## 9/28/12

### How the household split affects the household Gini ratio

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.
Figure 1. The distribution of households over sizes in 2011. Total number 121,084,00, with the average size 11.9 people in the 7+ group.
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.