Theoretical Economics Letters, 2011, 1, 38-40
doi:10.4236/tel.2011.12009 Published Online August 2011 (http://www.scirp.org/journal/tel)
Copyright © 2011 SciRes. TEL
A General Cournot-Bertrand Model with
Carol Horton Tremblay1, Mark J. Tremblay2, Victor J. Tremblay1
1Department of Economics, Oregon State University, Corvallis, USA
2Department of Economics, Michigan State University, East Lansing, USA
Received May 17, 2011; revised July 18, 2011; accepted July 25, 2011
We analyze a Cournot-Bertrand model where one firm competes in output and the other competes in price.
With general demand functions and perfectly homogeneous products, we show that the unique Nash equilib-
rium is the perfectly competitive equilibrium. Equilibrium price equals marginal cost, the Cournot-type firm
produces the perfectly competitive level of market output, and the Bertrand-type firm exits the market. Even
with just one firm in the market, the presence of a potential Bertrand-type competitor provides sufficient dis-
cipline to guarantee a competitive outcome.
Keywords: Cournot-Bertrand Model, Product Differentiation
There has been increasing interest in the static Cour-
not-Bertrand model in which one firm competes in out-
put (a la Cournot) and the other competes in price (a la
Bertrand). When two firms have the choice to compete in
output or in price, Singh and Vives  show that under
certain demand and cost conditions the dominant strategy
is for each firm to compete in output rather than price
(i.e., Cournot dominates Bertrand behavior and Cour-
not-Bertrand behavior). More recently, Tremblay et al.
 show how different institutional and technological
conditions can change firm payoffs so that Bertrand be-
havior or Cournot-Bertrand behavior becomes optimal.1
Given that there are many theoretical possibilities,
Kreps and Scheinkman  argue that whether firms
compete in output or in price is ultimately an empirical
question. In the real world, Cournot, Bertrand, and
Cournot-Bertrand behavior are observed. Vegetable
producers set quantities at local farmers’ markets, while
restaurants set prices. In the market for small cars, Saturn
and Scion dealers set prices and Honda and Subaru deal-
ers set quantities.2 Given this observation, additional
research on the Cournot-Bertrand model is warranted and
may further our understanding of oligopoly markets.
Tremblay and Tremblay  investigate the Cour-
not-Bertrand model when the degree of product differen-
tiation is allowed to vary. An interesting result emerges
from their work: when products are perfectly homoge-
neous, the perfectly competitive outcome results in
which the Cournot-type firm produces the competitive
level of market output and the Bertrand-type firm exits
the market. The mere threat of a Bertrand-type firm en-
sures the perfectly competitive outcome, demonstrating
that a potential competitor can have a dramatic effect on
market power. The main weakness with their work is that
demand functions are assumed to be linear. In this paper,
we show that their result holds with a general demand
2. The Model
Two firms, 1 and 2, compete in a static Cournot-Bertrand
game where firm 1 is the Cournot-type firm that com-
petes in output and firm 2 is the Bertrand-type firm that
competes in price. Firm i’s output level is qi and its price
is pi i = 1, 2. The goal of each firm is to maximize its
profit (πi). Information is complete.
*We would like to thank an anonymous referee for providing helpful
comments on an earlier version of the paper.
1That is, Tremblay et al.  show that when firms are given the choice
of competing in output or in price, cost asymmetries can lead to a Nash
equilibrium where one firm competes in output and the other competes
2See Palmeri  and Tremblay et al.  for further discussion of why
some small car dealers compete in price and others compete in output.
3Restaurants provide one example, where an Italian restaurant is darkly
lit and serves spaghetti, while a Chinese restaurant is brightly decorated
and serves stir-fried dishes.
C. H. TREMBLAY ET AL.39
Products are substitutes, and products may vary over a
variety of characteristics.3 Product differentiation of this
sort can be incorporated into a linear demand system, as
found in Dixit , Singh and Vives , and Beath and
Katsoulacos . The inverse demand function for firm i
, where j is firm i’s rival, a is a posi-
tive constant, and d is an index of product differentiation,
[0, 1]. Products 1 and 2 are perfectly homogeneous
when d = 1, and each firm is a monopolist when d = 0.
Thus, product differentiation diminishes as d → 1. In the
Cournot-Bertrand model, this system must be solved so
that demand is a function of the strategic variables, q1
and , where
α ≡ a – ad and b ≡ 1 – d2. Firms face the same linear cost
function, where c
(0, a) is defined as average and mar-
ginal cost. The profit equation for firm i is πi = (pi – c) qi.
Tremblay and Tremblay  investigate this model. In
Proposition 1, they show that the model has a stable
Nash equilibrium (NE) when there is sufficient product
differentiation (d is sufficiently close to 0). When d is
sufficiently close to 1, the NE becomes unstable. Once d
= 1, however, their Proposition 2 demonstrates that the
equilibrium becomes stable once again. In this case of
perfectly homogeneous goods, the equilibrium price
equals marginal cost, only firm 1 survives (firm 2 pro-
duces no output), and firm 1 produces the perfectly
competitive level of output (Qpc). Like a contestable
market (Baumol et al. ), this demonstrates how im-
portant a potential entrant can be to the level of price
The goal of this paper is to prove that the conclusion
in Proposition 2 is not conditional on the assumption that
demand functions are linear. Here, we consider a general
demand system: p1 = p1(q1, p2) and q2 = q2(q1, p2). Each
demand function is differentiable and has a negative
slope (∂p1/∂q1 < 0 and ∂q2/∂p2 < 0), and products are
substitutes (∂p1/∂p2 > 0 and ∂q2/∂q1 < 0).4 For notational
convenience, the demand price is defined as p(q1’) when
products are perfect substitutes, q = q1’ and q2 = 0.5 Un-
der these conditions, the following proposition holds.
Proposition: In this duopoly market with perfectly
homogeneous goods, there is a unique NE in which the
equilibrium price equals marginal cost, firm 1 produces
the perfectly competitive level of market output, and firm
2 produces zero output.
Proof: We investigate each of the possible strategy
1) First, we consider strategy profiles in which
a) For firm 1:
q1 > Qpc cannot be a NE strategy. At this level of out-
put and given a negatively sloped demand function, p(q1
> Qpc) < c and firm 1 earns negative profits. In this case,
firm 1 can earn zero profit by exiting the industry.
q1 < Qpc cannot be a NE strategy. If this is all that is
produced (q2 = 0), then p(q1 < Qpc) > c (given a nega-
tively sloped demand function). In this case, firm 2’s best
reply is to set p2 = p(q1 < Qpc) – ε > c for ε > 0. This en-
ables firm 2 to produce a positive level of output and for
both firms to earn a positive profit. Given p2 > c, how-
ever, firm 1 can earn greater profit by increasing its pro-
duction so that it is supplying all that is demanded at p2.
This leaves no residual demand for firm 2 (i.e., q2 = 0).
Thus, firm 2 has an incentive to lower p2 even further.
This process of lowering p2 and raising q1 will continue
until q1 = Qpc and p2 = c.
b) For firm 2:
p2 < c cannot be a NE strategy. At this price, firm 2
earns a negative profit and can earn zero profit by exiting
p2 > c cannot be a NE strategy. As demonstrated above,
firm 1’s best reply to p2 > c is to produce all that is
demanded at p2, such that q1 < Qpc. This in turn makes it
profitable for firm 2 to charge a lower price than p2. The
process of lowering p2 and raising q1 will continue until
q1 = Qpc and p2 = c.
2) Next, we consider the strategy profile q1 = Qpc and
p2 = c.6 This is a NE because a small deviation cannot
increase the profit of either firm. As shown above, firm 1
cannot increase its profit by increasing or decreasing its
output from Qpc, and firm 2 cannot increase its profit by
increasing or decreasing its price from c. These are the
only alternative strategy profiles. Thus, q1 = Qpc and p2 =
c is the only NE. Q.E.D.
This result demonstrates the dramatic effect that a po-
tential competitor can have on a market. In the Cournot-
Bertrand model, the threat of a price competitor that
produces a homogeneous good ensures that a monopolist
will behave as a perfectly competitive firm. In this case,
the potential entrant completely eliminates market power.
The Cournot-Bertrand model has several interesting
qualities and is receiving renewed interest in the litera-
ture. Previous theoretical studies show that technological
and institutional forces can make it profitable for firms
within the same industry to choose different strategic
variables. In addition, there is evidence that some firms
compete in output and others compete in price in the U.S.
market for small cars.
4The only restriction is that for the second-order conditions of profit
maximization to hold, demand functions cannot be too convex. That is,
the second derivative of each demand function with respect to its own
rice must be sufficiently small.
5Note that because products are perfect substitutes, p1 will equal p2in
6Because products are perfectly homogeneous, p2 = p1.
Copyright © 2011 SciRes. TEL
C. H. TREMBLAY ET AL.
Copyright © 2011 SciRes. TEL
In a model with general demand functions, we show
that the unique NE in the Cournot-Bertand model with
homogeneous goods is the perfectly competitive equilib-
rium. The equilibrium price equals marginal cost, the
Cournot-type firm produces the perfectly competitive
level of market output, and the Bertrand-type firm exits
the market. Even with just one firm serving the market,
the presence of a potential Bertrand-type competitor pro-
vides sufficient discipline to guarantee a competitive
 N. Singh and X. Vives, “Price and Quantity Competition
in a Differentiated Duopoly,” Rand Journal of Economics,
Vol. 15, No. 4, 1984, pp. 546-554.
 Tremblay, J. Victor, C. H. Tremblay and K. Isariya-
wongse, “Endogenous Timing and Strategic Choice: The
Cournot-Bertrand Model,” Bulletin of Economic Re-
 D. M. Kreps and J. A. Scheinkman, “Quantity Precom-
mitment and Bertrand Competition Yield Cournot Out-
comes,” The Bell Journal of Economics, Vol. 14, No. 2,
1983, pp. 326-337. doi:10.2307/3003636
 C. Palmeri, B. Elgin and K. Kerwin, “Toyota’s Scion:
Dude, Here’s Your Car,” Business Week, June 9, 2003.
 V. J. Tremblay, C. H. Tremblay and K. Isariyawongse,
“Cournot and Bertrand Competition when Advertising
Rotates Demand: The Case of Honda and Scion,” Work-
ing Paper, Oregon State University, 2010.
 C. H. Tremblay and V. J. Tremblay, “The Cournot-Bertrand
Model and the Degree of Product Differentiation,” Eco-
nomics Letters, Vol. 111, No. 3, 2011, pp. 233-235.
 A. Dixit, “A Model of Duopoly Suggesting a Theory of
Entry,” The Bell Journal of Economics, Vol. 10, No. 1,
1979, pp. 20-32. doi:10.2307/3003317
 J. Beath and Y. Katsoulacos, “The Economic Theory of
Product Differentiation,” Cambridge University Press,
Cambridge, 1991. doi:10.1017/CBO9780511720666
 W. J. Baumol, J. C. Panzar and W. D. Willig, “Cont-
estable Markets and the Theory of Industry Structure,”
Harcourt Brace Javonovich, New York, 1982.