Fertility, heterogeneity, and the Golden Rule

According to Phelps' Golden Rule, a rise in fertility decreases the optimal capital intensity, because a higher fertility increases the investment required to sustain a given capital intensity (the capital dilution effect). Using a matrix population model embedded in a two-period overlapping generation setting, we examine the robustness of that relationship to the partitioning of the population into two subpopulations having distinct fertility behaviors and entering the production process as distinct inputs. We show that, unlike what prevails under a homogeneous population, a rise in fertility (caused by a change in type-specific fertility) does not necessarily reduce the Golden Rule capital intensity. The intuition is that changes in type-specific fertility modify the composition of the labor force, which affects the marginal productivity of capital and the capital dilution effect. When the composition effect induced by the fertility change outweighs the standard capital dilution effect prevailing under a fixed partition of the population, a rise in fertility increases the optimal capital intensity. These results are robust to a finer description of heterogeneity, that is, a partitioning of the population into a larger number of subpopulations.