When Will Nepal’s Population Stop Growing?

A mathematical model of a population is a formula that gives its size for any year. To choose one, we draw the counts and try formulas through them. Each formula below is made to pass through the 2021 count exactly; you choose how it behaves before and after.

Notice the surprise. The formula Grade 10 used for population growth, the same percentage every year, is the worst of the three at describing Nepal. To see why, look at the counts more closely.

Here is each step from one census to the next, worked out as people added per year and as a percentage per year (the compound rate that turns one count into the next):

(The 1952/54 census was taken in two parts over two years; we place it at 1953.)

The two kinds of model test the two columns.

• A linear model, P = a + mt, adds the same number of people, m, every year. Its graph is a straight line, and m is the line’s slope. If Nepal were linear, the middle column would be the same all the way down.
• An exponential model, $P = a \times b^{t}$, multiplies by the same growth factor b every year: growth of 2% a year is b = 1.02. If Nepal were exponential, the last column would be the same all the way down.

Neither column is steady. The people added each year rose to a top in the 1990s and then fell; the percentage rose to a top in the 1970s, and since the 1990s it has fallen fast. That last column is the clue: whatever Nepal is doing, its rate of growth is falling.

Every model has numbers that mean something, its parameters. In the linear model, m is people a year. In the exponential, b − 1 is the rate, and it also fixes the doubling time: at 0.965% a year Nepal would double in about 72 years, since $1.00965^{72} \approx 2$. (A quick rule: 70 divided by the rate in percent, 70 ÷ 0.965 ≈ 73.) So “the same percentage” isn’t a mild assumption: it promises 5.8 crore people by about 2093, 11.7 crore by about 2165, and so on for ever.

How far off is a model?

At each census the gap between the count and the model is called a residual: count minus model, positive when the model is too low. The widget’s typical miss is the root-mean-square of the residuals: square each, average them, take the square root. Squaring stops a miss of +5 lakh and one of −5 lakh from canceling, and counts big misses heavily. The best fit is the setting that makes it smallest.

The residuals’ pattern matters as much as their size. For the best straight line, the residuals at 1961–2021 are +6.8, −5.8, −5.2, −4.6, +8.0, +7.3 lakh and 0: the line is too low at the start, too high in the middle and too low near the end. A model that fits well leaves residuals that look like random scatter; a run of misses all on one side means the model has the wrong shape.

In 1838 a Belgian mathematician, Pierre-François Verhulst, asked what happens if a population’s rate of growth falls as the population rises. The simplest such rule: the rate falls in a straight line, from some starting rate r when the population is tiny, down to zero at a ceiling K:

$$\text{rate of growth} = r\left(1 - \frac{P}{K}\right)$$

When P is small, the bracket is nearly 1 and the population grows almost exponentially. As P nears K, the bracket nears 0 and growth dies away. The population that results is the logistic model,

$$P = \frac{K}{1 + \left(\frac{K}{P_0} - 1\right) e^{-rt}}$$

where P₀ is the population at t = 0. Its graph is an S, flat at the start, steep in the middle, flat again at the top. Ecologists call K the carrying capacity, the most that a place can support. The widget’s S-curve is this formula, started from the 2021 count.

Verhulst’s idea also gives a way to read K straight from the data. If the rate really falls in a straight line as P rises, then a graph of rate against population is a straight line, and where it reaches zero is the ceiling.

So the S-curve says Nepal is about four-fifths of the way to a ceiling a little over 3.5 crore. The widget’s best S-curve, fitted to the counts themselves, had its ceiling at 3.86 crore. The two methods roughly agree, and the UN’s peak, 3.58 crore, is between them.

Why the steepest part is halfway up

In an S-curve the number of people added in a year is the rate times the population:

$$\text{people added a year} = r P \left(1 - \frac{P}{K}\right) = \frac{r}{K}\,P\,(K - P)$$

The two numbers P and K − P always add up to K. Of two numbers with a fixed sum, the product is largest when they are equal: with K = 100, 40 × 60 = 2{,}400, 45 × 55 = 2{,}475, 50 × 50 = 2{,}500. (In general $P(K - P) = \frac{K^2}{4} - \left(P - \frac{K}{2}\right)^2$, and a square is never negative.) So an S-curve adds the most people a year when it is halfway to its ceiling, at P = K/2, and two populations equally far below and above K/2 add the same number.

Check it against Nepal. The most people added a year was 466,033, from 1991 to 2001, when the population was about 2.07 crore. Half of a ceiling of 3.5 to 3.86 crore is 1.75 to 1.93 crore. Close enough to be a good sign for the S-curve, if a little late.

The S-curve fits Nepal’s past better than any other simple formula and gives a sensible-looking ceiling. Should we believe it? History says: be careful.

In 1920 two American scientists, Raymond Pearl and Lowell Reed, fitted a logistic curve to the US censuses of 1790 to 1910 and found that it described them “with a substantial degree of accuracy”. Their curve said the United States had passed its fastest growth in about 1914, and that its population could never pass a ceiling of 197,274,000. “While prophecy is a dangerous pastime”, they wrote, they believed the 1920 census would confirm it. The US census of 2020 counted 331,449,281 people, 68% more than the “ceiling”. Verhulst’s own curve, fitted to Belgium, gave a ceiling of about 9.4 million; Belgium had about 11.9 million people in 2026.

What went wrong? A curve that fits is only a description. The logistic’s ceiling is a guess at how many people the land can feed, but a country isn’t a jar of yeast: farming, trade, medicine and migration move the “ceiling” all the time.

And for Nepal there is a second problem. An S-curve can only level off: it never comes down. The UN expects Nepal’s population to fall after 2067. To see why, look at what really slowed Nepal down.

Nepal’s rate didn’t fall because the country ran out of room or food. It fell because families chose to have fewer children, and because many young people went abroad to work.

• Births. In the UN’s estimates a woman in Nepal had about 6 children on average in 1950, about 4 in 2000 and about 1.9 in 2025. This average is the total fertility rate. About 2.1 is the replacement level, the rate at which each generation of parents is replaced by one of children the same size.
• Migration. The UN estimates that from 2022 to 2025 about 4 lakh more people a year left Nepal than came in, more than births exceeded deaths, so by its count Nepal’s population actually shrank a little in those years. Its forecast assumes the outflow drops to about 30,000 a year from 2028; if it stays high, Nepal peaks sooner and lower.

So why should Nepal keep growing until 2067, when women already have fewer than 2.1 children? Because of who is alive now. The many children born when families were large are now young adults having children of their own. Even with small families, so many parents give more births than deaths, for decades. Demographers call this population momentum. In the UN’s forecast, births in Nepal stay ahead of deaths until about 2070; only then, as that big generation grows old, do deaths overtake births and the population begins to fall.

None of this is in the counts alone. No formula fitted to twelve numbers can see a change in how many children people choose to have.

The UN, and every other major forecaster, uses the cohort-component method. They don’t fit a curve to the total. They split the population by age and sex (all the girls aged 0–4, all the men aged 60–64, and so on) and step forward a year at a time: each group grows a year older, loses those expected to die at that age, gains or loses migrants, and the women of each age have the expected number of babies, who become the new 0-year-olds. The forecast is only as good as its guesses about future births, deaths and migration, so the UN gives a range as well as a middle forecast.

The fan in Figure 33.2 is the honest answer to the topic’s question. The middle forecast says the 2060s. But the range is wide: by 2100 the UN gives a 95% chance that Nepal will have between about 1.1 and 6.3 crore people. Much of that width comes from one number, the children per woman. In the UN’s “low” variant, where women have half a child fewer than in the middle forecast, Nepal peaks in 2048 at about 3.15 crore. In its “high” variant, half a child more, Nepal is still growing in 2100, at about 4.95 crore.

Nepal’s own National Statistics Office made its projection from the 2021 census, assuming 1.72 children per woman, and expects about 3.35 crore people by 2108 BS (2051–52 AD), thirty years after the census: about 13 lakh below the UN’s 3.48 crore for 2051. Two careful forecasts made with the same census differ by over a million people in thirty years: that is the size of the uncertainty.

• Planning a country. The UN’s World Population Prospects forecasts every country’s population. Its 2024 report calls an understanding of population trends “critical” for “designing policies that require a longer time horizon, such as those related to macroeconomic planning, social protection, national security and the environment”. The UN, the Wittgenstein Centre in Vienna and the Institute for Health Metrics and Evaluation in Seattle, the three groups that forecast the whole world’s population, all use the cohort-component method.
• Nepal’s own forecasts. The National Statistics Office made its own projection from the 2021 census, with its own assumptions about children per woman and Nepalis living abroad.
• The world. The UN’s middle forecast for the world peaks at about 10.3 billion in the mid-2080s, and it says 63 countries and areas, among them China, Japan and Germany, have already passed their peaks.

• What Is e, and Why Does Growth Keep Finding It? Where the e in the logistic formula comes from.
• How Many Tigers Can Chitwan Hold? (Going Further, Grades 9–12) The same logistic model, for Nepal’s tigers.
• Can 1,000 People Speak for 3 Crore? How surveys between censuses estimate a whole country.
• How Much Rain Falls on the Kathmandu Valley? Another model built from Nepal’s own numbers.
• How Much Rice Is on the Chessboard? (Grades 6–8) How quickly doubling runs away.
• How Does a Cold Spread Through a School? (Computing, Going Further, Grades 6–8) An outbreak’s S-curve, simulated.
• Grade 10 Math: Growth The same-percentage formula, and Nepal’s censuses.