This time I’d like to
present some features of household income distribution reported by the Census
Bureau for 2011. First, the size of mean household has been decreasing since
1967. Thus, we have to look into the evolution of each size independently, i.e.
one-person, two-person, … households. Moreover we have to present relative evolution
since the number of households of a given size does not change proportionally
to the total population. The falling mean size implies a lower and lower
portion of bigger households. Figure presents the income distribution of
households for all reported sizes. All distributions are normalized to the
total number of households of corresponding size.

Figure 1. Probability density
functions (PDFs) for income distribution of households from one- to seven+
persons.

We do not present here the
evolution of these distributions over time. It’s a task for a quantitative
study. There are two features deserving to be mentioned. The distributions for
all households with size of two and more persons are very similar. The one-person
households are distributed in a different way – the associated PDF fall much
faster. This implies a higher inequality. The Gini ratio calculated for the one-person
households is 0.479 with all other sizes characterized by Gini between 0.417
(7+) and 0.443 (5 people).

Another feature is associated
with the size of CPS universe. There are
around 75,000 households surveyed every March. This puts a severe constraint of
the accuracy of measurements in the higher income bins. The total number of
121,084,000 households is not a counted one but is projected from the CPS set
using the total population of 308,764,000 and the mean household size (2.55). Therefore, one household in the CPS is
multiplied by ~2500 to project to the total population reported by the CB. To
obtain a statistically reliable estimate one needs quite a few measurements (say
100) in any income bin. However, there are many bins with 5,000 to 10,000 households,
i.e. from 2 to 4 actually measured households in the CPS. This is inacceptable
for any reliable statistical estimate of income inequality. The incredibly high
uncertainty of the number of households in high-income bins is expressed in the
strong oscillations of the PDFs. All high
income estimates are biased and one should not calculate Gini ratio at all.

Figure 2 presents another puzzle.
The CB published income distributions depending on the number of earners in the
households. Figure 2 depicts the normalized curves (PDFs) for all categories in
the CPS report. There is no surprise in
the PDFs unless the failure to understand the households with “no earners” and
$200,000 income. I do not understand how a household can have $200,000 income
without people who earn money according to the CB definition.

Figure 2. PDFs for the income
distributions with various numbers of earners.

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