In an earlier paper (Hawkins & Veron, 1993MNRAS.260..202H) we presented quasar luminosity functions in three redshift bins, derived from a variability selected sample. Here we provide a major extension to this survey, with a view to improving statistics and completeness, and extending the redshift range. The luminosity functions for redshifts of less than 2.2 show a featureless power law of the form {phi}=10^{beta}(M-M_0_)^, with no sign of a 'break'. The quasar luminosity function is also derived by the redshift range 2.2<z<3.2, which again shows a featureless power law, but with a steeper slope. When the data are combined with a new sample of bright quasars, the quasar luminosity functions in all four redshift ranges are consistent with a single power law of index {beta}=0.63. Plots of quasar space density as a function of redshift in three luminosity bins are also presented and show strong evolution at low redshift but nearly constant space density beyond a redshift of 2.