  This is the fourth paper in a series aimed at finding high-redshift quasars
from five-color imaging data taken along the Celestial Equator by the SDSS.
during its commissioning phase. In this paper, we use the color-selected sample
of 39 luminous high-redshift quasars presented in Paper III to derive the
evolution of the quasar luminosity function over the range of 3.6<z<5.0, and
-27.5<M_1450<-25.5 (Omega=1, H_0=50 km s^-1 Mpc^-1). We use the selection
function derived in Paper III to correct for sample incompleteness. The
luminosity function is estimated using three different methods: (1) the 1/V_a
estimator; (2) a maximum likelihood solution, assuming that the density of
quasars depends exponentially on redshift and as a power law in luminosity and
(3) Lynden-Bell's non-parametric C^- estimator. All three methods give
consistent results. The luminous quasar density decreases by a factor of ~ 6
from z=3.5 to z=5.0, consistent with the decline seen from several previous
optical surveys at z<4.5. The luminosity function follows psi(L) ~ L^{-2.5} for
z~4 at the bright end, significantly flatter than the bright end luminosity
function psi(L) \propto L^{-3.5} found in previous studies for z<3, suggesting
that the shape of the quasar luminosity function evolves with redshift as well,
and that the quasar evolution from z=2 to 5 cannot be described as pure
luminosity evolution. Possible selection biases and the effect of dust
extinction on the redshift evolution of the quasar density are also discussed.
