ON A COURNOT DUOPOLY GAME WITH DIFFERENTIATED GOODS, HETEROGENEOUS EXPECTATIONS AND A COST FUNCTION INCLUDING EMISSION COSTS

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1 Sientifi Bulletin Eonomi Sienes, Volume 6/ Issue ON A COURNOT DUOPOLY GAME WITH DIFFERENTIATED GOODS, HETEROGENEOUS EXPECTATIONS AND A COST FUNCTION INCLUDING EMISSION COSTS Georges SARAFOPOULOS, Kosmas PAPADOPOULOS Department of Eonomis, Demoritus University of Thrae, Greee gsarafop@eon.duth.gr Department of Eonomis, Demoritus University of Thrae, Greee hurdlerofthrae@yahoo.gr Abstrat: In this study we investigate the dynamis of a nonlinear Cournot- type duopoly game with differentiated goods, linear demand and a ost funtion that inludes emission osts. The game is modeled with a system of two differene equations. Existene and stability of equilibria of this system are studied. We show that the model gives more omplex haoti and unpreditable trajetories as a onsequene of hange in the parameter of horizontal produt differentiation and a higher (lower) degree of produt differentiation (weaker or fierer ompetition) destabilize (stabilize) the eonomy. The haoti features are justified numerially via omputing Lyapunov numbers and sensitive dependene on initial onditions. Also, we show that in this ase there are stable trajetories and a higher (lower) degree of produt differentiation does not tend to destabilize the eonomy. Keywords: Cournot duopoly game, Disrete dynamial system, Heterogeneous expetations, Stability, Chaoti behavior. JEL Classifiation Codes : C6, C7, D43.. INTRODUCTION An Oligopoly is a market struture between monopoly and perfet ompetition, where there are only a few number of firms in the market produing homogeneous produts. The dynami of an oligopoly game is more omplex beause firms must onsider not only the behaviors of the onsumers, but also the reations of the ompetitors i.e. they form expetations onerning how their rivals will at. Cournot, in 838 has introdued the first formal theory of oligopoly. He treated the ase with naive expetations, so that in every step eah player (firm) assumes the last values that were taken by the ompetitors without estimation of their future reations. Expetations play an important role in modelling eonomi phenomena. A produer an hoose his expetations rules of many available tehniques to adjust his prodution outputs. In this paper, we study the dynamis of a duopoly model where eah firm behaves with heterogeneous expetations strategies. We onsider a duopoly model where eah player forms a strategy in order to ompute his expeted output. Eah player adjusts his outputs towards the profit maximizing amount as target by using his expetations rule. Some authors onsidered duopolies with homogeneous expetations and found a variety of omplex dynamis in their games, suh as appearane of strange attrators (Agiza, 999, Agiza et al., 00, Agliari et al., This is an open-aess artile distributed under the Creative Commons Attribution- NonCommerial 4.0 International Liense (

2 Georges SARAFOPOULOS, Kosmas PAPADOPOULOS 005, 006, Bishi, Kopel, 00, Kopel, 996, Puu, 998, Sarafopoulos, 05). Also models with heterogeneous agents were studied (Agiza, Elsadany, 003, 004, Agiza et al., 00, Den Haan, 003, Fanti, Gori, 0, Tramontana, 00, Zhang, 007). In the real market produers do not know the entire demand funtion, though it is possible that they have a perfet knowledge of tehnology, represented by the ost funtion. Hene, it is more likely that firms employ some loal estimate of the demand. This issue has been previously analyzed by Baumol and Quandt, 964, Puu 995, Naimzada and Rihiuti, 008, Askar, 03, Askar, 04. Bounded rational players (firms) update their prodution strategies based on disrete time periods and by using a loal estimate of the marginal profit. With suh loal adjustment mehanism, the players are not requested to have a omplete knowledge of the demand and the ost funtions (Agiza, Elsadany, 004, Naimzada, Sbragia, 006, Zhang et al, 007, Askar, 04). The paper is organized as follows: In Setion, the dynamis of the duopoly game with heterogeneous expetations, linear demand and a quadrati ost funtion inluding emission osts is analyzed. The existene and loal stability of the equilibrium points are also analyzed. In Setion 3 numerial simulations are used to show omplex dynamis via omputing Lyapunov numbers, and sensitive dependene on initial onditions.. THE GAME. The onstrution of the game The two firms offer their produts at disrete-time periods (t 0,,, ) on a ommon market. We onsider a simple Cournot-type duopoly market where firms (players) produe differentiated goods and their prodution deisions are taken at disrete-time periods (t 0,,, ). In this study we onsider heterogeneous players and more speifially, we onsider that the Firm hooses the prodution quantity in a rational way, following an adjustment mehanism (bounded rational player), while the Firm deides with naïve way by seleting a quantity that maximizes its output (naïve player). At eah period t, every firm must form an expetation of the rival s output in the next time period in order to determine the orresponding profit-maximizing quantities for period t+. We suppose that q, q are the prodution quantities of eah firm, then the inverse demand funtion (as a funtion of quantities) is given by the following equation: where p i is the produt prie of firm i. p a q dq, i j i i j So, we have for eah firm the following equations: p a q dq and p a q dq () where α is a positive parameter and d (,) is the parameter that reveals the differentiation degree of produts. For example, if d 0 then both produts are independently and eah firm partiipates in a monopoly. But, if d then one produt is a substitute for the other, sine the produts are homogeneous. It is understood that for positive values of the parameter d the larger the value, the less diversifiation we have in both produts. On the other hand negative values of the parameter d are desribed that the two produts are omplementary and when d then we have the phenomenon of full ompetition between the two ompanies.

3 On a Cournot Duopoly Game with Differentiated Goods, Heterogeneous Expetations and a Cost Funtion Inluding Emission Costs We suppose the following ost funtion: where C q C + C () i i p e C q q (3) pi i i is the quadrati prodution ost funtion for firm i, and > 0 is the same marginal ost for two firms. Also, C p ε q (4) ei i is the linear fumes emission ost funtion for eah player, where p is the emission liense prie whih is deided from the Government, ε [ 0,] is a positive oeffiient whih is ommon for two firms and when it is multiplied with eah prodution quantity, it gives the total emissions. With these assumptions, the profits of the firms are given by: and ( α ) Ρ (5) q, q pq C q q dq q q p ε q ( α ) Ρ (6) q, q pq C q q dq q q p ε q Then the marginal profits at the point of the strategy spae are given by: Ρ a p ε ( + ) q dq q To make the alulations easily we set: Ρ a p + q dq q and ε ( ) (7) Η a p ε (8) We suppose that first firm deides to inrease its level of adaptation if it has a positive marginal profit, or dereases its level if the marginal profit is negative (bounded rational player). If k > 0 the dynamial equation of the first player is: q t + q t k Ρ (9) q t q k is the speed of adjustment of player, it is a positive parameter whih gives the extent of prodution variation of the firm following a given profit signal. Moreover, it aptures the fat that relative effort variations are proportional to the marginal profit. 3

4 Georges SARAFOPOULOS, Kosmas PAPADOPOULOS The seond firm deides with naïve way by seleting a prodution that maximizes its profits (naïve player): arg max (, ) q t + Ρ q t q t (0) y The dynamial system of the players is desribed by: ( ) Η dq ( t) ( t + ) ( + ) q t + q t + kq t Η + q t dq t q () We will fous on the dynamis of this system to the parameter d.. Dynamial analysis.. The equilibriums of the game The equilibriums of the dynamial system Eq.() are obtained as nonnegative solutions of the algebrai system: kq Η + q dq 0 Η dq q ( + ) () whih obtained by setting q ( t + ) q ( t) q and q ( t ) q ( t) q +. If q 0, then q Η ( + ) and we have the boundary equilibrium: Η E 0 0, ( + ) (3) If ( ) q dq Η + 0, then we form the following system: Η dq q Η dq q + ( + ) ( ) (4) 4

5 On a Cournot Duopoly Game with Differentiated Goods, Heterogeneous Expetations and a Cost Funtion Inluding Emission Costs The system s solutions are: and the Nash equilibrium: q Η q + + d ( ), Η Η E ( q, q ),, ( + ) + d ( + ) + d (5).. Stability of equilibriums The study of the loal stability of the equilibrium is based on the loalization on the omplex plane of the eigenvalues of the Jaobian matrix of the dimensional map (Eq.()). In order to study the loal stability of equilibrium points of the model (), we onsider the Jaobian matrix, q, q : J ( q q ) along the variable strategy where (, ) g q q (, ) J q q ( + ) fq f q gq g q P f ( q, q ) q + k q q + k q Η ( + ) q dq q Η dq (6) (7) and we find the Jaobian matrix: (, ) J q q For equilibrium E 0 we have: J E 0 + k Η 4 + q dq dkq d 0 ( + ) d + kη 0 ( + ) d 0 ( + ) (8) (9) with d Tr J ( E0 ) + kη ( + ) and Det J E

6 Georges SARAFOPOULOS, Kosmas PAPADOPOULOS The harateristi equation of J ( E 0 ) is: l Tr l Det and the eigenvalues of the Jaobian matrix are: + 0 (0) Sine d + < l 0 and d l + kη ( + ), it s learly that l >, and the point E 0 is unstable. () In the Nash equilibrium point E the Jaobian matrix is: J E k + q kdq d 0 ( + ) () The equilibrium point is loally asymptotially stable if Sine and i) Det > 0 ii) Tr + Det > 0 iii) + Tr + Det > 0 d kη Det + q > 0 ( + ) (3), (4) kq Tr + Det ( ) d ( ) d > ( + ) (5) The onditions (i) and (ii) of Eq.(3) are always satisfied and then the ondition (iii) is the ondition for the loal stability of the Nash Equilibrium. This ondition beomes: + Tr + Det > 0 d kη d kh > 0 (6) The disriminant of Eq.(6) is positive if and only if: 6( + ) ( kη ) + kη + > 0 ( k ) k 0 Η + Η + > kη ( 0,+ ) (7) 6

7 On a Cournot Duopoly Game with Differentiated Goods, Heterogeneous Expetations and a Cost Funtion Inluding Emission Costs It follows that Eq.(6) is verified if and only if: where d, ( 0, ) kη + d ( d, d ) 4 + ± 4 + kη + kη + kη (8) (9) the real roots of Eq. (6). It follows that: Proposition: The Nash equilibrium of the dynamial system Eq. () is loally stable if and only if:..3 Numerial simulations ( 0, ) kη + d ( d, d ) To provide some numerial evidene for the haoti behavior of the system Eq.(), as a onsequene of hange in the parameter d of the produt differentiation degree. We present various numerial results to show the haotiity, inluding its bifurations diagrams, strange attrator, Lyapunov numbers and sensitive dependene on initial onditions (Kulenovi, M., Merino, O., 00). In order to study the loal stability properties of the equilibrium points, it is onvenient to take the parameters values as follows: Then,,6 ( 0, ) 7 a 0, k 0.7,, p 4, ε 0.5, Η 8. kη + and d 0.5, d 5.0 > and the stability ondition beomes: (, ) (, ) d (,) d d d d (30) Numerial experiments are omputed to show the bifuration diagram with respet to d, strange q, q and Lyapunov numbers. attrators of the system Eq.() in the phase plane Figure shows the bifuration diagrams with respet to the parameter d. In this figure, one observes omplex dynami behavior suh as yles of higher order and haos. Figure 3 shows the Lyapunov numbers of the orbit of (0.0,0.0) for a 0, k 0.7,, p 4, ε 0.5, Η 8, d 0.9 (left) and d 0.99 (right). If the Lyapunov number is greater of, one has evidene for haos. Figure 3 shows the graphs of the a 0, k 0.7,, p 4, ε 0.5, Η 8, d 0.9 same orbit (strange attrators) for (left) and d 0.99 (right). From these results when all parameters are fixed and only d is varied the struture of the game beomes ompliated through period doubling bifurations, more omplex bounded attrators are reated whih are aperiodi yles of higher order or haoti attrators. To demonstrate the sensitivity to initial onditions of the system Eq. () we ompute

8 Georges SARAFOPOULOS, Kosmas PAPADOPOULOS two orbits with initial points (0.0,0.0) and (0.00,0.00), respetively. Figure 5 shows sensitive dependene on initial onditions for x-oordinate of the two orbits, for the system Eq.(), plotted against the time with the parameter values a 0,, k 0.7, p 0.75, ε 0.5, Η 8, d 0.9. At the beginning the time series are indistinguishable; but after a number of iterations, the differene between them builds up rapidly. From Figure 4 we show that the time series of the system Eq.() is sensitive dependene to initial onditions, i.e. omplex dynamis behavior our in this model. If a 0,, k 0., p 4, ε 0.5, ( 8) Η (we hanged the value of parameter k), kη.6 ( 0, + ) and d <, d >. From Eq. (8) it follows that for eah d in the internal (-,) the Nash equilibrium is loally asymptotially stable (Figure 5). Therefore, in the ase of quadrati osts there are stable trajetories and a higher (lower) degree of produt differentiation does not to destabilize (stabilize) the market. Figure : Bifuration diagrams with respet to the parameter d against variable q (left) and q (right), a 0,, k 0.7, p 4, ε 0.5, Η 8. with 400 iterations of the map Eq. () for Figure : Lyapunov numbers of the orbit of the point A(0.,0.) versus the number of iterations a 0,, k 0.7, p 4, ε 0.5, Η 8, and for d 0.9 (left) and d 0.99 (right). for 8

9 On a Cournot Duopoly Game with Differentiated Goods, Heterogeneous Expetations and a Cost Funtion Inluding Emission Costs Figure 3: Phase portrait (strange attrators). The orbit of (0.,0.) with 000 iterations of the map a 0,, k 0.7, p 4, ε 0.5, Η 8, and for d 0.9 (left) and d 0.99 Eq.() for (right). Figure 4: Sensitive dependene on initial onditions for x-oordinate plotted against the time: the two orbits: the orbit of (0.0,0.0) (left) and the orbit of (0.00,0.00) (right), for the system Eq.(), with a 0,, k 0.7, p 4, ε 0.5, Η 8, d 0.9. the parameters values Figure 5: Bifuration diagrams with respet to the parameter d against variable q (left) and q (right), a 0,, k 0., p 4, ε 0.5, Η 8. with 400 iterations of the map Eq. () for 9

10 Georges SARAFOPOULOS, Kosmas PAPADOPOULOS 3. CONCLUSION In this paper we analyzed the dynamis of a nonlinear Cournot- type duopoly game with differentiated goods, linear demand and a ost funtion that inludes emission osts. Existene and stability of equilibria of this system are studied. We proved that the parameter of horizontal produt differentiation may hange the stability of equilibrium and ause a struture to behave haotially. For some values of this parameter there is a stable Nash equilibrium. Dereasing these values, the equilibrium beomes unstable, through period-doubling bifuration. The haoti features are justified numerially via omputing Lyapunov numbers and sensitive dependene on initial onditions. Also, we show that in our ase of quadrati osts there are stable trajetories for eah d in the internal (-,) and a higher (lower) degree of produt differentiation does not tend to destabilize the market. ACKNOWLEDGEMENTS The authors thank the two anonymous referees for interesting remarks and useful suggestions. REFERENCES. Agiza HN, (998), Expliit stability zones for Cournot games with 3 and 4 ompetitors. Chaos Solitons Frat. 9: Agiza, H.N., (999), On the analysis of stability, bifuration, haos and haos ontrol of Kopel map. Chaos, Solitons & Fratals 0, Agiza HN. (004) On the stability, bifurations, haos and haos ontrol of Kopel map. Chaos Solitons Frat. : Agiza HN, Elsadany AA, Chaoti dynamis in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput. 49: Agiza HN, Elsadany AA., (003) Nonlinear dynamis in the Cournot duopoly game with heterogeneous players. Physia A 30: Agiza HN, Hegazi AS, Elsadany AA. (00). Complex dynamis and synhronization of duopoly game with bounded rationality. Math. Comput. Simulat. 58: Askar, S.S., (03). On omplex dynamis of monopoly market, Eonomi Modelling, 3: Askar, S. S., (04) Complex dynami properties of Cournot duopoly games with onvex and log-onave demand funtion, Operations Researh Letters 4, Baumol, W.J., Quandt, R.E., (964). Rules of thumb and optimally imperfet deisions, Amerian Eonomi Review 54 (): Bishi GI, Kopel M. (00). Equilibrium seletion in a nonlinear duopoly game with adaptive expetations. J. Eonom Behav. Org. 46: Bishi GI, Lamantia F, Sbragia L. (004). Competition and ooperation in natural resoures exploitation: an evolutionary game approah. In: Cararro C, Fragnelli V, editors. Game pratie and the environment. Cheltenham: Edward Elgar; 87.. Bishi GI, Naimzada A. (000). Global analysis of a dynami duopoly game with bounded rationality. In: Filar JA, Gaitsgory V, Mizukami K, editors. Advanes in dynami games and appliations, vol. 5. Basel: Birkhauser;

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