DISCRETE TIME DYNAMIC OLIGOPOLIES WITH ADJUSTMENT CONSTRAINTS

. A classical n -ﬁrm oligopoly is considered ﬁrst with linear demand and cost functions which has a unique equilibrium. We then assume that the output levels of the ﬁrms are bounded in a sense that they are unwilling to make small changes, the output levels are bounded from above, and if the optimal output level is very small then the ﬁrms quit producing, which are realistic assumptions in real economies. In the ﬁrst part of the paper, the best responses of the ﬁrms are determined and the existence of inﬁnitely many equilibria is veriﬁed. The second part of the paper examines the global dynamics of the duopoly version of the game. In particular we study the stability of the system, the bifurcations which can occur and the basins of attraction of the existing attracting sets, as a function of the speed of adjustment parameter.

1. Introduction. Assume n firms produce the same product or offer identical service to a homogeneous market. Let x k denote the output of firm k, s = n k=1 x k the output of the industry. If p(s) is the inverse demand function and C k (x k ) is the production cost of firm k, then its profit is given as In this way a n-person game is defined, where the firms are the players, the set of non-negative real numbers is the strategy set of each player, and the payoff of player k is given by equation (1). This game is one of the most frequently studied models in mathematical economics. A comprehensive summary of the earliest results is given in [16], their multiproduct generalization with case studies are discussed in [17].

CHRYSTIE BURR, LAURA GARDINI AND FERENC SZIDAROVSZKY
The studied models were mainly linear making equilibrium and stability analysis relatively simple. The attention recently turned to the analysis of nonlinear models due to the drastically improved computational power. The authors of [3] offer a review of the most recent development in this area.
In this paper, we first consider the baseline Cournot model with no output adjustment constraints (called model I), in which case there is a unique equilibrium. Next, we consider the case (called model II) when it is difficult (costly) for firms to make output adjustments. Thus, we introduce the additional assumption that the firms are not willing to make very small output adjustments, since the small profit increase is insufficient to offset all required fixed adjustment costs in the production process such as difficulties in modifying input contracts or in coordination between divisions within a firm. This "stickiness" in outputs is analogous to the literature on "sticky prices" ( [9], [22]) whereas the sticky price is reflected in the speed of adjustment parameter and ours is reflected in the modified best response function. We further assume that the amount of output in each time period is bounded from above due to limited resources, such as physical capacity of the plant, so the output adjustments are also bounded from above. It's also assumed that firms are not willing to produce at a very low output level to avoid fixed operation and maintenance costs. These additional assumptions result in nonlinear, discontinuous best response functions and infinitely many steady states of the dynamic system.
We investigate in particular the dynamics in the duopoly case. We are particularly interested in the attractivity of the fixed point meaning that the trajectory converge to that fixed point. The interesting dynamics occur when the set of fixed points is no longer attracting, at this point, 2-cycles are the only possible attractors; in particular, at least one periodic point (or both) must take on the values introduced in the constrained model II.
The dynamics of model II are particularly interesting from a mathematical point of view as well. Indeed, the model is described by a discontinuous two-dimensional piecewise linear map, with several borders crossing which the system changes its definition, although limited to a rectangular absorbing region of the production phase plane (x 1 , x 2 ). The dynamics associated with piecewise smooth systems is a relatively new research area, and several papers have been dedicated to this subject in the last decade ( [27], [7]). This growing interest in nonsmooth dynamics is from both the new theoretical problems with constraints and the applied fields. In fact, many models are described by constrained functions, leading to piecewise smooth systems, continuous or discontinuous. We recall several oligopoly models with different kinds of constraints considered in the books [19] and [3], nonsmooth economic models in [6], [18], [10], [24], financial market modeling in [12], [25], [26], Shelling segregations models in [20], [21], and modeling of multiple-choice in [4], [11], [5].
Model II considered in the present paper is characterized by several constraints, leading to several different partitions of the phase plane in which the system changes definition. Moreover, the definitions in some regions are degenerate, as mapped into points or segments of straight lines. When the existing 2−cycle has a periodic point colliding with a border of the regions, then a border collision occurs, which in our case is always a persistent border collision, as it is simply transformed in a 2−cycle belonging to different partitions (see [14], [15], [23]).
The structure for the rest of this paper is as follows. Section 2 introduces the model set-ups and Section 3 analyzes the steady state. In Section 4 we investigate the dynamics of the affine duopoly model I, while Section 5 is devoted to the piecewise linear and discontinuous model II. In particular, in subsection 5.1 the symmetric case is fully described, while the proper argumentations are given for the general case in subsection 5.2. Section 6 concludes.
2.1. The baseline Model I. Assume linear demand, p(s) = a − bs and heterogeneous cost functions, C k (x k ) = c k x k + d k , k = 1, ..., n. The profit of firm k is therefore quadratic, which is a strictly concave function in x k . s k = =k x denotes the output of the rest of the industry from firm k 's perspective. The profit maximizing quantity x * k is derived by simple differentiation, leading to It is well known that there is a unique equilibrium of this static game. The dynamic extension of this model with discrete time scales and adjustments toward best responses can be modeled by the discrete system defined as: for k = 1, 2, ..., n where K k > 0 for any k is the speed of adjustment parameter of firm k. The usual assumption that K k < 1 models firms with "cautious" behavior, however, more "aggressive" firms may select K k > 1 counting for the continuation of the trend in rival firm's output levels. We also assume that the best response of firm k in the next period, x * k (t + 1) depends only on the aggregated output of the rest of the industry in the current period, s k (t): 2.2. Model II with output adjustment constraints (fixed capacity). Assume next that the firms do not want to make adjustment below a certain threshold k ("sticky output" constraint) from an initial output level, x k , and cannot produce outputs greater than a specified (fixed) positive threshold L k (capacity limit constraint), or smaller than a given small threshold, l k . That is, if the optimal output falls below this threshold, firm will simply choose not to produce. Based on these additional conditions, the modified best response of firm k becomes Fig.1 illustrates this function. It can be shown that the two restrictions on a firm's responsiveness to rivals' behavior imply that the firm's reaction curve is still downward sloping although it now has points of discontinuity, always in the form of downward jumps, and is thus no longer globally linear in rivals' output. It follows that the resulting oligopoly game is of strategic substitutes, so that a purestrategy Nash equilibrium always exists as a consequence of the general properties of submodular games that satisfy a common aggregation property, namely that each firm's profit function depends only on own output and on total rivals' output (see [13] and [1], [2]). We could develop a best response dynamic model based onR k , however, there is no guarantee that the new output levels x k (t + 1) would satisfy the constraints posed in (6). Instead we use a two step process. In the first step, we apply the best response dynamics based on the unconstrained reaction function (3), and then in the second step we enforce condition (6) on the computed output levels.
(8) Notice that the function defined in (7) is discontinuous and nonlinear making the equilibrium and stability analysis different with respect to the linear baseline model I.
3. Steady states. Consider first the baseline model I defined in (4): an output vector (x 1 , · · · ,x n ), wherex i = x i (t + 1) = x i (t), is a steady state if and only if for all k, Summing up equation (9) for all k = 1, 2, ..., n, we havē Then from (9) and (10) we can derive the equilibrium output levels of the firms: The attractivity analysis of the linear system in (4) will be given in the next section.
Here we only remark that in order to have a fixed point with positive coordinates, we assume that the following condition (h n ) holds: Due to the additional constraints, the steady states of model II is different from (11). An output vector (x 1 , · · · ,x n ) is a steady state of the system if and only if for all k, which can be rewritten as Note that the steady state of model I in (11) clearly satisfies these relations which means that the equilibrium quantities in the baseline model will always be one of the solutions in the models with the additional constraints. There are infinitely many steady states in model II since (14) is satisfied by a bounded polyhedron, denoted ER (Equilibria Region, or region of fixed points) as in Figure 2 for the case n = 2. In particular, ER belongs to the positive orthant assuming a small enough k and by condition (h n ) in (12). Example 1. Consider a Cournot duopoly, i.e. n = 2, then the relations in (14) can be rewritten as and  and The region ER includes the unique equilibrium, E, of the baseline model. From (11), Note that E belongs to the positive quadrant R 2 + as long as condition (h n ) is met, that is, In the next two sections, we consider in detail the dynamic behavior of the duopoly models I and II, respectively, assuming that condition (h 2 ) in (22) holds. We show that the system with the bounded output adjustments, due to the constraints of the model, is in some sense more stable than the baseline model. The intuition behind this observation is that despite the stability condition of E and of ER is the same in both models, model II has bounded dynamics even when ER is not attracting. 4. Stability analysis and dynamics of Model I. Let us explicitly rewrite the best response dynamics for n = 2: a point (x 1 , x 2 ) is mapped into (x 1 , x 2 ) by which is a linear, or more precisely, an affine model. We can refer to it as a linear model, following the common usage in dynamical systems, as it is topologically conjugate with a linear model after a change of variable setting the fixed point of (23) into the origin. Its fixed point, , has been given explicitly in (21). Notice that its position in the phase plane (x 1 , x 2 ) does not depend on K 1 and K 2 while the stability of the equilibrium E clearly depends on K 1 and K 2 . From the Jacobian determinant of the linear map in (23), we find that the characteristic polynomial is given by where The sufficient stability conditions of E are: These conditions can be summarized in Fig.3. We denote by S the stability region of the fixed point E, which is bounded by an arc of hyperbola (denoted F in Fig.3). At the bifurcation curve F of equation For values of (K 1 , K 2 ) outside of the stability region S, the fixed point is either a saddle or a repelling node. Recall that in a linear model when the equilibrium is or, equivalently, For the specific duopoly model, by substituting 2 and J 22 = 1 − K 2 , we obtain: It can be seen that becomes unstable when crossing the value λ − = −1 and the bifurcation occurrs for K 2 = 16−8K1 8−3K1 . If K 1 > 2, the system is unstable regardless of the value of K 2 .
If K 1 < 2 , the system is stable only if K 2 is small enough, i.e. when the point (K 1 , K 2 ) belongs to region S (the yellow region in Fig.3). If K 2 is large enough, i.e. (K 1 , K 2 ) is outside region S, then the fixed point E is a saddle since the eigenvalue never bifurcates.
For K 1 > 2, λ + also becomes negative and unstable when crossing the value This occurs on the right side of the vertical asymptote, that is, on the right branch of the hyperbola in Fig.3a. For (K 1 , K 2 ) in the gray region in Fig.3a, both eigenvalues are negative with λ ± < −1, and E is therefore a repelling node.
In the special case of equal speed of adjustments K = K 1 = K 2 , the eigenvectors e ± have constant slopes m ± , as in fact we have: 3 the eigenvalues are λ ± < −1, and E is a repelling node.

Stability analysis and dynamics of Model II.
In the duopolistic competition, a point ( and We can rewrite the piecewise linear map more explicitly as follows: As shown in the previous section there is no unique fixed point but instead a whole region ER of fixed points, which includes the equilibrium E of the baseline model. By its definition, the map takes on different values depending on the position of a point (x 1 , x 2 ) in the phase plane. We can thus subdivide the phase plane into regions R j j = 1, 2, ..., in each of which the map takes different definitions. These regions R j are bounded by segments of the straight lines satisfying the following equations: (r a ) : that is, in explicit form: Let us denote R 1 the region bounded by segments of all the four lines, in which the map is defined as in the linear model I except for the region ER of fixed points. Consider the regions of interests which are defined as 1 The map in each region is defined as From the definition of the map, the rectangle is absorbing, as it follows immediately from (32) and (33) that any point of the plane is mapped in D in one iteration and an orbit cannot escape from it. D is therefore the region of interest.
In general, depending on the values of the parameters, only a few of the regions R j for j = 1, 2, ... may have a portion, or subregion, present in D, or R j ∩ D = ∅, as shown in Fig.4. In Fig.4a only two regions have points in D, while in Fig.4b there are four regions.
Inside region R 1 segments of the 4 lines determining region of fixed points ER are also shown, together with the eigenvectors of the fixed point E inside ER.
Since we assume 1 and 2 are small enough, ER is internal to the positive quadrant and in the R 1 region. Therefore, the fixed points are surrounded by a region where the mapping is the same as in the baseline model. Fig.4 also shows the segments which are intersecting to give the region ER of fixed points (and E ∈ ER), and segments of the eigenvectors e + and e − of E, which are also invariant lines for the linear map defined in R 1 \ER.
A fixed point p is stable if there exist a neighborhood U 1 (p) and a neighborhood U 2 (p), U 1 (p) ⊂ U 2 (p) such that any initial condition (i.c. for short henceforth) in U 1 (p) has the trajectory inside U 2 (p). Meanwhile, p is attracting if it is stable and the trajectory with any i.c. in U 1 (p) converges to p. It is clear that each fixed point belonging to ER can be stable but not attracting. Since ER is surrounded by points with the same mapping as in model I, we have that as long as the K 1 and K 2 belong to the stability region S, determined for model I, the neighborhoods U 1 (p) and U 2 (p) exist for any p ∈ ER and any i.c. in U 1 (p) has the trajectory converging to a point in ER. Thus ER consists of points which are stable but not attracting, while the invariant set ER itself is an "invariant attracting set". We therefore have shown that the stability region of model II in the parameter space (K 1 , K 2 ) is the same as for model I, and we can state the following Proposition 1 (local stability). Let (K 1 , K 2 ) ∈ S(stability region of model I), then the invariant set ER of model II is attracting.
In addition, similar to the fixed point E being globally stable when (K 1 , K 2 ) is in S, ER is globally attracting for the same (K 1 , K 2 ) range. As remarked above, any point of the plane is mapped in one iteration in the rectangle D in (38). Clearly when E is stable then all the points in R 1 with a trajectory in R 1 will end in ER. When any of the regions R j satisfying R j ∩ D = ∅ exists, its points are mapped on the borders of D and the mappings on these border points are the same as in model I and therefore are contractions. For example, considering Fig.4a when ER is attracting and R 2 ∩ D = ∅. Then all the points of R 2 ∩ D are mapped into the x 1 −axis, on the border of D, obtaining a point (x 1 , x 2 ) = (f 1 (x 1 , x 2 ) , 0) which can either be in R 1 or in R 2 again. The contraction f 1 is applied until the point enters R 1 ; thus ER is globally attracting. We will not provide a rigorous proof of the global attractivity of ER when (K 1 , K 2 ) ∈ S because of the many possible cases to comment, but in all scenarios the reasoning is the same.
When the parameters K 1 and K 2 are outside of the stability region and E is a saddle, the points around ER behave as in the linear map thus the stable set of ER can be defined, and ER can declassified as an "unstable" invariant set having the same properties of a saddle. This means that there are points (a set of positive Lebesgue measure) in the phase plane whose trajectories ultimately end in ER, which constitute the stable set of ER. Meanwhile there are also points in a neighborhood of ER that diverge away from the neighborhood. Clearly when E is a repelling node then locally all the points around ER diverge, and ER can be defined as a locally repelling invariant set. Similarly we can explain the dynamic behavior at the bifurcation value, when the parameters K 1 and K 2 are on the boundary F of the stability region, and one eigenvalue is λ − = −1. The related eigenvector e − leads the points in R 1 to a segment of cycles of period 2 (while the other eigenvector e + is inside the stable set of ER). Fig.4b shows an example where E is a saddle. In this case, the eigenvector e − associated with the eigenvalue λ − < −1 plays an important role at the bifurcation value (when λ − = −1). That is, when λ − = −1, the points of the eigenvector e − belonging to R 1 \ER include two segments filled with 2-cycles (stable but not attracting). In general the 2-cycle at the external side of these segments has one point on some border, which can be either a border of the region D or a border line r k (k ∈ {a, b, c, d}). This determines which regions will contain the attracting 2-cycle which exists after the bifurcation (when λ − < −1).
It is worth noting that the map is defined either by constant values or by linear functions (as explicitly reported in (37)). Therefore, when the invariant set ER is unstable, the map cannot have an attracting set in one single region, that is, the possible attractors are obtained only by the piecewise definition of the map, and must necessarily have points in two different regions. In our specific model II, the system can only have 2-cycles as invariant stable sets. In addition, due to the particular structure of the functions in (37), at least one periodic point of the attracting 2-cycle must belong to the border of D (i.e. must include one of the constants x 2 = 0, x 2 = L 2 , x 1 = 0, x 1 = L 1 ), or both belong to the border of D.
As stated above, for λ − < −1 and λ + > −1 (i.e. when E is a saddle)), besides an attracting 2-cycle, there exists a set of points (of positive Lebesgue measure) which are mapped into ER in a finite number of iterations, which is called stable set of ER, W s (ER). Clearly, this stable set includes segments of the eigenvector e + of E, at least the portions in region R 1 , as well as all the other points in a neighborhood of e + belonging to the invariant curves of model I crossing ER, that is, points which are mapped (also in model II) into ER, where they will be fixed.
To comment on the existence of some border collision bifurcations which characterize the 2-cycles when ER is unstable, we start with the symmetric case and establish a few properties before analyzing the general case.

5.1.
Model II in the symmetric case.
That is, via the change of variable x 2 := x 1 and x 1 := x 2 we have the same dynamics when K 1 and K 2 are interchanged.
As a particular case of Property 1 we have another property when K 1 = K 2 (on the diagonal of the two-dimensional parameter plane in Fig.3

):
Property 2 (second symmetry). Let L ≡ L 1 = L 2 , l ≡ l 1 = l 2 , c ≡ c 1 = c 2 , ≡ 1 = 2 and K ≡ K 1 = K 2 . Then, besides the symmetry of the trajectories as stated in Property 1, we have that map T is invariant on the diagonal, d, of the phase plane, and its restriction to d is a one-dimensional system x (t + 1) = T d (x (t)) (x = x 1 = x 2 ) given in (39) and (40).
In fact, from the initial conditions x (0) = x 1 (0) = x 2 (0), we have x (t) = x 1 (t) = x 2 (t) for any integer t > 0. The iterates are given by the one-dimensional map x (t + 1) = T d (x (t)) defined as follows: where 5.1.1. Dynamics along the diagonal d. We can immediately see that as long as 0 < K ≤ 2 3 the straight line represented by x = f i (x, x) in (40) has a non-negative slope and smaller than 1. Therefore, the mapping T d has a segment of fixed points in where x E = a−c 3b is the same value occurring for the fixed point in model I, and the segment ER d is globally attracting. For K > 2 3 the straight line x = f i (x, x) has a negative slope, and the qualitative graph of the one-dimensional map T d is shown in Fig.5. The map T d still has the segment of fixed points the map is constant and equal to 0. Based on the position of the vertical intercept, we can further analyze two cases. One is associated with K( a−c 2b ) < L; here the production limit is not binding as shown in Fig.5a, while the second case refers to K( a−c 2b ) > L, so that the upper capacity limit constraint is binding as shown in Fig.5b. Therefore, the mapping is constant and equal to L for 0 < x ≤ p L , where The dynamic behavior of the one-dimensional map T d is straightforward to analyze.
Case (i) (Attractivity). For 0 < K < 4 3 , the segment ER d of fixed points is globally attracting -and independently of the shape of the map, that is, both for 0 < K ≤ 2 3 and for 2 3 < K < 4 3 . At the bifurcation value K = 4 3 we can have different dynamic behaviors, leading to two different attracting 2-cycles after the bifurcation, which can be completely described as follows.
. In this case the segment (x E + 2 , p l ] is filled with periodic points of period 2, and it is mapped in the segment [l, x E − 2 ). Moreover, from p l < T d (0) we have that at the bifurcation value the points 0 − T d (0), that is 0−2( a−c 3b ), belong to a 2−cycle, and it is the only one surviving after the bifurcation. In fact, for K > 4 3 the map has a global attractor, the unique 2−cycle with periodic points 0 and T d (0) = f i (0, 0) = K( a−c 2b ). This holds as K increases as long as a border collision bifurcation occurs (at K a−c 2b = L). When K > 2bL a−c , the globally attracting 2-cycle is then given by the two extrema: 0 − L.
In this case the segment (x E + 2 , L] is filled with periodic points of period 2, and it is mapped in the segment [2x E − L, x E − 2 ). In particular, the 2−cycle on the border having periodic points L and T d (L) = 2x E − L, is the only one which will survive after the bifurcation. For K > 4 3 the map has a global attractor, the unique 2−cycle with periodic points L and T d (L) = f i (L, L) = 2−3K 2 L + K a−c 2b . This holds as K increases until L = p l . Therefore, for K > 2b(L−l) 3bL−a+c , the globally attracting 2−cycle is given by the two extrema: 0 − L.
The transition case between (ii) and (iii) occurs when K a−c 2b = L at K = 4 3 , then at the bifurcation value K = 4 3 all the points not fixed belong to 2−cycles, that is (x E + 2 , L] is mapped into [0, x E − 2 ), and for any K > 4 3 the attracting 2−cycle is given by the two corner points (0, 0) − (L, L).

5.1.2.
Dynamics outside the diagonal d. When the initial conditions for the two firms are not the same, we need to consider the two-dimensional map T . The region ER of fixed points includes the segment ER d of T d on the diagonal of the phase plane (x 1 , x 2 ), inside which there is also E = (x E , x E ). Recall that in this symmetric case of equal speed of adjustments K = K 1 = K 2 the eigenvectors e ± have constant slopes m + = −1 (associated with the eigenvalue λ + = 1 − K 2 ) and m − = +1 (associated with the eigenvalue λ − = 1 − 3K 2 ), corresponding to the main diagonal d in the phase plane, which is invariant and on which the restriction is the one-dimensional map T d described above. The intersection point P l between the straight lines (r b ) and (r d ) is given by where p l has been defined for the one-dimensional map T d in (42), while the intersection point P L between the straight lines (r a ) and (r c ) is given by where p L has been defined for the one-dimensional map T d in (43).
Case (i) (Attractivity). For 0 < K < 4 3 , the region ER of fixed points is globally attracting, independently of the shape of the map T d on the diagonal, that is, for both 0 < K ≤ 2 3 and 2 3 < K < 4 3 . In fact, when K < 2 3 the points P l and P L are external to the rectangle D so that D belongs to R 1 ; for other ranges of K, the points P l or P L may be internal to the rectangle D. However, the points of D either belongs to R 1 or are mapped into R 1 in one iteration. For example, Fig.6a and Fig.8a show two examples in the different Cases (ii) and (iii). We can see that in both examples, the regions R j ∩ D = ∅ with R j = R 1 , are mapped on borders of D belonging to R 1 leading to the global attractivity of ER.
At the bifurcation value, K = 4 3 (λ − = −1), the existing 2−cycles belong to the main diagonal and thus are related to those determined for the map T d . For K > 4 3 and λ + > −1 the main diagonal is attracting, apart from ER d , and the existing attractor is the 2−cycle on the main diagonal determined for the one-dimensional map T d .
The intersection point between the straight lines (r b ) and (r d ) is on the diagonal, given by P l = (p l , p l ) where p l has been defined in (42) and At the bifurcation value K = 4 3 we have (f (0, 0), f (0, 0)) = (2 a−c 3b , 2 a−c 3b ) = 2E, and P = (p l , p l ) = (2 a−c 3b − l, 2 a−c 3b − l) which is smaller than (f i (0, 0), f i (0, 0)) on the diagonal (as (2 a−c 3b − l) < 2 a−c 3b ). Therefore the segment bounded by the points (x E + 2 , x E + 2 ) and P = (p l , p l ) is filled with periodic points of period 2, and it is mapped in the segment bounded by (l, l) and (x E − 2 , x E − 2 ), and (f i (0, 0), f i (0, 0)) belongs to region R 4 so that is a 2−cycle of the map belonging to the regions R 1 − R 4 , surviving after the bifurcation. That is, the attracting 2−cycle of the map after the bifurcation is (0, 0) − (f i (0, 0), f i (0, 0)) : Fig.6b shows an example of this. As discussed above, other than this attracting cycle on the diagonal, a stable set W s (ER) consisting of points of the plane whose trajectory ends in ER in a finite number of steps also exists (and thus W s (ER) includes the portion of e + in region R 1 ). A border collision takes place when (0, 0) merges with P L (i.e. when p L = 0) or, equivalently, when the other periodic point reaches the extremum of D, (K a−c 2b , K a−c 2b ) = (L, L). This occurs when K = 2bL a−c , after which the 2−cycle is given by the corner points (0, 0) − (L, L), see an example in Fig.7a. The stable set W s (ER) is larger than ER as long as E is a saddle. At K = 4 the second eigenvalue bifurcates, λ + = −1, so that for K > 4 the invariant set ER becomes repelling, and all the other points converge to the 2−cycle (0, 0) − (L, L), see an example in Fig.7b. The attracting 2−cycle always attracts all the points not belonging to W s (ER) or ER.  It is simple to comment on the transition between cases (ii) and (iii), when K a−c 2b = L at K = 4 3 . As we have already seen, at K = 4 3 all the points not fixed on the diagonal of D belong to 2−cycles, and then, for any K > 4 3 the attracting 2−cycle is given by the two corner points (0, 0) − (L, L).

5.2.
Model II in the general case. In the general case, the dynamics in the phase space are not symmetric with respect to the diagonal. However, as long as the point E of model I is attracting, the region ER appears to be globally attracting in the general case as well. At the bifurcation value when λ − = −1, we have segments of 2-cycles on the eigenvector e − , belonging to region R 1 , and the closest border determines which region will have the attracting 2−cycle when λ − < −1.
Let us describe, via a few examples, how the attracting 2-cycle and the related bifurcations can be determined.
However, in the asymmetric case this is clearly no longer true (For example, see Fig.11), but the local stability is not affected (as it depends only on K 1 and K 2 and the eigenvalues are unchanged unlike the eigenvectors). The reasoning on the bifurcations occurring to the periodic points of the attracting 2−cycle as a function of the parameters is similar to the one performed in the previous examples.
As remarked above, when the invariant set ER is not attracting the 2−cycle must have the two periodic points in two different regions, at least one periodic point must belong to the border of the region D (with one constant x i = L i or x i = 0) and whenever one of the periodic points crosses a border defined by the lines (r k ) the 2−cycle changes its definition. The number of lines which can be crossed depends on the regions R j with a portion in the absorbing rectangle D. Ultimately the 2−cycle must have both two periodic points on the border of the region D.
6. Conclusion. In this paper we have introduced a realistic assumption into a well-known oligopoly model, as defined in the baseline model. If the demand and all cost functions are linear, then the additional assumptions result in the loss of the linearity and the continuity of the dynamic model and the uniqueness of the steady state. Although in model II there are infinitely many steady states, defining a whole region ER, the invariant set ER is attracting under the same conditions of the baseline model. Clearly model II restricts and bounds the possible output values; therefore leads to a different behavior when the invariant region ER is not attracting. In the symmetric case all existing attracting 2−cycles are fully determined in Section 4.2. In the general case there are many possibilities, depending on the parameters. However, we have shown how to detect analytically the periodic points of the attracting 2−cycle, having at least one periodic point (or both) on the borders of the absorbing rectangle D.
The results show that with the added nonlinear constraints, we lost the ability to predict future outputs precisely at the unique steady state; however, for a moderately small speeds of adjustment K 1 and K 2 we can still expect the outputs in the long run to be in a steady state set, ER. When the speeds of adjustment are outside the stability region, the dynamics of model II become two-cyclical. For the symmetric duopoly case, we find that firms will either be both producing nothing or both producing at the amount that is equal to the monopoly output (in the static model) multiplied by the speed of adjustment or at the capacity limit. This result may first seem surprising but actually quite intuitive since firms are best responding to the rival firm's output level in the previous period. A firm will choose to produce at the monopoly level, while bounded by the capacity limit and adjusted by K, in the current period when the rival firm produced nothing in the last period. A similar result holds for the asymmetric case.
Without adaptive learning in the speed of adjustment parameters (as assumed in this paper), it may be best for the government to ensure that a small enough K is used by the firms to avoid the outcome swinging between nothing is produced or a total amount greater than the competitive equilibrium is produced.
For the future study, we plan to examine the dynamics of the system with a flexible capacity limit constraints, as a percentage of the current output levels. Interestingly, the dynamics and bifurcations become much richer, leading also to chaotic behaviors.