Chaos in adaptive expectation Cournot Puu duopoly model

Chaos in adaptive expectation Cournot Puu duopoly model Varun Pandit 1, Dr. Brahmadeo 2, and Dr. Praveen Kulshreshtha 3 1 Economics, HSS, IIT Kanpur(India) 2 Ex-Professor, Materials Science and Engineering, IIT Kanpur(India) 3 Associate Professor, Economics, HSS, IIT Kanpur(India) 1 varnp@iitk.ac.in 2 bdeo@iitk.ac.in 3 pravk@iitk.ac.in ABSTRACT Cournot model with isoelastic demand function and adaptive expectation of quantities is investigated and deterministic chaos is found for certain parameter values. Emergence of chaotic orbits and periodic windows are observed in bifurcation diagram. In bifurcation diagram the stretching of periodic windows are observed when adaptive weight factors are changed. Largest Lyapunov exponent is calculated and boundedness is proved to verify the existence of chaos. Kaplan-Yorke dimension of attractors are calculated to observe the complexity of chaotic attractor. JEL Classification: C61, C62, D43, D84 Keywords: Chaos, Adaptive expectation, Cournot duopoly, Isoelastic demand Introduction Finite suppliers participate in an economy and compete among themselves giving rise to oligopoly market structure. Oligopoly market structure lies between two extreme market situations, namely monopoly and perfect competition. Suppliers in perfect competition are considered to be Price takers while the monopolists achieve largest market power and reduce social surplus. Oligopoly market structure achieves intermediate market performance. Duopoly is the most studied oligopoly market structure where suppliers possess sufficient market power. Oligopoly market competition has been seen as a game where suppliers choose strategies based on quantity(cournot Oligopoly) or price(bertrand Oligopoly). Cournot Duopoly model, first introduced by Antoine Augustin Cournot in 1838, is a market structure where two firms produce homogeneous goods, donot collude and act rationally to gain the highest market power by strategically competing using quantity of goods. The Nash Equilibrium of the static, simultaneous competition in cournot model i

depends upon the structure of market demand and cost function of participating firms. Linear Market demand structure often simplifies the model giving explicit and unique Nash Equilibrium. One of the Cournot duopoly model, where market demand structure is considered to be iso-elastic, was analyzed by Puu in 1991. Cournot-Puu duopoly model showed existence of multi-periods and aperiodic NE leading to chaotic regime for certain values of parameters. The best response function of dynamical Cournot-Puu duopoly model was established on the naive expectation of quantities. However more reasonable expected quantity for the best response function are achieved using adaptive expectation of quantity. This paper uptakes the adaptive expectation in the best response function of Cournot-Puu duopoly model. It examines the bifurcation of adaptive expectation Cournot-Puu map and verifies existence of chaos in the model. Section 2 of the paper discusses the model, section 3 focuses on Nash Equilibria, section 4 discusses bifurcation of map, section 5 verifies existence of chaos by computing the Lyapunov exponent and section 6 computes the dimension of strange attractor. Model Assume an economy consisting of two firms(1 and 2) and large number of buyers with isoelastic demand function for a particular commodity. The isoelastic inverse demand function for the commodity is given as: P t = 1 Q 1,t + Q 2,t (1) where P t denotes the price of commodity for period t, Q 1,t and Q 2,t denote the quantities supplied by firm 1 and firm 2, respectively, in period t. The cost function is assumed to have a constant marginal cost for both firms (c i ;i = 1,2) with no fixed cost. Each firm rationally produces quantity Q i,t (i = 1,2) such that the respective profits are maximized for the period. i.e. maxπ i,t = P t Q i,t (Q i,t c i ) Now the i th firm expects the other firm to produce Q e i,t during period t (henceforth i denotes the other firm). First order condition of profit maximization gives following best response function of i th firm Q i,t = BRF i,t = max { Q e i,t c i Q e i,t,0 } (2) The best response function of i th firm is shown in graph 1

2.5 2 Q e 2,t c 1 Q e 2,t BRF1,t 1.5 1 0.5 0 0 2 4 6 8 10 12 14 Q e 2,t at c 1 = 0.1 Figure 1. Best Response Function of Firm 1 at its marginal cost 0.1 Units Adaptive expectation of quantity Hommes(1994) used adaptive expectation in price in nonlinear demand-supply setting, investigated the equilibrium and proved the existence of chaos in nonlinear set up. In this paper we have introduced adaptive expectation in quantities in the best response function of firms and observed variations due to change in expectation weight. As the quantity expected, Q e i,t, follows adaptive expectation, the expected quantity is given by Q e i,t = (1 λ i )Q e i,t 1 + λ iq i,t 1 (3) where λ i is the adaptive weight factor given to the actual quantity produced in previous period. Replacing the actual quantity produced, Q i,t 1 in equation (3) from equation (2), we get following set of difference equation in space of expected quantities. ( Q e 1,t = (1 λ 1 )Q e 1,t 1 + λ Q e ) 2,t 1 1max Q e 2,t 1 c,0 1 ( Q e 2,t = (1 λ 2 )Q e 2,t 1 + λ Q e ) 1,t 1 2max Q e 1,t 1 c,0 2 (4) (5) Above set of equations reflects the evolution of expected prices of both firms in

the expected price plane. Correspondingly, by substituting from equation (4) to equation(2), we can get the evolution of actual prices of both firms in the actual price plane. Nash Equilibrium Puu calculated the Nash equilibrium for the model with naive expectation. Nash equilibrium of a game appears when each player does not have any incentive to unilaterally deviate from the equilibrium strategy. In this model, Nash equilibrium arises when both firms simultaneously and rationally choose their respective quantities in expected quantity space (Q e 1,Qe 2 ), such that Q i = Q e i,t = Q i,t 1. The equilibrium quantity thus produced, same as equilibrium expected quantity, is given by Q i = c i ( ci + c i ) 2 i = 1,2 (6) Due to nonlinearity term in map(4), the above Nash Equilibrium does not remain globally stable. To analyze the stability of Nash Equilibrium, the linearized Jacobian[ matrix, J, is calculated about equilibrium points (6): ( (1 λ1 ) λ c2 c )] 1 1 2c J= ( 1 λ c1 c ) 2 2 2c 2 (1 λ 2 ) Following Routh Hurwitz criteria for two-dimensional linearized Jacobian, following conditions arises for the stability: 1. det(j) tr(j) 1 2. det(j) tr(j) 1 3. det(j) < 1 To analytically obtain the stability condition we assume symmetric expectation, i.e, λ 1 = λ 2 = λ and define c r := c 2 c 1. Thus parameter space reduces to (c r,λ). As the first two stability conditions are always met by Jacobian, J, the equilibrium quantity (5) loses stability when det(j) = 1 i.e. c r = ( 4 ( λ 1) ± 2 2 ) 2 ( λ 2 ) λ For c r > c 2 r, equilibrium (6) is unstable. It can be shown that at c r, period-2 equilibrium points come into existence. Thereby making c r a bifurcation point. Further change of parameter leads to rapid period doubling cascade. This leads to the loss of period-1 equilibrium point s stability and multi-period orbit comes into existence. The flexible stability criteria obtained for adaptive expectation is consistent with the work of Hommes(1994), where equilibrium had stringent stability for naive expectation and flexible stability for adaptive expectation.

λ 1 0.1716 5.8284 0.9 0.1484 6.7405 0.8 0.1270 7.8730 0.7 0.1073 9.3213 0.6 0.0889 11.2444 0.5 0.0718 13.9282 0.4 0.0557 17.9443 0.3 0.0406 24.6261 0.2 0.0263 37.9737 0.1 0.0128 77.9872 c 1 r Table 1. Bifurcation Point(Loss of period-1 NE stability) at different values of λ c 2 r Bifurcation Diagram Definition: Bifurcations in discrete dynamical systems are defined as the change in the topological property of phase portrait due to change in parameters. The bifurcation diagram for different values of λ are shown in diagram 1(a)- 1(d). It is observed that as λ increases, the euclidean distance increases between the bifurcation point c r and the limit point of period doubling cascade of 2-period window. For any given λ, further change of technical inefficiency ratio (c r ), beyond region of period-2 doubling bifurcation cascade, produces regions of aperiodic orbit and regions with new period windows. It can be observed that stable aperiodic orbits in the model are chaotic, although the acceptance of the proposition requires the proof that aperiodic orbits diverges in bounded space. Chaos Definition: For a given dynamical system, the system exhibits chaos (equivalently, orbits are chaotic) if the dynamical system satisfies following properties: 1. Sensitive dependence on initial condition 2. Dense aperiodic orbits 3. Bounded Sensitive dependence on initial conditions are captured using diverging property of close trajectories. If the bounded system having positive Lyapunov Exponent consists of aperiodic orbits, then the system is chaotic. Analysis of bifurcation diagram reflects the existence of aperiodic orbits in map(4-5).

a)λ = 1 b)λ = 0.9 c)λ = 0.8 d)λ = 0.7 e)λ = 0.6 f)λ = 0.5 g)λ = 0.4 b)λ = 0.3 i)λ = 0.2 j)λ = 0.1 Figure 2. Bifurcation diagram (Q 2 vs c r ): The set of graphs((a)-(j)) show the stretching of parameter range for given topologically equivalent phase portraits. As λ decreases, the range of c r for which period-1 NE is stable increases. With the decrement of λ new periodic windows and chaotic bands emerges

Boundedness Puu(1991) used inalienability condition and obtained the range of parameter for which quantities remain bounded. However, in this proposed model expected quantity space is explored. Using equation(2), we know that real quantities are non-negative and firms can undergo temporary shut down. Also from equation (3), the expected quantity is weighted average of previous period s expected quantity and actual quantity. Clearly, the expected quantities remain bounded if initial expected quantities are wisely chosen. The proof for the same is given in appendix. Lyapunov Exponent Lyapunov exponent characterizes the rate of separation of close trajectories. An orbit having positive Lyapunov exponent asymptotically diverges from all its neighbouring trajectories. Unlike the case of 1-dimensional map, 2-dimensional maps possess two lyapunov exponents. Thus the evolution of quantities in the model may exhibit expansion of neighbourhood volume in one eigen-direction while contraction in other such that overall volume of neighbourhood of an orbit is contracting. The expansion and contraction of volume can be charecterized by their lyapunov exponents while the overall change in neighbourhood volume of a trajectory is estimated as the average of determinant of Jacobian along the orbit. For difference equations, sum of lyapunov exponents determine the (exponential)rate of expansion/contraction of neighbourhodd area. As the change in area in linear transformations are determined by determinant of transformation matrix(jacobian matrix in case of nonlinear systems), the average rate of change of area of a map is calculated using the average of determinant over an orbit. Numerically calculated largest lyapunov exponent for some of the orbits are shown in table. As the lyapunov exponent are positive for certain parameter values, the orbits are thus chaotic. Attractor Definition: Attractors are invariant sets which cannot be further decomposed into attractors with separated basin of attraction. These are geometrical objects with certain well defined properties. Attractors consisting of periodic points (1-period, 2-period,...) and aperiodic attractors are found in some non-linear maps such as Logistic Map and Henon Map. Strange attractor is an attractor with fractal dimension. To estimate the complexity of map, we calculated the Kaplan-Yorke Dimension for chaotic attractors. Kaplan and Yorke (1979) conjectured that Haursdorff dimension of a strange attractor can be estimated by the dimension calculated from its lyapunov exponent. Definition: For a given map, if there exist possible k + 1 lyapunov characteristic exponents such that the sum of largest k lyapunov exponents is non-negative, then

λ c r Largest Lyapunov Exponent Kaplan-Yorke Dimension 0.9 7.5 0.045 1.19 0.9 8.0 0.017 1.09 0.9 8.5 0.079 1.27 0.8 7.9 0.016 1.05 0.8 10.7 0.014 1.03 0.7 14.3 0.056 1.13 Table 2. Largest Lyapunov Exponent and Kaplan Yorke dimension of attractors from the model Kaplan Yorke conjecture states that the fractal dimension of the attractor is given by: D KY = k + ε 1+ε 2..+ε k ε k+1 where ε i is the i th largest lyapunov characteristic exponent. Thus the Kaplan-Yorke dimension of chaotic attractor in our model, with one positive lyapunov exponent and contracting neighboorhood volume, can be estimated as: D KY = 1 + ε 1 ln(det(j)) ε 1 where ln(det(j)) is averaged over the orbit. The estimated value of Kaplan-Yorke dimension for attractors in given model are mentioned in table(2) while some of the attractors are shown in figure (3). As the fractal dimension of a geometrical object characterizes the complexity embodied in its nature, the dimensions obtained for attractors show variability in complexity of evolution of quantity. Thus the quantity evolution in Cournot-Puu adaptive model may possess sensitive dependence on measurement of their initial values. Control In deterministic model, chaotic systems do not allow long term predictions and therefore, control of chaotic models is brought under investigation to allow models to sustain along the predictable periodic trajectories in it. However in the adaptive Cournot-Puu model, one-period unstable periodic orbit remain extant but not embedded in chaotic attractor. Due to unavailability of analytical UPO embedded in chaotic attractor, OGY and DFC control methods fail to control the chaotic dynamics. Conclusion Puu(1991) showed that chaotic behavior arises in Cournot competition with adaptive expectation. This paper extends the result by investigating the dynamic be-

Figure 3. Chaotic attractor at λ = 0.8, c r = 7.9 havior of Cournot-Puu duopoly model with different adaptive weight factors. The bifurcation diagram showed existence of aperiodic orbits and expansion of periodic windows when weights on actual quantity are changed. Chaotic orbit emerges prior to emergence of new periodic window. Abrupt change from one-period to aperiodic orbit (rapid period doubling cascade) also lead to chaotic trajectory. So any attempt to predict the dynamic behavior of duopoly firms will be sensitive to the precision in measurement of quantities. For very large gap in cost efficiency parameter, the firms will undergo three period oscillation and firms will shut down at every third period. However for any intermediate value of cost efficiency the firms may undergo stable finite orbit cycle or may remain oscillating in stable aperiodic trajectory. It has been shown that chaotic attractors have different dimensions for different parameter values. However causal parameter of such complexity variation can further be studied. Reference Puu, Tönu. Chaos in duopoly pricing. Chaos, Solitons & Fractals 1.6 (1991): 573-581 Hommes, Cars H. Dynamics of the cobweb model with adaptive expectations and nonlinear supply and demand. Journal of Economic Behavior & Organization 24.3 (1994): 315-335. Bischi, Gian Italo, and Michael Kopel. Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior & Organization 46.1 (2001): 73-100 Sonis M. 1997. Linear Bifurcation Analysis with Applications to Relative Socio-Spatial Dynamics. Discrete Dynamics in Nature and Society, vol. 1, 45-56. Schuster, Heinz Georg, and Wolfram Just. Deterministic chaos: an introduction. John Wiley & Sons, 2006.

Sandri, Marco. Numerical calculation of Lyapunov exponents. Mathematica Journal 6.3 (1996): 78-84. Sprott, Julien C. Numerical calculation of largest Lyapunov exponent. URL http://sprott. physics. wisc. edu/chaos/lyapexp. htm (2004). Frederickson, Paul, et al. The Liapunov dimension of strange attractors. Journal of Differential Equations 49.2 (1983): 185-207. Agiza, H. N. On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons & Fractals 10.11 (1999): 1909-1916. Matsumoto, Akio. Controlling the cournot-nash chaos. Journal of optimization theory and applications 128.2 (2006): 379-392. Iwaszczuk, N., and I. Kavalets. Some features of application the delayed feedback controlmethod to Cournot-Puu duopoly model. ECONTECHMOD: an international quarterly journal on economics of technology and modelling processes 2 (2013): 29-37. Appendix A.1: Boundedness of Map At λ = 1, the map is bounded for c r (4/25,25/4) (Puu,1991). To Prove: For λ (0,1), if we choose any finite initial expected quantity then the map (4) and (5) remain bounded. Clearly, (0,0) is lower bound of expected quantities. The proof for upper bound follows. Proof(By Principle of Mathematical Induction): For any finitely chosen quantity Q e 1,0 and Qe 2,0, we have Qe 1,1 < 1 c 1 + 1 c 2 and Q e 2,1 < 1 c 1 + 1 c 2. Assume Q e 1,k < c 1 1 + c 1 2 and Q e 2,k < c 1 1 + c 1 2. Now, ( Q e 1,k+1 = (1 λ)q e 1,k + λmax Q e ) 2,k c 1 Q e 2,k,0 < (1 λ) ( 1 c1 + c 1 ) ( Q e ) 2 + λmax 2,k c 1 Q e 2,k,0 Q e 2,k c 1 Q e 2,k takes maximum value of 1 4c1 at Q e 2,k = 4c 1 1. Thus, Q e 1,k+1 < (1 λ) ( 1 c1 + 1 c 2 ) + λ 4c1 (1 λ) ( 1 c1 + 1 c 2 ) + λ c1 = (1 λ) c 2 + 1 c 1 < 1 c 2 + 1 c 1 Thus, by principle of mathematical induction, firm 1 s expected quantity, and similarly firm 2 s expected quantity, remain bounded.

A.2: Numerical calculation of maximum Lyapunov exponent and Kaplan- Yorke Dimension The method of calculating largest Lyapunov exponent and Kaplan-Yorke dimension of Henon Map has been explained by Sprott(2004). The steps to calculate largest LE and KY dimension for map (4-5) follows: 1. Take any initial set of quantity in the basin of attraction 2. Iterate the map so that trajectory is on the attractor at Q 0,t 3. Take a close nearby quantities, Q n,t separated by small distance δ 0 from Q 0,t 4. Iterate the two nearby quantity vectors in map to get the next iterates, f ( Q) and compute the distance δ 1 5. Calculate log δ 1 δ 0. 6. Calculate the determinant of Jacobain evaluated at Q 0,t and take its logarithm. 7. Find the next normalized neighbourhodd point Q n,t+1 using following transformation: Q n,t+1 = Q 0,t+1 + δ 0 ( f ( Q n,t ) Q 0,t+1 ) δ 1 8. Repeat step 4-7 and take arithmetic mean of step 5 with large number of orbit points to get largest Lyapunov exponent. Take arithmetic mean of values of step 6 to get the logarithm of exponential expansion/contraction. Using the above calculated values, the Kaplan Yorke dimension of chaotic attractor is evaluated as: D KY = 1 + ε 1 ln(det(j)) ε 1 where ln(det(j)) is averaged over the orbit and ε 1 is maximum Lyapunov exponent.