to the SDSS data [@Choi2007], i.e.$$\begin{aligned}
\phi_{\star} & = & 8.0\times10^{-3}h^{3}\rm{Mpc}^{-3} \nonumber \\ 
\sigma_{\star} & = & 161\,\rm{km}\,\rm{s}^{-1} \nonumber \\
\alpha & = & 2.32 \nonumber \\
\beta & = & 2.67.\end{aligned}$$ Following @Gold2017, we narrow the range of velocity dispersions to $(50\,\rm{km}\,\rm{s}^{-1}, 400\,\rm{km}\,\rm{s}^{-1})$ for galaxies which can effectively act as gravitational lenses. The lower limit coincides also with the minimum velocity dispersion measured from the SDSS spectroscopic observations.

We assume that the velocity dispersion function does not evolve with redshift. Although observational constraints on the velocity dispersion function are limited at high redshifts, this assumption is corroborated by existing data showing no evidence of redshift evolution at $z\lesssim1$, especially for the high velocity dispersion tail [@Mont2017; @Bez2011]. The assumption is also supported by theoretical arguments based on the standard model for halo formation, for which @Mitch2005 showed that the normalization of the velocity dispersion function can be higher only by $\sim15$ per cent at redshift $z=1$, at which only rare galaxies can generate strongly lensed images of high-redshift supernovae detectable in surveys with depths comparable to the LSST.

The number of lens galaxies in an observational cone as a function of redshift and velocity dispersion is given by $$\frac{dN_{\rm 