Malthus postulated that rapidly expanding populations will always experience premature death, since carrying capacity can only grow linearly. However population growth in industrialized countries is now curbed not by famine but by wealth, as fertility declines dramatically with rising per capita gross domestic product (GDP). Here, we show how wealth can be seen as a cause of density-dependent growth in the sense of classical population biology models. Starting from the Cobb-Douglas production function--which relates labor, technology and capital to GDP--and several empirical observations, we derive a population model with density-dependent growth that describes population dynamics in industrialized countries. Despite the power law relationship between fertility and GDP per capita, the model exhibits a globally stable population size for a given level of contribution to GDP from labor, technology and capital. Opening the population (to an inflow from other populations) indirectly suppresses the birth rate by significantly increasing GDP per capita. This exemplifies how human populations are subject to constraints that are similar, but not identical, to those experienced by other animal populations.