We present a multilevel algorithm to compute steady states of lattice Boltzmann models directly as fixed points of a time-stepper. At the fine scale, we use a Richardson iteration for the fixed point equation, which amounts to time-stepping towards equilibrium. This fine-scale iteration is accelerated by transferring the error to a coarse level. At this coarse level, one directly solves for the density (the zeroth moment of the lattice Boltzmann distributions), for which a coarse-level equation is known in some appropriate limit. The algorithm closely resembles the classical multigrid algorithm in spirit, structure and convergence behaviour. In this paper, we discuss the formulation of this algorithm. We give an intuitive explanation of its convergence behaviour and illustrate with numerical experiments.