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Technology adoption in a differentiated duopoly: Cournot versus Bertrand
Research in Economics 64 (2010) 128–136
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Technology adoption in a differentiated duopoly: Cournot versus Bertrand Rupayan Pal ∗ Indira Gandhi Institute of Development Research (IGIDR), India
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Article history: Received 6 May 2009 Accepted 16 November 2009 Keywords: Differentiated duopoly Limit-pricing Price effect Selection effect Technology adoption
abstract This paper shows that the cost as well as the effectiveness of technology has a differential impact on technology adoption under two alternative modes of competition. If the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt technology than Cournot competition unless the effectiveness of the technology is very low. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms. This demonstrates that the commonly subscribed assumption of ‘positive primary outputs’ restricts (inflates) the scope of higher degree of technology adoption under Bertrand (Cournot) competition. Moreover, in contrast to standard welfare ranking, it shows that Cournot competition leads to higher social welfare than Bertrand competition under certain situations. © 2009 University of Venice. Published by Elsevier Ltd. All rights reserved.
1. Introduction This paper analyses the incentives to adopt cost-reducing technology by firms in a horizontally differentiated industry under two alternative categories of product market competition, Cournot and Bertrand. When the cost of production is endogenously determined, whether Bertrand competition or Cournot competition provides the stronger incentive for technology adoption is an essential concern. Our framework allows us to address the question of how the cost of the technology as well as its effectiveness affects this comparison in a more general setting that does not rely on the commonly subscribed assumption of ‘positive primary outputs’—‘both firms sell positive outputs even if prices are set at respective marginal costs’.1 Whether higher or lower intensity of product market competition provides greater incentive to adopt cost-reducing technology is of perennial interest. A large literature, dating at least as far back as Schumpeter (1943), emphasizes the role of the intensity of competition on innovation. Schumpeter (1943) argues that, since the possibility to realize returns from technological advancement is higher in concentrated markets, market concentration stimulates innovation. In contrast, Arrow (1962) shows, comparing a perfectly competitive industry with a monopoly, that the gain from adopting cost-reducing technology is higher under a competitive environment. This indicates that a more competitive environment provides a higher incentive to innovate. Recently, attention has turned to the comparison of two oligopolistic industries. A number of recent studies, considering different scenarios, compare firms’ incentives to introduce cost-reducing technologies under alternative modes of product market competition. This helps us to understand a variety of issues: the role of the nature of product differentiation (Bester and Petrakis, 1993; Bonnano and Haworth, 1998), speed of technological
∗ Corresponding address: Indira Gandhi Institute of Development Research (IGIDR), Film City Road, Gen A. K. Vaidya Marg, Goregaon (East), Mumbai 400065, India. Tel.: +91 22 28416545; fax: +91 22 28402752. E-mail addresses: [email protected], [email protected]. 1 This assumption is crucial for ranking of the equilibrium outputs and profits under the two categories of competition (Zanchettin, 2006; Amir and Jin, 2001). 1090-9443/$ – see front matter © 2009 University of Venice. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.rie.2009.11.003
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progress (Aghion et al., 19997), conflict between static and dynamic efficiency (Delbono and Denicolo, 1990), impact of competition intensity (Boone, 2001), incentives in mixed oligopoly (Lin and Ogawa, 2005), etc. While characterizing equilibrium outcomes, these studies subscribe to the assumption of ‘positive primary outputs’ and thus restrict the parameter space, which is likely to distort the equilibrium outcomes. Also, to the best of our knowledge, existing studies do not analyse the impact of the cost of technology on technology adoption decision explicitly. This paper attempts to fill these gaps. We consider a two-stage non-cooperative game between two firms. Initially, both firms have symmetric cost functions. In the first stage, each firm simultaneously and independently decides whether to adopt a cost-reducing technology, by incurring some given cost, or not. In the second stage, firms engage either in Cournot competition or in Bertrand competition. The analysis shows that, if the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt technology than Cournot competition unless the cost-reducing effect of the technology is very low: an ‘Arrow-like’ result. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms: a ‘Schumpeter-type’ result. The intuition behind our result is as follows. Following technology adoption, Bertrand competition not only leads to lower prices (price effect), but also results in a lower market share of the non-adopting firm (selection effect) than Cournot competition. While the price effect generates more disincentive to adopt technology under Bertrand competition than under Cournot competition, the selection effect works in the opposite direction. When only one firm adopts the cost-reducing technology, the selection effect dominates the price effect, and the net effect is higher under Bertrand competition than under Cournot competition unless the cost-reducing effect of the technology is very low. On the other hand, if both firms adopt the technology, the selection effect vanishes and the gain from technology adoption becomes higher under Cournot competition than that under Bertrand competition. Needless to say, a firm’s gain from technology adoption is higher when only that firm adopts the technology. Therefore, when the cost of the technology is low, in equilibrium, both firms adopt the technology under Cournot competition whereas under Bertrand competition only one firm adopts the technology. This paper also shows that the scope for superior outcomes in terms of technology adoption under Bertrand (Cournot) competition is lower (higher) for lower level of effectiveness of the technology. If the cost-reducing effect of the technology is very low, the equilibrium technology adoption is never better under Bertrand competition than that under Cournot competition. This implies that the assumption of positive primary outputs is likely to restrict the scope of technology adoption under Bertrand competition and bolster that under Cournot competition. Comparison of social welfare under the two modes of product market competition reveals that, when technology adoption is higher under Cournot competition than that under Bertrand competition, Cournot competition may lead to higher social welfare than Bertrand competition. This result is in sharp contrast to the standard social welfare ranking of the two modes of competition. The rest of the paper proceeds as follows. The next section presents the model and characterizes Bertrand and Cournot equilibria. Section 3 presents the comparison of equilibrium outcomes under alternative modes of competition. Section 4 concludes. 2. The model Let us consider an economy with an oligopolistic sector, consisting of two firms, firm 1 and firm 2, that produce a differentiated good and a competitive numeraire sector. Initially, the marginal costs of production of firm 1 and firm 2 are equal to c¯ . That is, we start with a situation in which there is no asymmetry in terms of cost of production between firm 1 and firm 2. Now, before undertaking production decisions, firms can adopt the new technology by incurring an exogenously determined fixed cost r (>0) to reduce the marginal cost of production. If a firm adopts the technology, its marginal cost of production reduces to c (0 < c < c¯ = c + x), whereas the non-adopting firm’s marginal cost remains at c¯ . That is, the costreducing effect (effectiveness) of the technology is x = c¯ − c. Needless to say, such a type of discrete choice of technology is observed in many real-life situations and has received some attention in the literature recently (see, for example, Elberfeld and Nti (2004), Jensen (1992), Lahiri and Ratnasiri (2007), Khan and Ravikumar (2002), and Leung and Tse (2001), to name a few).2 1 We consider that, if there is positive demand for both goods, the direct demand function is qi = 1−γ 2 [(1 −γ )a − pi +γ pj ],
(a > c¯ ; i, j = 1, 2; i 6= j), where qi (pi ) is the quantity (price) of the product of firm i and γ (0 < γ < 1) is the product differentiation parameter.3 The corresponding inverse demand function is pi = a − qi − γ qj (i, j = 1, 2; i 6= j). However, if the prices are such that qj ≤ 0, the demand for good i is qi = a − pi , as in Zanchettin (2006).
2 Considering a dynamic framework, Lahiri and Ratnasiri (2007), Khan and Ravikumar (2002), Leung and Tse (2001) examine technology adoption and its impact on macroeconomic variables. Elberfeld and Nti (2004) and Jensen (1992) analyse technology adoption by firms under uncertain environment considering Cournot competition in the product market. We abstract away, for simplicity, from possible uncertainty about the cost-reducing effect of new technology. 3 The underlying utility function of the representative consumer is U = aq + aq − 1 (q2 + q2 + 2γ q q ) + m, where m is the quantity of the numeraire 1
good.
2
2
1
2
1 2
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R. Pal / Research in Economics 64 (2010) 128–136
Stages of the game involved are as follows. Stage 1 : Firm 1 and firm 2 simultaneously and independently decide whether to adopt the technology or not. Stage 2 : Firms engage either in Cournot competition or in Bertrand competition in the product market. Clearly, in stage 1, there are three possible cases for the decision to adopt the technology: (1) no firm adopts the technology, (2) only one firm, either firm 1 or firm 2, adopts the technology, and (3) both firm 1 and firm 2 adopt the technology.4 The mode of product market competition, Cournot or Bertrand, in the second stage is exogenously determined. We solve this game by the backward induction method. Note that when only one firm, say firm 1, adopts the technology, the mode of competition in the product market matters only if firm 1 cannot engage in monopoly pricing without bearing any competitive pressure from firm 2 (the non-adopting firm). Now, if the technology reduces marginal cost of firm 1 beyond a certain level, i.e., if the ex post difference between the firms’ marginal costs becomes sufficiently high, firm 2 is driven out of the market and firm 1 enjoys absolute monopoly a +c power irrespective of the mode of competition. Now, firm 1 sets the monopoly price pM , if at prices p1 = pM 1 = 1 and 2
¯−c ≥ p2 = c¯ the demand of firm 2’s product is zero, i.e., a(1 − γ ) − c¯ + γ pM 1 ≤ 0 ⇒ x = c of product market competition does not matter when x ≥
(a−¯c )(2−γ ) , γ
0 < γ < 1; 0 < x <
Since the mode
we consider that the reduction in marginal cost of
production through technology adoption is not drastic, i.e., we consider that x < parameter space, in which the mode of product market competition matters, is S=
(a−¯c )(2−γ ) 5 . γ
(a−¯c )(2−γ ) . γ
In other words, the relevant
(a − c¯ )(2 − γ ) . γ
Note that, if prices are set at respective marginal costs, p1 = c and p2 = c¯ , both firms sell positive outputs only if x < (a−¯c )(1−γ ) . Clearly, the assumption of positive primary outputs curtails the parameter space. γ 2.1. Product market equilibrium The second stage of the game is very similar to that in Zanchettin (2006). We, therefore, summarize the equilibrium outcomes of stage 2 without going into further details. We denote the equilibrium quantity of firm i (i = 1, 2) under Cournot competition by qCi (c1 , c2 ); c1 , c2 ∈ {c , c¯ }, where the first (second) argument is the marginal cost of production of firm 1 (firm 2). The superscript ‘C ’ denotes Cournot competition. We denote the equilibrium prices and profits similarly. To denote equilibrium outcomes under Bertrand competition, we replace the superscript ‘C ’ by ‘B’. Cournot competition: The equilibrium outcomes in stage 2 in the case of quantity competition in the product market, given the technology adoption decisions of firms, are as follows. (a−¯c )2
−¯c (Ca) When no firm adopts the technology, qCi (¯c , c¯ ) = pCi (¯c , c¯ ) − c¯ = 2a+γ and πiC (¯c , c¯ ) = (2+γ )2 , i = 1, 2. (a−c )2
a −c
(Cb) When both firms adopt the technology, qCi (c , c ) = pCi (c , c ) − c = 2+γ and πiC (c , c ) = (2+γ )2 − r, i = 1, 2. (Cc) When only one firm, say firm 1, adopts the technology, qC1 (c , c¯ ) = pC1 (c , c¯ )−c = (2−γ )(a−¯c )−γ x , 4−γ 2
π1C (c , c¯ ) =
[(2−γ )(a−¯c )+2x]2 (4−γ 2 )2
− r, π2C (c , c¯ ) =
[(2−γ )(a−¯c )−γ x]2 (4−γ 2 )2
(2−γ )(a−¯c )+2x C , q2 (c , c¯ ) 4−γ 2
= pC2 (c , c¯ )−¯c =
. Alternatively, when only firm 2 adopts the
technology, the equilibrium outcomes will be symmetric. It is evident that a higher reduction (x) in marginal cost of production through technology adoption leads to higher output, price-cost margin and profit of the technology-adopting firm, but lower output, price-cost margin and profit of the nonadopting firm. However, it is easy to check that the non-adopting firm remains active under Cournot competition for all x ∈ S. Bertrand competition: When firms compete in terms of price in the product market in stage 2, given the technology adoption decisions of firms of stage 1, the equilibrium outcomes are as follows. (Ba) When no firm adopts the technology, qBi (¯c , c¯ ) =
pBi (¯c ,¯c )−¯c
(Bb) When both firms adopt the technology, qBi (c , c ) =
1−γ 2
=
pBi (c ,c )−c 1−γ 2
(a−¯c ) (2−γ )(1+γ )
=
(a−¯c )2 (1−γ )
and πiB (¯c , c¯ ) = (2−γ )2 (1+γ ) , i = 1, 2.
(a−c ) (2−γ )(1+γ )
(a−c )2 (1−γ )
and πiB (c , c ) = (2−γ )2 (1+γ ) − r, i = 1, 2.
4 Apart from these, there is also a possibility that firms randomize between the strategies ‘adopt the technology’ and ‘don’t adopt the technology’. 5 Firm 2, i.e., the non-adopting firm, cannot engage in monopoly pricing unless the products are completely different (γ = 0).
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Fig. 1. Technology choice game under Cournot competition.
(Bc) If only one firm, say firm 1, adopts the technology and both firms are active in equilibrium, i.e., if 0 < x < (a−¯c )(2−γ −γ 2 ) , qB1 (c , c¯ ) γ 2 [(2−γ −γ )(a−¯c )+(2−γ 2 )x]2 (4−γ 2 )2 (1−γ 2 )
=
pB1 (c ,¯c )−c
pB (c ,¯c )−¯c (2−γ −γ 2 )(a−¯c )+(2−γ 2 )x , qB2 (c , c¯ ) = 21−γ 2 = (4−γ 2 )(1−γ 2 ) [(2−γ −γ 2 )(a−¯c )−γ x]2 B π2 (c , c¯ ) = . Alternatively, if only (4−γ 2 )2 (1−γ 2 )
1−γ 2
− r and
=
(2−γ −γ 2 )(a−¯c )−γ x , (4−γ 2 )(1−γ 2 )
π1B (c , c¯ ) =
firm 2 adopts the technology,
the equilibrium outcomes will be symmetric. (a−¯c )(2−γ −γ 2 )
(a−¯c )(2−γ )
(Bd) If the reduction in marginal cost due to technology adoption is sufficiently large ( ≤ x < ) γ γ and only one firm adopts the technology, limit-pricing occurs in equilibrium and the non-adopting firm is forced to stay out of the market. The corresponding (limit-pricing) equilibrium outcomes, when only firm 1 adopts the γ x−(a−¯c )(1−γ ) c technology, are as follows: qL2 (c , c¯ ) = pL2 (c , c¯ ) − c¯ = π2L (c , c¯ ) = 0, qL1 (c , c¯ ) = a−¯ , pL1 (c , c¯ ) − c = γ γ and π1L (c , c¯ ) =
(a−¯c ){γ x−(a−¯c )(1−γ )} γ2
− r, where the superscript L denotes limit-pricing under Bertrand equilibrium.
Note that the possibility of limit-pricing increases with a decrease in the degree of product differentiation (increase in
γ ).6 Nonetheless, even if the degree of product differentiation is high (γ is low), it is optimum for the technology-adopting (a−¯c )(2−γ −γ 2 ) firm to engage in limit-pricing, as long as ≤ x < (a−¯c )(γ 2−γ ) is satisfied, which keeps the non-adopting firm out γ
of the market.7 On the contrary, both firms, the technology-adopting firm as well as the non-adopting firm, remain active under Cournot competition for all x ∈ S (Zanchettin, 2006). 2.2. Equilibrium technology adoption Cournot competition: We now turn to the technology adoption decision in stage 1 of the game, when firms are engaged in Cournot competition in stage 2. The normal form of the stage 1 game is depicted in Fig. 1. The profit expressions are as given in (Ca)–(Cc), in Section 2.1. From Fig. 1, it is easy to observe that, if firm 2 does not adopt the technology, firm 1 adopts the technology provided that π1C (c , c¯ ) > π1C (¯c , c¯ ) ⇒ r <
4x2 +4(2−γ )(a−¯c )x (4−γ 2 )2
= r¯ C . On the other hand, if firm 1 adopts the technology, firm 2 does
not adopt the technology provided that π2C (c , c¯ ) > π2C (c , c ) ⇒ r >
4(1−γ )x2 +4(2−γ )(a−¯c )x (4−γ 2 )2 C
= r C . Clearly, r C < r¯ C . Also,
note that firms are ex ante symmetric in terms of cost of production. Therefore, if r ≥ r¯ , ‘Don’t Adopt’ is the dominant strategy of each firm and no firm adopts the technology in equilibrium. Alternatively, if r ≤ r C , ‘Adopt’ becomes the dominant strategy of each firm and both firms adopt the technology in equilibrium. If the cost of the technology lies in between these two extreme values, i.e., if r C < r < r¯ C , we have π1C (c , c¯ ) > π1C (¯c , c¯ ) > π1C (¯c , c ) > π1C (c , c ) and π2C (¯c , c ) > π2C (¯c , c¯ ) > π2C (c , c¯ ) > π2C (c , c ). This implies that, if r C < r < r¯ C , there are two pure-strategy Nash equilibria: (Adopt, Don’t Adopt) and (Don’t Adopt, Adopt). Therefore, if the cost of the technology is in the intermediate range (r C < r < r¯ C ), only one firm adopts the technology in equilibrium.8 Lemma 1. When firms are engaged in Cournot competition in the product market, the equilibrium technology adoption is as follows. 4(1−γ )x2 +4(2−γ )(a−¯c )x . (4−γ 2 )2 4x2 +4(2−γ )(a−¯c )x . (4−γ 2 )2
(a) If r ≤ r C , both firms adopt the technology, where r C = (b) If r ≥ r¯ C , no firm adopts the technology, where r¯ C = (c) If r C < r < r¯ C , only one firm adopts the technology.
6 The range of the limit-pricing region is (a − c¯ )γ , which is positively related to γ . 7 Note that the technology-adopting firm cannot engage in monopoly pricing without bearing any competitive pressure from the non-adopting firm, (a−¯c )(2−γ ) since we consider that the reduction in marginal cost through technology adoption is not drastic (i.e., x < ). That is, the technology-adopting γ firm cannot enjoy absolute monopoly power. In other words, though the non-adopting firm is driven out of the market, it exerts competitive pressure on the technology-adopting firm. 8 Other than these two pure-strategy Nash equilibria, none of which risk-dominates the other in the sense of Harsanyi and Selten (1988), there is also a C
mixed-strategy Nash equilibrium in which each firm chooses to adopt the technology with probability ρ C = r¯r¯C −−rrC , 0 < ρ C < 1.
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Fig. 2. Technology choice game when limit-pricing is possible.
Bertrand competition: Let us now analyse the technology adoption decision of firms in stage 1, when there is Bertrand competition in the product market. From (Ba)–(Bd), in Section 2.1, it is clear that there are two possible scenarios under Bertrand competition: (a)
(a−¯c )(2−γ −γ 2 ) γ
firm adopts the technology; and (b) 0 We depict the normal form of the (a−¯c )(2−γ ) ) γ
(a−¯c )(2−γ ) , i.e., limit-pricing occurs in equilibrium at stage 2 when only one γ (a−¯c )(2−γ −γ 2 )
≤x<
in Fig. 2. Now, we characterize the equilibrium outcomes of the above game following the same line as in case of Cournot competition. In this case, if firm 2 does not adopt the technology, firm 1 adopts the technology provided that
π1L (c , c¯ ) > π1B (¯c , c¯ ), i.e., r <
(a−¯c ){γ x−(a−¯c )(1−γ )} γ2
−
(a−¯c )2 (1−γ ) (2−γ )2 (1+γ )
= r¯ L . On the other hand, if firm 1 adopts the technology, (a−c )2 (1−γ )
firm 2 does not adopt the technology provided that π2L (c , c¯ ) > π2B (c , c ), i.e., r > (2−γ )2 (1+γ ) = r L . It is easy to check that r L < r¯ L .9 So, we have π1L (c , c¯ ) > π1B (¯c , c¯ ) > π1L (¯c , c ) > π1B (c , c ) and π2L (¯c , c ) > π2B (¯c , c¯ ) > π2L (c , c¯ ) > π2B (c , c ). Note that the rank of profits corresponding to different sets of strategies of firms are the same as in Cournot competition. Clearly, if r ≤ r L , both firms adopt the technology in equilibrium. But, if r ≥ r¯ L , in equilibrium, no firm adopts the technology. If r L < r < r¯ L , there are two pure-strategy Nash equilibria, (Adopt, Adopt) and (Don’t Adopt, Don’t Adopt).10 Therefore, if r L < r < r¯ L , only one firm adopts the technology in equilibrium. (a−¯c )(2−γ −γ 2 )
In the second scenario (0 < x < ), both firms remain active in the product market even when only one firm γ adopts the technology. We present the corresponding payoff matrix in Fig. 3. It is easy to check that in the second scenario the equilibrium outcomes are similar to those in the first scenario. To be explicit, if 0 < x <
(a−¯c )(2−γ −γ 2 ) , γ
in equilibrium, (a) both firms adopt the technology when r ≤ r B , (b) no firm
adopts the technology when r ≥ r¯ , and (c) only one firm adopts the technology when r B < r < r¯ B ; where r B = B
(a−c )2 (1−γ ) −γ 2 )(a−¯c )−γ x]2 − [(2−γ (2−γ )2 (1+γ ) (4−γ 2 )2 (1−γ 2 )
and r¯ B =
[(2−γ −γ 2 )(a−¯c )+(2−γ 2 )x]2 −¯c )2 (1−γ ) 11 − ((2a−γ . However, note that the critical values of the (4−γ 2 )2 (1−γ 2 ) )2 (1+γ )
cost of technology in the second scenario are lower than the corresponding critical values in the case of limit-pricing: r B < r L and r¯ B < r¯ L .12 This indicates that the possibility of both firms (no firm) adopting the technology is higher (lower) under limitpricing compared to that when limit pricing is not possible. Therefore, the assumption of positive primary outputs leads to underestimation of the possibility of technology adoption.13 Lemma 2. If firms are engaged in Bertrand competition in the product market, the equilibrium technology adoption is as follows. (a) When
(a−¯c )(2−γ −γ 2 ) γ
≤x<
(a−¯c )(2−γ ) , γ
(a−c )2 (1−γ )
(i) both firms adopt the technology, if r ≤ r L = (2−γ )2 (1+γ ) , (ii) no firm adopts the technology, if r ≥ r¯ L =
(a−¯c ){γ x−(a−¯c )(1−γ )} γ2
(iii) only one firm adopts the technology, if r L < r < r¯ L . (b) When x <
(a−¯c )(2−γ −γ 2 ) , γ
−
(a−¯c )2 (1−γ ) , (2−γ )2 (1+γ )
(a−c )2 (1−γ )
[(2−γ −γ 2 )(a−¯c )−γ x]2 , (4−γ 2 )2 (1−γ 2 ) [(2−γ −γ 2 )(a−¯c )+(2−γ 2 )x]2 (a−¯c )2 (1−γ ) − (2−γ )2 (1+γ ) , (4−γ 2 )2 (1−γ 2 )
(i) both firms adopt the technology, if r ≤ r B = (2−γ )2 (1+γ ) − (ii) no firm adopts the technology, if r ≥ r¯ B =
(iii) only one firm adopts the technology, if r B < r < r¯ B .
9 r L < r¯ L ⇒ [(a − c¯ )xγ {4 − γ (2 − γ )(1 + γ )} − (a − c¯ )2 (1 − γ )(4 − γ 2 + γ 3 ) − x2 γ 2 (1 − γ )] > 0, which is true for all x ∈ [ (a−¯c )(2−γ −γ 2 ) , (a−¯c )(2−γ ) ). γ γ 10 In this case, there is also a mixed-strategy Nash equilibrium in which each firm chooses to adopt the technology with probability ρ L = r¯ L −r , r¯ L −r L
0 < ρ L < 1. 11 There is also a mixed-strategy Nash equilibrium, if 0 < x < (a−¯c )(2−γ −γ 2 ) and r B < r < r¯ B , in which each firm chooses to adopt the technology with γ B probability ρ B = r¯r¯B −−rrB , 0 < ρ B < 1. 12 r B − r L = − [(2−γ −γ 2 )(a−¯c )−γ x]2 < 0 and π L (c , c¯ ) > π B (c , c¯ ) ⇒ r¯ B < r¯ L . 1 1 (4−γ 2 )2 (1−γ 2 )
13 Under the assumption of positive primary outputs we must have x < (a−¯c )(1−γ ) (< (a−¯c )(2−γ −γ 2 ) ). γ γ
R. Pal / Research in Economics 64 (2010) 128–136
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Fig. 3. Technology choice game when limit-pricing is not possible.
Fig. 4. Cost of technology and technology adoption.
3. Comparison of Cournot and Bertrand equilibria In this section, we compare Cournot and Bertrand equilibria. We focus on the technology adoption decision and on social welfare under alternative modes of product market competition. Technology adoption when limit-pricing is possible. Let us begin with the scenario in which the marginal cost reduction (x) (a−¯c )(2−γ −γ 2 )
(a−¯c )(2−γ )
through technology adoption is relatively high, that is, ≤x< , which leads to limit-pricing under γ γ Bertrand equilibrium (see (Bd) in Section 2.1). In this case, it is straightforward to observe that the relevant critical values of the cost of the technology, as given in Lemmas 1 and 2, satisfy the following relation: 0 < r L < r C < r¯ C < r¯ L .
(1)
The above inequality implies that, if the cost of the technology is high (r¯ < r < r¯ ), one firm adopts the technology under Bertrand competition whereas none adopts under Cournot competition. That is, we get very asymmetric outcomes under two alternative modes of product market competition. However, if the cost of the technology is moderate (r C < r < r¯ C ), under both Cournot and Bertrand competition only one firm adopts the technology in equilibrium. Nonetheless, the incentive to adopt technology by a single firm is higher under Bertrand competition than under Cournot competition, since π1L (c , c¯ ) − π1B (¯c , c¯ ) > π1C (c , c¯ ) − π1C (¯c , c¯ ). This is because, following technology adoption, Bertrand competition not only leads to lower prices (price effect), but also a lower market share of the non-adopting firm (selection effect) than Cournot competition. The selection effect dominates the price effect, and the net gain, from these two opposing effects, of the technology-adopting firm is higher under Bertrand competition than under Cournot competition. As a result, when the cost of the technology is high, firms do not find it profitable to adopt the technology under Cournot competition even when the other firm does not adopt the technology, whereas under Bertrand competition one firm gains by adopting the technology when the other firm does not opt to adopt it: an ‘Arrow-like’ result. However, if the cost of the technology reduces to a moderate level (r C < r < r¯ C ), the net gain from the two opposing effects, price effect and selection effect, becomes higher than the cost of the technology under Cournot competition also. As a result, when the cost of the technology is moderate, we get symmetric equilibrium outcomes in terms of technology adoption under Cournot and Bertrand competition: one firm adopts the technology irrespective of the mode of competition. If the cost of the technology reduces further to a low level, r L < r < r C , both firms find technology adoption to be gainful under Cournot competition whereas under Bertrand competition a firm gains by adopting the technology only if the other firm does not adopt the technology, since the price effect is smaller under Cournot competition than that under Bertrand competition. If under Bertrand competition both firms adopt the technology, each firm incurs losses. Therefore, when the cost of the technology is low, in equilibrium, both firms adopt the technology under Cournot competition whereas only one firm adopts the technology under Bertrand competition: a ‘Schumpeter-type’ result. The cost of the technology needs to be reduced even further to a very low level (r < r L ) to induce both firms to adopt the technology under Bertrand competition also. Fig. 4 depicts the pure-strategy Nash equilibrium choice of firms regarding technology adoption corresponding to different levels of the cost of the technology under alternative modes of competition. Clearly, the range of the cost of the technology in which both firms adopt the technology is higher under Cournot competition than that under Bertrand competition. On the contrary, the range in which no firm adopts the technology is lower under Bertrand competition than that under Cournot competition. C
L
Technology adoption when limit-pricing is not possible. We now turn to the scenario in which the marginal cost reduction (x) through technology adoption is lower than that in the earlier scenario; that is, now we consider that 0 < x <
(a−¯c )(2−γ −γ 2 ) . γ
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R. Pal / Research in Economics 64 (2010) 128–136
Fig. 5. Adoption of less superior technology.
In this scenario, as discussed in Section 2.1, both firms remain active in the market irrespective of the mode of product market competition and the cost of the technology. Therefore, under Bertrand competition, a firm’s gain from technology adoption when the other firm does not adopt the technology is lower than that in the previous scenario. Also, under Bertrand competition, when one firm adopts the technology, the other firm will also find it profitable to adopt the technology provided that the cost of the technology is less than what is required in the earlier case, since the cost-reducing effect of the technology is now lower than that in the earlier case. This implies that r B < r L and r¯ B < r¯ L , as noted in Section 2.2. Nonetheless, comparing critical values of the cost of the technology, as given in Lemmas 1 and 2, we get similar rankings as in (1): 2(1−γ )(a−¯c ) 2(1−γ )(a−¯c ) 0 < r B < r C < r¯ C < r¯ B unless x < . However, if x < , the relative position of r¯ B and r¯ C changes, γ γ i.e., we get 0 < r B < r C and r¯ B < r¯ C . This is because the unilateral gain from technology adoption by a single firm declines at a higher rate, due to a decrease in the cost-reducing effect of the technology, under Bertrand competition than that under
∂(π1B (c ,¯c )−π1B (¯c ,¯c )) ∂(π C (c ,¯c )−π C (¯c ,¯c )) > 1 ∂x 1 . ∂x 2(1−γ )(a−¯c ) (a−¯c )(2−γ −γ 2 )
Cournot competition:
Clearly, if However, the critical ranges of the cost of the technology are now quite different from the earlier scenario. Considering the entire range of the cost of the technology, we can say that the possibility of diverse outcomes in terms of technology adoption under alternative modes of product market competition is now higher (lower) for relatively lower (higher) cost of the technology, since r B < r L < r C and r¯ C < r¯ B < r¯ L . To be more specific, the scope for better outcomes under Cournot (Bertrand) competition is now higher (lower). 2(1−γ )(a−¯c ) , the comparison is as follows. (a) If r¯ B < r < r¯ C , no firm adopts the technology On the other hand, if x < γ under Bertrand competition whereas one firm adopts the technology under Cournot competition. (b) If r C < r < r¯ B , one firm adopts the technology irrespective of the mode of product market competition. (c) If r B < r < r C , both firms adopt the technology under Cournot competition whereas only one firm adopts the technology under Bertrand competition. (d) If 0 < r < r B (r > r¯ C ), both firms (no firm) adopt(s) the technology irrespective of the mode of product market competition. Fig. 5 depicts the choice of such technology, in equilibrium, under alternative modes of product market competition. 2(1−γ )(a−¯c ) Clearly, if the cost-reducing effect of the technology is low (x < ), the range of the cost of the technology in γ which no firm adopts the technology is higher under Bertrand competition than that under Cournot competition. Moreover, in this case, the equilibrium technology adoption is never better under Bertrand competition than that under Cournot 2(1−γ )(a−¯c ) competition.14 These findings are in sharp contrast to those in the case of x > . γ
Proposition 1. In the case of pure-strategy Nash equilibrium, the comparison of technology adoption under alternative modes of product market competition is as follows. (i) If the effectiveness of the new technology is high (
(a−¯c )(2−γ −γ 2 ) γ
≤x<
(a−¯c )(2−γ ) ), γ
Bertrand competition fares better than
Cournot competition in terms of technology adoption provided that the cost of the technology is high (r¯ C < r < r¯ L ). On the contrary, if the cost of the technology is low (r L < r < r C ), technology adoption is higher under Cournot competition than that under Bertrand competition. In all other ranges of the cost of the technology, technology adoption remains same under alternative modes of product market competition. (ii) The scope for better outcomes, in terms of technology adoption, under Cournot (Bertrand) competition is higher (lower), if the technology reduces the marginal cost of production to a lesser extent (i.e., if x <
(a−¯c )(2−γ −γ 2 ) ). γ
14 In the case of mixed-strategy Nash equilibrium, the comparison of Cournot competition and Bertrand competition in terms of technology adoption is similar to that in the case of pure-strategy Nash equilibrium. However, unlike in the case of pure-strategy Nash equilibrium, in this case identical outcomes in terms of technology adoption under Cournot competition and Bertrand competition are possible only for extreme values of the cost of the technology. If
(a−¯c )(2−γ −γ 2 ) γ
≤ x <
(a−¯c )(2−γ ) , γ
ρ C is greater (less) than ρ L , if the cost of the technology is less (greater) than r CL (=
r¯ C (¯r L −r L )−¯r L (¯r C −r C ) ), (¯r L −r L )−(¯r C −r C )
since 0 < r L < r C < r CL < r¯ C < r¯ L . Clearly, in this case, Cournot (Bertrand) competition fares better than Bertrand (Cournot) competition in terms of 2(1−γ )(a−¯c )
(a−¯c )(2−γ −γ 2 )
technology adoption by firms, if r L < r < r CL (r CL < r < r¯ L ): a ‘Schumpeter-type’ result (an ‘Arrow-like’ result). If
r¯ C (¯r B −r B )−¯r B (¯r C −r C ) . (¯r B −r B )−(¯r C −r C )
However, if x <
2(1−γ )(a−¯c )
γ
, we get 0 < r B < r C < r¯ B < r¯ C < r CB , which implies that
the probability of technology adoption is never higher under Bertrand competition than that under Cournot competition.
R. Pal / Research in Economics 64 (2010) 128–136
135
2(1−γ )(a−¯c )
(iii) If the technology is far less superior (i.e., if x < ), the equilibrium technology adoption is never better under γ Bertrand competition than that under Cournot competition, and the extent of technology adoption is strictly higher under Cournot competition provided that r B < r < r C or r¯ B < r < r¯ C . Clearly, in pure-strategy Nash equilibrium, the possibility of ex post cost asymmetry is higher under Bertrand competition than that under Cournot competition. It is evident that whether Bertrand competition or Cournot competition induces firms more towards technology adoption or not crucially depends on the level of effectiveness of the technology and its cost. Unless the effectiveness of the technology is very low, Bertrand competition leads to superior outcomes in terms of technology adoption than Cournot competition, if the cost of the technology is high. This result has implications to ‘technology subsidy’ policies. Moreover, note that the scope for superior outcome under Bertrand competition increases with the increase in cost-reducing effect of the technology. (a−¯c )(1−γ ) This implies that the assumption of positive primary outputs (0 < x < ) tends to restrict (inflate) the scope γ of higher degree of technology adoption under Bertrand competition (Cournot competition). Proposition 1 also suggests that the relation between intensity of competition and technology adoption is not necessarily monotonic, which is in line with Boone (2001). In order to compare our results with that of Bester and Petrakis (1993), note that, if γ is closer to one,
2−γ −γ 2
is closer
γ (a−¯c )(2−γ −γ 2 ) γ
to zero. That is, if the degree of substitutability between products is very high, the possibility of x > is high. Therefore, we can say that Bertrand competition fares better in terms of technology adoption than Cournot competition when the products are close substitutes provided that the cost of the technology is high. On the other hand, if the products are 2(1−γ )(a−¯c ) highly differentiated (γ is close to zero), is very large, and, thus, Cournot competition leads to superior/equivalent γ outcomes in terms of technology adoption compared to that under Bertrand competition. These results are similar to those in Bester and Petrakis (1993), which argues that the incentive to innovate is higher (lower) under Cournot competition than that under Bertrand competition, if the degree of substitutability is low (high). However, note that the results of Bester and Petrakis (1993) hold only for a selected range(s) of the cost of the technology. Therefore, in our set-up, the results of Bester and Petrakis (1993) emerge in special cases. Social welfare ranking. Finally, we turn to the social welfare (SW) ranking of the two modes of product market competition. It is evident that if technology adoption is identical under Cournot and Bertrand competition, the standard welfare ranking holds — social welfare under Bertrand competition is higher than social welfare under Cournot competition. To illustrate this further, if both firms adopt the technology or none adopts the technology, a switch from Cournot to Bertrand competition increases the consumer surplus by an amount that is more than the corresponding decrease in industry profits, resulting in higher social welfare. The same mechanism works also in the case when only one firm adopts the technology irrespective of the mode of competition, provided that the cost-reducing effect of the technology is sufficiently low; that is, if the possibility of limit-pricing does not arise. But, if limit-pricing under Bertrand competition is unavoidable, a switch from Cournot to Bertrand competition increases both the consumer surplus and industry profits, leading to an increase in social welfare (Zanchettin, 2006). Welfare ranking of the two modes of competition when they lead to asymmetric outcomes in terms of technology adoption draws special attention. If technology adoption is higher under Bertrand competition than that under Cournot competition, Bertrand competition leads to higher social welfare than Cournot competition, since (a) social welfare is higher under Bertrand competition than that under Cournot competition when technology adoption is identical under the two modes of competition and (b) higher technology adoption leads to higher total surplus irrespective of the mode of competition provided that the cost of the technology is not very high. However, if Cournot competition leads to higher technology adoption than Bertrand competition, the increment in total surplus due to higher technology adoption under Cournot competition may or may not be sufficient to outweigh the loss in total surplus due to a shift from Bertrand to Cournot competition. It depends on the relative magnitudes of these two opposing effects. Upon inspection, we find that, in the case of higher technology adoption under Cournot competition than that under Bertrand competition, Cournot competition leads to higher social welfare than Bertrand competition under plausible parametric configurations, though not always. To illustrate this further, let us consider that r L < r < r C and (2−γ −γ 2 )(a−¯c ) γ
(2−γ )(a−¯c ) . γ
< x <
Note that, in this case, both firms adopt the technology under Cournot competition, but only one firm adopts the technology under Bertrand competition and limit-pricing occurs under Bertrand competition. It is straightforward to check that the corresponding social welfare under Bertrand competition is less than that under Cournot competition unless the cost of the technology is more than a critical value (r1 ).15 Upon inspection, we find that r < r1 holds for many plausible parametric configurations. For example, if a = 40, c¯ = 30 and γ = 12 , the relevant range of x is 25 < x < 30 and r L < r1 holds for some plausible parametric configurations.16 This implies that, in this case, we can
15 In this case, social welfare under Cournot and Bertrand competition are, respectively, SW C (c , c ) (a−¯c ){2γ (a−c )−(a−¯c )} 2γ 2
C
(a−c )2 (3+γ ) (2+γ )2
− 2r and SW L (c , c¯ ) =
(a−c )2 (4−4γ −γ 2 )−2(a−c )xγ (4−2γ −γ 2 )+x2 2γ 2 (3+γ ) . 2γ 2 (2+γ )2
− r. SW (c , c¯ ) < SW (c , c ) ⇒r < r1 , where r1 = 16 It is easy to check that the equilibrium outputs and profit(s) are positive at a = 40, c¯ = 30 and γ = L
=
1 , 2
in all combination of ranges of x and r.
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have higher social welfare under Cournot competition than that under Bertrand competition. Similarly, we can demonstrate that Cournot competition can lead to higher social welfare than Bertrand competition in the remaining two cases, in which technology adoption is higher under Cournot competition than that under Bertrand competition, as well.17 The above finding is in sharp contrast to the standard welfare ranking of the two modes of product market competition. In other words, when the cost of production is endogenously determined through technology adoption by firms, the standard welfare ranking of the two modes of product market competition may be reversed. Proposition 2. When the cost of production is endogenously determined through technology adoption by firms, Cournot competition may lead to higher social welfare than Bertrand competition. 4. Conclusion In this paper we have compared technology adoption in a differentiated duopoly under two alternative modes of product market competition, Cournot and Bertrand. We have analysed how the quality of the technology and its cost affect this comparison in a more general set-up by enlarging the parameter space so as to relax the commonly subscribed assumption of positive primary outputs. We have shown that, if the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt the technology than Cournot competition unless the effectiveness of the technology is very low. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms. Moreover, in contrast to standard welfare ranking of the two modes of product market competition, we have shown that Cournot competition may lead to higher social welfare than Bertrand competition, if technology adoption is higher in the case of Cournot competition than that in the case of Bertrand competition. We have also shown that the scope for superior outcome in terms of technology adoption under Bertrand (Cournot) competition is lower (higher) for lower cost-reducing effect of the technology. If the cost-reducing effect of the technology is very low, the equilibrium technology adoption is never better under Bertrand competition than that under Cournot competition. This implies that the assumption of positive primary outputs restricts (inflates) the scope of higher degree of technology adoption under Bertrand (Cournot) competition. It seems to be interesting to extend the present analysis by considering a possible trade-off between product and process innovation. It might also be interesting to examine the implications of (semi)collusion on technology adoption and profitability in the present context. We leave these issues for future research. Acknowledgements I wish to thank the anonymous referees for detailed comments that have led to substantial improvements in the paper. I am solely responsible for any remaining errors. References Aghion, P., Harris, C., Vickers, J., 1997. Competition and growth with step-by-step innovation: An example. European Economic Review 41, 771–782. Amir, R., Jin, J., 2001. Cournot and Bertrand equilibria compared: Substitutability, complementarity and concavity. International Journal of Industrial Organization 19, 303–317. Arrow, K., 1962. Economic welfare and the allocation of resources for inventions. In: Nelson, R. (Ed.), The Rate and Direction of Inventive Activity. Princeton University Press, USA. Bester, H., Petrakis, E., 1993. The incentives for cost reduction in a differentiated industry. International Journal of Industrial Organization 11 (4), 519–534. Bonnano, G., Haworth, B., 1998. Intensity of competition and the choice between product and process innovation. International Journal of Industrial Organization 16, 495–510. Boone, J., 2001. Intensity of competition and the incentive to innovate. International Journal of Industrial Organization 19, 705–726. Delbono, F., Denicolo, V., 1990. R&D investment in a symmetric and homogeneous oligopoly. International Journal of Industrial Organization 8, 297–313. Elberfeld, W., Nti, K.O., 2004. Oligopolistic competition and new technology adoption under uncertainty. Journal of Economics 82 (2), 105–121. Harsanyi, J.C., Selten, R., 1988. A General Theory of Equilibrium Selection in Games. MIT Press, Cambridge. Jensen, R., 1992. Innovation adoption and welfare under uncertainty. Journal of Industrial Economics 40 (2), 173–180. Khan, A., Ravikumar, B., 2002. Costly technology adoption and capital accumulation. Review of Economic Dynamics 5 (2), 489–502. Lahiri, R., Ratnasiri, S., 2007. Concerning inequality, technology adoption, and structural change. International Advances in Economic Research 13 (4), 527–528. Leung, C.K.Y., Tse, C.Y., 2001. Technology choice and saving in the presence of a fixed adoption cost. Review of Development Economics 5 (1), 40–48. Lin, M.H., Ogawa, H., 2005. Cost reducing incentives in a mixed duopoly market. Economics Bulletin 12 (6), 1–6. Schumpeter, J., 1943. Capitalism. Socialism and Democracy. Allan and Unwin, London. Zanchettin, P., 2006. Differentiated duopoly with asymmetric costs. Journal of Economics and Management Strategy 15 (4), 999–1015.
17 This result goes through in the case of mixed-strategy Nash equilibrium also. Note that, if r B < r < r C and 0 < x < (a−¯c )(2−γ −γ 2 ) , both firms adopt γ
the technology under Cournot competition and each firm adopts the technology with probability ρ B under Bertrand competition. Now, for a = 40, c = 30 and γ = 1/2, the relevant range of x is 0 < x < 20. Let, x = 3. Then the relevant range of r is 13.5644 < r < 14.08. Now, if r = 14, ρ B = 0.708333, SW C (c , c ) = 66.64 > ESW B = 60.2474, where ESW B is the expected social welfare under Bertrand competition.
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Research in Economics journal homepage: www.elsevier.com/locate/rie
Technology adoption in a differentiated duopoly: Cournot versus Bertrand Rupayan Pal ∗ Indira Gandhi Institute of Development Research (IGIDR), India
article
info
Article history: Received 6 May 2009 Accepted 16 November 2009 Keywords: Differentiated duopoly Limit-pricing Price effect Selection effect Technology adoption
abstract This paper shows that the cost as well as the effectiveness of technology has a differential impact on technology adoption under two alternative modes of competition. If the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt technology than Cournot competition unless the effectiveness of the technology is very low. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms. This demonstrates that the commonly subscribed assumption of ‘positive primary outputs’ restricts (inflates) the scope of higher degree of technology adoption under Bertrand (Cournot) competition. Moreover, in contrast to standard welfare ranking, it shows that Cournot competition leads to higher social welfare than Bertrand competition under certain situations. © 2009 University of Venice. Published by Elsevier Ltd. All rights reserved.
1. Introduction This paper analyses the incentives to adopt cost-reducing technology by firms in a horizontally differentiated industry under two alternative categories of product market competition, Cournot and Bertrand. When the cost of production is endogenously determined, whether Bertrand competition or Cournot competition provides the stronger incentive for technology adoption is an essential concern. Our framework allows us to address the question of how the cost of the technology as well as its effectiveness affects this comparison in a more general setting that does not rely on the commonly subscribed assumption of ‘positive primary outputs’—‘both firms sell positive outputs even if prices are set at respective marginal costs’.1 Whether higher or lower intensity of product market competition provides greater incentive to adopt cost-reducing technology is of perennial interest. A large literature, dating at least as far back as Schumpeter (1943), emphasizes the role of the intensity of competition on innovation. Schumpeter (1943) argues that, since the possibility to realize returns from technological advancement is higher in concentrated markets, market concentration stimulates innovation. In contrast, Arrow (1962) shows, comparing a perfectly competitive industry with a monopoly, that the gain from adopting cost-reducing technology is higher under a competitive environment. This indicates that a more competitive environment provides a higher incentive to innovate. Recently, attention has turned to the comparison of two oligopolistic industries. A number of recent studies, considering different scenarios, compare firms’ incentives to introduce cost-reducing technologies under alternative modes of product market competition. This helps us to understand a variety of issues: the role of the nature of product differentiation (Bester and Petrakis, 1993; Bonnano and Haworth, 1998), speed of technological
∗ Corresponding address: Indira Gandhi Institute of Development Research (IGIDR), Film City Road, Gen A. K. Vaidya Marg, Goregaon (East), Mumbai 400065, India. Tel.: +91 22 28416545; fax: +91 22 28402752. E-mail addresses: [email protected], [email protected]. 1 This assumption is crucial for ranking of the equilibrium outputs and profits under the two categories of competition (Zanchettin, 2006; Amir and Jin, 2001). 1090-9443/$ – see front matter © 2009 University of Venice. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.rie.2009.11.003
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progress (Aghion et al., 19997), conflict between static and dynamic efficiency (Delbono and Denicolo, 1990), impact of competition intensity (Boone, 2001), incentives in mixed oligopoly (Lin and Ogawa, 2005), etc. While characterizing equilibrium outcomes, these studies subscribe to the assumption of ‘positive primary outputs’ and thus restrict the parameter space, which is likely to distort the equilibrium outcomes. Also, to the best of our knowledge, existing studies do not analyse the impact of the cost of technology on technology adoption decision explicitly. This paper attempts to fill these gaps. We consider a two-stage non-cooperative game between two firms. Initially, both firms have symmetric cost functions. In the first stage, each firm simultaneously and independently decides whether to adopt a cost-reducing technology, by incurring some given cost, or not. In the second stage, firms engage either in Cournot competition or in Bertrand competition. The analysis shows that, if the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt technology than Cournot competition unless the cost-reducing effect of the technology is very low: an ‘Arrow-like’ result. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms: a ‘Schumpeter-type’ result. The intuition behind our result is as follows. Following technology adoption, Bertrand competition not only leads to lower prices (price effect), but also results in a lower market share of the non-adopting firm (selection effect) than Cournot competition. While the price effect generates more disincentive to adopt technology under Bertrand competition than under Cournot competition, the selection effect works in the opposite direction. When only one firm adopts the cost-reducing technology, the selection effect dominates the price effect, and the net effect is higher under Bertrand competition than under Cournot competition unless the cost-reducing effect of the technology is very low. On the other hand, if both firms adopt the technology, the selection effect vanishes and the gain from technology adoption becomes higher under Cournot competition than that under Bertrand competition. Needless to say, a firm’s gain from technology adoption is higher when only that firm adopts the technology. Therefore, when the cost of the technology is low, in equilibrium, both firms adopt the technology under Cournot competition whereas under Bertrand competition only one firm adopts the technology. This paper also shows that the scope for superior outcomes in terms of technology adoption under Bertrand (Cournot) competition is lower (higher) for lower level of effectiveness of the technology. If the cost-reducing effect of the technology is very low, the equilibrium technology adoption is never better under Bertrand competition than that under Cournot competition. This implies that the assumption of positive primary outputs is likely to restrict the scope of technology adoption under Bertrand competition and bolster that under Cournot competition. Comparison of social welfare under the two modes of product market competition reveals that, when technology adoption is higher under Cournot competition than that under Bertrand competition, Cournot competition may lead to higher social welfare than Bertrand competition. This result is in sharp contrast to the standard social welfare ranking of the two modes of competition. The rest of the paper proceeds as follows. The next section presents the model and characterizes Bertrand and Cournot equilibria. Section 3 presents the comparison of equilibrium outcomes under alternative modes of competition. Section 4 concludes. 2. The model Let us consider an economy with an oligopolistic sector, consisting of two firms, firm 1 and firm 2, that produce a differentiated good and a competitive numeraire sector. Initially, the marginal costs of production of firm 1 and firm 2 are equal to c¯ . That is, we start with a situation in which there is no asymmetry in terms of cost of production between firm 1 and firm 2. Now, before undertaking production decisions, firms can adopt the new technology by incurring an exogenously determined fixed cost r (>0) to reduce the marginal cost of production. If a firm adopts the technology, its marginal cost of production reduces to c (0 < c < c¯ = c + x), whereas the non-adopting firm’s marginal cost remains at c¯ . That is, the costreducing effect (effectiveness) of the technology is x = c¯ − c. Needless to say, such a type of discrete choice of technology is observed in many real-life situations and has received some attention in the literature recently (see, for example, Elberfeld and Nti (2004), Jensen (1992), Lahiri and Ratnasiri (2007), Khan and Ravikumar (2002), and Leung and Tse (2001), to name a few).2 1 We consider that, if there is positive demand for both goods, the direct demand function is qi = 1−γ 2 [(1 −γ )a − pi +γ pj ],
(a > c¯ ; i, j = 1, 2; i 6= j), where qi (pi ) is the quantity (price) of the product of firm i and γ (0 < γ < 1) is the product differentiation parameter.3 The corresponding inverse demand function is pi = a − qi − γ qj (i, j = 1, 2; i 6= j). However, if the prices are such that qj ≤ 0, the demand for good i is qi = a − pi , as in Zanchettin (2006).
2 Considering a dynamic framework, Lahiri and Ratnasiri (2007), Khan and Ravikumar (2002), Leung and Tse (2001) examine technology adoption and its impact on macroeconomic variables. Elberfeld and Nti (2004) and Jensen (1992) analyse technology adoption by firms under uncertain environment considering Cournot competition in the product market. We abstract away, for simplicity, from possible uncertainty about the cost-reducing effect of new technology. 3 The underlying utility function of the representative consumer is U = aq + aq − 1 (q2 + q2 + 2γ q q ) + m, where m is the quantity of the numeraire 1
good.
2
2
1
2
1 2
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R. Pal / Research in Economics 64 (2010) 128–136
Stages of the game involved are as follows. Stage 1 : Firm 1 and firm 2 simultaneously and independently decide whether to adopt the technology or not. Stage 2 : Firms engage either in Cournot competition or in Bertrand competition in the product market. Clearly, in stage 1, there are three possible cases for the decision to adopt the technology: (1) no firm adopts the technology, (2) only one firm, either firm 1 or firm 2, adopts the technology, and (3) both firm 1 and firm 2 adopt the technology.4 The mode of product market competition, Cournot or Bertrand, in the second stage is exogenously determined. We solve this game by the backward induction method. Note that when only one firm, say firm 1, adopts the technology, the mode of competition in the product market matters only if firm 1 cannot engage in monopoly pricing without bearing any competitive pressure from firm 2 (the non-adopting firm). Now, if the technology reduces marginal cost of firm 1 beyond a certain level, i.e., if the ex post difference between the firms’ marginal costs becomes sufficiently high, firm 2 is driven out of the market and firm 1 enjoys absolute monopoly a +c power irrespective of the mode of competition. Now, firm 1 sets the monopoly price pM , if at prices p1 = pM 1 = 1 and 2
¯−c ≥ p2 = c¯ the demand of firm 2’s product is zero, i.e., a(1 − γ ) − c¯ + γ pM 1 ≤ 0 ⇒ x = c of product market competition does not matter when x ≥
(a−¯c )(2−γ ) , γ
0 < γ < 1; 0 < x <
Since the mode
we consider that the reduction in marginal cost of
production through technology adoption is not drastic, i.e., we consider that x < parameter space, in which the mode of product market competition matters, is S=
(a−¯c )(2−γ ) 5 . γ
(a−¯c )(2−γ ) . γ
In other words, the relevant
(a − c¯ )(2 − γ ) . γ
Note that, if prices are set at respective marginal costs, p1 = c and p2 = c¯ , both firms sell positive outputs only if x < (a−¯c )(1−γ ) . Clearly, the assumption of positive primary outputs curtails the parameter space. γ 2.1. Product market equilibrium The second stage of the game is very similar to that in Zanchettin (2006). We, therefore, summarize the equilibrium outcomes of stage 2 without going into further details. We denote the equilibrium quantity of firm i (i = 1, 2) under Cournot competition by qCi (c1 , c2 ); c1 , c2 ∈ {c , c¯ }, where the first (second) argument is the marginal cost of production of firm 1 (firm 2). The superscript ‘C ’ denotes Cournot competition. We denote the equilibrium prices and profits similarly. To denote equilibrium outcomes under Bertrand competition, we replace the superscript ‘C ’ by ‘B’. Cournot competition: The equilibrium outcomes in stage 2 in the case of quantity competition in the product market, given the technology adoption decisions of firms, are as follows. (a−¯c )2
−¯c (Ca) When no firm adopts the technology, qCi (¯c , c¯ ) = pCi (¯c , c¯ ) − c¯ = 2a+γ and πiC (¯c , c¯ ) = (2+γ )2 , i = 1, 2. (a−c )2
a −c
(Cb) When both firms adopt the technology, qCi (c , c ) = pCi (c , c ) − c = 2+γ and πiC (c , c ) = (2+γ )2 − r, i = 1, 2. (Cc) When only one firm, say firm 1, adopts the technology, qC1 (c , c¯ ) = pC1 (c , c¯ )−c = (2−γ )(a−¯c )−γ x , 4−γ 2
π1C (c , c¯ ) =
[(2−γ )(a−¯c )+2x]2 (4−γ 2 )2
− r, π2C (c , c¯ ) =
[(2−γ )(a−¯c )−γ x]2 (4−γ 2 )2
(2−γ )(a−¯c )+2x C , q2 (c , c¯ ) 4−γ 2
= pC2 (c , c¯ )−¯c =
. Alternatively, when only firm 2 adopts the
technology, the equilibrium outcomes will be symmetric. It is evident that a higher reduction (x) in marginal cost of production through technology adoption leads to higher output, price-cost margin and profit of the technology-adopting firm, but lower output, price-cost margin and profit of the nonadopting firm. However, it is easy to check that the non-adopting firm remains active under Cournot competition for all x ∈ S. Bertrand competition: When firms compete in terms of price in the product market in stage 2, given the technology adoption decisions of firms of stage 1, the equilibrium outcomes are as follows. (Ba) When no firm adopts the technology, qBi (¯c , c¯ ) =
pBi (¯c ,¯c )−¯c
(Bb) When both firms adopt the technology, qBi (c , c ) =
1−γ 2
=
pBi (c ,c )−c 1−γ 2
(a−¯c ) (2−γ )(1+γ )
=
(a−¯c )2 (1−γ )
and πiB (¯c , c¯ ) = (2−γ )2 (1+γ ) , i = 1, 2.
(a−c ) (2−γ )(1+γ )
(a−c )2 (1−γ )
and πiB (c , c ) = (2−γ )2 (1+γ ) − r, i = 1, 2.
4 Apart from these, there is also a possibility that firms randomize between the strategies ‘adopt the technology’ and ‘don’t adopt the technology’. 5 Firm 2, i.e., the non-adopting firm, cannot engage in monopoly pricing unless the products are completely different (γ = 0).
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Fig. 1. Technology choice game under Cournot competition.
(Bc) If only one firm, say firm 1, adopts the technology and both firms are active in equilibrium, i.e., if 0 < x < (a−¯c )(2−γ −γ 2 ) , qB1 (c , c¯ ) γ 2 [(2−γ −γ )(a−¯c )+(2−γ 2 )x]2 (4−γ 2 )2 (1−γ 2 )
=
pB1 (c ,¯c )−c
pB (c ,¯c )−¯c (2−γ −γ 2 )(a−¯c )+(2−γ 2 )x , qB2 (c , c¯ ) = 21−γ 2 = (4−γ 2 )(1−γ 2 ) [(2−γ −γ 2 )(a−¯c )−γ x]2 B π2 (c , c¯ ) = . Alternatively, if only (4−γ 2 )2 (1−γ 2 )
1−γ 2
− r and
=
(2−γ −γ 2 )(a−¯c )−γ x , (4−γ 2 )(1−γ 2 )
π1B (c , c¯ ) =
firm 2 adopts the technology,
the equilibrium outcomes will be symmetric. (a−¯c )(2−γ −γ 2 )
(a−¯c )(2−γ )
(Bd) If the reduction in marginal cost due to technology adoption is sufficiently large ( ≤ x < ) γ γ and only one firm adopts the technology, limit-pricing occurs in equilibrium and the non-adopting firm is forced to stay out of the market. The corresponding (limit-pricing) equilibrium outcomes, when only firm 1 adopts the γ x−(a−¯c )(1−γ ) c technology, are as follows: qL2 (c , c¯ ) = pL2 (c , c¯ ) − c¯ = π2L (c , c¯ ) = 0, qL1 (c , c¯ ) = a−¯ , pL1 (c , c¯ ) − c = γ γ and π1L (c , c¯ ) =
(a−¯c ){γ x−(a−¯c )(1−γ )} γ2
− r, where the superscript L denotes limit-pricing under Bertrand equilibrium.
Note that the possibility of limit-pricing increases with a decrease in the degree of product differentiation (increase in
γ ).6 Nonetheless, even if the degree of product differentiation is high (γ is low), it is optimum for the technology-adopting (a−¯c )(2−γ −γ 2 ) firm to engage in limit-pricing, as long as ≤ x < (a−¯c )(γ 2−γ ) is satisfied, which keeps the non-adopting firm out γ
of the market.7 On the contrary, both firms, the technology-adopting firm as well as the non-adopting firm, remain active under Cournot competition for all x ∈ S (Zanchettin, 2006). 2.2. Equilibrium technology adoption Cournot competition: We now turn to the technology adoption decision in stage 1 of the game, when firms are engaged in Cournot competition in stage 2. The normal form of the stage 1 game is depicted in Fig. 1. The profit expressions are as given in (Ca)–(Cc), in Section 2.1. From Fig. 1, it is easy to observe that, if firm 2 does not adopt the technology, firm 1 adopts the technology provided that π1C (c , c¯ ) > π1C (¯c , c¯ ) ⇒ r <
4x2 +4(2−γ )(a−¯c )x (4−γ 2 )2
= r¯ C . On the other hand, if firm 1 adopts the technology, firm 2 does
not adopt the technology provided that π2C (c , c¯ ) > π2C (c , c ) ⇒ r >
4(1−γ )x2 +4(2−γ )(a−¯c )x (4−γ 2 )2 C
= r C . Clearly, r C < r¯ C . Also,
note that firms are ex ante symmetric in terms of cost of production. Therefore, if r ≥ r¯ , ‘Don’t Adopt’ is the dominant strategy of each firm and no firm adopts the technology in equilibrium. Alternatively, if r ≤ r C , ‘Adopt’ becomes the dominant strategy of each firm and both firms adopt the technology in equilibrium. If the cost of the technology lies in between these two extreme values, i.e., if r C < r < r¯ C , we have π1C (c , c¯ ) > π1C (¯c , c¯ ) > π1C (¯c , c ) > π1C (c , c ) and π2C (¯c , c ) > π2C (¯c , c¯ ) > π2C (c , c¯ ) > π2C (c , c ). This implies that, if r C < r < r¯ C , there are two pure-strategy Nash equilibria: (Adopt, Don’t Adopt) and (Don’t Adopt, Adopt). Therefore, if the cost of the technology is in the intermediate range (r C < r < r¯ C ), only one firm adopts the technology in equilibrium.8 Lemma 1. When firms are engaged in Cournot competition in the product market, the equilibrium technology adoption is as follows. 4(1−γ )x2 +4(2−γ )(a−¯c )x . (4−γ 2 )2 4x2 +4(2−γ )(a−¯c )x . (4−γ 2 )2
(a) If r ≤ r C , both firms adopt the technology, where r C = (b) If r ≥ r¯ C , no firm adopts the technology, where r¯ C = (c) If r C < r < r¯ C , only one firm adopts the technology.
6 The range of the limit-pricing region is (a − c¯ )γ , which is positively related to γ . 7 Note that the technology-adopting firm cannot engage in monopoly pricing without bearing any competitive pressure from the non-adopting firm, (a−¯c )(2−γ ) since we consider that the reduction in marginal cost through technology adoption is not drastic (i.e., x < ). That is, the technology-adopting γ firm cannot enjoy absolute monopoly power. In other words, though the non-adopting firm is driven out of the market, it exerts competitive pressure on the technology-adopting firm. 8 Other than these two pure-strategy Nash equilibria, none of which risk-dominates the other in the sense of Harsanyi and Selten (1988), there is also a C
mixed-strategy Nash equilibrium in which each firm chooses to adopt the technology with probability ρ C = r¯r¯C −−rrC , 0 < ρ C < 1.
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Fig. 2. Technology choice game when limit-pricing is possible.
Bertrand competition: Let us now analyse the technology adoption decision of firms in stage 1, when there is Bertrand competition in the product market. From (Ba)–(Bd), in Section 2.1, it is clear that there are two possible scenarios under Bertrand competition: (a)
(a−¯c )(2−γ −γ 2 ) γ
firm adopts the technology; and (b) 0 We depict the normal form of the (a−¯c )(2−γ ) ) γ
(a−¯c )(2−γ ) , i.e., limit-pricing occurs in equilibrium at stage 2 when only one γ (a−¯c )(2−γ −γ 2 )
≤x<
in Fig. 2. Now, we characterize the equilibrium outcomes of the above game following the same line as in case of Cournot competition. In this case, if firm 2 does not adopt the technology, firm 1 adopts the technology provided that
π1L (c , c¯ ) > π1B (¯c , c¯ ), i.e., r <
(a−¯c ){γ x−(a−¯c )(1−γ )} γ2
−
(a−¯c )2 (1−γ ) (2−γ )2 (1+γ )
= r¯ L . On the other hand, if firm 1 adopts the technology, (a−c )2 (1−γ )
firm 2 does not adopt the technology provided that π2L (c , c¯ ) > π2B (c , c ), i.e., r > (2−γ )2 (1+γ ) = r L . It is easy to check that r L < r¯ L .9 So, we have π1L (c , c¯ ) > π1B (¯c , c¯ ) > π1L (¯c , c ) > π1B (c , c ) and π2L (¯c , c ) > π2B (¯c , c¯ ) > π2L (c , c¯ ) > π2B (c , c ). Note that the rank of profits corresponding to different sets of strategies of firms are the same as in Cournot competition. Clearly, if r ≤ r L , both firms adopt the technology in equilibrium. But, if r ≥ r¯ L , in equilibrium, no firm adopts the technology. If r L < r < r¯ L , there are two pure-strategy Nash equilibria, (Adopt, Adopt) and (Don’t Adopt, Don’t Adopt).10 Therefore, if r L < r < r¯ L , only one firm adopts the technology in equilibrium. (a−¯c )(2−γ −γ 2 )
In the second scenario (0 < x < ), both firms remain active in the product market even when only one firm γ adopts the technology. We present the corresponding payoff matrix in Fig. 3. It is easy to check that in the second scenario the equilibrium outcomes are similar to those in the first scenario. To be explicit, if 0 < x <
(a−¯c )(2−γ −γ 2 ) , γ
in equilibrium, (a) both firms adopt the technology when r ≤ r B , (b) no firm
adopts the technology when r ≥ r¯ , and (c) only one firm adopts the technology when r B < r < r¯ B ; where r B = B
(a−c )2 (1−γ ) −γ 2 )(a−¯c )−γ x]2 − [(2−γ (2−γ )2 (1+γ ) (4−γ 2 )2 (1−γ 2 )
and r¯ B =
[(2−γ −γ 2 )(a−¯c )+(2−γ 2 )x]2 −¯c )2 (1−γ ) 11 − ((2a−γ . However, note that the critical values of the (4−γ 2 )2 (1−γ 2 ) )2 (1+γ )
cost of technology in the second scenario are lower than the corresponding critical values in the case of limit-pricing: r B < r L and r¯ B < r¯ L .12 This indicates that the possibility of both firms (no firm) adopting the technology is higher (lower) under limitpricing compared to that when limit pricing is not possible. Therefore, the assumption of positive primary outputs leads to underestimation of the possibility of technology adoption.13 Lemma 2. If firms are engaged in Bertrand competition in the product market, the equilibrium technology adoption is as follows. (a) When
(a−¯c )(2−γ −γ 2 ) γ
≤x<
(a−¯c )(2−γ ) , γ
(a−c )2 (1−γ )
(i) both firms adopt the technology, if r ≤ r L = (2−γ )2 (1+γ ) , (ii) no firm adopts the technology, if r ≥ r¯ L =
(a−¯c ){γ x−(a−¯c )(1−γ )} γ2
(iii) only one firm adopts the technology, if r L < r < r¯ L . (b) When x <
(a−¯c )(2−γ −γ 2 ) , γ
−
(a−¯c )2 (1−γ ) , (2−γ )2 (1+γ )
(a−c )2 (1−γ )
[(2−γ −γ 2 )(a−¯c )−γ x]2 , (4−γ 2 )2 (1−γ 2 ) [(2−γ −γ 2 )(a−¯c )+(2−γ 2 )x]2 (a−¯c )2 (1−γ ) − (2−γ )2 (1+γ ) , (4−γ 2 )2 (1−γ 2 )
(i) both firms adopt the technology, if r ≤ r B = (2−γ )2 (1+γ ) − (ii) no firm adopts the technology, if r ≥ r¯ B =
(iii) only one firm adopts the technology, if r B < r < r¯ B .
9 r L < r¯ L ⇒ [(a − c¯ )xγ {4 − γ (2 − γ )(1 + γ )} − (a − c¯ )2 (1 − γ )(4 − γ 2 + γ 3 ) − x2 γ 2 (1 − γ )] > 0, which is true for all x ∈ [ (a−¯c )(2−γ −γ 2 ) , (a−¯c )(2−γ ) ). γ γ 10 In this case, there is also a mixed-strategy Nash equilibrium in which each firm chooses to adopt the technology with probability ρ L = r¯ L −r , r¯ L −r L
0 < ρ L < 1. 11 There is also a mixed-strategy Nash equilibrium, if 0 < x < (a−¯c )(2−γ −γ 2 ) and r B < r < r¯ B , in which each firm chooses to adopt the technology with γ B probability ρ B = r¯r¯B −−rrB , 0 < ρ B < 1. 12 r B − r L = − [(2−γ −γ 2 )(a−¯c )−γ x]2 < 0 and π L (c , c¯ ) > π B (c , c¯ ) ⇒ r¯ B < r¯ L . 1 1 (4−γ 2 )2 (1−γ 2 )
13 Under the assumption of positive primary outputs we must have x < (a−¯c )(1−γ ) (< (a−¯c )(2−γ −γ 2 ) ). γ γ
R. Pal / Research in Economics 64 (2010) 128–136
133
Fig. 3. Technology choice game when limit-pricing is not possible.
Fig. 4. Cost of technology and technology adoption.
3. Comparison of Cournot and Bertrand equilibria In this section, we compare Cournot and Bertrand equilibria. We focus on the technology adoption decision and on social welfare under alternative modes of product market competition. Technology adoption when limit-pricing is possible. Let us begin with the scenario in which the marginal cost reduction (x) (a−¯c )(2−γ −γ 2 )
(a−¯c )(2−γ )
through technology adoption is relatively high, that is, ≤x< , which leads to limit-pricing under γ γ Bertrand equilibrium (see (Bd) in Section 2.1). In this case, it is straightforward to observe that the relevant critical values of the cost of the technology, as given in Lemmas 1 and 2, satisfy the following relation: 0 < r L < r C < r¯ C < r¯ L .
(1)
The above inequality implies that, if the cost of the technology is high (r¯ < r < r¯ ), one firm adopts the technology under Bertrand competition whereas none adopts under Cournot competition. That is, we get very asymmetric outcomes under two alternative modes of product market competition. However, if the cost of the technology is moderate (r C < r < r¯ C ), under both Cournot and Bertrand competition only one firm adopts the technology in equilibrium. Nonetheless, the incentive to adopt technology by a single firm is higher under Bertrand competition than under Cournot competition, since π1L (c , c¯ ) − π1B (¯c , c¯ ) > π1C (c , c¯ ) − π1C (¯c , c¯ ). This is because, following technology adoption, Bertrand competition not only leads to lower prices (price effect), but also a lower market share of the non-adopting firm (selection effect) than Cournot competition. The selection effect dominates the price effect, and the net gain, from these two opposing effects, of the technology-adopting firm is higher under Bertrand competition than under Cournot competition. As a result, when the cost of the technology is high, firms do not find it profitable to adopt the technology under Cournot competition even when the other firm does not adopt the technology, whereas under Bertrand competition one firm gains by adopting the technology when the other firm does not opt to adopt it: an ‘Arrow-like’ result. However, if the cost of the technology reduces to a moderate level (r C < r < r¯ C ), the net gain from the two opposing effects, price effect and selection effect, becomes higher than the cost of the technology under Cournot competition also. As a result, when the cost of the technology is moderate, we get symmetric equilibrium outcomes in terms of technology adoption under Cournot and Bertrand competition: one firm adopts the technology irrespective of the mode of competition. If the cost of the technology reduces further to a low level, r L < r < r C , both firms find technology adoption to be gainful under Cournot competition whereas under Bertrand competition a firm gains by adopting the technology only if the other firm does not adopt the technology, since the price effect is smaller under Cournot competition than that under Bertrand competition. If under Bertrand competition both firms adopt the technology, each firm incurs losses. Therefore, when the cost of the technology is low, in equilibrium, both firms adopt the technology under Cournot competition whereas only one firm adopts the technology under Bertrand competition: a ‘Schumpeter-type’ result. The cost of the technology needs to be reduced even further to a very low level (r < r L ) to induce both firms to adopt the technology under Bertrand competition also. Fig. 4 depicts the pure-strategy Nash equilibrium choice of firms regarding technology adoption corresponding to different levels of the cost of the technology under alternative modes of competition. Clearly, the range of the cost of the technology in which both firms adopt the technology is higher under Cournot competition than that under Bertrand competition. On the contrary, the range in which no firm adopts the technology is lower under Bertrand competition than that under Cournot competition. C
L
Technology adoption when limit-pricing is not possible. We now turn to the scenario in which the marginal cost reduction (x) through technology adoption is lower than that in the earlier scenario; that is, now we consider that 0 < x <
(a−¯c )(2−γ −γ 2 ) . γ
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R. Pal / Research in Economics 64 (2010) 128–136
Fig. 5. Adoption of less superior technology.
In this scenario, as discussed in Section 2.1, both firms remain active in the market irrespective of the mode of product market competition and the cost of the technology. Therefore, under Bertrand competition, a firm’s gain from technology adoption when the other firm does not adopt the technology is lower than that in the previous scenario. Also, under Bertrand competition, when one firm adopts the technology, the other firm will also find it profitable to adopt the technology provided that the cost of the technology is less than what is required in the earlier case, since the cost-reducing effect of the technology is now lower than that in the earlier case. This implies that r B < r L and r¯ B < r¯ L , as noted in Section 2.2. Nonetheless, comparing critical values of the cost of the technology, as given in Lemmas 1 and 2, we get similar rankings as in (1): 2(1−γ )(a−¯c ) 2(1−γ )(a−¯c ) 0 < r B < r C < r¯ C < r¯ B unless x < . However, if x < , the relative position of r¯ B and r¯ C changes, γ γ i.e., we get 0 < r B < r C and r¯ B < r¯ C . This is because the unilateral gain from technology adoption by a single firm declines at a higher rate, due to a decrease in the cost-reducing effect of the technology, under Bertrand competition than that under
∂(π1B (c ,¯c )−π1B (¯c ,¯c )) ∂(π C (c ,¯c )−π C (¯c ,¯c )) > 1 ∂x 1 . ∂x 2(1−γ )(a−¯c ) (a−¯c )(2−γ −γ 2 )
Cournot competition:
Clearly, if However, the critical ranges of the cost of the technology are now quite different from the earlier scenario. Considering the entire range of the cost of the technology, we can say that the possibility of diverse outcomes in terms of technology adoption under alternative modes of product market competition is now higher (lower) for relatively lower (higher) cost of the technology, since r B < r L < r C and r¯ C < r¯ B < r¯ L . To be more specific, the scope for better outcomes under Cournot (Bertrand) competition is now higher (lower). 2(1−γ )(a−¯c ) , the comparison is as follows. (a) If r¯ B < r < r¯ C , no firm adopts the technology On the other hand, if x < γ under Bertrand competition whereas one firm adopts the technology under Cournot competition. (b) If r C < r < r¯ B , one firm adopts the technology irrespective of the mode of product market competition. (c) If r B < r < r C , both firms adopt the technology under Cournot competition whereas only one firm adopts the technology under Bertrand competition. (d) If 0 < r < r B (r > r¯ C ), both firms (no firm) adopt(s) the technology irrespective of the mode of product market competition. Fig. 5 depicts the choice of such technology, in equilibrium, under alternative modes of product market competition. 2(1−γ )(a−¯c ) Clearly, if the cost-reducing effect of the technology is low (x < ), the range of the cost of the technology in γ which no firm adopts the technology is higher under Bertrand competition than that under Cournot competition. Moreover, in this case, the equilibrium technology adoption is never better under Bertrand competition than that under Cournot 2(1−γ )(a−¯c ) competition.14 These findings are in sharp contrast to those in the case of x > . γ
Proposition 1. In the case of pure-strategy Nash equilibrium, the comparison of technology adoption under alternative modes of product market competition is as follows. (i) If the effectiveness of the new technology is high (
(a−¯c )(2−γ −γ 2 ) γ
≤x<
(a−¯c )(2−γ ) ), γ
Bertrand competition fares better than
Cournot competition in terms of technology adoption provided that the cost of the technology is high (r¯ C < r < r¯ L ). On the contrary, if the cost of the technology is low (r L < r < r C ), technology adoption is higher under Cournot competition than that under Bertrand competition. In all other ranges of the cost of the technology, technology adoption remains same under alternative modes of product market competition. (ii) The scope for better outcomes, in terms of technology adoption, under Cournot (Bertrand) competition is higher (lower), if the technology reduces the marginal cost of production to a lesser extent (i.e., if x <
(a−¯c )(2−γ −γ 2 ) ). γ
14 In the case of mixed-strategy Nash equilibrium, the comparison of Cournot competition and Bertrand competition in terms of technology adoption is similar to that in the case of pure-strategy Nash equilibrium. However, unlike in the case of pure-strategy Nash equilibrium, in this case identical outcomes in terms of technology adoption under Cournot competition and Bertrand competition are possible only for extreme values of the cost of the technology. If
(a−¯c )(2−γ −γ 2 ) γ
≤ x <
(a−¯c )(2−γ ) , γ
ρ C is greater (less) than ρ L , if the cost of the technology is less (greater) than r CL (=
r¯ C (¯r L −r L )−¯r L (¯r C −r C ) ), (¯r L −r L )−(¯r C −r C )
since 0 < r L < r C < r CL < r¯ C < r¯ L . Clearly, in this case, Cournot (Bertrand) competition fares better than Bertrand (Cournot) competition in terms of 2(1−γ )(a−¯c )
(a−¯c )(2−γ −γ 2 )
technology adoption by firms, if r L < r < r CL (r CL < r < r¯ L ): a ‘Schumpeter-type’ result (an ‘Arrow-like’ result). If
r¯ C (¯r B −r B )−¯r B (¯r C −r C ) . (¯r B −r B )−(¯r C −r C )
However, if x <
2(1−γ )(a−¯c )
γ
, we get 0 < r B < r C < r¯ B < r¯ C < r CB , which implies that
the probability of technology adoption is never higher under Bertrand competition than that under Cournot competition.
R. Pal / Research in Economics 64 (2010) 128–136
135
2(1−γ )(a−¯c )
(iii) If the technology is far less superior (i.e., if x < ), the equilibrium technology adoption is never better under γ Bertrand competition than that under Cournot competition, and the extent of technology adoption is strictly higher under Cournot competition provided that r B < r < r C or r¯ B < r < r¯ C . Clearly, in pure-strategy Nash equilibrium, the possibility of ex post cost asymmetry is higher under Bertrand competition than that under Cournot competition. It is evident that whether Bertrand competition or Cournot competition induces firms more towards technology adoption or not crucially depends on the level of effectiveness of the technology and its cost. Unless the effectiveness of the technology is very low, Bertrand competition leads to superior outcomes in terms of technology adoption than Cournot competition, if the cost of the technology is high. This result has implications to ‘technology subsidy’ policies. Moreover, note that the scope for superior outcome under Bertrand competition increases with the increase in cost-reducing effect of the technology. (a−¯c )(1−γ ) This implies that the assumption of positive primary outputs (0 < x < ) tends to restrict (inflate) the scope γ of higher degree of technology adoption under Bertrand competition (Cournot competition). Proposition 1 also suggests that the relation between intensity of competition and technology adoption is not necessarily monotonic, which is in line with Boone (2001). In order to compare our results with that of Bester and Petrakis (1993), note that, if γ is closer to one,
2−γ −γ 2
is closer
γ (a−¯c )(2−γ −γ 2 ) γ
to zero. That is, if the degree of substitutability between products is very high, the possibility of x > is high. Therefore, we can say that Bertrand competition fares better in terms of technology adoption than Cournot competition when the products are close substitutes provided that the cost of the technology is high. On the other hand, if the products are 2(1−γ )(a−¯c ) highly differentiated (γ is close to zero), is very large, and, thus, Cournot competition leads to superior/equivalent γ outcomes in terms of technology adoption compared to that under Bertrand competition. These results are similar to those in Bester and Petrakis (1993), which argues that the incentive to innovate is higher (lower) under Cournot competition than that under Bertrand competition, if the degree of substitutability is low (high). However, note that the results of Bester and Petrakis (1993) hold only for a selected range(s) of the cost of the technology. Therefore, in our set-up, the results of Bester and Petrakis (1993) emerge in special cases. Social welfare ranking. Finally, we turn to the social welfare (SW) ranking of the two modes of product market competition. It is evident that if technology adoption is identical under Cournot and Bertrand competition, the standard welfare ranking holds — social welfare under Bertrand competition is higher than social welfare under Cournot competition. To illustrate this further, if both firms adopt the technology or none adopts the technology, a switch from Cournot to Bertrand competition increases the consumer surplus by an amount that is more than the corresponding decrease in industry profits, resulting in higher social welfare. The same mechanism works also in the case when only one firm adopts the technology irrespective of the mode of competition, provided that the cost-reducing effect of the technology is sufficiently low; that is, if the possibility of limit-pricing does not arise. But, if limit-pricing under Bertrand competition is unavoidable, a switch from Cournot to Bertrand competition increases both the consumer surplus and industry profits, leading to an increase in social welfare (Zanchettin, 2006). Welfare ranking of the two modes of competition when they lead to asymmetric outcomes in terms of technology adoption draws special attention. If technology adoption is higher under Bertrand competition than that under Cournot competition, Bertrand competition leads to higher social welfare than Cournot competition, since (a) social welfare is higher under Bertrand competition than that under Cournot competition when technology adoption is identical under the two modes of competition and (b) higher technology adoption leads to higher total surplus irrespective of the mode of competition provided that the cost of the technology is not very high. However, if Cournot competition leads to higher technology adoption than Bertrand competition, the increment in total surplus due to higher technology adoption under Cournot competition may or may not be sufficient to outweigh the loss in total surplus due to a shift from Bertrand to Cournot competition. It depends on the relative magnitudes of these two opposing effects. Upon inspection, we find that, in the case of higher technology adoption under Cournot competition than that under Bertrand competition, Cournot competition leads to higher social welfare than Bertrand competition under plausible parametric configurations, though not always. To illustrate this further, let us consider that r L < r < r C and (2−γ −γ 2 )(a−¯c ) γ
(2−γ )(a−¯c ) . γ
< x <
Note that, in this case, both firms adopt the technology under Cournot competition, but only one firm adopts the technology under Bertrand competition and limit-pricing occurs under Bertrand competition. It is straightforward to check that the corresponding social welfare under Bertrand competition is less than that under Cournot competition unless the cost of the technology is more than a critical value (r1 ).15 Upon inspection, we find that r < r1 holds for many plausible parametric configurations. For example, if a = 40, c¯ = 30 and γ = 12 , the relevant range of x is 25 < x < 30 and r L < r1 holds for some plausible parametric configurations.16 This implies that, in this case, we can
15 In this case, social welfare under Cournot and Bertrand competition are, respectively, SW C (c , c ) (a−¯c ){2γ (a−c )−(a−¯c )} 2γ 2
C
(a−c )2 (3+γ ) (2+γ )2
− 2r and SW L (c , c¯ ) =
(a−c )2 (4−4γ −γ 2 )−2(a−c )xγ (4−2γ −γ 2 )+x2 2γ 2 (3+γ ) . 2γ 2 (2+γ )2
− r. SW (c , c¯ ) < SW (c , c ) ⇒r < r1 , where r1 = 16 It is easy to check that the equilibrium outputs and profit(s) are positive at a = 40, c¯ = 30 and γ = L
=
1 , 2
in all combination of ranges of x and r.
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have higher social welfare under Cournot competition than that under Bertrand competition. Similarly, we can demonstrate that Cournot competition can lead to higher social welfare than Bertrand competition in the remaining two cases, in which technology adoption is higher under Cournot competition than that under Bertrand competition, as well.17 The above finding is in sharp contrast to the standard welfare ranking of the two modes of product market competition. In other words, when the cost of production is endogenously determined through technology adoption by firms, the standard welfare ranking of the two modes of product market competition may be reversed. Proposition 2. When the cost of production is endogenously determined through technology adoption by firms, Cournot competition may lead to higher social welfare than Bertrand competition. 4. Conclusion In this paper we have compared technology adoption in a differentiated duopoly under two alternative modes of product market competition, Cournot and Bertrand. We have analysed how the quality of the technology and its cost affect this comparison in a more general set-up by enlarging the parameter space so as to relax the commonly subscribed assumption of positive primary outputs. We have shown that, if the cost of the technology is high, Bertrand competition provides a stronger incentive to adopt the technology than Cournot competition unless the effectiveness of the technology is very low. On the contrary, if the cost of the technology is low, Cournot competition fares better than Bertrand competition in terms of technology adoption by firms. Moreover, in contrast to standard welfare ranking of the two modes of product market competition, we have shown that Cournot competition may lead to higher social welfare than Bertrand competition, if technology adoption is higher in the case of Cournot competition than that in the case of Bertrand competition. We have also shown that the scope for superior outcome in terms of technology adoption under Bertrand (Cournot) competition is lower (higher) for lower cost-reducing effect of the technology. If the cost-reducing effect of the technology is very low, the equilibrium technology adoption is never better under Bertrand competition than that under Cournot competition. This implies that the assumption of positive primary outputs restricts (inflates) the scope of higher degree of technology adoption under Bertrand (Cournot) competition. It seems to be interesting to extend the present analysis by considering a possible trade-off between product and process innovation. It might also be interesting to examine the implications of (semi)collusion on technology adoption and profitability in the present context. We leave these issues for future research. Acknowledgements I wish to thank the anonymous referees for detailed comments that have led to substantial improvements in the paper. I am solely responsible for any remaining errors. References Aghion, P., Harris, C., Vickers, J., 1997. Competition and growth with step-by-step innovation: An example. European Economic Review 41, 771–782. Amir, R., Jin, J., 2001. Cournot and Bertrand equilibria compared: Substitutability, complementarity and concavity. International Journal of Industrial Organization 19, 303–317. Arrow, K., 1962. Economic welfare and the allocation of resources for inventions. In: Nelson, R. (Ed.), The Rate and Direction of Inventive Activity. Princeton University Press, USA. Bester, H., Petrakis, E., 1993. The incentives for cost reduction in a differentiated industry. 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17 This result goes through in the case of mixed-strategy Nash equilibrium also. Note that, if r B < r < r C and 0 < x < (a−¯c )(2−γ −γ 2 ) , both firms adopt γ
the technology under Cournot competition and each firm adopts the technology with probability ρ B under Bertrand competition. Now, for a = 40, c = 30 and γ = 1/2, the relevant range of x is 0 < x < 20. Let, x = 3. Then the relevant range of r is 13.5644 < r < 14.08. Now, if r = 14, ρ B = 0.708333, SW C (c , c ) = 66.64 > ESW B = 60.2474, where ESW B is the expected social welfare under Bertrand competition.