We study a non-parametric class of neoclassical trade models with global production networks. We characterize their properties in terms of sufficient statistics useful for growth and welfare accounting as well as for counterfactuals. We establish a formal duality between open and closed economies and use it to analytically quantify the gains from trade. Accounting for nonlinear (non-Cobb-Douglas) production networks with realistic complementarities in production significantly raises the gains from trade relative to estimates in the literature. We use our general comparative statics results to show how models that abstract away from intermediates, no matter how well calibrated, are incapable of simultaneously predicting the costs of tariff and non-tariff barriers to trade. Given trade volumes and elasticities, accounting for intermediates doubles the losses from tariffs. Better quantitative accuracy demands the use of more complicated, oftentimes computational, models. This paper seeks to help bridge the gap between computation and theory.