The analysis finds that in addition to U-shaped paths of environmental quality arising for growth in income per capita, growth in population can also produce socially efficient patterns that are U-shaped. Sufficient conditions for both types of paths are identified for a range of models and parameters, including symmetrical models with homothetic, constant-returns functions such as with CES functions. Similar results are also shown to arise in decentralized economies under either homogeneous or heterogeneous income levels.
