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# Structural Estimation of Non-Homothetic Demand Systems for Quantitative Trade Models

This thesis has three major chapters. Structural estimation of non-homothetic demands is the element that is the most common across the three papers in which structural parameters from the data.

**First Chapter**: Preference structures in applied general equilibrium models are commonly in favor of the family of linear expenditure system (LES) due to the desire for global regularity and applicability, while other emerging preference functions include the constant-elasticity-of-substitution (CES) forms that are used as sub-utility functions to fulfil regularity conditions with additional flexibilities. Hanoch (1975) introduces indirect, implicit additive relationships—a generalization of the CES—to obtain more flexible demand relationships that are globally regular. These preference relationships unlink substitution effects from income effects in ways that go beyond relaxation of homotheticity, and are more flexible than their direct dual. However, the estimation of these models as demand systems has proven to be challenging, and most published work in this area has focused on estimation approaches that involve approximations or that cannot fully identify parameter values in the preference relationships. Essay one introduces a direct approach which avoids approximations and allows parameters to be identified. We demonstrate the estimation using the readily accessible Global Trade Analysis Project (GTAP) and the confidential World Bank (International Comparison Program) databases, estimating the constant difference of elasticity or CDE directly in a maximum likelihood framework. In doing this, we show that the global regularity conditions stated in Hanoch (1975) can be slightly relaxed, and that the relaxed parametric conditions facilitate estimation. We introduce a normalization scheme that is beneficial for the scaling of the parameter values and which appears to have little impact on the economic performance of the estimated system. We develop a numerical test that justifies the normalization scheme. The series of procedures developed in this paper applied to this empirical example is generalized to solve many other econometric problems of general demand models of the Bergson family and those that are under-identified using reduced-form approaches.

**Second Chapter**: This paper presents a general equilibrium gravity model of trade based on the constant difference of elasticities of substitution preferences. Hanoch (1975) illustrates these preferences' advantages in terms of parsimony and flexibility. This paper introduces a parsimonious, non-homothetic and globally well-behaved demand model into the gravity model that both separates substitution effects from income effects and has non-constant substitution elasticities. These features of the demand model---together with the structural estimation procedure devised in this paper---allow nesting several prominent theoretical motivations for the gravity model, and exploring the merits of this more general model. They also allow identification of the elasticity of trade costs with respect to distance and asymmetric border coefficients from the elasticity of trade flows with respect to trade costs. Most previous studies cannot separately identify these structural parameters.

**Third Chapter**: The primary advantage of structural approaches to estimating the gravity model of trade is that they allow a transparent mapping of regression coefficients to structural parameters. Unfortunately, as shown in essay two, existing structural estimation methods are unable to separately identify trade costs and the trade elasticity without incorporating external data. We demonstrate that theoretical structure is alone sufficient for identifying all of the structural parameters of the canonical constant elasticity of substitution (CES) gravity model. We accomplish this by adopting an implicitly indirect representation of utility and estimating structurally using a mathematical program with equilibrium constraints. Our estimate of the elasticity of substitution is much smaller than in much of the rest of the literature, an outcome that we attribute to Pigou's Law, which ties income and substitution elasticities together in demand systems that assume additive preferences. This restriction is undesirable in demand systems, generally, and is a critical weakness for the canonical gravity model, a model that is commonly used to interpret the geographic trade pattern and to infer the welfare gains from trade. We demonstrate a non-homothetic CES model that both achieves identification and relaxes this restriction. Our counterfactual results based on the model suggest that the combination of a lower elasticity and lower trade costs generate a larger welfare change due to border removal compared to the CES model.