In Avengers: Infinity War, Thanos snaps his fingers and half of all life in the universe turns to dust. His reasoning is what he calls a simple calculus: the universe is finite, its resources are finite, and life left unchecked will eat through all of it. Take away the purple skin and the gauntlet and that's an economics argument. It's also a very old one.
In 1798 a 32 year old curate called Thomas Malthus published the same argument, short enough to fit on a napkin. Food grows by addition. People grow by multiplication. Multiplication always wins. Economists have been arguing about that napkin for 228 years.
So I put Malthus into equations and then ran Thanos's experiment on them: what happens if half the population disappears overnight? Do the survivors get rich? For how long? The answer turned out to be one tidy formula, and there's a half-life sitting inside it.
§ 1 Multiplication Beats Addition
Malthus took his growth rate from the American colonies, where land was cheap and people doubled every 25 years. For food he was generous: assume the harvest grows every 25 years by the same amount it produces today. Measuring time t in years,
and the ratio of mouths to food runs away:
His own numbers make the point better than the limit does. Britain had about 7 million people. A century of unchecked doubling gives 112 million, while the generous food line feeds 35 million. Push it to 225 years and people stand to food as 512 to 10.
Since that can't actually happen, something has to hold population down to the food supply. Malthus called these things checks. Preventive checks lower births (people marry later). Positive checks raise deaths (hunger, disease, child mortality). He described the positive ones; he never proposed them. His recommended fix, from the 1803 edition on, was fewer births.
The 1798 version is a cartoon, though. Food doesn't grow in a straight line, and population doesn't double forever. The version economists actually use is better, and a lot more fun.
§ 2 The Trap as a Steady State
Picture a farming economy. Output Y comes from labour L (which is just the population), a fixed amount of land X, and a technology level A:
Here 0 \lt \alpha \lt 1, and 1-\alpha is the share of output that goes to land. Income per head y falls as people crowd onto the same fields. That's diminishing returns doing its job.
Now the Malthusian part. Population grows when people live above subsistence and shrinks when they live below it:
Set y = y^{*} in (3) and solve for the population the land can hold:
Look at what's missing. Income in the long run is y^{*}, full stop. Better technology raises L^{*} and leaves living standards exactly where they were. Every new plough becomes more people, not richer people. That's the trap.
This isn't just a nice story. Ashraf and Galor tested it on data from 1 CE to 1500 CE and found precisely this pattern: places with better technology and more productive land had denser populations, but no significant difference in living standards.
§ 3 Delete Half the People
Start the economy at its steady state and halve L in one go. Land and technology don't change, so (3) says income per head jumps straight away:
With a land share of 0.4, that's 2^{0.4} = 1.32. Survivors are 32% richer the morning after. Nice for them. But now y \gt y^{*}, so by (4) the population starts growing again. How fast does the bonus disappear?
Work in logs. Let x = \ln(L/L^{*}) be how far below the old population we are. From (3), \ln(y/y^{*}) = -(1-\alpha)\,x. Near the steady state, approximate the growth response as linear in log income, n(y) \approx \eta \ln(y/y^{*}), where \eta measures how strongly births and deaths react to being richer. Then
Plug in x_0 = -\ln 2 and both paths drop out in closed form:
You don't even need to know \eta. Right after the shock the population grows at g_0 = k \ln 2, and that's something you can observe. The log gap shrinks exponentially, so it has a half-life, and so does the income bonus:
The time to close half of the log gap, and 90% of it, in terms of the growth rate just after the shock.
Say the survivors grow at 1% a year at first. Half the gap is gone in 48 years and 90% of it in 160. Within two lifetimes the economy is essentially back where it started, with the same population and the same subsistence income.
Figure 1. Halving the population in the model of § 2, with land share 0.4 and post-shock growth of 1% a year. Both solid lines are equation (8). The dashed line is the naive answer, population growing at a constant 1% forever, which gets back to full size in 69 years.
The dashed line is worth a second look. If you assume growth stays at 1%, you get the textbook doubling time \ln 2 / g_0 \approx 69 years, and the population sails right past where it started. The Malthusian model is slower and gentler, because growth itself fades as the extra income gets eaten. The destination doesn't change either way.
The steady state has no memory
Look at (5) again. L^{*} depends on land, technology and the birth response. The size of the shock appears nowhere. Cutting the population moves the economy along the curve; only changing A, X or n(\cdot) moves the curve.
Malthus knew this, by the way. In chapter 6 of the 1798 essay he points out that the 1666 plague in London left no visible mark on its population 15 or 20 years later, and that a plague killing millions would simply be followed by births running far ahead of deaths.
§ 4 A Real Experiment: the Black Death
History did run this experiment, once. The Black Death killed roughly 40% of Europe's population between 1347 and 1352. Plug a 40% loss into (6):
Historians reckon English survivors ended up with about 30% more disposable income than before the plague. Set the left side of (10) to 1.3 and solve:
That's back of the envelope stuff, but a land share around one half isn't a crazy number for a medieval farm economy. Gregory Clark's wage and price series for England show the same thing more carefully: a remarkably stable inverse relationship between real wages and population from 1250 to 1600. When the plague pushed population down, wages rose, and they kept rising into the 1440s.
Here's the catch. That bonus lasted about 150 years only because the plague kept coming back every few decades. It was a repeated shock, not a one-off. Even then, European cities were back to their pre-plague size on average by the 16th century. A single cull, by (9), gets undone much faster.
§ 5 How the Trap Actually Broke
If deleting people doesn't work, what did? Take growth rates of (3), writing g_z for the growth rate of z:
Inside the trap, population adjusts until g_y = 0, which means g_L = g_A/(1-\alpha). Faster technology just means faster population growth. There are two ways out, and the modern world used both.
Ideas come from people. Michael Kremer's 1993 paper on world population from one million BC to 1990 starts from the observation that an idea, once found, can be used by everyone. So more people means more inventors. Write that as g_A = \delta L and feed it into the trap condition:
The growth rate of population rises with its level. That's faster than exponential, and it's what world population actually did for most of history. Kremer also found that among regions with no contact with each other, the ones that started bigger had faster technological change. It also means a cull is doubly bad in this world: halve L and you halve the flow of new ideas too.
People stopped converting income into kids. The demographic transition breaks equation (4). Once incomes, schooling and cities pass a certain point, n(y) stops rising with y and starts falling. Malthus's preventive check turned up after all, chosen voluntarily, which is the version he hoped for and didn't expect. With g_L held down, (12) finally gives g_y \gt 0, year after year.
The numbers today look nothing like 1798:
| Indicator | Value |
|---|---|
| World population growth, peak (1963) | 2.3% a year |
| World population growth, 2026 | about 0.83% a year |
| Births per woman, 1990 | 3.31 |
| Births per woman, 2024 | 2.25 |
| Countries whose population has already peaked | 63 |
| UN projected peak, mid 2080s | about 10.3 billion |
A bet worth $576.07
In 1980 the economist Julian Simon bet the biologist Paul Ehrlich that five metals of Ehrlich's choosing would be cheaper in real terms ten years later. World population grew by more than 800 million over that decade. All five prices fell, and Ehrlich mailed Simon a cheque for $576.07. Fair warning: stretch the window to 1980 to 2011 and Simon would have lost on four of the five. Commodity prices are noisy. The long trend is the point, not any single decade.
§ 6 What I'm Taking Away
Malthus was right about the world he lived in. For most of history, technology really did turn into more people rather than richer people, and the data back him up.
But a one-off cut in population is a shock, not a policy. In his own model the bonus decays with a half-life of (\ln 2)^2/g_0, and the economy returns to exactly the same place, because nothing that sets L^{*} has changed. The only levers that move the destination are technology, land and the birth response, which is a fairly good description of what actually happened.
The trap didn't break because there were fewer of us. It broke because there were more of us thinking, and then because people decided, on their own, to have fewer kids.
Thanos would have had to snap again every couple of generations. The rest of us got out of the trap by thinking our way out, not by thinning ourselves out.
Refs Where the Numbers Come From
- Malthus, T. R. (1798). An Essay on the Principle of Population, chapters 1, 2, 6 and 7.
- Ashraf, Q. and Galor, O. (2011). Dynamics and Stagnation in the Malthusian Epoch. American Economic Review 101(5).
- Kremer, M. (1993). Population Growth and Technological Change: One Million B.C. to 1990. Quarterly Journal of Economics 108(3).
- Clark, G. (2016). Microbes and Markets: Was the Black Death an Economic Revolution? Journal of Demographic Economics 82(2).
- Jedwab, R., Johnson, N. and Koyama, M. Pandemics, places, and populations: Evidence from the Black Death. VoxEU.
- Stuttaford, A. (2025), citing James Belich. The Black Death's Economic Dividend. National Review.
- Our World in Data. World population growth, past and future.
- United Nations. World Population Prospects 2024, as reported by Dawn and Vision IAS.
- Statistics Times. World population 2026, based on UN projections.
- Wikipedia. The Simon and Ehrlich wager.