This paper quantitatively explores the role of the demand structure in explaining the relationship between an importer's per capita income and the extensive margin of bilateral trade. The underlying mechanism is based on the fact that agents expand the set of goods they consume with income. This in turn affects the structure of a country's import demand and therewith the extensive margin of trade. We formalize this intuition by incorporating preferences that allow for binding non-negativity constraints into an otherwise standard Ricardian multi-country model. We quantify the model using the data on US consumer expenditures and aggregate values of bilateral trade flows and find that the behavior of the model's extensive margin of bilateral trade is consistent with the data (as opposed to the standard model). Two popular counterfactual experiments - lower trade costs and the rise of China and India - demonstrate that the mechanism outlined in this paper is indeed quantitatively important.