In this paper, we propose a differential game model with a coupled constraint to represent the possible effects of climate agreements between industrialized, emerging and developing countries. Each group of countries is represented by an economic growth model where two different types of economies, called respectively 'low-carbon' and 'carbon' can co-exist, each of which having different productivities of capital and of emissions due to energy use. We assume that each group of countries participating in the negotiations has identified a damage function, which determines a loss of GDP due to warming and has also a possibility to invest in a capital permitting adaptation to climate changes. The climate agreements we consider have two main components: (i) they define a global emission budget for a commitment period and impose it as a limit on cumulative emissions during that period; (ii) they distribute this global budget among the different coalitions of countries taking part in the agreement. This implies that the game has now a coupled constraint for all participants in the negotiations. The outcome of the agreement is therefore obtained as a generalized or 'Rosen' equilibrium which can be selected among a whole manifold of such solutions. We show that the family of Nash equilibria in the games obtained through a distribution of the total budget among the different parties corresponds to the manifold of normalized equilibria. We then propose an equity criterion to determine a fair division of this total emission budget, or equivalently to select a proper weighting for a normalized equilibrium.