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We use a sample of 1669 quasars (r < 20.15, 3.6 < z < 4.0) from the Baryon Oscillation Spectroscopic Survey to study the intrinsic shape of their continuum and the Lyman continuum photon escape fraction (fesc,q), estimated as the ratio between the observed flux and the expected intrinsic flux (corrected for the intergalactic medium absorption) in the wavelength range 865-885 Å rest frame. Modelling the intrinsic quasar (QSO) continuum shape with a power law, Fλ ∝ λ-γ, we find a median γ = 1.30 (with a dispersion of 0.38, no dependence on the redshift and a mild intrinsic luminosity dependence) and a mean fesc,q = 0.75 (independent of the QSO luminosity and/or redshift). The fesc,q distribution shows a peak around zero and a long tail of higher values, with a resulting dispersion of 0.7. If we assume for the QSO continuum a double power-law shape (also compatible with the data) with a break located at λbr = 1000 Å and a softening Δγ = 0.72 at wavelengths shorter than λbr, the mean fesc,q rises to 0.82. Combining our γ and fesc,q estimates with the observed evolution of the active galactic nucleus (AGN) luminosity function (LF), we compute the AGN contribution to the UV ionizing background (UVB) as a function of redshift. AGN brighter than one-tenth of the characteristic luminosity of the LF are able to produce most of it up to z ~ 3, if the present sample is representative of their properties. At higher redshifts, a contribution of the galaxy population is required. Assuming an escape fraction of Lyman continuum photons from galaxies between 5.5 and 7.6 per cent, independent of the galaxy luminosity and/or redshift, a remarkably good fit to the observational UVB data up to z ~ 6 is obtained. At lower redshift, the extrapolation of our empirical estimate agrees well with recent UVB observations, dispelling the so-called Photon Underproduction Crisis. [ABSTRACT FROM AUTHOR]