Alumni refresher lecture series
2009 – A tipping point
Spaghetti unravelled: How income varies with age
Average income-age trajectories derived from longitudinal data look different from those derived from cross-sectional data
(pages 31-35 of printed journal)
My research provides new evidence about the shape of people's income-age trajectories - how income varies with age - and how these trajectories differ between individuals. I argue that income-age trajectories collectively look like cooked spaghetti - they are a complex mix of wiggly lines. My analysis is based on 17 years of data from the British Household Panel Survey (BHPS) - a survey which has reinterviewed the same set of individuals year after year. The BHPS is similar to the Australian panel survey run by the Melbourne Institute (HILDA).
Figure 1 provides some evidence of trajectory 'spaghetti': each line in the chart shows the evolution of hourly wages for men born in 1966 who have educational qualifications to at least A-level standard (the qualification for university entrance). There is one line for each of the 517 men in the sample. Similar spaghetti pictures arise for other groups too, including women, and for other definitions of income, including household income from all sources. It appears that, on average, wages rise with age but, even within this relatively homogenous group, there is substantial variation in wages at the start of the working life, and in wage growth thereafter.
Figure 1: How hourly wages (log scale) vary with age (men born 1966 with at least A-level educational qualification)
Source: British Household Panel Survey, waves 1-17
I argue that statistical models can be used to isolate some key patterns from this spaghetti - this is the unravelling of the title. I propose a framework that provides summary descriptions of not only the way in which incomes among groups of similar individuals change with age on average, but also the way in which trajectories for individuals diverge from the average trajectory of their group.
Most of the evidence currently available about the relationship between income and age is derived from cross-sectional data. The pictures of income-age trajectories are derived from survey data for a given year about a large sample of individuals of different ages. By contrast, my research uses longitudinal data that tracks the same people over time and accumulates information about the income-age trajectory for each person in the sample as each person ages. Data about how income varies between the age of 30 and 40 years (say) is derived by following 30 year olds over a decade until they are 40 rather than comparing today's 30 year olds with today's 40 year olds. If one is interested in documenting the nature of individual's income-age trajectories, including how income varies between one year and the next for each person, while also describing the heterogeneity across individuals in income-age trajectories, then a longitudinal approach is essential.
Relevance for social policymaking
Knowledge of how income varies with age on average, and the extent to which individual trajectories differ from an average profile, is relevant to many aspects of social policymaking. For example, how your income varies over your life is an important determinant of your spending possibilities - and hence consumption and economic well-being - at different ages, and your ability to save for old age, whether privately or through company, occupational, or state pension schemes. It is important to identify the characteristics of not only the groups who, on average, have persistently low incomes and hence low abilities to save, but also whether a 'group average' is potentially deceptive. Even if income increases with age on average, this is consistent with considerable year-on-year fluctuation in the incomes of a minority, or a mixture of subgroups with rising income and subgroups whose income is falling. These features complicate the design of effective policies for fostering saving by all.
Unravelling the spaghetti
The key ideas underpinning my approach are as follows. First, I differentiate twelve 'social groups', with group membership defined in terms of similarity of birth year, educational qualifications and sex. Second, within each group, I summarise income-age trajectories in terms of an average group profile combined with individual-specific divergences from the group average. Figure 2 helps explain the idea.
Figure 2. Stylised income-age trajectories for two individuals and the average trajectory
Note. Chart shows stylised income-age trajectories for two individuals (dashed lines) and an average trajectory (solid line): see text for further explanation.
The dashed lines show stylised income-age trajectories for two individuals from the same social group (men born in the same year who both left school with GSCEs but without any A-levels, say). John's profile is summarised by a relatively low income at the beginning of the working-life (taken to be 25 here) combined with a relatively large growth rate in income with age (long dashed line). The other profile (short dashed line) combines a relatively high initial income but a relatively low growth rate in income with age - the slope of the trajectory is less steep than in the first case. Think of the first situation as characterising someone who qualified as plumber. The starting salary is relatively low but increases over the working life, reflecting the return to the investment in training. The second situation represents Mike who instead trained as a motor mechanic. Initial earnings are higher than for John and remain so until both individuals are nearly 50 at which point, John's earnings are overtaken by Mike's. The solid line represents the average of the two individual profiles.
The key differences between John's and Mike's trajectories are, first, the difference in the initial incomes (one is below the average trajectory initially; the other is above the average initially) and, second, the difference in income growth with age (again one rate is above the average and the other is below it). A third important feature is that initial incomes and income growth rates are negatively correlated: John has a lower income than Mike to start with but experiences greater income growth. Their trajectories cross.
Now suppose that we wish to summarise the trajectories for all of the many individuals in this group, not only those for John and Mike. Given the average trajectory for the group as a whole, we can think of there being a distribution of initial incomes around the average and also a distribution of income growth rates, and some correlation between initial incomes and growth rates. Although most individuals within the group are located relatively close to the average, there are a few outliers either with relatively low or high initial incomes or growth rates. The relative frequencies of high and low deviations from the average initial income are illustrated in Figure 2 using the curvy solid line. Most people are located close to the average value at age 25 (the curve is 'higher'), with relatively small numbers with extreme values (where the curve is 'lower'). In my analysis, the joint distribution of initial incomes and growth rates with age is characterised using only three numbers - the standard deviation of initial incomes around the average, the standard deviation of growth rates around the average, and the correlation between initial income and growth rate - and these parameters can be estimated from longitudinal survey data along with the parameters that describe the group average income trajectory. This characterisation is consistent with both the trajectories increasing with age for a majority within the group, and declining with age for a minority.
The model implies that not only is there within-group inequality in income at each age, but also that this inequality varies with age. Intuitively, the less dispersion there is in initial incomes, or in income growth rates, the lower the within-group inequality at any age. Substantial dispersion in the income growth rate will tend to increase age-specific within-group inequality levels as the group members age. The cumulative effect of persistent differential income growth is to magnify initial income differences, providing an impetus for profiles to fan out with age.
There are additional features introduced into the model to make it more realistic. First, the group average trajectory is allowed to have more 'wiggles' than the stylised trajectory shown in Figure 2. Second, an additional year-by-year source of idiosyncratic variation in an individual's income from the group average is introduced to account for the substantial longitudinal variability in incomes that arises in real life. This variation might conceivably arise from several sources, including genuine transitory variation, measurement errors in income, or reflect the impact on income of major life events such as the birth of a child or divorce.
Estimates of income-age trajectories: group averages and individual divergences
The average trajectories for wages for employees of working age are shown for the twelve groups in Figure 3. The groups are characterised using information about individuals' sex, educational qualifications and year of birth. The trajectories are plotted using logarithmic scales, so that the slope of the trajectory shows how the proportionate growth rate of wages changes with age. (If wages increased at the same percentage rate each year, the profile would be a straight line.) The trajectories are shown only for the age ranges covered by the various estimation samples, so the pictures for the 1955+ birth cohort cover the age range 25-52 and those for the pre-1955 birth cohort cover ages ranging from 37 to 64 (men) or 59 (women).
Figure 3. Estimated average wage-age trajectories, by group, for employees of working age
Some clear patterns emerge. First, hourly wages increase with age from the beginning of the working life, but at a decreasing rate (with some anomalies that I return to shortly). On average, and regardless of group, men's wages grow continuously from the start of the working life but at a decreasing rate, peak in the late 40s and fall thereafter. In contrast, women's profiles do not have such a distinct peak - wage growth declines up until the late 30s but then appears to rise again. The growth slowdown for women is consistent with their greater prevalence of part-time work, which is less well paid, particularly over the ages when many have children.
Second, for both men and women, and for both birth cohorts, having higher educational qualifications is associated with higher wages, with the return to additional qualifications greater for women than for men up until middle age (women's trajectories appear more parallel than divergent). But, third, among persons with similar educational qualifications and birth cohort, men are paid more on average than women at every age. Fourth, individuals from the later-born birth cohort are on higher trajectories than those from the earlier-born cohort, other things equal.
The returns to different levels of education and between the sexes are substantial. For example, for men aged 40 from the 1955+ birth cohort, the difference on average between those with no qualifications and qualifications of at least A-level standard is around 50 per cent (just over £12 per hour compared with just under £8). For women, the corresponding difference is around 55 per cent. But the difference between the hourly wage of a 40 year old man and a 40 year old woman, both from the younger cohort, is more than one third in his favour on average (around 35 per cent). The average trajectory for women with A-level qualifications lies below that for men with some qualifications. The average trajectories for men with no qualifications lie almost everywhere above the average trajectories for women with some qualifications.
Potential anomalous aspectsThere are some potentially anomalous aspects to some profiles at the beginnings and ends of the working life, notably for the pre-1955 birth cohort: observe the upward twists in these cases. My explanation for these is that they reflect the impact of the selection effects cited earlier. For instance, arguably the women most likely to remain in the workforce as the state retirement age (60) approaches, are those for whom the pay rates are relatively high; those with relatively low pay rates retire. So, the pay rates used to estimate average trajectories over that age range are an over-estimate relative to the average that would be calculated were all women to have remained in work. Similar arguments can be made concerning older men, but it is a puzzle why the increase in the average is so pronounced for men with some qualifications but not for those with no qualifications. There is also a slight decline in average wages for men and women just prior to age among the pre-1955 birth cohort. Arguably this reflects a period effect. For this group, these years correspond to the recession years of 1991-1993 and, again, men with relatively low earnings propensities were less likely to work, thereby raising the average calculated from those who were in employment.
Within-and between-group differences
The nature of the within- and between-group differences in wage levels at different ages is illustrated by Figure 4, which focuses on the middle of the working life (age 40). The chart shows that there is substantial dispersion of wages within each group, and this implies substantial overlapping in the wage distributions of different groups. Among those born in or after 1955, the man at the 75th percentile of the group with no qualifications earns slightly more than the man at the middle of the group with some qualification. But the man at the middle of the no qualifications group earns more than the woman at the 75th percentile of the group with no qualifications. In additional analysis, I show how these patterns of inequality at each age exist right across the working life.
Figure 4. The distribution of log (hourly wage) at age 40, by group
Note: The line for each group shows the group-specific interquartile range (distance between the 25th and 75th percentiles). The filled circles show the group medians (50th percentile), which is the same as the mean.
Most descriptions of the income-age relationship are based on comparisons of income across age groups in a particular year and are based on cross-sectional data. In contrast, my research takes a longitudinal approach. Average income-age trajectories derived from longitudinal data look different from those derived from cross-sectional data. For hourly wages for instance, trajectories at the beginning of the working life are steeper - wage growth is greater - according to longitudinal data.
My analysis draws attention to the cooked spaghetti nature of income-age trajectories. I have argued that this pattern can usefully be summarised in terms of a number of factors. Looking at groups of individuals with similar observed characteristics, one can distinguish an average income-age trajectory for each group. Within groups, one can summarise differences across individuals in terms of: differences in incomes at the start of the working life; differences in income growth rates; and the association between initial incomes and income growth rates (they are negatively correlated). In addition, income-age trajectories differ because of substantial individual-specific transitory income changes from one year to the next.
My report argues that it is the transitory error component of income that cooks the spaghetti. These transitory changes may represent genuinely transitory effects on income, measurement error or, for broader measures of income, the effects of lifecourse events such as having children, and family formation or dissolution. A task for future research is to incorporate more sophisticated assumptions about its nature and persistence over time. This is likely to be facilitated by access to even longer panels than used in this study (perhaps from administrative record data, which may also have less measurement error than survey data). Long panels are necessary to help study the nature of income persistence in all its complex detail, including the extent to which observed short-run income changes for individuals are genuine.
1 The report was commissioned by the UK's National Equality Panel (www.equalities.gov.uk/national_equality_panel.aspx).