The average household
size has been decreasing since the start of measurements in 1967. Between 1994
and 2007, the average household size fell from 2.61 to 2.55, and the overall household
Gini ratio increased from 0.456 to 0.463. In 2009, a new measuring procedure was
introduced and all estimates of Gini ratio were subject to artificial corrections
due to the change in data granularity and the overall coverage by income bins.

Here we argue that the
change in Gini ratio results from the change in the household size distribution.
We demonstrate the effect of the average household size on Gini ratio using the
household size distribution measured in 2011.
This year is convenient since it covers with $5000-wide bins incomes up
to $200,000. This leaves 5,106,000
households from 121,084,000 in the income bin above $200,000. The average household size in 2011 was 2.55.
Figure 1 presents the distribution of households over sizes. One can calculate
that with the given size distribution and average size, the mean size in the 7+
(seven and more people) group is 11.9 people.

In order the average household
size to decrease, bigger households should split and create an excess of
smaller size households with lower incomes. As an alternative, a larger number
of smaller households (with lower mean income) should be created. Both processes
reduce the relative number of households with many people and increase the
number of small-size households.

Without loss of
generality, we split all six people households with incomes below $100,000 into
two equal households having a half-income. Therefore, instead of one six people
household with $50,001 income we have two three people households with
$25,000.5 income. These two households are now in the group of three people
households with incomes between $25,000 and $30,000. One can expand this procedure
to any household size and to any permutation of sizes. (For example, a six
people household might be split into two households of 2 and 4 people, or three
two-people households, etc.) The only
requirement is the same total income of the pieces. This process is linear and
the final mean size is a function of all splits. Here we just demonstrate the
principle. Figure 2 presents the original income distribution of six people
households. Figure 3 depicts the original income distribution of three people
households and that obtained after the split of all six people households in
Figure 2 into equal (size and income) pieces.

Figure 2. Income
distribution of six people households between $0 and $100,000.

Figure 3. Original (red) and
corrected (blue) income distribution of three people households between $0 and $50,000.

We have split
1,953,530 households and obtained extra 1,953,530 households with the total
number of 123,037,000 households. The
average household size decreased from 2.55 to 2.51 since bigger households were
replaced by a larger number of smaller ones.

The total income does
not change since all new households retain the income of split households. The
income distribution has changed, however. When calculating the Gini ratio for
the new income distribution we have to take into account the change in the mean
income in all income bins between $0 and $50,000 due to additional three people
households.

We have calculated the
Lorenz curve (Figure 4) and then estimated the Gini ratio for the new income distribution.
The original Lorenz curve (red) lies above the new one (blue). This is the
reason why the Gini ratio is higher for the new income distribution: it
increased from 0.4697 to 0.4746. This gives an increment
of 0.005 as related to the 0.04 fall in the average household size (2.55 to
2.51). Considering the overall decrease
in the average size by 0.06 between 1994 and 2007, one may expect the Gini
ratio rise by 0.0075. The actual figure is 0.007. Hence, the change in Gini
ration can be fully explained by the change in the average household size.

Figure 4. The Lorenz
curve for the original (red) and new (blue) income distribution.

## No comments:

## Post a Comment