Mathematical modeling is used to estimate the characteristics of the upconversion-pumped Er:YAG laser emitting in green function of the erbium concentration, reflectivity of the coupling mirror and co-doping. Besides the excited-state absorption, cooperative upconversion processes as well as cross-relaxation are included in our model. The main drawback of the Er:YAG system is the presence of a parasitic absorption at the emission wavelength, originating from the 4I13/2 level. The results of our simulations suggest that the highest efficiency could be obtained for an erbium concentration of 5 at.% and a reflectivity of the coupling mirror 0.94. Tentative reducing the population of the 4I13/2 level by co-doping would have a weak but negative influence.