1) The document discusses a model for constraining the evolution of supermassive black holes through solving the continuity equation, using observational data on quasar luminosity functions and scaling relations between black hole mass and galaxy properties. 2) The model assumes parametric forms for the duty cycle of active black holes, Eddington ratio distribution as a function of redshift, and solves the continuity equation backwards in time using the local black hole mass function as a boundary condition. 3) The best-fit model reproduces observational data on luminosity functions and mass functions of active black holes, and implies an anti-hierarchical growth of black holes with cosmic downsizing. The model can constrain the radiative efficiency and favors values around 5-10