Endogenous Price Leadership and Technological Differences Maoto Yano Faculty of Economics Keio University Taashi Komatubara Graduate chool of Economics Keio University eptember 3, 2005 Abstract The present study constructs a two-period duopoly model with a homogeneous product and examines endogenous price leadership. Two firms have technological differences in their marginal costs. In the first period, each firm chooses either to act as a price leader or to act as a price follower and, in the second period, engages in price competition. The present study predicts that the technologically superior firm becomes the price leader. Very preliminary draft. Not for circulation.
1 Introduction There have been an increasing number of studies that are concerned with endogenizing the leader-follower assignment in a tacelberg-type game. This study extends this line of research by focusing on the so-called price leadership game, in which the assignment of a price leader and a price follower is endogenously determined. It constructs a price leadership game with a duopoly maret for a homogeneous product and demonstrates that a firm with the more efficient technology tends to become a price leader or, in other words, that the state in which the technologically superior firm becomes the price leader can be supported as a unique strong Nash equilibrium of that game. In relating the determination of a price leader to the technology difference between duopolies, this study is closely related to the recent wor of Tasnádi (2003) and Dastidar and Furth (2005). The main differences between the present study and the other studies are in how to determine endogenous price leadership and in how to assume each firm s supply. For the determination of price leadership, the present study considers a game that follows Hamilton and lutsy (1990). 1 In contrast, Tasnádi (2003) follows Denecere and Kovenoc (1992); Dastidar and Furth (2005) follow Robson (1990). On a firm s supply, the present study assumes that a firm supplies so as to fulfill its demand given a price that the firm sets; the other studies assume that a firm choose its price and its supply. As mentioned above, the present study adopts a game theoretic structure by Hamilton and lutsy (1990). That is, in determining endogenous price leadership, each firm chooses either to act as a leader or to act as a follower, given payoffs in each role. This structure contains the Bertrand price competition if both firms choose the same role. Although their study does not describe how payoffs in each role are determined, the present study determines the payoffs in each role by examining a price competition explicitly. The present study s structure in determining endogenous price leadership is in sharp contrast to Dastidar and Furth (2005). Their study adopts the timing game by Robson (1990). Each firm chooses its timing of bidding a price from a continuous interval; the firm that bids a price before the other firm becomes the price leader. Moreover, the present study s structure is in sharp contrast to Tasnádi (2003). His study follows Denecere and Kovenoc (1992). In each period of the T-period game of his study, each firm chooses either to announce a price or to wait. 1 The present study is built on the price leadership game of Yano (2001), which points out the possibility that the price leadership can be determined in the Hamilton-lutsy framewor. 1
The firm that announce a price before the other firm becomes the price leader. Other related studies are as follows: The study by van Damme and Hurens (2004) also examines endogenous price leadership by focusing on technological differences. Their study considers the duopoly maret for differentiated products. Denecere and Kovenoc (1992) and Furth and Kovenoc (1993) are other studies that examine endogenous price leadership in the duopoly maret for a homogeneous product. Those studies focus on each firms size, i.e., capacity constraints. Amir and Grilo (1999), van Damme and Hurens (1999), and Matsumura (1999, 2002) also examine endogenous leadership. All of them focus on quantity competition. 2 Basic tructure of the Model There are two firms: and I. Both firms produce a homogeneous product. Let C j (y) be the total cost function of firm j =, I. It is continuous in y 0. There is no fixed cost, C j (0) = 0. The marginal cost, C j(y), is strictly increasing. Assume C (y) = αc I (y) for all y where 0 < α < 1. ince this implies that firm s marginal cost curve always lies below that of firm I on y > 0, it may be said that the technology of firm is superior to that of firm I. There are two periods. In the first period, the two firms play a game in which each firm chooses either to act as a price leader or to act as a price follower in the product maret that will open in the second period. If, in the first period, one of the firms chooses to act as a leader and if the other firm chooses to act as a follower, the firms will play a subgame of tacelberg price leadership, i.e., the firm that chooses to act as a leader will set its price before the other firm. Either if both firms choose to act as leaders or if they choose to act as followers, the firms will play a subgame of Bertrand price competition, in which they will choose their respective prices simultaneously. In describing the product demand in the second-period maret, we follow the specification of Dastidar (1995). 2 For the sae of explanation, let D(p) be the maret demand in the case in which both firms choose an identical price, p. Assume D (p) < 0. Denote as D (p ; p j ) the demand that firm faces in the case in which firms j and choose prices p j and p, respectively. Assume that if firm sets a price above firm j s price, firm s demand is zero. If firm sets a price equal to firm j s price, firm s demand is one half of the maret demand at that 2 Also see Yano (2001), who adopts Dastidar s specification. 2
price. If firm sets a price below firm j s price, firm s demand is equal to the maret demand at that price. This can be summarized as follows: 0 if p > p j 1 D (p ; p j ) = D(p 2 ) if p = p j (1) D(p ) if p < p j. Following Dastidar (1995), assume that each firm must satisfy completely the demand that comes to that firm. Let π (p) = pd(p) C (D(p)) and π (p) = 1 2 pd(p) C ( 1 2 D(p)), which are, respectively, firm s profit functions in the case in which it monopolizes the entire maret and in that in which it can eep one half of the entire maret. Assume that these profit functions, π (p) and π (p), are unimodal. In the tacelberg subgames, as in the standard Bertrand model, it sometimes become optimal for the follower to monopolize the maret by setting its price slightly below the leader s price. The closer to the leader s price the follower sets its price, the larger its profit becomes. trictly speaing, in that case, the follower s optimal strategy is not well defined. In order to overcome this difficulty, we assume that both firms must choose their respective prices from the set P = {p : p = nε, n N}, given that the minimum accounting unit ε is ignorably small. The best-response correspondence of the follower, say, is characterized by three critical values, p, p, and pm, defined as follows: (1) p P is the minimum of all prices p P satisfying (2) p π (p) 0. P is the maximum of all prices p N satisfying π (p) π (p ε). (3) p M P is the price p P that maximize π (p). ince the price is chosen from the set P, the maximum may be achieved at two prices. In that case, let p M be the smaller of those two prices. That is, p M = min [arg max p P 3 π (p)].
For the sae of explanation, Figure 1 depicts profit curves π (p) and π (p ε). It may be demonstrated that, as is shown in the figure, curve π (p) lies above curve π (p ε) to the left of p, and the former lies below the latter to the right of p. Moreover, pm lies to the right of p, given that the minimum accounting unit ε is ignorably small. The next lemma characterizes the best-response correspondence of firm. There are cases in which firm is indifferent between setting its price equal to the other firm s price and setting it ε below and in which it is indifferent between setting its price equal to the other firm s price and setting it ε above. In order to brea a tie, in those cases, we assume that firm sets its price equal to the other firm s price. Lemma 1 uppose that firm is the follower in a tacelberg subgame. Then, firm s best-response correspondence, P (p j ), and the associated supply correspondence, Q (p j ), are described by {p M } if p j > p M {p P (p j ) = j ε} if p < p j p M {p j } if p p j p (2) {p j + nε, n 1} if p j < p and respectively. D(p M ) if p j > p M D(p Q (p j ) = j ε) if p < p j p M 1 D(p 2 j) if p p j p 0 if p j < p, (3) Proof. ee the Appendix. As (3) shows, the follower (firm ) monopolizes the maret if the price of the leader (j) is above p. If the price of the leader (j) is between p and p, the follower () evenly shares the maret with the leader (j). If it is below p, the follower () chooses not to sell any. This in turn determines the demand that the price leader (j) faces. The demand for the leader Dj L can be characterized as follows: 4
Lemma 2 By setting its price at p j, the demand that the price leader, j, faces is characterized by 0 if p j > p Dj L 1 (p j ) = D(p 2 j) if p p j p j. (4) D(p j ) if p j < p, Proof. It is clear from Lemma 1. As (4) shows, given that is the price follower, the price leader (j) can monopolize the entire maret if it sets its price below p but cannot sell any if it sets its price above p. That the price leader (j) sets its price between p and p means that it shares the maret with firm j. For this reason, we call p and p the minimum and maximum, respectively, coexistable prices for firm j, acting as the price leader. Figure 2 illustrates the profit curves of firms and I. ince firm s marginal cost curve lies below that of firm I, the profit curves of, π (p) and π (p ε), lie above the corresponding profit curves of I, π I (p) and π I (p ε), respectively. This implies the following result. Lemma 3 p I > p and p I > p. Proof. Tae the value p R ( =, I) such that 1 2 p D(p ) C ( 1 2 D(p )) = 0. (5) ince the minimum accounting unit, ε, is ignorably small, p I > p can be proved by showing p I > p. Let y = 1D(p 2 ). This implies p = D 1 (2y ), where D 1 is the inverse of function D. Then, (5) is equivalent to D 1 (2y ) = C (y )/y. (6) ince C (y) = αc I (y) and 0 < α < 1 by assumption, (6) implies y I < y. Thus, p I = D 1 (2y I ) > D 1 (2y ) = p, since D 1 is decreasing. Next, tae the value p R ( =, I) such that 1 2 p D(p ) C ( 1 2 D(p )) = p D(p ) C (D(p )). (7) 5
As in the first part of the proof, p I > p can be proved by showing p I > p. Let y = D(p ). This implies that p = D 1 (y ). Then, (7) can be written as D 1 (y ) = 2 C (y ) C ( 1y 2 ) 1. (8) y ince C (y) = αc I (y) and 0 < α < 1 by assumption, and since 2C I (y)/y C I ( 1y)/ 1 y > 0, it holds that 2 2 2 C I(y) y C I( 1y) ( 2 1 y > α 2 C I(y) y 2 2 y C I( 1y) ) 2 1 y = 2 C (y) y 2 C ( 1 2 y) 1 2 y (9) for any y. ince D 1 is decreasing in y, the value y satisfying (9) is larger in the case of = than in the case of = I, i.e., y > y I. This implies p I > p. Figure 3 illustrates the leader s demands by D L (p ) and D L I (p I) for the cases in which is the leader and in which I is the leader, respectively. 3 econd-period Equilibrium In the second period, as is discussed at the beginning of the previous period, either a Bertrand equilibrium or a tacelberg equilibrium holds. 3.1 Bertrand Equilibrium First, suppose that the two firms engage in Bertrand competition. In this case, the best-responses of firms and I (their Nash strategies) are characterized by the best-response correspondences p = P (p I ) and p I = P I (p ); Figure 4 illustrates the best-response correspondences of and I. A Bertrand equilibrium is captured by the intersection of these best-response correspondences. The next lemma is obvious from Figure 4. Lemma 4 If p I < p, both firms sell positive amounts in a Bertrand equilibrium. In order to focus on a Bertrand equilibrium in which both firms sell positive amounts, we assume the following: 6
Assumption 1. Firm s minimum coexistable price, p I, is lower than firm I s maximum coexistable price, p, i.e., p I < p Let (p B, pb I ) be a Bertrand equilibrium, i.e., pb = P (p B I ) and pb I = P I (p B ). Under Assumption 1, the following result holds: Lemma 5 p I pb = pb I p. Proof. It is clear from Lemma 1. In a Bertrand equilibrium (p B, pb I ), each firm sets the same price and the price is lower than or equal to firm I s maximum coexistable price but higher than or equal to firm s minimum coexistable price, i.e., p I pb = pb I p. The profit of firm, π B, and that of firm I, πb I, are determined as follows: { π B = P (p B I )Q (p B I ) C (Q (p B I )) πi B = P I (p B )Q I(p B ) C I(Q I (p B )). (10) These profits are not uniquely determined because the Bertrand equilibrium is not a unique pair of prices. As is shown later, however, this does not affect the analysis of this study. This is because the profit of a Bertrand equilibrium is bounded above by the profits of a tacelberg equilibrium. 3.2 tacelberg Equilibrium Next, suppose that one of the two firms behaves as a tacelberg leader and that the other behaves as a tacelberg follower. ince, as is shown above, the demand that the leader, say firm j, faces is described as D L j (p j ), it chooses its price so as to maximize π L j (p j ) = p j D L j (p j ) C j (D L j (p j )). Let p j = p L j be the price that firm j chooses as the leader. Let p M the smaller of the prices p P that maximize π (p), i.e.,. P be p M = min [arg max p P π (p)]. We call p M firm s monopoly price in the half-maret. The next lemma characterizes the prices in the tacelberg equilibria, p L and pl I, in which is the leader and in which I is the leader, respectively. 7
Lemma 6 If p < pm Proof. ee the Appendix., then min{p M, p I } = pl > pl I = p. This lemma shows that when firm I is the leader, it chooses its maximum coexistable price, i.e., p L I = p. When firm is the leader, firm chooses its price above firm I s maximum coexistable price, i.e., p L > p. Moreover, it is shown that the follower chooses the same price as the leader. That is, P (p L I ) = pl I = p, and P I (p L ) = pl > p. Given that firm j is the leader and that firm is the follower, their profits are given as follows: { πj L = πj L (p L j ) π F = P (11) (p L j )Q (p L j ) C (Q (p L j )). 4 Price Leadership Game In this section, we demonstrate that the superior firm (firm ) becomes the price leader and the inferior firm (firm I) becomes the price follower. Price leadership is determined by a game in the first period, which we call a price leadership game. This game obeys the following rule: Each firm can choose either to act as a leader (strategy L) or to act as a follower (strategy F ); that is, x i = L, F, where x i is the strategy of firm i =, I. As is discussed above, if the two firms choose different strategies, i.e., if x x I, they will play a tacelberg subgame in the second period. If they choose the same strategy, i.e., if x = x I, they will play the Bertrand subgame in the second period. Denote by Π i (x, x I ) the payoff function. The price leadership game is summarized in Table 1., I L F L Π (L, L) = π B, Π I(L, L) = πi B Π (L, F ) = π L, Π I(L, F ) = πi F F Π (F, L) = π F, Π I(F, L) = πi L Π (F, F ) = π B, Π I(F, F ) = πi B Table 1 In order to determine a Nash equilibrium of the price leadership game, we need to characterize the payoff function Π i (x, x I ) for every possible strategy profile. If firm chooses x = L and if firm I chooses x I = F, they engage in the 8
tacelberg subgame in which firm is the leader. In this case the payoffs of firms and I are π L and πf I, respectively, as (11) shows. If firm chooses x = F and if firm I chooses x I = L, they engage in the tacelberg subgame in which I is the leader. In this case the payoffs of firms and I are π F and πl I, respectively, as (11) shows. If both firms choose x = x I = L or both firms choose x = x I = F, they engage in the Bertrand subgame. As is shown in Lemma 5, any p = p I [p I, p ] can be supported as an equilibrium in the Bertrand subgame; in contrast, the tacelberg subgame has a unique equilibrium no matter which firm acts as the leader. The next lemma shows that for any Bertrand equilibrium (p B, pb I ), the following relationship holds with respect to profit levels. Lemma 7 If p Proof. ee the Appendix. < pm, then { π L > πf πb πi F > πl I πb I. This lemma implies that each firm s profit in a Bertrand equilibrium is bounded above by each firm s profit in the tacelberg equilibrium in the case in which firm I is the leader though the former is not uniquely determined. This maes it possible to characterize the equilibrium of the price leadership game. Lemma 8 Let p < pm. The price leadership game has two Nash equilibria (x, x I ) = (L, F ) and (x, x I ) = (F, L). Proof. It is clear from Lemma 7. This lemma implies that both the state in which firm becomes the price leader and that in which firm I becomes the price leader can be supported as a Nash equilibrium. 3 The main result of this study, however, shows that the state in which becomes the price leader is a unique strong Nash equilibrium. 4 3 Yano (2001) demonstrates in a somewhat different setting of a model that (x, x I ) = (L, F ) is a Nash equilibrium. In that setting, the profits in (x, x I ) = (F, L) are not uniquely defined, and the possibility with which the latter strategy profile might become a Nash equilibrium is not examined. 4 A strong Nash equilibrium, which is defined by Aumann (1959), is a Nash equilibrium such that there is no coalition of which all its members can mae strictly better off by changing only their strategies. 9
Theorem 1 uppose that firm I s maximum coexistable price, p, is lower than firm s monopoly price in the half-maret, p M. Then, (x, x I ) = (L, F ) is the unique strong Nash equilibrium of the price leadership game. Proof. There are two Nash equilibria, (F, L) and (L, F ). By Lemma 7, there is no pair (x, x I ) such that π (x, x I ) > π (L, F ) and π I (x, x I ) > π I (L, F ). Hence, by definition, (L, F ) is a strong Nash equilibrium. By Lemma 7, we have π (L, F ) > π (F, L) and π I (L, F ) > π I (F, L), which imply that (F, L) is not a strong Nash equilibrium. Intuitively, this result stems from the fact that a firm acting as the price leader sets its price at the maximum coexistable price for that firm. The maximum coexistable price is higher in the case in which is the leader than in the case in which I is the leader. As a result, the profits of both firms become higher in the case in which is the leader. This implies that the state in which becomes the leader is Pareto superior to the state in which I becomes the leader. Moreover, in the case in which Bertrand price competition taes price, an equilibrium price must become below or equal to the maximum coexistable price for firm I. Therefore, the state in which becomes the leader is Pareto superior to the state of Bertrand price competition. This implies that the state in which becomes the leader is supported as a Pareto efficient Nash equilibruim of the price leadership game, which coincides with a strong Nash equilibrium of a two-player game. In this respect, it may be concluded from Theorem 1 that a more liely outcome of the price leadership game is that the technologically superior firm becomes the price leader. 5 Remars on Assumptions Remar 1: Yano (2001) focuses on the case in which Assumption 1 (p I < p ) is not satisfied; i.e., that study assumes that there is no Bertrand equilibrium in which both firms sell positive amounts. If, however, Assumption 1 is not satisfied, in the present setting as well as Yano s setting, the profit of firm is not uniquely determined in the case of (x, x I ) = (F, L), i.e., in the lower left-hand-side box of Table 1. For the sae of explanation, suppose p I p. In this case, firm I s optimal policy is not uniquely determined. That is, a possibility can be demonstrated such that firm I s profit is zero or negative. In that case, the optimal policy of I acting as the leader cannot uniquely be determined. ince the leader firm s price 10
is not determined uniquely, the profit of acting as the follower cannot uniquely be determined either. This study adopts Assumption 1 in order to determine the profits of the firms in all the boxes of Table 1. Remar 2: In the case in which the hypothesis of Theorem 1 (p < pm ) is not satisfied, the price leadership game fails to have a meaningful solution. If, in particular, p = pm, it holds that p L = pl I = p (see the Appendix for a proof). In this case, both and I have the same profits whether acts as the leader or I acts as the leader; it may be demonstrated that both states are supported as a strong Nash equilibrium. If p > pm, it holds that p pl I > pl (see the Appendix for a proof). In this case, the clear ordering cannot be established for a firm s profit between the case in which a tacelberg subgame is played in the second period and that in which the Bertrand subgame is played in the second period. This maes it difficult to characterize a Nash equilibrium of the price leadership game in a meaningful manner. Appendix Proof of Lemma 1 Note P (p j ) = {p M } if p j > p M, which proves the top expression of (2). This is because, if firm j sets p j above firm s monopoly price p M, it is optimal for firm to monopolize the maret by setting its price at the monopoly price. In this case, Q (p j ) = D(p M ), which establishes the top expression of (3). If p < p j p M, by the definition of p, it holds that and that (p j ε)d(p j ε) C (D(p j ε)) > p j D(p j )/2 C (D(p j )/2) (12) p j D(p j )/2 C (D(p j )/2) > 0. (13) These inequalities, (12) and (13), imply that it is better for firm to set p = p j ε and to sell D(p j ε) than to set p = p j and to sell D(p j )/2. Hence, if p < p j p M, firm sets p = p j ε and sells D(p j ε). This shows P (p j ) = {p j ε} and Q (p j ) = D(p j ε), which establish the second expressions of (2) and (3). Next, let p j = p. First, tae the case in which the following condition holds: (p ε)d(p ε) C (D(p ε)) p D(p )/2 C (D(p )/2). (14) 11
In this case, by the definition of p, it holds that (p and that ε)d(p ε) C (D(p ε)) < p D(p )/2 C (D(p )/2) (15) p D(p )/2 C (D(p )/2) > 0. (16) These inequalities, (15) and (16), imply that it is better for firm to set p = p and to sell 1 2 D(p ) than to set p = p ε and to sell D(p ε). Hence, we have P (p ) = {p } and Q (p ) = 1 2 D(p ) for the case of (14). econd, tae the case in which the following condition holds: (p ε)d(p ε) C (D(p ε)) = p D(p )/2 C (D(p )/2). (17) In this case, firm is indifferent between setting p = p ε and setting p = p By assumption, if firm j sets such p have P (p ) = {p } and Q (p Let p < p j < p ) = 1 2 D(p. In this case, it holds that, firm sets its price equal to p ) for the case of (17)... Hence we (p j ε)d(p j ε) C (D(p j ε)) < p j D(p j )/2 C (D(p j )/2) (18) and that p j D(p j )/2 C (D(p j )/2) > 0. (19) These inequalities, (18) and (19), imply that it is better for firm to set p = p j and to sell 1D(p 2 j) than to set p = p j ε and to sell D(p j ε). Hence, we have P (p j ) = {p j } and Q (p j ) = 1D(p 2 j). Let p j = p. First, tae the case in which the following condition holds: In this case, by the definition of p, it holds that p D(p )/2 C (D(p )/2) 0. (20) (p ε)d(p ε) C (D(p ε)) < p D(p )/2 C (D(p )/2) (21) and that p D(p )/2 C (D(p )/2) > 0. (22) These inequalities, (21) and (22), imply that it is better for firm to set p = p and to sell 1 2 D(p ) than to set p = p ε and to sell D(p ε). Hence, we have 12
P (p ) = {p } and Q (p ) = 1 2 D(p ) for the case of (20). econd, tae the case in which the following condition holds: p D(p )/2 C (D(p )/2) = 0. (23) In this case, firm is indifferent between setting p = p and setting p > p. By assumption, if firm j sets such p, firm sets its price equal to p. Hence we have P (p ) = {p } and Q (p ) = 1 2 D(p ) for the case of (23). These results establish the third expressions of (2) and (3). Finally, if p j < p, by the definition of p, it holds that p j D(p j )/2 C (D(p j )/2) < 0. This implies that, if p j < p, firm s profit when it sets p = p j becomes negative. Hence it is optimal for firm to set p > p j and not to sell the product. This shows P (p j ) = {p j + nε, n 1} and Q (p j ) = 0, which establish the last expressions of (2) and (3). Proof of Lemma 6 Consider the tacelberg subgame in which firm I is the leader. Firm I chooses its price p I so as to maximize its profit π L I (p I ) = p I D L I (p I ) C I (D L I (p I )). (24) Let p L I be the price maximizing the profit over P. By setting j = I in (4), firm I s demand, DI L(p I), can be written as 0 if p I > p DI L 1 (p I ) = D(p 2 I) if p p I p (25) D(p I ) if p I < p. We will demonstrate p L I = p. Towards this end, first, we will show that it is not optimal for firm I to set p I < p ε. The reason is as follows: If p I < p, (24) and (25) imply that π L I (p I ) = p I D(p I ) C I (D(p I )). (26) ince p < p I by Lemma 3 and p I < pm I by definition, it holds that p < pm I. Hence, by the definition of p M I, (26) is strictly increasing in p p. Hence, it holds that pd(p) C I (D(p)) < (p ε)d(p ε) C I (D(p ε)), 13
for all p < p ε. Thus, it is not optimal to set p I < p ε. Hence, pl I p ε. econd, we will show that it is not optimal to set p I = p ε. If p I = p ε, by (26), it holds that π L I (p ε) = (p ε)d(p ε) C I (D(p ε)). In contrast, by (24) and (25), if p I = p, it holds that ince p < p by definition and p Hence, by the definition of p I π L I (p ) = 1 2 p D(p ) C I ( 1 2 D(p )). < p I, it holds that by Lemma 3, it holds that p < p I. (p ε)d(p ε) C I (D(p ε)) < 1 2 p D(p ) C I ( 1 2 D(p )). This implies that it is better to set p I = p than to set p I = p ε. Hence, it is not optimal to set p I = p ε. Hence, it holds that pl I p. Third, we will show that it is not optimal to set p I > p. By (24) and (25), if p p I p, it holds that By the assumption of this lemma, p p M I. Hence, it holds that p π L I (p I ) = 1 2 p ID(p I ) C I ( 1 2 D(p I)). (27) < pm I < < pm and, by assumption, p M. This implies that (27) is strictly increasing in [p, p ]. Hence it is not optimal to set p I < p. Hence, it holds that p pl I p. Finally, we will show that it is not optimal to set p I > p. If p I = p, by (27), π L I (p ) = 1 2 p D(p ) C I ( 1 2 D(p )). (28) By Assumption 1, it holds that p I < p. Hence, by the definition of p I positive. In contrast, if p I > p, by (24) and (25), DL I (pl I ) = πl I (pl I it is not optimal to set p I > p. Hence we have pl I = p. Consider the tacelberg subgame in which firm is the leader. chooses its price p so as to maximize its profit, (28) is ) = 0. Thus Firm π L (p ) = p D L (p ) C (D L (p )). (29) 14
Let p L be the profit maximizing price. By (4), firm s demand, DL (p ), is given by 0 if p > p D(p L I 1 ) = D(p 2 ) if p I p p I (30) D(p ) if p < p I. We will demonstrate min{p M, p I } = pl will show that it is not optimal to set p < p I (29) and (30), if p < p I, > p. Towards this end, first, we ε. The reason is as follows: By π L (p ) = p D(p ) C (D(p )). (31) ince p I < p by Lemma 3 and p < pm by definition, it holds that p I < pm. Hence, by the definition of p M, it holds that pd(p) C (D(p)) < (p I ε)d(p I ε) C (D(p I ε)), for all p < p I ε. This implies that it is not optimal to set p < p I ε. Hence it holds that p L p I ε. econd, we will show that it is not optimal to set p = p I ε. The reason is as follows: If p = p I ε, by (31), it holds that π L (p I ε) = (p I ε)d(p I ε) C (D(p I ε)). By (29) and (30), if p = p I, it holds that π L (p I) = 1 2 p ID(p I) C ( 1 2 D(p I)). (32) By Assumption 1, it holds that p I < p. Hence, by the definition of p, it holds that (p I ε)d(p I ε) C (D(p I ε)) < 1 2 p ID(p I) C ( 1 2 D(p I)). This implies that it is better to set p = p I than to set p = p I ε. Hence, it is not optimal to set p = p I ε. Hence, it holds that pl p I. Third, we will show that it is not optimal to set p > p I. The reason is as follows: By Lemma 3, it holds that p < p I. Hence, by the definition of p, (32) is positive. In contrast, by (29) and (30), if p > p I, DL (p ) = π L(p ) = 0. Hence, it is not optimal to set p > p I. Hence it holds that p I pl p I. 15
Finally, we will show that p L and (30), By Assumption 1, p I < p Hence it holds that p I [p I p I, min{pm > p = min{pm, p I }. If p I p p I, by (29) π L (p ) = 1 2 p D(p ) C ( 1 2 D(p )). (33) < pm, p I }]. Hence, pl, we have min{pm Proof of Lemma 7 and, by the assumption of this lemma, p < pm.. This implies that (33) is strictly increasing in = min{pm, p I }. ince pm > p and, p I } = pl > p In order to prove Lemma 7, it suffices to prove π L > πf πb and πf I > πi L πi B. Recall that πb and πb I are the profits in a Bertrand equilibrium, (p B, pb I ). Then, by Lemma 5, p B = pb I [ p I, p ]. Thus, by (10), (2), and (3), it holds that { π B = 1 2 pb D(p B ) C ( 1 2 D(pB )) πi B = 1 2 pb D(p B ) C I ( 1 2 D(pB )), where p B = p B = pb I. Next, tae πi L and πf, i.e., the profits in the tacelberg subgame in which firm I is the leader. By Lemma 6, firm I s price and profit are characterized as p L I = p and πi L = 1 2 p D(p ) C I( 1 2 D(p )). ince p < pm by the assumption of this lemma and since p M < p M 0 < α < 1, it holds that p increasing in p p. Hence, p. I by the assumption of C = αc I and. Hence, 1pD(p) C 2 I( 1 D(p)) is strictly 2 pb implies that < pm I π L I = 1 2 p D(p ) C I ( 1 2 D(p )) 1 2 pb D(p B ) C I ( 1 2 D(pB )) = π B I. By (11), (2), and (3), the follower s profit is π F = 1 2 p D(p ) C ( 1 2 D(p )). ince p < pm by the assumption of this lemma, 1pD(p) C 2 ( 1 D(p)) is 2 strictly increasing in p p. Hence p pb implies that π F = 1 2 p D(p ) C ( 1 2 D(p )) 1 2 pb D(p B ) C ( 1 2 D(pB )) = π B. Finally, tae π L and πf, i.e., the profits in the tacelberg subgame in which firm is the leader. By Lemma 6, firm s price and profit are characterized as p < pl pm and π L = 1 2 pl D(pL ) C ( 1 2 D(pL 16 )). By the definition of
p M, 1 2 pd(p) C ( 1 2 D(p)) is strictly increasing in p pl. Hence, pl > p implies that π L = 1 2 pl D(p L ) C ( 1 2 D(pL )) > 1 2 p D(p ) C ( 1 2 D(p )) = π F. Moreover, by Lemma 6, it holds that p I pl p I the follower s profit is πi F = 1 2 pl D(pL ) C I( 1 2 D(pL by Lemma 6, and since p M < p M I p M I. By the definition of p M I p p L. Hence, pl > p. Hence, by (11), (2), and (3), )). ince p < pl pm as is noted above, it holds that p < pl <, 1pD(p) C 2 ( 1 D(p)) is strictly increasing in 2 implies that π F I = 1 2 pl D(p L ) C I ( 1 2 D(pL )) > 1 2 p D(p ) C I ( 1 2 D(p )) = π L I. Proof of the results in Remar 2 Assume that p = pm. In the tacelberg subgame in which firm I is the leader, by Lemma 6, we have p L I = p. In the tacelberg subgame in which firm is the leader, by Lemma 6, we have p L = min{pm, p I }. By assumption, this implies that p L = min{p, p I }. ince p < p I by Lemma 3, we have p L = p. This establishes p L = pl I = p. Assume that p > pm. In the tacelberg subgame in which firm I is the leader, we have p pl I p by Lemma 6. ince (27) is strictly increasing in [p, min{pm I, p }], it holds that pl I = min{p M I, p }. This implies that p pl I. In the tacelberg subgame in which firm is the leader, we have p I pl p I by Lemma 6. ince p I > p by Lemma 3 and p > pm by assumption, it holds that p I > p M. Hence (33) is strictly increasing in [p I, pm ]. Hence, we have p L = pm. ince p M I > p M and p > pm by assumption, we have p pl I = min{p M I, p } > pm = p L. This establishes the result that p pl I > pl. 17
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Tasnádi, A. (2003), Endogenous timing of moves in an asymmetric price-setting duopoly, Portuguese Economic Journal 2, 23 35. Van Cayseele, P., and D. Furth (2001), Two is not too many for monopoly, Journal of Economics 74, 231 258. Yano, M. (2001), Miuroeizaigau no Ouyou, Chapter 8, [Applications of Microeconomics], Toyo: Iwanami hoten, translated by T. Komatubara (Keio university, 2005). 19
(p- ) (p) p * p ** p M + p Figure 1
(p- ) I (p- ) (p) I (p) p * p I * p ** p I ** p Figure 2
p p I Maret Demand, D Maret Demand, D p I ** p I * p ** p * D L D I L The demand firm faces as the price leader. The demand firm I faces as the price leader. Figure 3
P I p p M P p I ** p I * p * p ** p I M p I Figure 4