International Journal of Industrial Organization
24 (2006) 785 – 793
www.elsevier.com/locate/econbase
Information disadvantage in linear Cournot duopolies
with differentiated products
Adi Chokler, Shlomit Hon-Snir, Moshe Kim *, Benyamin Shitovitz
Department of Economics, University of Haifa, Haifa 31905, Israel
Received 20 September 2004; received in revised form 30 May 2005; accepted 19 September 2005
Available online 5 December 2005
Abstract
We focus on a class of linear Cournot duopolies with differentiated products and prove that whether
there is an information advantage or disadvantage depends on firms’ information setup. Specifically, we
show that when the cross-effects are common value, the uninformed firm that commits to quantity will not
have lower ex ante profits than a firm that has complete information about its cross-effects. This result
contrasts to the information advantage that holds in the same duopolies with independent cross-effects.
D 2005 Elsevier B.V. All rights reserved.
JEL classification: C72; D43; D82
Keywords: Cournot duopolies; Cross-effects; Information disadvantage
1. Introduction
There is an extensive literature on the role of uncertain linear demand on firms’ profits. Most
of the research compares profits at different levels of uncertainty and answers the following main
questions: (i) How does more information for one or both firms change firms’ profits (see, for
example, Sakai, 1985; Vives, 1990); (ii) Is there any incentive to share information (see, for
example, Fried, 1984; Gal-Or, 1985; Raith, 1996).
* Corresponding author.
E-mail addresses: [email protected] (S. Hon-Snir), [email protected] (M. Kim),
[email protected] (B. Shitovitz).
0167-7187/$ - see front matter D 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.ijindorg.2005.09.009
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A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
The main question posed in this paper is whether the firm with more information or the firm
with less information earns more profits between two symmetric firms. In other words, is more
information reflected in higher profits? This situation is common in markets which are supplied
by both local and foreign firms. To answer this, we compare the equilibrium profits of two
different firms in the same game, which we call information advantage, as opposed to
comparing the equilibrium profits of one firm in two different games, which is termed
information value. That is, we compare the equilibrium profits of two firms, which ex ante have
the same demand and cost functions, and they differ by information level. More precisely, we
compare the profits of two firms in a Cournot game with differentiated products in which firm 1
(the informed firm) has full information about its demand function and firm 2 (the uninformed
firm) has no information about its demand function.
Einy et al. (2002) show that in a (one-stage) Cournot oligopoly with linear cost and uncertain
demand, the informed firm obtains higher profits. In their model, both firms have the same
demand function in each state of nature. Furthermore, they present an example for information
disadvantage in a Cournot duopoly with quadratic costs.
Gal-Or (1988) analyzed a two-stage game with a Cournot duopoly game in each stage. The
linear cost function has an unknown constant marginal cost and it is the same for both stages. In
the first stage, each firm chooses the level of production and receives a private signal about the
unknown cost. The level of production is higher when the signal is more accurate. In the first
stage, the less informed firm expands its production in order to learn more about the cost, and
causes its rival to reduce its production level. In the second stage, the firms choose the same
level of production. Therefore, the firm with less precise prior information earns more than the
rival firm. Moreover, the information disadvantage is a result of the information disclosure
process. Gal-Or (1987) analyzed the case of information advantage in a Stackelberg duopoly and
showed that the uninformed firm can earn higher profit. We discuss the information advantage in
a one-stage Cournot game.
We model Cournot competition with differentiated products between two firms, where the
uncertainty parameter of the linear demand function is the cross-effect. The uncertain crosseffect in a differentiated product setting means that the firm does not know the effect of a
change in a competitor’s price on its own demand. This is so because the uncertain crosseffect is reflected both in prices and product characteristics of the competitor’s product. A
straightforward example is the effect of firm entry on the incumbent’s demand, and vice versa.
Another reason the cross-effects may not exhibit symmetry is that different firms may exhibit
different characteristics; e.g., the firms’ clients may only partially overlap. Additionally, since
the cross-effects can be different and each firm collects its own information regarding the state
of nature, it is very likely that these firms are asymmetrically informed. To the best of our
knowledge, this is the first time that the cross-effect is modelled as the demand uncertainty
parameter. Most of the literature models uncertainty in the linear demand as an uncertain
intercept (a notable exception is Malueg and Tsutsui, 1996, who model a linear demand with
uncertain slope but analyze the incentive to share information). The uncertain cross-effect
implies that each firm affects the other firm’s demand, but the exact impact is a random
variable.
Here we prove a result contrary to that of Einy et al. (2002), namely, in Cournot
duopolies with differentiated products and linear demand (and cost) functions, the
uninformed firm earns higher level of profits, if both firms have symmetric demand
functions. In addition, we ask what might happen if the demand for each of the firms were
the same ex ante, but not ex post.
A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
787
Therefore we model two cases:
1. The common value case, in which each firm has the same effect on the other firm’s demand.
2. The independent private value case, in which the effect of each firm on the other firm’s
demand is the same ex ante, but the two variables are independently drawn.
As mentioned above, in the common value model there is information disadvantage and
the opposite holds for the independent private case. The key reason for information
disadvantage in the common value case is that lack of information allows the uninformed
firm to make a credible commitment to a higher level of output and, therefore, force the
informed firm to produce less. Since both firms face the same demand function, the
aggressive firm (the uninformed one) achieves higher profits compared to the informed firm.
By contrast, in the independent private value case, the uninformed firm commits to a less
aggressive output level; hence, it does not hurt the informed firm. More precisely, when the
correlation between the cross-effects decreases (from common value to independent private
value) the equilibrium expected demand of the uninformed firm decreases, and therefore, it
chooses lower output level and consequently obtains lower level of profits compared to the
informed firm’s profits.1
The rest of the paper is organized as follows: In Section 2 we introduce the model; Section 3
proves the existence and uniqueness of interior equilibrium; and Section 4 exhibits the
information advantage and disadvantage for the independent private value and common value
correspondingly.
2. The model
We consider two firms with linear costs which compete in quantities (Differentiated
Cournot model). The firms are ex ante symmetric, with equal constant marginal cost, c,
where the demand for each firm’s product is derived from the same (ex ante) linear demand.
The demand for each firm is uncertain since the cross-effect for each firm depends on the
state of nature. More precisely, the inverse demand (or price) for differentiated product i is
given by
pi qi ; qj ¼ A Bqi bi qj ; 1 V i p j V 2
ð2:1Þ
where q i is firm i quantity and b i is a random variable, depending on the state of nature. b i
measures the cross-effect, the change in firm i’s demand caused by a change in firm j’s
action. Therefore, for b i N 0(b i b 0) products are substitutes (complements).
In addition, we assume that
1. A N c z 0 and B N 0.
2. The own-price effect is greater than the cross-effect, that is: B V b i V B.
Assumption (2) implies that for substitute products, if the firm increases its output by one unit
and the other firm decreases its output by one unit then the price for the product falls. The
implication for the case of complement products is that, if both firms increase their outputs by
one unit, prices fall.
1
See the example in the next section.
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A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
The uncertainty with respect to the state of nature is described by a probability measure space
ðX; F ; l; Þ, where X is the set of states of nature, F is the set of events and l is a probability
measure. We assume measurability of b i :XY[0,B] for i = 1,2 substitute products and
b i :X Y [B,0] for i = 1,2 complement products.2
We consider two different information set-ups:
1. Independent private information where b 1(d ) and b 2(d ) are independent random variables but
have the same expected value.
2. Common value model in which b 1(d ) = b 2(d ).
The firms are asymmetrically informed about the realized state of nature. Firm 1 has full
knowledge about the true state of nature and, therefore, can choose any measurable nonnegative quantity q1 : X YRþ. We assume that q 1 has a bounded second moment. This
choice of q 1 captures firm 1 ability to choose its quantity contingent on the state of nature.
In contrast, the uninformed firm 2 has no information. Given ignorance regarding the true
state of nature, firm 2 can commit to a fixed nonnegative quantity q2 aRþ in all states of
nature. We shall consider here the Bayesian equilibria strategies based on expected profits.
Since the firms have constant marginal cost, c, their expected profits are given, respectively,
by
Ep1 ðd Þ ¼ E ½ð A c Bq1 ðd Þ b1 ðd Þq2 Þq1 ðd Þ
ð2:2Þ
Ep2 ðd Þ ¼ E ½ð A c Bq2 b2 ðd Þq1 ðd ÞÞq2 ð2:3Þ
Definition. The pair of strategies ( q*1 (d ),q*2 ) is an interior Bayesian equilibrium if
1. q*1 (d ) H 0 and q*2 N 0.
2. For firm 1: q*1 (x) is a best response to firm 2’s action q*2 , that is, for l-almost each xaX
ð A c Bq4
q 1 b1 ðxÞq42 Þq̃q 1 Þ for all q̃q 1 aRþ :
1 ðxÞ b1 ðxÞq4
2 Þq4
1 ðxÞzð A c Bq̃
*
3. For firm 2: q 2 is a best response to firm 1’s action q*1 (d ); that is, E½ð A c Bq42 b2 ðd Þq14ðd ÞÞq4
q 2 b2 ðd Þq4
q 2 for all q̃q 2 aRþ :
2 zE½ð A c Bq̃
1 ðd ÞÞq̃
The following example illustrates the effect of the main elements of symmetry, correlation
and product relationship on the results. In the example, the price for each product is given by
ð2:4Þ
pi qi ; qj ¼ 1 qi bi qj ; 1 V i p jV2:
Moreover, c = 0, and X={x 1,x 2,x 3,x 4}, where
State (x)
b 1(x)
b 1(x)
The probability l(x)
x1
x2
x3
x4
0
0
1
1
0
1
0
1
0.5 x
x
x
0.5 x
2
This assumption is necessary for an interior Bayesian equilibrium.
A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
789
Note that Eb i (d ) = 0.5, E(b i (d )b j (d ) )= 0.5 x and q(b 1,b 2) = 1 4x where q is the correlation
coefficient. For x = 0 (q(b 1,b 2) = 1) we obtain a common value information set-up, and for
x = 0.25 (q(b 1,b 2) = 0) we obtain an independent private value model. Recall that the two firms
are ex ante symmetric, but only for the common value model (x = 0), are the demand functions
faced by both firms the same for each state of nature. Equilibrium outputs are3
q4
1 ðx1 Þ ¼ q4
1 ðx 2 Þ ¼
1
2
q14ðx3 Þ ¼ q14ðx4 Þ ¼
4 þ 2x
2ð7 þ 2xÞ
q24 ¼
3
:
7 þ 2x
In addition, the expected profits of each firm, in equilibrium, are given by
Ep14ðd Þ ¼
Ep4
2 ðd Þ ¼
8x2 þ 44x þ 65
8ð7 þ 2xÞ2
9
ð7 þ 2xÞ2
:
Therefore, for x = 0 the uninformed firm (firm 2) earns more and for x = 0.25 the informed
firm (firm 1) earns more. Note that as the correlation between the cross-effects increases (x
decreases), q*2 increases, and therefore, the demand of firm 1 decreases, and hence, expected
profits decline. Moreover, note that for the bgoodQ state of nature (b 1 = 0), the informed firm
earns more than the uninformed firm, ex post, and for the bbadQ state of nature the opposite
holds. As one may conclude, for small x (x~0), there is an information disadvantage and for big
x (x~0.25), there is an information advantage. A similar intuitive argument may be found in Einy
et al. (2002). In order to simplify the proof, we consider the two extreme cases.
3. Information advantage vs. information disadvantage
We derive and characterize the unique interior Bayesian equilibrium for the linear duopoly for
both types of information setup.
Proposition 3.1. The linear duopoly admits a unique interior Bayesian equilibrium:
q14ðxÞ ¼
q4
2 ¼
A c b1 ðxÞq4
2
2B
ð A cÞð 2B Eb2 ðd ÞÞ
4B2 E ðb1 ðd Þb2 ðd ÞÞ
Proof. Suppose that there exists an interior equilibrium ( q*1 (d ),q*2 ). In nature state x, the profit
of firm 1 is p 1(x) = [A c Bq 1(x) b 1(x)q 2] q 1(x). Moreover, q*1 (d ) maximizes the profit of
3
It can be easily calculated from Proposition 3.1 in the next section.
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A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
firm 1 when firm 2 chooses q*2 . Since q*1 (x) H 0, for l-almost each x, it must satisfy the firstorder condition for profit maximization. That is,
Bp1 ðxÞ
¼ 0 ¼ A c 2Bq4
1 ðxÞ b1 ðxÞq4
2:
Bq1 ðxÞ
Therefore, for l-almost each xaX, we have
A c b1 ðxÞq24
q14ðxÞ ¼
2B
ð3:1Þ
and
2
p1 ðxÞ ¼ B½q4
1 ðxÞ :
ð3:2Þ
The expected profit of firm 2 is
Ep2 ðd Þ ¼ E ½ A c Bq2 b2 ðd Þq1 ðd Þq2 :
At the interior Bayesian equilibrium, the first-order condition of profit maximization for firm
2 yields
BEp2 ðd Þ
¼ 0 ¼ E ð A c 2Bq4
2 b2 ðd Þq4
1 ðd ÞÞ:
Bq2
Therefore,
q24 ¼
A c E ðb2 ðd Þq4
1 ðd ÞÞ
2B
and
2
Ep2 ðxÞ ¼ B½q4
2 :
Substituting (3.1) into the above expression, we get
q24 ¼
A c b1 ðd Þq4
2
Þ
ð A cÞð 2B Eb2 ðd ÞÞ þ q42 E ðb1 ðd Þb2 ðd ÞÞ
2B
¼
;
4B2
2B
A c Eðb2 ðd Þ
and, therefore,
q24 ¼
ð A cÞð 2B Eb2 ðd ÞÞ
:
4B2 E ðb1 ðd Þb2 ðd ÞÞ
ð3:3Þ
To summarize, we show that if an interior equilibrium exists it is the unique equilibrium
which satisfies (3.1) and (3.3). Next, we prove that the interior equilibrium candidate is indeed an
ð AcÞð 2BEb ðd ÞÞ
equilibrium. Since A c N 0, B z Eb 2(d ) and B 2 z E (b 1(d )b 2(d )) then q24 ¼ 4B2 Eðb1 ðd Þb22 ðd ÞÞ N0.
4
Acb
ð
x
Þq
2
1
In order to show that q*1 (x) N 0 for all x where q14ðxÞ ¼
, we consider two different
2B
cases: (1) If b 1(d ) V 0 then the result immediately follows. (2) For b 1(d ) z 0, since b 2(d ) z 0 as
well, we get
q4
2 ¼
ð A cÞð 2B Eb2 ðd ÞÞ ð A cÞ2B
2ð A c Þ
;
V
¼
4B2 E ðb1 ðd Þb2 ðd ÞÞ
3B2
3B
A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
791
and since b 1(d ) V B,
q4
1 ðxÞ ¼
A c b1 ðxÞq24 A c
N0:
z
6B
2B
Note that the profit function of firm 1, given q*2 , equals zero for q̃ 1 = 0 and for q̃q 1 ¼ Acb1 ðxÞq*2 in
B
state x. In addition, the profit function is concave and q*1 (x) N 0 satisfies the first-order
condition; therefore, q*1 (x) is a best response to q*2 and the equilibrium profits exceed zero. For
firm 2, we get the same result since the expected profit function, given q 1*(d ) is a concave
function which equals zero for q̃ 2 = 0. 5
We now show that for firm 1, the quantity and the profit for each state of nature is greater than
zero. Therefore, for each state, the price of product 1, p 1( q*1 (x),q*2 ) is greater than the marginal
cost, c.
Recall that the price of product 2 is
p2 ðq1*ðxÞ; q2*Þ ¼ A Bq2* b2 ðxÞq1*ðxÞ
and
q4
2 ¼
A c E ðb2 ðd Þq1*ðd ÞÞ
;
2B
q1*ðxÞ ¼
A c b1 ðxÞq2*
:
2B
Hence for b 2(x) N 0,
A c Eðb2 ðd Þq41 ðd ÞÞ
A c b1 ðxÞq42
b2 ð x Þ
2
2B
Ac
b ðxÞ
E ðb2 ðd Þq14ðd ÞÞ b1 ðxÞb2 ðxÞq24
þ
1 2
¼
N0:
þ
2
2
B
2B
p2 ð q4
1 ðxÞ; q4
2Þ c ¼ A c On the other hand, if b 2(x) V 0, then from (3.3),
Bq4
2 ¼
Bð A cÞð 2B Eb2 ðd ÞÞ
bA c;
4B2 E ðb1 ðd Þb2 ðd ÞÞ
hence, p 2( q*1 (x),q*2 ) N c.
We have shown that in equilibrium, each firm chooses positive quantity, the resulting prices
exceed marginal cost, c, and profits are positive. We should emphasize that according to (3.1),
strategies can be complements or substitutes depending on whether b i V 0 or b i z 0, respectively.
We will now show that in the Bayesian equilibrium of the common value model, the
uninformed firm has at least or greater expected profits than the informed firm.
Proposition 3.2. The linear duopoly with common value admits a unique interior Bayesian
equilibrium ( q*1 (d ),q*2 ). This equilibrium engenders an information disadvantage. That is,
E[p*2 (d )] z E[p*1 (d )].
Proof. In the common value case, b 1(x) = b 2(x) = b(x) for every xaX. Rewrite (3.2) as
2
A c bðd Þq4
2
2
Ep1 ðd Þ ¼ EBq14 ðd Þ ¼ BE
2B
"
#
2
ð A cÞ2 2ð A cÞbðd Þq24 þ b2 ðd Þq24
¼ BE
4B2
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A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
and
2
Ep2 ðd Þ ¼ Bq2* :
Thus,
"
#
2
2
4B2 q2* ð A cÞ2 2ð A cÞbðd Þq2* þ b2 ðd Þq2* Ep2 ðd Þ Ep1 ðd Þ ¼ BE
4B2
2
4B2 Eb2 ðd Þ q2* þ 2ð A cÞq2*Ebðd Þ ð A cÞ2
¼
4B
¼
q2*½ð A cÞð 2B Ebðd Þ þ 2ð A cÞq2*Ebðd Þ ð A cÞ2
4B
¼
q2*½ð A cÞð 2B þ Ebðd Þ ð A cÞ2
4B
½ð A cÞð 2B Ebðd Þ
½ðA cÞð 2B þ Ebðd Þ ð A cÞ2
4B2 Eb2 ðd Þ
¼
4B
2
ð A cÞ ð4B2 E 2 bðd ÞÞ ð A cÞ2 4B2 Eb2 ðd Þ
¼
:
4B 4B2 Eb2 ðd Þ
Therefore, since Eb 2zE 2b by the variance inequality, we have
ð A cÞ2 Eb2 ðd Þ E 2 bðd Þ
Ep2 ðd Þ Ep1 ðd Þ ¼
z0: 5
4B 4B2 Eb2 ðd Þ
Next, we show that in the Bayesian equilibrium of the independent private value model, the
informed firm has greater or equal expected profits than the uninformed firm, which is opposite
to the common value result.
Proposition 3.3. The linear duopoly with independent private value admits a unique interior
Bayesian equilibrium ( q*1 (d ),q*2 ). This equilibrium engenders an information advantage. That
is, E[p 1*(d )] z E[p 2*(d )].
In the independent private case, b 1(d ) and b 2(d ) are independent variables, but have the same
expected value Eb 1(d ) = b 2(d ) u K.
Rewrite (3.3) with Eb 1(d ) = Eb 2(d ) = K 2,
q2* ¼
ð A cÞð 2B Eb2 ðd ÞÞ
Ac
:
¼
4B2 Eb1 ðd ÞEb2 ðd Þ
2B þ K
Recall, q1*ðxÞ ¼
Eq1*ðd Þ ¼ E
*2
Acb1 ðxÞq
;
2B
hence,
A c b1 ðd Þq2*
2B
¼E
Þb1 ðd Þ
A c ð Ac
2BþK
2B
¼
Ac
¼ q2*:
2B þ K
A. Chokler et al. / Int. J. Ind. Organ. 24 (2006) 785–793
793
Regarding profits, we have,
2
2
2
Ep1*ðd Þ ¼ EBq1* ðd ÞzBð Eq1*ðd ÞÞ ¼ Bq2* ¼ Ep2 ðd Þ:
Therefore, we have an information advantage.
4. Concluding remarks
We have focused on a class of linear Cournot duopolies with differentiated products and
proved that whether there is an information advantage or disadvantage depends on the
information setup. Specifically, we show that in the common value case, the uninformed firm
that commits to a single quantity will have better or equal ex ante profits than to the firm with
complete information. This result contrasts to the information advantage that holds in the
duopoly game with independent values. One cannot therefore generally conclude that superior
information entails higher profits. Indeed, we present a family of games for which there is an
information disadvantage.
Acknowledgement
We thank S. Anderson (Editor) and two referees for very constructive suggestions. The
research of Chokler and Shitovitz was supported by the Israel Science Foundation.
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