A Centered Index of Spatial Concentration: Expected Influence Approach
Spatial Concentration, Expected Influence, Expected Utility, Population Concentration, Capital Cities, Gravity, CRRA, Harmonic Functions, Axiomatics
We construct a general axiomatic approach to measuring spatial concentration around a center or capital point of interest, a concept with wide applicability from urban economics, economic geography and trade, to political economy and industrial organization. By analogy with expected utility theory, we propose a basic axiom of independence (sub-group consistency) and continuity for a concentration order that ranks any two distributions relative to the capital point. We show that this axiom implies an expected influence representation of that order, conceptualizing concentration as an aggregation of the expected influence exerted by the capital on all points in the relevant space (or vice-versa). We then propose two axioms (monotonicity and rank invariance) and prove that they imply that the associated influence function must be a decreasing isoelastic function of the distance to the capital. We apply our index to measure the concentration of population around capital cities across countries and US states, and also in US metropolitan areas. We show its advantages over alternative measures, and explore its correlations with many economic and political variables of interest.