interior dispersion is an unstable equilibrium at the location stage. - 14
The detailed derivations are available from the authors upon request. To offer some explanation as to why maximum differentiation is the only possible equilibrium location for general demands in Cournot competition, it can be checked that the first-order condition of an interior symmetric equilibrium location never holds unless firms agglomerate at the centre. However, the possibility of spatial agglomeration in equilibrium is ruled out by the stability condition at the location stage.

- 15
We also try to extend and complete our investigation by involving the analysis with a constant elasticity demand (

*P*(*Q*) =_{j}*β*/*Q*) and a concave inverse demand () in Cournot competition. It can be shown that for a given linear transportation cost function (_{j}*t*(*d*) =*τd*), maximum differentiation is the unique Cournot equilibrium in each case. - 16
It can be shown that when , and when . Moreover, .

### References

- 1988) Equilibrium existence in the linear model of spatial competition. Economica 55: 479–491 (
- 1991) Cournot competition yields spatial agglomeration. International Economic Review 32: 793–808 , (
- 2006) Location as an instrument for social welfare improvement in a spatial model of Cournot competition. Investigaciones Economicas 30: 117–136 , (
- 2001) The intensity of competition and the incentive to innovate. International Journal of Industrial Organization 19: 705–726 (
- 1979) On Hotelling's ‘stability in competition’. Econometrica 47: 1145–1150 , , (
- 1990) R&D investment in a symmetric and homogeneous oligopoly. International Journal of Industrial Organization 8: 297–313 , (
- 1989) Symmetric equilibrium existence and optimality in a differentiated product market. Journal of Economic Theory 47: 178–194 (
- 2004) Where to locate in a circular city? International Journal of Industrial Organization 22: 759–782 , , , , (
- 2000) A note on price and quality competition in differentiated oligopolies. Journal of Economic Theory 93: 233–239 (
- 1989) Spatial discrimination: Bertrand vs. Cournot in a model of location choice. Regional Science and Urban Economics 19: 87–102 , , (
- 1929) Stability in competition. Economic Journal 39: 41–57 (
- 1990) Effects of spatial price discrimination on output, welfare, and location. American Economic Review 80: 567–575 , (
- 1983) Quantity precommitment and Bertrand competition yield Cournot outcomes. Bell Journal of Economics 14: 326–327 , (
- 2006) Spatial discrimination: Bertrand vs. Cournot with asymmetric demands. Regional Science and Urban Economics 36: 790–802 , , (
- 2005) A note on economic welfare in delivered pricing duopoly: Bertrand and Cournot. Economics Letters 89: 112–119 , (
- 2006) Cournot and Bertrand in shipping models with circular markets. Papers in Regional Science 85: 585–598 , (
- 1985) Two stage (perfect) equilibrium in Hotelling's model. Journal of Industrial Economics 33: 317–325 (
- 1998) Does Cournot competition yield spatial agglomeration? Economics Letters 60: 49–53 (
- 1997) On the dynamic efficiency of Bertrand and Cournot equilibria. Journal of Economic Theory 75: 213–229 (
- 1984) Price and quantity competition in a differentiated duopoly. Rand Journal of Economics 15: 546–554 , (
- 2009) A note on Arevalo-Tome and Chamorro-Rivas: Location as an instrument for social welfare improvement in a spatial model of Cournot competition. Investigaciones Economicas 33: 131–141 (
- 1985) Existence of spatial Cournot equilibria. Regional Science and Urban Economics 15: 219–227 (
- 2006) Differentiated duopoly with asymmetric costs. Journal of Economics and Management Strategy 15: 999–1015 (