We present a dynamic comparative advantage model in which moderate reductions in trade costs can generate sizable increases in trade volumes over time. A fall in trade costs has two effects on the volume of trade. First, for given factor endowments, it raises the degree of specialization of countries, leading to a larger volume of trade in the short run. Second, it raises the factor price of each country's abundant production factor, leading to diverging paths of relative factor endowments across countries and a rising degree of specialization. A simulation exercise shows that a fall in trade costs over time produces a non-linear increase in the trade share of output as in the data. Even when elasticities of substitution are not particularly high, moderate reductions in trade costs lead to large trade volumes over time. We present further empirical evidence in favour of our approach, documenting the link between trade liberalization and the cross-country divergence of investment shares.