We study a two-layer energy balance model, that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being captured by the other layer, and the effect of all non radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted ϵa. The value of ϵa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa. First we prove that global existence of solutions of the system holds if and only if ϵa∈(0,2), and blow up in finite time occurs if ϵa>2. (Note that the physical range of values for ϵa is (0,1].) Next, we explain the long time dynamics for ϵa∈(0,2), and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove in particular that the surface temperature increases with respect to ϵa. This is the key mathematical manifestation of the greenhouse effect.