I. Introduction

Forensic economists are frequently asked to estimate the present value of future earnings losses in cases involving personal injury or wrongful death. In providing such estimates, the economist generally must assume how the injured party's earnings would have grown from the date of injury until the end of work life. Surveys suggest that forensic economists use a variety of approaches for forecasting how earnings will grow.

This paper takes the position that the best forecast of future earnings should incorporate an age-earnings profile that allows earnings growth to vary with age. While there are several estimates available for the typical age-earnings profile, these are typically cross-sectional estimates that we will argue have shortcomings. Namely, cross-sectional estimates may not isolate the true effect of age on earnings. To isolate the effect of age on earnings, we provide synthetic cohort estimates of earnings growth that control for economic conditions using data from 2000 through 2015.

While most economists believe that wage growth is not constant throughout a worker's life, some forensic economists simplify their analysis by assuming a constant growth rate. We will also show how this might bias estimates of damages. We find that, particularly for younger workers, the estimate of lost earnings can be significantly understated when the forensic economist uses conventional methods for estimating a constant rate of earnings growth. For older workers, the reverse is true.

The paper is organized as follows. First, we review the theory and some of the empirical evidence supporting the view that earnings growth is greatest when a worker is young and gradually slows and may turn negative. We discuss some of the methods forensic economists use for estimating future wage growth for workers. We describe our data and the empirical approach we use for estimating age-earnings profiles, and the importance of sorting out age, period, and cohort effects. We close by examining how alternative assumptions about wage growth affect estimates of economic damages.

II. Background

Surveys of forensic economists indicate a variety of methods are used for forecasting earnings growth (e.g., Slesnick et al., 2012; Luthy et al., 2015). Some examples specifically mentioned in the surveys include 1% real growth based on average earnings growth over the past 30 years, a 5-year running average of the growth in the employment cost index; Congressional Budget Office (CBO) estimates of future growth in the Employment Cost Index; Social Security Administration estimates of the future growth in wages; or an age-earnings profile that allows wage growth to vary over the life-cycle.

While the assumption of constant wage growth for forecasting future earnings is frequently employed by forensic economists, the assumption is contrary to a wide body of literature showing that earnings growth is greatest at early ages and eventually turns negative (e.g., Mincer, 1974; Murphy and Welch, 1990; Heckman et al., 2006). Assuming a constant rate of wage growth that matched the average over the life-cycle will overstate losses for someone near retirement and understate losses for a young person. Nevertheless, the simplicity of assuming a constant rate of earnings growth is attractive. It is easier to explain to a jury and makes the calculations simpler for the economist. While the simplicity of assuming a constant growth rate may be tempting, it is important to know if such an assumption creates a significant bias in the estimated value of future earnings.

If an economist chooses to use an age-earnings profile to estimate future earnings, they have a few options. One option is to use Census or American Community Survey (ACS) data that provides estimates of earnings by age, gender, and education.¹ One could use a single year of such data and compute the average rate of earnings growth for a specific demographic group for different age ranges by comparing, for example, the average earnings of 25-34-year-olds with that of 35-44-year-olds. Alternatively, one could form synthetic cohorts by comparing age groups across years. For example, compare 25-34-year-olds in 1980 with 35-44-year-olds in 1990. Each of these approaches has shortcomings that we describe below.

The main problem economists face when estimating an age-earnings profile is that the real earnings path an individual will realize as they age will depend on (1) the rate of growth in the individual's productivity due to human capital accumulation/depreciation; (2) the state of the economy; and (3) the particular cohort that the person is born into. These three separate factors are frequently referred to as age, period, and cohort effects, respectively (e.g., Heckman and Robb, 1985; Hanoch and Honag, 1985). The age effect is the change in productivity that is specific to a given individual due to the accumulation or depreciation of human capital, and changes in work effort, over the life-cycle. Period effects are factors that increase or decrease the earnings of workers in a given year and might reflect either long-term trends in productivity or temporary shocks to the economy. Cohort effects impact a given birth cohort throughout their life-cycle. For example, the baby-boomers represented a particularly large cohort, and some have argued that their earnings were lower than smaller cohorts that preceded and followed them (e.g., Freeman, 1979; Berger, 1985, 1989).

Numerous authors discuss the identification problem inherent in separating these three effects in regression analysis (e.g., Glenn, 1976; Heckman and Robb, 1985). The problem is that a person's birth year, age, and, year are perfectly collinear. To illustrate the identification problem without a regression, consider the use of cross-sectional data to estimate earnings growth by, for example, comparing the earnings of 25-34-year-olds with that of 35-44-year-olds in the same year. This difference reflects both age effects and cohort effects. If the period effect is equal across cohorts, it is removed by differencing since the earnings are observed at the same point in time. If cohorts are gradually becoming more productive over time due to better schooling or training, or if the older cohort is especially large and has lower than usual wages, the cross-sectional comparison will understate the true age effect.

If using a synthetic cohort approach, the growth in earnings would be based on, for example, a comparison of 25-34-year-olds in 2000 with 35-44-year-olds in 2010. This comparison represents a combination of age effects and period effects. If cohort effects are equal across years of data, the cohort effects are differenced out of the comparison. The problem with this approach is that the period effects may be unusually large or small depending on the years chosen for the analysis. For example, if the economy is especially strong in 2000 and/or especially weak in 2010, the earnings growth obtained from the described synthetic cohort approach will understate the true growth that a worker is likely to realize.

To illustrate the identification problem in the context of a linear regression for the age-earnings profile, consider the following model:

Where y_it is real annual earnings for person i in period t, f(age) is a polynomial in age, P_t is a period fixed effect; C_i is a cohort-specific fixed effect, ε_it is a random error in the regression model. Given our particular case, the period fixed effects can be estimated by including dummies for the year of the survey, and cohort fixed effects by including dummies for the year of birth. The identification problem is that age_i, P_t, and C_i are perfectly collinear.

One possible solution to the identification problem is to assume cohort effects are equal across several adjacent birth years. For example, if all people born in the same decade experience the same cohort effect, dummies for decade of birth would be sufficient to capture the cohort effects and identification is possible. We experimented with this approach and found evidence against equal cohort effects across several birth years.²

The solution we choose for identification, suggested by Heckman and Robb (1985), replaces the year dummies with control variables that account for the period effects. Namely, if period effects are due to changing economic conditions that cause wages to rise or fall over time, the inclusion of these economic controls will negate the need for the year dummies and make it possible to separately identify cohort and age effects. Our controls for economic conditions are the state-specific unemployment rate and the coincident index.

III. Estimates of Age-Earnings Profiles

In this section, we provide estimates of age-earnings profiles using data from the 1% sample of the 2000 Census combined with data from the ACS administered between 2001 and 2015. We restrict the sample to civilian full-time, year-round wage and salary workers between the ages of 19 and 65. Annual earnings are converted to 2015 dollars using the Consumer Price Index. We estimate age-earnings profiles for seven education groups by gender. We match our education groups to those used by Skoog, Ciecka, and Kreuger (2011) so that we can use their measures of employment probabilities to estimate the expected present value of earnings losses. The one exception is that we combine those with a GED into the same group as high school graduates due to small sample sizes for the GED group.

After sample restrictions, our data set includes 12.5 million observations. The sample sizes are similar across the 16 years of data, except that the 2001 through 2004 ACS samples are about one-half the size of the other years. Other than the 2001-2004 samples, each year of data accounts for between 6.7% and 7.9% of the 12.5 million observations. The 2001-2004 samples account for between 2.6% and 2.9% of the data.

Using the pooled cross-sectional data, we estimate an age-earnings profile using a quartic in age for each of the seven education groups by gender. Based on research by Murphy and Welch (1990), the quartic is generally accepted as the functional form that best fits the age-earnings profile.³

For each education group, we include only those workers whose age is greater than or equal to the number of years typically required to complete that level of education and have a full year available for market work. For those with a high school degree or less, we assume their full-time work begins at age 19. The minimum age requirement required for inclusion in the sample for the other levels of educational attainment are as follows: (1) 21 for some college without a degree; 21 for associates degree 21; 23 for Bachelors degree; 25 for Masters degree; 27 for Professional degree or Ph.D. Since the sample sizes for workers become small at late ages and because of selection problems that arise due to retirement decisions, we exclude anyone over age 65 from the sample. The selection problems at retirement could bias our estimates of earnings growth. For example, if workers with higher income retire sooner, the selection bias would cause earnings growth to be understated. We should note that this bias could be present before age 65, but retirement rates are much lower before age 65 so that the bias would be smaller. This issue merits more study, but it is beyond the scope of this paper, and it is not easily addressed with pooled cross-sectional data.

Figures 1a-b show plots of three different estimates of age-earnings profiles for each of six education groups by sex up to age 65. As discussed earlier, the starting age differs depending on the level of educational attainment.⁴ The first age-earnings profile is a plot of the actual mean of real income by age. The second plot is the age-earnings profile estimated using a quartic in age. The first two plots are virtually identical and illustrate how well the quartic fits the cross-sectional age-earnings profile.

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The third age-earnings profile is the synthetic cohort model estimated by controlling for cohort fixed effects (i.e., a dummy for each year of birth in the sample) and economic conditions. As we noted earlier, a failure to control for cohort fixed effects could lead to either an over- or under-statement of the true effect of age on earnings. If workers born in more recent years have higher skills and/or face a stronger labor market than those born in earlier years, a failure to control for cohort effects would lead to an underestimate of the effect of age on earnings growth. The reverse is true if more recent cohorts are less skilled or face a weaker labor market. Since the synthetic cohort model isolates the effect of age on earnings from the period or cohort effects, it is the most suitable for estimating earnings growth as a worker ages.

For all but those with a Professional or Ph.D. degree, the estimated age-earnings profile is flatter after controlling for cohort effects and economic conditions for both men and women. For those with a Professional or Ph.D., the estimated age-earnings profile is steeper at later ages, particularly for women. As a result, using a typical cross-sectional age-earnings profile could generate systematic biases in estimated losses of future earnings and the direction of the bias will vary across education groups and by age.

Figures 2a-b provide plots of the estimated earnings growth by age using three different methods. The first is that implied by the synthetic cohort model estimated above. The second is the earnings growth implied by a quartic without controls for cohort effects or economic conditions. The third is that implied by using “age-bands” to calculate earnings growth the ACS data. As noted earlier, this third option for estimating wage growth is sometimes used by forensic economists since cross-sectional data is available from the Census Bureau for annual income by age and sex for different education groups. For example, if earnings for high school graduates are provided for 25-29 and 30-34-year-olds, the rate of growth is calculated as [(w₂/w₁)^1/5-1] where w₂ and w₁ are the estimated average wage for 25-29 and 30-34-year-olds. The comparison of the alternative methods for calculating the earnings growth rates establishes several points. First, comparing the earnings growth from the cross-sectional models versus those that control for cohort effects and economic conditions, it is clear that the cross-sectional estimates over-state earnings growth particularly below age 40 for both men and women with a Bachelor's degree or less. For both men and women with a Professional degree or Ph.D., cross-sectional estimates understate earnings growth. While the differences may appear modest in the plots, these effects compound over the worker's remaining work-life and could have a sizable impact on the overall estimate of the loss—especially for younger workers.

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The synthetic cohort estimates are also compared to earnings growth based on earnings by age-group described earlier. In Figures 2a-b, these growth rates look like a step function since the growth rate is assumed to be constant between the midpoints of two adjacent age bands (e.g., if comparing 25-29 with 30-34-year-olds, the wage growth estimate is applied to those between the ages of 27 and 32). See Appendix tables A1-A3 for age-specific wage growth rates by education and sex using synthetic cohort, unadjusted earnings, or cross-sectional data.

The estimates of earnings growth with age bands follow a path similar to the cross-sectional estimates from a quartic in age. However, it is fairly clear that the large drop in wage growth in the early years could lead to substantial differences in the estimates of earnings losses for someone whose age is immediately before versus immediately after an age where the wage growth estimates change.

IV. The Impact of Earnings Growth Assumptions on Projected Earnings Losses

Since the forensic economist typically starts with some base level of earnings and then forecasts forward, the effect of using alternative estimates of wage growth will vary depending upon the age at which the forecast begins. To illustrate the magnitude of these effects, we compute the expected present value of life-time earnings using alternative approaches. First, for a particular age, we start with base earnings of 100. We then grow these earnings using estimates of wage growth. To compute the expected present value of the loss in future years, we compute:

Where p_t is the probability that a worker who was active at age s is still active at age t, y_t is the projection of the nominal earnings for the worker at age t, r_t−s is the discount rate for (t − s) years in the future. The discount rate is specific to the number of years in the future that the earnings are to be realized and are drawn from the yield curve for zero coupon bonds. For example, forecasting earnings for a worker beyond age 25 (s = 25), the earnings that would be received at age 35 (t = 35) would be discounted using the yield on a 10-year zero-coupon bond. Since our estimates of earnings growth go up to age 65, we assume that nominal earnings growth beyond age 65 are zero. However, we include earnings from age 66 to 100. We ignore any chance that age exceeds 100.

To illustrate the effect of using cross-sectional versus synthetic cohort-age profiles, we compute the percentage change in the EPV associated with a change from the synthetic cohort approach to alternative assumptions forensic economists use for forecasting earnings growth (see Slesnick et al., 2012; Luthy et al., 2015). The first approach uses estimates of real earnings growth implied by using cross-sectional estimates of earnings by age-band that we plotted in Figures 2a-b. The second approach assumes a constant rate of earnings growth of 0.7% through age 65. This growth rate is based on CBO projections of the real rate of growth in the employment cost index.⁵ Since we will use the nominal interest rate to convert to present values, earnings growth is converted to nominal terms by adding the CBO projected inflation rate of 2.4% to all three measures of real earnings growth through age 65.

The results of the comparisons are provided in Figures 3a-b. Compared to the synthetic cohort estimates, the cross-sectional estimates of earnings growth lead to a fairly significant increase (up to 20%) in estimated damages for younger workers with less than a Bachelor's degree. For those with a Bachelor's or Master's degree, the effect of using cross-sectional estimates of wage growth is relatively small. For those with a Professional degree or Ph.D., cross-sectional earnings growth estimates reduce estimated losses fairly substantially (more than 20%) for those with a Professional degree or Ph.D. Overall, assuming earnings growth implied by cross-sectional data on earnings by age groups can create either an upward or downward bias in damage estimates depending on the specific age the damages begin and the sex and education of the damaged party.

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Assuming a constant rate of wage growth that matches the projected rate of increase in the employment cost index leads to a substantial understatement of earnings losses for young workers in all education groups, but eventually overstates losses for most groups of older workers. The understatement of damages for young workers is especially pronounced because wage growth is most rapid at early ages and the assumption of a 0.7% real growth rate over their career causes losses to be underestimated by more than 40% for some male education groups. In later years, the constant growth assumption can overstate damages because the synthetic cohort estimates for most sex/education groupings drop below 0.7% by age 50.

V. Summary and Conclusions

When a worker is injured or there is a loss of life, a forensic economist may be asked to forecast the present value of future earnings losses. There are several approaches that the economist may pursue to forecast future earnings. Typically, the economist starts with a base projection for earnings based on the worker's recent earnings history. From that base, the economist must forecast future earnings using some assumptions about earnings growth. Our paper provides estimates of earnings growth based on a synthetic cohort model that removes cohort and period effects. We argue that these estimates are superior to other commonly used methods for forecasting earnings growth.

When we compare estimates of economic damages based on other commonly used wage growth assumptions, we find that the other methods can lead to substantial positive or negative biases depending upon the age the damages begin as well as the specific sex and education group. In general, we believe that our synthetic cohort estimates of earnings growth provide a better forecast of earnings growth and economic damages. We also believe that more work in the area is warranted. For example, we do not provide estimates of earnings growth beyond age 65. This is partly because of small sample sizes, but also because there appears to be a selection issue in retirement behavior that may bias the estimates of earnings growth. For example, if high-income earners retire sooner, this will make it appear that earnings drop sharply at later ages. Research sorting out the selection effect from the true effect of age on earnings would be helpful—particularly for cases where the damaged party is older.

While the synthetic cohort estimates of age-earnings profiles have advantages relative to other common approaches, these estimates should not be used without caution. If an injured party has an earnings history that deviates substantially from the projections of earnings growth over the relevant age-period, one should question what is unusual about the case. For example, earnings growth could be affected by a change in career paths, interruptions due to family responsibilities, or other factors. One might also need to consider unusual features that could cause earnings growth to be different in the future for a person. Such issues might include health considerations, the likelihood of promotion in their job, or something that will likely affect the earnings of workers with a specific set of skills. Absent unusual circumstances about a person's likely earnings growth, however, the age-earnings profiles presented here are a defensible method for projecting future earnings.