Boltzmann-like income distribution in low and middle income classes: Evidence from the United Kingdom

We elucidate that a Boltzmann-like income distribution will emerge spontaneously in a long-run Arrow–Debreu economy, which is used to describe the well-functioning market economy. The emergence of such an income distribution can be regarded as a result of maximizing the entropy of the long-run Arrow–Debreu economy, which measures the extent of choice-freedom of permissible collective decisions offered to social members. By analyzing household income data of the United Kingdom from 2000 to 2015, we observe that the income structure of a market-economy country consists of three parts: super-low income class (i.e., unemployed households), low- and middle-income class, and top income class. The empirical analyses show that the low- and middle-income class (about 90%∼95% of populations) exactly obeys the Boltzmann-like income distribution. By contrast, top income class and super-low income class undermine the setting for Arrow–Debreu economy, and therefore do not conform to the Boltzmann-like distribution.