One can define an effective dissipation for highly iterated two-dimensional maps or recursion relations. The convergence rate ${\ensuremath{\delta}}_{\mathrm{eff}}$ of the sequence of period-doubling bifurcations is shown to depend in a universal way on this effective dissipation in the limit of long ${2}^{n}$ cycles. These conclusions are based on both a renormalization argument and numerical calculations. The paper concludes with a brief discussion of the implications for physical systems.