#### Abstract

In cases of nonpoint pollution sources, the regulator can observe the total emission but unable to distinguish between the firms. The regulator then selects an environmental standard. If the total emission level is higher than the standard, then the firms are uniformly punished, and if lower, then uniformly awarded. This environmental regulation is added to -firm dynamic oligopolies, and the asymptotical behavior of the corresponding dynamic systems is examined. Two particular models are considered with linear and hyperbolic price functions. Without delays, the equilibrium is always (locally) asymptotically stable. It is shown how the stability can be lost if time delays are introduced in the output quantities of the competitors as well as in the firms’ own output levels. Complete stability analysis is presented for the resulting one- and two-delay models including the derivations of stability thresholds, stability switching curves, and directions of the stability switches.

#### 1. Introduction

Oligopolies are among the most frequently examined models in mathematical economics. The early results up to the mid-70s are summarized in Okuguchi [1], and their multiproduct extensions are discussed in Okuguchi and Szidarovszky [2]. These models and the corresponding dynamic systems are linear, the asymptotical behavior of which is simple, since local asymptotical stability implies global asymptotical stability. From the early 90s, increasing attention was given to nonlinear oligopolies. The asymptotical stability of these models was examined by a variety of concepts including linearization, Lyapunov functions, and the critical curve methods among others. Bischi et al. [3] offer a comprehensive summary of these developments. In all previous models, instantaneous information was assumed about the actions of the competitors as well as about the own output selections of the firms. However, data collection, determining appropriate actions, and their implementations need time; therefore, delayed models describe reality more accurately. More recently, Matsumoto and Szidarovszky [4] offer a collection of delayed dynamic oligopolies with a brief summary of the used mathematical methodology as well as with discussions on different types of oligopoly models.

A large variety of oligopoly models consider environmental issues. The effects of different environmental regulation policies are examined by many authors including Downing and White [5], Segerson [6], Jung et al. [7], Montero [8], and Okuguchi and Szidarovszky [9, 10]. In the case of nonpoint source pollution in which neither the source nor the size of specific emissions can be observed, the government can define a cut-off value for the total emission level of the entire industry but cannot distinguish among the firms due to the information asymmetry. If the total emission level exceeds the cut-off value, then the firms are punished, otherwise rewarded. In the early stages, the existence of the Nash equilibrium was the main focus and how the governmental policy affects the total pollution level of the industry. Depending on the selected model, increased ambient charge in duopolies can lead to higher pollution levels [11], and in other cases, to lower pollution [12]. This result is generalized for -firm Cournot oligopolies by Matsumoto et al. [13]. Dynamic models are introduced, and their asymptotical behavior was examined without and with time delays. The corresponding Bertrand models are considered and investigated in Ishikawa et al. [14]. Hyperbolic duopolies and triopolies are studied in Matsumoto et al. [15] with static and dynamic analysis. Three-stage optimum models are introduced in Matsumoto et al. [16] for Cournot duopolies without product differentiation.

This paper extends and further generalizes the earlier methodology and stability results for two particular models. Linear and hyperbolic oligopolies are discussed with ambient charges or rewards, with selected cut-off pollution levels for the entire industry. As the dynamic models are very similar, we will present the complete analysis in detail for the general case including both particular models. The main results can be summarized as follows: (i)In the case of one delay, the model with symmetric firms has a critical value of the delay and stability is preserved for , is lost for via Hopf bifurcation, and cannot be regained for (ii)In the case of two delays, stability switching curves are constructed and the directions of the stability switches are analytically shown

Two notices of our methodology are given. The first one concerns whether dynamic consideration should be global or local. The asymptotic behavior of trajectories of differential equations without and with delays is usually examined by two different methods. One is the construction and investigation of the properties of an appropriate Lyapunov function. Unfortunately, there is no general tool to create Lyapunov functions in general, which makes the application of this methodology very difficult. In our future study, we will elaborate on the possibility of applying it to different extended versions of oligopolies. The other approach is based on local linearization. This is the method we use in this paper. The other concerns whether the model should be deterministic or stochastic. In our study, the dynamic model is given and the model parameters are considered deterministic. However, in many cases, there is uncertainty in the mathematical formulation and parameter values. This uncertainty can be modeled either by stochastic (e.g., [17], or [18]) or by fuzzy [19, 20] approach. We plan to extend our study into this direction in the future.

This paper is developed as follows. In Section 2, the mathematical models are introduced. In Sections 3 and 4, one-delay and two-delay models are examined. In both cases, two special cases are discussed in detail: symmetric firms and general duopolies. Conclusions and further research directions are outlined in Section 5.

#### 2. The Mathematical Models

An oligopoly of firm is considered. Let denote the output of firm , the output of the rest of the industry, and the industry output. Assume that the price function of the product by firm is given as . Firm emits pollutions in connection with its production, so the total amount of pollutions is . The government can measure this total amount and unable to distinguish between the firms. An exogenously determined environmental standard is selected by the government, and a is chosen to determine the emission penalty or award for the firms. If is the production unit cost of firm , then its payoff is given as

Assume that at time , each firm has only delayed information about the outputs of the competitors, so the payoff of firm is the following:

The gradient adjustment process of firm is driven by the delay differential equation where is the speed of adjustment of firm and the marginal profit is

Let

Then, equation (3) can be rewritten as

To linearize this equation, let and notice that for all , since by differentiation, the last term of cancels out. Then, the linearized equation has the form where is the difference between and its equilibrium level.

Before proceeding to the stability analysis of the system, two important cases are introduced.

*Case 1. *Assume differentiated products, when the price of the product of firm is as follows:
where is the maximum price and represents the substitutability of the products, . In this case,
therefore,
and so
Notice that for all ,

*Case 2. *Assume a hyperbolic oligopoly without product differentiation and common price function
where is a positive constant. In this case, the profit function is rewritten as
consequently,
Then at the equilibrium,
where and are the equilibrium levels of and . Similarly,
where is the equilibrium level of .

Notice that and if there is no dominant firm, then . In the rest of this paper, we will assume the absence of a dominant firm. So we will assume that

#### 3. Single Delay Stability

To determine the characteristic equation of model (5), assume exponential solution forms, , then substituting them into (5) gives

So the characteristic equation can be written as where the simplifying notations and are used. Let to have where the identity discussed in Bischi et al. [3] is used. So

We have now two possibilities. Consider the first equation Without delay and from (26), . Stability switch might occur if with some . Then, (26) implies that

Separation of the real and imaginary parts shows that

Adding the squares of these equations gives that since . There is no solution for .

Consider next the equation

Proposition 1. *System (5) is locally asymptotically stable if all roots of equation (30) are negative real numbers or complex with negative real parts.*

In Case 1, system (5) is linear, so the asymptotical stability is global. Equation (30) is very complicated in general, so two special cases are examined.

##### 3.1. Case of Symmetric Firms

Assume , then (30) is specialized as

Without delay, . Stability switching might occur with , and substituting it into (31) yields which implies that

By adding the squares of these equations, we have

If then the right-hand side is nonpositive with no solution and without stability switch. Otherwise,

From (33), it is clear that and ; furthermore, the critical values of the delays are

The direction of stability switches can be determined by using Hopf bifurcation. Let be selected as the bifurcation parameter and consider as a function of By implicitly, differentiating equation (31) with respect to shows that where equation (31) is used again. At the critical value the real part of which has the same sign as

Proposition 2. *In the symmetric case, the equilibrium is locally asymptotically stable if , otherwise if . At stability is lost via Hopf bifurcation and stability cannot be regained with larger values of .*

In the linear case,

Furthermore with so which is nonnegative if

One numerical example is given to confirm Proposition 2. Only for analytical simplicity, (that is, no product differentiation) and are assumed. From (36), the critical value of the delay at which stability is lost is given by

The down-sloping curve is illustrated in Figure 1(a) with It divides the parameter space of and into two subregions. Stability is held for and in the shaded region below the curve. For the critical value of the delay is denoted as the red point on the curve, and stability is lost. A resultant limit cycle is depicted in Figure 1(b). When increases, the curve shifts downward, leading to a smaller stability (shaded) region.

##### 3.2. General Duopolies

In the case of , equation (22) shows that which is a single-delay equation with :

Without delay, and (46) becomes

Since for both the linear coefficient and the constant term are positive implying that the roots are negative real values or complex with negative real parts. Stability switch might occur if with when from (46) we have implying that

By adding the squares of these equations and arranging the terms, we have

Since for all coefficients are positive showing that no stability switch can occur.

Proposition 3. *The equilibrium in a duopoly is always locally asymptotically stable with all *

#### 4. Two-Delay Stability

Assume next that firm has a delay in its output and delay in the outputs of its competitors. Then, the dynamic equation (9) modifies as follows:

Similarly to equations (22) and (25), the characteristic equation can be derived as

Proposition 4. *The equilibrium of system (51) is locally asymptotically stable if all roots of (52) are negative reals or complex with negative real parts.*

Similarly to the previous model, two special cases will be reexamined.

##### 4.1. Case of Symmetric Firms

Assume now that and . From (52), we have to consider two cases. In the first case, we examine the equation which can be rewritten as with

We will apply the method introduced by Gu et al. (2005) and discussed in detail in Matsumoto and Szidarovszky [4]. Notice that

The range of is determined by conditions, which simplify in our case as

So runs through interval . Moreover, by the law of cosine,

The stability switching curves are given by pairs with

In the second case, we consider the other equation which can be rewritten as with

Then,

The range of is determined again based on conditions (57), which are the following in this case: so a range of is in the interval,

By the law of cosines,

The stability switching curves are given by pairs with

Proposition 5. *The stability switching curves are formed by points with when runs through interval and points with when runs through *

We numerically confirm Proposition 5. To this end, we take and in the second case. The stability switching curve in the first case consists of the pairs of and illustrated as the dotted red-blue curve in Figure 2(a). It can be verified by equation (53) that the model with symmetric firms is stable without delays (i.e., ). Hence, stability is preserved for to the left of the curve. On the other hand, the stability switching curve in the second case, the locus of , is illustrated as the solid red-blue curve. Due to equation (61), the model in the second case is also stable without delays, implying that stability is preserved for below the curve. The lower-left corner surrounded by the dotted line is enlarged in Figure 2(b). The model is stable in the region including the origin and bounded by the solid red-blue curve and the dotted blue curve.

The directions of stability switching in the first case can be assessed by computing the following expressions: with real and imaginary parts, and finally, which has the same sign as

The directions of stability switches in the second case can be determined similarly to the first case. Notice that with real and imaginary parts, and therefore, which has the same sign as Then, we have the following:

Proposition 6. *Let be a point on the stability switching curve and assume that the curve is crossed at this point from right to left when we are looking forward to increasing values of on the curve. If is positive, then at least one pair of eigenvalues changes the sign of the real part from negative to positive. If is negative, then the sign change is in the opposite direction.*

##### 4.2. General Duopolies

In the case of from (52), we have which can be simplified as

This equation is analytically intractable since it has four delays, Therefore, we make the following simplifying assumption: when (76) becomes

By multiplying both sides by , we have

Without delay, a quadratic equation is obtained by

The linear coefficient and constant term are positive; the roots are negative real values since the discriminant is positive. Stability switch might occur if with then from (79),

By separating the real and imaginary parts, we have

We have to consider now two possibilities from the first equation of (82). then, and from the second equation of (82),

However, where the first term is positive and the second is being nonnegative. If it is positive, (85) has no solution. If it is zero, then implying that which is a contradiction. The other possibility is then, from the second equation of (82), and

The discriminant is furthermore, so both roots are positive,

Notice that if and one of and equals to zero. Then, . From the first equation of (82), the critical value of the delay is

The direction of the stability switches are assessed by considering as the function of the bifurcation parameter, and implicitly differentiating equation (79):

The multiplier of and the constant term are respectively, so

Notice that at the critical values of and implying that

At we are interested in the real part of (95), where the second term is pure complex. Therefore,

Substituting (91) with numerator into the denominator presents

We know however that so this expression is positive at and negative at . Hence, is always positive.

Proposition 7. *The equilibrium is locally asymptotically stable for , stability is lost at via Hopf bifurcation, and stability cannot be regained with larger values of *

#### 5. Conclusions

Environmental regulations were added to the classical -firm Cournot model. In cases of nonpoint source pollution, the regulator can measure only the total emission level without knowing the individual emissions of the firms. Therefore in the regulation, the firms are uniformly punished if the total emission is higher than a regulator selected standard, and awarded otherwise. In the dynamic extensions, we considered three cases. First, no delays were introduced about the information on the outputs of all firms; second, delayed data were assumed about the output levels of the competitors; and third, additional delays were added in the firms’ own output levels. The stability analysis was conducted under general conditions that are satisfied in cases of linear and hyperbolic price functions. Models without delays, with a single delay, and with two delays were analyzed in detail. It was demonstrated how the stability of the no-delay models can be lost by introducing delays. The stability thresholds, stability switching curves, and directions of stability switches were analytically derived.

It will be an interesting project to extend the results of this paper to more general cases including nonlinear Cournot models, multiproduct, labor-managed oligopolies, and rent-seeking games among others. This will be the subject of our continued research project. In addition, the possible uncertainty of model parameters will be considered in the extended oligopoly models with a fuzzy and stochastic approach.

#### Data Availability

We do not use data for our research.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

The first author highly acknowledges the financial supports from the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 20K01566) and Chuo University (Grant for Special Research). The usual disclaimers apply.