This paper presents a model of long-run growth where knowledge is treated as an input in production with increasing marginal productivity. Unlike models based on diminishing returns, this model allows for increasing growth rates over time, amplification of small disturbances by private agents, and larger countries growing faster than smaller ones. The model is a competitive equilibrium model with endogenous technological change, and it offers empirical evidence to support its theoretical framework. The paper discusses the historical origins of the idea of increasing returns and the conceptual difficulties that have impeded the development of formal models. It also presents empirical evidence from historical data on productivity growth and cross-country comparisons, which suggest that growth rates have been increasing rather than decreasing. The paper then introduces a simplified two-period model to illustrate the tools used to analyze a competitive equilibrium with externalities and increasing returns. Finally, it presents an infinite-horizon, continuous-time version of the model, characterizing the social optimum and the competitive equilibrium, both with and without optimal taxes. The key assumption is that knowledge has increasing marginal productivity, which allows for unbounded growth in per capita output.This paper presents a model of long-run growth where knowledge is treated as an input in production with increasing marginal productivity. Unlike models based on diminishing returns, this model allows for increasing growth rates over time, amplification of small disturbances by private agents, and larger countries growing faster than smaller ones. The model is a competitive equilibrium model with endogenous technological change, and it offers empirical evidence to support its theoretical framework. The paper discusses the historical origins of the idea of increasing returns and the conceptual difficulties that have impeded the development of formal models. It also presents empirical evidence from historical data on productivity growth and cross-country comparisons, which suggest that growth rates have been increasing rather than decreasing. The paper then introduces a simplified two-period model to illustrate the tools used to analyze a competitive equilibrium with externalities and increasing returns. Finally, it presents an infinite-horizon, continuous-time version of the model, characterizing the social optimum and the competitive equilibrium, both with and without optimal taxes. The key assumption is that knowledge has increasing marginal productivity, which allows for unbounded growth in per capita output.