Nonlinear Gravity

We extend the results of Arkolakis, Costinot and Rodriguez-Clare (2012) to environments with non-homotheticities and show how to measure the welfare gains from trade in a broad class of static, non-CES trade models. In these models, the elasticity of import intensity with respect to trade costs (trade elasticity) is a function, not a constant, implying a nonlinear version of the gravity relationship commonly studied in the empirical trade literature. We have two main results. First, we provide an explicit formula for the trade elasticity, which allows it to be computed in any particular non-homothetic model. Second, we prove that, even in this environment, the elasticity of welfare with respect to trade intensity (welfare elasticity) is equal to the reciprocal of the trade elasticity. We provide several examples of models that are impossible to solve analytically, yet where the welfare elasticity can be solved in closed form using our procedure allowing for one to get a closed form for the gains from trade. We also provide sufficient conditions to compare the gains from trade implied by non-homothetic models to those implied by CES models.