Since the 1982 publication of the Bureau of Labor Statistics updated

worklife tables, articles have appeared in the Monthly Labor Review and

several legal journals regarding the use of such tables in liability

proceedings. As stated in these articles expert witnesses in wrongful

death and injury litigation are interested primarily in using the

increment-decrement worklife tables to find the expected number of years

an individual would have been active in the work force had an injury or

death not occurred. This expected worklife is then used to calculate

the present value of “expected” earnings lost between the date

of death or injury and the date of expected final separation from the

work force.

It will be shown here that such methods do not yield a

mathematically defensible expectation of future earnings, because the

sum of earnings over the expected worklife need not equal the sum of

expected yearly earnings over life. A model based on the

increment-decrement worklife table is developed for calculations of

expected earnings in each year of possible life. This model is then

modified to obtain the discounted present value of expected future

earnings. The final section of this article presents our calculations

of expected earnings for representative individuals who die prior to age

85, and compares them with those reported by David Nelson and Kenneth

Boudreaux in past issues of the Review.

Expected earnings

It is a simple exercise to show that the sum of earnings over

expected worklife need not equal the sum of expected yearly earnings

over life. For instance, assume that a cohort of 1,000 people are

initially active in the work force but, at the end of the first year,

400 become inactive. Similarly, in the second and third years, 300

become inactive at the end of each year. The expected worklife for this

hypothetical cohort is 1.9 years. If individual earnings in each

subsequent year are projected to be $25,000, $30,000, and $35,000, then,

using current techniques, an expert witness would conclude that expected

earnings are $52,000 (= $25,000 + 0.9 X $30,000), ignoring discounting

and other adjustments. But such a calculation overlooks the interaction

of the probability of being active in each year and the earnings which

are projected for the year. The true mathematical mean, or expected

earnings, is $53,500 (= 0.4 X $25,000 + 0.3 X $55,000 + 0.3 X $90,000).

For pedagogical ease, the above example assumes that the

hypothetical cohort of 1,000 remains alive for all 3 years. It does not

allow for both movement into and out of the work force. These

complications affect the calculation of expected income. Using all the

information now available in the increment-decrement worklife tables,

the true mathematical expected earnings of an active individual at age x

can be derived in the following manner.

Let q.sub.x represent the probably (or more precisely, the relative

frequency) of death in the year following exact age x. Let l.sub.x

represent the number of survivors at age x. At each age, survivors can

be divided into those who are active in the work force and those who are

not. In addition, at each age, a survivor who is active may stay active

or leave the work force, while someone who is inactive may stay inactive

or move into the work force. Let the four relevant probabilities (or

relative frequencies) for work force transition be represented as

follows: I.sub.px.A = the probability that someone who is inactive at

age x will be active at age x + 1; I.sub.px.I = the probability that

someone who is inactive at age x will be inactive at age x + 1;

A.sub.px.I = the probability that someone who is active at age x will be

inactive at age x + 1; and, A.sub.px.A = the probability that someone

who is active at age x will be active at age x + 1

The above transitional probabilities are conditional on survival

from age x to age x + 1. Thus: I.sub.px.a + I.sub.px.I = 1, and

A.sub.px.I + A.sub.px.A = 1

Assuming that the probability of death and the probabilities of

transition between work force states are independent, the number of

inactive survivors at age x + 1 (that is, sup.1.l.sub.x+1) and the

number of active survivors at age x + 1 (sup.A.l.sub.x+1) can now be

defined as: sup.I.l.sub.x+1 = (1 – q.sub.x.)

(.sup.I.l.sub.x./sup.I.P.sup.I./.sub.x +

sup.A.l.sub.x./.sup.A.P.sup.I./.sub.x.); and sup.A.l.sub.x+1 = (1 –

q.sub.x.) (.sup.I.l.sub.x./.sup.I.P.sup.A./.sub.x +

sup.A.l.sub.x./.sup.A.P.sup.A./.sub.x.) where l.sub.x = sup.I.l.sub.x +

sup.A.l.sub.x., and l.sub.x+1 = l.sub.x (1 – q.sub.x.)

As in the published increment-decrement worklife tables, these

formulas yield: Expected worklife for persons = (1/.sup.A.l.sub.x) * [(1

– Q.sub.x+n) active at age x (sup.A.l.sub.x+n.sup.A.p.sup.A.sub.x+n +

0.5.sup.A.l.sub.x+n.sup.A.p.sup.I.sub.x+n +

0.5.sup.I.l.sub.x+n.sup.I.p.sup.A.sub.x+n) + 0.5.Sup.A.l.sub.x+n

q.sub.x+n] where M is the number of ages remaining after x until the

cohort is extinguished.

The above formula for expected worklife is based on a cohort that

dies out over x + M + 1 years. At age x, for each of the remaining M +

1 years, there are four terms over which yearly summation takes place.

The first three terms refer to persons who survive to the next year of

age. Among these survivors there are those who are active at the start

of the year and stay active for a full year. For this group. (1 —

q.sub.x+n)(.sup.A.l.sub.x+n.sup.A.p.sup.A.sub.x+n) is the total number

of active years accumulated between ages x + n and x + n + 1. Persons

who survive the year, but move from active to inactive or from inactive

to active status, are assumed to be active for one-half year. Thus (1 –

q.sub.x+n)(0.5.sup.A.l.sub.x+n.sup.A.P.sup.I.sub.x+n) and (1 –

q.sub.x+n)(0.5.sup.I.l.sub.x+n. sup.I.P.sup.A.sub.x+n) are the total

numbers of active years accumulated in year x + n by individuals who

live age x + n + 1, and who make midyear work force transitions from

either active to inactive or inactive to active status, respectively.

Persons who were active at the beginning of the year and die in the

interval are also considered active for one-half year. Thus,

0.5.sup.A.l.sub.x+n.q.sub.x+n is the total number of active years

between years x + n and x + n + 1 for individuals who are assumed to die

at age x + n + 0.5.

Unlike the simpler mortality tables, the increment-decrement model

poses an added complication in the formulation of expected worklife

which has implications for calculating expected earnings: The values for

survivors by age and work force status depend upon the age at which one

begins the computations and the distribution of persons by work force

status at that age. In a mortality table, any arbitrary value for

l.sub.o will yield the same expectation of life for each successive age.

In the increment-decrement table, one must set either the active or

inactive population to zero at the starting age. For example, the

expected working life for persons inactive at age 16 is computed by

setting sup.I.l.sub.16 = 1,000 and sup.A.l.sub.16 = 0. The associated

sup.I.l.sub.x and sup.A.l.sub.x can then be computed from these two

initial values. If one starts at age 17, or calculates the table for

persons out of the work force at age 16, all of the sup.I.l.sub.x and

sup.A.l.sub.x values will change.

Expected earnings at age x are calculated by introducing annual

earnings. Let total annual earnings in year x(y.sub.x.) be paid in two

equal biannual payments. The payments to persons changing work force

status during a year can be approximated by assuming that a person who

becomes inactive or dies is active for the first half of the year, and

that a person who is inactive and becomes active has earnings in the

last half of the year. Under these conditions:

Expected earnings for active person = (1/.sup.A.l.sub.x.) * [(1 –

q.sub.x+n.) at age x (.sup.A.l.sub.x+n.sup.A.P.sup.A.sub.x+n.y.sub.x+n +

0.5.sup.A.l.sub.x+n.sup.A.P.sup.I.sub.x+n.y.sub.x+n +

0.5.sup.I.l.sub.x+n.sup.I.P.sup.A.sub.x+n.y.sub.x+n.) +

0.5.sup.A.l.sub.x+n.q.sub.x+n.y.sub.x+n.]

As does the formula for expected worklife, this formula for

expected earnings describes four groups who work (or, more precisely,

are active) for different portions of the year between ages x + n and x

+ n + 1. Years of work, however, are now evaluated in terms of total

dollars earned by each of the four groups.

The above formula for calculating expected earnings involves an

assumption either that time has no value, or that productivity and

inflation gains are exactly offset by the market rate of interest. While

some expert witnesses still advocate the use of such a “total

offset method, “.sup.3 courts today will accept the discounting of

future earnings to reflect the net time value of money. The expected

earnings equation can be modified to accommodate discounting by defining

either a continuous compounding rate (r) or its annual discount rate

equivalent (d), that is, 1 + d).sup.-n = e.sup.nr.. (For instance, if

the annual rate of discount is 11 percent, its continuous compounding

equivalent is 10.44 percent.) The present value of expected earnings for

an active person at age x, in continuous discounting form is:

(1/.sup.A.l.sub.x.) * [left brace](1 – q.sub.x+n.)

[0.5.sup.A.l.sub.x+n.sup.A.P.sup.A sub.x+n.y.sub.x+n.(e.sup.(n+.5)r +

e.sup.(n+1)r.) + 0.5.sup.A.l.sub.x+n.sup.A.p.sup.I.sub.x+n.y.sub.x+n.e.sup.(n+.5)r + 0.5.sup.I.l.sub.x+n.sup.I.P.sup.A.sub.x+n.y.sub.x+n.e.sup.(n+1)r] + 0.5.sup.A.l.sub.x+n.q.sub.x+n.y.sub.x+n.e.sup.(n+1)r.[right

brace]

Corresponding expressions for the expected worklife, expected

earnings, and present value of expected earnings for persons inactive at

age x can be derived in a similar way.

Calculation procedures

In her comment on Boudreaux’s and Nelson’s methods for

adjusting the worklife tables to estimate lost earnings, Shirley Smith notes that “frequently, economists want to look past the

lifetime-worklife expectancy figure to study the timing of the potential

earnings stream.” Here we argue that in the calculation of

expected lost earnings it is not sufficient to know the “median

number of years until final separation,” as defined by Nelson; to

adjust this figure by assuming that activity is evenly spread over the

entire period until retirement, as suggested by Boudreaux; or to know

any other single number that represents the possible length of time that

a person will be active. A true mathematical expectation of lost

earnings requires knowledge of the timing of probable activity and of

the potential (nominal or discounted) earnings during the period of

probable activity.

Because the timing of probable activity is sensitive to both the

initial work force status and the age of an individual, our development

of the true mathematical expected earnings, unlike the approaches of

Nelson and Boudreaux, emphasizes an active or inactive starting point.

To assess the consequences of this distinction, consider the example

provided by Boudreaux. A man age 30 with annual earnings of $25,000

(using a current market interest rate of 11 percent and an annual

earnings increase of 4.5 percent) has a present value of

“expected” earnings of $332,913, by the worklife table

estimate of 29.2 years of remaining worklife for the entire population.

Using Nelson’s 31.5 years to final separation criterion, the

present value of “expected” earnings is $341,857.

Boudreaux’s 7.3-percentage reduction criterion drops this estimate

to $316,901. However, our calculations show that the true mathematical

present value of expected earnings for an active man at age 30 is

$319,397, and for an inactive man at age 30 it is $273,535.

In some cases, one might wish to ignore initial work force status.

A weighted average of our active and inactive estimates can be obtained

by using the proportions of men active and inactive at the initial age.

In the above example of a man at age 30, this average present value of

expected earnings is $316,502, which compares favorably to

Boudreaux’s estimate of $316,901. Given the ease of using

Boudreaux’s adjustment method, one might question the practical

value of using our more complicated true mathematical expectation

method.

Unfortunately, Boudreau’s approximation is close to the true

mathematical expectation only for younger men. His assumption that

inactivity is spread evenly over the entire period until retirement is

inappropriate at older ages, when the proportion that are inactive rises

rapidly. For a younger person, changes in expected earnings caused by

increasing probabilities of inactivity later in life are mitigated by

high discounting of expected earnings in distant years. For older

people, the mitigating effect of discounting is not present. Thus, for

a man age 45 with the same earnings stream used above, Nelson’s and

Boudreaux’s methods of estimating the present value of potential

earnings yield $256,044 and $242,217, respectively. Our mathematical

expectations are $236,626 for an active man, $155,310 for an inactive

man, and $231,325 for the weighted average of active and inactive

persons.

OUR METHOD OF CALCULATION requires two modifications of the

increment-decrement worklife tables published by BLS. First, the

probabilities of transition into and out of the work force at each age

must be converted to probabilities that are conditional on survival.

Second, conditional probabilities of transition between active and

inactive work force status must be added at age 76 to close the table.

The relevant probabilities of transition are provided in table 1. A

computer program for calculating the present value of expected earnings

based on these transitional probabilities is available from the authors.