  Urban scaling analysis, the study of how aggregated urban features vary with
the population of an urban area, provides a promising framework for discovering
commonalities across cities and uncovering dynamics shared by cities across
time and space. Here, we use the urban scaling framework to study an important,
but under-explored feature in this community - income inequality. We propose a
new method to study the scaling of income distributions by analyzing total
income scaling in population percentiles. We show that income in the least
wealthy decile (10%) scales close to linearly with city population, while
income in the most wealthy decile scale with a significantly superlinear
exponent. In contrast to the superlinear scaling of total income with city
population, this decile scaling illustrates that the benefits of larger cities
are increasingly unequally distributed. For the poorest income deciles, cities
have no positive effect over the null expectation of a linear increase. We
repeat our analysis after adjusting income by housing cost, and find similar
results. We then further analyze the shapes of income distributions. First, we
find that mean, variance, skewness, and kurtosis of income distributions all
increase with city size. Second, the Kullback-Leibler divergence between a
city's income distribution and that of the largest city decreases with city
population, suggesting the overall shape of income distribution shifts with
city population. As most urban scaling theories consider densifying
interactions within cities as the fundamental process leading to the
superlinear increase of many features, our results suggest this effect is only
seen in the upper deciles of the cities. Our finding encourages future work to
consider heterogeneous models of interactions to form a more coherent
understanding of urban scaling.
