The cavity model is a widespread empirical approach for the numerical simulation of patch antennas. An attempt to give a rigorous mathematical background for this way to proceed is presented. The justification is carried out in the framework of a two-dimensional representation of the underlying radiation problem. It is obtained by a suitable application of the method of matched asymptotic expansions. Furthermore, it is shown how to improve the cavity model by pushing the asymptotic expansion to the next order. A remarkable outcome of the asymptotic expansions is that they clearly confirm that the way to feed the antenna only determines the level of its radiation pattern but not its shape. Some numerical experiments are given to illustrate the theoretical developments.