The density limit observed in tokamak experiments is thought to be due to a radiative collapse of the current channel. A transport code coupled with a magnetohydrodynamic (MHD) equilibrium routine is used to determine the detailed, self‐consistent evolution of the plasma profiles in tokamak discharges with radiated power close to or equaling the input power. The present work is confined to Ohmic discharges in steady state. It is found that the shape of the density profile can have a significant impact on the variation of the maximum electron density with plasma current. Analytic calculations confirm this result.