This article proposes a theory of city size distribution via a hierarchy approach rather than the popular random growth process. It does so by formalising central place theory using an equilibrium entry model and specifying the conditions under which city size distribution follows a power law. The force driving the city size differences in this model is the heterogeneity in economies of scale across goods. The city size distribution under a central place hierarchy exhibits a power law if the distribution of scale economies is regularly varying, which is a general class that encompasses many well‐known, commonly used distributions.

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This paper develops an index of allocative efficiency that depends upon the distribution of mark-ups across goods. It determines how changes in trade frictions affect allocative efficiency in an oligopoly model of international trade, decomposing the effect into the cost-change channel and the price-change channel. Formulas are derived shedding light on the signs and magnitudes of the two channels. In symmetric country models, trade tends to increase allocative efficiency through the cost-change channel, yielding a welfare benefit beyond productive efficiency gains. In contrast, the price-change channel has ambiguous effects on allocative efficiency.

City-size distributions are known to be well approximated by power laws across a wide range of countries. But such distributions are also meaningful at other spatial scales, such as within certain regions of a country. Using data from China, France, Germany, India, Japan, and the United States, we first document that large cities are significantly more spaced out than would be expected by chance alone. We next construct spatial hierarchies for countries by first partitioning geographic space using a given number of their largest cities as cell centers and then continuing this partitioning procedure within each cell recursively. We find that city-size distributions in different parts of these spatial hierarchies exhibit power laws that are, again, far more similar than would be expected by chance alone—suggesting the existence of a spatial fractal structure.