We study the duopoly model proposed by Matsumoto and Nonaka (2006), in which two firms produce two complementary goods, and there are externalities of different signs. We analyze the topological complexity of the model by computing its topological entropy with prescribed accuracy, and, in addition, we show when such topological complexity is physically observable by characterizing the attractors. Finally, we exploit the fact that reaction maps have negative Schwarzian derivative to show the existence of absolutely continuous (with respect to Lebesgue measure) ergodic measures, and, as an economic application, we compute the average profit for almost all initial conditions.

1. Introduction: The Model and Our Aims

In [1] a two-market model consisting in two firms, which produce differentiated goods is introduced. The first firm produces good in the first market, and the second one produces good in the second market. It is assumed that externalities of different signs exist. An externality occurs when the actions of firms directly affect the production capabilities of other firms other than through the price mechanism of the market. In this case, the positive externality comes from the market demand; that is, the sales possibilities of one firm are positively influenced by the production of the other firm. The negative externality is due to the cost functions of each firm, since the cost depends not only on its own production but also on the other firm’s production.

In this frame, we are ready to introduce the equations of the model. Although in [1] a more general model is introduced, we are interested in the following particular case. Inverse demand functions are given by where and are the market prices of goods and , respectively, and and . Each firm decides future production depending on the other firm’s choice and production externalities. Cost functions are given by and thus, the profit function of each firm is given by It is assumed that each firm tends to maximize its profit. In order to get it, the firms can choose their production levels which would affect the other firm. The first firm maximizes the profit with respect to , and the same occurs for the second firm respect to , that is,

Hence, by solving (4), and under naive future expectations, the reaction functions of the firms are given by This is a static situation, but we are interested in the dynamic interactions between the two firms along the time. The function is the reaction function for the outcome . The iterations are given by that is, where and . The parametric region, , is not chosen in an arbitrary way. Observe that the reaction map is a map from into . Note that or, in other words, and therefore, one might expect that the dynamics of the whole system can be derived from the one-dimensional maps and given by

Our aims are, on one hand, to give an analytical proof of the existence of chaos in the model. This proof will be given by computations of topological entropy of the system with prescribed accuracy. As positive topological entropy will imply the existence of chaos in the sense of Li and Yorke, we will be able to give a proof of existence of chaos for a wide range of parameter values. On the other hand, we will be able to describe the nature of attractors of the system, and we will go beyond to the seminal work [1] proving the existence of absolutely continuous ergodic measures. The existence of such measures will allow us to obtain some consequences from both dynamic and economic points of view. Finally, we must emphasize that our approach is somewhat different from that made in [2]. Moreover, our scheme can be adapted to analyze the dynamics of models given by maps with the form of .

The paper is organized as follows. In Section 2 we study the complexity of the model. For that, we compute the topological entropy using different algorithms depending on the number of pieces of monotonicity of the functions involved in the model. As an approach to the attractors, their number is studied in Section 3. On one hand, the fact that the Schwarzian derivative of the function is negative allows us to know the possibilities of the metric attractors that can appear. On the other hand, a real computation of the attractors allows us to realize that topological chaos need not be physically observed, with the Lyapunov exponents being the key for analyzing these phenomena. The Lyapunov exponent gives us the key to study that situation. Invariant measures and average profits are considered in Section 4. Finally, Section 5 is devoted to conclusions.

2. Computing Topological Entropy

The goal of this section is to compute the topological entropy of the system. Topological entropy was introduced in [3] by Adler, Konheim, and McAndrew. When is a continuous piecewise monotone function, the Misiurewicz-Szlenk's theorem, [4], gives us the following characterization of the topological entropy  .

Theorem 1. Let be a piecewise continuous monotone functions and the number of pieces of monotonicity. Then,

Although our system is two-dimensional, the computation of topological entropy can be reduced to a one-dimensional problem. For that, we use the power formula [3], and the commutativity formula [5], to obtain

Next, we must recognize that Misiurewicz-Szlenk formula is not useful for computing topological entropy of our model, so we are forced to use numerical algorithms which are based on the number of monotone pieces of and . Therefore, we will make a precise description of monotonicity regions for these maps.

2.1. Monotonicity Regions of and

We study the shape of the composition map, where shape means in this case the number of pieces of monotonicity that each map has depending on the values of the parameters and . This will be used in the next section to compute the topological entropy.

The map , , in terms of monotonicity, has the following structure in the parametric space (see Figure 1).(i) is a monotone decreasing map if , where see example for and in Figure 2(a). (ii) is a unimodal map with a minimum for if , where see example for and in Figure 2(b). (iii) is a bimodal map with a minimum for and a maximum for if , where see example for and in Figure 2(c). (iv) is a trimodal map with two minima (for and ) and a maximum for if , where see example for and in Figure 2(d).

The map , has the following structure in the parametric space (see Figure 1).(i) is a unimodal map with a maximum (for ) if , where see example for and in Figure 3(a). (ii) is a bimodal map with a maximum for and a minimum for if , where see example for and in Figure 3(b). (iii) is a trimodal map with a maximum for and two minima in the points and if , where see example for and in Figure 3(c).

2.2. Practical Computation of Topological Entropy

Algorithms for computing the topological entropy for the special case of unimodal maps are given in [6, 7]. An algorithm for bimodal maps (with three pieces of monotonicity) is described in [8], and an algorithm for a particular class of piecewise continuous maps with four pieces of monotonicity is described in [9]. These three types of algorithms cover all the cases involved in the dynamical system that we are considering for the composition maps and . The results obtained for the model can be observed in Figure 4.

Figures 5(a), 5(b), 5(c), and 5(d) show the topological entropy with accuracy , where takes the values , , , and .

For , and the topological entropy with accuracy is given by Figures 6(a) and 6(b).

2.3. Existence of Chaotic Maps

Before stating the existence of chaos, we must recall what we understand for chaos. A continuous interval map is chaotic in the sense of Li and Yorke (LY-chaotic) if there is an uncountable set (called scrambled set of ) such that for any , , it is held that

Positive topological entropy implies chaos in the sense of Blanchard et al. [10], and therefore, the above computations of topological entropy are an analytical proof of the existence of chaos in the model. Even more, we can prove that positive topological entropy is equivalent to chaos in the sense of Li-Yorke. For that, we just need to notice that when the topological entropy is equal to zero, the map is not chaotic because the relative extreme points of the composition maps and are nonflat since the second derivative in these points is nonzero, (remember that a critical point is nonflat if some higher derivative is nonzero). Then, by [11, Theorem ], maps with nonflat turning points have no wandering intervals (i.e., for a continuous interval map , an interval is called wandering if , , are disjoint and no point is asymptotically periodic), and by [12, Lemma 2.7], a map with zero topological entropy is chaotic if and only if it has wandering intervals. Hence, our model is not chaotic when the topological entropy is zero.

Therefore, we have two regions the nonchaotic one where every trajectory has the property that for any there is a periodic trajectory which is close to it [13]. In practice, one might check that a computer simulation shows the convergence to a periodic orbit when parameters are in the nonchaotic region. For the second region one could expect a similar situation, but as we will show in the next section, the topological chaos we have shown to exist, could not be observed. For instance, for and , the topological entropy of is positive, but the time series of several orbits reveal a periodic motion (see Figure 7).

3. Chaos, Attractors, and Schwarzian Derivative

We start this section with some useful definitions. Let be a map. A probabilistic measure is said to be invariant for if for any Borel set . In addition, the measure is ergodic if the equality implies that or . Denote by and the set of invariant and ergodic measures of , respectively.

Given , define its -limit set as the set of limits points of its orbit. Recall that a metric attractor is a subset such that , has positive Lebesgue measure, and there is no proper subset with the same properties. is called the basin of the attractor.

By [14, Theorem 4.1], for a multimodal map without wandering intervals, there are three possibilities for its metric attractors.(A1)A periodic orbit (recall that is periodic if for some .(A2)A solenoidal attractor, which is basically a Cantor set in which the dynamics is quasiperiodic. More precisely, the dynamics on the attractor is conjugated to a minimal translation, in which each orbit is dense on the attractor. The dynamics of restricted to the attractor is simple; neither positive topological entropy nor Li-Yorke chaos can be obtained. Its dynamics is often known as quasiperiodic.(A3)A union of periodic intervals, , such that and , , and such that is topologically mixing. Topologically mixing property implies the existence of dense orbits on each periodic interval (under the iteration of ).

Moreover, if has an attractor of types (A2) and (A3), then it must contain the orbit of a turning point, and therefore its number is bounded by the turning points. In the case of our model, we can strengthen the above result by noticing that and have negative Schwarzian derivative. Namely, the Schwarzian derivative [1517] at a point is given by

Since and the maps and have negative Schwarzian derivative the same occurs to the composition maps and . Since attractors of type (A1) must also attract the orbit of a turning point when the map has negative Schwarzian derivative, we have that the number of attractors of and is bounded to at most 3. Even more, when and have three monotone pieces, then their value in two turning points agree, which implies that in fact, the number of attractors of and can be at most 2. The following result shows that the attractors of and are deeply connected.

Proposition 2. Let be maps such that is an attractor of . Then is an attractor of .

Proof. As usual, denote by the one-dimensional Lebesgue measure. Let with such that for any we have . Since is , we have that . Now, let , and fix such that . For any we have from which we conclude that is an attractor of .

The attractors of the map can be obtained by combining the attractors of and . Basically, we must use the inclusion [18, Theorem 2] and, in view of Proposition 2, check whether the attractors of and we are considering are linked or not. In any case, it should be noticed that even when and have just one attractor, the map may have several of them depending on their distribution in the plane (see, e.g., [18, 19] and the examples of attractors in Figure 12).

Obviously, if and have 2 attractors, the number of different attractors of increases. So, still we have a practical work to do to decide whether and have one or two attractors. We fix the map and realize that if it has three or four monotone pieces, then the image of its turning points is in . We take and converge it to where and are mutually independent ergodic measures of (see, e.g., [20]). Since different ergodic measures must be supported in different attractors, when the expression (27) tends to zero when tends to infinity we have an evidence that just one attractor exists. Hence, we make simulations to compute (27) showing the results in Figure 8. Note that we have to be cautious when or is eventually periodic, because in such case its trajectory ranges all the attractors if the periodic orbit is an attractor, which need not happen (e.g., we can address that the well-known example of the map has a dense orbit, and hence, its attractor is the whole interval, and the image of its turning point is 1, which is eventually the fixed point 0). Fortunately, the probability of finding such pathological examples is 0, but this is the case for and .

Taking into account the results obtained for generating Figure 8, we see that for there are values of around for which it seems that two different attractors of may coexist. Figure 9 shows a refinement of Figure 8 for and . The bifurcation diagrams (see Figure 10) also suggest the existence of two different attractors for but not for .

As an example, we show different types of attractors and limit sets (but nonmetric attractors) obtained for the parameter values and (see Figure 11 and the explanations therein).

When the map seems to have only one attractor, and, for , we see in Figure 12 that several of them may exist (see [18] for an analytical explanation of this fact). Figure 13 shows different types of attractors that can appear when we fix .

Once we have described the attractors of , we go back to the idea of explaining when topological chaos cannot be physically observed. To that end, it is worth to notice that for a interval map , a periodic point of period is an attractor when , and therefore, its Lyapunov exponent (see, e.g., [14]) is Since turning points characterize the attractors of and , we conclude that attractors of type (A3) may exist if and only if for some turning point of . The following graphic shows the maximum Lyapunov exponent for the parametric region. Figure 14 shows the maximum of , where and are the turning points that have different forward images by .

It is observed that there is a white band in the upper right side of the projection; this corresponds with the region where the fixed point is a local attractor. The topological entropy in these points is positive, but the chaos is not physically observable. Figure 14 explains the results obtained in Figure 7 because the parameter values and are clearly in a region with negative Lyapunov exponents.

4. On Invariant Measures and Average Profits

One of the main results in [1] states that for and the map has an invariant measure which is absolutely continuous with respect to the two-dimensional Lebesgue measure. Such invariant measure is supported on the whole square , and therefore, for almost every initial condition, (up to a set of zero Lebesgue measure) holds that where is the profit function. Therefore, even when its dynamics is chaotic, with probability one, the average profit can be estimated, and it decided whether the chaotic regime is more profitable than profits at equilibrium points or not. The same estimations can be done also for other economic quantities like production or costs.

The aim of this section is to prove that we can state the same result for a wide range of parameter values of and . Note that now we have to take into account several facts. (i)The map may have several attractors. Therefore, we can have several average values. (ii)When we are in the basis of a periodic attractor , the equality trivially holds. Therefore, we must concentrate on attractors constructed from types (A2) and (A3). (iii)Since, computationally, attractors of type (A2) are not easily detected and their probability is small, we must concentrate our efforts on deciding when attractors of type (A3) support an invariant ergodic measure which is absolutely continuous with respect to the Lebesgue measure. Note that, by, for example, [21], the existence of dense orbits is not enough to guarantee the existence of absolutely continuous ergodic measures.

We start with the next result, which states that absolutely continuous ergodic measures of and are deeply connected.

Proposition 3. Let be maps, and let be absolutely continuous with respect to the Lebesgue measure. Then defined by for every Borel set is an absolutely continuous ergodic measure of .

Proof. The fact that can be seen in [22]. Now, let be a density function for ; that is, for any Borel set . Since we conclude that is the density function of , and the proof ends.

Proposition 3 is a great help in detecting the absolutely continuous ergodic measures of ; we just consider one map or and decide whether they have absolutely continuous ergodic measures. The absolutely continuous ergodic measures of will be constructed by applying Proposition 3 and taking the product measure, which will be absolutely continuous but probably not ergodic. If we denote the product measure by , we take the ergodic decomposition , where , and is the number of different attractors constructed by the attractors of and supporting the measures and . The absolutely continuous ergodic measures are therefore . In the case studied in [1] we find that .

So, we fix a turning point of for instance and compute its Lyapunov exponent . If it is negative or zero, there is nothing to do. So, we assume that it is positive and let be periodic intervals such that is an attractor of type (A3). Any ergodic measure supported on must satisfy , , and then we can consider , the invariant subinterval for some , and the ergodic measure (for ) given by the formula . Since has negative Schwarzian derivative, by [23], we conclude that , and therefore, , can be chosen to be absolutely continuous with respect to the one-dimensional Lebesgue measure.

For instance, we consider the values and . The possible attractors were studied in detail in Figure 11, and we obtained the following average profits for the initial values as we showed in Table 1.

We compute the average profit for each firm. This result could be different if we change initial conditions when the number of attractors is different from one, as we studied in the previous section. Thus, for each firm we are going to present two different graphics in the square with two different initial conditions, and we observe the differences between them. In the square there is only at most one turning point. Then we study the number of fixed Cournot points depending on . When there are more than a fixed point we determine in which fixed point the profit is maximum for each firm. Finally we compare the average profit with the profit in the fixed point (in which it is maximum) and compute the residual profit.

Figure 15 shows the average profit of both firms. For that, we use orbits of length and, as initial conditions, , where is the minimum of when it exists and , otherwise, and is the maximum of . We will denote this election of initial conditions as min-max conditions. When is greater than one there are more turning points which eventually may produce other average profits as it is shown in Figure 16. Here, we show the profit of firm 1, taking as initial conditions , where is the maximum of and is the maximum of , which will be denoted as max-max conditions, and compare the situation with the initial conditions min-max previously computed. Figure 17 shows the same scenario for the second firm.

We are interested in analyzing whether the destabilization of Cournot points, even in the chaotic regime, is a good business for both firms. To this end, we compare the average profits of generic orbits with the expected profit at Cournot points. Firstly, we have to compute the number of fixed points of the composition , which can be seen in Figure 18. The parameter space can be divided into two regions, and . In there is one fixed point while in the number of fixed points is three, which will be ordered . As an exception, for , the number of fixed points are four. In [1, Theorems 1 and 2] the stability and bifurcation diagrams of stationary points is studied in detail.

In region we compute the profit of the first firm for each fixed point, and we observe that it is maximum for the fixed point and minimum for , obtaining the converse situation for the second firm. Figure 19 (resp., Figure 20) shows the profit of the first firm (resp., second firms) against the profit obtained by the second firm (resp., second firm) when the fixed point that maximizes the profit of firm 1 (resp., firm 2) is chosen.

The case was studied in Figure 11 and Table 1. We compute the profit of both firms in this case to compare the results see Figures 21 and 22.

5. Conclusions

A duopoly model depending on two parameters is studied in detail. By computing its topological entropy we characterize the parameter values which admit the existence of chaotic behavior of trajectories. We also analyze when the above mentioned chaotic trajectories either can be observed or remain on a set of zero Lebesgue measures and therefore are not physically observed in computer simulations. For that, we study the metric attractors of the model concluding that several of them can coexist. The existence of absolutely continuous ergodic measures is proved, and we showed that, even when chaotic maps are considered, the average of economic functions, like profit and production and along the orbits can attain a finite number of values. The existence of such values allows us to make decisions on whether the equilibrium points deserve to be stabilized by choosing strategies that are distinct from naive expectations.


This paper has been partially supported by Grants MTM2011-23221 from Ministerio de Ciencia e Innovación (Spain) and 08667/PI/08 from Programa de Generación de Conocimiento Científico de Excelencia de la Fundación Séneca, Agencia de Ciencia y Tecnología de la Comunidad Autónoma de la Región de Murcia (II PCTRM 2007-10), and COST Action IS1104 “The EU in the new economic complex geography: models, tools and policy evaluation.”