Mathematics of the Armington, Krugman and Melitz Models with Multiple Sectors and Heterogeneous Regions, 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 and the possibility of sector-specific inputs. Balistreri and Tarr (2019) apply these models to data where they assess the relative welfare impacts of trade cost reductions. We hope this will be a clear roadmap for understanding and constructing modern multi-sector, multi-region international trade models that must be fitted to data.