The expected evolution of quasars is calculated using the dynamics of merging galaxies and the assumption that the birth rate of quasars is proportional to the rate of mergers. The merger rate is described using a modification of the Press-Schechter theory. As a specific model, the luminosity of a quasar is assumed to be proportional to its host galaxy's halo mass, and the amount of gas supplied per merger is independent of the quasar's luminosity, implying a lifetime that is inversely proportional to luminosity. To be an acceptable description of the available quasar luminosity function data requires (1) that high-luminosity quasars, absolute magnitude ~ -28, have an effective lifetime of ~ 1% of the Hubble age; (2) that galaxies merge significantly less than their initially associated dark halos; (3) that the perturbation spectrum be similar to the cold dark matter spectrum on galaxy scales; and (4) not all mergers lead to quasar activity. The main strength of the model is its relation to the dynamics of clustering; its main weaknesses are the poorly constrained relations between host galaxies and quasars.