  We present the radio luminosity function (RLF) of optically-selected quasars
below 1~mJy, constructed by applying a Bayesian-fitting stacking technique to
objects well below the nominal radio flux-density limit. We test the technique
using simulated data, confirming that we can reconstruct the RLF over three
orders of magnitude below the typical $5\sigma$ detection threshold. We apply
our method to 1.4-GHz flux-densities from the Faint Images of the Radio Sky at
Twenty-cm survey (FIRST), extracted at the positions of optical quasars from
the Sloan Digital Sky Survey (SDSS) over seven redshift bins up to $z=2.15$
{and measure the RLF down to two orders of magnitude below the FIRST detection
threshold}. In the lowest redshift bin ($0.2<z<0.45$), we find that our
measured RLF agrees well with deeper data from the literature. The RLF for the
radio-loud quasars flattens below $\log_{10}[L_{1.4}/{\rm W\,Hz}^{-1}] \approx
25.5$ and becomes steeper again below $\log_{10}[L_{1.4}/{\rm W\,Hz}^{-1}]
\approx 24.8$, where radio-quiet quasars start to emerge. The radio luminosity
where radio-quiet quasars emerge coincides with the luminosity where
star-forming galaxies are expected to start to dominate the radio source
counts. This implies that there could be a significant contribution from star
formation in the host galaxies, but additional data is required to investigate
this further. The higher-redshift bins show a similar behaviour as for the
lowest-$z$ bin, implying that the same physical process may be responsible.
