Chapter 1

DISCRETE-TIME DYNAMICS OF ANOLIGOPOLY MODEL WITHDIFFERENTIATED 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 dependenceof our results on the degree of product differentiation. The duopoly employed to il-lustrate by way of example our findings consists of a firm following bounded rationalexpectations and a firm with so-called ‘naive’ expectations. We present our conclu-sions by inspecting the graphs of the profits of each firm as traced out by the evolutionof the corresponding dynamical systems. Finally, we pose some suitable models tostudy 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 iden-tical goods, when studied by certain classes of nonlinear difference equations, lead to somevery interesting conclusions concerning the types of Cournot equilibria achieved and theconditions for firm survival in a competitive market. In this Chapter, we follow an approachdeveloped for oligopolies operating in continuous time [4, 5] and introduce the importantelement of product differentiation within the framework of discrete-time dynamics.

Most markets produce differentiated goods. This differentiation is based primarily onthe 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

advertisement, of different location of the firms etc. However, it is reasonable to start byconsidering markets of identical goods as these are generally simpler and the results de-duced are more transparent.

The question that we address here and study in more detail in [6] is the evolution of aduopoly, which may gradually become a triopoly, quadrupoly etc (as more firms are allowedto enter the competition), selling identical goods in a market of differentiated products. Inthis Chapter we present a model showing how this is achieved and focus on the resultingdynamics within the economic relevance of our setting.

We follow the assumptions of Cournot [7] (this edition is hard to find; see [8] andparticularly Chapter 1 for a translation by Bacon) for the behavior of firms: firms actuallychoose their output level (productivity) first and then the market sets the prices based on ademand curve and the total quantity offered. Cournot used a general function, p = P (q1 +q2), for the price function, where q1 and q2 are the quantities produced by the two firms.The total supply is Q = q1 + q2. For many decades economists assumed a linear demandfunction the inverse of which, the market price, is P (Q) = a − bQ. Puu introduced theisoelastic price function, P (Q) = 1/Q, which is a direct consequence of the requirementthat consumers maximise a Cobb-Douglas type of utility function [9]. This is the pricefunction that we use throughout this Chapter1. Authors in the current scientific literature(see, for example, [2, 3, 4, 5, 9, 10, 12]) usually assume a (total) cost function of the formCi = ciqi which yields constant marginal costs, ∂Ci

∂qi= ci, i = 1, n. This is by no

means binding; on the contrary, cost functions are in reality different for short and longterm production. Obviously, such assumptions, as in [1] for technologies represented byCES functions, increase the complexity of the systems studied and the equations in theirvast majority become impossible to solve analytically.

Cournot was the first to describe explicitly the equilibrium achieved by firms competingin a market of identical goods as the set of quantities sold for which, holding the quantitiesof 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 suchequilibria within a game-theoretical framework. As the profits for each firm are the revenuesminus the costs, ui = Ri − Ci, the determination of the Cournot equilibrium is achievedthrough ∂ui

∂qi= 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 reactionfunctions for each firm. One could therefore say that in this way Cournot introduced theidea of ‘naive’ expectations, qi = f(Qi).

To make things precise we introduce the notation used throughout this Chapter assum-ing only two firms (a duopoly) with linear cost functions, Ci(qi) = ciqi, and isoelasticprice function, pi = 1/(qi + θiq3−i). The parameters θi represent the degree of productdifferentiation. If θ1 = θ2 = 1 then the products are identical. On the other hand, if θi = 0for some i, then firm i is actually considered as a monopolistic firm. When 0 < θi < 1for every i = 1, 2 then the duopoly produces and supplies different goods and the firmsinvolved ask for different prices. It is reasonable to claim that these (weight) parametersalso reflect the degree to which one firm takes into account the output level of the other. If

1Yet, note that in [10] we proposed a price function of the form P (Q) = 1/(Q2 + 1) and expressed theopinion 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.

θi is small then it may be argued that firm i is more monopolistic than its rival, or that itsproduct is of greater quality or more rare. In any case the corresponding quantity, qi, willbe less and its price, pi, greater.

Under these assumptions the profits for each firm are,

ui =qi

qi + θiq3−i− ciqi,

and the first-order variations give,

∂ui∂qi

=θiq3−i

(qi + θiq3−i)2 − ci.

In the duopoly case, setting these variations equal to zero, we define at equilibrium theso-called reaction functions, R1 and R2, by

q1 = R1(q2) =

√θ1q2c1− θ1q2

q2 = R2(q1) =

√θ2q1c2− θ2q1. (1)

Before any model can be constructed, one has to decide the appropriate time-scale forthe corresponding setting. Usually, in oligopoly models, discrete time is assumed. This ismainly because firms are not in a position to change their production levels continuouslyand time intervals are imperative for decision making and production processes. Howevercontinuous-time models do incorporate many important elements of the theory and the re-sulting dynamics is quite interesting. For an account on the subtle differences in theseapproaches consult [11].

Regarding discrete-time models, there are 3 well-studied and widely accepted expec-tation assumptions formulating the ways a firm is anticipated to adjust its output level atfuture times. These are:

1. The naive expectation of a reaction function (see (1) above), where each firm com-putes the production level in time t + 1 based on what the firms produced at time t,is written as

qi(t+ 1) = Ri(q3−i(t)) =

√θiq3−i(t)

ci− θiq3−i(t). (2)

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 weightsto last period’s output, qi(t), and its reaction function, Ri(q3−i(t)), as follows:

qi(t+ 1) = (1− λi)qi(t) + λiRi(q3−i(t)), (3)

where λi ∈ [0, 1] is the ‘speed of adjustment’ of the adaptive firm i.

3. The bounded rationality adjustment process is based on the idea that at each timeperiod a firm decides to increase its production if the marginal profit is positive, oth-erwise it considers it best to decrease the production level. This can be modelledusing the map,

qi(t+ 1) = qi(t) + αi(qi(t))∂ui∂qi

(t)

= qi(t) + αi(qi(t))

(θiq3−i(t)

(qi(t) + θiq3−i(t))2 − ci

), (4)

where αi(qi) is a positive function which reflects the extent of production variationof firm i following a given profit signal. In most research papers it is assumed thatαi(qi) = αiqi, with αi a constant. However in continuous-time dynamics one oftenconsiders the case, αi(qi) = αi [5, 10].

In this Chapter we study a duopoly with one firm following bounded rational expectationsand the other being ‘naive’. In fact our choice could have been any of the above mentionedpossibilities. Other choices are considered in [13].

2 From identical to differentiated products

In [12] the authors investigate a duopoly selling homogeneous goods. Their main assump-tions include a linear demand function, constant marginal costs and expectations that in-clude one firm behaving naively and the other bounded rationally. We summarise theirresults in Figure 1.

Insert Figure 1 about here

Our plan is the following: Firstly, we change the price function and from linear weinvestigate the isoelastic choice. This is already a more complex situation and fortunatelythe reaction functions can be found explicitly and therefore analytic results can be providedto support the numerical simulations. Next, we consider one firm switching its productstatus and producing slightly differentiated goods. This is naturally reflected in the demandfunction as explained in the Introduction. We then examine the effects of the same duopolyby 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 theisoelastic price function has been adopted.

Insert Figure 2 about here

Assume now that the first firm, that is the bounded rational one, produces a slightlydifferentiated product. This means that it employs a price function of the form p1 = 1/(q1+θ1q2), whereas the second firm’s price function remains p2 = 1/(q1 + q2). In Figure 3 weobserve that for a wide range of θ1-values the two firms reach the Cournot-Nash equilibriumpoint which is of period two. As the degree of product differentiation tends to zero theexpression for the profit of the first firm becomes poorly defined and this is due to theassumption of an isoelastic price function.

Insert Figure 3 about here

These results demonstrate that the second firm may indeed wish to differentiate its prod-uct. However, this is not a simple matter. In [6] we examine in detail the optimal degree ofdifferentiation 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 thewell-known results with a linear price function we applied the isoelastic price function tothe duopoly under investigation. As we are interested in markets with differentiated goodswe attempted to illustrate the dynamics of the duopoly when a firm begins to produce aslightly heterogeneous product. Yet, it should be pointed out that in reality, a firm wishingto differentiate its product would firstly determine if such a move would indeed increase itsprofits. Otherwise it is apparent that such a differentiation would not take place because itwould 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 θ1 = θ2 =1. Obviously the market would evolve until it reached an equilibrium state. Now, one firmcomputes its profit function and finds the optimal degree of product differentiation such thata product differentiation would increase its profits. The cost function for that firm changesand the product is sold at a different price. The market evolves again until it reaches anew equilibrium state. It is of the best interest for the second firm to do the same. Nowboth firms compute the optimal degree of differentiation at each time step and the duopolyevolves 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 Dy-namics, Jonkoping, Sweden for the financial support to present the work reflected by thisChapter. The State (Hellenic) Scholarships Foundation is also thanked. We appreciate theuseful 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:10.1016/j.chaos.2008.09.037).

[2] Bischi, G-I., Gallegati, M., & Naimzada, A. (1999). Symmetry-breaking bifurcationsand representative firm in dynamic duopoly games. Annals of Operations Research, 89,253–272.

[3] Elabbasy, E. M., Agiza, H. N., & Elsadany, A. A. (2009). Analysis of nonlinear triopolygame with heterogeneous players. Computers and Mathematics with Applications, 57,488–499.

[4] Matsumoto, A. & Szidarovszky, F. (2007). Delayed nonlinear Cournot and Bertrand dy-namics with product differentiation. Nonlinear Dynamics, Psychology and Life Sciences,11, 367–395.

[5] Andriopoulos, K. & Bountis, T. (2008). Dynamics of a Duopoly Model with PeriodicDriving. American Institute of Physics Conference Proceedings, 1076, 9–12.

[6] Andriopoulos, K., Bountis, T., & Dimas, S. (2009). Transition of a competitiveoligopoly from identical to differentiated goods. (In preparation)

[7] Cournot, A. A. (1838). Recherches sur les Principes Mathematiques de la Theorie desRichesses. 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, 573–581.

[10] Andriopoulos, K., Bountis, T., & Papadopoulos, N. (2008). Theory of Oligopolies:Dynamics and Stability of Equilibria. Romanian Journal of Applied and Industrial Math-ematics, 4 (1), 47–60.

[11] Yousefi, S. & Szidarovszky, F. (2009). A Chapter within this volume.

[12] Agiza, H. N. & Elsadany, A. A. (2003). Nonlinear dynamics in the Cournot duopolygame with heterogeneous players. Physica A, 320, 512–524.

[13] Andriopoulos, K. (2009). Mathematical Methods in Microeconomics and FinancialMathematics. Thesis, University of Patras, Patras, Greece.

Figure 1: For these graphs we have chosen a = 10, b = 0.5, c1 = 3 and c2 = 5 as in[12]. In that paper only the first and fourth graphs are given. The top left graph shows theevolution of the production outputs for the two firms. Note that firm 1 is assumed to bethe bounded rational one. The top right and bottom left graphs are drawn for the specificvalue of α = 0.41. Obviously the prices are the same for the two firms. However the profitversus time graph shows clearly that the profits for the first firm are greater than those forthe second, as expected. The bottom right graph is the usual bifurcation diagram for thequantities, q1 and q2 versus time, t. Chaos prevails for all α > 0.4.

Figure 2: Again we choose c1 = 3 and c2 = 5. The top graphs are derived for α = 0.75. Forα = 0.5678 the transition from a stable equilibrium point to period doubling occurs now ata higher α-value as shown clearly in the bifurcation diagram. More importantly, the secondfirm, 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, c1 = 3 and c2 = 5. The bifurcation diagram (bottom right) is sketchedfor α = 0.6 and shows the output levels, q, versus θ1. It is clearly seen that as the value forθ1 drops from 1 to 0 the two firms experience periods of stability and periods of obviousinstability. The three other graphs depict the change of q versus t (top left), p versus t (topright) and u versus t (bottom left) for θ1 = 0.8 (ie within the chaotic region). Once againthe second firm achieves greater profits every second period.

Figure 4: The bifurcation diagram, q versus θ1, for α = 0.6 and θ2 = 0.7 (left) and forα = 0.6 and θ2 = 0.3 (right). Note that for lower θ2-values chaos occurs at larger values ofθ1.