A Cournot duopoly game with heterogeneous players: nonlinear dynamics of the Gradient Rule versus Local Monopolistic Approach Fausto Cavallia,∗, Ahmad Naimzadaa a Department
of Economics, Management and Statistics, University of Milano-Bicocca, Milan, Italy
Abstract In this paper we analyze a duopolistic Cournotian game, where firms produce an homogeneous good, the demand function is isoelastic (Puu [1]) and total cost functions are linear. With these features the traditional dynamic adjustment based on the classical best replay mechanism is very demanding in terms of rationality and information set. We consider heterogeneous players with two different kinds of adjustment mechanisms based on a lower degree of rationality of the firms, on a reduced information set and reduced computational capabilities. By using the first adjustment process, that belongs to the class of “rules of thumb” mechanisms known in the literature as “Gradient Dynamics”, firms do not solve any optimization problem, but they adjust their production in the direction indicated by their (correct) estimate of the marginal profit. By using the second adjustment mechanism, known in the literature as “Local Monopolistic Approximation”, firms are able to get the correct local estimate of the demand function in order to get a global linear conjectured demand function. On the basis of this subjective demand function they solve their profit maximization problem. Both these repeated games can converge to a Cournot-Nash equilibrium, i.e. to the equilibrium of the best reply dynamics. Stability conditions of the Nash equilibrium and complex dynamics are studied. In particular we show two different routes to complicated dynamics: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation which originates an attractive invariant closed curve. The paper extends the results of other authors on duopolistic Cournotian games with heterogeneous adjustment processes by considering “Local Monopolistic Approximation” combined with the Gradient Rule. Keywords: Cournot duopoly game, Nash equilibrium, bounded rationality, heterogeneous players, bifurcation, complex dynamics
∗ Corresponding
author Email addresses:
[email protected] (Fausto Cavalli),
[email protected] (Ahmad Naimzada)
Preprint submitted to Elsevier
March 24, 2014
1. Introduction There are two extreme cases of market structure, monopoly and perfect competition. In the first situation there is only one firm providing the market, unlike in the second one in which the firms number is high and they are considerably smaller than the whole market size. Between these two situations, when the market is supplied by only a few firms, we have oligopoly. This case is more complex than monopoly and perfect competition, since the oligopolists have to take into account at the same time both their own actions and those of their competitors. The first formal theory of oligopoly with two sellers (duopoly) was introduced by Cournot in 1838 [2] and the Cournot model is still one of the most well known subjects of economic dynamics and game theory. The first setting in which the oligopolistic market model was studied was the static one, in which the main theoretical instrument is the notion of Nash equilibrium. This means that each firm has an elevated degree of rationality and is provided with cognitive and computational skills that allow to perfectly know the demand curve of the good it produces. Moreover, each firm is supposed to have perfect foresight of the next period production so that it knows exactly what the other firms operate. In this case the players are able to solve a one period optimization problem. The situation is different in the dynamic setting, which Rand [3] and Poston and Stewart [4] started studying in 1978. They showed that, under suitable conditions, a simple duopoly would give rise to complex dynamic phenomena. This is because of the unimodal character of the curve that shows how to react with respect to previous time strategy of the competitor (the reaction function). However, both in [3] and [4] the reaction functions are proposed using an abstract and exogenous framework and are not derived from the solution of an optimization problem. Conversely, Puu in 1991 [1] proposed a duopoly based on unimodal reaction functions derived solving an optimization problem for profit functions. Also in this situations complex phenomena arise. In [1] Puu studied the case of constant marginal costs and, starting from Cobb-Douglas type preferences for the consumer, isoelastic demand function. In this framework the outputs of each competitor can evolve showing a period doubling sequence of flip bifurcations leading finally to chaos. The work of Puu gave a boost to the study of duopolistic and oligopolistic markets and to develop the initial economic assumptions of the Cournot model. In the last decades many authors considered oligopoly games with other decisional mechanism, different from the best response rule, implying different degree of informational and computational abilities. In Bischi and Naimzada [5], Bischi et al. [6, 7] and Agiza et al. [8] a gradient-like mechanism is proposed, where players base the adjustment of their strategies on a local estimate of the marginal profit and players do not solve any optimization problem; for this kind of decisional mechanism the agents are not requested to have complete knowledge of the demand and cost function. Recently the same mechanism has been considered by Askar in [9, 10]. Another decisional mechanism is that called “Local Monopolistic Approximation” (LMA), where oligopolists do not 2
know the market demand function, they conjecture it is linear and they estimate such a linear function through the local knowledge of true demand curve and the knowledge of the current market state in terms of quantities and price. Such an adjustment process was introduced by Tuinstra [11], Bischi et al. [12], Naimzada and Sbragia [13] and was applied in a monopolistic setting by Naimzada and Ricchiuti [14]. A wide branch of the literature is focused on what happens if the firms are heterogeneous from the point of view of the decisional mechanisms. This approach characterizes the works by Leonard and Nishimura [15], Den-Haan [16], Agiza and Elsadany [17, 18], Angelini et al. [19], Tramontana [20]. This works are mainly focused on the study of the coupling of a best response decisional mechanism with the gradient like decisional mechanism, using different demand and cost functions. The main purpose of this paper is to analyze the dynamic behavior of a duopoly model where the two players adopt the gradient like and the Local Monopolistic approximations. In particular, we focus on a gradient mechanism with exogenous constant reactivity. The economic structure of our model is similar to those proposed by Angelini et al. [19] and Tramontana [20] with respect to the isoelastic demand function and to the constant marginal costs. However, in both these works one of the players adopts a best reply behavior whereas the other uses a gradient rule. Our work belongs to the research strand in which we are investigating several aspects of heterogeneous duopolies. In particular, we focus on the effect of different degrees of rationality in Cavalli and Naimzada [21], where we take into account mechanisms based on the gradient rule and on best response approaches with different level of informational and computational capabilities. Conversely, in Cavalli al. [22] we are investigating the effect of presence of endogenous reactivity. Our main result concerns the possible routes of destabilization for the CournotNash equilibrium point. Such scenarios include flip bifurcation, where loss of stability results in a stable cycle of period two, and the Neimark-Sacker bifurcation, where orbits spiral away from the equilibrium toward a stable invariant closed curve describing quasi-periodic motions. In particular, it is shown that sufficiently strong cost differences favorable to the firm using the gradient adjustment mechanism results in Neimark-Sacker bifurcation, whereas the flip bifurcation is a scenario associated with a costs structure favorable to the firm using linear approximation. Our result adds a new confirmation to recent and related studies, about oligopolies based on interactions involving gradient rule (see [6, 7, 19, 20]), where both Neimark-Sacker and flip bifurcations appear to be the possible destabilization routes. In this paper nonlinear, in particular isoelastic, inverse demand function and constant marginal cost function are considered. The paper is organized as follows. In Section 2 we introduce the model and the nonlinear system describing the dynamics of the productions of the firms. In Section 3 we determine the conditions under which the Nash equilibrium is locally stable. In Section 4 numerical efforts are devoted to the study of the local bifurcations and to the route to complex dynamics. 3
2. Model We suppose that the economy is populated by n agents with Cobb-Douglas preferences, so that the representative jth agent has the utility function Uj (qj ) = Qm 1 2 m k αk j k=1 (qj ) , where qj = (qj , qj , . . . , qj ) denotes the vector of the quantities of the m goods. The P previous utility function for the jth agent is subdued to the m budget constraint k=1 pk qjk ≤ y j , where pk is the price of commodity k and j y is the income of the jth agent. From the usual constrained maximization problem we obtain αkj y j qjk = . pk If we focus only on one market, we can suppress the index k. Summing over all the agents, P we consider the aggregated demand, from which, after normalizing so that j αj y j = 1, we have the inverse demand function p(Q) =
1 , Q
(1)
P where we set Q = j qj , which is a constant elasticity demand function. In the present work we suppose that the industry is composed by two firms (indexed by i = 1, 2) producing perfect substitute goods q1 and q2 and we consider linear cost functions Ci (qi ) = ci qi
i = 1, 2,
where ci > 0 represent the (constant) marginal costs of each duopolist. With the previous assumptions, we have that the profit of the ith firm is Πi (qi , q−i ) =
qi − ci qi , qi + q−i
(2)
where we use index −i to consider the other duopolist with respect to the ith one. This situation can be studied in the game theory context, where the players are the two dupolisits, the strategies are given by the set of all the production possibilities (qi ≥ 0) and the payoff functions are given by the profit functions (2). As in Puu [1], we have one Nash equilibrium given by c1 c2 . (3) E = (q1N , q2N ) = , (c1 + c2 )2 (c1 + c2 )2 We underline that, when the equilibrium (3) is achieved, the output produced by the most efficient firm is higher than the opponent’s one, as well as its profits. The Nash equilibrium notion is very demanding in terms of rationality and information set. In this work we consider two different mechanism which imply a lower degree of rationality. For the first firm, we consider a bounded rational adjustment process as in [12, 23], in which the firm has only a limited knowledge of the demand function. This is usually called “Local Monopolistic 4
Approximation” (LMA). We suppose that the firm knows at time t the market price p(t), the corresponding produced quantity Q(t), and has a local knowledge of the demand function in (p(t), Q(t)), which can be obtained for example through market experiments (see [12]). On the base of these informations, the firm conjectures the demand function through a linear approximation of the true demand function in (p(t), Q(t)) and optimizes the expected profits. The conjectured demand at time t + 1 is pe1 (t + 1) = p1 (t) + p′ (Q(t))(Qe (t + 1) − Q(t)),
(4)
where Qe = (q1 (t+1), q2e (t+1)) in which q2e (t+1) indicates the expected output, i.e. the output that the first oligopolist expects that its opponent i = 2 produces at time t + 1. We consider Cournotian expectations (q2e (t + 1) = q2 (t)) for the first firm, so we rewrite (4) as pe1 (t + 1) =
1 1 (q1 (t + 1) − q1 (t)), − q1 (t) + q2 (t) (q1 (t) + q2 (t))2
where we also used (1). The output assumed for time t + 1 is set by maximizing the expected profit (2) at time t + 1, obtaining q1 (t + 1) = arg max [pe1 (t + 1)q1 (t + 1) − c1 q1 (t + 1)], q1 (t+1)
which gives the maximum q1 (t + 1) =
1 1 1 − c1 (q1 (t) + q2 (t)) (q1 (t) + q2 (t)). q1 (t) + 2 2
(5)
We assume that also the second firm is a boundedly rational firm, following a gradient adjustment mechanism, analogous to that used in [5] q2 (t + 1) = q2 (t) + αφ2 (q1 (t), q2 (t)),
(6)
where φ2 represents the marginal profit of the ith duopolist q1 φ2 (q1 , q2 ) = − c2 . (q2 + q1 )2 The boundedly rational firm modifies (increasing or decreasing) its output according to the information given by the marginal profit of the last period, modulated by a positive parameter α that represents the speed of adjustment. The resulting duopoly game is described by the following discrete dynamical system 1 1 q1 (t + 1) = q1 (t) + 1 − c1 (q1 (t) + q2 (t)) (q1 (t) + q2 (t)), 2 2 T (q1 , q2 ) : q1 (t) q2 (t + 1) = q2 (t) + α − c 2 . (q1 (t) + q2 (t))2 (7) The main modeling novelty of the present work lies in considering an heterogeneous system where at least one of the boundedly rational players, in order to decide about its production strategy, uses the LMA. 5
3. Local analysis In this section we study the local stability of the fixed point of system (7) and its relation to the Nash equilibrium. We have the following proposition, which proof is straightforward. Proposition 1. The Nash equilibrium (3) is a fixed point for the iteration (7). Conversely, the only fixed point of (7) is (3). To investigate the local stability of the fixed point we introduce the Jacobian matrix of system (7) 1 1 − c1 (q1 + q2 ) − c1 (q1 + q2 ) 2 J(q1 , q2 ) = α(q1 − q2 ) 2αq1 , 1− − (q1 + q2 )2 (q1 + q2 )3 which evaluated at (3) becomes JE =
J(q1N , q2N )
c2 c2 1 − . = c1 + c2 c1 + c2 2 α(c21 − c22 ) −2αc22 − 2αc1 c2 + 1
We recall that a fixed point is locally stable if 1 − Tr(J) + det(J) > 0, 1 + Tr(J) + det(J) > 0, 1 − det(J) > 0. Let us introduce
αf =
(8)
4c1 + 8c2 , (c1 + c2 )(−c21 + 6c1 c2 + 7c22 )
(9)
2c1 . − 2c1 c2 − 3c22 )
(10)
and αns =
(c1 +
c2 )(c21
We have the following result concerning the stability of the Nash equilibrium. Proposition 2. Fixed point (3) is stable provided that 0
4 and α < αns , c2
(11)
Proof. We have that the first condition of (8) is always fulfilled. The second and the third condition of (8) are respectively equivalent to 2 c1 2c2 7c22 α + +2>0 (12) − 3c1 c2 − 2 2 c1 + c2 6
and
c1 α(−c21 + 2c1 c2 + 3c22 ) + > 0. c1 + c2 2
(13)
Condition (12) is verified for all α > 0 when 7c2 c21 − 3c1 c2 − 2 ≥ 0, 2 2 or equivalently, when c1 /c2 ≥ 7 and condition (13) is verified for all α > 0 when −c21 + 2c1 c2 + 3c22 ≥ 0 or equivalently, when 0 < c1 /c2 ≤ 3. So, when c1 /c2 ≥ 7 we have that condition (12) is automatically satisfied while it is easy to show that (13) is satisfied only if α < αns . Similarly, if c1 /c2 ≤ 3 we have that condition (13) is automatically satisfied while (12) is satisfied only if α < αf . On the contrary, when c1 /c2 ∈ (3, 7) neither condition (12) nor (13) are verified for all α > 0, so we need to impose α < min{αns , αf }. But we have that αns < αf when 2c1 4c1 + 8c2 4. Summarizing, we have that the fixed point is stable provided that c1 ≥ 7 and α < αns , c2 c1 < 7 and α < αns , 4< c2 c1 ≤ 3 and α < αf , c2 c1 < 4 and α < αf , 3< ≤ c2
(15)
which gives (11). When condition (11) is violated, we have that the Nash equilibrium becomes unstable. In particular, referring also to (15), when c1 /c2 < 3 but α = αf , a flip bifurcation occurs. Since for c1 /c2 ∈ [3, 4) we have that αns > αf , we again have that, for this range of the marginal costs ratio, a flip bifurcation starts when α = αf . Conversely, when c1 /c2 > 7 a Neimark-Sacker bifurcation takes place for α = αns , as well as when c1 /c2 ∈ (4, 7] since αns < αf . In all the situations, increasing the reactivity parameter α makes system (7) more unstable, in the sense that more reactive firms produce a more unstable evolution for the output level.
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4. Simulations In this section we perform several simulations on system (7) to confirm the results of Proposition 2 and to investigate in deep the different qualitative behaviors of the model, in particular when the equilibrium becomes unstable. In Figure 1 we show the bifurcation diagram, obtained evolving the initial datum (q10 , q20 ) = (3.6, 2.4) with c1 = 0.2 and c2 = 0.3, which portrays the characteristic behavior of a flip bifurcation. For these marginal costs, we have 1.25 1.2 1.15
q1
1.1 1.05 1 0.95 0.9 0.85 6
6.5
7
7.5 α
8
8.5
9
Figure 1: Bifurcation diagram for c1 /c2 < 3.
that (7) is stable for α < 6.7369, above which a period-2 cycle appears. The period is doubled when α ≈ 8.3385, and a cascade of flip leads to chaos α > 8.8. The time evolution plots for α = 6.8, 8.6, 9 are reported in Figure 2. In Figure 3 we show the evolution of the attractor in the phase space (q1 , q2 ) considering four different values of α = 8.76, 8.8, 9.1, 9.33. As we can see, the initial periodic attractor evolves into several unconnected regions which leads to a chaotic attractor. Conversely, when we take c1 = 0.75, c2 = 0.1, we have that c1 /c2 > 7, so (7) is stable for α < 4.6136. The bifurcation diagram for this choice of the marginal costs and for (q10 , q20 ) = (0.1246, 0.9343) is shown in Figure 4 and confirms the existence of a Neimark-Sacker bifurcation, according to Proposition 2. In Figure 5 we show q1 (t), t > 9900 for α = 4.65 which follows a quasi-periodic evolution. In this case, the attractors in the phase space (q1 , q2 ) are closed invariant curves, as reported in Figure 6 for α = 4.65, 4.7, 4.8, 4.878. The evolution for c1 /c2 ∈ (3, 4) is similar to that obtained with c1 /c2 < 3, and a flip bifurcation occurs, as well as the evolution for c1 /c2 ∈ (4, 7) is similar to that obtained with c1 /c2 > 7, and a Neimark-Sacker bifurcation occurs. It is worth underlining that, especially when c1 /c2 ≈ 4, it is very difficult to preserve positivity and avoid the blow-up of the solution of system (7) when the reactivity α increases over the critical values.
8
α=6.8
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Figure 2: Time evolution of q1 for three different values of the reactivity α for c1 /c2 < 3.
5. Conclusions In this paper we have analyzed two routes to complicated dynamics in a duopoly model where firms act with different decisional mechanisms, the gradient and the local monopolistic approximations. The demand function is assumed isoelastic, while the marginal cost is constant. We find that the speed of adjustment of the player adopting the gradient-like mechanism and the heterogeneity of the firms play a relevant role in the stability conditions of the Nash equilibrium and in the different dynamics emerging after its loss of stability. A high level of the speed of adjustment combined with a relatively unbalanced ratio in the marginal costs leads the system to instability via flip or Neimark-Sacker bifurcation of the Nash equilibrium. Our results, together with those recent of [19, 20], allow us to conjecture that decisional mechanisms based on gradient rule combined with an economic structure characterized by isoelastic demand function is responsible of the appearance of the two destabilization ways. In future research we aim to investigate this conjecture in more general settings. References [1] T. Puu, Chaos in duopoly pricing, Chaos, Solitons and Fractals 1 (6) (1991) 573–581. [2] A. A. Cournot, Reserches sur les Principles Mathematiques de la Theorie des Richesses., Paris, Hachette, 1838.
9
α=8.76
α=8.8
3
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2 q
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α=9.33
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1
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1 q
1
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Figure 3: Attractors for different values of the reactivity α when c1 /c2 < 3. The circle represents the Nash equilibrium (3).
[3] D. Rand, Exotic phenomena in games and duopoly models, Journal of Mathematical Economics 5 (2) (1978) 173–184. [4] T. Poston, I. Stewart, Catastrophe theory and its applications., London: Pitman Ltd., 1978. [5] G. Bischi, A. Naimzada, Global analysis of a duopoly game with bounded rationality, Adv. Dynam. Games Appl. 5 (1999) 361–385. [6] G. . Bischi, M. Gallegati, A. Naimzada, Symmetry-breaking bifurcations and representative firm in dynamic duopoly games, Annals of Operations Research 89 (1999) 253–272. [7] G. . Bischi, M. Kopel, A. Naimzada, On a rent-seeking game described by a non-invertible iterated map with denominator, Nonlinear Analysis, Theory, Methods and Applications 47 (8) (2001) 5309–5324. [8] H. N. Agiza, A. S. Hegazi, A. A. Elsadany, Complex dynamics and synchronization of a duopoly game with bounded rationality, Mathematics and Computers in Simulation 58 (2) (2002) 133–146.
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0.22 0.2 0.18
q
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0.16 0.14 0.12 0.1 0.08 0.06 4.5
4.55
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α
4.65
4.7
4.75
Figure 4: Bifurcation diagram for c1 /c2 > 7.
α=4.65
q1
0.16 0.14 0.12 0.1
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Figure 5: Time evolution of q1 for c1 /c2 > 7.
11
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Figure 6: Attractors for different values of the reactivity α when c1 /c2 > 7. The circle represents the Nash equilibrium (3).
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[9] S. S. Askar, Complex dynamic properties of cournot duopoly games with convex and log-concave demand function, Operations Research Letters 42 (1) (2014) 85–90. [10] S. Askar, The rise of complex phenomena in cournot duopoly games due to demand functions without inflection points, Communications in Nonlinear Science and Numerical Simulation 19 (6) (2014) 1918–1925. [11] J. Tuinstra, A price adjustment process in a model of monopolistic competition, International Game Theory Review 6 (3) (2004) 417–442. [12] G. I. Bischi, A. K. Naimzada, L. Sbragia, Oligopoly games with local monopolistic approximation, Journal of Economic Behavior and Organization 62 (3) (2007) 371–388. [13] A. K. Naimzada, L. Sbragia, Oligopoly games with nonlinear demand and cost functions: Two boundedly rational adjustment processes, Chaos, Solitons and Fractals 29 (3) (2006) 707–722. [14] A. Naimzada, G. Ricchiuti, Monopoly with local knowledge of demand function, Economic Modelling 28 (1-2) (2011) 299–307. [15] D. Leonard, K. Nishimura, Nonlinear dynamics in the cournot model without full information., Annals of Operations Research 89 (1999) 165–173. [16] W. Den-Haan, The importance of the number of different agents in a heterogeneous asset-pricing model., Journal of Economic Control 25 (2001) 721–746. [17] H. N. Agiza, A. A. Elsadany, Nonlinear dynamics in the cournot duopoly game with heterogeneous players, Physica A: Statistical Mechanics and its Applications 320 (2003) 512–524. [18] H. N. Agiza, A. A. Elsadany, Chaotic dynamics in nonlinear duopoly game with heterogeneous players, Applied Mathematics and Computation 149 (3) (2004) 843–860. [19] N. Angelini, R. Dieci, F. Nardini, Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules, Mathematics and Computers in Simulation 79 (10) (2009) 3179–3196. [20] F. Tramontana, Heterogeneous duopoly with isoelastic demand function, Economic Modelling 27 (1) (2010) 350–357. [21] F. Cavalli, A. Naimzada, Nonlinear dynamics and speed of convergence of an heterogeneous cournot duopoly with a local monopolistic approach versus a best-response approach with different degrees of rationality, forthcoming (2014).
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[22] F. Cavalli, A. Naimzada, F. Tramontana, Nonlinear dynamics and global analysis of an heterogeneous cournot duopoly with a local monopolistic approach versus a gradient rule with endogenuous reactivity., forthcoming (2014). [23] A. K. Naimzada, F. Tramontana, Controlling chaos through local knowledge, Chaos, Solitons and Fractals 42 (4) (2009) 2439–2449.
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