Using a complete, magnitude-limited sample of active galaxies from the Sloan Digital Sky Survey (SDSS) we show that the fraction of broad-line (Type 1) active galactic nuclei increases with luminosity of the isotropically-emitted [O III] narrow emission line. Our results are quantitatively in agreement with, and far less uncertain than, similar trends found from studies of X-ray and radio-selected active galaxies. While the correlation between broad-line fraction and luminosity is qualitatively consistent with the receding torus model, its slope is shallower and we therefore propose a modification to this model where the height of the torus increases slowly with AGN luminosity. We demonstrate that the faint-end slope of the AGN luminosity function steepens significantly when a correction for `missing Type 2 objects is made and that this can substantially affect the overall AGN luminosity density extrapolated from samples of more luminous objects.
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