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Transportation networks, such as railways, roads and highways provide standard examples of natural monopolies. Since the introduction of the term "natural monopoly" by T. Malthus in 1815, this concept has been defined in different ways by several authors (F. Bastiat, J. S. Mill or L. Walras). The current formal definition is due to Baumol (1977) and is based on the subadditivity of the cost functions. After estimating the cost functions, the researcher can test whether subadditive holds or not. Natural monopolies are associated to market efficiencies, which call for regulation (e.g., price cap regulation and Ramsey-Boiteux regulatory policy). As a key example, the econometric study of the British railways in the 19 th century shed light on the difficulty of regulating natural monopolies.

We develop a class of demand models for differentiated products. The new models facilitate the BLP method (Berry et al., 1995) while numerical inversion of the demand system is not required. They can accommodate rich patterns of substitution and complementarity while being easily estimated with standard regression techniques and allowing very large choice sets. We use the new models to describe markets for differentiated products that exhibit segmentation according to several dimensions and illustrate their application by estimating demand for cereals in Chicago.