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 [7], Singh and Vives [1], and Beath and

Katsoulacos [8]. The inverse demand function for firm i

is

qq

ii

ad

1

j

, where j is firm i’s rival, a is a posi-

tive constant, and d is an index of product differentiation,

d

[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 p2:12

bq pd

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.

22

qpad 1

q

Tremblay and Tremblay [6] 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. [9]), this demonstrates how im-

portant a potential entrant can be to the level of price

competition.

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

profiles.

1) First, we consider strategy profiles in which

1

qQ

c

and .

2

a) For firm 1:

pc

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

the industry.

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.

3. Conclusions

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

equilibrium.

6Because products are perfectly homogeneous, p2 = p1.

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