  Modeling the evolution of the number density distribution of quasars through
the Quasar Luminosity Function (QLF) is critical to improve our understanding
of the connection between black holes, galaxies and their halos. Here we
present a novel semi-empirical model for the evolution of the QLF that is fully
defined after the specification of a free parameter, the internal duty cycle,
$\varepsilon_{DC}$ along with minimal other assumptions. All remaining model
parameters are fixed upon calibration against the QLF at two redshifts, $z=4$
and $z=5$. Our modeling shows that the evolution at the bright end results from
the stochasticity in the median quasar luminosity versus halo mass relation,
while the faint end shape is determined by the evolution of the Halo Mass
Function (HMF) with redshift. Additionally, our model suggests the overall
quasar density is determined by the evolution of the HMF, irrespective of the
value of $\varepsilon_{DC}$. The $z\ge4$ QLFs from our model are in excellent
agreement with current observations for all $\varepsilon_{DC}$, with model
predictions suggesting that observations at $z\gtrsim7.5$ are needed to
discriminate between different $\varepsilon_{DC}$. We further extend the model
at $z\le4$, successfully describing the QLF between $1\le z\le4$, albeit with
additional assumptions on $\Sigma$ and $\varepsilon_{DC}$. We use the existing
measurements of quasar duty cycle from clustering to constrain
$\varepsilon_{DC}$, finding $\varepsilon_{DC}\sim0.01$ or
$\varepsilon_{DC}\gtrsim0.1$ dependent on observational datasets used for
reference. Finally, we present forecasts for future wide-area surveys with
promising expectations for the Nancy Grace Roman Telescope to discover
$N\gtrsim10$, bright, $m_{UV}<26.5$ quasars at $z\sim8$.
