Feenstra (1994) developed, and Broda and Weinstein (2006) refined, a structural estimator to estimate import demand and supply elasticities. Working through the first principles of the methodology from Leamer (1981), this paper analyzes and improves the technique to provide a unified estimator of import supply and demand elasticities. The proposed LIML routine corrects small sample biases and constrained search inefficiencies. Standard estimates are shown to overestimate the median elasticity of substitution by over 35%. Applied to U.S. import data from 1993-2007, the biases of the standard estimates translate into an understatement of consumer gains from product variety by a factor of 6. Lastly, I investigate the implications of violations to the underlying assumptions of the model.