Differentiated Bertrand duopoly with variable demand

Research in Economics (1997) 51, 85–100

Differentiated Bertrand duopoly with variable demand MARTIN PEITZ† Universidad de Alicante and IVIE (Received 20 March 1995, accepted 10 June 1996)

Summary Two one-product firms compete in prices on a market with differentiated products. Goods are differentiated because customers switch from one good to the other at different relative prices. With the specification that mean demand in the market is unit-elastic, I provide conditions on the shape of the customer density which guarantee the existence of a unique Bertrand equilibrium.  1997 Academic Press Limited J.E.L. Classification: D43, L13. Keywords: Duopoly equilibrium, price competition, product differentiation.

1. Introduction In an oligopolistic market with differentiated goods, firms compete in prices. Individual demand functions and the taste heterogeneity between the customers determine market demand. A large number of studies have looked at sufficient properties of the taste heterogeneity of customers, described by a customer density on a parameter space, in order to show the existence of equilibrium for particular specifications of individual demand. The literature on product differentiation started with models in which customers have unit demand and are uniformly distributed in a one-dimensional space (e.g. Hotelling, 1929; D’Aspremont, Gabszewicz & Thisse, 1979; Gabszewicz & Thisse, 1979; Salop, 1979). Another popular functional form on the taste heterogeneity is presented in the logit model (for an overview, see Anderson, de Palma & Thisse, 1992). To my knowledge the first articles dealing with shape assumptions on the customer density guaranteeing the existence of Bertrand equilibrium are Caplin and Nalebuff (1986) † Author for correspondence at: Departamento Fundamentos del Ana´lisis Econo´mico, Universidad de Alicante, Campus San Vicente, 03071 Alicante, Spain, e-mail: [email protected]. 1090–9443/97/020085+16 $25.00/0/re960038

 1997 Academic Press Limited

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and Neven (1986). Both assume that the customer density is concave, Neven (1986) in a Hotelling model of unit demand and Caplin and Nalebuff (1986) in a model with unit-elastic demand. Further work has focused on shape assumptions in models of unit demand which give rise to the existence of equilibrium (Champsaur & Rochet, 1988; Caplin & Nalebuff, 1991b; Allen & Thisse, 1992; Bester, 1992; Dierker & Podczeck, 1992). Ansari, Economides and Ghosh (1994) and Goeree and Ramer (1994) then studied locationthen-price games for non-uniform customer densities. Unit demand seems to be the adequate specification for many durables such as cars and microwaves. However, other goods such as soft drinks, beer, cigarettes and potato chips are bought in variable quantities. Models in which customers buy a variable quantity have received less attention (Caplin & Nalebuff, 1991b; Dierker, 1991; Anderson, de Palma & Thisse, 1992; Peitz, 1995). In this literature it is assumed that taste heterogeneity can be described by some logconcave customer density. I restrict attention to a duopoly with unit-elastic demand. The behaviour of customers and firms is presented in Section 2. Section 3 contains the existence and uniqueness results. I show the existence of a unique equilibrium for a class of “flat” density functions and for a class of log-concave density functions. All proofs are collected in the Appendix. 2. Behaviour of customers and firms Let me call the market under consideration the fruit market. I assume that there are only two kinds of fruit, apples and oranges. For each of the fruits there exists a positive price: p1 is the price of an apple, p2 is the price of an orange. There are other goods in the economy but their prices are fixed. These goods are summarized by a Hicksian composite commodity x0; its price p0 is normalized to 1. Hence, income and prices p1, p2 are measured in units of the composite commodity. 2.1.

CUSTOMERS

A customer has to make two decisions: in the first step, how much to spend on fruit and, in the second step, how to divide the fruit expenditure between apples and oranges. A customer with fixed income y>0 has a demand function for apples n1(p1, p2) and oranges n2(p1, p2) with the property that 0≤p1n1(p1, p2)+p2n2(p1, p2)≤y. I assume that a customer either buys apples or oranges. Expenditure on apples and oranges is fixed at b and it is assumed that the point pˆ12 at which the customer switches from one to the other is

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independent of expenditure in the market. Expenditure on apples is

G

p1n1(p1, p2)=

b 0

if p1/p2≤pˆ12 else

A customer only buys apples if p1/p2≤pˆ12.† pˆ12 is one of the characteristics of a customer. It will be convenient to use logs: ho−log pˆ12. A customer switches from oranges to apples when log p2−log p1≥h. Customers evaluate apples and oranges differently, i.e. they have different h. (A.1) There exists a continuous distribution function G over hvR with G(0)>0. G has a density g which is continuous on [h, h¯ ] and positive and continuously differentiable on (h, h¯ ). g has bounded support, i.e. hh¯ implies G(h)=1. Without loss of generality G(0)>0. g′(h) is defined as limh∏h g′(h) and g′(h¯ )olimh⊇h¯ g′(h). The assumption of a bounded support is taken because I want to include the uniform distribution. Also it facilitates the proof that firms choose out of a compact strategy set.‡ The assumption of a bounded support implies that for a given orange price one can always find apple prices sufficiently large such that mean apple expenditure is equal to zero. All customers buy only apples if log p2−log p1≥h¯ and only oranges if log p2−log p1
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B X1(p1, p2)= G(log p2−log p1), p1 B X2(p1, p2)= (1−G(log p2−log p1)) p2 where B>0 is a constant. Next, I present three examples which generate the mean demand functions from above. Customers maximize their utility function u subject to the budget constraint x0+p1x1+p2x2≤y. Example 1 in the goods-are-goods framework. Sattinger (1984) proposed a utility function which generates non-combinability and fixed individual expenditure b. u(x0, x1, x2)=(Ri=1,2 (xi /qi ))a x1−a 0 . At his critical price ratio (q1/q2) a customer is indifferent between apples and oranges. A distribution over (q1, q2) generates a distribution over h. Each customer spends ay on apples and oranges. Example 2 following the Lancastrian characteristics approach. Example 1 is slightly modified in the spirit of Lancaster (1979). Assume that each good can be described by a vector c=(c1, . . . , n of characteristics and that customers only derive utility cn)vR+ from the total amount in each characteristic, u(x0, x1, x2)=u˜(x0, Ri xi ci1, . . . , Ri xicin). One specification is that the characteristics are perfect substitutes and u(x0, x1, x2)=(Rk kk Ri cikxi )ax1−a 0 , where k defines the type of the customer and is an element of the ndimensional unit simplex. The distribution on h depends on the distribution over weights k and on the products’ characteristics c, h=log

Rk kkk1k Rk kkc2k

A different specification is provided by Caplin and Nalebuff (1986). Example 3 following the Hotelling approach. The utility function is defined as u=(Ri u˜i )ax01−a with u˜i=xi(1+d(|x−li |))−1 and 2 d(|x−li |)=e((x−li) )−1. xvR denotes the type of customer and livR the location of the good in the product space. The devaluation factor d(|x−li |) depresses the utility if the location of good i does not coincide with the location of the most preferred good of customer x. It plays the same role as the transportation cost function in the Hotelling model. The marginal customer x ˜ , who is indifferent between good 1 and good 2, is determined by log p2−log p1 l1+l2 + x ˜= 2(l2−l1) 2

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and there is a linear mapping from [x, x ¯ ] into [h, h¯ ]. Further examples in a one-dimensional product space are provided by Peitz (1996). 2.2.

FIRMS

Firm behaviour is standard: Good i, i=1, 2, is produced by firm i at constant marginal cost ci>0. Each firm faces a mean demand function Xi(p1, p2) depending on the prices for apples and for oranges. Firms maximize profits, i.e. maxpi pi(p1, p2), where pi(p1, p2)=(pi−ci )Xi(p1, p2), i=1, 2. Note that prices below marginal costs are weakly dominated. They are serially (strictly) dominated if assumption (A.2) is satisfied. (A.2) says that if each firm sets price equal marginal costs both firms have positive market shares. (A.2) h
LOG-CONCAVE DENSITIES

Since log-concavity of G, 1−G, implies that profit functions are quasi-concave, this property is important to show the existence of equilibrium. As shown in Lemma 1 in the Appendix, log-concavity of g carries over to G. The result corresponds to Lemma 1 in Dierker (1991). To prove the existence of an equilibrium it remains to be shown that the strategy set is compact. Quite a broad class of distributions has log-concave densities.† It includes such distributions as normal, exponential and Weibull (for a larger list, illustrations and the references, see Caplin & Nalebuff, 1991a). Truncated forms of the above-mentioned distributions also have log-concave densities for a convex support † In his Proposition 2, Dierker (1991) states that log-concavity of g is implied by Schur-concavity of g(h)g(h′), which expresses the idea that for a random draw out of the population more equal perturbations are more likely and shows the relationship between Schur-concavity and unimodality.

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and hence are examples of log-concave densities g with bounded support. To obtain uniqueness, two other properties, namely dominant diagonality under logarithmic transformation and log-supermodularity, are sufficient.† Proposition 1 Assume (A.1) and let G, 1−G be log-concave on [h, h¯ ]. Logarithmic profits satisfy the dominant diagonal property on the set of prices above marginal costs where demand is C 2, i.e.

K

KK

∂2 log pi ∂2 log pi (p , p ) > (p1, p2) 1 2 (∂ log pi )2 ∂ log pi ∂ log pj

K

for i, j=1, 2, j≠i The following proposition establishes that profits are log-supermodular.‡ Proposition 2 Assume (A.1) and let G, 1−G be log-concave on [h, h¯ ]. Profits are log-supermodular on the set of prices above marginal costs where demand is C 2. Theorem 1 Assume (A.1), (A.2), and let g be log-concave. There exists a unique pure strategy Bertrand–Nash equilibrium ( p∗1 , p∗2 ). The associated game is dominance solvable. The assumption on costs guarantees that the first-order condition of profit maximization is always satisfied in equilibrium and both firms make positive profits. If it is violated a firm can get the whole market with its price above marginal costs and, if this happens in equilibrium, the inactive competitor disciplines the firm which serves the whole market.

† In more than one dimension one can make use of Pre´kopa’s theorem (see Pre´kopa, 1973) to obtain log-concavity which was explored by Caplin and Nalebuff (1991a,b) and Dierker (1991). The existence result in the case of log-concave densities holds for any finite number of firms. ‡ Supermodularity has recently been analysed by Milgrom and Roberts (1990) and Vives (1990). It has already been applied by Caplin and Nalebuff (1991b). The fact that one can enhance the applicability by allowing for log-transformation is emphasized by Milgrom and Roberts.

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91

FLAT DENSITIES

I will now derive the same result as in Theorem 1 under the assumption that G is not too far from the uniform distribution. A density is called flat if

K

|q(h)|o

K

g′(h) ≤1 (g(h))2

on the support of g. Clearly, q(h)=0 for the uniform distribution on the interval [h, h¯ ]. But what does it mean that h is uniformly distributed? In contrast to Bester (1992), who looks at unit and not at unit-elastic demand, the distribution is related to relative and not to absolute price differences.† If one changes p2 from 1p1 to 2p1 and from 2p1 to 4p1 and if h is uniformly distributed, then the share of customers who switch to good 1 is the same for the price change from 1p1 to 2p1 and from 2p1 to 4p1. Note that multiple peaks are compatible with a flat density. Note also that a convex density can be flat. Hence, the customer mass can be more concentrated around the end points of [h, h¯ ] than in the middle. Theorem 2 is the analogue to Theorem 1. Here, flatness gives rise to the log-concavity of G. Theorem 2 Assume (A.1), (A.2), and let g be flat. There exists a unique pure strategy Bertrand–Nash equilibrium (p∗1 , p∗2 ). The associated game is dominance solvable. One can show that as (h¯ −h)→0, p∗2 →c2 and p∗1 →c1, i.e. when the interval shrinks to a single point the competitive outcome is reached. Without heterogeneity the standard undercutting argument is valid. The theorem, though, is only for non-degenerate intervals for h because, otherwise, continuity of mean demand is violated. Note that ∂2 log((pi−ci )/pi )/(∂ log pi )2=−pici /(pi−ci )2 which turns to infinity when price turns to marginal cost. Hence, for a small support of g and high marginal costs, possible equilibrium prices have to be quite close to marginal costs so that log G does not need to be concave to show the quasi-concavity of profit functions. Uniqueness can be shown without exploiting the log-concavity of G and as a consequence log-supermodularity. Even under a

† It is the advantage of the specification in logs that the distribution is scale independent.

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weaker condition on |q(h)| uniqueness holds. The proof of uniqueness is elementary because I can construct a decreasing function with a zero for the equilibrium. For 1<|q(h)|≤2, I was not able to show the existence of equilibrium. Proposition 3 Assume (A.1), (A.2), and let |q(h)|≤2 be flat. A pure strategy Bertrand–Nash equilibrium (p∗1 , p∗2 ) is unique for strategy sets Si=R+, i=1, 2. Under (A.2) one can compute equilibrium prices. For |q(h)|≤1 for hv[h, h¯ ], there exists a unique solution h∗ to c2 1+G′(h)−G(h) h =e c1 G′(h)+G(h) Equilibrium prices then are

A

B

G(h∗) p∗1 =c1 1+ G′(h∗) and

A

1−G(h∗) G′(h∗)

p∗2 =c2 1+

B

In the model the extent of taste heterogeneity is expressed by g. According to example 3 one can view an enlargement of the support of g from [h, h¯ ] to [(1+k)h, (1+k)h¯ ], k>0, as a higher degree of product differentiation. Under the assumption that g is uniform, the intuition that increased product differentiation leads to higher prices is correct. Log-profits of firm 1 are p1−c1 +log B p1 (log p2−log p1)/(1+k)−h +log h¯ −h

log p1(p, k)=log

Proposition 4 Assume (A.1), (A.2), and let g be uniform. Equilibrium prices p∗ are increasing functions of the measure of increased product differentiation k.

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4. Conclusion Two one-product firms compete in prices. Goods are differentiated because customers have different critical relative prices at which they revise their buying decision. I showed the existence of a unique equilibrium provided that the customer density is “flat” or log-concave. Equilibrium profits are computable and the model may be a building block for two-stage models with endogenous product specification (see Peitz, 1996).

Acknowledgements I wish to thank three anonymous referees for helpful comments. An earlier version of this paper has been circulated under the title “Differentiated duopoly in prices with variable demand”. Financial support by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303, Instituto Valenciano de Investigaciones Econo´micas (IVIE) and EU Human Capital and Mobility Program is gratefully acknowledged.

References Allen, B. & Thisse, J.-F. (1992). Price equilibria in pure strategies for homogeneous oligopoly. Journal of Economics & Management Strategies, 1, 63–81. Anderson, S. P., de Palma, A. & Thisse, J.-F. (1992). Discrete Choice Theory of Product Differentiation. Cambridge, MA: MIT Press. Ansari, A., Economides, N. & Ghosh, A. (1994). Competitive positioning in markets with nonuniform preferences. Marketing Science, 13, 248–273. Bester, H. (1992). Bertrand equilibrium in differentiated duopoly. International Economic Review, 33, 433–448. Caplin, A. & Nalebuff, B. (1986). Multi-dimensional product differentiation and price competition. Oxford Economic Papers, 38 (suppl.), 129–145. Caplin, A. & Nalebuff, B. (1991a). Aggregation and social choice: a mean voter theorem. Econometrica, 59, 1–23. Caplin, A. & Nalebuff, B. (1991b). Aggregation and imperfect competition: on the existence of equilibrium. Econometrica, 59, 25–59. Champsaur, P. & Rochet, J.-C. (1988). Existence of a price equilibrium in a differentiated industry. Discussion paper 8801. France: INSEE. D’Aspremont, C., Gabszewicz, J. J. & Thisse, J. F. (1979). On Hotelling’s “Stability in Competition”. Econometrica, 47, 1145–1150. Dierker, E. (1991). Competition for customers. In W. Barnett, B. Cornet, C. d’Aspremont, J. Gabszewicz & A. Mas-Colell, Eds. Equilibrium Theory and Applications. Cambridge: Cambridge University Press. Dierker, E. & Podczeck, K. (1992). The distribution of consumers’ tastes and the quasiconcavity of the profit function. Working Paper No. 9208, University of Vienna, Austria. Gabszewicz, J. J. & Thisse, J.-F. (1979). Price competition, quality and income disparities. Journal of Economic Theory, 20, 340–359.

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Goeree, J. & Ramer, R. (1994). Exact solutions of location games. Discussion paper TI94–70. The Netherlands: Tinbergen Institute. Hotelling. (1929). Stability in competition. Economic Journal 39, 41–57. Lancaster, K. (1979). Variety, Equity and Efficiency. New York: Columbia University Press. Milgrom, P. & Roberts, J. (1990). Rationalizability, learning and equilibrium in games with strategic complementarities. Econometrica, 58, 1255–1277. Neven, D. (1986). On Hotelling’s competition with non-uniform customer distribution. Economics Letters, 21, 121–126. Peitz, M. (1995). Aggregation and strategic complementarity in a model of price competition. Discussion Paper A-469, University of Bonn, Germany. Peitz, M. (1996). Two-stage models of product differentiation with unit-elastic demand. Mimeo: University of Alicante. Pre´kopa, A. (1973). On logarithmic concave measures and functions. Acta Scientifica Mathematics, 34, 335–343. Salop, S. C. (1979). Monopolistic competition with outside goods. Bell Journal of Economics, 10, 141–156. Sattinger, M. (1984). Value of an additional firm in monopolistic competition. Review of Economic Studies, 51, 321–332. Vives, X. (1990). Nash equilibrium with strategic complementarities. Journal of Mathematical Economics, 19, 305–321.

Appendix Proof of Proposition 1. Remark that prices are chosen from (×i=1,2[ci , ∞))i{p1, p2 |Xi(p1, p2)>0, i=1, 2}. On the interior of this set log pi , i=1, 2, are twice continuously differentiable in log pi , i=1, 2. I have to show that ∂2 log pi ∂2 log pi − > (∂ log pi )2 ∂ log pi∂ log pj for i, j=1, 2, j≠i p1c1 ∂2 log G ∂2 log G 2− 2> (p1−c1) (∂ log p1) ∂ log p1∂ log p2 =−

∂2 log G (∂ log p1)2

Similarly, for firm 2.

Ε

Proof of Proposition 2. For C 2 functions supermodularity is easily checked due to Topkis’ Characterization Theorem. ∂2 log pi (p1, p2)≥0 for i, j=1, 2, j≠i ∂ log pi∂ log pj as

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∂2 log G(log p2−log p1) ∂2 log G(log p2−log p1) =− ≥0 ∂ log p1∂ log p2 (∂ log p1)2 for G log-concave on [h, h¯ ] and in the same way for firm 2.

Ε

Lemma 1 If g is log-concave in h for hv[h, h¯ ] so are G, 1−G. Proof. (See Lemma 1 in Dierker, 1991.) log G(h˜ )=log

P

h˜

eh(h) dh

h

where h(h)olog g(h) ˜

eh(h) d 2 log G(h˜ ) h(h˜ ) =−e ˜ dh˜ 2 (/hh eh(h) dh)2 ˜

eh(h) ≤0 / eh(h) dh

+h′(h˜ ) ⇔h′(h˜ )

P

h˜ h

h˜ ˜

eh(h) dh≤eh(h)

h

As h is concave: h′(h˜ )≤h′(h), hv[h, h˜ ]. Hence, h′(h˜ )

P

h˜

h

Similarly, for 1−G.

eh(h) dh≤

P

h˜

˜

h′(h)eh(h) dh
h

Ε

Lemma 2 Assume (A.1) and let G, 1−G be log-concave on [h, h¯ ]. Profits pi are log-concave in log pi for all price combinations such that demand is positive and pi>ci . Proof

G

log p1=

log( p1−c1)+log B−log p1+log G (log( p2 /p1)) if log( p2 /p1)v[h , h¯ ] log( p1−c1)+log B−log p1 if log( p2 /p1)v(h¯ , ∞)

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log(p1−c1)+log B−log p1 is concave in log p1. What remains to be shown is that the last term is also concave in log p1 for (log p2−log p1)v[h, h¯ ]. Since h˜ (p1, p2)=log p2−log p1, d 2G(h˜ ) ∂2G(h˜ (p1, p2)) = dh˜ 2 (∂ log pi)2 Ε

Similarly, for firm 2.

Lemma 3 Assume that profits pi are log-concave in log pi for all price combinations where demand is positive and pi>ci . Profits are quasiconcave in its own price for pi≥ci . Proof. Quasi-concavity is violated if there exists a pi0, pi1 and pik with ci≤pi0(pik−ci )Xi(pik ) and (pi1−ci )Xi(pi1)>(pik−ci )Xi(pik ) Case (i) pi0=ci . The first inequality requires that pikci . Look at i=1, p2 fixed. For X1(p11=0) the second inequality requires p1k
BERTRAND DUOPOLY WITH VARIABLE DEMAND

G

ri(pj )=

+ z {p+ i |p(pi , pj )=maxpi p(pi , pj )}i[ci , e ] z {e }

97

if ≠£ else

Lemma 4 Let profit functions be quasi-concave in their own price. There exists a pure strategy Bertrand–Nash equilibrium for log ci≤log pi≤z. Proof. The best response correspondence R is a correspondence with compact convex domain into itself. Since profit functions are continuous the best response correspondence is upper-hemicontinuous. The quasi-concavity of the profit functions guarantees that R is convex-valued. Hence, one can apply Kakutani’s fixed point theorem. Ε Lemma 5 Assume (A.1) and (A.2). There is no log pi>z such that p1(p1, p∗2 )>p1(p∗1 , p∗2 ) or p2(p∗1 , p2)>p2(p∗1 , p∗2 ). Proof. Suppose there is log p1>z such that p1(p1, p∗2 )>p1(p∗1 , p∗2 ). Case (i) log p∗2 ≤z+h. Then p1(p1, p∗2 )=0 for all log p1>z contradicting p1(p1, p∗2 )>p1(p∗1 , p∗2 ). Case (ii) z+h≤log p∗2 . log p1>z≥log p∗2 leads to a profit for firm 1 of B G(log p∗2 −log p1) p1 B ≤(p1−c1) G(0) p1 (p1−c1)

because G(0)≥G(log p∗2 −log p1). For log p1≥log p∗2 −h one has G(log p∗2 −log p1)=0. Thus, one must have log p1z gives a payoff p1−c1 +log B p1

log p1(p1, p∗2 )=log

+log G(log p∗2 −log p1) p1−c1 +log B+log G(0) p1

≤log

A B


p∗2 −c1 +log B eh

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−log p∗2 +h+log G(0) because (p1−c1)/p1 is increasing in p1. Set log p′1=log p∗2 −h¯ . It gives firm 1 the profit

A B

log p1(p′1, p∗2 )=log

p∗2 ¯ −c1 eh

+log B−log

p∗2 ¯ eh

It is not in the interest of the firm to set log p1>z≥log p∗2 if p1(p1, p∗2 )
A B

⇐log

A B A B A B

p∗ p∗2 p∗ −c1 −log p∗2 +h+log G(0) < log h2 −c1 −log h2¯ eh e e G(0)

p∗2 2 h¯ −h p∗ ¯ −c1 h −c1 < e e eh ¯

⇐eh−hc1 < p∗2 (1−G(0))e−h (h¯ −h)−log(1−G(0))+h+log c1 < log p∗2 ⇐z+h < log p∗2 which is a contradiction. In summary, one has p1(p1, p∗2 )z. For firm 2 one obtains p2(p∗1 , Ε p∗2 )>p2(p∗1 , p2) for all log p2>z. Proof of Theorem 1. Lemmas 1, 2 and 3 imply that profit functions are quasi-concave. Equilibrium existence for given compact strategy sets follows from Lemma 4. Lemma 5 shows that firms will always choose out of strategy sets [0, ez]. Under (A.2) prices in ×i[0, ci ] are serially dominated and picj , j≠i. Hence, an equilibrium in ×i[ci , ez] is also an 2 . equilibrium in R+ Now I show uniqueness. Note that all strategy profiles ( p1, p2) with log p2−log p1v/ [h, h¯ ] are serially dominated by some strategy with log p2−log p1v[h, h¯ ]. Consequently, all prices for which Propositions 1 and 2 cannot be applied are serially dominated. By Proposition 1 log-profits satisfy the dominant diagonal property and there exists a unique pure strategy Bertrand–Nash equilibrium for

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the transformation (compare Milgrom & Roberts, 1990). Dominance solvability, which implies uniqueness, follows from the uniqueness under log-transformation and log-supermodularity. Ε Lemma 6 Assume (A.1) and let g be flat. Then G, 1−G are log-concave. Proof. h˜ (p1, p2) was defined as log p2−log p1. ∂(g(h˜ )/G(h˜ )) ∂2 log G(h˜ ) 2 =− (∂ log p1) ∂(log p1) =

1 (g′(h˜ )G(h˜ )−g2(h˜ )) G 2(h˜ )

Log-concavity of G is equivalent to g′(h˜ )G(h˜ )≤g2(h˜ ) which is implied by |g′(h˜ )|G(h˜ )≤g2(h˜ ). Hence, log-concavity of G follows from Ε |(g′(h˜ )/g2(h˜ ))|≤1. Similarly, for firm 2. Proof of Theorem 2. Lemma 6 establishes log-concavity for the domain of prices above marginal costs and where demand is positive. Lemmas 2, 3, 4 and 5 then give existence as in Theorem 1. Uniqueness and dominance solvability then follow from Propositions 1 and 2. Ε Lemma 7 Assume (A.1) and |q(h)|≤2. An equilibrium (p∗1 , p∗2 ) with prices above marginal costs is unique on ×i(ci , ∞). Proof. A situation in which both prices are above marginal costs and in which one firm has zero demand cannot be an equilibrium. Hence, only prices such that demand is positive for both firms need to be considered. First-order conditions of profit maximization can be written as c1G(log p∗2 −log p∗1 )−(p∗1 −c1)G′(log p∗2 −log p∗1 )=0 c2(1−G(log p∗2 −log p∗1 ))−(p∗2 −c2)G′(log p∗2 −log p∗1 )=0 Replacing log p∗2 −log p∗1 by h∗, rearranging and taking ratios gives c2 1+G′(h∗)−G(h∗) h∗ −e =0 c1 G′(h∗)+G(h∗) Define the function W with

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c2 1+G′(h)−G(h) h −e c1 G′(h)+G(h)

W(h)=

It has to be shown that W(h) has a unique zero h∗v[h, h¯ ]. W′(h)=D(2G″(h)G(h)−2G′(h)2−G″(h) −G′(h))−eh =DG′(h)2

C

D

1 g′(h)[2G(h)−1] −2− −eh 2 g (h) g(h)

where Do(c2/c1)(G′(h)+G(h))−2>0. Note that 0≤G≤1 implies (2G−1)v[−1, 1] and

K

K K K

g′(h) g′(h)[2G(h)−1] g′(h)[2G(h)−1] ≤ ≤ 2 ≤2 2 2 g (h) g (h) g (h) Then W′(h)≤DG′(h)2[2−2−(1/g(h))]−eh=−Dg(h)−eh<0. W is decreasing in h. Therefore, h∗ is the only zero of W. Ε Proof of Proposition 3. Because of (A.2), prices in [0, ci ] are serially dominated and the result follows from Lemma 7. Ε Proof of Proposition 4. For any k>0 one can find a z(k) such that there is no equilibrium outside [c1, ez(k)]×[c2, ez(k)]. Strategy sets are non-decreasing in k. The associated game C(k) with logarithmic profits as payoffs and logarithmic prices as strategic variables is dominance solvable (see proof of Theorem 1). Finally, I need to show that price elasticities of profits are increasing in the parameter k, 1 1 ∂2 log p1(p, k) = >0 2 ∂ log p1∂k (1+k) (log p2−log p1)/(1+k)−h Analogously, for firm 2.

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