  We present a joint analysis of the rest-frame ultraviolet (UV) luminosity
functions of continuum-selected star-forming galaxies and galaxies dominated by
active galactic nuclei (AGNs) at $z \sim$ 4. These 3,740 $z \sim$ 4 galaxies
are selected from broad-band imaging in nine photometric bands over 18 deg$^2$
in the \textit{Spitzer}/HETDEX Exploratory Large Area Survey (SHELA) field. The
large area and moderate depth of our survey provide a unique view of the
intersection between the bright end of the galaxy UV luminosity function
(M$_{AB}<-$22) and the faint end of the AGN UV luminosity function. We do not
separate AGN-dominated galaxies from star-formation-dominated galaxies, but
rather fit both luminosity functions simultaneously. These functions are best
fit with a double power-law (DPL) for both the galaxy and AGN components, where
the galaxy bright-end slope has a power-law index of $-3.80\pm0.10$, and the
corresponding AGN faint-end slope is $\alpha_{AGN} = -1.49^{+0.30}_{-0.21}$. We
cannot rule out a Schechter-like exponential decline for the galaxy UV
luminosity function, and in this scenario the AGN luminosity function has a
steeper faint-end slope of $-2.08^{+0.18}_{-0.11}$. Comparison of our galaxy
luminosity function results with a representative cosmological model of galaxy
formation suggests that the molecular gas depletion time must be shorter,
implying that star formation is more efficient in bright galaxies at $z=4$ than
at the present day. If the galaxy luminosity function does indeed have a
power-law shape at the bright end, the implied ionizing emissivity from AGNs is
not inconsistent with previous observations. However, if the underlying galaxy
distribution is Schechter, it implies a significantly higher ionizing
emissivity from AGNs at this epoch.
