A widely used class of quantitative trade models implicitly assumes that patterns of comparative advantage take a specific form such that they have no influence over the effect of trade barriers on aggregate trade flows and welfare. In this paper, I relax this assumption, developing a framework in which to analyze the role of interactions among countries' patterns of comparative advantage in determining the aggregate effects of trade barriers. My model preserves much of the tractability of standard aggregate quantitative trade models while allowing for the effects of any pattern of comparative advantage, across many products and countries, to be taken into account. After fitting my model to product-level trade data, I find that the composition of trade flows is quantitatively important in determining the welfare gains from trade and the aggregate effects of trade barriers. A key finding is that the welfare gains from trade tend to be larger and more skewed in favor of low-income countries than an aggregate model would suggest.