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.

Income inequality for households: a long biased history of Gini ratio. 2021 revision

The Census Bureau measures incomes and reports the estimates. One of the main questions is income inequality – personal and households. We published a post in 2012 on the bias in the household Gini ratio. Here we revise the previous study with new data.

In 2012, our first point was that the Gini ratio for personal incomes reported by the Census Bureau from the very same data set (CPS ASEC conducted every March) does not change much since 1994. The upper panel in Figure 1 reproduces the Gini ratio from the previous post, which varies from 0.494 to 0.512 – a relatively narrow window. In the lower panel, the dataset is extended to 2019 and the rise from 0.494 in 2007 to 0.524 in 2013 is a challenge for an economic explanation. This is a catastrophic and unexpected change in inequality. The years after the Great Recession (say, after 2010 with G=0.503) were not characterized by some outstanding economic processes or events. These were the years of President Obama.

In another post on January 17, 2021, we reported an unprecedented fall in the share of “compensation of employees” in the total personal income (PI) as reported by the Bureau of Economic Analysis. Figure 2 presents the corresponding curve, which demonstrates the accelerated decrease from 0.660 in 2006Q3 to 0.614 in 2016Q3. The 0.046 drop in the share of income from jobs reported by BEA is synchronized with the personal Gini rise by 0.03. The trough in 2020 related to the COVID-19 pandemic may be an interesting economic experiment for income distribution. Personal income in 2020 does not change much (even increase in Q2 and Q3 against pre-crisis expectations) despite the drop in compensation of employees. The question is where we will find the government social benefits to persons (+2.5 trillion in Q2 and +1 trillion in Q3 compared to the previous year). My current guess – stock market.

Figure 1.  Personal incomes:  Upper panel: Gini ratio evolution between 1994 and 2010 as presented in this post. Lower panel. Gini ratio evolution between 1994 and 2019. Between 2007 and 2013 the Gini ratio raised from 0.494 to 0.524, i.e. by 0.03.

Figure 2.  Ratio of compensation of employees and Personal Income (BEA. Table 2.1. Personal Income and Its Disposition). Quarterly data.  The fall 0.655 in the third quarter of 2006 to 0.614 in 2016Q3.

The upper panel in Figure 3 is borrowed from the previous post and shows the history of Gini ratio for households between 1967 and 2010. (The lower panel extends the period to 2019). We normalized the ratio to its maximum value (0.477 in 2011) in order to show that this inequality measure had risen by 20% since 1967. This dramatic increase was interpreted as harm for the US. In my view, this is just a misunderstanding of the income measurement procedures. Unlike personal incomes, the household income data are collected for entities that can evolve in size in all directions. There are two limit cases: 1) all households may have just one person and then the household Gini is fully equivalent to the personal Gini, which is higher as we can learn from Figure 1; 2) all people represent one household and then the Gini is 0 because there is no inequality for 1 object. For a given personal income distribution, any other combination of people gathering in households should give the Gini between 0 and the personal Gini. Reconfiguring the households’ sizes and personal content for the same population one may change the Gini for the household incomes without changing personal incomes. Therefore, the split of the population into households defines the Gini for a given population and time point. The distribution of the increasing number of people among households, i.e. the distribution of household sizes, and the personal income distribution are changing in time, and the household Gini is evolving in sync with these changes. The Census Bureau’s approach is straightforward – they measure the distribution of the household incomes and calculate the Gini ratio. This ratio is incompatible with the previous years since the distribution of household sizes is changing. Moreover, it is changing in the direction of the split of bigger households into smaller pieces, eventually into the single-person-households. Hence, the household income distribution approaches the personal income distribution and this must be accompanied by an artificial increase in the Gini ratio. This increase is reported as a big problem of American households. This is a definitional problem, however, and has no relevance to real changes in income distribution illustrated in the lower panel in Figure 1.   

The Census Bureau does not explicitly report the distribution of household sizes (in persons) and one has to make an own estimate, which is easy, however. Figure 4 presents (old and new) the total household population (different from the civil population or residential population) and the number of households reported by the CB.  Figure 5 depicts (old and new) the evolution of the average household size since 1967. Actually, it was quite spectacular: from 3.2 in 1967 to 2.49 in 2011. Between 2010 and 2019, the mean household size hovered around the 2.5 level. This constant mean size could be interpreted as the constant household size distribution between 2000 and 2019.

Does it matter for the household income inequality?  As we discussed above, the Gini ratio depends on the size distribution of objects if these are not indivisible persons. Intuitively, more low-income (e.g. one person) households result in a higher Gini ratio. The fall in average size indicates that one gets more and more small households over time and … the Gini ratio increases accordingly. The link between the average household size and the Gini ratio is not linear (as we discussed before, many household size distributions have the same average size) but Figure 6 shows the (old and new) product of the normalized Gini curve for households (see Figure 3) and the curve in Figure 5. This product is an approximation of the constant 1967 household size distribution as if all people in every year after 1967 were distributed in the same household size structure as in 1967. This product compensates the size distribution change but does not compensate the income change in the households, i.e. we do not compensate the process of income gain or loss in the households with time, and we do know that the income distribution for a given household size has been changing with time (see these posts). In Figure 6, we see a corrected (and likely closer to reality) Gini history.  This corrected normalized Gini is not fully compensated for the household size changeover time but tells a different story.

The original Gini ratio for households corrected to the change in the household size distribution is depicted in Figure 7. In 2019, the level is the same as in 1967 – 0.397. The positive shift from 1992 (0.358) to 1993 (0.378) is completely artificial. In 1993, there was a revision to income definition and all time series were subject to dramatic changes. Therefore, the current level is below that in 1967 if to use the 1967 household income definition.

Overall, the Gini ratio for households has not been changing as the CB estimate says because these estimates do not take into account the change in the household size distribution.

As we wrote in 2012, this is a methodological error.  The same logic must be applied to family income distribution.  Another sufferer is the mean income.  Since the size of households has been decreasing the number of households has been growing faster than the total household population.  The mean household income must also be corrected for the changing size.  Figure 8 shows the actual evolution of the mean income.  There was a period of constant mean income between 1996 and 2013 with no significant change in the average household size. Since 2014, the mean income curve has been demonstrating tangible growth.

Figure 3. The evolution of normalized Gini ratio for households. Old and new versions

Figure 4. The evolution of the total household population and the number of households (both in thousands)

Figure 5. The evolution of average household size.

Figure 6. Corrected normalized (see Figure 3) Gini ratio.

Figure 7. Original (Census Bureau) Gini ratio corrected to the change in household size distribution. In 2019, the level is the same as in 1967 – 0.397. The positive shift in 1992 (0.358) to 1993 (0.378) is completely artificial. Therefore, the current level is below that in 1967.

 

Figure 8. The growth of normalized (household) mean income and that corrected for the fall in the household average size.

The evolution of household size distribution and income inequality

There is an important problem raised by Coding Monkey in the comments to this post  on the evolution of household sizes (supported by the Arthurian).  With the mean size of household decreasing since 1967, who is responsible for the fall – poor or rich households?  I did not study this problem before and my first guess is that richer (and bigger) households have to split first. Their pieces are financially and logistically more viable than poor households. The latter have to retain their sizes in order to save money for living.

 

The Census Bureau provides some data to answer this question quantitatively. Unfortunately, the CB changes its rules and procedures as other statistical agencies. This makes impossible a direct comparison of data from different years. For example, the CB changed the bin size in 2009 to $5000 from $2500 between 1994 and 2008. It is difficult to compare mean household sizes in different bins and there is no possibility to merge two mean sizes in $2500 bins in one mean household size in $5000.  Thus we can directly compare only 1994 and 2008. However, the choice of 2007 seems more attractive because it provides the highest real GDP. 

 

Figure 1 directly compares mean household sizes in $2500 bins between $0 and $100,000 in 1994 and 2007.  One can see that the mean household size fell in all bins. A quick and wrong interpretation is that poor households merged and created bigger ones residing above $100,000.  This is not true because of several important changes between 1994 and 2007. The total number of households rose from 98,990 to 116, 783. The level of nominal GDP rose by a factor of 1.98, including real GDP increased by a factor of 1.49. 

All these changes are not taken into account in Figure 1. The total number of households may not affect the mean size when all newly created households repeat the overall distribution. This means that the mean size is retained the same in any income bin if the size distribution in this bin does not change.  

The change in nominal GDP does change the distribution in Figure 1. What we want to know is what did happen to the 2007 households that would occur in 1994 bins? One can imagine that $2500 in 1994 is not equal to $2500 in 2007. We have to scale the income axis according to the total change in GDP per one household. There are two components of the change – price inflation and real GDP growth per household. The former process shrinks the income scale by the factor of 1.30, i.e.  the overall change in prices between 1994 and 2007.  The growth in real GDP from 1994 to 2007 is 1.49.  If the number of households is the same, a 2007 household should have income by a factor of 1.98 higher than in 1994. However, there are 1.18 times more households in 2007 and an average household would have income by a factor of 1.68 larger than it would have in 1994.  All households with income $100,000/1.68= $59,523 in 1994 have to move above $100,000 in 2007 and to fall in the bin “$100,000 and above”.  

After scaling by a factor of 1.68, all bins in 2007 repeat the bins in 1994. Figure 2 displays the dependence of the mean household size on income with the scaled axis for 2007. Effectively, the 2007 curve in Figure 1 has been shrunk and shifted left.  As a result, one cannot distinguish between two curves except the very low income bins.  This is an obvious result that the low income bin is populated by one-person-households.  We again have a problem of the changing average household size. These estimates do not help much to resolve this problem.

Another way to address this problem is to estimate the density of households in all income bins.  Figure 3 displays the number of households in a given bin normalized to the total number of households in 1994 and 2007, respectively.  The income bins in 2007 are also scaled as discussed above.  Therefore, the graphs present the portion of household in a given bin.  The 2007 curve is below that of 1994.  The reason is simple – bins are different in 1994 and 2007. In order to compare curves in Figure 4 in an appropriate way, we have to calculate the distribution density, i.e. the portion of households per $1. In Figure 5 we normalized the curves in Figure 4 to their respective widths and obtained two density curves, which are very close.   The 2007 curve seems to be higher at lower incomes and lower at higher incomes. Therefore, the average size in 2007 has to be smaller than in 1994 because the density of households at higher incomes fell since 1994.  Economically, this is an expected result – when broken, high-income households create sustainable households. The assumption of the low-income households split due to poverty would result in the same portion of high-income households in 2007 and a sharp peak at very low incomes.  

Figure 6 shows cumulative curves from Figure 5. The deviation becomes higher with income and then the curves converge to 0.008 (1/$1250 the width of 1994 bin). This is a version of Lorenz curve which shows a higher Gini for 2007 because of lower density of the high-income households. We cannot continue the curves beyond $100,000 ($59,523 in 2007) since no size distributions are available.  (As always with the CB and other statistical agencies.)  This is one of the reasons for economics not to be a hard science. Measurements are made (or published) by a March hare.

Figure 1. The mean household size as a function of income for 1994 and 2007.

Figure 2.  Mean household size as a function of scaled bin width.

 Figure 3. Income distribution for households in 1994 and 2007.

 Figure 4. The portion of the households total number in a given bin.

Figure 5. Household distribution density, i.e. the normalized number of households per 1$ (in 1994), in 1994 and 2007.

Figure 6. Cumulative distributions from Figure 5. The 2007 curve is higher for lower incomes and lower for higher incomes. It has to intesect the red line at the level 0.008  at the highest income for one household, which is not reported by the CB.

The size of household and the rise in Gini ratio from 2010 to 2011

Yesterday, I showed that the average size of household in the US has been decreasing since the start of measurements in 1967. This is the reason behind the decreasing average household income and increasing Gini ratio. The Census Bureau (CB) should not publish these figures without correction for the average household size. The reported values are definitely biased and used for political games. This is unacceptable for a nonpartisan statistical agency.  

The CB does publish the size distribution of households and the mean household size. For 2011 and 2010, Figure 1 shows the number of households in the USA. It is worth noting that both numbers are obtained as a projection from the figures obtained during the CPS (around 75,000 households selected in a “scientific” way) with population controls taken from the 2010 census. The number of households is not a directly measured value!  From Figure 1, one can observe that the number of one- , two-, and three-person households increased from 2010 to 2011. Obviously, smaller households should be characterized by lower incomes. Therefore, more low-income households should produce higher inequality raising the share of low-incomers. 

However, the total number of households also grew from 2010 to 2011 and one needs relative values instead of absolute in order to estimate the input of household size. Figure 2 shows the probability distribution function for two distributions in Figure 1, i.e. the original distributions normalized to the associated total numbers. One can observe that the share of two and three-person households increased with the portion of one-person household slightly smaller in 2011.  

The Census Bureau also publishes the average household sizes. In 2010, it was 2.58 per household and only 2.55 in 2011. (In my previous pos , I used the total household population, and the CB likely used the civilian population to estimate the size. ) The mean size fell by 1.2% with the Gini ratio increased from 0.47 to 0.477, i.e. by 1.4%.  As we discussed before, the change in mean size should manifest itself in increasing Gini ratio. This is the reason for the step in the household Gini ratio as observed in 2011.

Figure 3 depicts two distributions of Gini ratio as a function of household size: for 2010 and 2011. These figures are borrowed from the CB.  Except the one-person households, Gini ratio increased for all household sizes in 2011.  Interestingly, the rise in Gini ratio in two groups with different average incomes does not necessary result in increasing Gini ratio for the joint group. The increasing inequality may be accompanied by decreasing difference between the average incomes and thus reduce the overall income dispersion.

Figure  1. The number of households (thousands) as a function of size. All households with seven and more people are gathered in one bin “7+”.

Figure 2.  Probability distribution function for the distributions in Figure 1.

Figure 3. Gini ratio as a function of household size.

Income inequality for households: a long biased history of Gini and mean income. The Census Bureau's failed again

The Census Bureau measures incomes and reports figures. Experts discuss and panic. The fame depends on the claim of disaster with inequality. It must grow; otherwise economic commenter would lose public power. Who is interested in the topic when no change is observed?  Let’s try to dig into raw data and find the reason for the observed tendency. Our first point is that the Gini ratio ( the most famous measure of inequality)  for personal incomes reported by the Census Bureau from the very same data set (CPS ASEC conducted every March) does not change much since 1994. Figure 1 reproduces the Gini ratio, which varies from 0.494 to 0.512 – a relatively narrow window.

Figure 1 .  Personal incomes:  Gini ratio evolution since 1994.

In Figure 2, we present  a sad history of Gini ratio for households.  We intentionally normalized the ratio to its maximum value (0.477 in 2011) in order to show that this inequality measure has risen by 20% since 1967. This dramatic increase is interpreted as harm for the US. Unlike personal incomes, the household data are collected for entities which can evolve in size. (A person always has a unit size.) The Census Bureau does not explicitly reports the distribution household sizes and one has to make an own estimate, which is easy, however. Figure 3 presents the total household population (different from civil population or residential population) and the number of households reported by the CB.  Figure 4 depicts the evolution of the average household size since 1967. Actually, it was quite spectacular: from 3.2 in 1967 to 2.49 in 2011.

Does it matter for the income inequality?  Sure - yes. The simplest ways is a household split - instead of one big household one gets two smaller households. The Gini ratio depends of the distribution of sizes. More low-income households result in a higher Gini ratio. The fall in  average size says that one gets more and more smaller households over time  and … the Gini ratio increases accordingly. There is no linear link between the average size and the Gini ratio but Figure 5 shows the product of the Gini curve for households and the curve in Figure 4. Now we see a corrected Gini history.  This corrected Gini is not fully compensated for the household size changeover time but  tells a different story to the educated audience: the Gini for households has not been changing since the 1970s. In 1993, there was a revision to income definition and all time series were subject to dramatic chances. This step is fully artificial.

Overall, the Gini ratio for households has not been changing as the CB estimate say because these estimates do not take into account the change in household size distribution.

This is a methodological (i.e. unprofessional) mistake. 

The same corerction logic must be applied to the family income distribution  - also biased in its current version.  Another sufferer is the mean (and aslo median) income.  Since the size of household has been decreasing the number of households has been growing faster than the total household  population.  The mean household income must also be corrected for the  changing size.  Figure 6 shows the actual evolution of the mean income (median income is harder to recover).  The history is much brighter than many experts would like to comment on. 

Figure 2. The evolution of normalized Gini ratio for households.

Figure 3. The evolution of total household population and the number of households (both in thousands)  

Figure 4. The evolution of an average household size.

Figure 5. Corrected Gini ratio.

Figure 6. The growth of normalized (household ) mean income and that corrected for the fall in the household average size.

What is the most efficient household size?

In my previous post, the evolution of mean family size was presented. In the same post, I also presented the long-term decrease in the mean household size from 2.89 in 1975 to 2.65 in 2011. The households break into smaller pieces as well as families. Here, we address the question of the most efficient size for a household. I propose to measure the efficiency in terms of income per person for a given household size. This measure might be not conventional but definitely explains the fall in the household size as the consequence of maximum income.

Figure 1 depicts seven curves of income per person (the average income for a given household size divided by the household size) for households of different sizes (10 people for the category 7+) normalized to the average household income (for all households) in a given year. The two people household is as efficient as one person household since 1998.  (Figure 2 depicts the same pattern for American families.)  Not surprisingly, the portion of one person and two people households increases rapidly since the start of measurement in 1975. These are two most efficient (in terms of personal income) household sizes. The portion of smallest households will rise in the future along their long term trends. Income rules!

Figure 1. Income per person for households of different sizes (10 people for the category 7+) normalized to the average household income in a given year.  

Figure 2. Income per person for families of different sizes (10 people for the category 7+) normalized to the average family income in a given year.

 

Figure 3. The portion of households of a given size since 1975.