New growth theories suffer from their counterfactual prediction that a higher (rate of) population (growth) generally implies a higher growth rate of per capita income. This population puzzle is solved by introducing imported intermediates and an exogenously growing world export demand creating a Keynesian balance of payments constraint in a neoclassical model of non-scale growth. It is shown among other things that the growth rate of per capita consumption may decline or increase in the rate of population growth depending on the relative magnitudes of particular elasticities. The growth rate of per capita income measured in terms of the intermediate produced abroad will fall in the population growth rate if world export demand is price-inelastic. This elasticity is also decisive in whether the implications of the model are more Keynesian or more neoclassical.