We present a measurement of the quasar luminosity function in the range 0.68 < z < 4 down to extinction corrected magnitude gdered = 22.5, using a simple and well understood target selection technique based on the time-variability of quasars. The completeness of our sample was derived directly from a control sample of quasars, without requiring complex simulations of quasar light-curves or colors. A total of 1877 quasar spectra were obtained from dedicated programs on the Sloan telescope (as part of the SDSS-III/BOSS survey) and on the Multiple Mirror Telescope. They allowed us to derive the quasar luminosity function. It agrees well with results previously published in the redshift range 0.68 < z < 2.6. Our deeper data allow us to extend the measurement to z = 4. We measured quasar densities to gdered < 22.5, obtaining 30 QSO per deg2 at z < 1, 99 QSO per deg 2 for 1 < z < 2.15, and 47 QSO per deg2 at z > 2.15. Using pure luminosity evolution models, we fitted our LF measurements and predicted quasar number counts as a function of redshift and observed magnitude. These predictions are useful inputs for future cosmology surveys such as those relying on the observation of quasars to measure baryon acoustic oscillations. © 2013 ESO.