Canada vs. USA
We finish our country analysis with Canada, which provides an extended set of income data. These data cover all persons who completed specific tax return forms for the year of reference. Therefore, the data are related to tax purposes only and some non-taxable income sources are missing from the dataset. In 2013, about 74.9% of Canadians (of all ages) filed tax returns. Most children do not file tax return as well some elder people. That may introduce age-specific bias in the estimates for children and pensioners. Unlike the CPS survey, all data are extracted from administrative files. The sample includes 100% of individuals who filed an individual tax return but not all population. Depending on age, from 89% to 96% of the population is covered by administrative records . Overall, the population and income source coverage is good for the purposes of our research.
Figure 19 depicts the evolution of real mean income since 1976 . Three years age shown for comparison – 1976, 1993 and 2011, i.e. with approximately a 17 year step. The real mean income estimates are given in 2011 constant Canadian dollars and calculated in 10-year bins except the two youngest age groups: under 20 years – the average age of 17 years is used, and between 20 and 24 years of age. The curve for 2012 presents real mean incomes only in 10-year bins; it was obtained from a different table provided by Statistics Canada.
Figure 19. The evolution of age-dependent real mean income (in 2011 constant Canadian dollars) in Canada from 1976 to 2012. All estimates are given in 10-year bins except the two youngest age groups (below 20, and between 20 and 24). The curve for 2012 presents mean incomes only in 10-year bins and was obtained from a different table provided by Statistics Canada. The highly unusual feature is the 1976 curve above the 1993 curve.
The first and highly unusual observation is that the 1976 curve is above the 1993 curve. The real GDP per capita curve in Figure 5 gives $18,138 (1990 US$) in 1992 and $14,902 in 1976. For comparison, the mean income for the whole population with income in Canada is presented in Figure 20. In fact, the average income in 1976 was $35,700 and only $33,300 in 1993. The contradiction between real GDP per capita and mean income is likely related to the difference between domestic currency and that converted at Geary Khamis PPPs. In our quantitative analysis, this discrepancy may introduce large errors in theoretical estimates of the peak age.
Following the procedure developed in two previous paragraphs, we normalize the curves in Figure 19 to their respective peak value and present the central segments of the obtained curves in Figure 21. The 1976 curve peaks between 41 and 42 years of age (27 to 28 years of work experience). In 1992, the peak age is about 45 years, and then it reaches 49 to 50 years in 2012. Overall, the increase in the age of peak mean income increases by about 8 years. Theoretically, the change in real GDP per capita from $14,902 to $25,629 should change by a factor of 1.31, i.e. the peak working age experience rises from 27.5 years to 36.1 years (50 years of age). This is an extremely accurate prediction, especially taking into account the problems with real GDP per capita estimates. The proportion of working age population does not change much in the USA since the mid-70s. It is likely not a big error if we assume that the share of working age population in three studied countries does not change since 1976. Then the correction for population does not affect the relative change in real GDP per capita and the estimates of the peak age increase in Canada, New Zealand and the UK are not biased.
Figure 20. The evolution of mean income (2011 CAD) in Canada between 1976 and 2011.
Figure 22 is similar to Figures 10 and 16 and displays the evolution of peak-normalized mean income in various age groups. The peak age resides in the bin from 35 to 44 years of age before 1990. Then the peak jumps into the elder group and will likely stay below 54 years before the group between 55 and 64 overtakes the lead. As in other three countries, the proportion of mean income has been increasing in all elder groups and decreasing in the younger groups. In Canada, fluctuation of the curves in Figure 22 is much lower than that for New Zealand and likely at the same level as in the UK. This is a consequence of income data quality – better coverage of population is directly translated into the accuracy of mean income estimates.
Figure 21. Three curves in Figure 19 normalized to their respective peak values. The age of peak mean income value increases with time from 42 years in 1976 to 49 years in 2012.
Figure 22. The evolution of mean income in all 10-year age groups normalized to the peak value in the same year. In the younger age groups (“18”, “22”, and ”30”) the proportion of mean income has been falling since 1976.
Figure 23 depicts two panels where the normalized mean income curves observed Canada in 2011 ($22,994) and 1980 ($12,931) are matched by U.S. curves for 1987 ($21,788) and 1962 ($11,904), respectively. The excellent match between Canada2011 and USA1987 curves well corresponds to GDP estimates, while the GDP levels for 1980 in Canada and 1960 in the USA differ more in relative terms. As we discussed above, the GDP values for the USA before 1975 have to be corrected for the effect of changing working age population. Between 1987 and 1962 the correction is approximately 7%. This makes the 1962 estimate to increase to $12,708. So, Canada provides the longest time series of mean income among three studied countries with the largest factor of peak age increase - 1.31. This makes our measurements more precise and allows better fir between observed and predicted change in the peak mean income.
Figure 23. The evolution of mean income in Canada in 2011 and 1980 is matched by the curves measured in the USA in 1987 and 1962, respectively. Age bins are 10 years for Canada. For the USA, we use microeconomic data with annual mean income smoothed with a MA(9).
Canada is the only country from the studied trio reporting age-dependent PIDs for the higher incomes. Figure 24 displays the number of people with income above a given income threshold as a function of age. We have selected different thresholds to retain the portion of population: $100,000 in 2000, $100,000 in 2006, and $150,000 in 2013. All age bins are 10 years, except the youngest between 0 and 24 years of age. The youngest bin is prone to strong bias because it includes children with incomes. The 2006 curve is higher than the 2000 curve because they have the same threshold but the total nominal income in 2006 is much larger than in 2000, and thus, more people have larger incomes. Moreover, the population pyramid changing with time may introduce a significant bias into the number of people of a given age. In Figure 24, three curves peak at different ages. To suppress the population effect we have scaled the portion of people with the highest income to the total population with income in the same age bin. Figure 25 shows three normalized curves. As one can see, the age pyramid effect is removed and all curves peak at the same age. Summing the number of people above the threshold in all age groups and dividing it by the total population with income, we calculate the total portion of people with incomes above the threshold. For the thresholds in Figure 24, this portion is 2.5% in 2000, 4.5% in 2006, and 2.8% in 2013.
Figure 24. The number of people with income above a given income threshold as a function of age. Thresholds are $100,000 in 2000, $100,000 in 2006 and $150,000 in 2013. Age bins are 10 years, except the youngest between 0 and 24 years of age. The number of people depends on threshold and population in each bin.
Figure 25. The portion of people with income above a given income threshold as a function of age: the number of people above the threshold in a given age bin is divided by the total number of people in the same bin. The age pyramid effect is suppressed. The cumulative portion of people above the threshold is 2.5% in 2000, 4.5% in 2006, and 2.8% in 2013. Notice the same bin and threshold settings as in Figure 24.
When total income and population are subject to significant changes with time the best way to compare age-dependent income distributions in different years is to normalize them to their respective peak values. Then direct comparison is possible which shows the relative rate of income/population change with age. Figure 26 depicts three peak-normalized curves from Figure 25. These normalized curves practically coincide with just small differences in the age groups 25 to 34 years and 55 to 64 years. The age of the largest portion of people with the highest incomes resides in the bin between 45 and 54 years. Because of the width of this bin, which includes the true peak year for all curves, it is difficult to resolve any change.
Figure 26. The curves in Figure 25 are normalized to their respective peak values. The age of largest portion of people with the highest incomes resides in the bin between 45 and 54 years. It is difficult to estimate the change in peak age. Notice the same bin and threshold settings as in Figure 24.
One problem in Figure 26 is that the 2006 curve lies above the 2000 curve, while our model and experience suggest the opposite situation. This controversy is actually related to the difference in the total portion of people with the highest incomes: 2.5% in 2000 and 4.5% in 2006. All PIDs available for Canada between 2000 and 2013 are characterized by $50,000 income bins above $100,000. The choice of threshold is limited to the boundaries of these bins. The year of 2000 is characterized by the lowermost number of people and gross nominal income, and thus, by the lowermost threshold of the Pareto distribution. We use it to illustrate the change in the age-dependent portion of people with the highest incomes for several thresholds between $50,000 and $250,000. Figure 27 illustrates the change in the age-dependent curve. The peak age grows with the threshold, i.e. more and more time is needed to reach higher incomes. This observation puts an important constraint on the direct cross comparison of the age-dependent portion of people with the highest incomes. One should use thresholds, which retain the total portion of people at the same level.
In Figure 28 we match the 2013 Canada curve and the 1995 US curve. The coincidence between the portions of total population with incomes above $150,000 (CAD) in Canada and $82,500 (USD) in the USA, both in current dollars, is striking. We retain the total portion of population above these thresholds at approximately 2.7%. That eliminates the threshold dependent bias. Total Economy Database reports $26,000 (1990 US$) for Canada in 2013 and $24,712 for the USA in 1995. The difference is not large and a 4% correction for the change in working age population makes the 1995 U.S. estimate to rise to $25,643. Hence, the fit between two curves proves that the portion of people with the highest incomes is likely a country-independent variable. In other words, it is likely a universal variable which depends only of real GDP per capita. It is important to extend the set of countries in order to support this finding.
Figure 27. Canada 2000. The age-dependent portion of people with incomes above five thresholds - from $50,000 to $250,000. The peak age depends of threshold. Therefore, one has to compare curves with thresholds giving the same portion of people.
Figure 28. Comparison of the age-dependent portion of people with incomes above given threshold. The 2013 Canada curve (>$150,000, current CAD) is best fit by the 1995 US curve (>$82,500, current US$). TED reports for Canada $26,000 (1990 US$) in 2013 and $24,712 for the USA in 1995. The total portion of population above the thresholds in both cases is approximately 2.7%.
We have studied two specific features of personal income distribution in three countries: Canada, New Zealand, and the UK, and compared them with the USA. The dependence of mean income and the portion of people with the highest incomes on age are both characterized by varying length of the involved time series and their accuracy. Canada provides a set of mean income data covering practically the whole population and the period since 1976, but the data on personal income evolution with age are limited to the period between 2000 and 2013. Statistics New Zealand also reports tax-related income as obtained from a survey covering a small portion of the whole population. Higher amplitude fluctuations, likely induced by underrepresentation of the highest incomes, limit the usefulness of these estimates for the purposed of our study. In addition, there is some controversy between real GDP and income estimates reported by New Zealand and the Conference Board. The UK provides the shortest but useful time series collected by HMRC.
All data corroborate our main assumption – the evolution of income distribution is universal, follows a unique trajectory, and depends only on real GDP per capita converted at PPP exchange rates. We have proved quantitatively that the dependence of mean income on age in Canada, New Zealand and the UK as well as the age-dependent portion of people with the highest incomes in Canada do reproduce similar dependencies observed in the USA, but many years before, when the level of real GDP per capita was the same. Since the U.S. outpaces three studied countries by several thousand dollars per head, the lag reaches 20 to 25 years. For example, the mean income dependence measured in the UK in 2012-2013 one-to-one repeats that observed in the USA in 1992. The age-dependent portion of rich people in Canada in 2013 reproduces that measured in the USA in 1995. The time dependence of the studied characteristics is just parametric, however.
We have proven that the growth of work experience corresponding to the peak mean income is accurately described by the square-root function of real GDP per capita. This feature corresponds to the key assumption of our model, which predicts both studied features precisely. The coherence of theoretical predictions and long-term observations in four countries proves that the evolution of personal income (at least in these four countries) is a physical process described by a simple relationship.
The results of measurements carried out in this study well match the prediction of our microeconomic model. A unified quantitative description in four countries is useful not only from theoretical point of view as a possibility to mathematically describe the process of personal income distribution as the evolution of a physical system. The universal character of personal income evolution as a unique function of real GDP per capita allows accurate forecasting of very specific income characteristics related to fiscal, monetary and other types of socio-economic policies. Extending the observed linear trends of real GDP growth into the future (see Figure 5), one can be use the past U.S. PIDs as templates for the future PIDs in three countries at a time horizon from 15 year (Canada) to 50 years (New Zealand).