In modern industrialized countries, human birth rates have been declining persistently for decades. In many cases they have now fallen below the replacement threshold. However, unlike in natural populations where population growth is constrained by limited resources, birth rates in modern industrialized countries are negatively correlated with resource availability. Here, declining birth rates in human populations are shown to be a manifestation of density-dependent population growth brought on by socioeconomic development. This is demonstrated by combining empirical power law relations between population size, gross domestic product (GDP) per capita, and fertility in a simple theoretical model describing population dynamics in developed countries. For a closed population, the model exhibits growth to a globally stable equilibrium population size, for both national and city populations. A version of the model that is open with respect to immigration and the influence of foreign technology and capital exhibits a good fit to long-term time series data on population size, GDP per capita, and birth rates for the United States, France and Japan.