We build into a Ricardian model sectoral linkages and differing productivity levels across sectors to understand how the gains from tariff reductions in a given sector spread to the rest of the economy. We also propose a new method to estimate sectoral trade elasticities consistent with any trade model that delivers a gravity trade equation. We apply our model and use our estimated elasticities to identify the impact of NAFTA's tariff reductions on exports and imports of all members. We find that the trade and welfare effects of tariff reductions are reduced by more than 40% when the structure of production does not take into account intermediate goods in production and input-output linkages. We then decompose the effects of tariff changes for NAFTA members and find that 93% of Mexico's, 58% of Canada's and 55% of the United States' trade effects due to tariff reductions can be attributed to NAFTA's tariff reductions.