Mixed Oligopoly: analysis of oligopoly models considering social welfare

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(2) INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY ENGINEERÍNG & ARCH1TECTURE DIVISIÓN ENGINEERÍNG GRADUATED PROGRAM. Mixed Oügopoly: Analysis of Oligopoly Modela Considering Social Welfare". DISERTATION DOCTOR OF PHILOSOPHY ENGINEERÍNG SCIENCES. BY: ALVARO EDUARDO CORDERO FRANCO. MONTERREY, N. L.. DECEMBER, 2009.

(3) INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY. CAMPUS MONTERREY ENGINEERING & ARCHITECTURE DIVISIÓN ENGINEERING GRADUATED PROGRAM. "Mixed Oligopoly: Analysis of Oligopoly Models Considering Social Welfare". DISERTATION DOCTOR OF PHILOSOPHY ENGINEERING SCIENCES BY:. Alvaro Eduardo Cordero Franco. MONTERREY, N.L.. DECEMBER, 2009.

(4) INSTITUTO TECNOLÓGICO Y DE ESTUDIOS SUPERIORES DE MONTERREY CAMPUS MONTERREY. DIVISIÓN DE INGENIERÍA Y ARQUITECTURA PROGRAMA DE GRADUADOS EN INGENIERÍA Los miembros del comité de tesis recomendamos que el presente proyecto de tesis presentado por el L.M. Alvaro Eduardo Cordero Franco sea aceptado como requisito parcial para obtener el grado académico de: Doctor en Ciencias de la Ingeniería. Comité de Tesis:. Director de Doctorado en Ciencias de Ingeniería Diciembre, 2009.

(5) Dedicatorias. Para mi familia que quiero mucho. Gracias por su apoyo y amor.. Para mi Mamá Eulalia Franco, le agradezco su amor, apoyo, comprensión y entusiasmo. Este trabajo es el reflejo de su constante apoyo y sus enseñanzas.. Para mi Papá Felizardo Cordero, a quién le agradezco su apoyo, amor, consejos y el estar conmigo en todo momento.. Para mi hermano Hid, mi mejor amigo, una persona que admiro y aprecio. Le agradezco sus consejos, su cariño y su gran apoyo.. Para mi hermana Dinorah, ejemplo de tenacidad y perseverancia. Le agradezco su cariño y su apoyo. Me ha enseñado con su ejemplo que la base del éxito es el trabajo constante.. Para mi hermana Diana, gracias por su invaluable cariño, por estar siempre conmigo y apoyarme en los momentos más difíciles.

(6) Agradecimientos. Para el Dr. Vyacheslav Kalashnikov, mi asesor, le agradezco la oportunidad de trabajar con él, su confianza para mi trabajo y sus enseñanzas. Para el Dr. Mario Beruvides, mi asesor, le agradezco el tiempo dedicado a esta investigación, los consejos brindados a lo largo de estos años y la oportunidad de haber sido parte de su grupo de trabajo. Para mis sinodales Dr. Neale Smith, Dr. Milton Smith y Dr. José Manuel Sánchez, muchas gracias por el tiempo dedicado a esta investigación. A mis profesores en el Tecnológico de Monterrey y en Texas Tech University por sus enseñanzas dentro y fuera de clase. En especial a la Maestra María del Carmen Temblador, muchas gracias por su invaluable amistad, por su apoyo y por tantos consejos y enseñanzas; y a la Dra. Natalya Kalashnikova, gracias por su apoyo, cariño y. confianza. A mis compañeros del doctorado Gerardo (gracias por apoyarme durante todo el doctorado), Cecy, Fer, Rosy, Julio, Abril, Eileen, Pepe y Krystel con quienes compartí invaluables experiencias a lo largo de este camino. A mis compañeros del grupo de Seis Sigma Samuel, Margie, Elena, Sara, Rodrigo, Fernando, Víctor, Bertha, Erika, Civy, Mayra, Carlos, Gustavo, Lucy, Pilar y Diana; por su increíble amistad y apoyo. En especial al Dr. Alberto Hernández por sus valiosos consejos y por darme la oportunidad de aprender y convivir en este equipo de trabajo. A mis amigos Daniela, Alicia, Delia, Andrés, Gil, Erick, Alex, Jesús, Héctor, Ernesto, Gildardo, Rubén, Sergio, Yenny, Celina, Xóchitl, Aydee, Flor y Minerva; les agradezco el estar conmigo en cada momento. Este logro ha sido posible gracias a su invaluable apoyo y amistad..

(7) Contents CHAPTER 1. - 1 -. 1.1 History and Background. -1 -. 1.2 Problem Statement. -2-. 1.3 Research Question. -4-. 1.4 General Hypothesis. -5-. 1.5 Research Purpose. •6-. 1.6 Research Objective. ~ 7-. 1.7 Delimitations. ~ 7-. 1.8 Relevance of this Study. - 8-. 1.9 Research Outputs and Outcomes. - 8-. CHAPTER 2. - 10 -. 2.1 Introduction and Background. - 10 -. 2.2 Oligopoly Models and Game Theory. - 11 -. 2.3 The electricity Market. - 15 -. 2.4 Conclusión. - 17 -. CHAPTER 3. - 19 -. Research 1. - 19 -. Cournot and Stackelberg Equilibrium. - 21 -. in Mixed Duopoly Models. - 21 -. Abstract. - 21 -. Introduction. - 22 -. l.TheModel. -24-. 2. Domestic Firm Leader. - 41 -. 3. Prívate Firm Leader. - 61 -. 4. Solution of the Observable Delay Game. - 70 -. 5. Examples. - 78 -. 6. Conclusión. - 84 -. References. - 86 i.

(8) CHAPTER 4. - 88 -. Research 2. - 88 -. Conjectural Variation Equilibrium in an Oligopoly Electricity Markets. - 90 -. Abstact. - 90 -. Introduction. - 90 -. l.Model. -93 -. 2. Duopoly Case. - 101 -. 3. Numerical Results. - 101 -. 4. Conclusions. • 104 -. References. - 105 -. C H A P T E R 5. C O N C L U S I O N S. - 109 -. 5.1. Conclusions. - 109 -. 5.2 Future Researches. -112-. REFERENCES. - 114 -. ii.

(9) L I S T OF T A B L E S. Table 4.1. Payoff Table. 76. Table 4.2. T h e o r e m 4.3 (i). 77. Table 4.3. T h e o r e m 4.3 (ii). 77. Table 4.4. T h e o r e m 4.3 (iii). 78. Table 5 . 1 . E x a m p l e 5.1. 82. Table 5.2. E x a m p l e 5.2. 84. T a b l e 1. C o s t F u n c t i o n P a r a m e t e r s. 102. Table 2. E q u i l i b r i u m. 102. CV Case (Mixed and Prívate O l i g o p o l i e s ). Table 3. C o u r n o t Case (Mixed and Prívate Oligopolies). 103. T a b l e 4. P e r f e c t C o m p e t i t i o n C a s e. 104. iii.

(10) CHAPTER 1. INTRODUCTION TO RESEARCH. 1.1 H i s t o r y and B a c k g r o u n d This r e s e a r c h is i m p o r t a n t in the área of e n g i n e e r i n g e c o n o m i c s because this study f o c u s e s on an a n a l y s i s of o l i g o p o l y m o d e l s c o n s i d e r i n g not only the p r o f i t s for c o m p e t i n g firms, but also the benefits to s o c i e t y . Oligopoly m o d e l s have been a n a l y z e d for more than 200 y e a r s , but recently t h e s e m o d e l s have c o n s i d e r e d the b e n e f i t s to s o c i e t y , in an approach n a m e d m i x e d o l i g o p o l y , in which the e c o n o m y of t h e s e e n t e r p r i s e s f o c u s e s not only on the profits of the f i r m s , but also on societal w e l f a r e .. The i n t e r e s t in o l i g o p o l y m o d e l s , that a n a l y z e not only the profits for the firms but also the welfare to society (mixed d u o p o l i e s ) , is high because of t h e i r i m p o r t a n c e in the e c o n o m i e s of E u r o p e ( G e r m a n y , England and o t h e r s ) , C a n a d á and Japan (see M a t s u s h i m a and M a t s u m u r a , 2003 for a n a l y s i s of " h e r d b e h a v i o r " by p r i v a t e firms in many b r a n c h e s of the e c o n o m y in J a p a n ) , among o t h e r s . There are e x a m p l e s of mixed o l i g o p o l i e s in U n i t e d States such as the p a c k a g i n g and o v e r n i g h t - d e l i v e r y i n d u s t r i e s . M i x e d o l i g o p o l i e s are also common in the East E u r o p e a n and former Soviet U n i o n t r a n s i t i o n a l e c o n o m i e s , in which c o m p e t i t i o n among public and p r i v a t e firms existed or still exists in many i n d u s t r i e s such as banking, h o u s e l o a n , a i r l i n e , t e l e c o m m u n i c a t i o n , n a t u r a l g a s , e l e c t r i c power, h o s p i t a l , h e a l t h c a r e , r a i l w a y s and o t h e r s . These s i t u a t i o n s have been i n v e s t i g a t e d in different w a y s . Many p r e v i o u s s t u d i e s a n a l y z e d Cournot and S t a c k e l b e r g m o d e l s with the role of each firm a s s i g n e d exogenously.. E x a m i n a t i o n s of m i x e d o l i g o p o l i e s , in which s o c i a l s u r p l u s - m a x i m i z i n g public firms c o m p e t e a g a i n s t p r o f i t - m a x i m i z i n g p r i v a t e f i r m s , have become i n c r e a s i n g l y p o p u l a r in r e c e n t y e a r s . For p i o n e e r i n g works on mixed o l i g o p o l i e s , see M e r r i l l and S c h n e i d e r ( 1 9 6 6 ) , H a r r i s and Wiens -1-.

(11) 1.2 Problem Statement. CHAPTER 1.. (1980), and Bós ( 1 9 8 6 , 1991). E x c e l l e n t s u r v e y s can be found in Vickers and Y a r r o w ( 1 9 8 8 ) , De Fraja and D e l b o n o ( 1 9 9 0 ) , N e t t ( 1 9 9 3 ) . This r e s e a r c h l o o k s at two studies to further the k n o w l e d g e of mixed oligopolies c o n s i d e r i n g the benefits to s o c i e t y . F i r s t , based in a model which a n a l y z e d a S t a c k e l b e r g mixed duopoly model ( M a t s u m u r a 2 0 0 5 ) , a duopoly m o d e l is d e v e l o p e d with different a s s u m p t i o n s that a l l o w s a more complete a n a l y s i s w h i c h i n c l u d e s the s t r e n g t h of the firms. The second study is an e x a m p l e of a p r a c t i c a l o l i g o p o l y m o d e l ; the e l e c t r i c i t y market is a n a l y z e d . This s e c o n d study is based on a model of o l i g o p o l y applied in the e l e c t r i c i t y m a r k e t ( Y o u f e i , 2 0 0 5 ) , this r e s e a r c h e x t e n d s its r e s u l t s to the c o n t e x t of a m i x e d o l i g o p o l y c o n s i d e r i n g c o n j e c t u r a l v a r i a t i o n s and with different a s s u m p t i o n s in order to c o n s i d e r more c o m p l e x m o d e l s .. In t h e s e s t u d i e s , an a n a l y s i s of different o l i g o p o l y m o d e l s will be done in order to d e t e r m i n e the c o n d i t i o n s in w h i c h a mixed o l i g o p o l y maximizes the b e n e f i t s to s o c i e t y ( s o c i e t a l w e l f a r e ) .. 1.2 P r o b l e m S t a t e m e n t Some c o u n t r i e s have sold some d o m e s t i c p u b l i c firms to p r i v a t e agents, for i n s t a n c e in M é x i c o banks and the p h o n e i n d u s t r y have been p r i v a t i z e d . In some c a s e s some c o u n t r i e s have a c l a s s i c a l o l i g o p o l y in some i n d u s t r i e s , and in cases like health care and school s e r v i c e s many countries still has a m i x e d o l i g o p o l y (these s e r v i c e s are p r o v i d e d by b o t h : private firms and d o m e s t i c f i r m s ) .. In some c a s e s , the m o n o p o l i s t i c firm cannot s u p p o r t the full demand of the s o c i e t y . For t h i s r e a s o n a c o m p e t i t i o n model with more than one firm could h e l p to satisfy the total d e m a n d . One case of t h i s p r o b l e m is Federal C o m m i s s i o n of E l e c t r i c i t y ( C F E ) , the m o n o p o l i s t i c firm in México in the e l e c t r i c i t y m a r k e t , who has a l l o w e d the p r o d u c t i o n of electricity by e x t e r n a l p r o d u c e r s in order to satisfy the t o t a l d e m a n d . The price and the q u a n t i t y p r o d u c e d by these firms d e p e n d on the d e c i s i o n s of CFE, t h u s m a k i n g it not a real d u o p o l i s t i c s i t u a t i o n : the e x t e r n a l producers are j u s t s u p p l i e r s of CFE. -2-.

(12) CHAPTER 1.. 1.2 Problem Statement. The p a r t i c i p a t i o n of p r i v a t e firms in such i n d u s t r i e s as electric energy and the p r i v a t i z a t i o n of firms like " F e d e r a l C o m m i s s i o n of E l e c t r i c i t y " or " P E M E X " (the state managed M e x i c a n Oil C o m p a n y ) are frequent t h e m e s of d i s c u s s i o n in M é x i c o . This d i s c u s s i o n often focuses on the r e s o u r c e s t h a t the g o v e r n m e n t will r e c e i v e in a short t e r m , but few times this d i s c u s s i o n a d d r e s s e s the social w e l f a r e .. This r e s e a r c h will e v a l ú a t e the e c o n o m i c s of welfare for society under c o n d i t i o n s of m i x e d o l i g o p o l y . This will be a c h i e v e d by c o m p a r i n g the welfare to s o c i e t y (in t h i s c a s e , the d o m e s t i c s o c i a l s u r p l u s for a domestic p u b l i c firm) and the profits of a p r i v a t e firm in a mixed oligopoly m o d e l in four different s c e n a r i o s : 1. The d o m e s t i c p u b l i c firm is a m o n o p o l i s t i c firm. The o p t i m i z a t i o n f u n c t i o n is to m a x i m i z e the d o m e s t i c social s u r p l u s function. 2. The p r i v a t e firm is a m o n o p o l i s t i c firm. The o p t i m i z a t i o n is to m a x i m i z e the p r o f i t s function.. function. 3. The p r i v a t e o l i g o p o l y . The o p t i m i z a t i o n function is to m a x i m i z e the profits function. 4. The m i x e d o l i g o p o l y . O p t i m i z a t i o n will a d d r e s s m a x i m i z i n g two different f u n c t i o n s , d o m e s t i c social s u r p l u s function and profits function simultaneously.. All t h e s e s c e n a r i o s will be a n a l y z e d in order to d e c i d e the c o n d i t i o n s or c h a r a c t e r i s t i c s under which each s c e n a r i o is o p t i m a l to society.. The e l e c t r i c i t y p r o d u c t i o n market is studied as an o l i g o p o l y e x a m p l e under t h e s e s c e n a r i o s . This p r o b l e m is m o d e l e d u n d e r a s s u m p t i o n s of p r o d u c t i o n c a p a c i t y and flow c a p a c i t y . A l s o , this model is a n a l y z e d under the c o n j e c t u r a l v a r i a t i o n s a p p r o a c h , which c o v e r s C o u r n o t model and Perfect C o m p e t i t i o n m o d e l s .. -3 -.

(13) 1.3 Research Question. CHAPTER 1. 1.3 R e s e a r c h Q u e s t i o n. The a n a l y s i s of o l i g o p o l i e s is in g e n e r a l r e l a t e d j u s t with the profits of p r i v a t e c o m p e t i t o r s . But in a lot of c o u n t r i e s a d o m e s t i c firm c o m p e t e s in a o l i g o p o l i s t i c m a r k e t . In c o n s e q u e n c e , t a k i n g into account that domestic firm s h o u l d try to m a x i m i z e the welfare to s o c i e t y , then previous c o n c l u s i o n s about p r i v a t e o l i g o p o l i e s are not valid in these mixed o l i g o p o l i e s m o d e l s . Some i n v e s t i g a t i o n s have tried to a n a l y z e these mixed m o d e l s but m o s t of them with some r i g o r o u s a s s u m p t i o n s , for example a s s u m p t i o n s a b o u t the c o n c a v i t y of the p r i c e f u n c t i o n s . In this research two i n v e s t i g a t i o n s are done with r e s p e c t to mixed m o d e l s . In the first one an a n a l y s i s of a t h e o r e t i c a l mixed d u o p o l y m o d e l is done with assumptions t h a t a l l o w c o n c a v e and some convex p r i c e function. Also an analysis of the s t r e n g t h of the firms and t h e i r role in the S t a c k e l b e r g model is d e v e l o p e d . In the second r e s e a r c h a p r a c t i c a l mixed o l i g o p o l y model is d o n e in the e l e c t r i c i t y m a r k e t s o l v i n g the c o n j e c t u r a l v a r i a t i o n model and c o m p a r i n g t h i s model with e q u i l i b r i u m s in different g a m e s . In both i n v e s t i g a t i o n s c o m p a r i s o n of d o m e s t i c social s u r p l u s , p r o f i t s , price and total v o l u m e p r o d u c e d are c o n d u c t e d and based on t h e s e c o m p a r i s o n s next q u e s t i o n s are a d d r e s s e d by the r e s u l t s that will be o b t a i n e d in the researches s t u d i e s p r o p o s e d for this d i s s e r t a t i o n :. 1.3.1 Research 1: Cournot and Stackelberg Equilibrium in Mixed Duopoly Models. Questions. In the r e s e a r c h of M a t s u m u r a 2 0 0 5 , in a S t a c k e l b e r g mixed duopoly model, the a u t h o r s p r o v e d that the role of the d o m e s t i c firm should always be the l e a d e r in the m a r k e t . This c o n c l u s i ó n is based on the assumptions made by the a u t h o r s , w h i c h a l l o w j u s t c o n c a v e price functions. In t h i s r e s e a r c h these a s s u m p t i o n s are m o d i f i e d in order to allow not only c o n c a v e functions but also convex f u n c t i o n s . With this new model the r o l e for each firm is analyzed a c c o r d i n g to the s t r e n g t h of the firms. In t h i s addressed. •. Stackelberg. mixed. duopoly. research,. next. questions. are. In the S t a c k e l b e r g model under the a s s u m p t i o n of mixed d u o p o l y , w h i c h firm s h o u l d be the l e a d e r ?. -4-.

(14) CHAPTER 1. •. C o n s i d e r i n g the welfare for society ( c o m p a r i n g the d o m e s t i c social s u r p l u s ) , w h i c h model m a x i m i z e s the d o m e s t i c social surplus among the m o n o p o l y , c l a s s i c a l duopoly or mixed d u o p o l y o p t i o n s ?. 1.3.2 Research 2: Conjectural Variation Equilibrium in an Oligopoly Electricity Markets. Questions. Based on the r e s e a r c h of Youfei 2 0 0 5 , in w h i c h the a u t h o r found the existence and u n i q u e n e s s of the e q u i l i b r i u m in the o l i g o p o l y m o d e l . He concluded t h a t the e q u i l i b r i u m in the c o n j e c t u r a l v a r i a t i o n s a p p r o a c h is more c o m p e t i t i v e than the C o u r n o t m o d e l . In t h i s new r e s e a r c h , the c o n c l u s i o n s of Youfei are e x t e n d e d to the mixed o l i g o p o l y model of e l e c t r i c i t y , and the r e s u l t s from this second r e s e a r c h will be useful to a new e l e c t r i c i t y m o d e l w h i c h will be a n a l y z e d in a future r e s e a r c h . The following q u e s t i o n s are a n a l y z e d in this r e s e a r c h : • •. Is the e q u i l i b r i u m in c o n j e c t u r a l v a r i a t i o n model more c o m p e t i t i v e than the C o u r n o t s o l u t i o n or the Perfect C o m p e t i t i o n s o l u t i o n ? Is the d o m e s t i c s o c i a l surplus higher in this s o l u t i o n than the d o m e s t i c s o c i a l s u r p l u s in m o d e l s as p r i v a t e m o n o p o l y , d o m e s t i c m o n o p o l y or p r i v a t e o l i g o p o l y ?. 1.4 General H y p o t h e s i s A c c o r d i n g to the l i t e r a t u r e r e v i e w , c o m p a r i n g p r i v a t e o l i g o p o l y , mixed o l i g o p o l y and m o n o p o l y , the general h y p o t h e s i s is the next: H l : The d o m e s t i c social surplus is m a x i m i z e d in a mixed o l i g o p o l y model. This h y p o t h e s i s will be a d d r e s s e d s o l v i n g the f o l l o w i n g h y p o t h e s e s :. 1.4.1 Research 1: Cournot and Stackelberg Equilibrium in Mixed Duopoly Models. Hypothesis. B a s e d in M a t s u m u r a ( 2 0 0 5 ) , with some c h a n g e s made in the model in order to o b t a i n r e s u l t s i n c l u d i n g the s t r e n g t h of the firms, the following h y p o t h e s e s are f o r m u l a t e d :. -5-.

(15) 1.5 Research Purpose. CHAPTER 1.. H l . l : In the S t a c k e l b e r g m o d e l , the l e a d e r firm is not always the domestic firm. A l s o , the p r i v a t e firm (or a c o m b i n a t i o n of d o m e s t i c and private firms) can be l e a d e r under some c o n d i t i o n s of the m a r k e t . H 1 . 2 : A m i x e d m o d e l m a x i m i z e s the d o m e s t i c social s u r p l u s .. 1.4.2 Research 2: Conjectural Variation Equilibrium in an Oligopoly Electricity Markets. Hypothesis. Based in Youfei 2005 and the r e s u l t s of Isaac 2002 in c o n j e c t u r a l variations r e s e a r c h e s , the following is h y p o t h e s i z e d : H 2 . 1 : This s o l u t i o n is more Perfect c o m p e t i t i o n s o l u t i o n s .. competitive. than. the. Cournot. and. H 2 . 2 : A m i x e d m o d e l m a x i m i z e s the d o m e s t i c s o c i a l s u r p l u s .. 1.5 R e s e a r c h P u r p o s e The main p u r p o s e of t h i s r e s e a r c h is the b e t t e r u n d e r s t a n d the different oligopoly m o d e l s . T h i s r e s e a r c h p r o v i d e s more b a s e s to d e c i d e what the best model of o l i g o p o l y ( p r i v a t e o l i g o p o l y , mixed o l i g o p o l y , m o n o p o l y ) is for a s o c i e t y (in the c o n t e x t of d o m e s t i c social s u r p l u s ) , in the case that the best m o d e l e x i s t s . If t h e r e is no a best m o d e l , this r e s e a r c h will provide c o n d i t i o n s u n d e r w h i c h each of the m o d e l s a n a l y z e d is the best for b e n e f i t i n g s o c i e t y .. Another p u r p o s e for this study is to a n a l y z e the p r o b l e m s studied in the c l a s s i c a l o l i g o p o l y in the c o n t e x t of mixed o l i g o p o l y . The e l e c t r i c i t y market has been a n a l y z e d under c o n d i t i o n s of p r i v a t e o l i g o p o l y . In this research, t h i s p r o b l e m is a n a l y z e d c o n s i d e r i n g a mixed o l i g o p o l y , b e c a u s e some c o u n t r i e s own m o n o p o l i s t i c firms that c o n t r o l the total e l e c t r i c i t y market.. -6-.

(16) 1.6 Research Objective. CHAPTER 1. 1.6 R e s e a r c h O b j e c t i v e. The t h e o r e t i c a l o b j e c t i v e s of this r e s e a r c h a r e : 1. To r e s e a r c h and e x p a n d the theory of o l i g o p o l y m o d e l s , c l a s s i c a l and m i x e d . O l i g o p o l y m o d e l s have been a n a l y z e d u n d e r C o u r n o t , S t a c k e l b e r g and C o n j e c t u r a l V a r i a t i o n s models among o t h e r s . In this r e s e a r c h t h e s e t h r e e m o d e l s will be e x p a n d e d to an a n a l y s i s of mixed o l i g o p o l y , by a d d i n g a d o m e s t i c c o m p e t i t o r m a x i m i z i n g d o m e s t i c s o c i a l s u r p l u s . This a n a l y s i s is m o t i v a t e d by the d i s c u s s i o n of the a d d i t i o n of a d o m e s t i c c o m p e t i t o r in a p r i v a t e o l i g o p o l y . U s i n g different models and e x a m p l e s some c o n s e q u e n c e s of the a d d i t i o n of a d o m e s t i c firm in the m a r k e t will be a n a l y z e d for the s o c i e t y and for the p r i v a t e c o m p e t i t o r s .. Practical objectives:. 2. To d e v e l o p a n u m e r i c a l p r o g r a m to solve different o l i g o p o l y m o d e l s in order to c o m p a r e t h e s e models ( m o n o p o l y , m i x e d o l i g o p o l y and p r i v a t e o l i g o p o l y ) , in the e l e c t r i c i t y m a r k e t in order to o b t a i n under what c o n d i t i o n s each model is the best for the s o c i e t y .. 1.7 D e l i m i t a t i o n s This r e s e a r c h a s s u m e s that the c o m p e t i t o r s are r a t i o n a l , that is, in this r e s e a r c h is a s s u m e d that the c o m p e t i t o r s are l o o k i n g for m a x i m i z e their p r o f i t s or the d o m e s t i c social s u r p l u s , in a c o m p l e t e i n f o r m a t i o n game.. M o d e l s u n d e r c o n s i d e r a t i o n in this r e s e a r c h are t h e o r e t i c a l models which, b e c a u s e of the c o m p l e x i t y of the p r o b l e m , did not p r o v e d with real data. These m o d e l s w e r e a n a l y z e d under t h e o r e t i c a l c a s e s solved with numerical a l g o r i t h m s .. -7-.

(17) CHAPTER 1.. 1.8 Relevance of this Study 1.8 R e l e v a n c e of this Study. This r e s e a r c h will be c o n d u c t e d b e c a u s e in several c o u n t r i e s Europe and A m e r i c a t h e m e s like p r i v a t i z e a d o m e s t i c firm, m o n o p o l y mixed o l i g o p o l y are a c o m m o n theme of d i s c u s s i o n . In the study of Stackelberg m o d e l for the mixed o l i g o p o l y this r e s e a r c h tries to solve observable delay g a m e , b e c a u s e this is an e n d o g e n o u s form of decide role of each firm in the case of a s e q u e n t i a l g a m e .. of and the the the. In a d d i t i o n , t h i s r e s e a r c h a n a l y z e s a p r a c t i c a l p r o b l e m that has been studied in different c o u n t r i e s ( S p a i n , J a p a n , E n g l a n d and W a l e s , among others) u n d e r a c l a s s i c a l o l i g o p o l y c o n d i t i o n s using s i m u l a t i o n and others methods, the p r o d u c t i o n e l e c t r i c i t y m a r k e t , which has been analyzed always u n d e r the a s s u m p t i o n of j u s t p r i v a t e firms in c o m p e t i t i o n . In some former c o u n t r i e s , like M é x i c o , there e x i s t s a d o m e s t i c m o n o p o l i s t i c firm, by this r e a s o n is c o n v e n i e n t the study of this m a r k e t c o n s i d e r i n g a mixed oligopoly and c o m p a r e which model b r i n g s more b e n e f i t s to s o c i e t y .. 1.9 R e s e a r c h Outputs and O u t c o m e s The p r i n c i p a l o u t p u t of this r e s e a r c h is the a n a l y s i s of the a d d i t i o n of a d o m e s t i c firm in a p r i v a t e m a r k e t . This will be done c o m p a r i n g prices, t o t a l o u t c o m e , p r o f i t s and social s u r p l u s in the e q u i l i b r i u m for mixed and p r i v a t e o l i g o p o l i e s . in a d d i t i o n , with t h i s a n a l y s i s the r e s u l t s in p r i v a t e o l i g o p o l y m o d e l s will be e x p a n d e d to a mixed o l i g o p o l y models considering three oligopolistic models: Cournot, Stackelberg and Conjectural V a r i a t i o n m o d e l s . These m o d e l s d e s c r i b e the o l i g o p o l y model in a c o n t e x t of h o m o g e n e o u s p r o d u c í s and perfect i n f o r m a t i o n . By these reason t h e s e m o d e l s have been used for m o d e l i n g e l e c t r i c i t y o l i g o p o l y models. A m o d e l for a mixed e l e c t r i c i t y m a r k e t will be d e v e l o p e d and solved and c o m p a r i s o n s of the p r o f i t s , social s u r p l u s , total o u t c o m e and price will be done for different o l i g o p o l i s t i c s i t u a t i o n s .. One i m p o r t a n t o u t c o m e is the c l a s s i f i c a t i o n of firms in strong and weak, and the c o m p a r i s o n of the d o m e s t i c social s u r p l u s in the c l a s s i c a l (Cournot and S t a c k e l b e r g models with p r i v a t e firms) and mixed -8-.

(18) CHAPTER 1.. 1.9 Research Outputs and Outcomes. oligopolies ( C o u r n o t and S t a c k e l b e r g m o d e l s with one p r i v a t e firm and one d o m e s t i c f i r m ) .. In the r e s e a r c h that i n c l u d e s the e l e c t r i c i t y m a r k e t , a numerical program that s o l v e s t h i s m u l t i - l e v e l p r o b l e m will be d e v e l o p e d and an analysis of the d i f f e r e n t c o m p e t i t i o n s c e n a r i o s will be d o n e .. -9-.

(19) CHAPTER 2.. 2.1 Introduction and Background. CHAPTER 2. LITERATURE REVIEW. 2.1 I n t r o d u c t i o n and B a c k g r o u n d This s t u d y is focused in the Cournot m o d e l , which is a game where two firms p r o d u c e a h o m o g e n e o u s p r o d u c t and each firm is trying to maximize its own p r o f i t . Each firm c h o o s e s its v o l u m e s i m u l t a n e o u s l y . The s o l u t i o n for t h i s p r o b l e m was a n a l y z e d by C o u r n o t in 1 838; this solution is e q u i v a l e n t to the Nash e q u i l i b r i u m for this p r o b l e m .. A l s o , t h i s r e s e a r c h c o n s i d e r s a second model d e r i v e d by the Cournot mode, the S t a c k e l b e r g m o d e l . This model is a s e q u e n t i a l game in which one c o m p e t i t o r c h o o s e s its v o l u m e after o b s e r v i n g the v o l u m e of the other firm. The firm that d e c i d e s first is called the Leader and the other is Follower.. This r e s e a r c h a n a l y z e s the models d e s c r i b e d above in a different context; m i x e d o l i g o p o l y is studied instead of c l a s s i c a l o l i g o p o l y . The concept of m i x e d o l i g o p o l y is an o l i g o p o l y in w h i c h a d o m e s t i c p u b l i c firm m a x i m i z i n g d o m e s t i c social surplus (a welfare for the s o c i e t y ) and a foreign firms s e a r c h i n g to m a x i m i z e their own p r o f i t s , c o m p e t e . C o n s u m e r surplus is the d i f f e r e n c e b e t w e e n the p r i c e c o n s u m e r s are w i l l i n g to pay and the a c t u a l p r i c e . If s o m e o n e is w i l l i n g to pay more than the actual price (the p r i c e at w h i c h the supply curve i n t e r s e c t the d e m a n d c u r v e ) , their benefit in a t r a n s a c t i o n is how much they saved when they d i d n ' t pay that p r i c e . The a g g r e g a t e c o n s u m e r s ' surplus is the sum of the c o n s u m e r ' s surplus for each i n d i v i d u a l c o n s u m e r .. Also in t h i s r e s e a r c h an e x a m p l e of an o l i g o p o l y p r o b l e m is analyzed. B e c a u s e of some i n h e r e n t c h a r a c t e r i s t i c s of the p r o b l e m of production and d i s t r i b u t i o n of e l e c t r i c i t y in the e l e c t r i c i t y m a r k e t , this problem has been m o d e l e d as a Cournot game by many a u t h o r s . In this research a o l i g o p o l y p r o b l e m of g e n e r a t i o n and of e l e c t r i c i t y will be modeled. - 10-.

(20) 2.2 Oligopoly Models and Game Theory. CHAPTER 2.. In next s e c t i o n , some r e l e v a n t o l i g o p o l y model and some i m p o r t a n t aspects of game t h e o r y in w h i c h this r e s e a r c h is based will be d e s c r i b e d .. 2.2 O l i g o p o l y Models and Game T h e o r y The first of t h e m a t h e m a t i c a l a n a l y s e s of the o l i g o p o l y p r o b l e m s appeared in the 1 9 C e n t u r y by A n t o i n e A u g u s t i n C o u r n o t ( 1 8 0 1 - 1 8 7 7 ) , who p r o p o s e d a m o d e l of duopoly in 1838, named " T h e C o u r n o t M o d e l " in which both firms p r o d u c e a h o m o g e n e o u s p r o d u c t and the price of the product is a f u n c t i o n of the v o l u m e s p r o d u c e d by both f i r m s . In C o u r n o t ' s analysis, both firms c h o o s e t h e i r r e s p e c t i v e v o l u m e at the same time in order to m a x i m i z e t h e i r own p r o f i t s , k n o w i n g that the d e c i s i ó n of each firm will affect the p r o f i t s of the second firm. With this idea, C o u r n o t contrasted the idea of Adam Smith about the i n v i s i b l e h a n d , b e c a u s e the profit of each firm d e p e n d s not only of its own d e c i s i ó n but also in the decisión of the o t h e r f i r m s . t h. In 1883 a n o t h e r model of duopoly was p r o p o s e d by J o s e p h Louis Francois B e r t r a n d ( 1 8 2 2 - 1 9 0 0 ) , he s u g g e s t e d that the firms a c t u a l l y choose p r i c e s i n s t e a d of v o l u m e , as first s t i p u l a t e d in the C o u r n o t m o d e l . In this m o d e l the d e m a n d d e p e n d s on the price set by b o t h firms, for which the profit of e a c h firm d e p e n d s , like in the C o u r n o t m o d e l , on the decisión of both f i r m s . This model was named "The B e r t r a n d M o d e l " .. The C o u r n o t and B e r t r a n d m o d e l s reflect p r o b l e m s in which the payoff of each firm is d e t e r m í n a t e by the c o m b i n a t i o n of the d e c i s i ó n of both f i r m s , u n d e r the a s s u m p t i o n that each firm m a k e s a h o m o g e n e o u s product. A l s o , the d e c i s i ó n is taken at the same t i m e and i n d e p e n d e n t l y with c o m p l e t e i n f o r m a t i o n (all the firms know the f u n c t i o n costs and the price or v o l u m e f u n c t i o n ) . These k i n d s of m o d e l s were the first mathematical a n a l y s e s of m u l t i - p e r s o n d e c i s i ó n p r o b l e m s , in which the profit of each p e r s o n ( c o m p e t i t o r ) is d e t e r m i n e d by the c o m b i n a t i o n of the decisión of all the c o m p e t i t o r s .. - 11 -.

(21) 2.2 Oligopoly Models and Game Theory. CHAPTER 2.. In the 20 C e n t u r y a new theory that a n a l y z e s these k i n d s of m u l t i person d e c i s i ó n p r o b l e m s was d e v e l o p e d mainly by John von N e u m a n n (1903-1957) who in his book " T h e o r y of Games and E c o n o m i c B e h a v i o r " published in 1944 w i t h Oskar M o r g e n s t e r n , p r o v i d e d the first d e f i n i t i o n s of game and gave the s o l u t i o n s for p a r t i c u l a r t y p e s of g a m e s . The a u t h o r s focused p r i m a r i l y on s u m - z e r o g a m e s , in which the sum of the benefits of all p l a y e r s is a l w a y s equal to z e r o .. F o r m e r l y , Game T h e o r y is the study of m u l t i - p e r s o n d e c i s i ó n problems, in w h i c h each p e r s o n ( p l a y e r ) c h o o s e s ( s i m u l t a n e o u s l y or s e q u e n t i a l l y ) a s t r a t e g y ( d e c i s i ó n ) and the c o m b i n a t i o n of all the strategies c h o s e n by the p l a y e r s d e t e r m i n e s the payoff for each p l a y e r . The n o r m a l r e p r e s e n t a t i o n a l form of a game s p e c i f i e s the number of players, the s t r a t e g i e s a v a i l a b l e for each player and the payoff r e c e i v e d for each p l a y e r for each p o s s i b l e c o m b i n a t i o n of s t r a t e g i e s . Let S¡ be the set of s t r a t e g i e s for p l a y e r / and s¡ a m e m b e r of t h i s set. Let be the payoff. function. of the p l a y e r s . Then the game. u ,u ,...,u x. 2. n. is d e n o t e d. by. A g e n e r a l s o l u t i o n for the games a p p e a r e d in 1950 and was p r o p o s e d by John F o r b e s N a s h , Jr. His s o l u t i o n of a g a m e , k n o w n as the Nash e q u i l i b r i u m , is the set of s t r a t e g i e s in which no p l a y e r has a n y t h i n g to gain by c h a n g i n g o n l y his or her own s t r a t e g y u n i l a t e r a l l y . That means that if each p l a y e r has c h o s e n a strategy and no p l a y e r can benefit by changing his or her s t r a t e g y w h i l e the other p l a y e r s keep t h e i r s unchanged, t h e n the c u r r e n t set of s t r a t e g y c h o i c e s and the c o r r e s p o n d i n g payoffs c o n s t i t u t e the N a s h e q u i l i b r i u m . This c o n c e p t has been widely used and has c o n t r i b u t e d to the d e v e l o p m e n t of the Game T h e o r y . The formal d e f i n i t i o n of the N a s h e q u i l i b r i u m is as f o l l o w s :. - 12-.

(22) 2.2 Oligopoly Models and Game Theory. CHAPTER 2. (1). (2). This c o n c e p t is decisión p r o b l e m s as solution p r o p o s e d by applied in t h i s s p e c i f i c. used to p r o v i d e a s o l u t i o n for the m u l t i - p e r s o n the C o u r n o t and B e r t r a n d m o d e l s ( a c t u a l l y the C o u r n o t for his model is the N a s h e q u i l i b r i u m problem).. T h a n k s to the d e v e l o p m e n t of game t h e o r y , the N a s h e q u i l i b r i u m and the d u o p o l y m o d e l s of C o u r n o t and B e r t r a n d ; new o l i g o p o l y m o d e l s based in t h e s e c l a s s i c a l m o d e l s with different a s s u m p t i o n s in order to represent the d i f f e r e n t t y p e s of o l i g o p o l y p r o b l e m s have been d e v e l o p e d and solved.. Heinrich Freiherr von S t a c k e l b e r g (1905-1946) developed a sequential game in w h i c h each firm plays the role e i t h e r being leader or being f o l l ó w e r . The l e a d e r firm d e c i d e s its m o v e m e n t first and then the follower firm m o v e s s e q u e n t i a l l y after k n o w i n g the d e c i s i ó n of the leader firm. The c o n d i t i o n s for t h i s game are the same c o n d i t i o n s as that for the Cournot game (the p r o d u c t is h o m o g e n e o u s and the d e c i s i o n s and the information is c o m p l e t e ) . In t h i s game the r o l e s , l e a d e r and f o l l o w e r , are given e x o g e n o u s l y . H a m i l t o n and S l u t s k y ( 1 9 9 0 ) p r o p o s e d an e n d o g e n o u s s o l u t i o n for the roles of the S t a c k e l b e r g game using the N a s h e q u i l i b r i u m where the strategies for each firm is either being l e a d e r or f o l l o w e r . This game is named o b s e r v a b l e d e l a y g a m e .. The p r i c e s e t t i n g p r o b l e m is a p r o b l e m d e r i v e d of the B e r t r a n d model in w h i c h each firm c h o o s e s the p r i c e of its p r o d u c t s in a s e q u e n t i a l way (as in S t a c k e l b e r g game) for a d e t e r m i n e d n u m b e r of m o v e m e n t s . -13 -.

(23) 2.2 Oligopoly Models and Game Theory. CHAPTER 2.. There are w o r k s d e r i v e d of this model for e x a m p l e in 1998 M a t s u m u r a considered t h i s p r o b l e m in a s e q u e n t i a l form with two s t a g e s .. Recently new o l i g o p o l y m o d e l s in which a s o c i a l d o m e s t i c firm competes a g a i n s t p r i v a t e foreign firm has been a n a l y z e d . These models are called m i x e d o l i g o p o l y m o d e l s and have been s t u d i e d b e c a u s e of their importance in c o u n t r i e s in which the c o m p e t i t i o n among p u b l i c and private firms e x i s t s . For e x a m p l e s of t h e s e t y p e s of s t u d i e s see Net (1993), M e r r i l l and S c h n e i d e r ( 1 9 9 6 ) , M a t s u m u r a ( 1 9 9 8 ) , and M a t s u m u r a (2003). The p r i n c i p a l a d d i t i o n to t h e s e m o d e l s is the c h a n g e of the objective f u n c t i o n of the d o m e s t i c firm. In t h e s e m o d e l s the o b j e c t i v e of the d o m e s t i c firm is to m a x i m i z e the social s u r p l u s in order to m a x i m i z e the benefits to s o c i e t y . These models have been a n a l y z e d under the assumptions of the C o u r n o t model ( m a x i m i z i n g the firms p r o f i t s given the output q u a n t i t y t h a t the other firm p r o d u c e s ) by M a t s u m u r a (2003) and under the a s s u m p t i o n s of B e r t r a n d model (each firm m a x i m i z i n g its own profit given the e s t a b l i s h e d p r i c e set by the other firm) by M a t s u m u r a (1998). Also N e t t a n a l y z e d this model under the a s s u m p t i o n s of Cournot and B e r t r a n d m o d e l s . The main p r o b l e m w i t h t h o s e p r e v i o u s s t u d i e s is that t h e i r a s s u m p t i o n s are based in the C o u r n o t and B e r t r a n d m o d e l s , and the s u b s e q u e n t models analyze t h e . o l i g o p o l y p r o b l e m in a s i m p l i s t i c m a n n e r . These m o d e l s do not c o n s i d e r the w h o l e o l i g o p o l y p r o b l e m in w h i c h the firms should decide the q u a n t i t y of g o o d s p r o d u c e d , the p r i c e of the g o o d s and also the firms can c o n s i d e r a g r e e m e n t s with their c o m p e t i t o r s in order to maximize t h e i r p r o f i t s .. Also the m i x e d o l i g o p o l y m o d e l s c o n s i d e r the w e l f a r e for the society as just the d o m e s t i c s o c i a l s u r p l u s ; which is the difference b e t w e e n the price c o n s u m e r s are w i l l i n g to pay and the actual m a r k e t p r i c e . But price is not the only c h a r a c t e r i s t i c of the p r o d u c t (or the m a r k e t ) that can be analyzed as the w e l f a r e for society. For e x a m p l e the q u a l i t y of the product, the q u a n t i t y of p r o d u c t in the market or the s a t i s f i e d d e m a n d , the jobs c r e a t e d by the firms and the salary of their e m p l o y e r s , the t a x e s paid by the firms to the g o v e r n m e n t (taxes being c o n s i d e r e d a benefits to society), as well as o t h e r s . - 14-.

(24) 2.3 The electricity Market.. CHAPTER 2.. Recently s o m e a u t h o r s have a n a l y z e d o l i g o p o l y m o d e l s under the Conjectural V a r i a t i o n s (CV) m e t h o d , which i n c l u d e s the C o u r n o t model and Perfect C o m p e t i t i o n . The ñame of t h i s model is Conjectural Variations m o d e l . The first n o t i o n of this m e t h o d was i n t r o d u c e d by Frish in 1933 The C o n j e c t u r a l V a r i a t i o n is defined as o n e ' s e x p e c t a t i o n on the rival's r e a c t i o n to his d e c i s i ó n . H e n e e , it can be e x p e c t e d that, after taking into a c c o u n t the c o n j e c t u r e about the r i v a l ' s r e s p o n s e , the Conjectural V a r i a t i o n s m o d e l will p r o v i d e a b e t t e r way to study the supplier's s t r a t e g i c b e h a v i o r s . M a t h e m a t i c a l l y , the p a r a m e t e r i z e d first order c o n d i t i o n ( F O C ) for this are:. Where v¡j are the c o n j e c t u r a l v a r i a t i o n s . If t h e s e v a l ú e s are all equal to zero t h i s c o r r e s p o n d s to the first order c o n d i t i o n s to C o u r n o t m o d e l . If these v a l ú e s are all equal to - 1 , then this c o r r e s p o n d s to the perfect competition m o d e l . For different v a l ú e s of the CV different c o m p e t i t i v e models are r e p r e s e n t e d .. Some of t h e s e o l i g o p o l y m o d e l s , as C o u r n o t , S t a c k e l b e r g and Conjectural V a r i a t i o n s m o d e l s , have been used to model the e l e c t r i c i t y production m a r k e t . In t h e next s e c t i o n , some r e s e a r c h e s that a n a l y z e this problem are r e v i e w e d , in order to d e v e l o p a new mixed o l i g o p o l y model in this m a r k e t .. 2.3 The e l e c t r i c i t y M a r k e t . R e c e n t l y , many a u t h o r s have a n a l y z e d the e l e c t r i c i t y p r o d u c t i o n market as an e x a m p l e of o l i g o p o l y p r o b l e m , and t h e n they have used Cournot and B e r t r a n d m o d e l s to s i m ú l a t e and find the c o m p e t i t o r ' s optimal p r o d u c t i o n of e l e c t r i c i t y , b e c a u s e of some c h a r a c t e r i s t i c s of the electricity m a r k e t as long c o n s t r u c t i o n p e r i o d and huge c a p i t a l i n v e s t m e n t (Liu Y o u f e i ) . By t h i s r e a s o n many a u t h o r s have a n a l y z e d the c o m p e t i t i o n. - 15-.

(25) 2.3 The electricity Market.. CHAPTER 2.. in e l e c t r i c i t y m a r k e t u n d e r C o u r n o t , B e r t r a n d or different c o m p e t i t i o n models and they h a v e founded the Nash e q u i l i b r i u m s in these g a m e s .. In 1995 B o r e n s t e i n , B u s h n e l l , Kahn and Stoft a n a l y z e d the Cournot equilibrium in the e l e c t r i c i t y m a r k e t in C a l i f o r n i a , b e c a u s e of the emerging new e l e c t r i c i t y i n d u s t r y in that place d u r i n g t h o s e y e a r s . They analyzed two d i f f e r e n t c a s e s , the first o n e , a two node m o n o p o l y competition and the s e c o n d one the two s y m m e t r i c n o d e s and s u p p l i e r s . They h i g h l i g h t e d some of the p o t e n t i a l m a r k e t p o w e r c o n c e r n s in two of the most s i g n i f i c a n t m a r k e t s and some m e t h o d o l o g i e s for e x a m i n i n g the potential s e v e r i t y of t h e s e c o n c e r n s .. In 1 9 9 8 , R u d k e v i c h , D u c k w o r t h and Rosen a n a l y z e d the efficiency of the c o m p e t i t i o n of a small number of firms in the m a r k e t of e l e c t r i c i t y . They used in t h e i r a n a l y s i s a c o m p e t i t i o n model n a m e d P o o l c o , which was developed by G e r b e r in 1994. They found the N a s h e q u i l i b r i u m of this game under some c o n s t r a i n t s with respect to c a p a c i t y of p r o d u c t i o n of the firms, d e m a n d and cost f u n c t i o n s of p r o d u c e e l e c t r i c i t y .. C a t h e r i n e W o l f r a m in 1999 a n a l y z e d the m a r k e t e l e c t r i c i t y in England and W a l e s after the p r i v a t i z a t i o n on this m a r k e t . She a n a l y z e d this model c o n s i d e r i n g a game with g e n e r a t o r s and d i s t r i b u t o r s firms. She also c o n s i d e r s r e g u l a t o r y c o n s t r a i n t s and financial c o n t r a c t s b e t w e e n the suppliers and t h e i r c u s t o m e r s . She found the C o u r n o t e q u i l i b r i u m s for this game.. In 2 0 0 2 , B e s s e m b i n d e r and Lemmon p r e s e n t e d an e q u i l i b r i u m model in the e l e c t r i c i t y m a r k e t . In t h e i r s i m u l a t i o n they found that the forward power p r i c e will e x c e e d e x p e c t e d . spot p r i c e s when e i t h e r e x p e c t e d demand or d e m a n d v o l a t i l i t y are high.. G a r c í a , V e n t o s a , R i v i e r , Ramos and R e l a ñ o in 2002 a n a l y z e d the electricity m a r k e t u n d e r a c o n j e c t u r a l v a r i a t i o n s a p p r o a c h . They also -16-.

(26) 2.4 Conclusión. CHAPTER 2.. analyzed a case of study of the Spanish E l e c t r i c i t y M a r k e t with several firms c o m p e t i n g in the m a r k e t . They found the different b e h a v i o r of the firms r a n g i n g from Perfect C o m p e t i t i o n to C o u r n o t m o d e l .. C o n t r e r a s , K l u s c h and Krawczyk in 2004 found n u m e r i c a l s o l u t i o n s to N a s h - C o u r n o t e q u i l i b r i u m s and used t h e s e s o l u t i o n s for two cases of study r e l a t e d to the e l e c t r i c i t y m a r k e t . The first one i n c l u d e s a IEEE 30bus system and a n o t h e r one i n c l u d i n g line m o d e l i n g and flow c o n s t r a i n t s .. Youfei Liu, Ni Y.X, a n a l y z e d the c o m p e t i t i o n in the e l e c t r i c i t y market u s i n g a c o n j e c t u r a l v a r i a t i o n a p p r o a c h , which c o n s i d e r s w i t h i n this model the p e r f e c t c o m p e t i t i o n and the C o u r n o t model in 2 0 0 5 . They analyzed the s t r a t e g i c b e h a v i o r in a d e r e g u l a t e d e l e c t r i c i t y m a r k e t , and found the e x i s t e n c e and u n i q u e n e s s of their s o l u t i o n .. N o w , b a s e d in t h e s e o l i g o p o l y m o d e l s and r e s e a r c h e s of e l e c t r i c i t y production m a r k e t p r e s e n t e d in s e c t i o n s 2 . 2 and 2.3 r e s p e c t i v e l y ; next section d e s c r i b e s the p a r t s in which this r e s e a r c h , w h o s e aim is obtain a better u n d e r s t a n d i n g of m i x e d o l i g o p o l y m o d e l s , is c o m p o s e d .. 2.4 C o n c l u s i ó n In t h i s r e s e a r c h an a n a l y s i s of the mixed o l i g o p o l y p r o b l e m will be done in order to d e v e l o p a better u n d e r s t a n d i n g of t h e s e c o m p e t i t i o n systems a n a l y z e d p r e v i o u s l y as p r i v a t e o l i g o p o l y m o d e l s in the c o n t e x t of mixed o l i g o p o l y m o d e l s . This r e s e a r c h will be helpful for u n d e r s t a n d the implications of a d o m e s t i c firm in a p r i v a t e m a r k e t , the c o n s e q u e n c e s for society and for p r i v a t e c o m p e t i t o r s . In order to d e v e l o p t h a t , the r e s e a r c h is divided in two r e s e a r c h e s d e s c r i b e d next w h i c h a n a l y z e t h e s e m o d e l s :. In C h a p t e r 3 a t h e o r e t i c a l mixed duopoly model was a n a l y z e d under the p e r s p e c t i v e of C o u r n o t and S t a c k e l b e r g c o m p e t i t i o n . The o b j e c t i v e of this part is to a n a l y z e the c o n d i t i o n s under w h i c h each firm, either domestic or p r i v a t e , should be the leader in t h i s c o m p e t i t i o n t y p e . - 17-.

(27) CHAPTER 2.. 2.4 Conclusión. Comparisons of p r o f i t s , p r i c e s , v o l u m e s of p r o d u c t i o n and d o m e s t i c social welfare were done in different c a s e s . This r e s e a r c h was p u b l i s h e d in the Journal O p t i m i z a t i o n , and its c o n c l u s i o n s are the base to the next research.. In C h a p t e r 4 an e l e c t r i c i t y model based in the model p r e s e n t e d by Youfei in 2005 is a n a l y z e d with a mixed o l i g o p o l y i n s t e a d of a p r i v a t e oligopoly. T h i s r e s e a r c h c o n s i d e r s the c o n j e c t u r a l v a r i a t i o n s method and proves the e x i s t e n c e and u n i q u e n e s s of e q u i l i b r i u m u n d e r this m e t h o d . A comparison of the e q u i l i b r i u m s founded in different o l i g o p o l y m o d e l s , such as C o u r n o t m o d e l , perfect c o m p e t i t i o n m o d e l , c o n j e c t u r a l v a r i a t i o n method will be d o n e in order to compare these m o d e l s . Also a c o m p a r i s o n of the r e s u l t s of t h e n e w e q u i l i b r i u m s and the e q u i l i b r i u m s founded in Youfei 2 0 0 5 will be done will be done in o r d e r to c o m p a r e mixed oligopolies w i t h p r i v a t e o l i g o p o l i e s .. -18-.

(28) CHAPTER 3.. CHAPTER 3. Research 1. Cournot and Stackelberg Equilibrium in Mixed Duopoly Models. In t h i s r e s e a r c h a t h e o r e t i c a l mixed d u o p o l y m o d e l is a n a l y z e d under the p e r s p e c t i v e of C o u r n o t and S t a c k e l b e r g c o m p e t i t i o n . In this duopoly m o d e l one d o m e s t i c p u b l i c firm m a x i m i z i n g d o m e s t i c social surplus, and a p r i v a t e firm s e a r c h i n g to m a x i m i z e its own profits c o m p e t e . This model is d e r i v e d by the Cournot and S t a c k e l b e r g m o d e l s was are classic m o d e l s in p r i v a t e o l i g o p o l y with s o l u t i o n s k n o w n for t h e s e k i n d s of m o d e l s . U n d e r t h e s e new c o n d i t i o n s of m i x e d c o m p e t i t i o n , new solutions ( e q u i l i b r i u m s ) are founded.. The o b j e c t i v e of this r e s e a r c h is to a n a l y z e the c o n d i t i o n s under which each firm, e i t h e r d o m e s t i c or p r i v a t e , should be the leader in this competition t y p e . This o b j e c t i v e was d e r í v a t e b e c a u s e of in p r e v i o u s analysis, u n d e r some c o n d i t i o n s ; the d o m e s t i c p u b l i c firm always is the leader.. C o m p a r i s o n s of p r o f i t s , p r i c e s , v o l u m e s of p r o d u c t i o n and d o m e s t i c social w e l f a r e w e r e d o n e in different c a s e s . Based on t h e s e c o m p a r i s o n s following c o n c l u s i o n s w e r e o b t a i n e d : •. In the S t a c k e l b e r g model under the a s s u m p t i o n of m i x e d o l i g o p o l y , e i t h e r d o m e s t i c or p r i v a t e firm can be the l e a d e r of the g a m e , the role of each firm d e p e n d s on its r e a c t i o n function. •. The m o n o p o l i s t i c model is not the model t h a t m a x i m i z e s the d o m e s t i c s o c i a l s u r p l u s . In order to m a x i m i z e the w e l f a r e to society it is n e c e s s a r y to have a model in which two firms c o m p e t e .. - 19-.

(29) Research 1. CHAPTER 3.. This p a p e r was p u b l i s h e d in the J o u r n a l of O p t i m i z a t i o n . This r e s e a r c h is presented as f o l l o w s :. -20-.

(30) CHAPTER 3.. Cournot and Stackelberg Equilibrium Cournot and Stackelberg in M i x e d Duopoly. Equilibrium Models. 1. Vyacheslav Kalashnikov ,. 2. Tecnológico de Monterrey (ITESM), Campus Monterrey, México; kalash@itesm.mx Alvaro Eduardo Cordero Tecnológico de Monterrey (ITESM), Campus Monterrey, México; a00791629(a),itesm.mx Vitaly Kalashnikov Department of Civil Engineering and Architecture (FICA), Durango State University (UJED), Gómez Palacio, México; kalashnikov_de(5),vahoo,de. Abstract We investígate Cournot and Stackelberg. mixed duopoly models where a state-owned. public firm. maximizing. domestic social surplus, and a foreign firm searching to maximize its own profit, compete. First we establish the existence and uniqueness results for the Cournot scheme, and propose the agents' classification as strong or weak, according to the agent's optimal reaction function properties desirable role (either leader or follower) domestic social surplus and the production. at the Cournot equilibrium.. Then we examine a. of both firms in the Stackelberg schemes and compare the profits and volumes at each type of Stackelberg equilibrium. Finally, we provide. examples of each type of model, with different inverse. demandfunctions.. Keywords: mixed duopoly, Cournot equilibrium, Stackelberg equilibrium AMSSubject Classification codes: 91B52, 91B54, 91B60, 91B68. 1. Corresponding author On leave from the Central Economics and Mathematics Institute (CEMI) of the Russian Academy of Sciences, Nakhimovsky prospekt 47, Moscow 117418, Russian Federation 2. -21 -.

(31) CHAPTER 3.. Introduction Introduction Examinations. of. mixed. oligopolies,. in. which. social. surplus-. maximizing p u b l i c firms c o m p e t e against p r o f i t - m a x i m i z i n g p r i v a t e firms, have b e c o m e i n c r e a s i n g l y p o p u l a r in recent y e a r s . For p i o n e e r i n g works on mixed o l i g o p o l i e s , see M e r r i l l and S c h n e i d e r ( 1 9 6 6 ) , H a r r i s and Wiens (1980), and Bós ( 1 9 8 6 , 1991). E x c e l l e n t s u r v e y s can be found in Vickers and Y a r r o w ( 1 9 8 8 ) , De Fraja and D e l b o n o ( 1 9 9 0 ) , Nett ( 1 9 9 3 ) .. The importance. interest in. in. mixed. economies. of. oligopolies Europe. is. high. (Germany,. because. England. and. of. their. others),. Canadá and J a p a n (see M a t s u s h i m a and M a t s u m u r a , 2 0 0 3 ) for a n a l y s i s of "herd b e h a v i o u r " by p r i v a t e firms in many b r a n c h e s of the economy in Japan). T h e r e are e x a m p l e s of mixed o l i g o p o l i e s in U n i t e d States such as the p a c k a g i n g and o v e r n i g h t - d e l i v e r y. i n d u s t r i e s . Mixed o l i g o p o l i e s. are. also c o m m o n in the East E u r o p e a n and former Soviet U n i o n t r a n s i t i o n a l economies, in w h i c h c o m p e t i t i o n among p u b l i c and p r i v a t e firms existed or still e x i s t s in many i n d u s t r i e s such as b a n k i n g , h o u s e loan, a i r l i n e , telecommuñication,. natural. gas, e l e c t r i c. power,. hospital,. health. care,. in different. ways.. Many. railways and o t h e r s .. These s i t u a t i o n s have been i n v e s t i g a t e d. works a n a l y z e d C o u r n o t and S t a c k e l b e r g m o d e l s with the role of each firm a s s i g n e d e x o g e n o u s l y . H o w e v e r , it is r e a s o n a b l e to a s s u m e that each firm d e c i d e s what a c t i o n s to t a k e , and when to take t h e m .. DeFraja and D e l b o n o (1989) are p i o n e e r s in t h e s e. investigations.. They showed t h a t in s i m u l t a n e o u s - m o v e g a m e s , p r i v a t i z a t i o n of the p u b l i c -22-.

(32) Introduction. CHAPTER 3.. firm may i m p r o v e w e l f a r e . In 1989 M a t s u m u r a showed that under certain conditions, the p a r t i a l p r i v a t i z a t i o n of the p u b l i c firm i m p r o v e s w e l f a r e . Pal (1998) found that the p u b l i c firm can be f o l l o w e r , but he assumed that private firms are d o m e s t i c .. In the p a p e r by M a t s u m u r a ( 2 0 0 3 ) , the a u t h o r i n v e s t i g a t e s. mixed. duopoly and a n a l y z e s a d e s i r a b l e role (either l e a d e r or f o l l o w e r ) of the public firm, w h e n t h e i n v e r s e demand function is c o n c a v e . Under these conditions, M a t s u m u r a founds that the role of the p u b l i c firm should be that of the. leader. (however. the. author. make. assumptions. about. the. concavity of d o m e s t i c social surplus and profits function with r e s p e c t to volume of d o m e s t i c p u b l i c firm and p r i v a t e foreign firm Matsumura. also. establishes. the. domestic. social. respectively).. surplus. in. a. mixed. duopoly is g r e a t e r t h a n in a m o n o p o l i s t i c m a r k e t .. In t h i s p a p e r , foreign. private. we also e x a m i n e the d e s i r a b l e. agent. and. the. domestic. public. roles. firm.. In. of both contrast. the to. Matsumura ( 2 0 0 3 ) , h e r e we do not demand the i n v e r s e d e m a n d function to be c o n c a v e . H e n e e the m o d e l d e s c r i b e s more g e n e r a l s i t u a t i o n s , and the role of firms. in the o b s e r v a b l e delay game could be e i t h e r leader. or. follower.. An. extended. Proceedings. of. of. abstract the. Computing, I n f o r m a t i o n. 2nd. of. this. work. International. will. be. published. Conference. on. -23 -. the. Innovative. and Control ( I C I C I C ' 2 0 0 7 ) , K u m a m o t o ,. September 05 - 0 7 , 2 0 0 7 (cf. V.V. K a l a s h n i k o v et a l . , 2 0 0 7 ) .. in. Japan,.

(33) l.The Model. CHAPTER 3.. The p a p e r is o r g a n i z e d as follows. In S e c t i o n 1 we d e s c r i b e the model and e s t a b l i s h e x i s t e n c e and u n i q u e n e s s t h e o r e m s for the Cournot e q u i l i b r i u m . After a n a l y z i n g the a g e n t s ' o p t i m a l r e s p o n s e functions at the Cournot e q u i l i b r i u m we define the c o n c e p t of a s t r o n g firm and a weak firm. S e c t i o n 2 d e a l s w i t h the game where the d o m e s t i c p u b l i c firm is the leader and the p r i v a t e foreign firm is the f o l l o w e r . S e c t i o n 3 c o n s i d e r s the game w h e r e the d o m e s t i c p u b l i c firm is the follower and the p r i v a t e foreign. firm. is the. leader.. The. domestic. public. firm. may. have. two. different t y p e s of o p t i m a l r e a c t i o n at the C o u r n o t e q u i l i b r i u m , and as a c o n s e q u e n c e t h i s firm could be weak or s t r o n g . In S e c t i o n 4, we make comparisons private. between. firm's. the. profits. domestic. at v a r i o u s. social. surplus. Stackelberg. and. quantities Cournot. and. the. equilibrium. states, and we e x a m i n e the o b s e r v a b l e delay game when p r i v a t e firm is strong and when it is weak.. F i n a l l y , in S e c t i o n 5, we p r e s e n t e x a m p l e s of. these games w i t h t h e i r s o l u t i o n s . In E x a m p l e 5.1 we c o n s i d e r a convex inverse d e m a n d f u n c t i o n , and in E x a m p l e 5.2 we use a c o n c a v e one.. 1. The Model Consider. two. firms. producing. a homogeneous. product.. represent the t o t a l o u t p u t , and p{(j) denote an i n v e r s e demand. Let. G. function,. i.e. the p r i c e of a unit of the p r o d u c t . The goods p r o d u c e d by the two firms are sold at the d o m e s t i c m a r k e t . L e t o , z = l,2, d e n o t e the o u t p u t of. -24-.

(34) CHAPTER 3.. 1. The Model. Firm 1 is a foreign p r í v a t e firm, which m a x i m i z e s its own p r o f i t s , and firm 2 is a d o m e s t i c p u b l i c firm that m a x i m i z e s d o m e s t i c. social. surplus. D o m e s t i c s o c i a l s u r p l u s S is the sum of c o n s u m e r surplus and profits of firm 2, and is given by:. The profit of firm 1 is given by:. We also w a n t to solve an o b s e r v a b l e delay g a m e . This game c o n s i s t s of three. stages.. At. the. first. stage,. each. firm. / , z" = l,2. independently. chooses e,e(2,3), i = 1,2, w h e r e e, i n d i c a t e s when to p r o d u c e the output. q. r. Namely, e¡=2 i m p l i e s that firm / p r o d u c e s at the second s t a g e , and e,=3 means that firm ' p r o d u c e s at the third s t a g e . In the end of the first s t a g e , each firm select its e,. e (2,3). At the second s t a g e , each firm. i. choosing. e¡=2 a s s i g n s its o u t p u t ^ , . At the third s t a g e , each firm / c h o o s i n g. e¡=3. selects its o u t p u t ^ , . In the end of the g a m e , the m a r k e t o p e n s and each firm / sells its o u t p u t .. We a c c e p t the f o l l o w i n g a s s u m p t i o n s c o n c e r n i n g the i n v e r s e demand (price) function and cost f u n c t i o n s :. -25 -.

(35) 1. The Model. CHAPTER 3.. whereas for i = 2, t h e r e e x i s t s a n i / > 0 , such t h a t : 2. A4. P r i n c i p i e of P o t e n t i a l P a r t i c i p a t i o n . For i = \ t h e r e e x i s t <7,°>0 such. that. G<G. 0. implies. that. inequality h o l d s :. -26-. forg,^,. 0. the. G >0. following. 0. and. (strict).

(36) CHAPTER 3.. 1. The Model Remark 1.1.. E x a m p l e s of functions that satisfy A l a r e : p{G) = AG. 7. with. i4>0and 0 < / < l , a m o n g o t h e r s . ». Remark 1.2. The P r i n c i p i e of P o t e n t i a l P a r t i c i p a t i o n given in A4 e x e l u d e s the p o s s i b i l i t y of t h e t r i v i a l (zero) e q u i l i b r i u m . •. Henee, t h e r e l a t i o n s h i p. of a s s u m p t i o n A3 y i e l d s that t h e r e exists an H >0. such that. 3. 1.4c)i. To make optimality. it p o s s i b l e. conditions,. to define. we first. have. equilibrium to verify. with that. only. first-order. t h e profit. and/or. domestic s o c i a l s u r p l u s f u n c t i o n s be c o n c a v e over t h e i r d o m a i n s . We do that by e s t a b l i s h i n g t h e f o l l o w i n g a u x i l i a r y r e s u l t s .. -27-.

(37) CHAPTER 3.. 1. The Model. Lemma 1.1. U n d e r a s s u m p t i o n s A l and A2, the firm l ' s profits U.(G,q ) is c o n c a v e w i t h r e s p e c t x. function. toq . x. Proof. We c a l c ú l a t e the first and second c o m p l e t e d e r i v a t i v e s of the firm 1 nrnfit. fnnrtion. with. resnect. toa,:. H e r e , we t a k e into a c c o u n t the b a l a n c e r e l a t i o n s h i p ( 1 . 1 ) . Due to .. rs.. •. ,. 1. .. _. _ 1_. (1.5). which c o m p l e t e s the proof. • -28-.

(38) CHAPTER 3.. l.The Model. Lemma 1.2. U n d e r a s s u m p t i o n s A l and A 2 , the d o m e s t i c social. surplus. function S(G,q ) is c o n c a v e with respect to q . 2. 2. Proof. We c o m p u t e the first and second c o m p l e t e d e r i v a t i v e s of the firm 2 domestic s o c i a l s u r p l u s function S(G,q ) with r e s p e c t to q : 2. here, we again. take. into. account. the b a l a n c e. 2. equality 2. assumption A2 (the c o n v e x i t y of the cost f u n c t i o n ' ). ( 1 . 1 ) . Due to. we need to show. only that. due to r e l a t i o n s h i p ( 1 . 4 a ) in A l and thus c o m p l e t e s the p r o o f . • -29-.

(39) CHAPTER 3.. 1. The Model Remark. 1.4.. It is easy to see that if one a s s u m e s the cost. f¡,i = 1,2, to be s t r i c t l y. c o n v e x , then both Lemnia. 1.1. functions. and Lemma. 1.2. guarantee the s t r i c t c o n c a v i t y of the r e s p e c t i v e o b j e c t i v e functions n and S.i. Now we are in a p o s i t i o n to define different kinds of e q u i l i b r i u m States and c o m p a r e the e q u i l i b r i u m v o l u m e s for v a r i o u s s c e n a r i o s . First we. consider. Z = (G,q q )e v. 2. the. classical. Cournot. equilibrium,. i.e.,. a. vector. Rl, such t h a t :. Problem. (1.7). -. (1.9). is. a standard. complementarity. problem.. Therefore, to e s t a b l i s h the e x i s t e n c e of s o l u t i o n to the l a t t e r , we can use powerful t h e o r e t i c a l t o o l s d e v e l o p e d e.g. in the book I s a c , B u l a v s k y and Kalashnikov ( 2 0 0 2 ) .. Theorem 1.3 ( E x i s t e n c e T h e o r e m ) . Let a s s u m p t i o n s A l - A4 be valid. Then the C o u r n o t e q u i l i b r i u m p r o b l e m (1.7) - (1.9) has a ( n o n t r i v i a l ) «ni íiti r»n. -30-.

(40) CHAPTER 3.. 1. The Model Proof. The e x i s t e n c e of s o l u t i o n s to p r o b l e m (1.7) - ( 1 . 9 ) the. next. result.. This. Cournot. equilibrium. problem. is. follows from a. standard. complementarity p r o b l e m and can be r e w r i t t e n in the f o l l o w i n g form: Find 2. a vector x e i ? s u c h t h a t :. here. As. it. follows. from. assumption. A l , the m a p p i n g. 2. 2. F: R -> R +. is. continuous over the n o n n e g a t i v e ortant R^. We may use the next t h e o r e m to establish the e x i s t e n c e of s o l u t i o n s :. Theorem. 6.8. [Isac,. Bulavsky. and. Kalashnikov. (2002)].. Consider. continuous m a p p i n g F: R" —» R"and a n o n e m p t y b o u n d e d subset that for every xeR"and. a. Cci?"such. I Í C , the i n e q u a l i t y. is valid for at least one of the Índices i = \,...,n. solution, and all the s o l u t i o n s b e l o n g to C .. -31 -. Then p r o b l e m (1.10) has a.

(41) CHAPTER 3.. 1. The Model Coming. b a c k to the proof of T h e o r e m. 1.3,. select a. (nonempty). subset (1.13). with H. 3. defined in R e m a r k 1.3 (see i n e q u a l i t y ( 1 . 4 c ) ) . N o w we prove that. inequality ( 1 . 1 2 ) h o l d s for at least one index at any p o i n t x o u t s i d e the subset C . C o n s i d e r an a r b i t r a r y x<£C, that is, at least one of the following conditions is v a l i d : (i) q >H . x. X. In t h i s c a s e , a c c o r d i n g to a s s u m p t i o n s A l - A 4 , we get. the i n e q u a l i t y (1.14). Now recall that. which, t o g e t h e r with ( 1 . 1 4 ) , i m m e d i a t e l y i m p l i e s that F (x)>0, x. henee (1.15). I n e q u a l i t y ( 1 . 1 5 ) i m p l i e s i n e q u a l i t y (1.12) in case ( i ) .. -32-.

(42) CHAPTER 3.. l.The Model. (1.17). N o w we t u r n to e x a m i n i n g u n i q u e n e s s p r o p e r t i e s. of the. Cournot. equilibrium defined by (1.7) - ( 1 . 9 ) . First we are e n g a g e d in d e t e r m i n i n g the u n i q u e n e s s of a n o n - m o n o p o l i s t i c e q u i l i b r i u m v o l u m e . To do that, we need to. involve. an. extra. assumption. function p . -33 -. concerning. the. inverse. demand.

(43) CHAPTER 3.. 1. The Model. which, t o g e t h e r w i t h a s s u m p t i o n A l , i m p l i e s the c o n v e x i t y of the inverse demand f u n c t i o n p . •. Theorem 1.4 ( T h e o r e m of U n i q u e n e s s ) . Under a s s u m p t i o n s A l , A2 and A5, the c l e a r e d m a r k e t q u a n t i t y G is the same at each n o n - m o n o p o l i s t i c equilibrium.. Proof.. Suppose. on. the. contrary. that. monopolistic equilibria:. -34-. there. exist. two. different. non-.

(44) CHAPTER 3.. 1. The Model. with G <G . In o t h e r w o r d s , for each e q u i l i b r i u m , the b a l a n c e e q u a l i t i e s X. 2. are v a l i d :. and the r e m a i n i n g r e l a t i o n s h i p s (1.8) - (1.9) also hold.. Observe. that D e f i n i t i o n. clearly i m p l i e s that. 1.1. of the n o n - m o n o p o l i s t i c. all the c o m p o n e n t s. of. Z, (. equilibrium. i = l,2, are p o s i t i v e ; in. particular,. Now. we. establish. that. for. each. Í' = 1,2, the. following. inequality. holds: (1.18). I n d e e d , c o n s i d e r first z' = l and s u p p o s e , on the c o n t r a r y , that. Then o b v i o u s a l g e b r a i c m a n i p u l a t i o n s yield the i n e q u a l i t i e s : -35-.

(45) 1. The Model. CHAPTER 3.. N o w m a k i n g use of ( 1 . 1 9 a ) and (1.19b) we d e d u c e the. following. chain of r e l a t i o n s h i p s :. Since Z implies z < 1 . 2. 2. is a n o n - m o n o p o l i s t i c e q u i l i b r i u m we have q\'<G , 2. which.

(46) CHAPTER 3.. 1. The Model. which a l l o w s one to omit the last terms in all p a r t s of the p r e v i o u s cháin to obtain the key i n e q u a l i t y : (1.19c). N o w s u b t r a c t s from ( 1 . 1 9 c ) the i n e q u a l i t y. b r i n g s us to the. which is valid inequality. (1.19d). which c o n t r a d i c t s to the a s s u m e d (cf. A l ) c o n c a v i t y of the function P(G)G , as. G < G. X. 2. -37-.

(47) CHAPTER 3.. 1. The Model. Thus we have p r o v e d r e l a t i o n s h i p (1.18) to be valid for / = 1. The proof for. i-2. differs. from the j u s t d e s c r i b e d one only in the way to. obtain an i n e q u a l i t y s i m i l a r to ( 1 . 1 9 b ) . T o w a r d t h i s end, a s s u m e , on the contrary, t h a t. and use the j u s t o b t a i n e d i n e q u a l i t y ( 1 . 1 8 ) :. -38-.

(48) CHAPTER 3.. 1. The Model. w h i c h y i e l d an i m p o s s i b l e i n e q u a l i t y 1 < 1 . The latter m e a n s that the assumption of e x i s t e n c e of two n o n - m o n o p o l i s t i c e q u i l i b r i u m states with distinct c l e a r e d m a r k e t v o l u m e s was w r o n g , which c o m p l e t e s the p r o o f . a. Remark 1.6. When the e q u i l i b r i u m is n o n - m o n o p o l i s t i c , then a c c o r d i n g to Theorem. 1.4,. However,. in. the some. cleared. market. cases. both. volume. G is. monopolistic. determined and. uniquely.. non-monopolistic. equilibrium s t a t e s w i t h d i s t i n c t v o l u m e s can occur. For i n s t a n c e , that may happen if the cost f u n c t i o n s. f¡ and the p r o d u c t. linear f u n c t i o n s . ». -39-. p(G)G. are. piece-wise.

(49) CHAPTER 3.. 1. The Model. Corollary 1.5.. U n d e r a s s u m p t i o n s A l - A 5 , the e q u i l i b r i u m state Z exists. uniquely.. Proof.. F i r s t , u n d e r a s s u m p t i o n s Al - A 4 , t h e r e are e q u i l i b r i u m s t a t e s .. Moreover, as a s s u m p t i o n A4 i m p l i e s , no agent can have zero p r o d u c t i o n volume, h e n e e the e q u i l i b r i u m states are all n o n - m o n o p o l i s t i c . Therefore, Theorem. 1.4 i m p l i e s that the e q u i l i b r i u m c l e a r e d m a r k e t volume. G is. uniaue. N o w s u n n o s e t h a t for the same v o l u m e G. t h e r e exist two distinct. and the r e s p e c t i v e p a i r of c o m p l e m e n t a r i t y r e l a t i o n s h i p s (1.8) and ( 1 . 9 ) . Without affecting g e n e r a l i t y assume that (1.24). H e n e e , a s s u m p t i o n A2 i m p l i e s that (1.25). -40-.

(50) CHAPTER 3.. 2. Domestic Firm Leader. which. implies. an. impossible. in. inequality. 0<0.. This. contradiction. completes the p r o o f of the u n i q u e n e s s of the e q u i l i b r i u m c o n t r i b u t i o n by the foreign agent q . F i n a l l y , the b a l a n c e e q u a l i t y ( 1 . 2 3 ) g u a r a n t e e s the x. uniqueness of the d o m e s t i c f i r m ' s e q u i l i b r i u m p r o d u c t i o n volume q. 2. as. well.a. 2. D o m e s t i c Firm L e a d e r F i r s t , in t h i s S e c t i o n , we examine the game w h e r e firm 2 ( p u b l i c one) is the l e a d e r . Firm 2 c h o o s e s its output v o l u m e (prívate one) c h o o s e s ^ after having o b s e r v e d net profit. q, 2. q, 2. and firm 1. so as to m a x i m i z e its.

(51) CHAPTER 3.. 2. Domestic Firm Leader. On the o t h e r h a n d , when. G <G. X. then the d e r i v a t i v e. q\ (G) can be. found m a k i n g use of the s e c o n d - o r d e r e q u a t i o n o b t a i n e d by d i f f e r e n t i a t i n g the left-hand side of e q u a t i o n (2.3) with r e s p e c t to G:. whence. -42-.

(52) CHAPTER 3.. 2. Domestic Firm Leader. As the d e n o m i n a t o r in (2.6) is always p o s i t i v e (by A l and A 2 ) , then. (2.7). N o w we i n t r o d u c e the following c l a s s i f i c a t i o n of a g e n t s , a c c o r d i n g to their o p t i m a l r e a c t i o n f u n c t i o n ' s p r o p e r t i e s at the C o u r n o t e q u i l i b r i u m . Such a c l a s s i f i c a t i o n was c o n s i d e r e d first in K a l a s h n i k o v ( 1 9 9 5 ) and can be also found in the b o o k by. I s a c , B u l a v s k y and K a l a s h n i k o v ( 2 0 0 2 ) . To. do that, we d e n o t e the ( u n i q u e ) s o l u t i o n of the C o u r n o t game (1.7) - (1.9) c. total v o l u m e by G .. Definition. 2 . 1 . A firm. is called strong if the d e r i v a t e of its. reaction. function is n o n n e g a t i v e at the C o u r n o t e q u i l i b r i u m c l e a r e d m a r k e t v o l u m e c. G,. that. i s , if. c. <7',(G )>0. R e s p e c t i v e l y ,. a firm. is c a l l e d. weak. if. its. reaction f u n c t i o n ' s d e r i v a t i v e is n e g a t i v e at the C o u r n o t e q u i l i b r i u m , that. Remark 2 . 2 . D e f i n i t i o n 2.1 is based upon the p o t e n t i a l r e a c t i o n of firm i when the l a t t e r a s s u m e s that the rival firm j i n c r e a s e s its o u t p u t . If the reaction of firm But if firm. / is not to d e c r e a s e its o u t p u t , firm / is a strong firm.. i d e c r e a s e s its o u t p u t , firm. / is a weak firm. For e x a m p l e ,. (2.7) i m p l i e s t h a t , u n d e r a s s u m p t i o n s A l - A4, if the i n v e r s e function p is c o n c a v e t h e n the p r i v a t e firm is a l w a y s w e a k . -43 -. demand •.

(53) CHAPTER 3.. 2. Domestic Firm Leader. N o w we r e a l i z e a c o m p a r a t i v e a n a l y s i s for v a r i o u s s t r a t e g i e s of the firms. We are going to c o m p a r e the v o l u m e of the C o u r n o t e q u i l i b r i u m with t h o s e of S t a c k e l b e r g e q u i l i b r i u m states when the d o m e s t i c F,L. firm is the l e a d e r G ,. c. G. public. L,F. and when the p r i v a t e firm is the l e a d e r G .. Also. we c o m p a r e the d o m e s t i c s o c i a l s u r p l u s of the d o m e s t i c firm and the profits of the p r i v a t e firm at t h e s e v a r i o u s e q u i l i b r i u m s t a t e s .. Lemma 2 . 1 . U n d e r a s s u m p t i o n s A l - A 4 ,. (2.8). Proof. B e c a u s e of c o n c a v i t y of the function. p(G)G. a s s u m e d in A l , one. has:. (2.9) which can be r e w r i t t e n it as follows (2.10). -44-.

(54) 2. Domestic Firm Leader. CHAPTER 3.. which, t o g e t h e r with r e l a t i o n s h i p s ( 2 . 8 ) , c o m p l e t e s the proof. •. -45 -.

(55) CHAPTER 3.. 2. Domestic Firm Leader. (2.14). H e n e e , the f u n c t i o n. G = G(Q) is d i f f e r e n t i a b l e. at every p o i n t 0 * G. 1;. with. or. (2.15). L e m m a 2.1 g u a r a n t e e s that (2.16). As. for. the. point. Q =G, X. the. function. G(Q) has. only. one-side. derivatives t h e r e :. (2.17). (2.18). -46-.

(56) CHAPTER 3.. 2. Domestic Firm Leader. In. a. similar. manner. we. obtain. the. formulas. for. the. one-side. derivatives of the d o m e s t i c social s u r p l u s at the p o i n t Q = G when it is X. finite:. As the o n e - s i d e d e r i v a t i v e s are e q u a l , we can c o n c l u d e that. the. consumer s u r p l u s is d i f f e r e n t i a b l e at the point Q = G, as w e l l , with. Now we are in a p o s i t i o n to recall a m a t h e m a t i c a l l y r i g o r o u s d e f i n i t i o n of the S t a c k e l b e r g e q u i l i b r i u m state with the d o m e s t i c l e a d e r and. foreign. follower. Definition 2 . 2 . A S t a c k e l b e r g e q u i l i b r i u m (with the d o m e s t i c firm as a leader. and. the. foreign. firm. as. -47-. a. follower). is. the. vector.

(57) CHAPTER 3.. 2. Domestic Firm Leader. (2.22) (2.23) (2.24). Next we e s t a b l i s h r e l a t i o n s h i p s to c o m p a r e the p r o d u c t i o n v o l u m e s of the firms at the S t a c k e l b e r g e q u i l i b r i u m state (2.22) - (2.24) to those at the C o u r n o t e q u i l i b r i u m defined by the c o m p l e m e n t a r i t y p r o b l e m (1.7) L. - (1.9). B e s i d e s , it is i n t e r e s t i n g to compare the v a l ú e s Q° and Q domestic. firm's. equilibrium,. that. optimum is,. output volume. when. the. domestic. p. Q &\ the p e r f e c t producer. competition. ignores. the. variation and s o l v e s the f o l l o w i n g c o m p l e m e n t a r i t y p r o b l e m : Find a such that:. Proof. By a s s u m p t i o n s A l - A4 the following i n e q u a l i t y h o l d s : í. p{G)-H. x. u. 1+. \. p'{H )-f' (q )<0 2. 2. 2. -48-. for. G>q >H . 2. 2. to the. price Q>0.

(58) CHAPTER 3.. 2. Domestic Firm Leader. 49-.

(59) 2. Domestic Firm Leader. CHAPTER 3.. due to a s s u m p t i o n s A 1 - A 2 and the p r o p e r t y G'(Q)>0. following e s t í m a t e is v a l i d :. -50-. But in this case, the.