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Comparing Bertrand and Cournot Equilibria: A Geometric Approach
Author(s): Leonard Cheng
Reviewed work(s):
Source: The RAND Journal of Economics, Vol. 16, No. 1 (Spring, 1985), pp. 146-152
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Rand Journal of Economics
Vol. 16, No. 1, Spring 1985
ComparingBertrandand Cournot equilibria:
a geometric approach
Leonard Cheng*
By presenting a geometric analysis of a duopoly, this article attempts to provide some
insights into recent results obtained by Singh and Vives regarding Bertrand and Cournot
equilibria and the relative dominance of price and quantity strategies. It shows that under
fairly general and reasonable assumptions (a) Cournot equilibrium prices (quantities) are
higher than Bertrand equilibrium prices (quantities) and (b) a quantity (price) strategy
dominates a price (quantity) strategy if the goods are substitutes (complements).
1. Introduction
* In a recent article Singh and Vives (1984) have shown in the case of a duopoly that
Bertrand (price strategy) equilibrium is more efficient than Cournot (quantity strategy)
equilibrium, in the sense that the consumer surplus and total surplus are higher at the
former equilibrium than at the latter, regardless of whether the goods are substitutes or
complements. An important underlying reason is that prices are lower and quantities
greater at the Bertrand equilibrium.' They have also shown that in a symmetric duopoly
a firm will choose a quantity strategy over a price strategy if the goods are substitutes,
but it will choose a price strategy over a quantity strategy if the goods are complements.
This article attempts to provide some insights into these results by presenting a geometric
analysis of the duopoly problem. In particular, we shall show that under fairly general
and reasonable assumptions (a) Cournot equilibrium prices (quantities) are higher than
Bertrand equilibrium prices (quantities) and (b) a quantity (price) strategy dominates a
price (quantity) strategy if the goods are substitutes (complements).
The geometric approach adopted in this article has several advantages. First, unlike
in some of the related literature, we do not have to impose special functional forms on
the demand structure. Second, being geometric in nature, the analysis can be visualized
more easily. Specifically, we can identify the region in the price space in which a Cournot
* University of Florida.
I would like to thank Roger Blair, Thomas Cooper, Stephen Cosslett, Avinash Dixit, James Friedman,
Edward Golding, Tom Lee, Lionel McKenzie, Koji Okuguchi, Richard Romano, Martin Shubik, Xavier Vives,
and Edward Zabel for their valuable comments on a related paper from which this article draws. I am grateful
to Stephen Salant and to a referee for pointing out several mistakes and making some helpful suggestions.
'There is a literature on the comparison of Bertrand and Cournot equilibrium prices when firms produce
differentiated products. Some authors rely more heavily on numerical examples (Shubik, 1959, 1968, 1980),
while others use more analytical methods (Bramness, 1979; Cheng, 1984; Hathaway and Rickard, 1979;
Okuguchi, 1984; Singh and Vives, 1984; Vives, 1984). The general conclusion, however, is that Bertrand
equilibrium prices tend to be lower than Cournot equilibrium prices.
146
CHENG /
147
equilibrium must lie. Finally, our approach reveals in a natural way a fundamental
relationship between the two equilibria: for substitutes a Cournot equilibrium is equivalent
to a Bertrand equilibrium with positive conjectural variations.2Thus, a firm with Bertrand
conjectures about its rival will have an incentive to supply more than what it would with
Cournot conjectures. Consequently, Bertrand equilibrium tends to yield larger quantities
and lower prices than Cournot equilibrium.
In this article we shall focus on a duopoly, as in Singh and Vives (1984), although
the geometric analysis can be generalized to the case of any finite number of firms
(Cheng, 1984). Moreover, we shall deal only with the case of substitutes. By virtue of the
"duality" relationship between substitutes and complements discovered by Sonnenschein
(1968) and used by Singh and Vives (1984), the same results apply to complements,
provided that the roles of prices and quantities are interchanged.
2. Demand
specification
* Let the demand functions facing the duopoly be given by qi = Fi(pl, P2), i = 1, 2.
We make the following standard assumptions regarding differentiated products or gross
substitutes:3
Fi is twice continuously differentiable with bounded derivatives;
(1a)
aFi/api-< 0 with strict inequality if qi > 0;
(1b)
aFi/apj>_ (j = 1, 2, j * i), with strict inequality if both ql and q2 are positive;
(aFI lap1)(aF2/4lp2)> (aFIlap2)(aF2/4lp1).
(1c)
(id)
Firm i's profits are given by
iri(p) = (pi
-
ci)Fi(p),
(2)
where ci is firm i's constant marginal cost,4 and p = (PI, P2).
We assume further that
<0 with strict inequality if qi >
a%-1iadp2
d
d
0
0;
(3a)
(j # i) with strict inequality if q1 > 0 and q2> 0;
(ar/ap~l)(a-72/ap)2>
if ql
(a%-1/ap9ap2)(a%22/ap2apI)
>
0 and q2
> 0.
(3b)
(3c)
Assumptions (3a) and (3b) say that a firm's marginal profit with respect to its own
price (d-ri1/pi)goes down with its own price, but goes up with its rival's price. Assumption
(3c) says that the own effects dominate the cross effects. These conditions are satisfied if
the demand functions are linear, and they all seem reasonable for gross substitutes.
Firm i's Bertrand reaction function is given by Ri = {pip 0_ 0. ariapi = 0}. From
the first-order condition of profit maximization and (lb), we see that the optimal price
pi* exceeds ci as long as qi > 0. Assumptions (3a) and (3b) imply that pi* is a nondecreasing
function of pj, j * i. Assumption (3c) implies that RI is steeper than R2 in the (PI, P2)space. Thus a Bertrand equilibrium exists and is unique. (See Figure 1, where B denotes
the Bertrand equilibrium.)
2 This equivalence relationship has been discussed in Bramness (1979) and more recently in Kamien and
Schwartz (1983).
3 Similar assumptions are made, for instance, in Friedman (1977, pp. 50-55).
4 Constant marginal costs are assumed to avoid discontinuity of the firms' reaction functions (through the
violation of (3a) below) and thus the nonexistence of a Bertrand equilibrium. Notice, however, that for
differentiated products, constant marginal costs are not strictly necessary for continuous reaction functions or
for the existence of a Bertrand equilibrium. For an example, see Bramness (1979).
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THE RAND JOURNAL OF ECONOMICS
FIGURE 1
BERTRAND AND COURNOT EQUILIBRIA
P
P
3. Characterization of a Cournot equilibrium in the price space
* Let lli(b) be firm i's isoprofit curve in the price space yielding profits equal to
b, ie., lli(b) = {PIP_ 0; 7rj(P) = b}. Because iri(p) is continuously differentiable, Hl
is smooth and differentiable..The slope of this curve, dp1/dPi|I11 (j 7&i), is given by
-(diri/dpi)/(diri/dp1). By definition (diri/dp>) = (Pi - ci)(dFi/dp1), which is positive
(negative) if Pi > (<) c1. But from the first-order condition of profit maximization we
know that Pi > c, as long as qi > 0. Therefore, we need to focus only on the region in
the price space in which dwri/dpy
> 0. From (3a) we know that there exists a unique Pi*
for each value of pi such that d~r/dpi 2 0 if and only if Pi 5 Pi*. This implies that a
typical isoprofit curve ll1(p) in the relevant region of the (Pi, p>)-space is first declining
and then rising. Moreover, because diri/dp1> 0 for j #&i when neither firm prices itself
out of the market, an isoprofit curve Hi further away from the pi-axis yields higher profits.
Define Qi(qi) as firm i's isoquantity curve in the price space which yields the same
quantity qi, i.e., Qi(qi) = {plp > 0; F.(p) = qj}. Since dpj/dPilQ,= -(dFi/dpi)/(dFi/dp1),
we know from (ib) and (1c) that a typical isoquantity curve is upward sloping in the
price space. This reflects the fact that when a firm sets its quantity, the price it can get
for its product depends positively on its rival's price. Assumption (1c) also implies that
the closer is Qj~q1)to the pr-axis, the larger is the value of qi. Finally, assumption (id)
implies that Qi is steeper than Q2 in the (Pi, P2)-space. (See Figure 2.)
Using the isoprofit and isoquantity curves defined above, we can now characterize a
Cournot equilibrium in the price space. Given the value of qj, the maximum 7riis given
by firm i's isoprofit curve that is as far away from the pi-axis as possible and still has at
least one point in common with firm j's isoquantity curve Qj(qj).Point C in Figure 2 is
a Cournot equilibrium because qc7is firm l's best reply in quantity to q~jand q~jis firm
2's best reply in quantity to q?. The Cournot equilibrium prices are given by p? and psi.
CHENG
/
149
FIGURE 2
COURNOT EQUILIBRIUM IN THE PRICE SPACE
/C)
P2
<4
'2
IT
2~~~~~~~~~~~~~~~
0
CTpi
4. A comparison of Bertrand and Cournot equilibria
We have demonstrated in Section 2 that a unique Bertrand equilibrium exists. To
make our comparison meaningful, we assume directly that a Cournot equilibrium exists.
As we shall see shortly, the uniqueness of a Cournot equilibrium is not essential to the
result that Bertrand equilibrium prices are lower than Cournot equilibrium prices.
From the properties of the isoprofit curves, firm 1's reaction function (R1) divides
the (PI, p2)-space into two parts. To its right R1I has positive slopes and to its left R1Ihas
negative slopes. Similarly, firm 2's reaction function (R2) divides the same space into two
parts. Above it 112 has positive slopes and below it 112 has negative slopes. But we have
already seen in Figure 2 that at a Cournot equilibrium both R1I and 112 have positive
slopes. That is to say, a Cournot equilibrium must lie above R2 and to the right of R1.
Also, if a Cournot equilibrium is interior (i.e., qC > 0 and qc > 0), it must lie below
Q2(0) and to the left of Q1(O). In Figure 1 a Cournot equilibrium can only be found in
the interior of the dotted region NBYM (such as C1, C2, and C3). Regardless of the
number of Cournot equilibria and their exact location, it is obvious that pf < pc and
U
pA < pc
for every Cournot
equilibrium.
What can we say about the firms' profits at a Cournot equilibrium compared with
those at the Bertrand equilibrium? In Figure 1 the firms' profits at the Bertrand
equilibrium are given by their isoprofit curves R1B and 11k, respectively. It is obvious that
at least one of the firms must achieve larger profits at the Cournot equilibrium, and in
the hatched area (the intersection of the area above B11 and the area to the right of 1kB)
both firms achieve larger profits. Thus, provided that the demand structure is not
excessively asymmetric, the profits of both firms are larger at a Cournot equilibrium (e.g.,
at C1 or C2). On the other hand, it seems that if demand is sufficiently asymmetric, then
the possibility that one of the firms may achieve lower profits at a Cournot equilibrium
(e.g., C3) cannot be eliminated.5
5 I am indebted to the referee for pointing
out this possibility to me.
150
/
THE RAND JOURNAL OF ECONOMICS
In the case of differentiatedproducts, the formal equivalence of a Cournot equilibrium
with no conjectural variations and a Bertrand equilibrium with positive conjectural
variations has been pointed out by Bramness (1979), and more recently by Kamien and
Schwartz (1983). Figure 2 serves to bring out this relationship graphically.
5. Price strategies
versus
quantity strategies
* Using the simple device of isoquantity curves, we can also address the issue of strategy
dominance in a framework slightly more general than that of Singh and Vives (1984).'
Below we shall demonstrate their result that a quantity (price) strategy dominates a price
(quantity) strategy if the goods are substitutes (complements).
Before we proceed, let us take note of the fact that a firm, by choosing to set its
price, induces its rival to have Bertrand conjectures. Similarly, a firm, by choosing to set
its quantity, induces its rival to have Cournot conjectures. That is to say, a firm's choice
of a price or quantity strategy does not change its own reaction function, but that of
its rival.
By definition, firm i's Bertrand reaction function, Ri, is derived on the assumption
that firm i (j # i) chooses a price strategy. Using firm j's isoquantity curves, we may
derive firm i's Cournot reaction function in the price space. In Figure 3 these functions
are denoted by R' and R', the broken lines. From the properties of isoquantity curves
and isoprofit curves, it follows that R' lies to the right of RI and that R'2 lies above R2.
By definition the intersection of R' and R', denoted by C, yields a Cournot equilibrium.
The intersections of R'1 and R2, A, A', and A", are the Nash equilibria with firm l's
choosing a price strategy and firm 2's choosing a quantity strategy. The intersections of
R1 and R', D, D', and D", are the Nash equilibria with firm l's choosing a quantity
strategy and firm 2's choosing a price strategy.
FIGURE 3
PRICE STRATEGY VERSUS QUANTITY STRATEGY
P22i
\R
0-
~~~~~~~~~~~~~~~~
6 Specifically, our demand functions are not necessarily symmetric (i.e., we permit
(OF11/P2)
and we allow for multiple Cournot as well as price-quantity Nash equilibria.
)),
=A (OF2/0p1
CHENG
/
151
Now consider the strategy choice of firm 1. First suppose firm 2 chooses a price
strategy. As a result, firm 1's reaction function is given by RI. If firm 1 also chooses a
price strategy, firm 2's reaction function is given by R2, and the resultant equilibrium is
given by B, the Bertrand equilibrium. If firm 1 chooses the quantity strategy instead, firm
2's reaction function is given by R', and the resultant equilibria are given by D, D', and
D" (and possibly others). From the properties of the isoprofit curves, we know that as we
move up RI, firm 1's profits increase. Therefore, firm 1's profits at D, D', and D" must
be higher than those at B. Thus, from firm l's point of view, a quantity strategy is more
profitable than a price strategy if firm 2 chooses a price strategy.
What if firm 2 chooses a quantity strategy instead? Firm l's reaction function is then
given by R'. If firm 1 chooses a price strategy, firm 2's reaction function is given by R2,
and the resultant equilibria are A, A', and A" (and possibly others). If firm 1 chooses a
quantity strategy instead, firm 2's reaction function is given by R', and the resultant
equilibrium is given by the Cournot equilibrium, C. Again, from the properties of the
isoprofit curves, we know that as we move down R' (toward the pi axis), firm l's profits
fall. But we already know that R'2 lies above R2. Therefore, as we move down R'1, it
must first intersect R'2 every time it intersects R2. That is to say, for each price-quantity
strategies Nash equilibrium, there is always a Cournot equilibrium which gives firm 1
higher profits. More strongly, if the Cournot equilibrium is unique, as is C in Figure 3,
then firm l's profits are higher at the Cournot equilibrium than at any one of the pricequantity strategies Nash equilibria (A, A', or A"). In this latter case, a quantity strategy is
unambiguously more profitable than a price strategy when firm 2 chooses a quantity
strategy. Therefore, from the point of view of firm 1, a quantity strategy dominates a
price strategy. By symmetry, from the point of view of firm 2, a quantity strategy also
dominates a price strategy.
6. Concluding
remarks
* As in Singh and Vives (1984), we have attempted to answer two related questions:
(1) If a firm can choose either a price or quantity strategy, which one will it adopt?
(2) Are Bertrand equilibrium prices lower than Cournot equilibrium prices? These
questions are interesting because the price and quantity strategies are most widely used
in oligopoly models. When additional strategies are considered, one would like to raise
similar questions as well. It seems that our approach can be generalized to the case of
several alternative strategies.
With more than two strategies, it is conceivable that none will dominate the others.
Nevertheless, the question as to which strategy will be adopted remains interesting. In
this article, under the special assumption of constant marginal cost, a firm's strategy
choice depends only on how its alternative strategies affect its own equilibrium profits by
affecting the other firm's conjecture about itself. In the general case one would expect
that the shape of the firm's marginal cost curve could also play an important role.
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/
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