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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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.
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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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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 ˜ 1 b1 ðxÞq42 Þq˜ 1 Þ for all q˜ 1 aRþ : ð A c Bq4 1 ðxÞ b1 ðxÞq4 2 Þq4 1 ðxÞzð A c Bq * 3. For firm 2: q 2 is a best response to firm 1’s action q*1 (d ); that is, E½ð A c Bq42 ˜ 2 b2 ðd Þq4 ˜ 2 for all q˜ 2 aRþ : b2 ðd Þq14ðd ÞÞq4 2 zE½ð A c Bq 1 ðd ÞÞq 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.
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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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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
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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 q˜ 1 = 0 and for q˜ 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 q˜ 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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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
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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. References Einy, E., Moreno, D., Shitovitz, B., 2002. Information advantage in Cournot oligopoly. Journal of Economic Theory 106, 151 – 160. Fried, D., 1984. Incentives for information production and disclosure in duopolistic environment. The Quarterly Journal of Economics 99, 367 – 381. Gal-Or, E., 1985. Information sharing in oligopoly. Econometrica 53, 329 – 343. Gal-Or, E., 1987. First mover disadvantages with private information. Review of Economic Studies 54, 279 – 292. Gal-Or, E., 1988. The advantages of impreciser information. Rand Journal of Economics 19, 266 – 275. Malueg, D., Tsutsui, S., 1996. Duopoly information exchange: the case of unknown slope. International Journal of Industrial Organization 14, 119 – 136. Raith, M., 1996. A general model of information sharing in oligopoly. Journal of Economic Theory 71, 260 – 288. Sakai, Y., 1985. The value of information in a simple duopoly model. Journal of Economic Theory 36, 36 – 54. Vives, X., 1990. Information and competitive advantage. International Journal of Industrial Organization 8, 17 – 35.
