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.

10% return fixed .. What next?

I've sold S&P 500  in view of the political and economic turbulence around the corner. Fixed the 10% profit since May and now consider new opportunities. I expect the S&P 500 to fall below 1375  during the next few months. Then  another rally is coming to 1500 and even higher by the end of 2013.
Meanwhile, cash is a safe haven with bonds as an alternative. Will go for 10-year notes when yield is above 2%. Do not like TIPS with price deflation (since the market falls) as an immediate danger.

Statistical question

The importance of income inequality for public, politics, and economy is high. There are many surveys and estimation programs to measure incomes and inequality, including the level of poverty. There is no one complete income survey, however. Figures 1 and 2 show the shares of GDP and total population covered by major statistical agencies: the Census Bureau, the IRS, and the Bureau of Economic Analysis. No one is complete.
Why so? Why to have many incomplete data sets instead of one complete?
Looks incompetent.

Figure 1. Shares of personal income estimated by the CB (MI – money income), IRS, and BEA.

Figure 2. Shares of population reporting income covered by the CB and IRS.

A surprise from the Census Bureau - 'no-earners' with $200,000 income

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.

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.

Income inequality: a sharp fall for the youngest age group

The Census Bureau reported an overall increase in personal income inequality as expressed by Gini ratio jump from 0.503 to 0.510. This is a significant change considering the uncertainty of ~0.002 as reported by the Bureau. It is interesting to know how this rise is distributed over age pyramid. Figure 1 displays the change in Gini from 2010 to 2011 in various age groups from 15 to 75 years. The overall increase of 0.007 is unevenly distributed over the pyramid. The youngest age group between 15 and 24 years, is the gainer  - the Gini has dropped by 0.015. This is a significant drop since the uncertainty of the measurement is 0.006. Another age group characterized by a fall in income inequality is between 65 and 69 years of age. The Gini has dropped by 0.012 (±0.006). 

 

An intriguing observation is the difference in the change in Gini ratio between adjacent age groups 60-64 and 65-69. The difference has changes by 0.032 – a huge amount which needs special explanation from the Census Bureau. This difference looks spurious but it is likely responsible for the overall rise.