Demographic Flux
Lexis surface · observed HMD/HFD starting populations
What it shows
The piece shows birth cohorts aging through calendar time on a Lexis surface, with five strip panels below tracking crude birth, death and growth rates, the total fertility rate, life expectancy at birth, and the first demographic dividend as the population age structure shifts.
The method
The piece runs cohort-component (Leslie) projection on a Lexis diagram, seeded from the observed Human Mortality Database (HMD) and Human Fertility Database (HFD) profile at the base year, with sex-specific Lee–Carter mortality. The forward path is a user-controlled scenario, not any country’s recorded history after the base year.
How to read it
The main panel is a Lexis surface: calendar year runs left to right, age runs upward from 0 to 100. The run spans the profile base year through 2100.
Each vertical slice shows how many people are at each age that year. Stroke weight and brightness track each age group’s share of the total population. The encoding compares that share to the largest share seen anywhere in the run so far, not to the peak within a single year. A broad young pyramid therefore reads heavier than a narrow aging column or a thin cohort born under low fertility.
Color marks age, not size: shading from blue at the young ages to rust at the older ages.
At the left edge, the population already alive at the base year enters at its observed ages. Cohorts born later enter from the bottom as the period clock advances.
Horizontal reference lines at ages 20 and 65 mark the working-age band: cohorts crossing 20 enter the producer group that drives the demographic dividend; cohorts crossing 65 join the older dependent group. Both boundaries are labeled on the right margin.
Five strip panels below trace the scenario in sequence: crude birth and death rates together (per 1,000 population), crude growth rate (births minus deaths), the period total fertility rate (TFR) along its logistic trajectory, life expectancy at birth for each sex along its Lee–Carter trajectory, and the first demographic dividend.
Each panel’s vertical scale expands with the run so extreme sidebar settings stay in frame. A vertical sweep marks the current calendar year on the strips only. Sidebar Live Statistics badges echo each strip’s live readout; values adopt the same color as the corresponding strip trace.
The visual
Data & sources
Empirical starting populations draw on HMD and HFD at each profile’s base year. The table separates what is observed at the base year from the forward scenario set in the sidebar.
| Component | Base year (observed) | Forward path (sidebar scenario) |
|---|---|---|
| Population | HMD Population 1×1 age–sex counts | Closed population; no migration after base year; sex ratio at birth (SRB) fixed per profile |
| Mortality | HMD period life tables; Lee–Carter schedule at base year | Mortality improvement pace scaled from HMD history (Mortality improvement) |
| Fertility | HFD period age-specific fertility rate (ASFR) shape and opening TFR | Logistic TFR path: start, end, transition midpoint (slider from base year to 2100) |
What it is — and isn’t
The piece traces how a starting population evolves under a stylized demographic transition, driven by cohort-component (Leslie) projection with sex-specific Lee–Carter mortality and a logistic fertility path — the standard machinery of formal demography.
The starting stock and base-year vital schedules are drawn from observed data. The forward trajectory is a sidebar scenario: a stylized path the user controls, distinct from any country’s recorded demographic history after the base year.
Methods
Mortality
Sex-specific Lee–Carter mortality: \(\ln \mu(x,t) = a(x) + b(x)\,\kappa(t)\).
\(a(x)\) is the observed base-year schedule from sex-specific HMD period life tables: \(a(x) = \ln m(x,t_0)\), estimated separately for females and males. At the profile base year, \(\kappa=0\) and the model matches observed mortality.
\(b(x)\) is the age pattern of mortality change from a one-factor singular value decomposition (SVD) over HMD history between the base year and the last available mortality year, normalized so \(\sum_x b(x) = 1\).
Survival in the Leslie step uses period-life-table survivorship ratios rather than point-hazard probabilities. Each year the model integrates the current \(\mu(x,t)\) into a survivorship curve \(l(a)\) (radix \(l(0)=1\), trapezoidal rule on a fine age grid) and forms person-years \(_1L_x = \int_x^{x+1} l(a)\,\mathrm{d}a\). Cohorts age with \(_1L_{x+1}/{_1L_x}\), births enter age 0 through \(_1L_0/l_0\) (carrying infant mortality), and the open age group persists through its \(T_x\) ratio.
\(\kappa\) advances linearly from zero at the base year:
\[\kappa(t) = (t - t_0)\,\lambda\,\omega,\]
where \(\lambda\) is the sex-specific average annual drift estimated from the observed HMD window and \(\omega \ge 0\) is a user-selected scale factor. At \(\omega = 0\) mortality holds at the base-year schedule; at \(\omega = 1\) the historical pace of improvement continues unchanged.
Life expectancy at birth updates each calendar year as the model integrates the current period \(\mu(x,t)\) (trapezoidal rule on a fine age grid). It tracks the modeled period schedule throughout the run; the static HMD \(e_0\) in the build JSON provides only a base-year verification anchor.
Population
The starting age structure is the observed profile population (HMD Population 1×1) at the base year: female and male counts by single year of age.
Each calendar year, the stylized population advances by two-sex cohort-component (Leslie) projection: cohorts age one year using period-life-table survivorship ratios \(_1L_{x+1}/{_1L_x}\), survivors remain on the Lexis surface, and new births enter at age 0 carrying first-year survival \(_1L_0/l_0\). Births follow the female ASFR schedule (Fertility); a fixed sex ratio at birth (SRB) divides them into girls and boys. Deaths use the sex-specific Lee–Carter hazards (Mortality).
Crude rates (sidebar and strip panels, per 1,000): \(\mathrm{CBR} = 1000\,B/\mathrm{PY}\), \(\mathrm{CDR} = 1000\,D/\mathrm{PY}\), \(\mathrm{CGR} = \mathrm{CBR} - \mathrm{CDR}\), where \(B\) and \(D\) count births and deaths over the year and \(\mathrm{PY}\) is person-years lived, approximated by the mean of the start- and end-of-year populations. The closed population carries no migration term.
Fertility
Fertility follows the observed period ASFR shape (HFD) at the profile base year, which the model applies to the female population. TFR follows a logistic path from the user-selected opening level \(\text{TFR}_\text{start}\) toward a 2100 endpoint \(\text{TFR}_\text{end}\):
\[\text{TFR}(t) = \text{TFR}_\text{end} + (\text{TFR}_\text{start} - \text{TFR}_\text{end})\,\frac{1}{1 + e^{k(t - t_\text{mid})}}, \qquad k = 0.06,\]
where \(t_\text{mid}\) is the transition midpoint, the year TFR crosses halfway between the two endpoints. The observed base-year ASFR shape then scales proportionally each year,
\[f(x,t) = f(x,t_0)\,\frac{\text{TFR}(t)}{\text{TFR}(t_0)},\]
holding the relative age pattern of childbearing fixed throughout the run.
Dividend
In economic demography, the first demographic dividend is the temporary economic advantage a population gains when its age structure tilts toward the working ages. As fertility falls during the demographic transition, the large cohorts born before the decline enter the working ages while fewer children arrive behind them. For several decades the working-age population grows faster than the dependent population, so each worker supports fewer non-workers and output per person can rise without any gain in productivity (Lee & Mason, 2006). The effect is transitory: the window opens as fertility declines and closes as those same cohorts reach old age. The first dividend is distinct from a second — the later rise in savings and capital that longer lives encourage as households prepare for retirement.
The support ratio measures this balance directly, comparing the working-age population (ages 20–64) with the dependent population (ages 0–19 and 65 and older):
\[\mathrm{SR}(t) = \frac{N_{20\text{–}64}(t)}{N_{0\text{–}19}(t)+N_{65+}(t)}\]
where \(N_{20\text{–}64}\), \(N_{0\text{–}19}\), and \(N_{65+}\) are the total (female and male) population counts in each age group at calendar year \(t\).
A high support ratio is not, by itself, the dividend. The first demographic dividend is the phase of positive growth in the support ratio: the span when the working-age population grows faster than the dependent population, and therefore faster than the total population, so the working-age share keeps rising. A population can sit at a high support ratio and gain no dividend once that growth stops.
The bottom strip and the sidebar Demographic dividend badge report this growth rate, the log growth rate of the support ratio, in percent per year, measured between consecutive simulated years. The base year has no predecessor, so the dividend is undefined there and the curve begins one year later. Values above zero lie inside the dividend window; the curve crossing below zero marks its end:
\[\frac{\mathrm{d}}{\mathrm{d}t}\ln \mathrm{SR}(t) \approx \ln \mathrm{SR}(t) - \ln \mathrm{SR}(t-1).\]
The series is a structural index of age composition. The support ratio and its growth rate track demographic shifts only; income, wages, and gross domestic product lie outside the model’s scope.
Controls
The Period Clock sets animation speed and replays the run from the base year via Restart. The Transition Regime steers the forward scenario, restarting the clock on any change: Fertility start is the base-year TFR (pre-filled from HFD), Fertility end the 2100 target, and Fertility transition midpoint the year \(t_\text{mid}\) at which TFR crosses halfway between them. Mortality improvement sets the scale factor \(\omega\) on the historical pace of change: 1 continues the observed annual rate, 0 freezes the base-year life table, and values above 1 accelerate improvement. The Live Statistics panel reports period TFR, life expectancy at birth, support ratio, and first dividend alongside the vital rates as the clock advances.
Assumptions
The model runs a closed population (migration = 0) under a two-sex Leslie structure with female-driven fertility and SRB-split births.
Profiles: Each row in the build registry supplies one HMD and HFD starting population.
Time stepping: The period clock advances continuously to animate the sweep, but all demographic statistics update on integer calendar years only.
Simplifications
Lexis display: Age shares on the main panel come from the stylized population counts at each calendar year (density strokes by age), not microsimulated cohort lifelines. Sidebar readouts (CBR, CDR, CGR, \(e_0\), support ratio) use the same Leslie step.
References
Lee, R. D., & Mason, A. (2006). What is the demographic dividend? Finance and Development, 43(3). International Monetary Fund. (See also Lee & Mason, 2006, Population and Development Review 32(s1), 69–83.)
Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(419), 659–671. https://doi.org/10.1080/01621459.1992.10475265
Preston, S. H., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and Modeling Population Processes. Blackwell.
Human Mortality Database. Max Planck Institute for Demographic Research (Germany), University of California, Berkeley (USA), and French Institute for Demographic Studies (France). www.mortality.org
Human Fertility Database. Max Planck Institute for Demographic Research (Germany) and Vienna Institute of Demography (Austria). www.humanfertility.org
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