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Consistent Conjectures in A Mixed Oligopoly Model Dr. Vyacheslav Kalashnikov ITESM, Campus Monterrey Mexico Consistent Conjectures in A Mixed Oligopoly Model Dr. Vyacheslav Kalashnikov ITESM, Campus Monterrey Mexico Seminar at the Department of Optimization UTIA, Prague May 04, 2010

Agenda • Introduction • Model Specification • Exterior Equilibrium • Consistency Criterion • Interior Agenda • Introduction • Model Specification • Exterior Equilibrium • Consistency Criterion • Interior Equilibrium • Existence and Uniqueness • Numerical Experiments • Conclusions

Introduction In recent years, investigation of behavioral patterns of agents in mixed markets, in Introduction In recent years, investigation of behavioral patterns of agents in mixed markets, in which state-owned (public, domestic, etc. ) welfare-maximizing companies compete with profit-maximizing (private, foreign, etc. ) firms has become more and more widespread. Among the pioneering works one can see: • Merrill and Schneider (1966) • Ruffin (1971) • Harris and Wiens (1980) • Bös (1986, 1991)

Introduction (cont-ed) Excellent surveys can be found in: • Vickers and Yarrow (1988) • Introduction (cont-ed) Excellent surveys can be found in: • Vickers and Yarrow (1988) • De Fraja and Delbono (1990) • Nett (1993) • Pal (1998) The interest to mixed oligopolies is high because of their importance to the economics of Europe, Canada and Japan (see Matsushima and Matsumura, 2003, for an analysis of “herd behavior” by private firms in many branches of the economy in Japan).

Introduction (cont-ed) There also examples of mixed oligopolies in the United States such as Introduction (cont-ed) There also examples of mixed oligopolies in the United States such as the packaging and overnight-delivery industries. Mixed oligopolies are also common in the East European and former USSR transitional economies, in which competition among public and private firms existed or still exists in many industries such as banking, house loan, life insurance, airline, telecommunication, natural gas, electric power, automobile, steel, education, hospital, health care, broadcasting, railways, and overnight-delivery.

Introduction (cont-ed) In the majority of the above-mentioned papers, the mixed oligopoly is studied Introduction (cont-ed) In the majority of the above-mentioned papers, the mixed oligopoly is studied in the framework of classical Cournot, Hotelling or Stackelberg models: • Matsushima and Matsumura (2003) • Cornes and Sepahvand (2003) • Figuères et al. (2004)

Introduction (cont-ed) Conjectural variations equilibrium (CVE) was introduced by Bowley (1924) and Frisch (1933) Introduction (cont-ed) Conjectural variations equilibrium (CVE) was introduced by Bowley (1924) and Frisch (1933) as another possible solution concept in static games. According to this concepts, agents behave as follows: each agent chooses her/his most favorable action taking into account that every rival’s strategy is a conjectured function of her/his own strategy.

Introduction (cont-ed) In the works: • Bulavsky and Kalashnikov (1994, 1995) • Isac, Bulavsky Introduction (cont-ed) In the works: • Bulavsky and Kalashnikov (1994, 1995) • Isac, Bulavsky and Kalashnikov (2002) a new gamma of conjectural variations equilibria (CVE) was introduced and applied to the classical oligopoly. We assumed that each producer used the conjectural variations of the total market volume as below: Here G is the current total quantity of the product cleared in the market, and are, respectively,

Introduction (cont-ed) the present and the expected supplies by agent i, whereas is the Introduction (cont-ed) the present and the expected supplies by agent i, whereas is the total cleared market volume conjectured by agent i as a response to the variation of her own supply from to. The conjecture function was referred to as the i-th agent´s influence coefficient. • Cournot model: • Perfect competition: • CVE: Under general enough assumptions, new existence and uniqueness results for the CVE were obtained.

Introduction (cont-ed) This approach was further developed in Kalashnikov et al. (2009, 2010) with Introduction (cont-ed) This approach was further developed in Kalashnikov et al. (2009, 2010) with application to the mixed oligopoly model. Here again, all agents (both public and private) make their decisions based upon the model’s data (inverse demand cost functions) and their influence coefficients (conjectures) Under general enough assumptions, new existence and uniqueness theorems for the CVE were deduced.

Introduction (cont-ed) As is mentioned in: • Figuières et al. (2004) • Giocoli (2005) Introduction (cont-ed) As is mentioned in: • Figuières et al. (2004) • Giocoli (2005) • Lindh (1992) the concept of conjectural variations has been the subject of numerous theoretical controversies. The literature on conjectural variations has focused mainly on two-players games. Usually, the variational conjecture rj describes player j’s reaction, as anticipated by player i, to an infinitesimal variation of player i’s strategy. This leads to the notion of a conjectured reaction function of the opponent.

Introduction (cont-ed) Given these conjectured reactions on part of the rivals, each agent optimizes Introduction (cont-ed) Given these conjectured reactions on part of the rivals, each agent optimizes her/his perceived payoff. This yields the concept of conjectural best response function. An equilibrium is obtained when no player has an interest in deviating from her/his strategy, i. e. her/his conjectural best response to the strategies of the other player. The consistency (“rationality”) of the equilibrium is defined as the coincidence between the conjectural best response and the conjectured reaction function of the same.

Introduction (cont-ed) A conceptual difficulty arises in the case of many agents: when n Introduction (cont-ed) A conceptual difficulty arises in the case of many agents: when n agents are present, there are n best response functions and n(n – 1) conjectures. Therefore, if n > 2, an equilibrium is consistent only if all players have the same conjectures about player i. One finds this approach in: • Başar and Olsder (1982) • Fershtman and Kamien (1987) • Laitner (1980) • Bresnahan (1981) • Novshek (1985)

Introduction (cont-ed) A bit different scheme has been applied in: • Perry (1982) for Introduction (cont-ed) A bit different scheme has been applied in: • Perry (1982) for oligopoly • Corner and Sandler (1984) for public goods • Sugden (1985) for public goods, who considered a class of games where for each agent, the contributions of all other players to her/his payoff are aggregated (it is as if each agent plays against a unique (virtual) player representing the remaining agents. )

Introduction (cont-ed) A new approach was proposed by Bulavsky (1997): he assumed that each Introduction (cont-ed) A new approach was proposed by Bulavsky (1997): he assumed that each player makes conjectures not about the (optimal) response functions of the other players but only about the variations of the market price depending upon her/his infinitesimal output variations. Knowing the rivals’ conjectures (influence coefficients), each agent can realize certain verification procedures and check out if her/his influence coefficient is consistent with the others. If all the influence coefficients are mutually consistent, we call the corresponding equilibrium interior (or consistent) one.

Introduction (cont-ed) Exactly the same verification formulas were obtained independently by Liu et al. Introduction (cont-ed) Exactly the same verification formulas were obtained independently by Liu et al. (2007) in their paper establishing the existence and uniqueness of a consistent CVE in an electricity market. However, they applied a much more complicated optimal control technique, searching for steady states as a final result (a similar technique was used by Driskill and Mc. Cafferty, 1989. ) Moreover, Liu et al. used only linear inverse demand function and quadratic cost functions in their model, whereas Bulavsky (1997) allows nonlinear (even nondifferentiable) demand functions and not necessarily quadratic (however convex) cost functions.

Introduction (cont-ed) In this paper, we extend the results of Bulavsky (1997) to a Introduction (cont-ed) In this paper, we extend the results of Bulavsky (1997) to a mixed oligopoly model. After having introduced the mathematical model of a homogeneous single product market, we define a concept of exterior CVE and prove its existence and uniqueness for any collection of influence coefficients. Then we define a consistency verification procedure and the related interior (consistent) CVE. The consistent CVE existence theorem is also established, together with the characteristics of the consistent influence coefficients’ behavior as functions of a parameter representing G´(p). Results of numerical experiments with an electricity market model finish the talk.

Consistent Conjectural Equilibrium Consistent Conjectural Equilibrium

Assumptions Assumptions

Assumptions (cont-ed) Assumptions (cont-ed)

Model Specification Model Specification

Model Specification (cont-ed) Model Specification (cont-ed)

Model Specification (cont-ed) Model Specification (cont-ed)

Exterior Equilibrium Exterior Equilibrium

Assumptions (cont-ed) Assumptions (cont-ed)

Assumptions (cont-ed) Assumptions (cont-ed)

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

G Remark 1 (cont-ed) concave point convex point O L p G Remark 1 (cont-ed) concave point convex point O L p

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Remark 1 (cont-ed) G concave point G(p) convex point g(p) O p Remark 1 (cont-ed) G concave point G(p) convex point g(p) O p

Verification Procedure Verification Procedure

Verification Procedure (cont-ed) Verification Procedure (cont-ed)

Verification Procedure (cont-ed) Verification Procedure (cont-ed)

Consistency Criterion Consistency Criterion

Consistency Criterion (cont-ed) Consistency Criterion (cont-ed)

Interior Equilibrium Interior Equilibrium

Existence of Interior Equilibrium Existence of Interior Equilibrium

Quadratic Cost Functions Quadratic Cost Functions

Existence of Interior Equilibrium Existence of Interior Equilibrium

Behaviour of Consistent Conjectures Behaviour of Consistent Conjectures

Consistent Conjectures As Equilibrium Strategies It is also interesting to consider the following game: Consistent Conjectures As Equilibrium Strategies It is also interesting to consider the following game: all producers treat their conjectures (influence coefficients) as their strategies, with each player’s payoff value equal to the (external) equilibrium profit , which exists uniquely, by Theorem 1, for any conjectures vector. It turns out that the solution (Nash equilibrium) in this game coincides with the consistent conjectures set obtained from (6) with.

Consistent Conjectures As Equilibrium Strategies Theorem 4. A Nash equilibrium in the above-described game Consistent Conjectures As Equilibrium Strategies Theorem 4. A Nash equilibrium in the above-described game exists, and it’s components , , satisfy equations (6) with , where is the equilibrium price of the interior equilibrium containing the consistent conjectures.

Consistent Conjectural Equilibrium in Mixed Oligopoly Consistent Conjectural Equilibrium in Mixed Oligopoly

Consistent Conjectural Equilibrium in Mixed Oligopoly Consistent Conjectural Equilibrium in Mixed Oligopoly

Assumptions Assumptions

Assumptions (cont-ed) Assumptions (cont-ed)

Model Specification Model Specification

Model Specification (cont-ed) Model Specification (cont-ed)

Model Specification (cont-ed) Model Specification (cont-ed)

Model Specification (cont-ed) Model Specification (cont-ed)

Optimality Conditions Optimality Conditions

Optimality Conditions (cont-ed) Optimality Conditions (cont-ed)

Optimality Conditions (cont-ed) Optimality Conditions (cont-ed)

Exterior Equilibrium Exterior Equilibrium

Assumptions (cont-ed) Assumptions (cont-ed)

Assumptions (cont-ed) Assumptions (cont-ed)

Assumptions (cont-ed) Assumptions (cont-ed)

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Existence of Exterior Equilibrium Existence of Exterior Equilibrium

Consistency Criterion Consistency Criterion

Consistency Criterion (cont-ed) Consistency Criterion (cont-ed)

Interior Equilibrium Interior Equilibrium

Existence of Interior Equilibrium When proving Theorem 6, the following construction was used. First, Existence of Interior Equilibrium When proving Theorem 6, the following construction was used. First, we introduced a parameter α so that for an appropriate.

Existence of Interior Equilibrium Then we established that the mapping with the components: , Existence of Interior Equilibrium Then we established that the mapping with the components: , and

Existence of Interior Equilibrium is continuous and maps the compact set into itself, which Existence of Interior Equilibrium is continuous and maps the compact set into itself, which allows one to apply Brouwer Fixed Point Theorem; here Moreover, for any fixed value of α, the mapping is strictly contracting on a corresponding compact set, which allows one to find the consistent conjectures by a simple iteration process:

Simple Iteration Mehods Simple Iteration Mehods

Behaviour of Consistent Conjectures In our forthcoming research, we are going to extend the Behaviour of Consistent Conjectures In our forthcoming research, we are going to extend the obtained results to the case of nondifferentiable demand functions. To develop a technique similar to that of Theorem 3, we denote the value of the demand function‘s derivative by and rewrite the consistency criterion (14) – (15) in the following form:

Behaviour of Consistent Conjectures (cont-ed) Behaviour of Consistent Conjectures (cont-ed)

Behaviour of Consistent Conjectures Behaviour of Consistent Conjectures

Numerical Experiments To illustrate the difference between the mixed and classical oligopoly cases related Numerical Experiments To illustrate the difference between the mixed and classical oligopoly cases related to the conjectural variations equilibrium with consistent conjectures (influence coefficients), we applied formulas (16) – (17) to the simple example of an oligopoly in the electricity market from Liu et al. (2007).

Numerical Experiments (cont-ed) The only difference in our modified example from the instance of Numerical Experiments (cont-ed) The only difference in our modified example from the instance of Liu et al. (2007) is in the following: in their case, all six agents (suppliers) are private companies producing electricity and maximizing their net profits, whereas in our case, we assume that Supplier 0 (or Supplier 5) is a public enterprise seeking to maximize domestic social surplus. All the other parameters involved in the inverse demand function p and the producers’ cost functions, are exactly the same.

Numerical Experiments (cont-ed) So, following Liu et al. (2007), we select the IEEE 6 Numerical Experiments (cont-ed) So, following Liu et al. (2007), we select the IEEE 6 -generator 30 -bus system to illustrate the above analysis. The inverse demand function in the electricity market is given in the form: The cost functions parameters of suppliers (generators) are listed in Table 1.

Numerical Experiments (cont-ed) Table 1. Cost functions’ coefficients Supplier i 0 ci bi ai Numerical Experiments (cont-ed) Table 1. Cost functions’ coefficients Supplier i 0 ci bi ai 0 2. 0 0. 02 1 0 1. 75 0. 0175 2 0 3. 0 0. 025 3 0 3. 0 0. 025 4 0 1. 0 0. 0625 5 0 3. 25 0. 00834

Numerical Experiments (cont-ed) To find the consistent influence coefficients in their classical oligopoly market Numerical Experiments (cont-ed) To find the consistent influence coefficients in their classical oligopoly market (Case 1), Liu et al. (2007) use formulas (14) for all six suppliers, whereas for our mixed oligopoly model (Case 2), we exploit formula (14) for Supplier 0 and formulas (15) for Suppliers 1 through 5. With the thus obtained influence coefficients, the (unique) equilibrium is found for each of Cases 1 and 2. The equilibrium results (influence coefficients, production outputs, equilibrium price, and the objective functions’ optimal values) are presented in Table 2.

Numerical Experiments (cont-ed) Table 2. Computation results in consistent CVE: vi, generation, profits vi Numerical Experiments (cont-ed) Table 2. Computation results in consistent CVE: vi, generation, profits vi Case 1 Supplier 0 Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 0. 19635 qi qi (MWH) Case 1 Case 2 0. 18779 353. 405 626. 006 0. 16674 405. 120 358. 138 0. 18759 0. 15887 258. 436 220. 451 1082. 9 761. 90 0. 17472 0. 14761 142. 898 125. 462 707. 48 538. 37 0. 22391 0. 19270 560. 180 488. 905 2709. 8 1917. 98 0. 19275 vi Case 2 Profits ($/H) Case 1 1727. 4 Profits ($/H) Case 2 595. 77 2076. 6 1550. 04

Numerical Experiments (cont-ed) As it could be expected, the equilibrium price in Case 1 Numerical Experiments (cont-ed) As it could be expected, the equilibrium price in Case 1 (classical oligopoly) turns out to be higher: p 1=10. 4304, than in Case 2 (mixed oligopoly): p 2=9. 2118. On the contrary, the total electricity power generation is higher: 2039. 412 MWH, - in the second case (mixed oligopoly) than in Case 1: 1978. 475 MWH. In Case 2, profit is minimal in the cell of Supplier 0, as her objective function is not the profit but domestic social surplus: $42, 187. 8/H.

Numerical Experiments (cont-ed) It is also interesting to compare the results in CVE with Numerical Experiments (cont-ed) It is also interesting to compare the results in CVE with consistent conjectures (Cases 1 and 2) against the production volumes and profits obtained for the same cases at the classic Cournot equilibrium (i. e. , with all ) Table 3 illustrates the yielded results, with in the classical oligopoly (Case 1) much higher than the market equilibrium price in the mixed oligopoly (Case 2).

Numerical Experiments (cont-ed) Table 3. Computation results in Cournot model: vi, generation, profits vi Numerical Experiments (cont-ed) Table 3. Computation results in Cournot model: vi, generation, profits vi Case 1 Supplier 0 Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 1. 0 vi Case 2 qi (MWH) Case 1 1. 0 319. 06 qi (MWH) Case 2 Profits ($/H) Case 1 3054. 0 Profits ($/H) Case 2 – 5358. 1 3461. 7 1239. 02 1. 0 347. 00 1200. 00 207. 597 1. 0 261. 39 145. 220 2220. 5 685. 38 1. 0 166. 82 103. 453 1426. 2 548. 51 1. 0 406. 23 221. 767 3988. 5 1188. 70

Numerical Experiments (cont-ed) Again, the total electricity power generation is higher: 2023. 256 MWH, Numerical Experiments (cont-ed) Again, the total electricity power generation is higher: 2023. 256 MWH, – in the second case (mixed oligopoly), than in Case 1: 1761. 9 MWH. Both results are more advantageous for consumers. Simultaneously, the private producers’ net profit values are observed to be much lower in the mixed oligopoly (Case 2) than those in the classical oligopoly (Case 1. ) In Case 2, profit is even negative in the cell of Supplier 0, as her objective function is not the profit but domestic social surplus defined above; in this example, it is equal to $35, 577. 50/H. The latter data, together with the market price values, suggest that the mixed oligopoly with consistent conjectures is more advantageous to consumers than the Cournot model.

Numerical Experiments (cont-ed) Table 4. Computation results in perfect competition model: vi, generation, profits Numerical Experiments (cont-ed) Table 4. Computation results in perfect competition model: vi, generation, profits vi Case 1 Supplier 0 Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 0. 0 vi Case 2 qi (MWH) Case 1 0. 0 348. 43 qi (MWH) Case 2 Profits ($/H) Case 1 1214. 00 Profits ($/H) Case 2 1214. 00 1488. 80 0. 0 412. 49 348. 43 412. 49 0. 0 238. 74 712. 47 0. 0 127. 50 507. 98 0. 0 685. 68 1960. 50

Numerical Experiments (cont-ed) Of course, the perfect competition model (see Table 4) with is Numerical Experiments (cont-ed) Of course, the perfect competition model (see Table 4) with is the best for consumers in both Case 1 and 2: with and the total produce MWH. Domestic social surplus is also higher in this case than in all the previous ones: $43, 303. 52/H. It is curious to note (cf. Tables 2 – 4) that in the classical oligopoly (Case 1), the Cournot model demonstrates to be the most profitable for the producers, whereas it is not the case for the mixed oligopoly: here, the existence of a public enterprise with a distinct utility function makes the consistent CVE more advantageous for the rest of suppliers than the Cournot one.

Numerical Experiments (cont-ed) Finally, we may compare the consistent CVEs (Table 5), Cournot equilibria Numerical Experiments (cont-ed) Finally, we may compare the consistent CVEs (Table 5), Cournot equilibria (Table 6) and the perfect competition for the above-defined Case 2 (mixed oligopoly with Supplier 0 being a public company) against a similar Case 3, in which not Supplier 0 but the (much stronger) Supplier 5 is the public producer.

Numerical Experiments (cont-ed) Table 5. Computation results in consistent CVE: vi, generation, profits vi Numerical Experiments (cont-ed) Table 5. Computation results in consistent CVE: vi, generation, profits vi Case 2 Supplier 0 Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 0. 16674 qi qi (MWH) Case 2 Case 3 0. 13208 626. 006 259. 480 0. 13497 358. 138 303. 229 0. 15887 0. 12803 220. 451 176. 884 761. 90 471. 22 0. 14761 0. 11843 125. 462 105. 984 538. 37 377. 63 0. 19270 0. 21584 488. 905 1083. 785 1917. 98 114. 52 0. 18779 vi Case 3 Profits ($/H) Case 2 595. 77 Profits ($/H) Case 3 851. 16 1550. 04 1052. 75

Numerical Experiments (cont-ed) With the market price even lower and domestic social surplus $44, Numerical Experiments (cont-ed) With the market price even lower and domestic social surplus $44, 477. 30/H even higher than those in the perfect competition model, this consistent CVE may serve as a good example of the strong public company realizing the implicit price regulation within an oligopoly.

Numerical Experiments (cont-ed) A bit curious are the results reflected in Table 6: comparing Numerical Experiments (cont-ed) A bit curious are the results reflected in Table 6: comparing the Cournot oligopoly in Cases 1, 2, and 3, one may see that with a weaker public firm (Case 2), the private producers may incline to the Cournot model of behavior (cf. Table 3). However, with stronger public supplier, as it is in Case 3, private companies would rather select the consistent CVE: in the Cournot model, the strong public company subdues all the rivals to the minimum levels of production and profits. Nevertheless, the Cournot model with stronger public firm provides for the very low market price: , even though at the cost of a somewhat lower domestic social surplus: $41, 111. 59/H.

Numerical Experiments (cont-ed) Table 6. Computation results in Cournot model: vi, generation, profits vi Numerical Experiments (cont-ed) Table 6. Computation results in Cournot model: vi, generation, profits vi Case 2 Supplier 0 Supplier 1 Supplier 2 Supplier 3 Supplier 4 Supplier 5 1. 0 qi qi (MWH) Case 2 Case 3 1. 0 1200. 00 122. 612 1. 0 207. 597 137. 452 1. 0 145. 220 86. 766 685. 38 244. 67 1. 0 103. 453 71. 569 548. 51 265. 51 1. 0 221. 767 1649. 612 1188. 70 – 5319. 0 1. 0 vi Case 3 Profits ($/H) Case 2 – 5358. 1 Profits ($/H) Case 3 451. 01 1239. 02 543. 18

Numerical Experiments (cont-ed) As it could be expected, in the perfect competition model, both Numerical Experiments (cont-ed) As it could be expected, in the perfect competition model, both Cases 2 and 3 give exactly the same results, albeit different domestic social surplus values: $43, 303. 52/H in Case 2 against a bit higher $44, 050. 04/H in Case 3 with stronger public company.

Conclusions In this work, we consider a model of mixed oligopoly with conjectural variations Conclusions In this work, we consider a model of mixed oligopoly with conjectural variations equilibrium (CVE). The agents’ congectures concern the price variations depending upon their producton outputs’ variations. We establish existence and uniqueness results for the CVE (called an exterior equilibrium) for any collection of feasible conjectures. To introduce the concept of an interior equilibrium, we develop a consistency criterion for the conjectures and prove the existence theorem for the interior equilibrium.

Conclusions (cont-ed) To prepare a base for an extension of our results to the Conclusions (cont-ed) To prepare a base for an extension of our results to the case of non-differentiable demand functions, we also investigate the behaviour of the consistent conjectures in dependence upon a parameter repesenting the demand function’s derivative with respect to the equilibrium price. Also, more general than quadratic cost functions case will be examined.

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