  We investigate the constraints on models of supermassive black hole (SMBH)
and quasar formation obtainable from two recent observational developments: the
discovery of luminous quasars at z~6, and estimates of the local mass density
of SMBHs. If ~90 per cent of this mass was accreted at redshifts z<3, as
suggested by the observed quasar luminosity functions, these joint constraints
pose a challenge for models, which must account for the observed luminous
quasar population at z~6 within a very limited `mass budget'. We investigate a
class of models based within the hierarchical structure formation scenario, in
which major mergers lead to black hole formation and fuelling, and the
resulting quasars shine at their Eddington-limited rate until their fuel is
exhausted. We show that the simplest such model, in which a constant fraction
of the gas within the halo is accreted in each major merger, cannot satisfy
both constraints simultaneously. When this model is normalized to reproduce the
number density of luminous quasars at z~6, the mass budget is grossly exceeded
due to an overabundance of lower mass SMBHs. We explore a range of
modifications to the simple model designed to overcome this problem. We show
that both constraints can be satisfied if the gas accretion fraction scales as
a function of the halo virial velocity. Similar scalings have been proposed in
order to reproduce the local M-sigma relation. Successful models can also be
constructed by restricting the formation of seed black holes to redshifts above
z~11.5 or to haloes above a velocity threshold ~55 km/s, or assuming that only
a fraction of major mergers result in formation of a seed SMBH. (abridged)
