Is housing price distribution across cities, scale invariant? Fractal distribution of settlements' house prices as signature of self-organized complexity

Locations and sizes of settlements within regions often exhibit a self-organized central place pattern, where larger settlements are encircled by smaller ones, with the former more set apart. This hierarchical spatial pattern is displayed by the fractal signature of the power law typical of the rank-size distribution of settlements. Is a similar pattern also revealed in the housing prices across settlements? An empirical analysis on the almost 8000 Italian municipalities, whose residential population varies between 33 and similar to 3 million, shows that, for 15 out of 20 regions, the upper tails rank-house price distribution of settlements is plausibly following a power law. This result discloses a universality in the function behaviour: a scale invariance. The average scaling parameters have been estimated via Maximum Likelihood Estimator and Kolmogorov-Smirnov statistics, and the goodness-of-fit assessed via bootstrapping. Ordinary Least Squares regressions have also been used for comparative purposes. In average, Maximum Likelihood Estimator found a scaling parameter of 4.4. Namely, doubling the rank of the settlement (in terms of housing price), its housing price decreases by 15 %: between the settlement with the highest average house prices (ranked 1), and the settlement with the second highest average house prices (ranked 2), there is a 15 % difference in the average house prices. Idem between the ranked 5 and 10, 10 and 20 ... it does not matter; it is a universal behavioural pattern whose scale invariance suggests a self-organized complexity in the phenomena behind it.