Mathematics of Generalized Versions of the Melitz, Krugman and Armington Models with Detailed Derivations

We provide detailed textbook style mathematical derivations of an extended version of the heterogenous firms model of Melitz (2003), as well as the Armington (1969) and Krugman (1980) models. Our model of heterogeneous firms extends the model of Melitz (2003) by allowing multiple sectors, intermediates, heterogeneous regions based on data, labor-leisure choice, initial heterogeneous tariffs as well as iceberg trade costs, multiple factors of production, the possibility of sector-specific inputs and trade balances based on data, and we incorporate global and unilateral tariff policy shocks. Although the models in this paper are extensions in numerous directions of the Melitz trade model of heterogeneous firms, the pedagogical approach in this paper should substantially facilitate the accessibility of the applied heterogenous-firms model of international trade for students and reviewers. Balistreri and Tarr (2020) apply these models to GTAP data where they assess the relative welfare impacts in the Armington, Krugman and Melitz style models of trade cost reductions in eighteen model variants. This paper documents the equations of those models, and we hope it will be a clear roadmap for understanding and constructing modern multi-sector, multi-region international trade models that must be fitted to data.