بررسی توزیع درآمد شخصی در استرالیا

We analyze the data on personal income distribution from the Australian Bureau of Statistics. We compare fits of the data to the exponential, log-normal, and gamma distributions. The exponential function gives a good (albeit not perfect) description of 98% of the population in the lower part of the distribution. The log-normal and gamma functions do not improve the fit significantly, despite having more parameters, and mimic the exponential function. We find that the probability density at zero income is not zero, which contradicts the log-normal and gamma distributions, but is consistent with the exponential one. The high-resolution histogram of the probability density shows a very sharp and narrow peak at low incomes, which we interpret as the result of a government policy on income redistribution.

The study of income distribution has a long history. More than a century ago, Pareto [1] proposed that income distribution obeys a universal power law, valid for all time and countries. Subsequent studies found that this conjecture applies only to the top 1–3% of the population. The question of what is the distribution for the majority (97–99%) of population with lower incomes remains open. Gibrat [2] proposed that income distribution is governed by a multiplicative random process resulting in the log-normal distribution. However, Kalecki [3] pointed out that such a log-normal distribution is not stationary, because its width keeps increasing with time. Nevertheless, the log-normal function is widely used in literature to fit the lower part of income distribution [4], [5] and [6]. Yakovenko and Drăgulescu [7] proposed that the distribution of individual income should follow the exponential law analogous to the Boltzmann–Gibbs distribution of energy in statistical physics. They found substantial evidence for this in the statistical data for USA [8], [9], [10] and [11]. Also widely used is the gamma distribution, which differs from the exponential one by a power-law prefactor [12], [13] and [14]. For a recent collection of papers discussing these distributions, see the book [15]. Distribution of income x is characterized by the probability density function (PDF) P(x), defined so that the probability to find income in the interval from x to x+dx is equal to View the MathML source. The PDFs for the distributions discussed above have the following functional forms:

All three functions in Eq. (1) are the limiting cases of the generalized beta distribution of the second kind (GB2), which is also discussed in econometric literature on income distribution [16]. GB2 has four fitting parameters, and distributions with even more fitting parameters are considered in literature [16]. Generally, functions with more parameters are expected to fit the data better. However, we do not think that increasing the number of free parameters gives a better insight into the problem. We think that a useful description of the data is the one that has the minimal number of parameters, yet reasonably (but not necessarily perfectly) agrees with the data. From this point of view, the exponential function has the advantage of having only one parameter T over the log-normal, gamma, and other distributions with more parameters. Fig. 1(a) shows that logC vs. x is approximately a straight line for about 98% of population, although small systematic deviations do exist. The log-normal and gamma distributions do not improve the fit significantly, despite having more parameters, and actually mimic the exponential function. Thus we conclude that the exponential function is the best choice. The analysis of PDF shows that the probability density at zero income is clearly not zero, which contradicts the log-normal and gamma distributions, but is consistent with the exponential one, although the value of P(x=0) is somewhat lower than expected. The coarse-grained P(x) is monotonous and consistent with the exponential distribution. The high resolution PDF shows a very sharp and narrow peak at low incomes, which, we believe, results from redistribution of probability density near the income threshold of a government policy. Technically, none of the three functions in Eq. (1) can fit the complicated, three-peak PDF shown in Fig. 2. However, statistical physics approaches are intended to capture only the baseline of the distribution, not its fine features. Moreover, the deviation of the actual PDF from the theoretical exponential curve can be taken as a measure of the impact of government policies on income redistribution.