G-Cubed Labor-force growth rates
Table of contents
G-Cubed uses labor_force_growth_rates.csv to describe the demographic contribution to growth in effective labor. The standard age-adjusted construction combines United Nations population projections with National Transfer Accounts (NTA) labor-income profiles by age. It captures changes in both population size and age composition, then smooths the resulting regional growth rates and brings demographic growth to zero in the long run.
The resulting measure is age-income-weighted effective labor supply. It is an index of potential labor contribution, rather than a count of employed workers or a measure of technological progress. A country’s population can grow while its effective labor supply falls if its population shifts toward ages with lower labor-income weights.
This guide describes the standard age-adjusted construction and its model interface. Individual model builds may use separately maintained growth assumptions, so confirm the method used for the model being studied.
Construction at a glance
| Stage | Calculation or decision | Result |
|---|---|---|
| Source preparation | Read UN population by country, year and single age, and NTA labor income per person by country and age | Population and age-income tables |
| Profile preparation | Replace blank NTA age cells with zero, normalize each profile, and apply the country proxy mapping | Fixed relative age-income weights for each country |
| Country totals | Multiply each age’s population by its assigned weight and sum over ages | Effective labor supply by country and year |
| Regional totals | Map countries to GTAP regions and then to the selected G-Cubed regions, summing levels | Effective labor supply by model region and year |
| Raw growth | Take annual log differences of regional totals | Growth rates labelled by the transition’s starting year |
| Smoothing and continuation | Apply the multi-stage smoother, convergence window and common tail rule | Model-ready percentage log growth rates |
| Model integration | Combine demographic growth with separate sectoral technical change | Baseline effective-labor and potential-growth adjustments |
The sources and missing-data guide explains input formats, normalization, country proxies and coverage limits. The smoothing and long-run continuation guide gives the complete algorithm, including the terminal-year and regional-composition effects.
How age composition changes effective labor
Let $P_{c,a,t}$ be the population of country $c$, age $a$, in year $t$, and $w_{c,a}$ its assigned relative labor-income weight. The weights are held fixed through time. Country effective labor supply is
\[E_{c,t}=\sum_{a=0}^{100} P_{c,a,t}w_{c,a}.\]The final UN age category is 100+, which the loader treats as age 100. In the actual calculation, only age columns shared by the two inputs are used. The coverage qualifications matter when an age cell, an entire age column, or a country is absent.
NTA labor income per person reflects participation, hours worked and earnings as well as differences in productivity across ages. These weights are a proxy for relative labor contribution. They do not separately estimate those components, and the projection does not update them for future retirement policy, education, participation or hours worked.
Writing total population as $N_{c,t}$ and the population share of each age as $s_{c,a,t}$ makes the two demographic effects explicit:
\[E_{c,t}=N_{c,t}\bar w_{c,t}, \qquad \bar w_{c,t}=\sum_a s_{c,a,t}w_{c,a}.\]Therefore, before smoothing,
\[100\log\frac{E_{c,t+1}}{E_{c,t}} =100\log\frac{N_{c,t+1}}{N_{c,t}} +100\log\frac{\bar w_{c,t+1}}{\bar w_{c,t}}.\]The first term is population growth. The second is the change in the average age-income weight caused by the changing age distribution. An aging population need not always reduce effective labor: the result depends on which cohorts grow or shrink and the weights assigned to their ages.
A small example
The following is a synthetic illustration using three age groups; the real calculation uses single ages and empirical profiles.
| Illustrative group | Relative weight | Population in $t$ | Population in $t+1$ | Contribution in $t$ | Contribution in $t+1$ |
|---|---|---|---|---|---|
| Younger | 0 | 20 | 20 | 0 | 0 |
| Prime working ages | 2 | 40 | 30 | 80 | 60 |
| Older | 1 | 40 | 50 | 40 | 50 |
| Total | 100 | 100 | 120 | 110 |
Population is unchanged, but effective labor falls from 120 to 110. Its raw percentage log growth rate is $100\log(110/120)=-8.70$. This is an 8.33 percent decline in the level. The difference illustrates why log growth and ordinary percentage change should not be interchanged. Smoothing would subsequently modify the raw annual rate.
Regional aggregation and annual timing
Country effective-labor totals are summed through the ISO3-to-GTAP and GTAP-to-G-Cubed mappings. For model region $r$,
\[E_{r,t}=\sum_{c\in r}E_{c,t}, \qquad g_{r,t}=100\left[\log E_{r,t+1}-\log E_{r,t}\right].\]Country age-income weights are applied before regional aggregation. There is no simple average of country growth rates and no GDP weighting in this step. Countries’ contributions depend on their effective-labor levels, so the mix of countries inside a region matters.
The column labelled 2023 represents the transition from the 2023 level to the 2024 level. A raw rate of 1.0 means a log increase of 0.01, equivalent to an ordinary level increase of approximately 1.005 percent. Negative rates are allowed. Positive effective-labor totals are needed to obtain finite log growth rates; missing coverage does not become economically valid merely because a CSV can be written.
Output file and use in the model
The generated CSV has one row per G-Cubed region in SYM region order, with gcubed_region_code as its index label and every configured projection year as a column. Its cells contain the smoothed percentage log rates, rather than the unsmoothed rates in the equations above. See the file-format reference.
G-Cubed divides these values by 100 and selects the configured projection years. It requires every region and year; it does not interpolate gaps, extend a short input file, impute missing countries or perform another smoothing pass.
For sector $s$ in region $r$, let $a_{s,r,t}$ be the separate annual log rate of labor-augmenting technical change and $n_{r,t}$ the demographic CSV rate divided by 100. The change in the combined effective-labor log index is
\[\ell_{s,r,t+1}-\ell_{s,r,t}=a_{s,r,t}+n_{r,t}.\]The same regional demographic rate applies to every sector. The catch-up growth model supplies technological progress from frontier growth, technology gaps and catch-up rates; it is separate from demographic age weighting.
Starting with a zero log index in the first projection year $t_0$, G-Cubed accumulates these rates and subtracts the common configured growth trend $G$ (in decimal log units). The generated baseline SHL adjustment is
Changes in this adjustment also contribute to the generated ROGY potential-growth adjustment, using fixed initial sector output shares. These baseline adjustments differ from the absolute growth rates reported for build-200 ROGY. The baseline guide explains the surrounding workflow.
Zero demographic growth in the tail means that effective-labor growth can continue through technical change. It does not impose zero productivity or GDP growth. Actual GDP and employment also depend on the rest of the model; this subsystem is not a standalone GDP or employment forecast.
Reproducibility and interpretation
Record the UN extract and its metadata, NTA extract and profile reference years, country/proxy mapping, GTAP-to-model mapping, model horizon, and data-code revision when producing a new file. Preserve the raw inputs: final smoothing does not preserve every source annual growth rate or necessarily the cumulative source level change.
The source guide records important properties of the inspected inputs, including the current common proxy profile and the treatment of old-age groups. The smoothing guide explains the long-run continuation and how the model’s set of regions affects the common zero-tail rule.
After changing growth inputs, regenerate the dependent baseline inputs and rebuild the solved model and dependent projections through the supported regeneration workflow.