A Complex Analysis On A Dynamic Duopoly Game With Heterogeneous Players And Emission Trading

An oligopoly is a market structure which is controlled by only a few numbers ofproducers. As it is known, the interreactional behaviors between oligopolists makeoligopoly the most complicated one amongst all kinds of market structrures. It iscommon to observe the existence of oligopoly market in real economic life, due to theconcentration of capital, the closed nature of areas and policy constraints, etc. As anemerging economy, China has transformed to socialist market economy from plannedeconomy by carrying out the reform and openning policies. Many areas in China’seconomy have the characteristics of an oligopoly market.Emission trading system, as a market-based environmental protection practice, hasa lower level of managing costs compared with traditional Pollution Discharge FeeSystem and it has economic stimulating ability in the aspect of total pollution control.However, many literatures has pointed out that the efficiency of emission tradingsystem largely depends on market structure. For the case of perfect-competition,emission trading is able to show its advantages. But for a non-idealized market withmonopoly power, emissions trading system may lead to a distortion of the market,which will reduce the effect of pollution control.In this paper we develop a micro-founded duopoly game model with differentiatedproducts. Polluting behaviors and emission trading system are introduced under theframework of Bowley duopoly game. We analyze the impacts of emission tradingsystem to the Bowley-Nash equilibrium. The results show that the differentiation levelof products, the price of tradable permits inside the emission trading market and theduopolist’s respectivecoefficients ofpollutiongenerationhaveimpacts onthechangingin equilibrium output, while the duopolist’s coefficients of pollution treatment and the initial level of emission permits do not have influence on it. Besides, for the case thatsubstitutability between goods exists (in fact this case is more common in an oligopolymarket), if one polluting firm’s coefficient of pollution generation is sufficiently smallcompared to its competitor’s, the pollution control policy and emission trading systemwill cause increase in its equilibrium output. This indicates that emission trading canimpel polluting firms to improve its production technology so as to reduce thecoefficient of pollution generation and gain more profit, which will also improve theenvironment.Then we abandon the classic assumption of fully rational behaviors and introduceheterogeneous bounded-rational mechanisms of output adjustment, which are morerealistic, to each duopolist and created the two-dimensional nonlinear dynamic systemdescribing the dynamic game between bounded-rational firms. Mathematical studiesshow that the dynamic system has an unstable boundary equilibrium and a locallystable interior equilibrium, which is totally the same with the Bowley-Nashequilibrium of the static fully-rational game we discussed before. However, the stabilityconditions of will propose stricter requirements to the parameters of the model.When the parameters cannot satisfy the stability conditions (though compatible to thebasic model), the dynamic behaviors of the duopoly may be complex.Then we use numerical simulations to analyze the complex dynamic behaviors ofthe nonlinear system that may arise when the values of the two parameters of boundedration (the speed of adjustment and adaptive adjustment parameter) and the priceof emission permits change. Numerical simulations show that:(1) The speed ofadjustment based on marginal profit has a strong disturbing impact on the stabilityof equilibrium. When the value of increases, bifurcations and chaos will be observed;(2) Adaptive adjustment parameter may also destabilize the market equilibrium.However its destabilize power is not that strong as;(3) Under different conditions ofthe other model parameters, the price of tradable emission permits has different impactson the stability of the system’s equilibrium. For a certain case of market mechanismdetermined by a type of parameter condition, the increase in the price of permits willalso cause chaotic behaviors in the system, thus causing unpredictable distortions and damaging effects to the market.