Discussion Paper No. 776

EVOLUTION
OF COMPETITIVE EQUILIBRIUM
WITH ENDOGENOUS PRODUCTDIFFERENTIATION

Toshihiro Matsumura
Noriaki Matsushima
Tetsuo Yamamori

May 2010

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Evolution of Competitive Equilibrium with Endogenous
Product Diﬀerentiation

Toshihiro Matsumura
Institute of Social Science, University of Tokyo
Noriaki Matsushima
Institute of Social and Economic Research, Osaka University
Tetsuo Yamamori∗
Department of General Education, Gunma National College of Technology
May 21, 2010

Abstract
Previous theoretical researches show that learning from good performers yields intensecompetition and results in the low proﬁtability of ﬁrms. These researchers do not take into
account diﬀerentiation strategies being referred as a useful strategic tool to mitigate competition. We introduce an evolutionary (learning) game into a duopoly model with productdiﬀerentiation on the Hotelling line. We ﬁnd that central agglomeration appears in theunique stochastically stable state in which the equilibrium price is equal to the marginal costof ﬁrms. This implies that perfectly competitive equilibrium appears even when ﬁrms havean opportunity to diﬀerentiate themselves through product diﬀerentiation and to mitigate
competition.

JEL classiﬁcation: C73, D41, D43, R32
Key words: product positioning, two dimensional choice, Walrasian, evolutionally stable,
spatial agglomeration.
∗

Corresponding author: Tetsuo Yamamori, Gunma National College of Technology, 580, Toriba, Maebashi,
Gunma, 371-8530, Japan phone: +81-27-254-9100, fax: +81-27-254-9022, email:yamamori@gen.gunma-ct.ac.jp1

1

Introduction

Learning from good performers seems to be a reasonable strategy to improve performance inmany situations. For instance, ﬁrms often adopt a ”me-too” strategy by following market leaders. In the literature of evolutionary learning, however, this “me-too” strategy often leads to
low performance under strategic environments. For instance, Vega-Redondo (1997) investigatesa Cournot model where ﬁrms learn through the imitation of success and shows that Walrasianequilibrium appears in the unique stochastically stable state.1 That is, each ﬁrm equates its
marginal cost to the market price (price taking behavior) in the stable state. This outcome,obviously, means that each ﬁrm’s performance is low from the viewpoint of proﬁtability. To
escape this kind of ﬁerce competition, we need to note that product diﬀerentiation is an important strategic tool of ﬁrms.2 When ﬁrms can choose their product positions endogenously, arethey able to diﬀerentiate their products by the imitation of success? We show that the answer
is “No”.
In this paper, we introduce an evolutionary game into an endogenous product diﬀerentiation
model where ﬁrms compete in terms of price and product position on the Hotelling line. We
adopt the idea of Vega-Redondo (1997) and investigate the stochastically stable state in the
dynamic stochastic framework. We ﬁnd that central agglomeration (no product diﬀerentiation)
appears in the unique stochastically stable state. At the equilibrium, the price is equal to themarginal cost of ﬁrms. The competitive equilibrium appears in the long run even when ﬁrms
1

Starting with the analysis of Vega-Redondo (1997), evolution dynamics have been applied to Cournot

oligopolies (Schenk-Hoppé (2000), Alós-Ferrer (2004), Alós-Ferrer and Ania (2005), Schipper (2009)); Bertrand
oligopolies (Alós-Ferrer et al. (2000), Hehenkamp (2002), Ania (2008)); and diﬀerentiated product oligopolies(Tanaka (2000, 2001) and Tasnádi (2006)).
2

Since the seminal work of Hotelling (1929), the spatial model has become one of the most important methodsof analyzing product diﬀerentiation. For instance, d’Aspremont et al. (1979) formulate a two-stage location-pricegame on the Hotelling line. They show that the products are maximally diﬀerentiated under quadratic transport
costs. As a result, the equilibrium prices exceed marginal costs; this is unlike in a perfectly competitive equilibrium.

2

have an opportunity to diﬀerentiate themselves through product diﬀerentiation and to relax thecompetition. This is quite diﬀerent from the results in the literature on spatial models withprice competition. Some researchers have already attempted to explain central agglomeration.
de Palma et al. (1985) show that suﬃcient heterogeneity between ﬁrms induces central agglomeration. Price collusion after ﬁrms have made location choices is considered in Friedman andThisse (1993) and Jehiel (1992). Cooperation between ﬁrms is considered in the form of information exchange through communication by Mai and Peng (1999). In all works, the equilibriumoutcomes are not Walrasian.
Our result strengthens the result of Vega-Redondo (1997). As mentioned above, he investigates a quantity competition in homogeneous product markets and shows that the Walrasianequilibrium appears in the unique stochastically stable state where each ﬁrm equates its marginal
cost with the market price (price taking behavior). This result is in sharp contrast to the Cournot
limit theorem since his result holds true as long as the number of ﬁrms is equal or more than two.
However, his result depends on the assumption of homogeneous product markets. If products
are diﬀerentiated, the equilibrium prices exceed marginal costs. Our result indicates that his
result is quite robust and his basic principle is applicable to broader situations. Firms dare not
diﬀerentiate their products in the unique stable state.
The remainder of this paper is organized as follows. Section 2 formulates the model. Section
3 presents the main result. Section 4 includes the concluding remarks.

2

The Model

2.1

Location and Price Competition

Consider a model of diﬀerentiation on the Hotelling (1929) line. A linear city of length 1 lieson the abscissa of a line, and consumers are uniformly distributed with density 1 along thisinterval. There are two ﬁrms, and each of them locates at some point in the city and sells the3same physical product. When a consumer moves a distance d to buy the product, he/she incurs
a transportation cost t(d). We assume that t(d) is continuous, strictly increasing, and strictly
convex.3 Each consumer buys one unit if and only if the minimum generalized price (price plustransportation cost) for the two ﬁrms does not exceed u, the reservation value of the product.
Two ﬁrms compete in terms of their locations and prices. Each ﬁrm chooses its price froma common countable set P = {δ, 2δ, ..., vδ}, where δ > 0 and v is a positive integer. To simplify
the analysis, we assume that u − t(1/2) > vδ. Firm 1 locates at a point in the left-side of the cityand ﬁrm 2 locates at a point in the right-side of the city. Each ﬁrm chooses how far it locatesfrom the center of the city. The set of locations of each ﬁrm is given as A = {0, 1/n, 2/n, ..., 1},
where n is a positive integer. When ﬁrm 1 (ﬁrm 2) chooses action a1 (a2 )∈ A, it is located at(1 − a1 )/2 ((1 + a2 )/2). A strategy si of ﬁrm i identiﬁes its price and location. Both ﬁrms havethe same strategy set S = A × P .
Given a strategy proﬁle (s1 , s2 ) = ((a1 , p1 ), (a2 , p2 )) where ai ̸= 0 for some i, let x(s1 , s2 ) bea real number satisfying
¯)
¯)
(¯
(¯
¯
¯
¯1
¯1
¯
¯
¯
p1 + t ¯ (1 − a1 ) − x(s1 , s2 )¯ = p2 + t ¯ (1 + a2 ) − x(s1 , s2 )¯¯ .
2
2
If 0 ≤ x(s1 , s2 ) ≤ 1, it is the location of the consumer who is indiﬀerent as to from where he/shebuys the product. For a strategy proﬁle (s1 , s2 ) = ((a1 , p1 ), (a2 , p2 )) where ai ̸= 0 for some i,the demand X1 (s1 , s2 ) for ﬁrm 1 is given by
if x(s1 , s2 ) < 0 0x(s1 , s2 ) if 0 ≤ x(s1 , s2 ) ≤ 1
X1 (s1 , s2 ) =1
if 1 < x(s1 , s2 )For a strategy proﬁle (s1 , s2 ) = ((a1 , p1 ), (a2 , p2 )) where a1 = a2 = 0, the demand for ﬁrm 1 is

3

The assumption of convexity of the transport cost is just for simplicity. Our theorem holds even without

this assumption.

4

given by
if p1 > p2
 0
1/2 if p1 = p2
X1 (s1 , s2 ) =1
if p1 < p2

The demand X2 (s1 , s2 ) for ﬁrm 2 is given by (1 − X1 (s1 , s2 )). The payoﬀs of the ﬁrms 1 and 2
are given by π1 (s1 , s2 ) = p1 X1 (s1 , s2 ) and π2 (s1 , s2 ) = p2 X2 (s1 , s2 ), respectively.
Lemma 1. Given a strategy proﬁle (s1 , s2 ) = ((a1 , p1 ), (a2 , p2 )),(i) if p1 = p2 and aj > ai , then πi (s1 , s2 ) > πj (s1 , s2 ),(ii) if pj > pi and a1 = a2 = 0, then πi (s1 , s2 ) > πj (s1 , s2 ).
Hotelling has already shown a result similar to Lemma 1(i). If both ﬁrms name the same
price, the ﬁrm closer to the central point obtains a larger market share, which results in higher
proﬁts for that ﬁrm. Lemma 1(ii) discusses the case with no product diﬀerentiation. Without
product diﬀerentiation, the ﬁrm naming a higher price obtains zero market share, which resultsin zero proﬁts.

2.2

Imitation Dynamics

Evolutionary dynamics are taken to proceed in discrete time, which is indexed by t = 0, 1, 2, ....
At each t, the state of the system may be identiﬁed using st = (st1 , st2 ) = ((at1 , pt1 ), (at2 , pt2 )).
Thus, the state space of the system is the set of strategy proﬁles Θ = S 2 . Associated to any
such st , the induced proﬁt π t = (π1t , π2t ) is deﬁned by πit = πi (st ) for each i. At every time t,each ﬁrm is assumed to enjoy a common and independent probability r > 0 of being able torevise its former strategy. When ﬁrm i revises its strategy, it imitates ﬁrm j’s strategy if thepayoﬀs of j are higher than those of i. Otherwise, it continues with its earlier strategy.4 This4

Our results hold true even if the revision probabilities of location and price are diﬀerent as long as the revisionoccurs at the same time with a positive probability.

5

imitation process deﬁnes a Markov process T0 = {T0 (s, s′ )}s,s′ ∈Θ on the state space Θ, where
T0 (s, s′ ) is a transition probability from s to s′ .
A nonempty set Q ⊆ Θ is absorbing if T0 (s, s′ ) = 0 for all s ∈ Q and s′ ̸∈ Q, and no proper
subset of Q has this property. The basin of attraction of an absorbing set Q is the set of statesfrom which there exists a positive probability that the imitation process moves the system to Q
in ﬁnite time.
It is evident that any state s where πi = πj constitutes a singleton absorbing set. Note thatif πi > πj , the event that ﬁrm j imitates ﬁrm i occurs with a positive probability. Thus, fromany state with πi ̸= πj , there exists a positive probability that the imitation process moves thesystem to a singleton absorbing set in which the two ﬁrms have the same payoﬀs.
Lemma 2.

(i) For any state s such that πi (s) ̸= πj (s), there exists an absorbing set {θ},whose basin of attraction contains s.
(ii) It follows from (i) that any absorbing set is a singleton in which the payoﬀs of the two
ﬁrms are the same.
Hereafter, an absorbing set is referred to as an absorbing state.

2.3

Mutations

At the end of each period t, the price and location of each ﬁrm mutate by a common independent
probability ϵ > 0. In this event, all of the prices or all of the locations are chosen with a positive
probability. Note that the probability of mutating both price and location of a ﬁrm is ϵ2 . Itis crucial for this paper that two or more mutations are required for a price and location tochange simultaneously. An interpretation of this mutation process is as follows. There are two
managers in each ﬁrm: a pricing manager who decides the ﬁrm’s price and a location manager
who decides the ﬁrm’s location. With a small probability, each manager experiments with new
choices or is replaced by a new manager.
6

The imitation process with mutations deﬁnes a Markov process Tϵ = {Tϵ (s, s′ )}s,s′ ∈Θ on
the state space Θ, where Tϵ (s, s′ ) is a probability that the combination of the imitation andmutation processes moves the system from s to s′ . Note that Tϵ (s, s′ ) > 0 for all s, s′ ∈ Θ.
Since all elements in Tϵ are strictly positive, Tϵ is irreducible and aperiodic. Therefore, Tϵ
has the following properties:5 (i) there exists a unique stationary distribution µ∗ϵ satisfyingµ∗ϵ Tϵ = µ∗ϵ ; (ii) the system converges to µ∗ϵ from any initial condition; (iii) in any suﬃciently
lengthy time period, the cumulative relative frequency of the time the system spends on eachstate is approximately given by µ∗ϵ ; (iv) the limiting distribution, limϵ→0 µ∗ϵ , exists.
A state in support of the limiting distribution is known as a stochastically stable state inYoung (1993). When the mutation rate is small, the system spends most of the time in stochastically stable states.

3

Results

3.1

Existing results

In order to investigate stochastically stable states, we rely on the mutation-counting arguments
employed by Kandori et al. (1993) and Young (1993). For each s ∈ Θ, a s-tree h is a binary
relation on Θ such that (i) for every state s′ ∈ Θ \ {s}, there exists one and only one transitionof the form (s′ → s′′ ); (ii) from every state s′ ∈ Θ \ {s}, there exists a sequence of the form(s′ → k 1 ), (k 1 → k 2 ), . . . , (k l → s); (iii) there exists no transition of the form (s → s′ ).
The cost ψ(s, s′ ) of the transition of the form (s → s′ ) is deﬁned by the least number ofmutations required to move the system from s to s′ . Note that for any absorbing state θ,ψ(θ, s) ≥ 1 for any s ∈ Θ \ {θ}, since it takes at least one mutation to move from any absorbing
state. On the other hand, for any non-absorbing state s, there is an absorbing state θ withψ(s, θ) = 0.
5

See Freidlin and Wentzel (1984).

7

The cost of a tree h is given by∑′(s→s′ )∈h ψ(s, s ). It is well known that state s is a stochas-tically stable state if and only if the minimum cost among all s-trees is also the minimum among
all states in Θ. We now claim the following lemma without proof.
Lemma 3. State s∗ is a stochastically stable state if and only if∑ψ(s, s′ ) ,s∗ ∈ arg min  min
s∈Θ

h∈Hs

(s→s′ )∈h

where Hs is the set of s-trees.
It is easy to see that s∗ is a stochastically stable state only if it is an absorbing state.

3.2

Stochastically stable stateLet θ∗ = ((a∗1 , p∗1 ), (a∗2 , p∗2 )) be an absorbing state, where a∗1 = a∗2 = 0 and p∗1 = p∗2 = δ. The
following lemma implies that θ∗ is a stochastically stable state.
Lemma 4. There exists a θ∗ -tree h whose cost is exactly λ − 1, where λ is the number ofabsorbing states.
Proof. Consider a θ∗ -tree h that satisﬁes the following conditions. (1) For any absorbing stateθ = (s1 , s2 ) with s1 ̸= s2 , h contains the transition of the form (θ → θ′ ), where θ′ = ((a, p), (a, p))and ψ(θ, θ′ ) = 1. (2) For any absorbing state θ = ((a, p), (a, p)) with a ̸= 0, h contains (θ → θ′ ),where θ′ = ((0, p), (0, p)). (3) For any absorbing state θ = ((0, p), (0, p)) with p ̸= δ, h contains(θ → θ∗ ). (4) For any non-absorbing state s, h contains (s → θ) such that s ∈ B(θ).
Note that in any absorbing state θ = (s1 , s2 ) with s1 ̸= s2 , π1 (s1 , s2 ) = π2 (s1 , s2 ) by Lemma 2(ii). Note also that the payoﬀs of two ﬁrms will diﬀer if at least one mutation occurs. Therefore,by Lemma 2 (i), there exists an absorbing state θ′ = ((a, p), (a, p)) with ψ(θ, θ′ ) = 1. Thus, we
can construct a tree h that satisﬁes condition (1). Furthermore, it follows from Lemma 1 thatthe cost of each transition that appeared in the conditions (2) and (3) is 1. Therefore, the costof the above tree is exactly λ − 1.
8

Lemma 5. For any absorbing state θ ̸= θ∗ , the cost of any θ-tree h is larger than λ − 1.
Proof. We show that more than one mutation is required for going from θ∗ to any other absorbing
state. Consider a state s = ((a∗1 , p′1 ), (a∗2 , p∗2 )), where p′1 ̸= p∗1 . Since p′1 > p∗2 = δ, π2 (s) > π1 (s)by Lemma 1 (ii). Therefore, from state s, the imitation process without mutations convergesto θ∗ . Next, consider a state s = ((a′1 , p∗1 ), (a∗2 , p∗2 )), where a′1 ̸= a∗1 . Since a′1 > a∗2 = 0,
π2 (s) > π1 (s) by Lemma 1 (i). Therefore, from state s, the imitation process without mutations
converges to θ∗ . Similarly, the imitation process without mutations converges to θ∗ from any
state where either the price or the location of ﬁrm 2 is not θ∗ .
Note that the assumption that either a price or a location changes in one mutation is crucial
for this Lemma. If a price and a location change simultaneously with probability ϵ, only one
mutation may move the system from θ∗ to another absorbing state.
Lemmata 4 and 5 imply that θ∗ is a unique stochastically stable state. Thus, we have thefollowing theorem.
Theorem . There exists a unique stochastically stable state θ∗ = ((a∗1 , p∗1 ), (a∗2 , p∗2 )), wherea∗1 = a∗2 = 0 and p∗1 = p∗2 = δ.
We now explain the intuition behind the result. In the long run, ﬁrms name the same
price through the imitation process. Given the common price, the ﬁrm located closer to thecentral point obtains larger proﬁts (Lemma 1(i)). Thus, through the imitation process, theﬁrms agglomerate at the central point.

4

Concluding Remarks

In this paper, we investigate evolutional dynamics in an endogenous product diﬀerentiation
model. Firms compete in terms of price and location on the Hotelling line. We adopt the idea of
Vega-Redondo (1997) and investigate the stochastically stable state in the dynamic stochastic
9

framework. We ﬁnd that ﬁrms produce homogeneous products in the unique stochasticallystable state. The competitive equilibrium appears in the long-run even when ﬁrms have an
opportunity to relax the competition through product diﬀerentiation.
We adopted the standard assumption of the Hotelling model that each consumer consumes
one unit of product. However, our result does not depend on this assumption. Even if we
introduce the model of Anderson and de Palma (2000) with elastic demand for each consumer,
our result still holds true.
Our result is very closely related to the discussions on the relative payoﬀ approach.6 If theﬁrms care more about their relative proﬁts, ﬁrms have an incentive to choose smaller productdiﬀerentiation. Investigating the relationship between the relative proﬁt approach and evolutionary dynamics in more general contexts remains for the future.

6

One of the most fundamental assumptions in economics is that ﬁrms maximize absolute proﬁts. However,already Alchian (1950) suggested that ﬁrms may maximize relative proﬁts in the long run rather than absolute
proﬁts.

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