Economic Modelling 39 (2014) 204-212 Contents lists available at ScienceDirect Economic Modelling journal homepage: www.elsevier.com/locate/ecmod Analysis on the dynamics of a Cournot investment game with bounded rationality (D CrossMark Zhanwen Ding *, Qiao Wang, Shumin Jiang Faculty of Science, jiangsu University, Zhenjiang 212013, PR China ARTICLE INFO Article history: Accepted 21 February 2014 Available online 28 March 2014 Keywords: Cournot game Bounded rationality Investment Dynamic system Chaos control ABSTRACT In this work, a dynamic system of investment game played by two firms with bounded rationality is proposed. It is assumed that each firm in any period makes a strategy for investment and uses local knowledge to make investment strategy according to the marginal profit observed in the previous period. Theoretic work is done on the existence of equilibrium solutions, the instability of the boundary equilibriums and the stability conditions of the interior equilibrium. Numerical simulations are used to provide experimented evidence for the complicated behaviors of the system evolution. It is observed that the equilibrium of the system can loose stability via flip bifurcation or Neimark-Sacher bifurcation and time-delayed feedback control can be used to stabilize the chaotic behaviors of the system. © 2014 Elsevier B.V. All rights reserved. 1. Introduction Cournot (1838) introduced earliest the mathematical model which describes production competitions in an oligopolistic market. In a classical Cournot model each participant uses a naive expectation to guess that opponents' output remains at the same level as in the previous period and adopts an output strategy which solves the corresponding profit maximization problem. Since then, a great number of literatures have been devoted to enrich and expand the Cournot oligopoly game theory. Much work has paid attention to the stability and the complex phenomena in a dynamical Cournot game with this kind of naive expectation [e.g., (Teocharis, 1960; Puu, 1991; Puu, 1996; Puu, 1998; Agiza, 1998; Kopel, 1996; Ahmed & Agiza, 1998; Agliari et al., 2000; Rosser, 2002)]. As a more sophisticated kind of learning rule with respect to naive expectations, adaptive expectations or adaptive adjustments have been proposed in other dynamical models [e.g., (Okuguchi, 1970; Bischi & Kopel, 2001; Agiza et al., 1999; Rassenti et al., 2000; Szidarovszky & Okuguchi, 1988)]. In recent years, many researchers have paid attention to a kind of bounded rationality, with which a player (without complete information of the demand function) uses local knowledge to update output by the marginal profit. Bischi and Naimzada (1999) gave a general formula of the dynamical Cournot model with this form of bounded rationality, assuming that producers behave as local profit maximizers in a local adjustment process, Corresponding author. E-mail address: dgzw@ujs.edu.cn (Z. Ding). "where each firm increases its output if it perceives a positive marginal profit and decreases its production if the perceived marginal profit is negative (Bischi & Naimzada, 1999)". Much work has been done on the dynamical Cournot games performed by players with this kind of marginal profit method. The models with homogeneous players (all players are boundedly rational players and use the marginal profit method to adjust strategies) are considered in Agiza et al. (2001), Agiza et al. (2002), Ahmed et al. (2000), and Bischi and Naimzada (1999). Some other work has focused on modeling the system with heterogeneous expectations. Agiza and Elsadany (2003) and Zhang et al. (2007) studied the dynamics of a Cournot duopoly game with one bounded rationality player and one naive player. Agiza and Elsadany (2004) and Dubiel-Teleszynski (2011) considered a duopoly model in which one player has bounded rationality and the other has adaptive expectation. In the model by Fan etal. (2012), there is one player using the marginal profit method and one player adjusting production in terms of the market price in the previous period. Ding et al. (2009) studied the dynamics of a two-team Cournot game with heterogeneous players. In these models for dynamical Cournot game, output is a key variable and each player is able to take any needed output updating for the purpose of local profit maximization; thus it is based on an implicit assumption that all players could provide sufficient quantity of products on the market. However, this implicit assumption may be impractical in an economy market where investment plays the most important role. For instance, in an emerging industry with immature development (e.g., a new energy market), it is unlikely for a firm to hold productivity large enough due to its lack of investment accumulation. As a strategic http://dx.doi.org/l 0.1016/j.econmod.2014.02.030 0264-9993/© 2014 Elsevier B.V. All rights reserved. Z. Ding et al. / Economic Modelling 39 (2014) 204-212 205 behavior in these economic activities, investment accumulation plays a significant role in achieving a good production level. Moreover, we know that even in a mature industry, the production capacity of a firm is greatly dependent on its large-scale investment stock. Only when the investment comes up to a certain level can a firm provide as much goods as the market demands. Therefore, during the developing period of an infant industry, the competition among producers lies mainly in their investment strategy. To obtain a competitive market share and get superiority over opponents, producers must consider their investment strategies in successive periods. The main purpose of our work is to formulate a novel model, which puts investment decision as a substitute for output adjustment into the dynamical Cournot game. In our model, all producers are also assumed to have bounded rationality and make their investment decisions in line with the marginal profit in the previous period. That is to say, each firm will increase its investment if it perceives a positive marginal profit and decrease its investment if the perceived marginal profit is negative. It is analyzed as to how this novel dynamical Cournot game, in a local adjustment process, evolves to equilibrium or exhibits complicated dynamic behaviors. This article is organized as the following. In Section 2, we model the dynamical investment game played by players with bounded rationality. In Section 3, we discuss the existence and local stability of the equilibrium points for the system. In Section 4, we show the dynamic features of this system with numerical simulations, including bifurcation diagram, phase portrait, stable region and sensitive dependence on initial conditions. In Section 5, time-delayed feedback control is used to stabilize the chaotic behaviors of the system. 2. The model Agiza et al., 2002; Agliari et al., 2000; Ahmed et al., 2000; Bischi & Naimzada, 1999; Rassenti et al., 2000): CiiQiit)) = ctqt(t), 1 = 1,2, (4) where c\ and c2 are both positive. With all the assumptions above, we get firm i's profit in period t as follows: 7T1-(x1(t),x2(t)) = qi(t)p(t)-Q(qi(t))-if(0 iB- BjCjWK^t-V+Xj -bBtBj(0Kt(t-1) + x,-(t)) (öKj(t-l) + xj(t)) -Mt) (aß t-Btq)(0Kt(t-1) + xi(t))-bB2(9Ki(t-\) +xi(t))2 ijtj, i,j = l,2. (5) By differentiating 7r,(xi(t),x2(t)) (i = 1,2), we obtain firm i's marginal profit with respect to its investment in period t(i = 1,2, respectively): 2(t) = 9rr2(xi(t),x2(t)) : aßn-1- dx2(t) -öß1ß2(e/c1(t-i) + x1(t)) (6b) -2öß^(e/c2(t-i) + x2(t)). Our work focuses on firms' investment competition rather than their output competition. We pay attention to a duopoly investment game, where producers' investment choices are substituted for their output decisions discussed in classic Cournot games. We consider a competition between two firms (players), labeled by i = 1,2, producing homogeneous goods. The strategy of each firm is to choose an investment in every period. Both players make their decisions in discrete periods t = 0,1,2 -. We write /<",-( t — 1) for firm i's capital stock in period t — 1, and x,(t) for its investment in period t. We pay our attention mainly to relatively long economic periods and assume that the previous accumulated capital Kt(t — 1), after depreciating in a period, keeps a residual value S/C,(t — 1) (0 < 9 < 1, identical for both firms). Then the capital stock /<",(t) for firm i is formulated as Ki(t) = 0Ki(t--[)+xi(t), i = 1,2. (1) We suppose that for firm i, its totally accumulated capital J(,(t) determines its potential production capacity in period t and it produces at full capacity to make its output q,(t). Namely, q,(t) is assumed to be a function of /<",( t). For simplicity, we consider a linear form of output function: q,(t) = Bj/C,(t), where B, is a positive constant. From Eq. (1) we have We suppose both players have bounded rationality and use the marginal profit method to update their investment strategy in the next period, as supposed in the existing work on the classical Cournot games for output competition [e.g., (Agiza et al., 2001; Agiza et al., 2002; Agiza & Elsadany, 2003; Agiza & Elsadany, 2004; Ahmed et al., 2000; Bischi & Naimzada, 1999; Ding et al., 2009; Dubiel-Teleszynski, 2011; Fan et al., 2012; Zhang et al., 2007)]. It means that each firm will increase its investment flow in period t + 1 if the marginal profit in the current period t is positive; otherwise the firm will decrease its investment. Then the investment adjustment mechanism for player i can be modeled as: xt(t + 1) = xt(t) + a,.(x,.(t))^(t), i = 1,2, (7) where a;(x;(t)) is a positive function which gives the extent of firm i's investment variation based on its marginal profit. For simplicity, we also put the function a,-(Xj(t)) in a linear form (Agiza & Elsadany, 2003; Agiza & Elsadany, 2004; Agiza et al., 2001; Agiza et al., 2002; Bischi & Naimzada, 1999; Ding et al., 2009; Dubiel-Teleszynski, 2011; Fan et al., 2012; Zhang et al., 2007): a;(x;(t)) = a;X;(t), where at is a positive constant representing the relative adjustment speed. Then the dynamics (7) takes its form as: qt(t)=Bt(0Kt(t-1)+xt(t)), 1 = 1,2. (2) xi(t + \)=xi(t) + aixi(t)i(t), i = 1,2. (8) For the price in the market, we consider a linear inverse demand function (Agiza & Elsadany, 2003; Agiza & Elsadany, 2004; Agiza et al., 2001; Bischi & Naimzada, 1999; Ding et al., 2009; Dubiel-Teleszynski, 2011; Rassenti et al., 2000; Szidarovszky & Okuguchi, 1988; Zhang et al., 2007): p(t) = a-bQ(t), (3) where a > 0,0 > 0, and 0_(t) = qx(t) + q2(t) is the total supply by both firms. We also suppose that the production cost function of each firm takes a linear form (Agiza & Elsadany, 2003; Agiza & Elsadany, 2004; From Eqs. (1), (6a)-(6b) and (8), we obtain a four-dimensional discrete dynamic system: Xi (t + 1) = Xi (t) + axxx (t)(aßi -1 -ßjCi -2ößi (Ö/Ci (t-1) + x1(t))-öß1ß2(e/c2(t-i)+x2(t))), x2(t + 1) = x2(t) + a2x2(t)(aB2-1-B2c2-bB1B2(6K1(t-l) + x1(t))-2bß2>K2(t-l)+x2(t))), K1(t) = e/C1(t-l)+x1(t), [K2(t) = 0K2(t-1)+x2(t). (9) 206 Z. Ding et all Economic Modelling 39(2014) 204-212 If we denote K,-(t - 1) by /,(t) (and hence /C,(t) by /,(t + 1)), i = 1,2, then we can rewrite system (9) as the following standard dynamics In the following, all the nonnegativity conditions (12a)-(12d) are assumed. xx (t + 1) = *i (t) + ajXi (t)(oB! -1 -BjCi -2bB2 (ffli (t) + Xj (t)) -bBiB2(0/2(t)+x2(t))), J x2(t + l)=x2(t) + a2x2(t)(aB2-l-B2c2-2faB^(e/2(t)+x2(t)) (10) -faB1B2(ffl1(t)+x1(t))), /1(t + l) = ffl1(t)+x1(t), [/2(t + l) = 0/2(t)+x2(t). System (10) describes a duopoly game played by two boundedly rational players making decision in a process of dynamical investment. In the following sections, we are to investigate the dynamical properties of this model. 3. Equilibrium points and stability Letx,(t + 1) = x,(t) and /,(t + 1) = /,(t) (i = 1,2) in system (10), then we get 'x1(0(aB1-l-B1c1-2bB?(ffl1(0+x1(0)-M1B2(e/2(0+x2(t)))=0, x2{t)(aB2-l-B2c2-2bBl{ei2{t) +x2{t))-bB1B2{0I1{t) +x1{t))^ =0, (0-1)/! (0+^(0=0, (0-l)/2(t)+x2(t)=O. (11) Solving equations in Eq. (11), we obtain four equilibrium states of dynamics (Eq. (10)), which are listed as follows: £0 = (0,0,0,0), Q (l-8)(aB2-l-B2c2) Q aB2-l-B2c2 2bB\ 2bB\ (l-e)(aßi-l-ßiCi) 0 aßi-1-ßiCi 0 2bB\ 2bB\ e* = (x;,x2,/;,/2), where ?[t _(1-e)(BlB2(a + c2-2cl)+Bl-2B2) 1 3bB\B2 . (l-e)(ßiß2(a + ci-2c2)+ß2-2ßi) 2 3bBlBj , ^BiB2(a + c2-2ci)+Bi-2B2 1 ~ 3bBJB2 , ^BiB2(a + Ci-2c2)+B2-2Bi 2 ~ 3faBifi2 £0,£i and e2 are all boundary equilibriums and e" is a unique interior equilibrium. In order to make these equilibrium points have economic meaning, we only consider the nonnegative cases. Since b, B\, B2 and 0 are positive parameters, £i,£2 and £" are all positive provided that aßi-1-ßiCi > 0, aß2-l-ß2c2 > 0, ßiß2(a + c2-2ci) +ßi-2ß2 > 0, ßiß2(a + Ci-2c2) +ß2-2ßi>0. (12a) (12b) (12c) (12d) 3.1. Stability of the boundary equilibriums To investigate the local stability of an equilibrium (x!,x2,/!,/2) of system (10), we work out its Jacobian matrix J: y(xi,x2,/i,/2) I Ax -ößiß2aiXi -2eöß2aiX1 -ößjß2a2x2 A2 —0bBxB2a2x2 1 0 9 V 0 1 0 where -0bßiß2aiX, -2ööß2a2x2 0 (13) = 1 +ai(aBi-l-BiCi)-2aibB2(0/i + 2xi)-faaiBiB2(e/2 +x2), A2 = 1 +a2(aB2-l-B2c2)-2a2faB2(e/2+ 2x2)-faa2BiB2(e/i +x1). An equilibrium (X1.X2.W2) w'" be locally asymptotically stable if all the eigenvalues (real or complex) of the Jacobian matrix J{x\,x2,h,I2) lie inside the unit disk, i.e. |A| < 1 holds for any eigenvalue A. of_/(xi,x2, /i,/2). An equilibrium (x-1.X2.W2) will be unstable if there is an eigenvalue A. ofy(xi,x2,/i,/2) such that |\| > 1. Proposition 1. The boundary equilibrium e0 is an unstable equilibrium. Proof. Taking the expression of the equilibrium £0 into Eq. (13), we get the Jacobian matrix at £0 as the following: ](ßo) (\ +al(aBl-1-Blcl) 0 0 0\ 0 1+a2(aß2-l-ß2c2) 0 0 1 0 e 0 0 10 6 J which has four eigenvalues: \j = \2 = 9, A3 = 1 +oii(aBi— 1— BiCi),A4 = 1 + a2(aB2 — 1— B2c2). From the nonnegativity conditions (12a)-(12b) and the positivityofthe parameter a„ it follows that |A34| > 1. So the equilibrium £0 is unstable. Proposition 2. The boundary equilibriums £1 and e2 are both unstable. Proof. At the boundary equilibrium point £1, the Jacobian matrix (Eq. (13)) is given by 1 9-\)a2BxV 0 V 2B2 1 0 1 + 1 -l)a,V 0 -l)a2BiV 2B, 0 0 -l)a2V 0 I where U = a1{B1B2{a + c2 - 2cO + Bi - 2B2),V = aB2 - 1 - B2c2. By simple calculation, we get four eigenvalues of the matrix J(£i): \i =0, A? = 1 + a, (B,B2 (a + c2 -2q) + B, -2B2) 2B, A3i4=l(l+0 + a2( -l)(aß2-l-ß2c2) ± V-40 + (1 + 0 + a2(0-l)(aB2-l-B2c2))2). According to the inequality (Eq. (12c)) and the condition that B2 and ai are both positive, we see that A2 > 1 and hence conclude that the equilibrium £1 is unstable. A similar approach shows that £2 is unstable too. Z. Ding et al. / Economic Modelling 39 (2014) 204-212 207 A) 0 = 0.35 B) e = 0.5 1.15 I-1- 0.95 0.75 0.55 0.35 0.15 -i-r- C) e = 0.69 0.65 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 0.55 0.45 0.35 0.25 1.95 h-r Fig. 1. Bifurcation diagrams with respect to the adjustment rate 0 must hold. In addition, Det{Mf) = 1 ± p4 = 1 ± 02 > 0 also holds since 0 < 0 < 1. Consequently, we conclude that the interior equilibrium point E" of system (10) is asymptotically locally stable if it meets the following conditions P(-l) = l-pj +p2-p3 +p4>0, (14a) where X = aB2 + B2c2 - 2B2cx + 1, Y = aB2 + B2cx - 2B2c2 - 2. For all the roots of the polynomial P(\) (the eigenvalues of the Jacobian matrixJ{E")) to lie inside the unit disk, Schur-Cohn Criterion [e.g., (Elaydi, 2005)] gives the necessary and sufficient conditions as: (i) P(1)>0; (ii) (-1)4P(-1)>0; (iii) The determinants of the lxl matrices Mf and the 3x3 matrices Mf are all positive, where Mf = l±p4, M3 0 0 1 0 P2 Pi 1 0 0 p4 I 0 p4 p3 Va Pí p2 Det^Mi)>0,Det(M3 )>0, (14b) where Det(M) represents the determinant of the matrix M. 4. Numerical simulation In this section, we show by numerical simulations how the system evolves under different levels of parameters, especially of the capital residual rate 0 and the adjustment speed a. In all the numerical simulations, the other parameters are fixed: a = 5, b = 1, Ci = 0.3, c2 = 0.5, B-i = 0.6 and B2 = 0.8. For three cases of the capital residual rate 0, Fig. 1 is about bifurcation diagrams of system (10) with respect to the adjustment rate 04 while Z. Ding et al. / Economic Modelling 39 (2014) 204-212 A) 9=0.35 B) 9=0.5 209 0 2 4 6 8 10 Fig. 3. Stability region in ( 0 is a controlling coefficient. Then the controlled system is given by It is easy to see that the new system (15) has the same equilibriums as system (10) and it takes the following equivalent form: x1(t + 1)=x1 (t) + y-j^«!*! (t)(aß! -1 -ß! q -2bB] (91, (t) + xx (t)) -bBxB2{9l2{t)+x2{t))), {x2(t + \)=x2(t) + a2x2{t) {aB2 -1 -ß2c2 -2bß2 (9\2 (t) + x2 (t)) -M1B2(0/1(t)+x1(t))), /1(t + l) = 0/1(t)+x1(t), L/2(t + i) = e/2(t) + x2(t). (16) Thejacobian matrix of the controlled system (16) is given by *i (t + 1) = *i (t) + «1*1 (t) (oßj -1 -ßi Ci - 2bB\ (0/i (t) + *i (t)) -ößiß2(0/2(t) +x2(t))) + k(x(t)-x(t + 1)), l x2(t + l)=x2(t) + a2x2(t)(aß2-l-ß2c2-2öß^(0/2(t)+x2(t)) (15) -öß1ß2(0/1(t)+x1(t))), í1(t + i) = e/1(t) + x1(t), [i2(t + í) = ei2(t) + x2(t). J(XUX2,IUI2) 1 + k -bBxB2a2x2 1 0 bßiß2oiiXi 20bßiO!iXi 0bßiß2oiiXi \ 1+fe 1 + a2A2 0 1 1+fe -dbBxB2a2x2 0 0 1+fe -20bß2a2x2 0 (17) Z. Ding et al. / Economic Modelling 39 (2014) 204-212 211 A) a, = 1.98 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.5 B) a, = 9.305 0.25 0.687 0.692 0.697 0.702 0.707 0.712 0.717 0.722 0.727 0.687 0.692 0.697 0.702 0.707 0.712 0.717 0.722 0.727 Fig. 6. Bifurcation diagram with respect to the controlling factor k. In Fig. 6, it is actually observed that with the control coefficient fc increasing, the system gradually gets out of chaos and periodic windows and achieves to stability when k > 0.2005. For k = 0.25, Fig. 7 shows the stable behaviors of the orbits of the controlled system beginning from the initial state (x1(0),x2(0),Ji(0),J2(0)) = (0.94,0.87,1.56.1.44). 6. Conclusion In this work we have taken into consideration firms' investment decisions as substitute for the output choices considered by the existing work on classic Cournot games. We have formulated a novel Cournot form of investment game played by two players with bounded rationality. The main idea in our model is that each firm's decision is to choose its investment in each period according to the marginal profit observed from the previous period. We have established a corresponding Fig. 5. Bifurcation diagrams with respect to the residual rate 6. where Ai = aBi - 1 - - IbBftOh + 2x0 - bB^B2(012 + x2) and A2 = aB2 - 1 - B2c2 - 2bB2{9I2 + 2x2) - bBxB2{9h + *i). As has been shown in Fig. 1(B), chaotic behavior of system (10) occurs when all the model parameters take their values as {a,b,B1,B2,c1,c2,a1,a2,0) = (5,1,0.6,0.8,0.3,0.5,1.426,2.2,0.5). Using this group of parameters values, we obtain the Jacobian matrix (Eq. (17)) at the interior equilibrium as the following ,x2,/i, fM 1.22408 V 1+fc 0.91872 1 0 0.643411 1+fc -2.02368 0 1 0.482558 1+fc -0.45936 0.5 0 0.321706 1+fc -1.22496 0 0.5 (18) From the stability conditions (14a)-(14b), we get that all the eigenvalues of the matrix (18) will lie inside the unit disk provided that fc > 0.2005. That is, when k > 0.2005 the controlled system (16) will be asymptotically locally stable. Fig. 7. Stability of equilibrium with k = 0.25. 212 Z. Ding et all Economic Modelling 39(2014) 204-212 dynamics of players' investment adjustment and done a detailed dynamic analysis for it. There are three boundary equilibriums and a unique interior equilibrium in this system. We have shown the instability of the boundary equilibriums and found the conditions for local stability of the interior equilibrium by Schur-Cohn Criterion. We have made similar numerical simulations for the system evolution as done in other existing work on classic Cournot games for output competition, including bifurcation diagrams, phase portraits, stable regions and sensitivity to initial state. It is shown that a relatively high residual rate (or low depreciation rate) can strengthen the system stability. It is observed that the equilibrium of the system may loose stability via different bifurcations, either flip bifurcation or Neimark-Sacker bifurcation. It is also shown that time-delayed feedback control can be used to stabilize the chaotic behaviors of the system. Acknowledgements We are very grateful to the anonymous referees for the valuable comments and suggestions that greatly help us to improve the paper. 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