Demography + Economics Lab

Change demographic and economic assumptions and watch population, GDP per capita, labor force, and fiscal pressure evolve together.

Starting values — Demography + Economics Lab

Country baseline	Custom
Initial population	15.2
Birth rate	0.95
Death rate	1.06
Net migration	0.24
Working-age share	0.64
Labor-force participation	0.73
Unemployment	0.062

Change any control above and the results recompute from these.

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Lesson

The theory — Demography + Economics Lab

The dependency ratio asks how many people each worker supports: the young and the old divided by those of working age. It matters because tax revenue scales with the working share while pensions, schooling and healthcare scale with the dependent share — so the same population can be growing and getting fiscally harder at the same time.

What each symbol means

DR: the dependency ratio, printed as DR = (old share + young share) / working share with the substitution shown, and reaching 0.525 in the first row here.
labour: the labour force in millions — the part of the population the tax base is computed from.
GDP/capita: output per person, which is not output per worker: a rising dependency ratio can pull it down while every worker gets more productive.
net fiscal: the tax base minus social cost — the surplus or deficit the projection implies for each year.

How to read what you see

One row per projected year, reading left to right from cause to consequence: population, then labour force, then output, then the dependency ratio, then the tax base and social cost it implies, and finally the net fiscal position. Following a single column down the years shows a trend; reading one row across shows how that year’s demography became that year’s budget.

Assumes: Constant rates once you set them — birth, death, migration and productivity all held flat for the whole projection. That is what makes it a scenario rather than a forecast: real rates respond to the very pressures the table is projecting.
Breaks when: The defaults show the trap plainly: the birth rate (0.95) is below the death rate (1.06), yet the population still rises — because migration is making up the difference. A headline population that looks stable can hide a shrinking native cohort, and the dependency ratio is the column that notices. Country baselines come from World Bank data loaded live (217 countries here), so the starting point is real even though everything after year one is an assumption.

Why this feels like math magic 🖖

The key is decomposition: the same final GDP path can come from very different drivers. This lab separates demography, labor, productivity, and fiscal load so tradeoffs become visible. Population dynamics compound slowly but dominate over 45 years — a 0.3% annual net migration rate doubles the effect of a large productivity boost.

Total output versus output per person 🖖

This lab always tracks GDP per capita, not just headline GDP. Output is produced only by workers, yet every resident shares it — so the total and the per-person figure can drift in opposite directions. A country whose population outpaces its economy can look richer in total while each person grows poorer. Watch the two curves split whenever the working-age share shrinks: that gap is what living standards actually feel.

A shrinking nation has a half-life 🖖

The population step P_t+1 = P_t(1 + b − d + m) is pure geometric growth — the same equation behind compound interest and radioactive decay. When births plus migration fall short of deaths, the net rate turns negative and population decays exponentially, half-life and all. A country contracting 1% per year has a population half-life near 70 years (the rule of 70) — the demographic twin of an isotope's decay curve.

Problems solved in full

Migration rate at which a population with 0.95 births stops growing 5 steps
This baseline records 0.95 births and 1.06 deaths per 100 people per year, plus net migration of 0.24. Deaths outnumber births. Project the population to 2070 anyway, then find the migration rate at which the total stops growing.
1. Take births and deaths on their own first. They subtract to −0.11% a year, so left to itself this population shrinks, and nothing further down the model changes that sign.
  
  $r_{nat} = b - d = 0.95% - 1.06% = - 0.11% y r^{- 1}$
2. Migration enters the same sum with the same weight — one arrival counts exactly as one birth here — and flips it to +0.130% a year. The country grows, and it grows entirely on the third term.
  
  $r = b - d + m = 0.95% - 1.06% + 0.24% = + 0.130% y r^{- 1}$
3. Fixed rates compound, so the projection is one power rather than a column of additions. Raise 1.0013 to the 25th and the 45th and you have 2050 and 2070 without simulating the years in between.
  
  $P_{t} = P_{0} (1 + r)^{t} : P_{2050} = 15.20 (1.0013)^{25} = 15.70 M P_{2070} = 15.20 (1.0013)^{45} = 16.12 M$
4. Now delete the migration and rerun the same power. 14.47 M: instead of gaining 0.92 M over the 45 years, the country loses 0.73 M.
  
  $m = 0 : P_{2070} = 15.20 (0.9989)^{45} = 14.47 M$
5. Between those two runs sits a rate at which the population neither grows nor shrinks, and it is not zero migration. It is the migration that exactly refills the natural deficit: m = d − b = 0.11 pp.
  
  $m^{*} = d - b = 0.11 pp, \frac{m ^{*}}{m} = \frac{0.11}{0.24} = 45.8%$
Answer

+0.130% a year and 16.12 M by 2070 — with the sign owned by migration alone. The counterfactual is the part the tool will not run for you: strip the 0.24 out and the same model walks the same country down to 14.47 M, a 1.65 M swing worth 10.9% of where it started, off one input that most people file under social policy rather than arithmetic. The break-even is sharper than the headline. Growth stops at m = 0.11 pp, which is 45.8% of the 0.24 assumed here, so nearly half the inflow buys no growth at all — it only cancels the deficit. In the first year that is 16,720 arrivals out of 36,480 standing still, and the remaining 19,760 people are the entire growth of the country. One caution the projection keeps quiet about: d is pinned at 1.06% for all 45 years while the tool's own dependency ratio climbs from 0.525 to 0.606. A crude death rate is an average weighted by age structure, and it rises as a population ages, so +0.130% is the optimistic end of this model rather than its centre.

$Δ P_{45} = 16.12 - 14.47 = 1.65 M = 10.9% \times P_{0}$
Fiscal pressure of 0.409 for a country of 152 M people 7 steps
The 2025 row puts fiscal pressure at 0.409 — the age-related bill takes 40.9% of the tax base. That was computed for 15.20 M people. Rebuild it from the inputs, then work out how much of it a country of 152 M would inherit.
1. One number has to be settled before anything else moves. You typed a 64% working-age share, but the printed dependency ratio is 0.525, and that ratio is the two dependent shares over the working-age share. The dependent shares are yours and fixed, so the divisor is forced: 0.365 ÷ 0.525 = 0.695. That is the model widening the working-age band by a fifth of its gap to the dependent shares, 0.64 + 0.2 × (0.64 − 0.365) = 0.695. Every line below runs on 0.695, not on 0.64.
  
  $DR = \frac{s _{old} + s _{young}}{s _{w}} = 0.525 ⟹ s_{w} = \frac{0.21 + 0.155}{0.525} = 0.695$
2. Employment is three multiplications from population: that share is of working age, 73% of them participate, and 93.8% of those have a job.
  
  $L_{0} = P_{0} s_{w} LFPR (1 - u) = 15.20 \times 0.695 \times 0.73 \times 0.938 = 7.234 M$
3. Output is the working headcount times output per worker, and nothing else — this model has no capital in it.
  
  $GDP_{0} = L_{0} A_{0} = 7.234 \times $74, 000 = $535.3 B$
4. The tax base is the government's slice of that output, 36 cents in the dollar.
  
  $TB_{0} = τ GDP_{0} = 0.36 \times 535.3 = $192.7 B$
5. The bill is charged per dependent head: $18,500 for each of the 21.0% who are old, $8,400 for each of the 15.5% who are young. Together that is $5,187 per head of the entire population, children and pensioners and workers alike.
  
  $C_{0} = P_{0} (s_{old} c_{old} + s_{young} c_{young}) = 15.20 \times (3, 885 + 1, 302) = $78.8 B$
6. Divide the bill by the base and you have the tool's fiscal pressure, built from inputs rather than read off the panel.
  
  $FP = \frac{C _{0}}{TB _{0}} = \frac{78.8}{192.7} = 0.409$
7. Now look back at what you multiplied. The 15.20 M appears once in the bill and once in the tax base. Cancel it and fiscal pressure turns out to be a per-head ratio: $5,187 of obligations against $12,678 of tax base for every person in the country.
  
  $FP = \frac{s _{old} c _{old} + s _{young} c _{young}}{τ s _{w} LFPR ( 1 - u ) A _{0}} = \frac{5 , 187}{12 , 678}$
Answer

0.409, and the size of the population never entered it. A country of 152 M with the same shares, the same participation, the same output per worker and the same tax rate lands on 0.409 as well — ten times the pensioners, ten times the schoolchildren, and ten times the workers paying for both. So migration, the one term that decided the first problem, does nothing whatever to this ratio unless the arrivals move the shares, and shares are the only thing the ratio is made of. Take the tax rate out too and it gets blunter: $5,187 of bill against $35,216 of output per head is 14.7% of GDP, and that figure does not know what tax rate you picked. Fiscal pressure is simply 0.147 ÷ τ. At the 36% assumed here it reads 0.409 and looks comfortable; halve the rate and the identical population with the identical pensions and the identical schools reads 0.82. Nothing demographic moved. The reach of the state did.

$\frac{$5 , 187}{$35 , 216} = 14.7% ⟹ FP = \frac{0.147}{τ}$

References (1)

The country baselines the projection starts from, loaded live for 217 countries: World Bank, World Development Indicators. Washington, DC: The World Bank.

By the 42Math team · Updated 2026-05-31

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Estonia - Small open economy with mild net outflow pressure and aging dynamics.
Japan - Japan's demography arrives live from the World Bank, elderly and youth shares included, so the projection starts from a country where three people in ten are over sixty-five. What does not arrive is the price of them. Per-pensioner spend, per-child spend and the tax take all stay at the European defaults the tool opens with, so the fiscal-pressure column is Japan's population costed at Europe's rates.
Sweden - Switching country changes nine World Bank series and nothing else. Sweden and Japan are costed at the same 36.0% tax take, the same $19K a year per pensioner and $8K per child, and the same 1.30% productivity growth. So the gap between their fiscal-pressure columns is demography and output alone. Not one policy difference is in the model.