ENDOGENOUS TIMING IN A MIXED DUOPOLY: WEIGHTED WELFARE AND PRICE COMPETITION

Transcription

1 ENDOGENOU TIMING IN A MIXED DUOPOY: WEIGHTED WEFARE AND PRICE COMPETITION y Juan Carlos Bárcena-Ruiz and Máximo edano 0 Working Paper eries: I. 6/ Departamento de Fundamentos del Análisis Económico I Ekonomi Analisiaren Oinarriak I aila University of the Basque Country

2 ENDOGENOU TIMING IN A MIXED DUOPOY: WEIGHTED WEFARE AND PRICE COMPETITION* JUAN CARO BÁRCENA-RUIZ a* Universidad del País Vasco MÁXIMO EDANO Universidad del País Vasco a Departamento de Fundamentos del Análisis Económico I Facultad de Ciencias Económicas y Empresariales Universidad del País Vasco Avenida ehendakari Aguirre Bilao pain. Departamento de Economía Aplicada III Facultad de Ciencias Económicas y Empresariales Universidad del País Vasco Avenida ehendakari Aguirre Bilao pain. In this paper we analyse the endogenous order of moves in a mixed duopoly for differentiated goods. Firms choose whether to set prices sequentially or simultaneously. The private firm maximises profits while the pulic firm maximises the weighted sum of the consumer and producer surpluses (weighted welfare. It is shown that the result otained in equilirium depends crucially on the weight given to the consumer surplus in weighted welfare and on the degree to which goods are sustitutes or complements. We also analyse whether the equiliria otained maximise the sum of the consumer and producer surpluses or not. Finally we study whether the nationality of the private firm influences the results. Keywords: Mixed duopoly; Price competition; Endogenous timing; Weighted welfare. JE Classification: 00; 0 * Corresponding author. Departamento de Fundamentos del Análisis Económico I Facultad de Ciencias Económicas y Empresariales Universidad del País Vasco Avenida ehendakari Aguirre Bilao pain. Tel.: 96089; fax: address: juancarlos.arcena@ehu.es.

3 . Introduction The literature on mixed markets has analysed whether firms take decisions (on quantities or prices sequentially or simultaneously assuming that private firms maximise profits and pulic firms maximise social welfare. In this regard Pal (998 shows that when firms produce a homogeneous good and the pulic firm is less efficient than the private firms they decide quantities sequentially in a mixed oligopoly. Matsumura (00 and u (006 extend this analysis to consider that the pulic firm competes with foreign private firms. They also find that decisions are taken sequentially. The papers cited aove consider that firms compete in quantities and produce a homogeneous good. Under Bertrand competition and assuming that the pulic firm is as efficient as the private firm Bárcena-Ruiz (007 shows that when firms produce a heterogeneous good they decide prices simultaneously in a mixed duopoly. The literature on mixed markets usually assumes that pulic firms maximise social surplus defined as the sum of the consumer and producer surpluses. However White (00 p. 89 argues that while the standard equally-weighted welfare function may e desirale for normative reasons ased on utilitarianism or fairness doctrines (as in Harsanyi 955 it may e restrictive for purposes of predicting the ehavior of actual pulic firms and the resulting market outcomes. Thus the ojective function of the pulic firms can give a different weight to the consumer surplus than to the producer surplus which permits different political interpretations ranging from pro-consumer (when the weight given to the This literature also analyses whether other variales are decided sequentially or simultaneously. For example Matsumura and Matsushima (00 analyse the sequential choice of location in a mixed duopoly assuming a Hotelling-type spatial model. Bárcena-Ruiz and Garzón (00 consider that the private firms compete with a semipulic firm rather than a pulic firm; the semipulic firm is jointly owned y the pulic sector and private domestic shareholders. Assuming that all firms are equally efficient they show that in equilirium firms take production decisions simultaneously. The aove cited papers need to assume that the pulic firm is less efficient than the private firm to avoid the pulic firm exiting the market. However when there is a semipulic firm rather than a private firm this assumption is not necessary and all firms can e equally efficient. Matsumura and Ogawa (00 investigate a cost difference version where the private firm is more efficient than the semipulic firm. White (00 considers a quantity-setting mixed duopoly and a weighted welfare function to analyse how the pulic firm responds to different ojective functions and how this impacts on the industry as a whole. He shows that a government can maximise its true ojective y assigning the pulic firm a different ojective function.

4 consumer surplus is high to pro-usiness (when the weight given to producer surplus is low. In this way the ojective function of the government (social surplus can differ from that of the pulic firm (weighted welfare. This assumption is important since it can affect the result otained in equilirium. Moreover attaching different weights to the components of weighted welfare enales different political interpretations to e given to the ojective function of the government and thus different results can e otained corresponding to the different political interpretations. In this paper we analyse the endogenous order of moves in a mixed duopoly under price competition 5 assuming that the ojective function of the pulic firm (denoted as weighted welfare can give a different weight to consumer surplus than to producer surplus and that firms produce heterogeneous goods. 6 The ojective function of the government is the sum of the consumer and producer surpluses (social surplus. This paper addresses the issue of the endogenous order of moves y considering the oservale delay game of Hamilton and lutsky (990. We consider a mixed duopoly with a domestic-private firm and a pulic firm that produce heterogeneous goods where firms decide the timing of their choosing of prices. It is shown that the result otained in equilirium depends crucially on the weight given to consumer surplus in weighted welfare and on the degree to which goods are sustitutes or complements. We otain that when goods are sustitutes if the weight given to consumer surplus in weighted welfare is high enough firms decide prices simultaneously. 7 When this weight takes an intermediate value we otain a sequential equilirium in which the pulic firm is the leader in prices and the private firm the follower. When the weight given to It should e noted that papers analysing a mixed oligopoly usually assume that the pulic firm and the government have the same ojective function which restricts the results. 5 Other papers analysing a mixed market under price competition include Anderson et al. (997 Ogawa and Kato (006 and Bárcena-Ruiz (009. However it should e noted that the literature on endogenous timing in mixed oligopoly with price setting competition is relatively small. 6 We do not consider in the paper that the government manipulates the ojective function of the pulic firm. In this regard see Matsumura (998 and Matsumura and Kanda (005 who assume that the ojective function of the pulic firm is a linear comination of the ojective of the government and its profits and that the government can manipulate this ojective function through its shareholding. 7 This result is also otained y Bárcena-Ruiz (007 y assuming that the consumer and producer surpluses have the same weight in social welfare. In fact the results of the paper contain those of Bárcena-Ruiz (007 as a special case when consumer s surplus and producer s surplus have the same weight in weighted welfare.

5 consumer surplus is low enough there are two sequential equiliria: the pulic firm is the leader in prices in one of them and the follower in the other. 8 Finally when goods are complements in equilirium firms decide prices simultaneously. 9 Comparing the social surplus otained in the different cases considered in the paper we otain that the lowest social surplus is otained when the private firm is the leader. Thus the government does not want the private firm to e the leader in prices. The maximum social surplus is otained when firms make decisions simultaneously or when the pulic firm is the leader. When goods are complements social surplus is maximum when prices are set simultaneously. However when goods are sustitutes the result depends on the weight given to the consumer surplus in the ojective function of the pulic firm. In this case if this weight takes an intermediate value social surplus is maximum when the pulic firm is the leader in prices. For the remaining values of this weight social surplus is greatest under simultaneous decisions. Next we consider the comparison of the three outcomes corresponding to the three games with an exogenously given order of moves. We find that when goods are complements prices are set simultaneously in equilirium which maximises social surplus. When goods are sustitutes if the weight given to the consumer surplus in the ojective function of the pulic firm is high enough prices are set simultaneously in equilirium which maximises social surplus. If the weight takes an intermediate value prices are set simultaneously and the maximum social surplus is otained when the pulic firm is the leader in prices. Finally if the weight is low enough prices are set sequentially and social surplus is maximised when prices are set simultaneously. 8 It can e shown that in the limiting case of 0 two sequential equiliria are found to exist when products are sustitutes: the pulic firm is the leader in prices in one of them and the follower in the other. When products are complements firms set prices simultaneously. These results correspond to the case in which the two firms are privately owned. 9 Analyzing an n private firm model is eyond the scope of this paper. However we have checked that the main results of the paper hold when n is and 5 (see Häckner 000 for the case of n private firms competing under Bertrand competition. This is due to the fact that the strategic effects that arise when there is only one private firm are still present when there is more than one private firm. It has to e noted that when considering the oligopoly game we assume that all private firms take decisions simultaneously. Thus we are considering a two stage game as in the duopoly case.

6 In the literature on mixed oligopoly it is well known that nationality of the private firms plays a crucial role. 0 Thus we extend the model to consider that the private firm is foreign-owned. We otain a similar result to when the private firm is domestic-owned except for the case of sustitute goods when the weight given to the consumer surplus in the ojective function of the pulic firm takes an intermediate value. In this case when the private firm is foreign-owned it is found to e the leader in prices. However when private firm is domestic-owned it is found to e the follower in prices The rest of the paper is organised as follows. ection presents the model. ection analyses whether firms set prices sequentially or simultaneously. ection considers that the private firm is foreign-owned and conclusions are drawn in ection 5.. The model We consider an economy comprising two firms that produce a heterogeneous good. One firm is pulicly owned and the other is privately owned; we denote these firms y 0 and respectively. On the consumption side there is a continuum of consumers of the same type. The representative consumer maximises U(q 0 q p 0 q 0 p q where q i is the amount of the good i and p i is its price (i 0. The function U(q 0 q is assumed to e quadratic strictly concave and symmetric in q 0 and q : U(q 0 q a ( q0 q (( q0 q0q ( q >> 0 where parameter measures the degree to which goods are sustitutes/complements. If >0 (<0 goods are sustitutes (complements. Demand functions are thus given y: a( pi p j qi i j; i j 0. ( 0 For example Fjell and Pal (996 analyse a mixed oligopoly in the presence of foreign private firms Matsumura (00 studies tackelerg mixed duopoly with a foreign competitor u (006 analyses endogenous timing in a mixed oligopoly with foreign competitors Matsushima and Matsumura (006 study the spatial location of firms in a mixed oligopoly when there are foreign firms and Matsumura et al. (009 investigate whether or not privatization is eneficial from the view-point of social welfare in a monopolistic competition model. We assume that < to ensure that the function U(q 0 q is strictly concave (see Vives 98. Besides 0 means two independent monopoly markets so the results in this case are ovious. 5

7 The marginal cost of production of oth firms is c. The profit of firm i is given y: π i (p i c q i i 0 ( where q i is given y (. The private firm chooses the price p which maximises its profit. The pulic firm chooses the price p 0 which maximises the weighted sum of the consumer surplus (C and the producer surplus (P denoted as weighted welfare: W C P << H ( where max{0 ( ( } and H min{ }. As usual the ( ( producer surplus is the sum of the profits of the firms P π 0 π and the consumer surplus is given y: a( ( a p C U(q 0 q p 0 q 0 p q 0 p ( p ( 0 p 0 p ( p. ( We refer to the sum of the consumer and producer surplus as social surplus: C P. We consider the oservale delay game of Hamilton and lutsky (990 in the context of a price-setting mixed duopoly where the firms choose whether to set prices at t or at t. We propose a two-stage game with the following timing. In the first stage the firms decide whether to set prices at t or at t. If one firm sets its price at t and the other at t the game is sequential. If oth firms choose their prices in the same period firms take decisions simultaneously. In the second stage the firms decide their prices either sequentially or simultaneously. To otain a sugame perfect equilirium the game is solved ackwards. The assumption << H ensures that the pulic firm otains a positive profit: > ensures that the pulic firm otains a positive profit when it is the leader in prices and < H ensures that the pulic firm otains a positive profit when prices are set sequentially. The proof is relegated to the appendix. These ounds on parameter also assure that the output level of the two firms is positive in all cases. It can e shown that H for 0 H 8/7 for H 0 for 0.70 and 0 for 0.90; esides H < for <0 H > for >0 and < for all. 6

8 . Results Given that firms may decide prices sequentially or simultaneously we have three cases. First the pulic firm and the private firm decide prices simultaneously. econd the pulic firm decides its price efore the private firm does. Finally the private firm decides its price efore the pulic firm does. We first solve the case in which the prices are set simultaneously denoted y superscript. The private firm chooses the value of p that maximises ( for i and j0. The pulic firm chooses the value of p 0 that maximises (. olving these two prolems simultaneously we otain the reaction functions in prices of the two firms: p ( a( c p 0 (5 ( ( c a( p 0 p. (6 From expressions (5 and (6 we otain that when >0 prices are strategic complements ( p / p > 0 i j; ij0. Thus if one firm raises (lowers its price the i j other firm reacts y raising (lowering its price too. When <0 prices are strategic sustitutes ( p / p < 0 i j; ij0 and thus if one firm raises (lowers its price the other i j firm reacts y lowering (raising its price. From (5 and (6 the following is otained: a( (( ( c( p 0 ( ( a( ( ( c( ( ( ( p ( ( a c (( ( π 0 ( ( ( ( a c (8 ( 6 C ( ( ( ( ( a c ( ( π ( ( ( ( a c ( (8 8 ( ( P ( ( ( ( a c ( 0 ( 6 ( ( ( ( ( (6 ( ( 7

9 8 ( ( ( (9 ( (8 ( c a W. Next we consider the case in which the pulic firm decides its price efore the private firm does. We denote the leader firm y superscript and the follower firm y superscript F. The private firm chooses the value of p that maximises ( for i and j0. olving this prolem we otain equation (5. The pulic firm chooses the value of p 0 that maximises ( taking into account equation (5. olving this prolem we get: ( ( ( ( (( ( 0 c a p ( ( ( ( ( (( ( c a p F ( ( ( ( (( ( ( ( ( 0 π c a ( ( ( ( (( ( ( π c a F ( ( ( ( 8 (8 8 (8 ( ( c a P ( ( ( ( (8 ( c a C ( ( ( ( (7 ( 0 ( ( c a. ( ( ( (7 ( (8 ( c a W Finally we consider the case in which the private firm decides its price efore the pulic firm does. The pulic firm chooses the value of p 0 that maximises (. olving this prolem we otain equation (6. The private firm chooses the value of p that maximises ( for i and j0 taking into account equation (6. olving this prolem we get: ( ( ( ( ( ( ( (( 0 c a p F ( ( ( ( ( ( c a p

10 π ( a c ( ( ( ( π F 0 ( a c (( ( ( ( ( C F P F F W F ( a c (8 (6 5 8( ( ( a c (8 8 ( ( ( ( ( a c ( 0 ( ( ( ( a c (8 ( 8( ( (5 6 (. ( et. From the results otained in the three cases the following result is otained. Proposition. In equilirium: i If for >0: p 0 p 0 F p 0 > p 0 p > p F p p P P P F >P C C C >C F ; ii If < for >0: iii If <0: p i > p i p i > p i > p i F p i > p i P >P P F >P C >C F C >C i0 ; p F i P >P P >P F C >C F C >C i0. Proof. ee Appendix. We first analyse the case in which the weight given to the consumer surplus in the ojective function of the pulic firm is great enough when goods are sustitutes ( for >0. In this case when the private firm is the leader in prices it sets a higher price than in the simultaneous case since as prices are strategic complements this firm knows that the follower (the pulic firm will also set a higher price. The private firm wants to increase prices to reduce market competition in order to increase profits. When the pulic firm is the leader in prices it sets a price lower than or equal to that of the simultaneous case since as In a private duopoly when prices are set sequentially oth the leader and the follower set higher prices than when prices are set simultaneously (see Gal-Or 985; Dowrick 986; Hamilton and lutsky 990. This is due to the fact that reaction functions in prices are upward sloping and thus if one firm raises its price the other firm reacts y doing likewise. 9

11 prices are strategic complements this firm knows that the follower (the private firm will also set a lower price. The pulic firm wants to reduce prices in order to increase market competition and thus consumer surplus since as parameter is great enough the consumer surplus has a greater effect than the producer surplus in the ojective function of the pulic firm. As a result when the private firm is the leader (follower it sets a higher (lower or equal price than in the simultaneous case: p p > ( F p p ; when the pulic firm is the follower (leader it sets a price that is higher (lower or equal to that of the simultaneous case: F p p0 0 > ( p0 p0. Market competition is greater when the pulic firm is the leader than in the simultaneous case ecause oth firms set lower or equal prices in the first case. Thus the consumer surplus is greater or equal and the producer surplus is lower or equal in the first case: C C and P P. By contrast as market competition is lower when the private firm is the leader than in the simultaneous case the consumer surplus is lower and the producer surplus is greater in the first case: C >C F and P F >P. When the weight given to the consumer surplus in the ojective function of the pulic firm is low enough and goods are sustitutes (< for >0 the incentive of that firm to ehave aggressively in the product market is lower than when. As in the preceding case when the private firm is the leader in prices it sets a higher price than in the simultaneous case since as prices are strategic complements the pulic firm also sets a higher price thus reducing market competition: p p > and F p0 p0 >. When the pulic firm is the leader in prices it sets a higher price than in the simultaneous case since as the weight of the consumer surplus is low enough the profit of the firms (the producer surplus has a greater effect on the ojective function of the pulic firm than the consumer surplus. Thus in this case the pulic firm seeks to reduce market competition in comparison with the simultaneous case which means that oth firms set higher prices: 0 > and F p > p. p p0 Under sequential decisions independently of which firm is the leader market competition is lower than when decisions are taken simultaneously since prices are lower in the first case. As a result the consumer surplus is greater and the producer surplus is lower under simultaneous decisions than under sequential decisions: C >C C >C F P >P and P F >P. 0

12 Finally we consider that products are complements (<0. When the private firm is the leader in prices it sets a higher price than in the simultaneous case which reduces its production. As prices are strategic sustitutes the follower (the pulic firm sets a lower price which as products are complements increases the output of the private firm. This last effect implies that if the private firm increases its price the reduction in its output level is small. As a result when the private firm is the leader in prices it sets a higher price than in the simultaneous case and the follower sets a lower price: p p > and F p0 > p0. When the pulic firm is the leader in prices it sets a price higher than in the simultaneous setting which reduces its production. However this firm knows that the follower (the private firm will set a lower price thus increasing its production which as products are complements increases the output of the pulic firm. This last effect implies that if the pulic firm increases its price the reduction in its output level is small and the private firm produces more. As a result when the pulic firm is the leader in prices it sets a higher price than in the simultaneous case and the follower sets a lower price: 0 > and F p > p. As in the p p0 simultaneous case when prices take an intermediate level we otain that C >C and C >C F. Finally when the pulic firm is the leader (follower in prices it sets a higher (lower price than in the simultaneous case implying that P >P (P >P F. Next we compare the weighted welfare and the profit of the private firm otained in the sequential and simultaneous cases. We denote as Zone I the values of parameter such that H > for >0 as Zone II the values of parameter such that > for >0 as Zone III the values of parameter such that >> for >0 and finally as Zone IV the values of parameter such that H >> for <0 where ( ( (. Zones I to IV are shown in Figure.

13 Proposition. In equilirium: F F i W W W > W π > and π if H > for >0; F F ii W > W W W π > and π > if > for >0; F F iii W > W W > W π > and π > if >> for >0; F F iv W > W W > W π > and π > if H >> for <0. Proof. ee Appendix. H Zone IV Zone I Zone II - Zone III 0 FIG.. IUTRATION OF PROPOITION We consider first the case in which the weight given to the consumer surplus is great enough when products are sustitutes: H > for >0 (Zone I in Figure. Proposition shows that in this zone when the private firm is the leader in prices oth firms set higher prices than in the simultaneous case. This reduces market competition and increases the profit of the private firm ( π >. By contrast when the pulic firm is the leader in prices in this zone oth firms set lower prices than in the simultaneous case. This increases market F competition and reduces the profit of the private firm ( π. Proposition also shows that consumer surplus when the pulic firm is the leader is greater than or equal to that of the simultaneous case (C C and in this last case is greater than if the pulic firm is the follower (C >C F. Moreover the producer surplus is greater when the pulic firm is the follower than in the simultaneous case (P F >P and in this last case is greater than or equal

14 to the level otained if the pulic firm is the leader (P P. Given that in this zone is great enough ( the consumer surplus has a great enough weight in the ojective function of the pulic firm. As a result when that firm is the leader the greater consumer surplus has a stronger effect on weighted welfare than the lower producer surplus implying that W >W. Moreover when the pulic firm is the follower the lower consumer surplus has a stronger effect on weighted welfare than the greater producer surplus implying that W >W F. Next we consider the case in which the weight given to the consumer surplus is low enough when products are sustitutes: >> for >0 (Zone III in Figure. Proposition shows that independently of which firm is the leader in prices in this zone oth firms set higher prices than in the simultaneous case. This reduces market competition and increases the profit of the private firm: F π > and π >. Proposition also shows that the consumer surplus is greater and the producer surplus is lower in the simultaneous case than in the sequential cases: C >C C >C F P >P and P F >P. Given that in this zone is low enough (< the consumer surplus has a low enough weight in weighted welfare. This means that independently of which firm is the leader in prices the lower consumer surplus has less effect than the greater producer surplus implying that W >W and W F >W. Next we consider the case in which the weight given to the consumer surplus takes an intermediate value when products are sustitutes: > for >0 (Zone II in Figure. Proposition shows that in the sequential cases independently of which firm is the leader oth firms set higher prices than in the simultaneous case. This reduces market competition and increases the profit of the private firm: F π > and π >. Proposition also shows that in this zone the consumer surplus is greater and the producer surplus is lower in the simultaneous case than in the sequential cases: C >C C >C F P >P and P F >P. In this zone takes an intermediate value ( > and the consumer (producer surplus has a greater effect on weighted welfare when the pulic firm is the follower (leader. As a result when the pulic firm is the leader in prices the fact that the greater producer surplus has a greater effect on weighted welfare than the lower consumer surplus means that W >W. When the pulic firm is the follower in prices the fact that the

15 lower consumer surplus has a greater effect on weighted welfare than the greater producer surplus means that W W F. Finally we consider the case in which products are complements: H >> for <0 (Zone IV in Figure. Proposition shows that in this zone when the private (pulic firm is the leader in prices it sets a higher price than in the simultaneous case while the pulic (private firm sets a lower price. This increases (reduces the profit of the private firm: π > π ( π > π F. Proposition also shows that consumer surplus is greater in the simultaneous case that when one of the firms is leader in prices (C >C and C >C F. Moreover the producer surplus is smaller (greater when the pulic firm is the follower (leader than in the simultaneous case: P >P F (P >P. As oth the consumer and producer surpluses are greater in the simultaneous case than when the pulic firm is the follower it is otained that W >W F. When the pulic firm is the leader in prices the producer surplus has a greater effect on weighted welfare than the consumer surplus (since in this case H < implying that W >W. Taking into account Proposition the following result is otained. Proposition. In equilirium firms decide prices simultaneously at t if H > max{0 }. For the remaining values of parameter firms decide prices sequentially. In this case if > the pulic firm set prices at t and the private firm at t. However if > > there are two sequential equiliria: the pulic firm is the leader in prices in one of them and the follower in the other. It would e interesting to compare this result with that otained in the context of endogenous timing with quantity-setting competition when the ojective of the pulic firm is a weighted welfare function and firms produce heterogeneous goods. However no such analysis has een made in the literature on mixed oligopoly. In can e shown (the proof is availale from the authors on the request that when goods are complements in equilirium quantities are decided sequentially. When goods are sustitutes in equilirium quantities are decided sequentially if the weight attached to the consumer surplus in weighted welfare is high enough. If the weight takes an intermediate value the pulic firm is the leader in quantities. Finally if the weight is low enough quantity decisions are made simultaneously. Therefore the result otained is the opposite of that under Bertrand competition except for the case of sustitute goods when the weight takes an intermediate value since the pulic firm is the leader under oth types of market competition.

16 From Proposition it is otained that if H > for >0 oth firms want to e the F leader in prices and neither firm wants to e the follower ( π > π > W >W and W >W F. Thus in equilirium they set prices simultaneously at t. In this zone it is a dominant strategy for oth firms to set prices at t. The same result is otained when > > for <0. H If > for >0 the pulic firm wants to e the leader in prices ut does not want to e the follower (W >W and W >W F. Moreover the private firm otains greater F profit when prices are set sequentially rather than simultaneously ( π > and π >. As a result in this zone the pulic firm sets prices at t and the private firm at t. In this way the pulic firm ecomes the leader and prices are set sequentially. Finally if > > for >0 the two firms prefer to set prices sequentially rather F than simultaneously ( π > π > W >W and W F >W. This implies that there are two sequential equiliria: the pulic firm is the leader in prices in one of them and the follower in the other. Comparing the social surplus otained in the different cases we otain the following 8 5 result (see Figure. et >. 8 5 Proposition. In equilirium social surplus is maximum when the pulic firm is the leader in prices if > ; for the remaining values of parameters social surplus is maximum when firms decide prices simultaneously. When >0 the lowest social surplus is otained when the private firm is the leader in prices since oth firms raise prices thus decreasing consumer surplus and social surplus. As a result the maximum social surplus is otained under simultaneous decisions or when the pulic firm is the leader. It remains to e analysed which of the two cases generates the greater social surplus. 5

17 H - 0 FIG.. IUTRATION OF PROPOITION When for >0 it is otained that C C and P P. In this case if is great enough (> the producer surplus has a greater effect on social surplus than the consumer surplus implying that social surplus is greater under simultaneous decisions. However if is low enough ( the consumer surplus has a greater effect on social surplus than the producer surplus implying that social surplus is greater when the pulic firm is leader in prices. As a result when > for >0 social surplus is maximum when the pulic firm is the leader in prices; when H > for >0 social surplus is maximum when firms decide prices simultaneously. When < for >0 it is otained that C >C and P >P. As the consumer surplus has a greater effect than the producer surplus social surplus is maximum when prices are set simultaneously. Finally we consider that products are complements (<0. In this case C >C and P >P. Although the producer surplus is greater than in the simultaneous case the lower consumer surplus has a stronger effect on social surplus. As a result social surplus is maximum when prices are set simultaneously. Next we analyse whether the equiliria otained when firms decide whether to set prices sequentially or simultaneously maximises social surplus or not. 6

18 Proposition 5. In equilirium if H > for >0 and if <0 in equilirium prices are set simultaneously which maximises social surplus. If > for >0 prices are set simultaneously and the maximum social surplus is otained when the pulic firm is the leader in prices. If > > for >0 prices are set sequentially and the maximum social surplus is otained when prices are set simultaneously. This result is due to the fact that the pulic firm can give different weights to the consumer surplus and to the producer surplus while social surplus attaches the same weight to oth surpluses. When <0 under simultaneous decisions the consumer surplus is greater than under sequential decisions for all implying that weighted welfare and social surplus are greater in the first case. When for >0 as is great enough the weight given to the consumer surplus in weighted welfare implies that prices are set simultaneously. However in this case social surplus is maximum only if ; if < social surplus is maximum when the pulic firm is the leader in prices since now the effect of the consumer surplus is not strong enough. When < for >0 as is low enough (< prices are set sequentially ut the maximum social surplus is otained when prices are set simultaneously. In this case as social surplus attaches the same weight to the consumer surplus as to the producer surplus the greater consumer surplus otained in the simultaneous case explains the result. However weighted welfare attaches less weight to the consumer surplus than to the producer surplus which explains why in equilirium prices are set sequentially.. Extension: the private firm is foreign-owned We now consider that the private firm is foreign-owned. Thus weighted welfare is now: ( WCπ 0 0<< H where H min{ }. 5 imilarly social surplus is: Cπ 0. We denote as f the value of parameter such that W F >W if and only if 5 The assumption 0<< H ensures that the two firms otain a positive profit and produce a positive output in all cases. 7

19 < f ; similarly f is the value of parameter such that F π > if and only if < f f < f. olving as for when the private firm was domestic-owned we otain the following result (see Figure. Proposition 6. In equilirium firms decide prices simultaneously at t if H > max{0 f }. For the remaining values of parameter firms decide prices sequentially. In this case the pulic firm sets prices at t and the private firm at t if f > f. However if f >>0 there are two sequential equiliria: the pulic firm is the leader in prices in one of them and the follower in the other. Proof. ee Appendix. H simultaneous decisions simultaneous decisions private leader f f - sequential decisions 0 FIG.. IUTRATION OF PROPOITION 6 The result shown in Proposition 6 is illustrated in Figure. As weighted welfare does not comprise the profit of firm since this firm is foreign-owned the effect of the producer surplus on the decisions of the pulic firm is weaker than when firm was domestic-owned. In oth cases when <0 firms decide prices simultaneously. imilarly in oth cases when >0 firms decide prices simultaneously (sequentially if is great (low enough. The main difference etween propositions 6 and arises when parameter takes an intermediate value for >0. In this case when firm is domestic-owned the pulic firm is 8

20 the leader in prices (if < ; however when firm is foreign-owned the pulic firm is the follower in prices (if f < f. It can e shown that when firm is foreign-owned (see Appendix: W > W F F W > W if and only if > f π > and π > if and only if > f. Therefore if f < f the pulic firm prefers to decide prices sequentially rather than simultaneously ut the private firm prefers to e the leader ecause the effect of the consumer surplus on weighted welfare is now stronger so the pulic firm is more aggressive in the product market. When the pulic firm is the leader in prices its aggressiveness increases since prices are strategic sustitutes which reduces the profit of the private firm. As a result the private firm does not want to e the follower. If the private firm is the leader in prices oth firms raise prices which reduces weighted welfare. Next we analyse which of the three cases generates the greatest social surplus when the private firm is foreign-owned (see Figure. Proposition 7. In equilirium social surplus is maximum when firms decide prices simultaneously if f >0 for <0 and if f >0 for >0; for the remaining values of the parameters social surplus is maximum when the pulic firm is the leader in prices. Proof. ee Appendix. H f - 0 FIG.. IUTRATION OF PROPOITION 7 f 9

21 In this case the result otained is similar to that found when firm was domestic-owned. This is due to the fact that in oth cases the lowest social surplus is otained when the pulic firm is the leader since oth firms rise prices. Thus the greatest social surplus is otained under simultaneous decisions or when the pulic firm is the leader. This last takes place when parameter is low enough. Next we analyse whether the equiliria otained when firms decide whether to set prices sequentially or simultaneously maximises social surplus or not. Comparing Propositions 6 and 7 we otain that if f >0 for <0 in equilirium prices are set simultaneously which generates the maximum social surplus. If H >> f for <0 in equilirium prices are set simultaneously and the maximum social surplus is otained when the pulic firm is the leader in prices. If H > f for >0 in equilirium prices are set simultaneously and the maximum social surplus is otained when the pulic firm is the leader in prices. If f > f for >0 in equilirium the private firm is the leader in prices and the maximum social surplus is otained when the pulic firm is the leader in prices. Finally If f >0 for >0 in equilirium prices are set sequentially and the maximum social welfare is otained when prices are set simultaneously. 5. Conclusions This paper studies a setting where firms decide aout prices and the ojective function of the pulic firm is the weighted sum of consumer and producer surpluses. A mixed duopoly is considered where the firms decide the timing of their choice of prices. Firms produce heterogeneous goods. It is shown that whether firms take decisions sequentially or simultaneously depends crucially on the ojective function considered y the pulic firm and on the degree to which goods are sustitutes or complements. We otain that if the weight given to the consumer surplus in weighted welfare is high enough when goods are sustitutes firms decide prices simultaneously; however when that weight is low enough firms take decisions sequentially. When goods are complements firms decide prices simultaneously independently of the weight allocated to the consumer surplus in weighted welfare Comparing the figures for social surplus otained in the different cases considered in the paper we find that the lowest social surplus is otained when the private firm is the 0

22 leader. We also otain that when goods are complements in equilirium prices are set simultaneously which maximises social surplus. When goods are sustitutes the result depends on the weight given to the consumer surplus in the ojective function of the pulic firm. In this case if the weight is high enough in equilirium prices are set simultaneously which maximises social surplus. If the weight takes an intermediate value prices are set simultaneously and the maximum social surplus is otained when the pulic firm is the leader in prices. Finally if the weight is low enough prices are set sequentially and the maximum social surplus is otained when prices are set simultaneously. Finally we consider that the private firm is foreign-owned. We otain a result similar to that otained when the private firm is privately-owned except for the case of sustitute goods when the weight given to the consumer surplus takes an intermediate value. In this case when the private firm is foreign-owned it is otained that this firm is the leader in prices. However when the private firm is domestic-owned it is otained that this firm is the follower in prices. APPENDIX The profit otained y the pulic firm when it is the leader in prices π 0 is positive if ( ( <<. The profit otained y the pulic firm when it is the follower in F prices π 0 is positive if <. Therefore it is necessary to assume << H ( ( ( where max{0 } and H min{ ( profit of the pulic firm is positive in all cases. } to assure that the ( ( It can e shown that the output level of the firms in the different cases considered is: 0 a c ( ( q ( a c( ( q ( a c( ( q 0 ( ( ( ( ( ( ( a c( ( q F a c q F ( a c( ( 0 q. ( ( ( ( ( ( ( It is easy to see that the output level of the two firms is positive in all cases when << H.

23 Proof of Proposition. By comparing the prices otained in the different cases we otain: ( ( a c( i p0 p0 0 ( ( ( if and only if and p0 p0 <0 if <0; F ( a c( ( ii p0 p0 > 0 ( ( ( ( a c( ( iii p p > 0 ( ( ( if >0 and if >0 and F p0 p0 <0 if <0; p p >0 if <0; F ( a c( ( iv p p 0 ( ( ( if and only if and F p p >0 if <0; By comparing the consumer and producer surpluses otained in all cases we otain: i P P ( ( a c ( (( ( ( ( ( and P P <0 if <0; ( 0 if and only if ii P F P ( a c ( ( ( ( ( > 0 ( ( ( if >0 and P F P <0 if <0; iii C C ( ( a c ( (8 ( ( 0 ( ( ( if and only if and C C <0 if <0; iv C C F and C C F <0 if <0. ( a c ( ( (8 ( 8( ( ( ( (5 >0 if >0 Proof of Proposition. i W W ( ( a c ( > 0 ( ( ( if and only if > for >0 W W 0 for and W W >0 if <0. ii W W F and W W F >0 if <0. ( a c ( ( ( ( 8( ( ( ( ( 0 if and only if

24 F ( (8( 8( ( 5 iii π π ( a c ( ( ( ( 5 ( 0 if and only if for >0 and π >0 if <0. F ( a c ( ( iv π π > 0 ( ( ( for all. The private firm is foreign-owned. When prices are set simultaneously we otain: 0 p ( a( ( c( /( ( p ( a( ( ( c( /( ( 0 π ( ( ( ( D /( π ( ( ( ( D /( C (8 (8 6 ( ( D /(( P ( (8 8 ( 5 ( ( D /( (6 (8 6 ( ( ( D /(( W (8 (8 6 ( ( ( D /(( where D ( a c /( (. When the pulic firm is the leader in prices we get: 0 ( a( (( ( c( ( /(8 ( p p F ( a( ( ( c( π ( /(8 ( 0 ( ( ( (( ( ( D /( F π ( (( ( ( D /( P ( ((8 8 (8 0 C (( (8 (6 8 (8 ( 6 ( ( D /( ( 6 8 (0 8 ( D 5 ((6 (8 ( (8 6 5 ( 8 6 (7 6 ( D /(( 5

25 W ( ( ( (8 ( ( D /(( where D ( a c /(8 (. Finally we consider that the private firm is the leader in prices. This firm chooses the value of p 0 that maximises weighted welfare. olving this prolem we otain the reaction function in prices of the pulic firm: p ( p ( c a( /(. The 0 private firm chooses the value of p that maximises its profits taking into account the reaction function in prices of the pulic firm. olving this prolem we get: 0 ( a( ( ( ( ( c(( ( p F ( ( /(( ( ( p ( a( ( ( c( ( ( /(( ( π π ( ( ( ( ( ( ( D /(( F 0 ( ( ( ( ( ( D /(( C F (( (8 ( ( (( (5 5 5 ( ( ( ( ( ( D /(8( P F ( (( (8 8 ( ( 5 ( ( ( (6 5 ( ( D /(( F (( (6 ( ( ( 0 (5 ( ( ( ( 8 ( ( D /(8( 5 W F ( (( ( ( ( (0 (6 6 ( (5 ( ( D /(8( where D ( a c /(( ( (. Next we provide an outline of the proof of Proposition 6. i W W >0 for all. ii W W F >0 for <0; W W F >0 if and only if < f for >0 where f is the value of such that: ( ( ( (- ( ( 5 ( (6 85 ( (6 5 ( 0. 5

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