Kinked-Demand Equilibria and Weak Duopoly in the Hotelling Model

The B.E. Journal of Theoretical Economics Contributions Volume 10, Issue 1

2010

Article 12

Kinked-Demand Equilibria and Weak Duopoly in the Hotelling Model of Horizontal Differentiation Pierre R. Mérel∗

∗ †

Richard J. Sexton†

University of California, Davis, [email protected] University of California, Davis, [email protected]

Recommended Citation Pierre R. Mérel and Richard J. Sexton (2010) “Kinked-Demand Equilibria and Weak Duopoly in the Hotelling Model of Horizontal Differentiation,” The B.E. Journal of Theoretical Economics: Vol. 10: Iss. 1 (Contributions), Article 12. Available at: http://www.bepress.com/bejte/vol10/iss1/art12 c Copyright 2010 The Berkeley Electronic Press. All rights reserved.

Kinked-Demand Equilibria and Weak Duopoly in the Hotelling Model of Horizontal Differentiation∗ Pierre R. Mérel and Richard J. Sexton

Abstract The Hotelling model with finite consumer reservation price is, in its various forms, perhaps the canonical model of horizontal product differentiation. Yet the following key aspects of this model are little understood: (i) the existence of asymmetric price equilibria when consumers have unit demands and (ii) for a very broad set of model specifications, the non-monotonicity of price as a function of consumers’ transportation cost, i.e., the degree of product differentiation in the market. We provide a complete characterization of the asymmetric equilibria, show that they exist for a comparatively “wide” range of markets, and argue that their existence is robust to various extensions of the prototype model. Introducing elasticity into consumer demands suppresses the kink in the firms’ demand functions and ensures the uniqueness (and symmetry) of the price equilibrium. However, the key perverse comparative static of the symmetric kinked equilibrium, decreasing price as a function of transport cost, survives relaxation of the unit-demand assumption. Indeed the symmetric kinked equilibrium of the unit-demand case is a special case of a general but largely unexplored set of equilibria in the Hotelling model we call weak duopoly. Weak-duopoly equilibria exist for intermediate values of transportation cost for a very broad class of consumer demands. They occupy a comparatively wide range of the model parameter space, and firm behavior in weak duopoly differs fundamentally from that which occurs for lower values of the transportation cost, or when consumers have no reservation price. KEYWORDS: duopoly, Hotelling, kinked-demand equilibrium, asymmetric equilibria, transportation cost

∗

We are grateful to the editor and two anonymous referees for helpful comments.

Mérel and Sexton: Weak Duopoly in the Hotelling Model

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Introduction

The canonical model of horizontal product differentiation is due to Hotelling (1929), whose original model involved duopoly price competition for the patronage of consumers distributed uniformly on a line with respect to their preferences for an attribute of the product. All consumers purchase one unit from the firm which offers them the lower total price, defined as the mill price set by the firm plus the consumer’s linear-in-distance costs of “traveling” to the firm’s location, or more generally, his disutility from not consuming his ideal product variety. This extreme form of demand inelasticity was questioned by Lerner and Singer (1937), who introduced a reservation price to the otherwise unitary consumer demands. Salop (1979) provided a formal justification for the reservation price based upon the presence of an outside good that can be purchased at a fixed price in lieu of the good produced by the spatially differentiated sellers. Hotelling’s framework is utilized in three broad classes of models: (i) those where both prices and locations are chosen (Hinloopen and van Marrewijk, 1999), (ii) those where only price is chosen and locations are fixed (Tirole, 1988), and (iii) those where only locations are chosen and prices are fixed (Anderson and Engers, 1994, Hinloopen, 2002) or absent, as in Downsian models of political competition (Aldrich, 1983).1 Our focus is price competition; we take firms’ locations as given in the base model. Given that Hotelling models of price and location choice are often solved sequentially in two stages (location being the first stage), our results are directly relevant to the pricing stage of those models. When Hotelling’s model is extended to include an outside good, the consumers’ reservation price adds a participation constraint to the sellers’ pricing decisions, and this affects the nature of price competition significantly. The importance of product differentiation is then determined by the magnitude of the market length times transport cost per unit length relative to the reservation price net of any unit production cost (Hinloopen and van Marrewijk, 1999)—a ratio we call the normalized transportation cost. First, a local-monopoly equilibrium arises for sufficiently large values of the normalized transportation cost. Second, a transition region appears between the “competition region” (Salop, 1979), where sellers compete in prices and the participation constraint is not binding for any consumer, and the monopoly region.2 Here each seller sets price at the “kink” of her demand function. The kink occurs because the demand curves facing a seller in the monopoly and competition regions have different slopes, the latter being steeper because a price decrease is less effective at attracting customers in this region. 1 We 2 The

thank a referee for suggesting this characterization of the literature. competition region occupies the entire parameter space in the original Hotelling model.

Published by The Berkeley Electronic Press, 2010

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The B.E. Journal of Theoretical Economics, Vol. 10 [2010], Iss. 1 (Contributions), Art. 12

The focus of this paper is on this competition-to-monopoly transition region which we call weak duopoly. We show that the well-known kinked-demand equilibrium of the inelastic-demand case is in essence a special case of the more general concept of weak duopoly, which prevails for intermediate values of the normalized transportation cost. The weak-duopoly region occupies a substantial range of the available parameter space,3 and there key comparative statics are reversed relative to the competition region. Of particular importance is that the price set by either seller is inversely related to the magnitude of consumers’ transportation costs (i.e., the extent of product differentiation in the market) in this range.4 Yet, this facet of the Hotelling model with reservation price has received relatively little attention and seems often misunderstood. In what follows we establish and explain several propositions pertaining to equilibrium prices in the weak-duopoly region of the Hotelling model with reservation price and the important special case of kinked-demand equilibria. First, in section 2 for the prototype model with unitary consumer demands we show that in the parameter region where firms set price at the kink of their demand curves, a continuum of asymmetric Nash equilibria exist besides the well-known symmetric one explicated by Salop (1979) (Proposition 1). Although the existence of multiple kinked-demand equilibria has been known at least since Economides (1984), their properties have not been fully explicated and the assumptions that cause their existence have not been identified. We show that for each possible equilibrium, the sum of prices is the same, and prices and firm market radii are maintained within a certain bracket. Hence, even though an infinity of equilibria exist, they remain “close” to the symmetric equilibrium. We trace the existence of asymmetric equilibria to the kink in the demand functions facing each seller, which leads to reaction functions that partially overlap. This result provides a guide to identifying which properties of the model are crit3 It

occurs in the inelastic-demand version of our model for half of the range of values of the normalized transportation cost that support the competition regime. In the related long-run spatial model of Hinloopen and van Marrewijk (1999), type II equilibria (conceptually similar to what we call the transition region) actually occupy a much larger portion of the parameter space than type I equilibria (conceptually similar to Salop’s competition region). 4 A few papers have recently drawn attention to implications of this result. Cowan and Yin (2008) make the point that entry in a horizontally differentiated market can harm consumers if it moves the market from monopoly to the transition region. In a broader context where consumer preferences for two product varieties are described by the statistical distribution of their net valuations for each variety, rather than their distribution on a line, Chen and Riordan (2010) derive a necessary and sufficient condition for the duopoly price to be higher than the monopoly price. In a duopsony context, Mérel, Sexton, and Suzuki (2009) emphasize that investments to reduce transportation costs in the transition region will harm farmers distributed on the line and, thus, fail as an economicdevelopment strategy.

http://www.bepress.com/bejte/vol10/iss1/art12

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ical to the existence of the asymmetric equilibria. We show in section 3 that such equilibria continue to exist for quadratic transportation costs (Proposition 2), and also that their existence is robust to locating sellers in the interior of the market. The reason in each case is that the adaptation to the base model does not eliminate the kinked seller demands. However, we show in section 4 that relaxing the unit-demand assumption by allowing the quantity demanded by each consumer to decrease with price does suppress the continuum of asymmetric Nash equilibria (Proposition 3). Comparative statics at the symmetric kinked-demand equilibrium are said to be “perverse” (Salop, 1979). In light of the greater realism of elastic consumer demands,5 it is worth asking whether any of these perverse comparative statics survive in this generalized setting with smooth seller demands. The answer, as shown in section 5, is yes, but only as it pertains to the key result that the equilibrium price is a non-monotonic function of the transportation cost (Proposition 4). It is increasing in the transportation cost for its low values, but decreasing in the transportation cost in the weak-duopoly region. The robustness of this key comparative static result affirms that weak duopoly is a very general spatial oligopoly concept of which Salop’s kinked-demand equilibrium represents a special case. In conjunction with the competition and monopoly regions, it completes the characterization of market conditions in the Hotelling model when consumers have elastic demands subject to a choke price. Section 6 explores the three regimes for a broad class of consumer demands of the power form. There we introduce a metric, which we call relative net price, defined as output price net of unit production costs, expressed relative to buyer reservation price, also net of unit production costs. Thus, relative net price is closely analogous to the traditional Lerner index of relative price markup. All comparative statics for the relative net price are reversed in the weak-duopoly region compared to the competition region (Proposition 5). In addition, the weak-duopoly region occupies a significant portion of the parameter space relative to the competition region—an equal amount in the limit as consumer demands become very elastic.

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Kinked-Demand Equilibria in the Hotelling Model with Reservation Price

Consumers are distributed uniformly along a segment of length one. Their total number is normalized to one. One seller of a good is located at each end of the 5 This

point was made as early as Smithies (1941), and several subsequent authors have investigated extending the Hotelling model to this case.


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segment, and denoted a and b, respectively. The products are identical except for their location on the segment. Sellers set mill prices so that consumers bear the cost of transporting the product to their respective location. Each consumer purchases at most one unit of the good subject to a reservation price R. The cost of producing the good is constant and equal to c < R. Transportation costs are linear per unit of γ the normalized transportation cost. distance, and denoted γ. We denote by t ≡ R−c Setting the location of firms at the endpoints of the market is not crucial to our argument, as we show in section 3. But this modeling choice allows us to ignore the complications that arise, when firms are located within the segment, from the presence of some consumers in the “hinterlands” between a firm and a market endpoint, and the related incentive of each firm to undercut the other firm’s price.6 Let Qa (Pa , Pb ; R, γ) be the quantity demanded from seller a under prices Pa and Pb . Pa and Pb must be lower than R in order to attract any buyer at all. Further, it is not optimal for seller a to charge a price strictly lower than Pb − γ, because all consumers already purchase from him at Pa = Pb − γ. Similarly, seller a will not charge a price greater than Pb + γ, because then all consumers purchase from seller b. Therefore, we take for granted that Pa ≤ R, Pb ≤ R, Pa ≥ Pb − γ and Pa ≤ Pb + γ. The demand function facing seller a has the form7 ( P −P +γ a b if Pb − γ ≤ Pa ≤ min(γ + Pb , 2R − γ − Pb ) 2γ (1) Qa (Pa , Pb ; R, γ) = R−Pa if 2R − γ − Pb < Pa ≤ min(R, γ + Pb ) γ and is kinked for values of Pb such that γ + Pb > R. Demand on the segment [Pb − γ, min(γ + Pb , 2R − γ − Pb )] depends on the price charged by the other seller and corresponds to the “competitive” demand. In this range, a decrease in price by ∆ ∆ leads to a rise in sales by 2γ . On the segment (2R − γ − Pb , min(R, γ + Pb )] demand only depends on Pa and corresponds to the monopoly demand. In this range, a decrease in price by ∆ leads to a rise in sales by ∆γ , that is, twice as much as in the competitive range. By symmetry, the demand facing seller b is of the same form as Qa , with the indices a and b reversed. Sellers’ reaction functions, and the resulting pure-strategy Nash equilibria are derived in Appendix A.8 Proposition 1 In Hotelling’s model with fixed locations at the endpoints and unit demands subject to reservation price, three competition regimes obtain. 6 For a discussion of the consequences of undercutting in a model with endogenous locations and symmetric price equilibria, see Hinloopen and van Marrewijk (1999). 7 See Appendix A for the derivation. 8 Due to the strict quasi-concavity of sellers’ profit functions, mixed-strategy Nash equilibria can be ruled out for this variant of the Hotelling model with fixed locations at the endpoints.


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1. A “competitive” regime obtains for t ∈ (0, 23 ]. The Nash equilibrium is unique and symmetric, characterized by P¯a = P¯b = c + γ. The market is covered and the indifferent consumer is located at Y¯ = 12 . For 0 ≤ t ≤ 12 , the demand facing each seller consists only of the competitive segment. 2. Multiple kinked-demand equilibria arise for t ∈ ( 23 , 1]. Each firm prices at the kink of its demand function. There are 2 subcases: (a) 23 < t ≤ 56 . Equilibrium prices are characterized by (P¯a , P¯b ) ∈ [ 43 R − 3c − ¯ ¯ γ, 2R+c 3 ] and Pa + Pb = 2R − γ. The market is covered and the indifferent ¯ consumer is located at Y¯ = R−γ Pa , that is, within the segment [ 3t1 , 1 − 3t1 ]. 3 (b) 65 < t ≤ 1. Equilibrium prices are characterized by (P¯a , P¯b ) ∈ [ R+c 2 , 2 R− c ¯ ¯ 2 − γ] and Pa + Pb = 2R − γ. The market is covered and the indifferent ¯ consumer is located at Y¯ = R−γ Pa , that is, within the segment [1 − 2t1 , 2t1 ]. 3. When t > 1, each firm is a monopolist; the desired market radii of the two firms do not overlap and each sets price independently of the other. The equilibrium price is P¯a = P¯b = R+c 2 . Proposition 1 establishes the existence of an infinity of kinked-demand Nash equilibria in the Hotelling model with reservation price for values of the normalized transport cost in the range ( 23 , 1).9 Their existence is a direct consequence of the fact that the reaction curves of the two sellers overlap in (Pa , Pb ) space. Given the symmetry of the game, the curves can overlap only if the reaction curve of a given firm is symmetric with respect to the line Pa = Pb in (Pa , Pb ) space. This is made possible because there exists a range of values of Pb for which the optimal response of seller a is to set price at the kink of her demand curve (the location of which is determined by Pb ), a direct consequence of the discontinuity in the marginal revenue of seller a at the kink. Setting price at the kink of the demand curve, which separates the monopoly and competition portions of demand, implies that the a +γ , indifferent buyer earns zero surplus. Since she is located at the point Y = Pb −P 2γ ∗ ∗ this condition implies that the optimal response Pa is such that Pa + Pb = 2R − γ in this range of Pb . It is then apparent that this portion of a’s reaction curve is symmetric with respect to the line Pa = Pb , allowing the two reaction curves to overlap for intermediate values of the normalized transportation cost t. 9 To

our knowledge, the prior published papers that acknowledge and attempt to characterize the set of asymmetric kinked-demand equilibria are Economides (1984) (for a model where sellers are located within the market segment), Yin (2004) and Cowan and Yin (2008) (for a model similar to ours). In contrast to these papers, we explicitly derive the range of values that prices can take on for each of the two subcases in point 2 of Proposition 1.


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Yet, asymmetric equilibria remain “close” to the symmetric one in the sense that possible prices are bounded from below and above, and the indifferent consumer is located relatively close to the mid point of the market, meaning that firms’ market shares are also bounded.10 The equilibrium price schedule is depicted in figure 1. In terms of stability, the Nash equilibria arising at the kink in firms’ demand curves differ from those in the competitive range in the sense that, starting from one possible kinked-demand equilibrium, small departures from Nash behavior by one seller will not trigger a series of myopic responses that converge to the initial equilibrium. Instead, the immediate myopic price response of the other seller will lead to a new kinked-demand equilibrium different from the initial one. In this respect, kinked-demand equilibria are not stable when considered individually, although the set of all possible such equilibria itself is stable.

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Extensions to the Base Model

We now show the robustness of the existence of asymmetric kinked-demand equilibria to extensions of the prototype Hotelling model. We first examine the case of quadratic transport costs and then allow sellers to be located in the interior of the market segment. Our purpose is simply to establish the existence of asymmetric kinked-demand equilibria, and we do not provide an exhaustive characterization of the set of price equilibria that arise for all values of the model parameters.

3.1

Quadratic Transport Costs

Assume that transportation costs have the form C(x, xi ) = γ|x − xi |2 , where x denotes the consumer’s location and xi denotes the seller’s location. Quadratic transportation costs constitute a popular modeling choice in a Hotelling model (e.g., d’Aspremont, Gabszewicz, and Thisse (1979), Friedman and Thisse (1993), Rath and Zhao (2001), Lambertini (2002)). They are particularly intuitive when costs represent the disutility incurred by consumers from not being able to purchase their preferred product variety. A continuum of equilibria may still arise in this case.11 10 The

exact bounds on the location of the indifferent consumer depend on t and are derived in Proposition 1, but the indifferent consumer never moves outside the segment [ 52 , 53 ]. 11 This result comes in contrast to Economides (1989)’s assertion that for quadratic costs the kinked-demand equilibrium is unique (see his Theorem 3 and proof of Lemma 1). The error can be traced to his assertion that when the equilibrium is at the kink of the demand curve, the absolute value of the slope of the reaction function is strictly smaller than one. Using his notation with k


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P¯ 6 HH @ H @ @ HH @ @ HH@ @ H@ H H @ @

competitive equilibrium

2 3

kinked-demand equilibrium

1

monopoly equilibrium

-

γ

¯ Figure 1: Symmetric equilibrium price schedule P(γ) for (R, c) = (1, 0). The maximum and minimum kinked-demand equilibrium prices (P¯a , P¯b ) are also represented ¯ ¯ for γ ∈ ( 23 , 1]. Kinked-demand price pairs satisfy Pa +2 Pb = R − 2γ , and this mean price is equal to the symmetric price. These equilibria also correspond to kinked-demand equilibria in that each firm sets price at the kink of its demand function. 3.1.1

The Demand Function

As in the linear case, we know that Pa ≤ R, Pb ≤ R, Pa ≤ γ + Pb and Pa ≥ Pb − γ. We show in Appendix B-1 that the demand function to seller a is ( Pb −Pa +γ if Pb − γ ≤ Pa ≤ min(γ + Pb , Pa+ ) 2γ q Qa (Pa , Pb ; R, γ) = R−Pa if Pa+ < Pa ≤ min(R, γ + Pb ) γ

p denoting the reservation price, the ratio d p j /d p j−1 is equal to 1 − (x j − x j−1 )/ k − p j−1 , but there is no reason why this last expression should be less than one.


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The B.E. Journal of Theoretical Economics, Vol. 10 [2010], Iss. 1 (Contributions), Art. 12 Qa

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

1.0

Pa

Figure 2: Demand curve facing each seller in the case of unit demands and quadratic costs. p where Pa+ (Pb ) = Pb − γ + 2 γ(R − Pb ) defines the kink in seller a’s demand funca +γ tion. When γ + Pb < R, Qa = Pb −P on the relevant range, but when γ + Pb ≥ R, 2γ Qa has two different expressions on [Pb − γ, Pa+ ] and (Pa+ , R] and is kinked. Figure 2 depicts Qa in the latter case. From the symmetry of sellers, the demand function Qb has a similar expression, with the roles p of Pa and Pb reversed and the function + + Pb (Pa ) defined as Pb (Pa ) = Pa − γ + 2 γ(R − Pa ). Note that the functions Pa+ (Pb ) and Pb+ (Pa ) overlap in (Pa , Pb ) space. 3.1.2

Condition for Seller a to Set Price at the Kink of His Demand Curve

We now derive the conditions under which seller a chooses to set price at the kink of his demand curve, that is, chooses Pa∗ = Pa+ . For ease of presentation, we assume that c = 0, in which case t = Rγ . A first condition necessary for seller a to set price at Pa+ is that Qa be kinked at Pa+ , and this requires that Pb ≥ R − γ. In this case, the profit function of seller a, Πa (Pa , Pb ; R, γ), is concave on each of the subintervals [Pb − γ, Pa+ ] and (Pa+ , R]. Seller a will choose Pa∗ = Pa+ if and only if the leftward derivative of his profit function is positive at Pa+ and the rightward derivative of his profit function is negative at Pa+ , two conditions that imply that p Pb − 3γ + 4 γ(R − Pb ) < 0 (2) p 2 Pb − γ + 2 γ(R − Pb ) − R > 0. (3) 3 http://www.bepress.com/bejte/vol10/iss1/art12

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Condition (2) is shown in Appendix B-2 to be equivalent to p Pb ≤ 3γ and Pb > −5γ + 4 γ(γ + R) which can only be satisfied if R < 3γ, a condition on consumers’ utility of consuming their preferred product relative to their rate of disutility from consuming an alternative to their preferred variety. Since we know that Pb ≤ R, the condition Pb ≤ 3γ becomes redundant p once we impose that R < 3γ. In addition, under R < 3γ the interval (−5γ + 4 γ(γ p + R), R) is always nonempty, so that the conditions Pb ≤ R and Pb > −5γ + 4 γ(γ + R) are compatible. Therefore, under the maintained assumptions that Pb ≥pR − γ and Pb ≤ R, condition (2) is satisfied if and only if R < 3γ and Pb > −5γ + 4 γ(γ + R). Under the condition that R 2R − γ − Pb . Since Pa ≤ R, this may happen only when 2RR − γ − Pb < R, that is, γ + Pb > R. The demand for seller a’s product is then Qa = 0X φ (Pa + γX)dx = R−Pa 1 γ [Φ(Pa + γX) − Φ(Pa )], where X solves Pa + γX = R, that is, X = γ . Therefore, Qa = 1γ [Φ(R) − Φ(Pa )]. Summarizing the above results, the demand function facing seller a is  Pa +P +γ b −Φ(Pa )  Φ 2 if Pb − γ ≤ Pa ≤ min(γ + Pb , 2R − γ − Pb ) . γ Qa =  Φ(R)−Φ(Pa ) if 2R − γ − Pb < Pa ≤ min(R, γ + Pb ) γ 1 1 γ [Φ(Pa + γY ) − Φ(Pa )] = γ

The interval [Pb − γ, min(γ + Pb , 2R − γ − Pb )], where Qa depends on the rival’s price Pb , corresponds to the competitive portion of demand. The interval (2R − γ − Pb , min(R, γ + Pb )], which corresponds to the monopoly demand, is nonempty if and only if γ + Pb > R. 14 The

alternative assumption is that the transportation cost is borne only once, i.e., it is a fixed cost of consumption as in Stahl (1987), Anderson, de Palma, and Thisse (1992), and Rath and Zhao (2001). Our assumption that the cost is borne on each unit is more plausible when transportation costs represent the disutility from not consuming one’s preferred product or when shipping charges are proportional to weight or volume.


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Assume that γ + Pb > R, so that Qa has two different expressions on the subintervals [Pb − γ, 2R − γ − Pb ] and (2R − γ − Pb , R]. The leftward derivative of the profit function Πa (Pa , Pb ) = (Pa − c)Qa (Pa , Pb ) at Pa = 2R − γ − Pb is equal to Qa (2R − γ − Pb , Pb ) + (2R − γ − Pb − c) ∂∂QPaa (2R − γ − Pb , Pb ). Similarly, the rightl

ward derivative of Πa at Pa = 2R − γ − Pb is equal to Qa (2R − γ − Pb , Pb ) + (2R − γ − Pb − c) ∂∂QPaa (2R − γ − Pb , Pb ). These two derivatives are equal if and only if r h i ∂ Qa ∂ Qa 1 φ (R) (2R−γ −P , P ) = (2R−γ −P , P ), that is, − φ (2R − γ − P ) = b b b b b γ 2 ∂ Pa ∂ Pa l 1 [−φ (2R − γ − Pb )], γ

r

which is always true since φ (R) = 0.

Lemma 1 The demand function facing seller a is differentiable at Pa = 2R − γ − Pb . Lemma 1 implies that the demand function for seller a is no longer kinked at Pa = 2R − γ − Pb .15 This result eliminates the possibility of asymmetric Nash equilibria. To see why, first note that there can only be one (symmetric) price equilibrium when sellers behave as isolated monopolies. The demand facing each seller, and therefore each seller’s profit function, does not depend on the price charged by the other seller in this case. On the relevant range, the reaction functions of sellers are flat and intersect at a single point. Second, there cannot exist a continuum of price equilibria when sellers set prices on the competitive portion of their demand curves. The reason is that a seller’s best response on this range is a function that is generally not symmetric with respect to the line Pa = Pb in (Pa , Pb ) space, and therefore the reaction curves of the two sellers do not overlap.16 Proposition 3 When consumer demands are of the form q = φ (p), for a nonincreasing, continuous function φ satisfying φ (p) = 0 if p ≥ R > 0, there does not exist a continuum of price equilibria for any value of the model parameters. 15 This result corrects a misconception in the literature,

namely that the kink survives the introduction of elasticity into consumer demands. See for example Greeenhut, Norman, and Hung (1987, p. 92). 16 See Appendix C. Ensuring that the equilibrium is indeed unique on the competitive range (that is, that the reaction curves intersect at a single point) would require further assumptions on the second derivative of the payoff function, to ensure that the best response mapping is a contraction. Since we are only interested in ruling out the continuum of asymmetric equilibria arising from the kink in the demand function, we abstain from deriving the sufficient conditions under which the reaction functions intersect only once. See, for instance, Economides (1989).


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5

Weak Duopoly

Salop (1979) derived the comparative statics properties of symmetric kinked-demand equilibria for his circular-market model, where firms locate symmetrically. He concluded that they were all perverse: price (i) decreases with transportation costs, (ii) is insensitive to increases in marginal cost in the short run, and (iii) rises with the reservation price of consumers (whereas price is insensitive to the reservation price in the competitive region). These properties also hold for the symmetric kinkeddemand equilibrium in the Hotelling model with linear transportation costs, where price is P¯ = R − 2γ . All three results follow as a consequence that the market is fully covered, but the indifferent consumer obtains zero net surplus. Thus, the entirety of an increase in R is transmitted to price, but no part of an increase in marginal cost, c, can be transmitted. Similarly, an increase in γ requires a decrease in P¯ to maintain market coverage. Here we ask if any of the perverse comparative statics associated with the kinked-demand case survive extension to elastic consumer demands and, hence, smooth seller demands. We conclude in the affirmative but only as it pertains to the key result that equilibrium price is inversely related to consumers’ transportation costs for a range of model parameters that characterize the market in a transition region between competition and monopoly.17 This transition region between spatial monopoly and “ordinary” spatial competition, which we call weak duopoly and which was associated with kinked seller demands in the inelastic-consumer-demand case, thus appears to be a quite general phenomenon of spatial product differentiation models. We establish the key nonmonotonicity result regarding the effect of transportation costs on equilibrium price in this section and fully characterize the properties of the weak-duopoly region for demand functions of the power form in the next section. The effects of c and R on P¯ are investigated in Appendix D. There, we argue that a marginal increase in c unambiguously raises the equilibrium price ¯ while the effect of R on P¯ in the region where sellers compete is ambiguous. P, Assume that individual consumer demands are of the form q = φ (p), with 0 if p ≥ R φ (p) = ψ(p) otherwise where ψ is a decreasing function that is continuously differentiable on the segment [0, R] and satisfies ψ(R) = 0. The above specification ensures that φ itself is continuous on [0, R] and continuously differentiable at least on the segment [0, R). In addition, φ admits a leftward derivative at R, equal to ψ 0 (R). 17 This

non-monotonicity has been characterized in the studies by Nero (1999) in a duopoly context and Zhang and Sexton (2001) in a duopsony context. However, those papers only solve the price equilibrium in the special case of linear demand or supply functions, respectively.


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¯ To show that the equilibrium price schedule P(γ) is non-monotonic, we ¯ ¯ ˜ < 0, where γ˜ denotes the value of transportaprove that ddγP (0) > 0 and ddγP (γ) γ≤γ˜

tion costs that separates the monopoly regime from the regime where the market is covered and firms compete. The equilibrium price under monopoly, P¯m , satisfies −(P¯m − c)φ (P¯m ) + Φ(R) − Φ(P¯m ) = 0.

(6)

Equation (6) corresponds to the first-order condition for profit maximization by one seller on the monopoly portion of its demand curve, and together with the secondorder condition characterizes the equilibrium price under monopoly. ¯ satisfies The equilibrium price under duopoly competition, P, 1 γ γ ¯ ¯ = 0. ¯ ¯ ¯ − φ (P) + Φ P + − Φ(P) (7) (P − c) φ P + 2 2 2 Equation (7) corresponds to the first-order condition for profit maximization by one seller on the competitive portion of its demand curve, with the symmetry condition Pa = Pb = P¯ imposed after taking the derivative. Together with the second-order condition for a seller’s profit maximization, evaluated at the equilibrium prices Pa = ¯ equation (7) implicitly defines the equilibrium price P¯ under duopoly. Pb = P, The value of transportation costs separating the monopoly equilibrium from ˜ is such that the desired market radius of each the duopoly equilibrium, denoted γ, monopolistic seller is equal to one half of the total market length. Since the market γ ¯ ˜ radius of a seller who charges price P is equal to R−P γ , γ must solve Pm = R − 2 , where P¯m denotes the monopoly equilibrium price and is an implicit function of R and c defined by (6). Therefore, γ˜ is also an implicit function of R and c. ˜ c), so that the Note that equations (6) and (7) become identical for γ = γ(R, ¯ ˜ Totally differentiating equation (7) yields18 price schedule P(γ) is continuous at γ. ¯ (P−c) 0 P¯ + γ + 1 φ P¯ + γ ψ d P¯ 4 2 2 2 =− ¯ + 3 φ P¯ + γ − 2φ (P) ¯ dγ (P¯ − c) 12 ψ 0 P¯ + 2γ − ψ 0 (P) 2 2 which, evaluated at γ = 0, yields ˜ we obtain: derivative of P¯ at γ,

d P¯ dγ (0) = 1

(8)

¯ since P(0) = c. Evaluating the leftward ¯

(Pm −c) 0 d P¯ 4 ψ (R) ˜ =− 1 (γ) dγ γ≤γ˜ (P¯m − c) 2 ψ 0 (R) − ψ 0 (P¯m ) − 2φ (P¯m ) 18 The expression in (8) is defined since on the duopoly range the equilibrium price P ¯ must satisfy γ ¯ P + 2 ≤ R for the market to be covered.


15


which is negative given the second-order condition for profit maximization on the monopoly range, −(P¯m − c)ψ 0 (P¯m ) − 2φ (P¯m ) ≤ 0.19 Proposition 4 With elastic consumer demands, the price schedule is non-monotonic in the transportation cost γ. It is increasing in γ on a neighborhood of γ = 0 and ˜ decreasing in γ on a neighborhood of γ ≤ γ. To understand the non-monotonicity of P¯ in γ, note that there are offsetting effects from an increase in transport costs. An “elasticity effect” causes either seller to wish to absorb a portion of an increase in γ in the form of a lower price to limit the reduction in sales. In the unit-demand case, this effect is solely due to the participation constraint of the marginal consumer, while in the elastic demand case it is due to each consumer demanding less as γ increases. Second is a “competition effect” due to the increase in product differentiation between firms caused by an increase in γ, which promotes higher prices. The competition effect dominates for small values of γ; in fact, it is the only effect in the unit-demand case until the participation constraint starts binding for the marginal consumer. The elasticity effect becomes more important relative to the competition effect for larger values of γ because the profit margin on sales is higher, due to high differentiation between the firms. Proposition 4 demonstrates that for any smooth downward-sloping consumer demand with choke price, a range of values of γ exists in the region of competition for which the elasticity effect dominates the competition effect, such that P¯ is decreasing in γ. Proposition 4 further implies that such values of γ can be found in the vicin ¯ d P ˜ < 0 while ddγP¯ (0) > 0, there exists at least one ˜ Formally, because dγ (γ) ity of γ. γ≤γ˜

˜ such that ddγP¯ (γ) = 0. Denote by γ¯ the largest of these values. value of γ in (0, γ) ¯ γ) ˜ the price schedule is decreasing in γ, and it is precisely Then, on the range (γ, this region of the parameter space that we call weak duopoly.

6

Weak Duopoly with Power Demands

In this section, we derive the entire price schedule and therefore extend the local analysis of section 5 for the case where ψ(p) = (R − p)ε , for ε > 0.20 This parame19 If

the function ψ is not differentiable at R but satisfies lim p→R ψ 0 (p) = −∞, as is the case with p 1).


16


terization allows more insight to be obtained into the roles that parameters γ, R and c play in determining price in the three regimes, competition, weak duopoly, and monopoly, of a Hotelling model with reservation price. These regimes, in essence, replace the competitive, kinked-demand equilibria, and monopoly regimes that exist for the special case of inelastic consumer demands. The typology of regimes is based on the comparative static properties of the ¯ measure of relative price markup we call relative net price, ρ¯ = P−c R−c , with respect to γ, R and c. The effect of space on competition in this case is fully captured by γ the normalized transportation cost t = R−c , and ρ¯ itself can be written as a function of t only. The following proposition is proven in Appendix E. Proposition 5 Denote by tˆ the positive solution to the equation 1 − 2t t)(2 + ε) = 0, and define t˜ ≡ 2(1+ε) 2+ε .

1+ε

− 1 + εt

ε

(1−

• For 0 ≤ t < tˆ, ρ¯ is an increasing function of t (competition). • For tˆ ≤ t < t˜, ρ¯ is a decreasing function of t (weak duopoly). 1 • For t ≥ t˜, ρ¯ = ε+2 and is independent of t (monopoly). Proposition 5 establishes the relevance of the normalized transportation cost as an indicator of the effect of space on competition in the Hotelling model with power demands.21 The competitive and weak duopoly regimes correspond to cases where the market is covered, but they differ fundamentally in terms of how the relative net price ρ¯ varies with t. For low values of t (competitive regime), the “competition effect” dominates and ρ¯ is an increasing function of t, and thus an increasing function of γ, a decreasing function of R and an increasing function of c. In contrast, for larger values of t (weak duopoly) the “elasticity effect” dominates so that ρ¯ is a decreasing function of t. Since γ does not appear in the definition of ¯ Proposition 5 implies that in the competitive range P¯ is an increasing function of ρ, γ, while on the weak-duopoly range P¯ is a decreasing function of γ. With elastic demands there are no “perverse” comparative statics of P¯ itself with respect to parameters c and R. But the reversal of sign of the comparative statics for c and R once ρ¯ is considered helps to clarify the nature of competition in the weak-duopoly region. Consider an increase in consumers’ reservation price, R. It represents a parallel shift upward of a consumer’s inverse demand curve, which, for given values of γ and c, is sufficient to increase P¯ for each of the three ranges of 21 Although

the relevance of using t as an indicator of the importance of space has been noted in previous studies (Hinloopen and van Marrewijk, 1999, Zhang and Sexton, 2001), to our knowledge the literature has only established this relevance in the special cases ε → 0 and ε = 1. Note also that the relevance of this metric does not extend to arbitrary demand functions.


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competition. However, it also reduces product differentiation, as represented by the magnitude of t. In the competitive region, the fact that ρ¯ is increasing in t implies that an increase in R has the effect of increasing the net price P¯ − c by a smaller percentage than the net value of output R − c, that is, d ln(P¯ − c) < d ln(R − c). The result is opposite if sellers initially set price in the weak-duopoly region. In the competitive region, where the competition effect dominates, the reduction in product differentiation due to higher R works to reduce price, offsetting the price increase commanded by the increase in consumers’ willingness to pay. But in the weak-duopoly region, the elasticity effect dominates and thus a reduction in the degree of product differentiation tends to increase price, which reinforces the effect of the increase in consumers’ willingness to pay, causing the relative increase in the net price to exceed the relative increase in the net value of output.

Figure 4: Equilibrium price schedule for ψ(p) = (1 − p)ε , ε > 0, p ≤ 1, and c = 0. ¯ Figure 4 displays the price schedule P(γ) for several demand functions of ε ¯ ¯ the form ψ(p) = (1 − p) and for c = 0, in which case P(γ) = ρ(γ). As ε → 0, the price schedule approaches the symmetric price schedule of the unit-demand case. Figure 5 illustrates the size of the parameter region that supports the weak duopoly regime, as a function of the elasticity parameter ε. This size is measured relative to that of the competitive region as t˜−tˆ tˆ . It varies from about 0.31 for ε = 0.3 to 1 for http://www.bepress.com/bejte/vol10/iss1/art12

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ε → ∞. Somewhat interestingly, the relative size of the weak-duopoly range is not a monotonic function of ε.

Figure 5: Size of the weak-duopoly region relative to that of the competitive region, as a function of the elasticity parameter ε. Proposition 5 provides a new perspective on the economic significance of the perverse comparative statics identified by Salop (1979) in the case of kinkeddemand equilibria. While the non-monotonicity of the equilibrium price P¯ with respect to the transport cost extends to elastic demands, and appears to be a fundamental characteristic of weak-duopoly, the perverse short-run comparative statics of P¯ with respect to c and R are a mere by-product of the unit-demand simplification. ¯ rather However, once the analysis is refocused to consider the relative net price ρ, than P¯ itself, for consumer demand functions of the power form all comparative statics of ρ¯ with respect to γ, c and R are indeed reversed in weak duopoly.

7

Conclusion

This paper has investigated two little-understood and sometimes confusing aspects of the canonical Hotelling duopoly model with finite consumer reservation price: Published by The Berkeley Electronic Press, 2010

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the existence and characteristics of asymmetric kinked-demand equilibria and the non-monotonicity of price as a function of consumers’ transportation cost. We characterized the continuum of asymmetric kinked-demand equilibria in the prototype Hotelling model and showed that their existence survives various extensions. Introducing elasticity into consumer demands suppresses the kink in the demand function, and thereby eliminates the continuum of asymmetric equilibria. The key perverse comparative static of the symmetric kinked-demand equilibrium, the non-monotonicity of price with respect to transport cost, survives relaxation of the unit-demand assumption. This result holds for a broad class of demand curves and prevails over a comparatively wide range of values for normalized transportation costs. Our result showing that introducing even a “small” degree of elasticity to consumers’ demands eliminates the kink in firms’ demands and the attendant continuum of asymmetric price equilibria provides a justification for researchers to assume unit demands for simplicity and limit their focus to the symmetric equilibrium. However, the facts that the non-monotonicity of the price schedule in transportation cost is preserved through the extension of the Hotelling model to elastic demands, and occupies a comparatively large range of the parameter space, mean that this key result cannot be dismissed as a conceptual anomaly. We proposed the concept of weak duopoly to characterize those market settings for which price is a decreasing function of the degree of product differentiation in the market. The standard model with inelastic consumer demands subject to reservation price and kinked-demand equilibria represents a special case of this generalized concept.

Appendix A: Proof of Proposition 1 The equilibria are derived by first computing the demand function facing each seller, then a seller’s reaction function. The intersections of the two reaction functions determine the set of Nash equilibria.

The Demand Function We can distinguish two cases: either there exists a consumer who is indifferent between purchasing from one seller or the other, or such a consumer does not exist. Suppose there exists such a consumer. She must be located at a point Y ∈ [0, 1] a +γ . In addition, this such that Pa + γY = Pb + γ(1 −Y ), which implies that Y = Pb −P 2γ consumer must obtain nonnegative utility from purchasing from either seller, which implies that Pa + γY ≤ R, or equivalently that Pa ≤ 2R − γ − Pb . The constraint that


20


Y ∈ [0, 1] further implies that Pa ≤ γ + Pb and Pa ≥ Pb − γ. The demand for seller a’s product is then Qa = Y . Suppose now that an indifferent buyer does not exist. Since we have already ruled out the case where either seller captures the entire market by setting a delivered price strictly lower than that of its competitor on the entire segment, each seller must sell to a nonempty set of customers. Further, since no indifferent buyer exists, it must be that all consumers purchasing from seller a would not purchase from seller b. A necessary and sufficient condition for this is that the consumer a located at the market boundary of seller a, X = R−P γ will not purchase from b, i.e., Pb + γ(1 − X) > R, or, equivalently, Pa > 2R − γ − Pb . The demand for seller a’s product is then Qa = X. Summarizing the above results, we can write the demand for seller a as: ( P −P +γ a b if Pb − γ ≤ Pa ≤ min(γ + Pb , 2R − γ − Pb ) 2γ . (9) Qa (Pa , Pb ; R, γ) = R−Pa if 2R − γ − Pb < Pa ≤ min(R, γ + Pb ) γ Note that either γ + Pb ≥ R, and the demand function Qa has a kink at Pa = 2R − γ − Pb (the derivative of Qa with respect to Pa is different for the two expressions of a +γ Qa ), or γ + Pb < R and then Qa = Pb −P on the relevant range. In all cases, Qa is 2γ continuous in Pa .

The Reaction Function Seller a maximizes its profit, given price Pb , by solving: max Pa

Πa (Pa , Pb ; R, γ) =

(Pa − c)Qa (Pa , Pb ; R, γ). Given the expression for Qa , we need to distinguish two cases according to whether the demand facing seller a is kinked or not. 1. γ + Pb ≥ R. The profit function has two different expressions according to whether Pa ≤ 2R − γ − Pb or Pa ≥ 2R − γ − Pb , and each expression is a polynomial of degree 2 in Pa that is bell-shaped and single-peaked. The leftward derivative of Πa evaluated at Pa = 2R − γ − Pb is equal to 2γ1 (3Pb + 3γ − 4R + c), while the rightward derivative is equal to 1γ (2Pb + 2γ − 3R + c). It is straightforward to see that the sign of those derivatives will depend on the relative positions of Pb , 43 R − 3c − γ, and 32 R − 2c − γ. Since R > c, we have that 23 R − 2c − γ > 43 R − 3c − γ and therefore we can distinguish between three cases: • Strict duopoly: Pb ≤ 34 R − 3c − γ: the peak occurs on the range [Pb − γ, 2R − γ − Pb ]. On this range seller a competes with seller b to attract the marginal buyer. Published by The Berkeley Electronic Press, 2010

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• Monopoly: Pb ≥ 32 R − 2c − γ: the peak occurs on [2R − γ − Pb , R]. • Kinked-demand: 43 R − 3c − γ < Pb < 32 R − 2c − γ: the peak occurs at Pa = 2R − γ − Pb . On this range, the optimal strategy is to price at the kink of the demand function. In the strict-duopoly regime, the optimal price is given by Pa∗ = max Pb − γ, Pb +γ+c . 2 In the monopoly regime, the optimal price is Pa∗ = R+c 2 . In the kinked-demand ∗ regime, the optimal price is Pa = 2R − γ − Pb . a +γ 2. γ + Pb < R. In this case, Qa = Pb −P on the range [Pb − γ, γ + Pb ]. We need 2γ to distinguish two cases: • If Pb ≥ c + 3γ, then Pa∗ = Pb − γ. • If Pb < c + 3γ, then Pa∗ = Pb +γ+c . 2 Note that we must have Pb ≥ c − γ, otherwise seller b would make a negative margin. This implies that Pb +γ+c ≤ γ + Pb . 2 Therefore, we have the following reaction function: • For γ + Pb ≥ R: – If Pb ≤ 43 R − 3c − γ and Pb ≤ c + 3γ, then Pa∗ = Pb +γ+c (competitive por2 tion of demand) – If Pb ≤ 43 R− 3c −γ and Pb > c+3γ, then Pa∗ = Pb −γ (competitive portion of demand) – If Pb ≥ 32 R − 2c − γ, then Pa∗ = R+c 2 (monopoly portion of demand) 4 c 3 c – If 3 R− 3 −γ < Pb < 2 R− 2 −γ, then Pa∗ = 2R−γ −Pb (kink in demand) • For γ + Pb < R: – If Pb ≥ c + 3γ, then Pa∗ = Pb − γ (competitive portion of demand) – If Pb < c + 3γ, then Pa∗ = Pb +γ+c (competitive portion of demand) 2 Due to the symmetry of sellers, the reaction function Pb∗ is similar to Pa∗ .

The Nash Equilibria The Nash equilibria (P¯a , P¯b ) are determined by the intersection of the reaction curves Pa∗ (Pb ; c, R, γ) and Pb∗ (Pa ; c, R, γ). These reaction curves are piecewise affine and have different shapes according to the values of (c, R, γ). First, note that Pb always lies between c and R. In addition, since R ≥ c we always have that 43 R − 3c − γ ≤ 32 R − 2c − γ and that 43 R − 3c − γ ≥ R − γ. The following relationships are used in determining the shape of the function Pa∗ : • c ≤ R−γ ⇔ t ≤ 1 http://www.bepress.com/bejte/vol10/iss1/art12

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• • • • • •

c ≤ 43 R − 3c − γ ⇔ t ≤ 43 c ≤ 32 R − 2c − γ ⇔ t ≤ 32 R ≥ 32 R − 2c − γ ⇔ t ≥ 12 R ≥ 43 R − 3c − γ ⇔ t ≥ 13 c + 3γ ≤ R ⇔ t ≤ 13 c + 3γ ≤ R − γ ⇔ t ≤ 14

All instances can be characterized in terms of the value of the normalized transportation cost t. We can distinguish the following cases. 1. t ≤ 41 : In this case, c + 3γ and R − γ are inside the acceptable range for Pb , the segment [c, R]. The values of 43 R − 3c − γ and 32 R − 2c − γ are above R. The reaction function Pa∗ consists of two upward-sloping segments: on [c, c + 3γ], Pa∗ is increasing with slope 21 , while on [c + 3γ, R] it is increasing with slope 1. At the breakpoint Pb = c + 3γ, the profit-maximizing constraint Pa∗ ≥ Pb − γ starts to be binding, i.e., seller a charges the same price as seller b minus the transport cost needed to travel from b to a, and therefore captures the entire market (or would do so if it was in b’s best interest to price in this range of Pb ). The reaction functions Pa∗ and Pb∗ are depicted in figure 6, where the symbols R 4 and R 3 denote the expressions 43 R − 3c − γ and 32 R − 2c − γ, respectively. 3 2 There is a unique symmetric equilibrium located on the segment [c, c + 3γ], at the intersection point (P¯a , P¯b ) = (c + γ, c + γ). 2. 14 < t ≤ 31 : This case differs from the previous one in the sense that the relative positions of R − γ and c + 3γ on the segment [c, R] have been reversed, but the reaction function has the same shape, with the breakpoint still occurring at Pb = c + 3γ. There is a unique symmetric equilibrium at (P¯a , P¯b ) = (c + γ, c + γ). γ 3. 13 < t ≤ 12 : Because now R−c > 13 , c+3γ is no longer on the segment [c, R], but 4 c 1 4 c ∗ 3 R − 3 − γ is. The function Pa is increasing with slope 2 on [c, 3 R − 3 − γ] and decreasing with slope −1 on [ 43 R − 3c − γ, R]. The graphs of the two reaction functions Pa∗ and Pb∗ will intersect on their increasing part if and only if the kink of the function Pa∗ lies below the 45 degree line, that is, γ Pa∗ ( 43 R − 3c − γ) < 34 R − 3c − γ. This last condition is satisfied iff R−c ≤ 23 , which holds on this range of parameter values. Therefore, there is a unique symmetric equilibrium on this range as well, (P¯a , P¯b ) = (c + γ, c + γ). 4. 21 < t ≤ 1: This case differs from the previous one in that 23 R − 2c − γ now lies within the acceptable segment [c, R]. Therefore, on this range the reaction function Pa∗ is composed of three affine segments. It is increasing on [c, 43 R − c 1 4 c 3 c 3 − γ] with slope 2 , decreasing on [ 3 R − 3 − γ, 2 R − 2 − γ] with slope −1, Published by The Berkeley Electronic Press, 2010

23


Pa 6

Pb∗ @

R−γ

@ R @

c + 2γ

c+γ c + 2γ

r

@ I @ @ P∗

a

0 c

c + 3γ

R

R4

R3

3

2

-

Pb

Figure 6: Equilibrium prices when 0 < t ≤ 13 . The solution is P¯a = P¯b = P¯ = c + γ. and flat on [ 32 R − 2c − γ, R] at the monopoly price Pa∗ = R+c 2 . We already know γ 2 that as long as R−c ≤ 3 , the intersection of the two reaction functions will occur on their (strictly) increasing part, and in this case there will be a unique γ symmetric equilibrium (P¯a , P¯b ) = (c + γ, c + γ). However, for values of R−c greater than 23 the graphs will intersect on their decreasing portion and will do so as long as the breakpoint ( 32 R − 2c − γ, Pa∗ ( 32 R − 2c − γ)) lies below the 45 γ degree line, that is, R−c ≤ 1. In the case where the reaction functions intersect on their decreasing portions, we can distinguish two subcases according to whether the value of 43 R − 3c − γ exceeds the monopoly price R+c 2 (see figures 7 and 8). This leads to the following equilibria: • If 12 < t ≤ 23 , there is a unique symmetric equilibrium (P¯a , P¯b ) = (c + γ, c + γ). http://www.bepress.com/bejte/vol10/iss1/art12

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Pa 6

2R+c 3

R+c 2

c + 2γ

@ @ @ @ @ @ @ @ @ @ R @ @ @ @ I @ @ P∗ b

Pa∗

0 c

Figure 7: Equilibrium prices when ¯ ¯ γ, 2R+c 3 ], with Pa + Pb = 2R − γ.

2 3

R4

R3

3

2

R

-

Pb

< t ≤ 56 . The solution is P¯a , P¯b ∈ [ 43 R − 3c −

• If 23 < t ≤ 56 , there is an infinity of Nash equilibria characterized by ¯ ¯ (P¯a , P¯b ) ∈ [ 43 R − 3c − γ, 2R+c 3 ] and Pa + Pb = 2R − γ. 5 • If 6 < t ≤ 1, there is an infinity of Nash equilibria characterized by 3 c ¯ ¯ (P¯a , P¯b ) ∈ [ R+c 2 , 2 R − 2 − γ] and Pa + Pb = 2R − γ. 4 5. 1 < t ≤ 3 : On this range, the overall shape of the reaction function is the same as in the previous case, except that now the 2 reaction functions intersect on their flat (monopoly) range. There is a unique symmetric equilibrium R+c (P¯a , P¯b ) = ( R+c 2 , 2 ). 6. 43 < t ≤ 32 : On this range, the increasing segment of the reaction function Pa∗ no longer occurs on the acceptable range of Pb , and therefore Pa∗ is composed of one decreasing segment (on [c, 32 R − 2c − γ]) and one flat segment (on [ 32 R − Published by The Berkeley Electronic Press, 2010

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Pa 6

2R+c 3

R+c 2 c + 2γ

Pa∗ @ @ @ @ @ R @ @ @ @ @ @ @ @ @ I @ @ P∗ b

0 c

Figure 8: Equilibrium prices when c ¯ ¯ 2 − γ], with Pa + Pb = 2R − γ.

5 6

R4

R3

3

2

R

-

Pb

3 < t ≤ 1. The solution is P¯a , P¯b ∈ [ R+c 2 , 2R −

c 2

− γ, R]). As in the previous case, the intersection occurs on the flat segment R+c and there is a unique equilibrium (P¯a , P¯b ) = ( R+c 2 , 2 ). 7. t > 32 : On this range, the only relevant segment of the reaction curve Pa∗ is the R+c flat one, and therefore there is a unique equilibrium (P¯a , P¯b ) = ( R+c 2 , 2 ). Although the reaction functions have different shapes according to the values taken by t, some of the cases overlap in terms of the type of Nash equilibrium they generate. The four relevant cases, competition, weak-duopoly (with two subcases), and monopoly are summarized as Proposition 1.


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Appendix B: Kinked-Demand Equilibria with Quadratic Transport Costs B-1: The Demand Function Assume first that an indifferent buyer does not q exist. The demand facing seller a is a then X, where X satisfies Pa + γX 2 = R, or X = R−P γ . It must be that the marginal buyer located at X would not buy from seller b, i.e., we must have Pb + γ(1 − X)2 > 2 Pb −Pa +γ R−Pa R, which can be shown to be equivalent to Pa ≤ γ + Pb and γ < . 2γ Now assume that there exists an indifferent buyer. She must be located at a a +γ point Y satisfying Pa + γY 2 = Pb + γ(1 −Y )2 , or Y = Pb −P . In addition, the price 2γ net of transportation paid by the buyer at Y must be smaller than her reservation 2 Pb −Pa +γ a price R, that is, R−P ≥ . γ 2γ 2 Pb −Pa +γ a a It is easily shown that the condition R−P < (resp. R−P ≥ γ 2γ γ 2 p Pb −Pa +γ + (resp. P ≤ P+ ), with P+ = P −γ +2 γ(R − P ). ) is equivalent to P > P a a b b a a a 2γ Therefore, we can write the demand function to seller a as ( Pb −Pa +γ if Pb − γ ≤ Pa ≤ min(γ + Pb , Pa+ ) 2γ q Qa (Pa , Pb ; R, γ) = . R−Pa if Pa+ < Pa ≤ min(R, γ + Pb ) γ

B-2: Condition Equivalent to Condition (2) Condition (2) can be rewritten

p γ(R − Pb ) < 14 (3γ − Pb ), which is equivalent to

and Pb 2 + 10Pb γ + 9γ 2 − 16γR > 0. p The last condition is equivalent to P > −5γ + 4 γ(γ + R). The interval (−5γ + b p 4 γ(γ + R), 3γ] is nonempty if and only if R < 3γ. Pb ≤ 3γ

B-3: Condition Equivalent to Condition (3) p Condition (3) can be rewritten γ(R − Pb ) > R3 + 2γ − P2b , which implies that ei 2 ther R3 + 2γ − P2b < 0 or R3 + 2γ − P2b ≥ 0 and γ(R − Pb ) > R3 + 2γ − P2b . The first alternative implies that Pb > γ + 23 R, which under the maintained assumption that Published by The Berkeley Electronic Press, 2010

27


Pb ≤ R also implies that Pb > 3γ, a condition incompatible with the fact that R < 3γ, itself a consequence of condition (2). Therefore, if condition (2) is satisfied, condition (3) can only be satisfied under the second alternative, and therefore we must 2 γ Pb R have that γ(R − Pb ) > 3 + 2 − 2 , which can be rewritten 9Pb 2 + Pb (18γ − 12R) + 4R2 + 9γ 2 − 24γR < 0. This last condition is equivalent to Pb ∈ (−γ + 23 (R − √ √ 3γR), −γ + 23 (R + 3γR)). Under the condition R < 3γ, the lower bound of this interval is a negative real number,√and therefore the essential condition for (3) to be satisfied is that Pb < −γ + 23 (R + 3γR).

p √ B-4: Proof that (−5γ +4 γ(γ + R), −γ + 23 (R+ 3γR)) is Nonempty We need to show that p p 2 γ(γ + R) < −γ + (R + 3γR). (10) 3 p √ Condition (10) is equivalent to 4 γ(γ + R) < 4γ + 23 (R+ 3γR). Taking the square of each side of the inequality and rearranging, we √ obtain that condition (10)√is 8 √ 16 satisfied if and only if 49 R2 − 28 γR + R 3γR + γ 3γR > 0. Defining r = R 3 9 3 √ 2 and g = 3γ and dividing by r, this last condition can be rewritten 49 r3 − 28 9 g r+ 8 2 16 3 9 gr + 9 g > 0. The polynomial on the left-hand-side of this inequality is equal to 4 2 9 (r − g) (r + 4g) and is therefore strictly positive under the condition R < 3γ. −5γ + 4

B-5: Condition Equivalent to Condition (5) p Consider first the inequality −5γ + 4 γ(γ + R) < R − 4γ . Isolating the square root term on the left-hand side of the inequality, taking squares and collecting terms, one 105 2 obtains that the polynomial R2 − 13 2 γR + 16 γ must be positive. This implies that 5γ R must lie outside of the roots, R+ = 21γ 4 and R− = 4 . Under the caveat R < 3γ, only the condition R < 5γ4 becomes relevant. √ The second part of condition (5), R − 4γ < −γ + 23 (R + 3γR), is equivalent √ √ √ to 4R + 9γ − 8 3γR < 0. Defining r = R and g = 3γ, this is equivalent to 3γ 27γ 4r2 − 8gr + 3g2 < 0, which is satisfied whenever r ∈ ( g2 , 3g 2 ), that is, R ∈ ( 4 , 4 ). Since the first part of condition (5) implies that R < 5γ4 , only the condition R > 3γ4 is relevant. Therefore, it must be that R ∈ ( 3γ4 , 5γ4 ), or, equivalently, that t ∈ ( 45 , 43 ).


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Appendix C: Proof of Proposition 3 Denote U = [c, R]. Suppose that both sellers set price on the competitive portion of their demand curve. The first-order condition for profit maximization for seller a is then: 1 Pa + Pb + γ Pa + Pb + γ − Φ(Pa ) + (Pa − c) φ − φ (Pa ) = 0 Φ 2 2 2 which we can rewrite as Ψ(Pa , Pb ) = 0. This relationship between Pa and Pb , together with the second-order condition, implicitly defines the reaction function Pa∗ (Pb ) in its competitive portion. A continuum of equilibria can obtain only if Pa∗ (Pb ) and Pb∗ (Pa ) overlap. Given the symmetry of sellers, a necessary condition for this overlap is that {Ψ(Pa , Pb ) = 0} ⇒ {Ψ(Pb , Pa ) = 0} on a subset of U2 consistent with duopoly competition and not reduced to a discrete set of points. Given that the above expression for Ψ is not symmetric in Pa and Pb , this last condition will generally fail to be satisfied. Therefore, there will generally be no continuum of solutions when sellers set price in the competitive portion of demand.

Appendix D: Effects of c and R on P¯ for Elastic Demands The first-order condition maximization i by seller a in the competitive h for profit P +P +γ P +P +γ − φ (Pa ) + Φ a 2b − Φ(Pa ) = 0. Differregime is: (Pa − c) 12 φ a 2b entiating this condition with respect to to c and Pa , keeping Pb constant, gives the shift in the reaction function Pa∗ (Pb ; c): Pa +Pb +γ 1 − φ (Pa ) 2φ 2 ∂ Pa∗ h i = Pa +Pb +γ 1 0 Pa +Pb +γ ∂c 0 φ − 2φ (P ) + (P − c) ψ − ψ (P ) 2

a

a

4

2

a

¯ yields which, evaluated at the equilibrium where Pa = Pb = P, 1 ¯ φ P¯ + 2γ − φ (P) ∂ Pa∗ 2 1 . = γ ¯ + (P¯ − c) ψ 0 P¯ + γ − ψ 0 (P) ¯ ∂c φ P¯ + 2 − 2φ (P) 4 2 The numerator is negative since φ is decreasing, and the denominator is negative due to the second-order condition for profit maximization. Therefore, an increase in c causes the reaction function Pa∗ (Pb ) to shift upwards. A similar argument can be made regarding Pb∗ (Pa ). This will rise equilibrium price if and only if the slope Published by The Berkeley Electronic Press, 2010

29


of the reaction function Pa∗ is smaller than 1. Since this is a necessary condition for a stable Nash equilibrium, we have the following proposition: Provided that a stable Nash equilibrium exists, an increase in the marginal production cost raises equilibrium price. Analyzing the effect of R on P¯ requires specializing ψ to a function of the form ψ(p) = χ(R − p), for R > 0 and χ an increasing function continuously differ¯ entiable on [0, R]. Yet, it is not possible to unambiguously sign the derivative ∂∂ PR on the duopoly range without further assumptions on the form of χ.

Appendix E: Proof of Proposition 5 First consider the region of the parameter space for which monopoly occurs. The ˜ solves critical value of transportation cost that separates duopoly and monopoly, γ, γ P¯m = R − 2 , where P¯m is the function of R and γ implicitly defined by equation (6). Specializing to the power demand function, we obtain γ˜ = 2(1+ε) 2+ε (R −c). For values R+c(1+ε) ˜ the monopoly price is P¯m = 2+ε . of γ above γ, ˜ the duopoly price equilibrium is defined by (7) and the secondFor γ < γ, order condition for profit maximization. Under the assumed demand specification, equation (7) becomes ε 1+ε 1 γ 1 γ ε 1+ε ¯ ¯ + ¯ (P−c) R − P¯ − − (R − P) (R − P) − R − P¯ − = 0. 2 2 1+ε 2 ¯

γ Defining ρ¯ = P−c R−c and t = R−c , we can rewrite this equation as 1 t ε 1 t 1+ε ε 1+ε ¯ ¯ ρ¯ 1 − ρ¯ − − (1 − ρ) + (1 − ρ) − 1 − ρ¯ − = 0. 2 2 1+ε 2 (11) ¯ we obtain the condition for the Totally differentiating (11) with respect to t and ρ, 1− 2t ¯ ¯ derivative ddtρ to be zero on the range [0, 2(1+ε) ), ρ = . Substituting into (11), 2+ε 1+ ε2 1+ε ¯ becomes flat must solve 1 − 2t the critical value tˆ at which the schedule ρ(t) − 2(1+ε) t ε 1 + ε (1−t)(2+ε) = 0. This equation has a unique root on the interval [0, 2+ε ). Proposition 5 follows.


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