Bertrand Equilibria in Markets with Avoidable Fixed Costs
Germán Coloma (*) and Alejandro Saporiti (**)
Abstract
This paper extends Dastidar’s (1995) analysis of Bertrand equilibria in homogenousproduct markets to the case where firms face avoidable fixed costs. Looking at a symmetric
duopoly, we first show that the existence of a price-taking equilibrium (PTE) is sufficient, but not
necessary, for the existence of pure-strategy Bertrand equilibria (PBE) where all firms are active.
Moreover, there could exist multiple PBE even if the PTE does not exist. Finally, we prove that,
in this framework, a PBE exists if and only if the total cost function is superadditive at the output
level corresponding to the duopoly break-even price.
JEL Classification Number: L13, D43.
Keywords: Bertrand equilibrium, price-taking equilibrium, avoidable fixed costs, superadditivity.
Equilibrios de Bertrand en mercados con costos fijos evitables
Germán Coloma y Alejandro Saporiti
Resumen
Este trabajo extiende el análisis del equilibrio de Bertrand en mercados con productos
homogéneos, efectuado originalmente por Dastidar (1995), para el caso en el cual las empresas
tienen funciones de costos que incluyen costos fijos evitables. Tomando el caso de un duopolio
simétrico, se muestra primero que la existencia de un equilibrio tomador de precios (PTE) es
suficiente, pero no necesaria, para que existan equilibrios de Bertrand en estrategias puras
(PBE) en los cuales todas las empresas proveen cantidades positivas. Más aún, múltiples PBE
pueden existir aun cuando el PTE no exista. Finalmente, el trabajo prueba que, en este modelo,
el PBE existe si y solo si la función de costo total es superaditiva para el nivel de producción
correspondiente al precio para el cual las empresas obtienen beneficios nulos.
Clasificación del JEL: L13, D43.
Descriptores: Equilibrio de Bertrand, equilibrio tomador de precios, costos fijos evitables,
superaditividad.
(*)
Department of Economics, CEMA University; Av. Córdoba 374, Buenos Aires, C1054AAP, Argentina;
Telephone: (54-11)6314-3000; E-mail: gcoloma@cema.edu.ar.
(**)
School of Economics, University of Manchester; Oxford Road, Manchester, M13 9PL, United Kingdom;
Telephone: (44-161)275-4865; E-mail: Alejandro.Saporiti@manchester.ac.uk.
1
Bertrand Equilibria in Markets with Avoidable Fixed Costs
Germán Coloma and Alejandro Saporiti
1. Introduction
In this paper, we analyze necessary and sufficient conditions for the existence of a pure
strategy Bertrand equilibrium (PBE) in a symmetric and homogeneous-product price-competition
game with avoidable fixed costs.
Our work is closely related to Dastidar’s (1995) analysis of equilibrium existence in
Bertrand games with strictly decreasing returns to scale and (possible) completely unavoidable
fixed costs. Apart from the fact that we focus on a symmetric duopoly, our model differs from
Dastidar’s in that we assume that firms have a costless exit option, in the sense that they can
supply zero output and avoid the fixed production costs. As we will see, this small departure
from Dastidar’s analysis has important and interesting consequences for the existence of
Bertrand equilibria in pure strategies in such kind of markets.
Indeed, if fixed costs are fully unavoidable, then increasing and strictly convex variable
costs are sufficient to ensure strictly decreasing returns to scale. Thus, independently of whether
there exists an exit option, there is always a price-taking equilibrium (PTE) and a Bertrand
equilibrium in pure strategies. This result comes out immediately from Dastidar (1995), who has
also shown that this kind of games have typically multiple PBE, being the PTE price an element
of the set of equilibrium prices1.
On the contrary, with partially or completely avoidable fixed costs, the total cost function
is not longer convex around the origin. Under increasing and strictly convex variable costs, the
market exhibits variable returns to scale. Consequently, neither a PTE nor a PBE is anymore
guaranteed.
A related result to ours is the one derived by Telser (1991), who showed that markets
with non-decreasing returns to scale may have an empty core and, therefore, that a PTE may
not exist. However, his framework is different, and there is no reference to whether Bertrand
equilibria exist in such industries.
In this paper, we deal with the problem of equilibrium existence. To simplify the analysis,
we focus on the extreme, albeit interesting, case where fixed costs are completely avoidable.
This situation can also be thought of as a long-run extension of Dastidar’s work, whose model
can be seen as a short-run version where fixed costs have been already incurred and are
therefore unavoidable.
Our two theoretical results appear in lemma 1 and proposition 1. In lemma 1 we show
that the existence of a price-taking equilibrium is sufficient to guarantee that the market has a
Bertrand equilibrium in pure strategies where all firms are active. Furthermore, as a numerical
example illustrates, if fixed costs are avoidable, this condition is not necessary, and the duopoly
could exhibit multiple PBE even if a PTE does not exist.
Finally, in proposition 1, we formulate and prove the main result of the paper, namely that
a symmetric and homogenous-product duopoly has a Bertrand equilibrium in pure strategies
where all firms supply positive quantities if and only if the total cost function is superadditive at
the output level corresponding to the duopoly break-even price (which is the largest traded
quantity in any PBE candidate). As we argue, this result can be easily generalized to the case of
oligopolies with any finite number of symmetric firms. Hence, it can be interpreted as a
necessary and sufficient condition for the existence of a “natural oligopoly” where all the existing
firms are active2.
1
In a posterior paper, Dastidar (2001) has also shown that, under certain conditions, this set may even include the
perfectly collusive price.
2
We do not know yet whether this can be extended to a situation where fixed costs are partially avoidable. However,
2
The rest of this article is organized as follows. Section 2 introduces the duopoly game
with avoidable fixed costs and strictly convex variable costs. The formal definitions of the pricetaking and the pure-strategy Bertrand equilibria are given in section 3, while section 4 presents
the results and their proofs. Section 5 provides a numerical example, and some final remarks
appear in section 6.
2. The model
Let us assume a market with two firms. Each of them has the following cost function:
C(Qi) = VC(Qi) + F
(for i = 1, 2)
;
where Qi is the quantity supplied by the ith firm, VC is a continuous, differentiable, increasing
and strictly convex function of Qi, for which it holds that VC(0) = 0, and F is a non-negative
parameter that represents the avoidable fixed cost3.
The product traded in this market is homogeneous, with total demand equal to:
Q = D(P)
;
where Q is total quantity, P is the price paid by consumers, and D is a continuous, differentiable
and decreasing function of P, with limP→∞D(P) = 0.
In a situation of price competition, each of the two firms faces the following individual
demand function:
0
 D( Pi )
Di ( Pi , Pj ) = 
 2
 D( Pi )
(if Pi > Pj )
(if Pi = Pj )
;
(if Pi < Pj )
where Pi is the ith firm’s price and Pj is the price chosen by its competitor4.
The ith firm’s profit, therefore, can be defined as:
Πi(Pi, Pj) = Pi⋅Di(Pi, Pj) – VC(Di(Pi, Pj)) – F
.
Alternatively, profit can also be expressed as a function of output, leaving implicit the
price vector and the corresponding individual demand. This implies that:
Πi(Qi) = Pi⋅Qi – VC(Qi) – F
.
3. Equilibrium definitions
Definition 1 (Price-taking equilibrium): Given a non-negative price Pc, a price-taking
equilibrium (PTE) is a pair (Q1, Q2) ∈ℜ2+ such that, for each i = 1, 2:
Qi = arg max Qi∈ℜ+ {Pc ⋅ Qi − VC(Qi ) − F}
(C1) ;
Pc⋅Qi – VC(Qi) – F ≥ 0
(C2) ;
it is clear that Dastidar’s (1995) does not longer apply, so the question of equilibrium existence remains open.
3
Using the terminology adopted by Telser (1991), this is a “Viner cost function”, because it is the sum of a constant
plus a strictly convex function. The average cost curve implied by this total cost function is U-shaped, and it has a
unique minimum at the point where it crosses the marginal cost curve.
4
Note that this definition of the individual demand of the ith firm assumes an “equal sharing rule”. For other
alternative rules applicable to situations of price competition, see Hoernig (2006).
3
Q1 + Q2 = D(Pc)
(C3) .
Note that C3, together with the sharing rule implicit in the definition of individual
demands, implies that, if (Q1, Q2) is a PTE for a given Pc ≥ 0, then Q1 = Q2 = D(Pc)/2. We can
therefore refer to (Pc, Qi) as a PTE, understanding that this means that (Q1, Q2) = (Qi, Qi)
satisfies conditions C1-C3 above under the price Pc.5
The assumptions about D and VC guarantee that it is always possible to find a unique
pair of positive values of Pc and Qi that satisfies C1 and C3. C2, conversely, may fail to hold if F
is sufficiently large for the pair (Pc, Qi) implied by C1 and C3, and in that case the PTE does not
exist6. This occurs if the market-clearing equality between price and marginal cost holds for a
value of Qi for which the average cost is larger than the marginal cost. This is a case of a market
with an empty core, since it is impossible to find an efficient allocation in which none of the two
firms has incentives to form “partial coalitions” with some consumers.
Definition 2 (Bertrand equilibrium): A pure-strategy Bertrand equilibrium (PBE) is a pair (P1,
P2) ∈ℜ2+ such that, for each i ≠ j:
Π i ( Pi , Pj ) ≥ Π i ( P̂, Pj ) (for all P̂ ∈ ℜ+ )
(E1) ;
Π i ( Pi , Pj ) ≥ 0
(E2) ;
Qi ( Pi , Pj ) = Di ( P̂, Pj )
(E3) ;
where Qi(Pi, Pj) is the output supply of the ith firm at prices (Pi, Pj).
It is relatively easy to show that, if a PBE exists, then P1 = P2 = Pb. As the market-sharing
rule assumed implies that D1(Pb, Pb) = D2(Pb, Pb) = D(Pb)/2, then E2 can be re-written as:
Pb ⋅
D( Pb )
 D( Pb ) 
− VC
−F≥0
2
 2 
(E4) ;
while E1 simply requires that:
Pb ⋅
(
)
D( Pb )
 D( Pb ) 
− VC
 − F ≥ P̂ ⋅ D( P̂) − VC D( P̂) − F (for all P̂ < Pb )
2
 2 
(E5) .
When E4 is satisfied as a strict equality, we obtain the minimum price Pmin that can be
supported as a PBE. Similarly, when E5 is satisfied as a strict equality we get the maximum
price Pmax that can be supported as a PBE. For the set of PBE to be non-empty, it is necessary
that Pmax ≥ Pmin. In fact, if Pmax > Pmin, there exists a continuum of Bertrand equilibria (P1, P2), with
the property that in each of them it holds that P1 = P2 ∈ [Pmin, Pmax] 7.
4. Results
We begin this section showing that the existence of a PTE is sufficient for a PBE to exist.
5
Note that our definition of PTE refers to a situation that is also called, by part of the literature, a “perfectly
competitive equilibrium”. We prefer the expression “price-taking equilibrium” because “perfectly competitive
equilibrium” is also used to denote the equilibrium of a perfectly competitive market (which is not the case here).
6
Note that, if F were fully unavoidable, then the PTE would always exist. In that situation C2 would be restated as
Pc⋅Qi – VC(Qi) ≥ 0, and that condition always holds if C1 is fulfilled and VC is increasing and convex.
7
Note that all these are “Bertand equilibria” and not “Bertrand-Edgeworth equilibria”, since E3 requires that firms
meet all the demand at the equilibrium prices. Their only strategic choice, therefore, is the price that they charge and
not the quantity that they sell. For an explanation of the difference between Bertrand and Bertrand-Edgeworth
equilibria, see Vives (1999), chapter 5.
4
Remarkably, the proof of this claim does not depend on neither the number of firms in the
industry nor the nature of the fixed costs. Thus, the result would go through to any homogenousproduct oligopoly, under similar demand and cost conditions that our model and a finite number
of symmetric firms.
Lemma 1: If (Pc, Qi) is a PTE, then (Pc, Pc) is a PBE, and Pc ∈ [Pmin, Pmax].
Proof: Assume, by contradiction, that (Pc, Pc) is not a PBE. Note first that, since (Pc, Qi) is a
PTE, then C2 and C3 imply that E3 and E4 are satisfied at (Pc, Pc). Hence, there must exist a
price Pi such that:
Πi(Pi, Pc) > Πi(Pc, Pc) ≥ 0
.
That means that:
Πi(Pi, Pc) = Pi⋅Di(Pi, Pc) – VC(Di(Pi, Pc)) – F > 0
;
and, therefore, Pi < Pc. In equilibrium, then:
Qi = Di(Pi, Pc) = D(Pi)
.
Totally differentiating Pi⋅Qi – VC(Qi) – F > 0 with respect to Qi, we have:
Pi > ∂C(Qi)/∂Qi
.
But we know, by C1, that Pc = ∂C(Qc)/∂Qi. Moreover, since Pi < Pc, D’ < 0 and C’’ > 0, it follows
that:
∂C(Qc)/∂Qi < ∂C(Qi)/∂Qi
.
Therefore, Pi > ∂C(Qi)/∂Qi implies that Pi > Pc, and this is a contradiction. Hence, (Pc, Pc) is a
PBE, i.e., Pc ∈ [Pmin, Pmax], qed.
Note that, although the existence of a price-taking equilibrium is a sufficient condition for
the existence of pure-strategy Bertrand equilibria, the reverse is not true. If it simultaneously
holds that Pmin > Pc and Pmax ≥ Pmin, then the price-taking equilibrium does not exist but the set of
pure-strategy Bertrand equilibria is not empty8. The existence of a non-empty set of Bertrand
equilibria is in fact determined by the fact that the market is not a natural monopoly. If, following
Baumol (1977), we identify a natural monopoly with the idea of subadditivity of the individual firm
cost function, then we can show that Pmax ≥ Pmin if and only if C(Q) is not subadditive when Q =
D(Pmin)9. To prove this, we have to introduce the formal definition of subadditivity:
Definition 3 (Subadditivity): A real valued function f: ℜ → ℜ is subadditive at z ∈ ℜ if and only
if f(z) < f(x) + f(y), for all x, y ∈ ℜ such that x + y = z.
It is easy to show that, if f is continuous and differentiable, then f(x)+f(y) subject to x + y =
z (for a given z ∈ ℜ) has a unique minimum x = y = z/2. Therefore, f is subadditive at z only if
8
A similar idea appears in the literature on oligopoly but associated to a different equilibrium concept. Specifically,
using a model similar to ours, Grossman (1981) has shown that the price-taking equilibrium, when it exists, is one of
the possible “supply function equilibria” for an oligopoly with fixed avoidable costs and variable convex costs.
Moreover, supply function equilibria may exist even when the set of price-taking equilibria is empty.
9
This result is linked to the concept of Bertrand equilibrium that we use in this paper. Chowdhury (2002), for
example, proves the existence of a class of Bertrand equilibria for natural monopolies with increasing returns to
scale, but that class of equilibria implies that only one firm supplies a positive quantity.
5
f(z) < 2⋅f(z/2). Or, equivalently, f is superadditive at z if f(z) ≥ 2⋅f(z/2). Applying superadditivity to
our framework, it follows that the cost function C(Q) is superadditive at Q if:
C(Q) ≥ 2⋅C(Q/2)
⇒
VC(Q) ≥ 2⋅VC(Q/2) + F
.
Let us now state and prove the main result of this paper.
Proposition 1: The set of PBE is not empty if and only if C(Q) is superadditive at Q = D(Pmin).
Proof: In order to prove proposition 1, we will divide the proof into a sufficiency condition and a
necessity condition. Therefore we will first prove that, it C(Q) is superadditive at D(Pmin), then
(Pmin, Pmin) is a PBE. To do that, let us define the function H: ℜ → ℜ:
H(x) = x ⋅
D( Pmin )
 D( Pmin ) 
− VC
−F
2
 2 
.
Note that H is continuous in x, because D and VC are continuous functions. Furthermore, H(0) <
0 and H(Pmin) = 0, where the last equality follows from the definition of Pmin.
We want to show, by contradiction, that if E5 is not satisfied for some Pi < Pmin, then H(Pi) > 0
and, therefore, there must exist Pk ∈ (0, Pi) such that H(Pk) = 0, contradicting in this way the
definition of Pmin.
Suppose, indeed, that there exists Pi < Pmin such that E5 is not satisfied. That is, assume that for
some Pi < Pmin:
Pi ⋅ D( Pi ) − VC(D( Pi ) ) − F > Pmin ⋅
D( Pmin )
 D( Pmin ) 
− VC
−F=0
2
 2 
.
Then it must hold that:
Pi ⋅ D( Pi ) − F > VC(D( Pi ) )
.
If C(Q) were superadditive at D(Pi), then VC(D(Pi)) ≥ 2⋅VC(D(Pi)/2) + F. Therefore it should hold
that:
 D( Pi ) 
Pi ⋅ D( Pi ) − F > 2 ⋅ VC
+F
 2 
.
On the other hand, note that Pi < Pmin, D’ < 0 and C’’ > 0 imply that:
 D( Pmin ) 
VC(D( Pi ) ) > VC(D( Pmin ) ) ≥ 2 ⋅ VC
+F
 2 
;
where the last inequality follows from the fact that C(Q) is superadditive at D(Pmin). Thus,
combining all these, we have that:
 D( Pmin ) 
 D( Pi ) 
2 ⋅ VC
 + F < VC(D( Pi ) ) < 2 ⋅ VC
+F
 2 
 2 
.
However, as figure 1 illustrates, this cannot be true under our assumptions, because VC is
continuous and strictly convex and, therefore, VC(Q) converges to 2⋅VC(Q/2)+F as Q rises.
Hence, (Pmin, Pmin) is a PBE, qed.
6
Figure 1
2VC ( y ) + F
(
)
VC ( y )
2VC Dˆ / 2 + F
VC Dˆ
( )
VC (DL )
2VC (DL / 2 ) + F
F
DL / 2
D̂ / 2
DL
D̂
Let us now turn to the necessity condition. We want to prove that, if the set of PBE is not
empty, then C(Q) is superadditive at D(Pmin). First of all, note that, if the set of PBE is not empty,
then (Pmin, Pmin) is a PBE, since by definition Pmin satisfies E3 and E4, and E5 is verified for all P
≤ Pmax, including Pmin. Now assume, by contradiction, that C(Q) is not superadditive at D(Pmin).
That means:
 D( Pmin ) 
VC(D( Pmin ) ) < 2 ⋅ VC
+F
 2 
.
But, by definition:
Pmin ⋅
D( Pmin )
 D( Pmin ) 
− VC
−F=0
2
 2 
;
which can be rewritten as:
 D( Pmin ) 
Pmin ⋅ D( Pmin ) − VC(D( Pmin ) ) − F = 2 ⋅ VC
 + F − VC(D( Pmin ) )
 2 
.
But if C(Q) is not superadditive at D(Pmin), then the right-hand side of this equation has to be
strictly greater than zero. Therefore it must hold that:
Pmin ⋅ D( Pmin ) − VC(D( Pmin ) ) − F > 0
.
Moreover, note that x⋅D(x)-VC(D(x))-F is continuous in x, since D and VC are continuous
functions on ℜ. Hence, by definition of continuity, there exists ε > 0 small enough, and Pi < Pmin
sufficiently close to Pmin, such that:
0 < Pmin ⋅ D( Pmin ) − VC(D( Pmin ) ) − F − ε < Pi ⋅ D( Pi ) − VC(D( Pi ) ) − F
.
This, however, enters in contradiction with the fact that, by the definition of Pmin:
7
Pi ⋅ D( Pi ) − VC(D( Pi ) ) − F ≤ 0
;
for all Pi < Pmin. Therefore C(Q) has to be superadditive at D(Pmin), qed.
Why the necessary and sufficient condition for a PBE to exist is superadditivity of C(Q) at
D(Pmin), and not at any other output Q ∈ [Pmin, Pmax]? The reason is simply because Pmin is the
lower bound of that interval and, if C(Q) is not superadditive at D(Pmin), we already know that
(Pmin, Pmin) cannot be a PBE where both firms are active. Hence, in that case, the set of PBE
must necessarily be empty. However, note that this is not necessarily true if superadditivity fails
at any other output lower than D(Pmin).
A possible extension of the results of proposition 1 is its application to a market with more
than two symmetric firms. Our conjecture is that proposition 1 holds, because of the following
argument. If F is an avoidable fixed cost, then the necessity part of the demonstration is easily
derived. If the total cost function (the same for everybody) is not superadditive at D(Pmin) then,
following Baumol (1977), there must exist a natural monopoly at that output level. Because
Πi(Pmin, …, Pmin) = 0 for all i, then monopoly’s profits at D(Pmin) must be strictly positive. Thus,
invoking continuity, at least one firm should find profitable to deviate from (Pmin, …, Pmin).
However, this contradicts the statement that the previous strategy profile is a PBE where all
firms are active.
What about the sufficiency part? Is it possible that the total cost function were
superadditive at D(Pmin) and, nevertheless, that there were no PBE where all firms supply a
positive output? With think that the strictly convexity of variable costs rules out this possibility.
To illustrate the reasoning, consider the case with three identical firms. Suppose, by
contradiction, there is no PBE where all firms are active. Since the market is not a natural
monopoly at D(Pmin), there must exist a Bertrand equilibrium where, say, firm 1 and 2 charge Pmin
and firm 3 is not in the market.
Now assume that monopoly profits at D(Pmin) are nonnegative. Then it is easy to check
that the situation described above requires that:
 D(Pmin ) 
 D(Pmin ) 
3 ⋅ VC
+F
 + 2 ⋅ F < VC(D(Pmin ) ) < 2 ⋅ VC
 3 
 2 
.
However, as figure 2 illustrates, this cannot be true under the assumptions that VC(Q) is
continuous, increasing and strictly convex, because all these three functions, 3·VC(Q)+2F,
2·VC(Q)+F and VC(Q), converge as Q raises. Hence, (Pmin, Pmin, Pmin) must be a PBE, and we
obtain the desired result.
Note that the argument above seems to carry over every time that the monopoly is
ranked, according with the profits that it generates, in between two market structures with
different number of active firms. For instance, in our previous example, it is ranked between the
case of 2 and 3 firms. A detailed proof of this intuition is, however, still in progress.
8
Figure 2
3VC (q ) + 2 F
2VC (q ) + F
3VC (DL / 3) + 2 F
VC (DL )
VC (q )
2F
2VC (DL / 2 ) + F
F
D L / 3 DL / 2
DL
5. Numerical example
To illustrate the results of the previous sections, consider the following example where
D(P) = 2/P and C(Qi) = Qi2 + F. If the PTE exists, then Pc = 1.4142 and Qc = 0.7071, but this
occurs only if F ≤ 1/2. If, conversely, F > 1/2, then no PTE exists, because charging Pi = 1.4142
and selling Qi = 0.7071 generates negative profits to the supplying firms. The interval [Pmax, Pmin],
however, is in this example determined by the values Pmax = 1.7321 and Pmin = 1/(1-F)1/2. If F ≤
2/3, this interval exists, and there is a range of multiple symmetric PBE.
For C(Q) to be superadditive, it must hold that Q2 ≥ Q2/2 + F or, equivalently, that Q ≥
1/2
(2·F) . When F ≤ 2/3, C(Q) is superadditive for Q ≥ 1.1547, and this condition assures that the
market is not a natural monopoly for the quantity demanded in the PBE allocations.
If F ≤ 1/2, the range of PBE includes the PTE outcome as a possible equilibrium price,
but if 1/2 < F ≤ 2/3, then there are multiple Bertrand equilibria, although no PTE exists. Finally, if
F > 2/3, then the market is a natural monopoly and no PBE exists, because C(Q) is subadditive
for Q = 1.1547 (which is the total quantity demanded at Pmax).
The average cost, marginal cost and demand functions that correspond to the numerical
example referred to in the previous paragraphs are graphically represented on figure 3. In it we
see the total demand curve that we have postulated (D) and the portion of that demand that
corresponds to each of the two firms that operate in the market (D/2). We have also depicted the
individual marginal cost curve (MC), and three alternative specifications of the average cost
curve (AC). These specifications correspond to cases where F is respectively equal to 1/5, 3/5
and 4/5.
9
Figure 3
AC (3 / 5)
MC
D
AC (1 / 5)
AC (4 / 5)
pL (4 / 5)
pH
pc
pL (1 / 5)
•
•
•
•
D/2
0
Q
The PTE of this market, when it exists, corresponds to a situation where the curves D/2
and MC cross. This is a feasible allocation when F = 1/5, because the curves AC(1/5) and D/2
cross at a break-even price (PL(1/5)) that is smaller than the price-taking equilibrium price (Pc).
When F = 3/5 or F = 4/5, conversely, the price-taking equilibrium is not feasible, because the
corresponding break-even prices are larger than Pc. However, the maximum Bertrand
equilibrium price (PH) is greater than PL(3/5), so in that situation the set of Bertrand equilibria is
not empty. When F = 4/5, conversely, the set of Bertrand equilibria is empty, since the breakeven price is larger than PH.
Figure 4
P
pm (F)
pL (F )
pH
pc
0
•
•
1/ 2
2/3
F
The relationships between PH, PL and Pc can also be observed in figure 4, where all
those concepts have been depicted as functions of F. We see that, if F < 1/2, then PL < Pc < PH.
For 1/2 < F < 2/3, conversely, PL > Pc, and therefore no PTE exists. However, for that range of
10
values of F it holds that PL < PH, and that implies multiple PBE for prices between PL and PH.
When F > 2/3, finally, it holds that PL > PH, and therefore the set of PBE is empty. The price for
which PL = PH, however, is the same price for which it holds that PL = Pm(F), where Pm(F) =
(2/F)1/2 is the minimum “natural monopoly price” (that is, the demand price that corresponds to
the maximum quantity for which the cost function is subadditive). For all the values of F < 2/3, it
holds that PL < Pm(F), and therefore the market is not a natural monopoly at Q(PL). When F >
2/3, conversely, it holds that PL > Pm(F), and this implies that the market is a natural monopoly at
Q(PL) and no PBE exists.
6. Final remarks
In this paper, we have analyzed the existence of a Bertrand equilibrium in pure strategies
in a symmetric and homogenous-product duopoly with increasing and strictly convex variable
costs and avoidable fixed costs, which implies that firms can, in the long run, produce zero
output at zero cost. In this regard, the paper constitutes an extension of Dastidar (1995), which
has studied the existence of a PBE in oligopolies where variable costs are increasing and strictly
convex, but fixed costs -if they exist- are unavoidable, so that the industry exhibits strictly
decreasing returns to scale.
Interestingly enough, while in Dastidar (1995) a price-taking equilibrium and purestrategy Bertrand equilibria can always be found10, in our model both may fail to exist. The
reason for this is that, in our case, returns to scale may be strictly increasing around the origin.
This result is linked to the one found by Telser (1991) for markets where avoidable fixed costs
generate non-convex total cost functions, that may create a situation of empty core and,
consequently, the non-existence of a PTE.
In proposition 1 we have proved that a necessary and sufficient condition to guarantee
pure-strategy Bertrand equilibria where both firms are active is that the total cost function is
superadditive at the output level corresponding to the duopoly break-even price. Additionally, we
have also showed in lemma 1 that the existence of a price-taking equilibrium is sufficient, but not
necessary, for such a PBE to exist, and that the market could have multiple Bertrand equilibria
even if a PTE does not exist. These results can easily be generalized to oligopolies with any
finite number of symmetric firms.
There are, of course, several extensions to look at, such as oligopolies where fixed costs
are partially avoidable and partially unavoidable, asymmetric variable costs, capacity constraints
(i.e., Bertrand-Edgeworth price competition), differentiated products, etc. As always, we hope to
deal with some of these problems in a future research.
Finally, it is worth to mention that what we find really appealing of the necessary and
sufficient condition found to guarantee pure-strategy Bertrand equilibria is that it is a property of
the total cost function that is very well known in economics. Moreover, it has received a lot of
attention in the literature on regulation of natural monopolies. Thus, we hope our findings in this
article may have not only some theoretical relevance, but also practical importance as well.
References
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Dastidar, Krishnendu (1995). “On the Existence of Pure Strategy Bertrand Equilibrium”;
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In fact, under Dastidar’s assumptions, there also typically are multiple mixed-strategy Bertrand equilibria. See
Hoernig (2002).
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