We provide a tractable, quantitatively-oriented theory of innovation and technology diffusion to explore the role of international trade in the process of development. We model innovation and diffusion as a process involving the combination of new ideas with insights from other industries or countries. We provide conditions under which each country's equilibrium frontier of knowledge converges to a Fréchet distribution, and derive a system of differential equations describing the evolution of the scale parameters of these distributions, that is, countries' stocks of knowledge. The model remains tractable with many asymmetric countries and generates a rich set of predictions about how the level and composition of trade affect countries' frontiers of knowledge. We use the framework to quantify the contribution of bilateral trade costs to long-run changes in TFP and individual post-war growth miracles. For our preferred calibration, we find that both gains from trade and the fraction of variation of TFP growth accounted for by changes in trade more than double relative to a model without diffusion. ECONOMIC MIRACLES are characterized by protracted growth of per-capita income and productivity as well as increases in trade flows. The experiences of South Korea in the postwar period and the recent performance of China are prominent examples. These experiences suggest an important role played by openness in the process of development. 1Yet quantitative trade models relying on standard static mechanisms imply relatively small gains from openness and, therefore, cannot account for growth miracles. 2 These findings call for alternative channels through which openness can affect development. In this paper, we present and analyze a tractable, quantitatively-oriented model of an alternative suggested a strong relationship between openness and growth, although Rodriguez and Rodrik (2001) subsequently argued that many estimates in the literature suffered from econometric issues including omitted variables, endogeneity, and lack of robustness. More recent contributions to the literature have developed strategies to overcome some of these issues. To estimate the impact of trade on growth, Feyrer (2009a,b) studied the natural experiments of the decade-long closing of the Suez Canal in the 1970s and the long run decline in the cost of shipping goods by air, each of which had larger impacts on some pairs of countries than others, and Pascali (2017) studied the introduction of the steamship which affected some trade routes more than others. See also Lucas (2009b) and Wacziarg and Welch (2008), and Donaldson (2015) for reviews of the literature.2 See Connolly and Yi (2015) for a quantification of the role of trade on Korean's growth miracle. Atkeson and Burstein (2010) also found relatively small effects in a model with innovation.