DISCRETE-TIME DYNAMICS OF AN

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1 Chapter 1 DISCRETE-TIME DYNAMICS OF AN OLIGOPOLY MODEL WITH DIFFERENTIATED GOODS K. Andriopoulos, T. Bountis and S. Dimas * Department of Mathematics, University of Patras, Patras, GR-26500, Greece Abstract We investigate the dynamics of a competitive market and examine the dependence of our results on the degree of product differentiation. The duopoly employed to illustrate by way of example our findings consists of a firm following bounded rational expectations and a firm with so-called naive expectations. We present our conclusions by inspecting the graphs of the profits of each firm as traced out by the evolution of the corresponding dynamical systems. Finally, we pose some suitable models to study the dynamics of a competitive oligopoly with heterogeneous products. 1 Introduction It was recently shown by several authors [1, 2, 3] that the dynamics of oligopolies with identical goods, when studied by certain classes of nonlinear difference equations, lead to some very interesting conclusions concerning the types of Cournot equilibria achieved and the conditions for firm survival in a competitive market. In this Chapter, we follow an approach developed for oligopolies operating in continuous time [4, 5] and introduce the important element of product differentiation within the framework of discrete-time dynamics. Most markets produce differentiated goods. This differentiation is based primarily on the different quality and production cost of the goods, but may also be a consequence of * kand@aegean.gr, bountis@math.upatras.gr, spawn@math.upatras.gr

2 advertisement, of different location of the firms etc. However, it is reasonable to start by considering markets of identical goods as these are generally simpler and the results deduced are more transparent. The question that we address here and study in more detail in [6] is the evolution of a duopoly, which may gradually become a triopoly, quadrupoly etc (as more firms are allowed to enter the competition), selling identical goods in a market of differentiated products. In this Chapter we present a model showing how this is achieved and focus on the resulting dynamics within the economic relevance of our setting. We follow the assumptions of Cournot [7] (this edition is hard to find; see [8] and particularly Chapter 1 for a translation by Bacon) for the behavior of firms: firms actually choose their output level (productivity) first and then the market sets the prices based on a demand curve and the total quantity offered. Cournot used a general function, p = P (q 1 + q 2 ), for the price function, where q 1 and q 2 are the quantities produced by the two firms. The total supply is Q = q 1 + q 2. For many decades economists assumed a linear demand function the inverse of which, the market price, is P (Q) = a bq. Puu introduced the isoelastic price function, P (Q) = 1/Q, which is a direct consequence of the requirement that consumers maximise a Cobb-Douglas type of utility function [9]. This is the price function that we use throughout this Chapter 1. Authors in the current scientific literature (see, for example, [2, 3, 4, 5, 9, 10, 12]) usually assume a (total) cost function of the form C i = c i q i which yields constant marginal costs, = c i, i = 1, n. This is by no means binding; on the contrary, cost functions are in reality different for short and long term production. Obviously, such assumptions, as in [1] for technologies represented by CES functions, increase the complexity of the systems studied and the equations in their vast majority become impossible to solve analytically. Cournot was the first to describe explicitly the equilibrium achieved by firms competing in a market of identical goods as the set of quantities sold for which, holding the quantities of all other firms constant, no firm can obtain a higher profit by choosing a different quantity. This equilibrium is called the Cournot-Nash equilibrium because Nash later unified all such equilibria within a game-theoretical framework. As the profits for each firm are the revenues minus the costs, u i = R i C i, the determination of the Cournot equilibrium is achieved through u i q i = 0, i = 1, n. One obtains a simultaneous system of equations, the solution of which reveals the best-responses for each firm. This system, in fact, represents the reaction functions for each firm. One could therefore say that in this way Cournot introduced the idea of naive expectations, q i = f(q i ). To make things precise we introduce the notation used throughout this Chapter assuming only two firms (a duopoly) with linear cost functions, C i (q i ) = c i q i, and isoelastic price function, p i = 1/(q i + θ i q 3 i ). The parameters θ i represent the degree of product differentiation. If θ 1 = θ 2 = 1 then the products are identical. On the other hand, if θ i = 0 for some i, then firm i is actually considered as a monopolistic firm. When 0 < θ i < 1 for every i = 1, 2 then the duopoly produces and supplies different goods and the firms involved ask for different prices. It is reasonable to claim that these (weight) parameters also reflect the degree to which one firm takes into account the output level of the other. If 1 Yet, note that in [10] we proposed a price function of the form P (Q) = 1/(Q 2 + 1) and expressed the opinion that one should use a price function appropriate to the market under consideration. This, of course, would require the empirical data for that specific market to support such a choice. C i q i

3 θ i is small then it may be argued that firm i is more monopolistic than its rival, or that its product is of greater quality or more rare. In any case the corresponding quantity, q i, will be less and its price, p i, greater. Under these assumptions the profits for each firm are, and the first-order variations give, u i = u i q i = q i q i + θ i q 3 i c i q i, θ i q 3 i (q i + θ i q 3 i ) 2 c i. In the duopoly case, setting these variations equal to zero, we define at equilibrium the so-called reaction functions, R 1 and R 2, by q 1 = R 1 (q 2 ) = q 2 = R 2 (q 1 ) = θ 1 q 2 θ 1 q 2 c 1 θ 2 q 1 θ 2 q 1. (1) c 2 Before any model can be constructed, one has to decide the appropriate time-scale for the corresponding setting. Usually, in oligopoly models, discrete time is assumed. This is mainly because firms are not in a position to change their production levels continuously and time intervals are imperative for decision making and production processes. However continuous-time models do incorporate many important elements of the theory and the resulting dynamics is quite interesting. For an account on the subtle differences in these approaches consult [11]. Regarding discrete-time models, there are 3 well-studied and widely accepted expectation assumptions formulating the ways a firm is anticipated to adjust its output level at future times. These are: 1. The naive expectation of a reaction function (see (1) above), where each firm computes the production level in time t + 1 based on what the firms produced at time t, is written as θ i q 3 i (t) q i (t + 1) = R i (q 3 i (t)) = θ i q 3 i (t). (2) c i 2. The adaptive expectation assumption is actually a slight modification of the naive case 1 above: a firm computes the productivity level at time t+1 by assigning weights to last period s output, q i (t), and its reaction function, R i (q 3 i (t)), as follows: q i (t + 1) = (1 λ i )q i (t) + λ i R i (q 3 i (t)), (3) where λ i [0, 1] is the speed of adjustment of the adaptive firm i.

4 3. The bounded rationality adjustment process is based on the idea that at each time period a firm decides to increase its production if the marginal profit is positive, otherwise it considers it best to decrease the production level. This can be modelled using the map, q i (t + 1) = q i (t) + α i (q i (t)) u i (t) q ( i ) θ i q 3 i (t) = q i (t) + α i (q i (t)) (q i (t) + θ i q 3 i (t)) 2 c i, (4) where α i (q i ) is a positive function which reflects the extent of production variation of firm i following a given profit signal. In most research papers it is assumed that α i (q i ) = α i q i, with α i a constant. However in continuous-time dynamics one often considers the case, α i (q i ) = α i [5, 10]. In this Chapter we study a duopoly with one firm following bounded rational expectations and the other being naive. In fact our choice could have been any of the above mentioned possibilities. Other choices are considered in [13]. 2 From identical to differentiated products In [12] the authors investigate a duopoly selling homogeneous goods. Their main assumptions include a linear demand function, constant marginal costs and expectations that include one firm behaving naively and the other bounded rationally. We summarise their results in Figure 1. Insert Figure 1 about here Our plan is the following: Firstly, we change the price function and from linear we investigate the isoelastic choice. This is already a more complex situation and fortunately the reaction functions can be found explicitly and therefore analytic results can be provided to support the numerical simulations. Next, we consider one firm switching its product status and producing slightly differentiated goods. This is naturally reflected in the demand function as explained in the Introduction. We then examine the effects of the same duopoly by gradually increasing the degree of differentiation among the goods sold. In Figure 2 we plot the equivalent graphs as compared with Figure 1 only this time the isoelastic price function has been adopted. Insert Figure 2 about here Assume now that the first firm, that is the bounded rational one, produces a slightly differentiated product. This means that it employs a price function of the form p 1 = 1/(q 1 + θ 1 q 2 ), whereas the second firm s price function remains p 2 = 1/(q 1 + q 2 ). In Figure 3 we observe that for a wide range of θ 1 -values the two firms reach the Cournot-Nash equilibrium point which is of period two. As the degree of product differentiation tends to zero the expression for the profit of the first firm becomes poorly defined and this is due to the assumption of an isoelastic price function.

5 Insert Figure 3 about here These results demonstrate that the second firm may indeed wish to differentiate its product. However, this is not a simple matter. In [6] we examine in detail the optimal degree of differentiation and only present in Figure 4 a flavor of the dynamics. Insert Figure 4 about here 3 Conclusion In this Chapter we have studied two firms operating in a market. After overviewing the well-known results with a linear price function we applied the isoelastic price function to the duopoly under investigation. As we are interested in markets with differentiated goods we attempted to illustrate the dynamics of the duopoly when a firm begins to produce a slightly heterogeneous product. Yet, it should be pointed out that in reality, a firm wishing to differentiate its product would firstly determine if such a move would indeed increase its profits. Otherwise it is apparent that such a differentiation would not take place because it would not be of the firm s benefit to alter its current status. In [6] we begin with two firms that produce and sell an identical product, ie θ 1 = θ 2 = 1. Obviously the market would evolve until it reached an equilibrium state. Now, one firm computes its profit function and finds the optimal degree of product differentiation such that a product differentiation would increase its profits. The cost function for that firm changes and the product is sold at a different price. The market evolves again until it reaches a new equilibrium state. It is of the best interest for the second firm to do the same. Now both firms compute the optimal degree of differentiation at each time step and the duopoly evolves likewise. Our main concern is the resulting dynamics as compared to, for example, the case of homogeneous products reported in [1]. Acknowledgements KA thanks the organisers of the 6th International Conference on Nonlinear Economic Dynamics, Jönköping, Sweden for the financial support to present the work reflected by this Chapter. The State (Hellenic) Scholarships Foundation is also thanked. We appreciate the useful remarks and suggestions from some of the attendants of that Conference. References [1] Puu, T., & Panchuk, A. (2008). Oligopoly and stability. Chaos, Solitons and Fractals (doi: /j.chaos ). [2] Bischi, G-I., Gallegati, M., & Naimzada, A. (1999). Symmetry-breaking bifurcations and representative firm in dynamic duopoly games. Annals of Operations Research, 89,

6 [3] Elabbasy, E. M., Agiza, H. N., & Elsadany, A. A. (2009). Analysis of nonlinear triopoly game with heterogeneous players. Computers and Mathematics with Applications, 57, [4] Matsumoto, A. & Szidarovszky, F. (2007). Delayed nonlinear Cournot and Bertrand dynamics with product differentiation. Nonlinear Dynamics, Psychology and Life Sciences, 11, [5] Andriopoulos, K. & Bountis, T. (2008). Dynamics of a Duopoly Model with Periodic Driving. American Institute of Physics Conference Proceedings, 1076, [6] Andriopoulos, K., Bountis, T., & Dimas, S. (2009). Transition of a competitive oligopoly from identical to differentiated goods. (In preparation) [7] Cournot, A. A. (1838). Recherches sur les Principes Mathématiques de la Théorie des Richesses. Paris: Hachette. [8] Daughety, A. F. (1988). (Ed.) Cournot oligopoly; Characterization and applications. New York: Cambridge University Press. [9] Puu, T. (1991). Chaos in duopoly pricing. Chaos, Solitons and Fractals, 1, [10] Andriopoulos, K., Bountis, T., & Papadopoulos, N. (2008). Theory of Oligopolies: Dynamics and Stability of Equilibria. Romanian Journal of Applied and Industrial Mathematics, 4 (1), [11] Yousefi, S. & Szidarovszky, F. (2009). A Chapter within this volume. [12] Agiza, H. N. & Elsadany, A. A. (2003). Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. Physica A, 320, [13] Andriopoulos, K. (2009). Mathematical Methods in Microeconomics and Financial Mathematics. Thesis, University of Patras, Patras, Greece.

7 Figure 1: For these graphs we have chosen a = 10, b = 0.5, c 1 = 3 and c 2 = 5 as in [12]. In that paper only the first and fourth graphs are given. The top left graph shows the evolution of the production outputs for the two firms. Note that firm 1 is assumed to be the bounded rational one. The top right and bottom left graphs are drawn for the specific value of α = Obviously the prices are the same for the two firms. However the profit versus time graph shows clearly that the profits for the first firm are greater than those for the second, as expected. The bottom right graph is the usual bifurcation diagram for the quantities, q 1 and q 2 versus time, t. Chaos prevails for all α > 0.4. Figure 2: Again we choose c 1 = 3 and c 2 = 5. The top graphs are derived for α = For α = the transition from a stable equilibrium point to period doubling occurs now at a higher α-value as shown clearly in the bifurcation diagram. More importantly, the second firm, although having greater marginal costs, achieves greater profits every second period (top right graph). Finally observe a phenomenon of chaos recurrence at about α = 0.8, following a window of α-values where stable equilibria exist. Figure 3: Again, c 1 = 3 and c 2 = 5. The bifurcation diagram (bottom right) is sketched for α = 0.6 and shows the output levels, q, versus θ 1. It is clearly seen that as the value for θ 1 drops from 1 to 0 the two firms experience periods of stability and periods of obvious instability. The three other graphs depict the change of q versus t (top left), p versus t (top right) and u versus t (bottom left) for θ 1 = 0.8 (ie within the chaotic region). Once again the second firm achieves greater profits every second period. Figure 4: The bifurcation diagram, q versus θ 1, for α = 0.6 and θ 2 = 0.7 (left) and for α = 0.6 and θ 2 = 0.3 (right). Note that for lower θ 2 -values chaos occurs at larger values of θ 1.