Cross-Sectional Average Length of Life (CAL)

Definition

CAL (cross-sectional average length of life) is the mean length of life in a population with constant births, computed using cohort survival probabilities rather than period death rates. It is formally equivalent to the "generalized tempo-adjusted period life expectancy" proposed by Bongaarts and Feeney (2002) (BF). CAL is a measure of population dynamics — specifically, the relative size of different age cohorts in a stationary population — not a correction to period life expectancy.

Key Ideas

Population-dynamics interpretation. CAL measures what period life expectancy would be if the population had constant births — i.e., it represents the life expectancy implicit in the current cross-sectional age distribution. It answers: "given the current age structure, what is the typical lifespan of those alive today?" This is distinct from the synthetic-cohort question period $e_0$ answers.
Equivalence to tempo-adjusted $e_0$ . Bongaarts and Feeney (2002) proposed adjusting period $e_0$ for "tempo distortions." Wilmoth (2005) shows that their generalized adjustment, applied exactly, yields a quantity mathematically identical to CAL as defined by Brouard (1986) and Guillot (2003). The BF approximation ( $e_0^{BF} \approx e_0/(1-\dot{r})$ ) is a first-order approximation to CAL.
CAL $< e_0$ under improving mortality. Period survival $\ell_p(x,t)$ conditions on current (favorable) death rates. CAL uses cohort survival $\ell_c(x,t-x)$ , which conditions on past (higher) rates. Under sustained improvement, CAL is systematically lower than $e_0$ . Guillot (2003) estimated the gap at approximately 2–4 years for contemporary high-income countries.
CAL $\approx \tfrac{1}{2}(e_0' + e_0'')$ . Under the linear shift model (Wilmoth 2005), CAL lies approximately halfway between $e_0'$ (mean age at death in the constant-birth population under period mortality) and $e_0''$ (achieved cohort life expectancy (LE) for the cohort whose mean age at death is attained at t). This places CAL between the period measure and the cohort projection.
Independent discovery. CAL was independently derived by Brouard (1986, unpublished lecture notes) and Guillot (2003). Guillot (2003) showed the BF generalized formula equals CAL; Guillot (2005) further examined the relationship. The concept predates Bongaarts-Feeney's "tempo adjustment" framing.
CAL is sensitive to tempo effects (Schoen and Canudas-Romo 2005): Despite BF's intent to use CAL as a tempo-robust alternative to period LE, Schoen and Canudas-Romo show it is not. In a pure-tempo scenario where cohort LE is held fixed but the timing of deaths within cohorts shifts, CAL fluctuates by more than 2 years — nearly as much as period LE. This is because CAL integrates cohort survival probabilities across all ages, inheriting the full history of past timing shifts. By contrast, Average Cohort Life Expectancy (ACLE) (which depends only on completed cohort LEs, not intra-cohort timing) remains flat in this scenario. CAL also lies consistently below period LE under sustained improvement, making it a downward level adjustment rather than a neutral correction. See Average Cohort Life Expectancy (ACLE).
Smoothing requirement. Computing CAL requires cohort survival probabilities for all cohorts currently alive, including younger cohorts whose full mortality experience is not yet observed. Any empirical CAL calculation requires smoothing over adjacent cohort years, introducing a data dependency absent from period $e_0$ .

How It Works

For a population with constant births $B$ , the number alive at age $x$ at time $t$ is $B \cdot \ell_c(x, t-x)$ , where $\ell_c$ is cohort survival. The total population is:

$N(t) = B \cdot \int_0^\infty \ell_c(x,\, t-x)\, dx$

The mean age in this population (= CAL, denoted $e_0^*$ by Wilmoth) is:

$e_0^*(t) = \frac{\int_0^\infty x \cdot \ell_c(x,\, t-x)\, dx}{\int_0^\infty \ell_c(x,\, t-x)\, dx}$

This uses cohort survival probabilities $\ell_c(x,t-x)$ evaluated along the cohort diagonal of the Lexis surface — i.e., at the age-time combinations actually experienced by cohorts alive at time $t$ .

Why It Matters

Resolves the BF debate. The apparent conflict between period $e_0$ and "tempo-adjusted $e_0$ " dissolves when CAL is recognized as a different measure with a valid but distinct interpretation, not a corrected version of period $e_0$ .
Population age structure. CAL directly relates to the age structure of the current population in a way period $e_0$ does not. Under constant births, the population share over age $x$ is $\ell_c(x,t-x)/e_0^*(t)$ . This makes CAL useful for studying population momentum and age-structural dependency.
Tempo effects in fertility vs. mortality. The Bongaarts-Feeney (1998) quantum/tempo framework legitimately identifies distortions in count fertility measures like Total Fertility Rate (TFR). CAL's equivalence to tempo-adjusted $e_0$ makes clear that no analogous distortion affects period $e_0$ (which is already a timing measure). See Tempo Effects in Mortality.

Open Questions

The empirical CAL series for high-income countries shows a gap of ~2–4 years below period $e_0$ . It is not settled whether this gap is widening or stable as mortality improvement continues.
Whether CAL or period $e_0$ is more appropriate for actuarial projections (e.g., Old-Age, Survivors, and Disability Insurance (OASDI) cost forecasts) depends on whether one is projecting the fate of a synthetic cohort (period $e_0$ ) or the longevity of current survivors (CAL). The question is rarely made explicit in applied practice.