"Life expectancy" can be defined in two quite different ways, such that social policies that increase a society's average life expecancy by one definition may well lower it by the other. Of course "life expectancy" is one way we measure the success of a society, so the precise definition used may influence a society's direction. It is not clear to me that today's standard definition of life expectancy is really the one we want to use for measuring the overall health or success or our society. When I say "not clear", I mean just that; it's also not clear to me that the alternate definition is superior. The purpose of this memo is to explain the difference between the two definitions.

When you read a headline saying for instance that American life expectancy has
increased but is below Japan's, it is always using what might be called "life
expectancy at birth", or more precisely the expected age at death for a
newborn infant. Let's call this definition of life expectancy LE1, and call the
alternate definition LE2. LE2 is the average expected age at death *for
people now living*. To see the difference between LE1 and LE2,
consider that a table of 1989 life expectancies shows the life expectancy for a
newborn American white female to be 79.2. But a white female who is
already age 80 has an expected future life expectancy of 8.9 years, and thus an
expected age at death of 88.9 years. One's future life expectancy of course
falls as one ages, but one's expected age at death only rises--slowly at first and
then more rapidly. For American white females of ages 0, 20, 40, 60, and
80, expected age at death is respectively 79.2, 80.2, 80.9, 82.9, and 88.9.
Therefore LE2, defined as the mean expected age at death for people now
living, always exceeds LE1.

To see the difference between LE1 and LE2 more clearly, consider a simple artificial example involving a shorter-lived species such as a squirrel. Suppose you had noted the life spans of 5 squirrels, and had observed that one had lived exactly one year, one exactly two, and one each exactly 3, 4, and 5 years. If you used this sample of 5 squirrels to compute LE1 for squirrels, you would note that the mean age at death was 3, so LE1 = 3. However, to compute LE2 you would reason as follows. Based in this sample, you assume that in a typical population of squirrels at any given moment, for every 5 squirrels aged 0 to 1 (that is, they have not yet reached their first birthday), there are another 4 aged 1 to 2, another 3 aged 2 to 3, another 2 aged 3 to 4, and another 1 aged 4 to 5. These numbers sum to 15 (that is, 5+4+3+2+1 = 15), so the assumed proportions in the various age groups are respectively 5/15, 4/15, 3/15, 2/15, and 1/15.

Based on the sample data the squirrels aged 0 to 1 are considered to have 20% chances of dying at each of the ages 1, 2, 3, 4, 5, for an expected age at death of 3. Those aged 1 to 2 are considered to have 25% chances of dying at each of the ages 2, 3, 4, 5, for an expected age at death of 3.5. By similar calculations those in the remaining three age groups have expected ages at death of 4, 4.5, and 5 respectively.

We thus have 5 age-groups comprising respectively 5/15, 4/15, 3/15, 2/15, and 1/15 of the assumed population, with expected ages at death of 3.0, 3.5, 4.0, 4.5, and 5.0. LE2 is a weighted average of these latter 5 values, using the former 5 values as weights. Therefore

LE2 = 1/15*(3.0*5+3.5*4+4.0*3+4.5*2+5.0*1)

= (15+14+12+9+5)/15 = 55/15 = 3.67.

As already mentioned, LE1 computed from the same data is 3.0.

Some algebra shows that if we let A denote the ages at death of the cases in a sample from a population, then an estimate of LE2 based on that sample is

LE2 = SUM(A^{2})/SUM(A)

where SUM denotes summation. For the current example the 5 values of A
are 1, 2, 3, 4, 5, so SUM(A^{2}) = 55 and SUM(A) = 15, and
LE2 = 55/15 = 3.67 as already calculated.

LE2 is closely related to future life expectancies, since LE2 is an average of expected ages at death, and everyone's expected age at death is their current age plus their future life expectancy. A person's future life expectancy is one determinant of their commitment to society. It is often noted that younger people are more likely to vote for schools, roads, conservation measures, and other political measures that are essentially investments in the future, since they personally expect to be alive to reap those future benefits, while older people tend to favor current expenditure over investments in the future. Therefore by adopting policies that maximize future life expectancy, society encourages longer-term thinking. If a society measured its life expectancy as LE2 rather than LE1, that would contribute toward such thinking, since LE2 is more closely related to future life expectancy than LE1 is.

An odd and perhaps troubling feature of LE2 is that if some individual
person's expected age at death is estimated to be less than half LE1, then LE2
is *increased* by terminating that person's life as early as possible.
Thus use of LE2 would tend to encourage such policies. The algebra behind
this claim is beyond the scope of this note, but I can use my 5-squirrel
example to illustrate it. Recall the 5 squirrels were observed to die at ages 1,
2, 3, 4, and 5, so LE1 was 3.0. Suppose that the first of these squirrels had
died at birth (age 0) instead of age 1. Then using the algebraic formula above,
we have SUM(A^{2}) = 0+4+9+16+25 = 54, SUM(A) =
0+2+3+4+5 = 14, LE2 = 54/14 = 3.86. The earlier death thus raised
LE2 from 3.67 to 3.86, while at the same time lowering LE1 from 3.0 to 14/5
or 2.8.

Actually something like this is built into our current definition of life expectancy, which is not quite LE1 as I have defined it. Under LE1 strictly defined, an infant who dies after 3 days lowers computed life expectancy very substantially. Many people consider that unreasonable, so that infants dying very young (usually under 1 year) are not even counted in the computations. Thus an infant dying after 370 days lowers the society's computed life expectancy, while an infant dying after 360 days doesn't because they aren't even counted. That's somewhat like the feature of LE2 just discussed.

In general, if two societies are equal on LE1, then the one with the more unequal distribution of ages at death--with some people dying very early and others living very long-- will have the higher value of LE2.