In this work, we study a game in which the players are the governments, the action of a player is the carbon emission level, and the utilities implement a tradeoff between the benefit for emitting which is the gross domestic product and the damage caused by climate changes. We discuss the choice of the benefit function, the damage function, and the choice of the geophysical model. Compared to existing works, our approach integrates the existence of a dynamics for the global earth temperature and CO2 concentrations and considers the damage functions obtained from the DICE model.