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 n-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.
Highlights
Oligopolies are among the most frequently examined models in mathematical economics
The stability switching curves are formed by points ðτ±1kðωÞ, τ∓2mðωÞÞ with k = m = 0 when ω runs through interval 1⁄2B − A,−ðA + BÞ and points ð~τ±1kðωÞ, ~τ∓2mðωÞÞ with k = m = 0 when ω runs through 1⁄2jA − ðn − 1ÞBj,−ðA + ðn − 1ÞBÞ: We numerically confirm Proposition 5
Environmental regulations were added to the classical n-firm Cournot model
Summary
The early results up to the mid70s 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. 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. 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 n-firm Cournot oligopolies by Matsumoto et al [13].
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