J curve for abatement with transboundary pollution

In this note an alternative framework to Selden and Song's (1995) work is proposed in order to clarify some of their results and verify whether the relationship between pollution abatement and national income can be drawn as a J curve when there is more than one country involved. As Selden and Song do, pollution is considered as a flow externality. Their model is extended to a two country case which requires the use of differential games. The optimisation problem is decomposed into two stages, the pollution abatement and the accumulation problem. A J curve for pollution abatement is replicated and a static comparative analysis confirms that the smaller the rate of discount and/or the less polluting a technology is, the higher the stock of capital, current expenditure and pollution abatement will be at the steady state.