Economic dynamics with differential fertility

Abstract: We characterize the outcomes of a canonical deterministic model for the intergenerational transmission of capital that features differential fertility. A fertility function determines the relationship between parental capital and the number of children, and a transmission function determines the relationship between the capital of a parent and that of their children. Together these functions generate an evolving cross-sectional distribution of capital. We establish easy-to-verify conditions on the fertility and transmission functions that guarantee (a) that the dynamical system has a steady state distribution that is either atomless (exhibiting inequality) or degenerate (not exhibiting inequality), and (b) that the system converges to such states from essentially any initial distribution. Our characterization provides new insights into the link between differential fertility and long-run cross-sectional inequality, and it gives rise to novel comparative statics relating the two. We apply our results to several parametric examples and to a model of economic growth that features endogenous differential fertility.