Institute of Social Science, University of Tokyo
Discussion Paper Series No. F148

On the uniqueness of Bertrand equilibrium
Daisuke Hirata∗
Graduate School of Economics, University of TokyoandToshihiro Matsumura†Institute of Social Science, University of Tokyo
Discussion Paper Series No. F-148
July 2009

Abstract
We introduce product diﬀerentiation in Dastidar’s (1995) model of price competition withstrictly convex costs. We ﬁnd that even a small degree of product diﬀerentiation resolves the
problem of the indeterminacy of equilibria. Although a continuum of equilibria exists in a homogeneous product market, the competitive price equilibrium is the only robust one. Speciﬁcally,
as long as the equilibrium correspondence is nonempty, the equilibrium price converges to thecompetitive price when the degree of product diﬀerentiation shrinks to zero.

JEL classification numbers: C72, L13
Key words: price competition, strictly convex cost, uniqueness of equilibrium, increasing
marginal costs, product diﬀerentiation∗

Corresponding author: Graduate School of Economics, University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo
113-0033, Japan. Phone and Fax: (81)3-5841-4932. E-mail: dhirata@fas.harvard.edu†
E-mail: matsumur@iss.u-tokyo.ac.jp1

1

Introduction

Bertrand competition is one of the most important models in the literature on oligopoly, and it
has been intensively discussed in various contexts. In a homogeneous product market, when ﬁrms
have identical constant marginal costs, Bertrand competition yields a unique equilibrium pricethat is equal to the perfect-competition price. However, this clear result does not hold in thecase of strictly convex costs. Since Edgeworth (1897), many researchers have demonstrated thata pure strategy equilibrium does not exist under strictly convex costs. The non-existence of purestrategy equilibrium can be a serious obstacle in using Bertrand models. First, the property of
mixed strategy equilibrium is generally complicated and it causes models that involve Bertrand
competition to become intractable. Second, more importantly, it is doubtful whether decision
makers in ﬁrms shoot dice for price selection.1
Dastidar (1995) introduces an assumption that a ﬁrm always supplies for all of the demands
it faces, and shows the existence of pure strategy equilibrium in such a market.2 However, his
setting leads to another problem: the indeterminacy of equilibria. Speciﬁcally, a continuum of purestrategy equilibria exists in his model.
In this paper, we introduce product diﬀerentiation in Dastidar’s (1995) model. We ﬁnd that evena small degree of product diﬀerentiation resolves the problem of the non-uniqueness of equilibria
in his model: the equilibrium price converges to the competitive price when the degree of productdiﬀerentiation shrinks to zero. Our result highlights the utility of Dastidar’s approach. Althougha continuum of equilibria exists in his setting, we can focus on the Walrasian outcome because allof the other equilibria are unstable and vulnerable to product diﬀerentiation.

1

See Friedman (1988) and Martin (2001).
Vives (1999, Chapter 5) explains the industries in which this assumption is plausible. Dastidar’s (1995) modelis extensively used in various contexts. See Abbink and Brandts (2008) for experimental approach, Ania (2008) for
evolutionary game, Dastidar (1997, 2001) for competition policy implications, Hoernig (2002) for mixed strategy
equilibria, and Ogawa and Kato (2006) for public policy and privatization.
2

2

2

The Model

Suppose that there are n symmetric ﬁrms in an industry. All ﬁrms have an identical cost function
C : R+ → R+ , which is assumed to be increasing, continuously diﬀerentiable, and strictly convex.
Without loss of generality, we assume C(0) = C  (0) = 0. In what follows, C  (·) and its inverse aredenoted by M C(·) and Y (·), respectively.
We consider a sequence of markets in which the ﬁrms’ products are diﬀerentiated, but thedegree of production diﬀerentiation shrinks to zero. Then, we analyze the limit of equilibria in thediﬀerentiated markets. Let d ∈ R+ denote a degree of product diﬀerentiation: d = 0 means that theﬁrms produce homogeneous commodities and d > 0 indicates that the products are diﬀerentiated.
For each d ∈ R+ , let Did : Rn+ → R+ denote the contingent demand function for ﬁrm i under
product diﬀerentiation d. We impose the following assumptions on Did .
Assumption 1. For all d > 0, (a) Did is continuous, and (b) Did is decreasing in pi and increasingin pj (j = i) at any point (p1 , . . . , pn ) such that Did (p1 , . . . , pn ) > 0.
Assumption 2. For all d > 0 and (p1 , . . . , pn ), if pi = pj , then Did (p1 , . . . , pn ) = Djd (p1 , . . . , pn ).
Assumption 3. For all d > 0, D d (p) :=di Di (p, . . . , p) is decreasing in p.

Assumption 4. As d  0, Did converges pointwise to Di0 deﬁned by

Di0 (p1 , . . . , pn ) =⎧⎪⎪⎨D(p )/| arg min p |if pi = minj pj ,⎪⎪⎩0otherwise,

i

j

j

where D : R+ → R+ is a continuous and bounded function such that (a) ∃P [p ≥ P ⇔ D(p) = 0],
and (b) D is decreasing on [0, P ].
Let us explain Assumptions 1–4. Assumption 1 represents our intended interpretation thatd > 0 refers to diﬀerentiated products. Assumption 2 means that ﬁrms setting an identical priceface an identical demand under any degree of production diﬀerentiation. Assumption 3 indicates,
roughly speaking, that the aggregate demand is decreasing in price under any degree of production
diﬀerentiation. Assumption 4 indicates that as d  0, the demand function converges to that of the3homogeneous product market analyzed by Dastidar (1995). In the homogeneous product market
with Di0 , the ﬁrm setting the lowest price obtain the entire demand. In case of a tie, the ﬁrms
setting the lowest price obtain an equal share.3
Assumptions 1–4 imply the following two lemmata.
Lemma 1. Suppose that {dk }k∈ is a sequence such that dk → 0. For any p and  > 0, there exist
δ > 0 and K ∈ N such that |Didk (p, . . . , p) − Didk (p , . . . , p )| <  if |p − p| < δ and k ≥ K.
Proof. Fix arbitrary p and , and take a, b ∈ R+ such that p ∈ (a, b) and D(a) − D(b) < n. By
Assumptions 2 and 4, D dk (a) and Ddk (b) converge to D(a) and D(b), respectively. Then, there
exists K ∈ N such that |Ddk (a) − Ddk (b)| < n for all k ≥ K. By Assumption 3, for all p ∈ (a, b)
and k ≥ K, |Ddk (p) − Ddk (p )| < n, which is equivalent to |Didk (p, . . . , p) − Didk (p , . . . , p )| <  byAssumption 2.
Lemma 2. Suppose that {(dk , pk )}k∈ is a sequence such that dk → 0 and pk → p. If Di0 (p, . . . , p)is strictly greater (resp. smaller) than Y (p), then there exists K ∈ N such that Didk (pk , . . . , pk ) isstrictly greater (resp. smaller) than Y (pk ) for all k ≥ K.
Proof. By Lemma 1 and the assumption that pk → p, it can be easily checked that as k → ∞,Didk (pk , . . . , pk ) → Di0 (p, . . . , p). By the continuity of M C, it also follows that Y (pk ) → Y (p).
These directly imply the statement.
We formulate price competition games corresponding to each d ∈ R+ as follows. Following Dastidar(1995), we assume that a ﬁrm always supplies for all of the demands it faces even if that quantity
is not optimal.4 Speciﬁcally, given a degree of production diﬀerentiation d, i’s payoﬀ (or proﬁt)function πid is deﬁned by πid = pi Did −C(Did ). Each ﬁrm i simultaneously and independently choosespi ∈ R+ so as to maximize πid .

34For a discussion on a more general sharing rule in homogeneous good markets, see Hoernig (2007).
Note that when a ﬁrm has a strict convex cost, an increase in supply does not necessarily raise its proﬁt.

4

3

Results

Let p∗ be the competitive price in the limit homogeneous good market, i.e.,
Y (p∗ ) = Di0 (p∗ , . . . , p∗ ).

(1)

Under the assumption that C is strictly convex, p∗ uniquely exists.
We deﬁne symmetric Nash equilibrium correspondence S as follows: p ∈ S(d) if and only if(p, . . . , p) is a Nash equilibrium in the price game under d. Dastidar (1995) shows that there exists a continuum of equilibrium when products are homogeneous, i.e., S(0) = [pm , pM ] for certain
pm , pM ∈ R++ . However, the following proposition states that a small degree of production differentiation excludes all equilibria, except for the competitive price one. In other words, S is not
lower hemi-continuous at d = 0.
Proposition 1. Under Assumptions 1–4, there is no sequence {(dk , pk )}k∈ such that dk > 0 and
pk ∈ S(dk ) for all k, and limk→∞ pk = p = p∗ .
Proof. By way of contradiction, assume that there exists such a sequence {(dk , pk )}k∈.
First, suppose that p ≡ lim pk < p∗ . Then, it follows from Lemma 2 that for suﬃciently large k,
Didk (pk , . . . , pk ) > Y (pk ). Then, by Assumption 1, there exists p̂ > pk such that Didk (p̂; pk , . . . , pk ) =Y (p̂). That is, each ﬁrm can proﬁtably deviate from (pk , . . . , pk ) by charging p̂, which is a contradiction to the assumption that pk ∈ S(dk ).
Second, suppose that p ≡ lim pk > p∗ and D(p) < Y (p). Then, Lemma 1 implies that
πidk (pk , . . . , pk ) → pD(p)/n − C(D(p)/n).5 On the other hand, by Assumptions 1 and 4, for anyp < p, Didk (p ; pk , . . . , pk ) → D(p ) and thus πidk (p ; pk , . . . , pk ) → p D(p ) − C(D(p )). Then, for pthat is suﬃciently close to p, lim πidk (pk , . . . , pk ) < lim πidk (p ; pk , . . . , pk ), which is a contradiction
to the assumption that pk ∈ S(dk ) for all k ∈ N.6
Finally, suppose that p ≡ lim pk > p∗ and D(p) ≥ Y (p). It follows that D(p ) > Y (p ) for anyp < p. On one hand, since Didk (p ; pk , . . . , pk ) → D(p ) as k → ∞, Didk (p ; pk , . . . , pk ) > Y (p ) for
5
6

Note that by Assumption 2, Di0 (p, . . . , p) = D(p)/n.
Note that D(p ) < Y (p ) for p that is suﬃciently close to p.

5

suﬃciently large k. On the other hand, by Lemma 2, Didk (pk , . . . , pk ) < Y (pk ) for suﬃciently largek. Thus, by Assumption 1, as long as k is suﬃciently large, there exists p̂k ∈ (p , pk ) such that
Didk (p̂k ; pk , . . . , pk ) = Y (p̂k ). Since p is arbitrary, it can be easily checked that p̂k → p. Therefore,
Didk (p̂k ; pk , . . . , pk ) → Y (p), and thus,
lim πidk (p̂k ; pk , . . . , pk ) = pY (p) − C(Y (p)) > lim πidk (pk , . . . , pk ),(2)

which is a contradiction to the assumption that pk ∈ S(dk ) for all k ∈ N.
Proposition 1 states that the equilibrium price cannot converge to p = p∗ as d → 0, whereas acontinuum of equilibria exists when d = 0. Let us explain the intuition.
When d = 0, ﬁrm i’s demand is discontinuous at a point (p, . . . , p), as shown in Figure 1. If
ﬁrm i sets a slightly higher price than p, it does not obtain any demand. Thus, ﬁrm i has no
incentive for upward deviation as long as its proﬁt before deviation is non-negative. The conditionof non-negative proﬁt before deviation yields the lower bound pm of equilibrium prices. If ﬁrm
i sets a slightly lower price than p, it obtains all of the demands. However, since marginal costis increasing, the deviation reduces proﬁts, unless p is suﬃciently high. This condition yields theupper bound pM of equilibrium prices. Since it is obvious that pm < p∗ < pM , a continuum ofequilibria exists when d = 0.
When d > 0, ﬁrm i’s demand is continuous. Even though the elasticity of demand is large at(p, . . . , p), it is ﬁnite, as seen in Figure 2. If p < p∗ , each ﬁrm i supplies less than it wants to at(p, . . . , p). Thus, ﬁrm i can increase its proﬁt by setting a slightly higher price than p by continuity,and a price below the Walrasian one cannot be an equilibrium under product diﬀerentiation. If
p > p∗ , i supplies more than it wants to at (p, . . . , p). Thus, by continuity, ﬁrm i can increase
its proﬁt by setting a slightly lower price than p when d is suﬃciently close to zero but is strictly
positive. Hence, the equilibrium price cannot converge to p = p∗ as d → 0.
Furthermore, imposing another moderate assumption, we can show that the equilibrium pricemust converge to p∗ as d  0 as long as we can choose a sequence of equilibrium prices.
Assumption 5. There exists P  such that Dd (P  ) = 0 for all d > 0.
6

Proposition 2. Under Assumptions 1–5, if {(dk , pk )}k∈ is a sequence such that dk > 0 and
pk ∈ S(dk ) for all k, and limk→∞ dk = 0, then pk → p∗ .
Proof. First, suppose that lim inf pk < p∗ . Then, there exists a subsequence {(dki , pki )}i∈ suchthat dki  0 and pki → p < p∗ as i → ∞. However, this yields a contradiction as in the proof of
Proposition 1.
Second, suppose that lim sup pk > p∗ . Note that Assumption 5 implies that pk must be bounded
and thus, lim sup pk < ∞. Then, there exists a subsequence {(dki , pki )}i∈ such that dki  0 and
pki → p > p∗ as i → ∞. This yields a contradiction as shown in the proof of Proposition 1.
We have shown that lim sup pk ≤ p∗ ≤ lim inf pk . Since lim inf pk ≤ lim sup pk by deﬁnition, we
can conclude that lim sup pk = lim inf pk = p∗ .
Proposition 2 guarantees the robustness of the competitive equilibrium under d = 0, while Proposition 1 states that the other equilibria are fragile. These results highlight the signiﬁcance ofDastidar’s (1995) model. When we regard a homogeneous good market model as an approximationof a market with a small degree of product diﬀerentiation, the disadvantage of using Dastidar’s
model (i.e., equilibrium multiplicity) vanishes, and its advantage (i.e., existence of pure strategy
equilibrium) is retained.

4

Concluding Remarks

In this paper, we introduce product diﬀerentiation in Dastidar’s (1995) model of price competition
with strictly convex costs. Dastidar (1995) discussed a Bertrand model where a ﬁrm always supplies for all of the demands it faces and showed that in a homogeneous good market, a continuumof equilibria exists. We ﬁnd that even a small degree of product diﬀerentiation resolves the problem of the indeterminacy of equilibria. The equilibrium price converges to the competitive priceequilibrium when the market becomes close to the homogeneous good market, and thus only one
equilibrium is robust with respect to product diﬀerentiation.
Our result depends on the assumption that a ﬁrm always supplies for all of the demands it faces.
If we discard this assumption, a pure strategy equilibrium does not exist in a homogeneous product7

market, and a small degree of production diﬀerentiation does not yield a pure strategy equilibrium,
although a large degree of diﬀerentiation usually yields a pure strategy equilibrium. This factindicates that our result regarding the signiﬁcance of a small degree of product diﬀerentiation is
not obvious in a general context of price competition, although the intuition behind it is natural.

Reference
Abbink, K., Brandts, J.: 24. Pricing in Bertrand competition with increasing marginal costs.
Games and Economic Behavior 63(1), 1–31 (2008)
Ania, A.B.: Evolutionary stability and Nash equilibrium in ﬁnite populations, with an application
to price competition. Journal of Economic Behavior and Organization 65(3-4), 472–488(2008)
Dastidar, K.G.: On the existence of pure strategy Bertrand equilibrium. Economic Theory 5(1),19–32 (1995)
Dastidar, K.G.: Comparing Cournot and Bertrand in a homogeneous product market. Journal of
Economic Theory 75(1), 205–212 (1997)
Dastidar, K.G.: Collusive outcomes in price competition. Journal of Economics 73(1), 81–93(2001)
Edgeworth, F.: La teoria Pura del monopolio. Giornale degli Economisti 40, 13–31 (1897) (InEnglish: The pure theory of monopoly. In: Edgeworth, F. (ed.) Papers relating to political
economy 1, London: Macmilan 1925)
Friedman, J.W.: On the strategic importance of prices versus quantities. Rand Journal of Economics 19(4), 607–622 (1988)
Hoernig, S.H.: Mixed Bertrand equilibria under decreasing returns to scale: an embarrassment of
riches. Economics Letters 74(3), 359–362 (2002)
Hoernig, S.H.: Bertrand games and sharing rules. Economic Theory 31(3), 573–585 (2007)
Martin, S.: Advanced industrial economics (2nd ed.), Blackwell publisher (2001)
Ogawa, A., Kato, K.: Price competition in a mixed duopoly. Economics Bulletin 12(4), 1–5 (2006)
Vives, X.: Oligopoly Pricing: Old Ideas and New Tools, MIT Press, London (1999)

8