What is the relationship between trade and current account openness and growth? Can a catching-up economy borrow like Argentina or Spain and grow like China? To address these questions, we develop a model of endogenous converge growth, which we study under various policy regimes regarding trade and capital account openness. In the model, entrepreneurs adopt heterogenous projects based on their profitability. Trade openness has two effects on the relative profitability of tradable projects. First, the foreign competition effect unambiguously discourages tradable innovation. Second, the relative market size effect may favor or discourage tradable innovation. We show that balanced trade ensures that the two effects exactly offset each other, while trade deficits unambiguously favor non-tradable innovation. The increase in domestic consumption associated with international borrowing results in a relative market size effect that reinforces the foreign competition effect to discourage tradable innovation, as well as the aggregate innovation rate and the pace of productivity convergence. We further show that net exports relative to domestic absorption is a sufficient statistic for the feedback effect from aggregate allocation into sectoral productivity growth, and we find empirical support for the predictions of the model in the panel of sectoral productivity growth rates in OECD countries. A sudden stop in capital flows during the transition phase results simultaneously in a recession due to a fall in local demand and a sharp rebound in tradable productivity growth, provided the labor market can adjust flexibly via a sharp decline in the wage rate.