  Quasars are widely believed to be powered by accretion onto supermassive
black holes and there is now considerable evidence for a link between mergers,
quasars and the formation of spheroids. Cattaneo, Haehnelt & Rees (1999) have
demonstrated that a very simple model in which supermassive black holes form
and accrete most of their mass in mergers of galaxies of comparable masses can
reproduce the observed relation of black hole mass to bulge luminosity. Here we
show that this simple model can account for the luminosity function of quasars
and for the redshift evolution of the quasar population provided a few
additional assumptions are made. We use the extended Press-Schechter formalism
to simulate the formation of galaxies in hierarchical models of the formation
of structures and we assume that, when two galaxies of comparable masses merge,
their central black holes coalesce and a fraction of the gas in the merger
remnant is accreted by the supermassive black hole over a time-scale of about
10^7 yr. We find that the decrease in the merging rate with cosmic time and the
depletion in the amount of cold gas available due to the formation of stars are
not sufficient to explain the strong decline in the space density of bright
quasars between z=2 and z=0, since larger and larger structures form, which can
potentially host brighter and brighter quasars. To explain the redshift
evolution of the space density of bright quasars between z=2 and z=0 we need to
assume that there is a dependence on redshift either in the fraction of
available gas accreted or in the time-scale for accretion.
