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Unexploited comparative advantages in a differentiated duopoly
Ricerche Economiche (1996) 50, 183–191
Unexploited comparative advantages in a differentiated duopoly PAOLO G. GARELLA† Dipartimento di Scienze Economiche, Universita` di Bologna, Strada Maggiore 45, 40123-Bologna, Italy (Received 10 November 1992, accepted for publication 17 February 1995)
Summary The present paper analyses the question whether firms choose product varieties for which they enjoy a comparative advantage with respect to their rivals. In a limited set-up, that of a vertically differentiated duopoly, it is here found that firms may not choose in such an optimal way, but rather end up in “perverse” equilibria where the firm most efficient in producing a high quality variant of a product produces instead the low quality one, and leaves to the less efficient rival the high quality position. 1996 Academic Press Limited J.E.L. Classification: L13. Keywords: Oligopoly theory, vertical differentiation, product quality.
1. Introduction An obvious conjecture to put forward for the behaviour of firms in oligopoly is that when a firm decides to select a product design, it will choose the one where it has a competitive advantage over the rivals. It is my purpose here to put under scrutiny this idea for a duopoly model of vertical product differentiation. In the case studied below the firms choose positions in the space of characteristics. Only two such positions are available: high and low quality. The model is based on Bonanno (1986), while a version in the set-up of Gabszewicz and Thisse (1979) is available upon request. The decision process is represented by a two-stage game. At the first stage the firms choose the quality level; at the second they compete in prices. The firms are asymmetric in the cost function. The cost of producing the low quality is the same for both (to simplify the algebra), while the cost of producing the high † The author wishes to thank two anonymous referees for helpful comments. He also wishes to thank I.G.I.E.R. (Milano) for generous hospitality. Mailing address: Dipartimento di Scienze Economiche, Strada Maggiore 45, 40123-Bologna (e-mail: [email protected]). 183 0035–5054/96/020183+09 $18.00/0
1996 Academic Press Limited
184
P. G. GARELLA
quality is different. A firm that selects the expensive variety, at any rate, incurs higher costs, but enjoys the advantage of selling a superior product. The one that chooses the low quality has the advantage of cheap production, with an inferior product. As for the high quality, if the cost disadvantage is low enough, the premium for quality commanded by the equilibrium price may make the choice worth the cost. This difference between cost and benefit may induce even the firm that is comparatively less able to choose the high quality. The main result found in the present paper is that the exploitation of comparative advantages in production is not the only possible outcome of competition between the two firms. This is due to the existence of two equilibria in the analysed game; one where the comparative advantages dictate the choice of product, and the other where the choice is made so as to counter the application of the comparative advantages. Furthermore, it is found that the sum of profits may be higher at the “perverse” than at the “natural” equilibrium, so that even if firms coordinate their choices they may not behave as the pursuit of the social optimum suggests.
2. The model The present model is taken from Bonanno (1986). Consider a market for an indivisible good. Each consumer has the choice whether to buy one unit or none of the good. The income of a consumer is denoted by t, where tv[0, 1], and with one consumer for each value of t, so that consumers form a continuum. The consumer’s utility has the form Ut=h(t−ph) if she buys quality “high” and Ut=l(t−pl) if she buys quality “low”, with h1l11, and where ph, and pl represent the prices for high and low quality respectively. If she buys nothing her utility is t. The decision whether to buy quality high instead of buying nothing is “yes” if t≤h(t−ph), and it is “no” otherwise. Similarly for quality low. By the same reason, consumer t buys quality low instead of quality high if l(t−pl)≥h(t−ph). The consumer indifferent between the two qualities, t′, is then identified by t′(ph, pl)=(phh−pll)/(h−l), and the consumer who is indifferent between quality jv{h, l} and the status quo is identified by tj=jpj /(j−1). The inequality ph[(h/(h−1)]1pl[l/(l−1)] ensures that t′1th1tl, while its violation ensures that t′0th0tl. Note also that since h1l11, then it is assured that for any positive price couple one has that th≥0 and tl≥0 are verified. Therefore it is possible to
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conclude that if ph[h/(h−1)]1pl[l/(l−1)], then the demand† to firms h and l are given by Dh(ph, pl)=1−t′(ph,pl) and Dl(ph, pl)=t′(ph, pl)−tl. On the other hand, when the inverse inequality, ph[h/ (h−1)]1pl[l/(l−1)] holds, the demand to h is given by 1−th and that to l is null. (Therefore for any positive price of firm l as small as desired, it is always possible for firm h to get the whole market at price ph0=pl[l(h−1)]/[h(l−1)].) Assume now that both firms have zero costs for producing the low quality, while they incur a positive constant marginal cost for producing the high quality, and let these costs be denoted by a for firm A and b for firm B, with a1b. It would be natural to expect that free competition between the two firms leads the more efficient of the two, here the firm B, to produce the high quality and become the firm labelled h in our notation. What follows is an investigation on whether this expectation is true or false. The firms’ decision process is represented as a two-stage game: at the first stage firms decide what to produce, i.e. if they produce the high or the low quality. At the second stage they compete in prices (Bertrand game) or in quantities (Cournot game). I shall first analyse the Bertrand and then the Cournot game. 2.1.
THE BERTRAND GAME
The solution criterion here adopted is one of backward induction, where the choices at earlier stages are made in the expectation of equilibrium behaviour at later stages. At the beginning of the second stage the firms already know their ”labels”, i.e. firm A may be firm l, and firm B may be firm h, or any other of the four possible combinations. (1) Suppose that both firms have chosen quality l. Then Bertrand competition between two zero-cost rivals ends up in pl=ph=0. Profits are pA=pB=0. (2) Suppose both firms have chosen quality h. Then Bertrand † The complete description of demand is: Dh=Min{1−t′(ph, pl), 1} if ph≤(h−l−lpl)/h, and Dh=0 otherwise. And similarly for the demand function Dl.
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competition stabilizes at prices pl=ph=a if a is smaller than the monopoly price, i.e. if a0phMo((h−1+bh)/2h): firm A produces nothing and firm B captures the “monopolistic” demand 1−th. Profits are pA=0 and pBM=a[1−ah/(h−1)]. Otherwise, if a1((h−1+bh)/2h) prices are pl=a and ph=phMo((h−1+bh)/2h), and profits are pA=0, and pBM=((h−1−bh)2/4h(h−1)). (3) Suppose that firm A has chosen quality l, and that firm B has chosen quality h (the “natural choice”). Then prices stabilize at the prices: (l−1)z 2(h−1)z and pBn = . l h
pAn =
where D=h−l, and z=(D+bh)/(4h−l−3). Therefore profits are pAn =pAn (t′−tl) and pBn =(pBn −b)(1−t′), that is:
A
B
(h−1)(l−1) Dl
pAn =z2 and
C
DA B 2
2D(h−1)−bh(2h−l−1) (4h−l−3)
pBn =
1 . Dh
(4) Suppose that firm B has chosen quality l, and that firm A has chosen quality h. Then prices stabilize at the prices: (l−1)x 2(h−1)x , and p˜A= , l h
p˜B=
where x=(D+ah)/(4h−l−3). Therefore profits are
C
p˜ A=
DA B
2D(h−1)−ah(2h−l−1) (4h−l−3)
and
A
p˜ B=x2
B
(h−1)(l−1) . Dl
2
1 Dh
UNEXPLOITED COMPARATIVE ADVANTAGES
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Consider the following matrix of payoffs to firms A and B of a game where A picks a row and B picks a column.
C
D
(0, 0) (pAn , pBn ) . (p˜ A, p˜ B) (0, pBM)
Where the payoffs in the upper right-hand corner of the matrix are those obtained at the natural choice of qualities (case 3), while the payoffs in the lower left-hand corner are those corresponding to the “perverse” choice (case 4). The choices at the first stage are governed by the magnitudes of the entries in the matrix above. Obviously, the first stage choice of quality h for firm B and quality l for firm A always represents a subgame perfect Nash equilibrium.† Interestingly, however, the “perverse” (case 4) choice may also be a subgame perfect equilibrium if (i) p˜ B≥pBM, and (ii) p˜ A≥0. In the simple case where b=0, i.e. for an extremely efficient firm B, the condition (i) writes as: (i.1)
C
D A 2
B
2(h−1) ah[1−ah/(h−1)] ≥ 4h−3−l D
if a0phM,
and it writes as: (i.2)
C
D
2
(h−1−bh)2 2(h−1) if a1phM. ≥ 4h−3−l 4h(h−1)
Simulation results It is not possible to solve analytically for the range of parameters over which the conditions (i.1) and (i.2) are satisfied (recall that (i.1) and (i.2) assume b=0). The present section, therefore, resumes some results obtained for fixed values of a and l. I have chosen to fix l=1·1 throughout, and compute the range of h values for which the conditions hold corresponding to various values of a. Computing condition (i.2), for av[0·4, 0·9], it is found that, for all these values of a, the condition itself is satisfied for all values of h belonging to the range [1·2, 2·5]. On the other hand, for av[0·4, 0·9], the monopoly price is lower than a provided h02·5, therefore condition (i.2) is the condition that guarantees existence of the perverse equilibrium. † The conditions are simply: (i) pnA≥0, and (ii) pnB≥0.
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As another instance, a=0·1 and for hv[1·7, 3], one has that the relevant condition is (i.1), and that this is satisfied for hv[1·5, 3], therefore the perverse equilibrium also exists for a=0·1 and hv [1·7, 3]. Finally, it must be noted that the sum of profits at the perverse equilibrium, (p˜ A+p˜ B), may exceed the sum of profits at the natural equilibrium, (pA+pB). In particular, and as an example,† this occurs for a=0·5 over the range of h-values [1·2, 1·5], and for a=0·4 over the range of h-values [1·2, 1·6]. This implies that, since there is no Pareto improving allocation of profits if firms decided to coordinate and move from the bad to the good equilibrium, there is no guarantee that cooperation at the first stage leads to the socially desirable choice of qualities. 3. The Cournot game The analysis of quantity competition requires the inversion of the system of demands for the duopoly in the cases (1)–(4) above. As for case (1), where both firms choose the low quality, the inverse demand function is the inverse of ql=1−pl/(l−1). As for case (2), where both firms choose the low quality, the inverse demand function is the inverse of qh=1−ph/(h−1). Accordingly, if both firms choose quality jv{h, l}, the inverse demand is given by p=(1−q)(j−1)/j, where q=qA+qB. The calculations of the Cournot profits for cases (1) and (2) are standard and are not repeated here. In case (1) profits are pcl=(l−1)/9l for both firms, and in case (2) they 2 for firm B, and are pch B =(h−1+ah−2bh) /[9h(h−1)] ch 2 pA =(h−1−2ah+bh) /[9h(h−1)] for firm A. As concerns cases (3) and (4), the demand functions Dh(ph, pl), and Dl(ph, pl) are continuous and can be inverted. In particular, inversion of Dh(ph, pl)=1−t′(ph, pl) gives phh=D(1−qh)+pll, where ql denotes the quantity produced by the firm that has chosen low quality. Furthermore, inversion of Dl(ph, pl)=t′(ph, pl)−tl gives pll=(l−1)/(h−1)hph−qlD. The two equations form a system that, once inverted, gives l−1 (1−qh−ql) l
pl(qh, ql)=
† Computer simulations using “Mathematica” software show that the difference (p˜ A+p˜ B)−(p˜ A+p˜ B) is an increasing and concave function of h, positive for values of h as given in the text.
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and ph(qh, ql)=
h−1 l−1 (1−qh)− ql. h h
It is then easy to write the two maximization programs relative to the solution of the Cournot equilibrium for the cases (3) and (4) where the qualities chosen at the first stage differ. For the firm that has chosen quality h, the program is max[ph(qh, ql)−c]qh, where c is equal to a or b, according to whether it is firm A or firm B that has chosen quality h. For the other firm it is max pl(qh, ql)ql. Let g=4h−l−3; the simultaneous solution of the first order conditions provides the Cournot quantities qnc B o(2h−l−1−2bh)/ c ˜ cAo(2h−l−1−2ah)/g and g and qnc A =(1−qh)/2 for case (3), and q q˜cB=(1−qch)/2 for case (4). This allows the calculation of the profits for cases (3) and (4). cn The expressions for pcn A and pB are: nc nc pcn B =(2h−l−1−2hb)ph(qA , qB )/g, nc nc pcn A =(h−1+b)pl(qA , qB )/g.
Therefore, p˜ cA=(2h−l−1−2ha)ph(q˜cA, q˜cB)/g, is the profit of firm A when it chooses the high quality against the rival choosing low, and p˜ cB is given by p˜ cB=(h−1+a)pl(q˜cA, q˜cB)/g. This completes the analysis of the second stage under Cournot competition. The first stage choices, accordingly, can be made as in a two by two matrix form game. The relevant matrix is
C
pcl, pcl p˜ cA, p˜ cB
D
cn pcn A , pB , ch pA , pch B
cn where the meaning of pcn A , pB is the same as for the Bertrand case: the profits of case (3) correspond to the natural choice of qualities. Similarly, the notation p˜ cA, p˜ cB corresponds to the “perverse” choices. The conditions ensuring that the perverse choice is an equilibrium are apparent from inspection of the entries in the matrix for the Cournot case. They are: (AA) p˜ cA≥pcl, and (BB) p˜ cB≥pch B . The first writes as
(AA)
(2h−l−1−2ha)ph(q˜cA, q˜cB)/g≥(l−1)/9l
and the second as
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(BB)
P. G. GARELLA
(h−1+a)pl(q˜cA, q˜cB)/g≥(h−1+ah−2bh)2/[9h(h−1)].
Simulation results The evaluation of conditions (AA) and (BB) shows that they can be simultaneously verified. For instance, for values a=0·4 and b= 0·35 both conditions are verified if hv[1·2, 1·48]. Or for values a= 0·3 and b=0·2 both conditions are verified if hv[1·2, 1·25]. The range of values for h over which the “perverse” case 4 is an equilibrium seems to be smaller than for the Bertrand game. One should not forget that the remaining strategy combinations (both choose high and both choose low) at the first stage might also constitute an equilibrium here.† Unlike the case of Bertrand competition, in fact, there is nothing to prevent a less efficient firm making positive profits at a Cournot equilibrium. Note finally that, always for a=0·4 and b=0·35, the sum of profits at the perverse equilibrium, when it exists, is higher than the sum of profits at the natural equilibrium.‡ This means, as discussed above for the Bertrand game, that firms, since there must be at least one loser, may not find a way to cooperate and find an agreement to select the “good” equilibrium. 4. Conclusion The above analysis provides a negative answer to the question whether firms always choose their product design following the “natural” dictates of comparative advantages. In particular, a firm that produces a high quality at a cost that exceeds that of a rival may acquire for itself a high quality position in the spectrum of differentiated products. This is made by speculating on the unwillingness of its rival to contest the acquisition. Otherwise stated, a more efficient firm may refuse to contend the position of high quality acquired by a less efficient rival, in order to enjoy the profits acquired through product differentiation. In a sense, using the title of Shaked and Sutton (1982), product differentiation may relax price competition too much: it may lead to an insufficient incentive for firms to engage in battles for the acquisition of the privilege of producing the best quality. One could be tempted to interpret the model in the preceding † Bonanno (1986) also finds that firms with symmetric cost functions may not differentiate at a subgame perfect equilibrium of the Cournot game. ‡ Computer graphics created through the “Mathematica” program show that the difference between profits in case 4 and those in case 3 is an increasing function of h, positive valued for h11·22.
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sections as a model for testing the robustness of the Ricardian comparative advantage theory as it is usually applied to the advantages from trade in general. While the temptation is strong, one should refrain from it and consider that the Ricardian theory is intended to apply to perfectly competitive markets. References Bonanno, G. (1986). Vertical differentiation with Cournot competition. Economic Notes, 15, 68–91. Gabszewicz, J.J. & Thisse, J.-F. (1979). Price competition, quality, and income disparities. Journal of Economic Theory, 20, 340–359. Gabszewicz, J.J. & Thisse, J.-F. (1986). Spatial competition and the location of firms. In H. Sonnenschein & J. Lesournes, Eds. Fundamentals of Pure and Applied Economics: “Location Theory”. London: Harwood Academic. Shaked, A. & Sutton, J. (1982). Relaxing price competition through product differentiation. Review of Economic Studies, 49, 3–13.
Unexploited comparative advantages in a differentiated duopoly PAOLO G. GARELLA† Dipartimento di Scienze Economiche, Universita` di Bologna, Strada Maggiore 45, 40123-Bologna, Italy (Received 10 November 1992, accepted for publication 17 February 1995)
Summary The present paper analyses the question whether firms choose product varieties for which they enjoy a comparative advantage with respect to their rivals. In a limited set-up, that of a vertically differentiated duopoly, it is here found that firms may not choose in such an optimal way, but rather end up in “perverse” equilibria where the firm most efficient in producing a high quality variant of a product produces instead the low quality one, and leaves to the less efficient rival the high quality position. 1996 Academic Press Limited J.E.L. Classification: L13. Keywords: Oligopoly theory, vertical differentiation, product quality.
1. Introduction An obvious conjecture to put forward for the behaviour of firms in oligopoly is that when a firm decides to select a product design, it will choose the one where it has a competitive advantage over the rivals. It is my purpose here to put under scrutiny this idea for a duopoly model of vertical product differentiation. In the case studied below the firms choose positions in the space of characteristics. Only two such positions are available: high and low quality. The model is based on Bonanno (1986), while a version in the set-up of Gabszewicz and Thisse (1979) is available upon request. The decision process is represented by a two-stage game. At the first stage the firms choose the quality level; at the second they compete in prices. The firms are asymmetric in the cost function. The cost of producing the low quality is the same for both (to simplify the algebra), while the cost of producing the high † The author wishes to thank two anonymous referees for helpful comments. He also wishes to thank I.G.I.E.R. (Milano) for generous hospitality. Mailing address: Dipartimento di Scienze Economiche, Strada Maggiore 45, 40123-Bologna (e-mail: [email protected]). 183 0035–5054/96/020183+09 $18.00/0
1996 Academic Press Limited
184
P. G. GARELLA
quality is different. A firm that selects the expensive variety, at any rate, incurs higher costs, but enjoys the advantage of selling a superior product. The one that chooses the low quality has the advantage of cheap production, with an inferior product. As for the high quality, if the cost disadvantage is low enough, the premium for quality commanded by the equilibrium price may make the choice worth the cost. This difference between cost and benefit may induce even the firm that is comparatively less able to choose the high quality. The main result found in the present paper is that the exploitation of comparative advantages in production is not the only possible outcome of competition between the two firms. This is due to the existence of two equilibria in the analysed game; one where the comparative advantages dictate the choice of product, and the other where the choice is made so as to counter the application of the comparative advantages. Furthermore, it is found that the sum of profits may be higher at the “perverse” than at the “natural” equilibrium, so that even if firms coordinate their choices they may not behave as the pursuit of the social optimum suggests.
2. The model The present model is taken from Bonanno (1986). Consider a market for an indivisible good. Each consumer has the choice whether to buy one unit or none of the good. The income of a consumer is denoted by t, where tv[0, 1], and with one consumer for each value of t, so that consumers form a continuum. The consumer’s utility has the form Ut=h(t−ph) if she buys quality “high” and Ut=l(t−pl) if she buys quality “low”, with h1l11, and where ph, and pl represent the prices for high and low quality respectively. If she buys nothing her utility is t. The decision whether to buy quality high instead of buying nothing is “yes” if t≤h(t−ph), and it is “no” otherwise. Similarly for quality low. By the same reason, consumer t buys quality low instead of quality high if l(t−pl)≥h(t−ph). The consumer indifferent between the two qualities, t′, is then identified by t′(ph, pl)=(phh−pll)/(h−l), and the consumer who is indifferent between quality jv{h, l} and the status quo is identified by tj=jpj /(j−1). The inequality ph[(h/(h−1)]1pl[l/(l−1)] ensures that t′1th1tl, while its violation ensures that t′0th0tl. Note also that since h1l11, then it is assured that for any positive price couple one has that th≥0 and tl≥0 are verified. Therefore it is possible to
UNEXPLOITED COMPARATIVE ADVANTAGES
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conclude that if ph[h/(h−1)]1pl[l/(l−1)], then the demand† to firms h and l are given by Dh(ph, pl)=1−t′(ph,pl) and Dl(ph, pl)=t′(ph, pl)−tl. On the other hand, when the inverse inequality, ph[h/ (h−1)]1pl[l/(l−1)] holds, the demand to h is given by 1−th and that to l is null. (Therefore for any positive price of firm l as small as desired, it is always possible for firm h to get the whole market at price ph0=pl[l(h−1)]/[h(l−1)].) Assume now that both firms have zero costs for producing the low quality, while they incur a positive constant marginal cost for producing the high quality, and let these costs be denoted by a for firm A and b for firm B, with a1b. It would be natural to expect that free competition between the two firms leads the more efficient of the two, here the firm B, to produce the high quality and become the firm labelled h in our notation. What follows is an investigation on whether this expectation is true or false. The firms’ decision process is represented as a two-stage game: at the first stage firms decide what to produce, i.e. if they produce the high or the low quality. At the second stage they compete in prices (Bertrand game) or in quantities (Cournot game). I shall first analyse the Bertrand and then the Cournot game. 2.1.
THE BERTRAND GAME
The solution criterion here adopted is one of backward induction, where the choices at earlier stages are made in the expectation of equilibrium behaviour at later stages. At the beginning of the second stage the firms already know their ”labels”, i.e. firm A may be firm l, and firm B may be firm h, or any other of the four possible combinations. (1) Suppose that both firms have chosen quality l. Then Bertrand competition between two zero-cost rivals ends up in pl=ph=0. Profits are pA=pB=0. (2) Suppose both firms have chosen quality h. Then Bertrand † The complete description of demand is: Dh=Min{1−t′(ph, pl), 1} if ph≤(h−l−lpl)/h, and Dh=0 otherwise. And similarly for the demand function Dl.
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P. G. GARELLA
competition stabilizes at prices pl=ph=a if a is smaller than the monopoly price, i.e. if a0phMo((h−1+bh)/2h): firm A produces nothing and firm B captures the “monopolistic” demand 1−th. Profits are pA=0 and pBM=a[1−ah/(h−1)]. Otherwise, if a1((h−1+bh)/2h) prices are pl=a and ph=phMo((h−1+bh)/2h), and profits are pA=0, and pBM=((h−1−bh)2/4h(h−1)). (3) Suppose that firm A has chosen quality l, and that firm B has chosen quality h (the “natural choice”). Then prices stabilize at the prices: (l−1)z 2(h−1)z and pBn = . l h
pAn =
where D=h−l, and z=(D+bh)/(4h−l−3). Therefore profits are pAn =pAn (t′−tl) and pBn =(pBn −b)(1−t′), that is:
A
B
(h−1)(l−1) Dl
pAn =z2 and
C
DA B 2
2D(h−1)−bh(2h−l−1) (4h−l−3)
pBn =
1 . Dh
(4) Suppose that firm B has chosen quality l, and that firm A has chosen quality h. Then prices stabilize at the prices: (l−1)x 2(h−1)x , and p˜A= , l h
p˜B=
where x=(D+ah)/(4h−l−3). Therefore profits are
C
p˜ A=
DA B
2D(h−1)−ah(2h−l−1) (4h−l−3)
and
A
p˜ B=x2
B
(h−1)(l−1) . Dl
2
1 Dh
UNEXPLOITED COMPARATIVE ADVANTAGES
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Consider the following matrix of payoffs to firms A and B of a game where A picks a row and B picks a column.
C
D
(0, 0) (pAn , pBn ) . (p˜ A, p˜ B) (0, pBM)
Where the payoffs in the upper right-hand corner of the matrix are those obtained at the natural choice of qualities (case 3), while the payoffs in the lower left-hand corner are those corresponding to the “perverse” choice (case 4). The choices at the first stage are governed by the magnitudes of the entries in the matrix above. Obviously, the first stage choice of quality h for firm B and quality l for firm A always represents a subgame perfect Nash equilibrium.† Interestingly, however, the “perverse” (case 4) choice may also be a subgame perfect equilibrium if (i) p˜ B≥pBM, and (ii) p˜ A≥0. In the simple case where b=0, i.e. for an extremely efficient firm B, the condition (i) writes as: (i.1)
C
D A 2
B
2(h−1) ah[1−ah/(h−1)] ≥ 4h−3−l D
if a0phM,
and it writes as: (i.2)
C
D
2
(h−1−bh)2 2(h−1) if a1phM. ≥ 4h−3−l 4h(h−1)
Simulation results It is not possible to solve analytically for the range of parameters over which the conditions (i.1) and (i.2) are satisfied (recall that (i.1) and (i.2) assume b=0). The present section, therefore, resumes some results obtained for fixed values of a and l. I have chosen to fix l=1·1 throughout, and compute the range of h values for which the conditions hold corresponding to various values of a. Computing condition (i.2), for av[0·4, 0·9], it is found that, for all these values of a, the condition itself is satisfied for all values of h belonging to the range [1·2, 2·5]. On the other hand, for av[0·4, 0·9], the monopoly price is lower than a provided h02·5, therefore condition (i.2) is the condition that guarantees existence of the perverse equilibrium. † The conditions are simply: (i) pnA≥0, and (ii) pnB≥0.
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As another instance, a=0·1 and for hv[1·7, 3], one has that the relevant condition is (i.1), and that this is satisfied for hv[1·5, 3], therefore the perverse equilibrium also exists for a=0·1 and hv [1·7, 3]. Finally, it must be noted that the sum of profits at the perverse equilibrium, (p˜ A+p˜ B), may exceed the sum of profits at the natural equilibrium, (pA+pB). In particular, and as an example,† this occurs for a=0·5 over the range of h-values [1·2, 1·5], and for a=0·4 over the range of h-values [1·2, 1·6]. This implies that, since there is no Pareto improving allocation of profits if firms decided to coordinate and move from the bad to the good equilibrium, there is no guarantee that cooperation at the first stage leads to the socially desirable choice of qualities. 3. The Cournot game The analysis of quantity competition requires the inversion of the system of demands for the duopoly in the cases (1)–(4) above. As for case (1), where both firms choose the low quality, the inverse demand function is the inverse of ql=1−pl/(l−1). As for case (2), where both firms choose the low quality, the inverse demand function is the inverse of qh=1−ph/(h−1). Accordingly, if both firms choose quality jv{h, l}, the inverse demand is given by p=(1−q)(j−1)/j, where q=qA+qB. The calculations of the Cournot profits for cases (1) and (2) are standard and are not repeated here. In case (1) profits are pcl=(l−1)/9l for both firms, and in case (2) they 2 for firm B, and are pch B =(h−1+ah−2bh) /[9h(h−1)] ch 2 pA =(h−1−2ah+bh) /[9h(h−1)] for firm A. As concerns cases (3) and (4), the demand functions Dh(ph, pl), and Dl(ph, pl) are continuous and can be inverted. In particular, inversion of Dh(ph, pl)=1−t′(ph, pl) gives phh=D(1−qh)+pll, where ql denotes the quantity produced by the firm that has chosen low quality. Furthermore, inversion of Dl(ph, pl)=t′(ph, pl)−tl gives pll=(l−1)/(h−1)hph−qlD. The two equations form a system that, once inverted, gives l−1 (1−qh−ql) l
pl(qh, ql)=
† Computer simulations using “Mathematica” software show that the difference (p˜ A+p˜ B)−(p˜ A+p˜ B) is an increasing and concave function of h, positive for values of h as given in the text.
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and ph(qh, ql)=
h−1 l−1 (1−qh)− ql. h h
It is then easy to write the two maximization programs relative to the solution of the Cournot equilibrium for the cases (3) and (4) where the qualities chosen at the first stage differ. For the firm that has chosen quality h, the program is max[ph(qh, ql)−c]qh, where c is equal to a or b, according to whether it is firm A or firm B that has chosen quality h. For the other firm it is max pl(qh, ql)ql. Let g=4h−l−3; the simultaneous solution of the first order conditions provides the Cournot quantities qnc B o(2h−l−1−2bh)/ c ˜ cAo(2h−l−1−2ah)/g and g and qnc A =(1−qh)/2 for case (3), and q q˜cB=(1−qch)/2 for case (4). This allows the calculation of the profits for cases (3) and (4). cn The expressions for pcn A and pB are: nc nc pcn B =(2h−l−1−2hb)ph(qA , qB )/g, nc nc pcn A =(h−1+b)pl(qA , qB )/g.
Therefore, p˜ cA=(2h−l−1−2ha)ph(q˜cA, q˜cB)/g, is the profit of firm A when it chooses the high quality against the rival choosing low, and p˜ cB is given by p˜ cB=(h−1+a)pl(q˜cA, q˜cB)/g. This completes the analysis of the second stage under Cournot competition. The first stage choices, accordingly, can be made as in a two by two matrix form game. The relevant matrix is
C
pcl, pcl p˜ cA, p˜ cB
D
cn pcn A , pB , ch pA , pch B
cn where the meaning of pcn A , pB is the same as for the Bertrand case: the profits of case (3) correspond to the natural choice of qualities. Similarly, the notation p˜ cA, p˜ cB corresponds to the “perverse” choices. The conditions ensuring that the perverse choice is an equilibrium are apparent from inspection of the entries in the matrix for the Cournot case. They are: (AA) p˜ cA≥pcl, and (BB) p˜ cB≥pch B . The first writes as
(AA)
(2h−l−1−2ha)ph(q˜cA, q˜cB)/g≥(l−1)/9l
and the second as
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(BB)
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(h−1+a)pl(q˜cA, q˜cB)/g≥(h−1+ah−2bh)2/[9h(h−1)].
Simulation results The evaluation of conditions (AA) and (BB) shows that they can be simultaneously verified. For instance, for values a=0·4 and b= 0·35 both conditions are verified if hv[1·2, 1·48]. Or for values a= 0·3 and b=0·2 both conditions are verified if hv[1·2, 1·25]. The range of values for h over which the “perverse” case 4 is an equilibrium seems to be smaller than for the Bertrand game. One should not forget that the remaining strategy combinations (both choose high and both choose low) at the first stage might also constitute an equilibrium here.† Unlike the case of Bertrand competition, in fact, there is nothing to prevent a less efficient firm making positive profits at a Cournot equilibrium. Note finally that, always for a=0·4 and b=0·35, the sum of profits at the perverse equilibrium, when it exists, is higher than the sum of profits at the natural equilibrium.‡ This means, as discussed above for the Bertrand game, that firms, since there must be at least one loser, may not find a way to cooperate and find an agreement to select the “good” equilibrium. 4. Conclusion The above analysis provides a negative answer to the question whether firms always choose their product design following the “natural” dictates of comparative advantages. In particular, a firm that produces a high quality at a cost that exceeds that of a rival may acquire for itself a high quality position in the spectrum of differentiated products. This is made by speculating on the unwillingness of its rival to contest the acquisition. Otherwise stated, a more efficient firm may refuse to contend the position of high quality acquired by a less efficient rival, in order to enjoy the profits acquired through product differentiation. In a sense, using the title of Shaked and Sutton (1982), product differentiation may relax price competition too much: it may lead to an insufficient incentive for firms to engage in battles for the acquisition of the privilege of producing the best quality. One could be tempted to interpret the model in the preceding † Bonanno (1986) also finds that firms with symmetric cost functions may not differentiate at a subgame perfect equilibrium of the Cournot game. ‡ Computer graphics created through the “Mathematica” program show that the difference between profits in case 4 and those in case 3 is an increasing function of h, positive valued for h11·22.
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sections as a model for testing the robustness of the Ricardian comparative advantage theory as it is usually applied to the advantages from trade in general. While the temptation is strong, one should refrain from it and consider that the Ricardian theory is intended to apply to perfectly competitive markets. References Bonanno, G. (1986). Vertical differentiation with Cournot competition. Economic Notes, 15, 68–91. Gabszewicz, J.J. & Thisse, J.-F. (1979). Price competition, quality, and income disparities. Journal of Economic Theory, 20, 340–359. Gabszewicz, J.J. & Thisse, J.-F. (1986). Spatial competition and the location of firms. In H. Sonnenschein & J. Lesournes, Eds. Fundamentals of Pure and Applied Economics: “Location Theory”. London: Harwood Academic. Shaked, A. & Sutton, J. (1982). Relaxing price competition through product differentiation. Review of Economic Studies, 49, 3–13.