More than two hundred years ago Malthus stated a law: population grows by a fixed proportion every generation, while the harvest from the land can rise only slowly, so sooner or later population outruns the number the land can feed. When it does, famine, plague and war bring it down. Before the Industrial Revolution the law broadly held. By the economic historian Maddison's estimate, world income per head rose by less than half in the more than eight hundred years from AD 1000 to 1820; the extra grain that better techniques produced went mostly to feeding more people, not to making each person better off.
After the Industrial Revolution things changed. Between 1820 and 2010 world population grew from about 1 billion to about 6.9 billion, while income per head rose about tenfold. Population became nearly seven times larger, and each person was better off. In the rich countries Malthus's law no longer holds. Many people have concluded that human society has escaped this kind of mathematical law for good: as long as technology keeps advancing, economic growth will make life better for most people.
Since the Industrial Revolution, capital and the creation of wealth have both grown exponentially, but for most of the past two hundred years the first has grown faster than the second. The root cause of this imbalance is that owners of capital always want a return higher than the average rate at which wealth grows. The Industrial Revolution let them obtain it by pursuing technical progress; capitalism let them obtain it by squeezing the proletariat through relations of debt. Once the gap between the two rates of growth has accumulated far enough, the contradiction breaks out as it did in the Malthusian age. At bottom it is still a mathematical problem.
Each year a country's income is divided between capital and labour. Capital's share equals the rate of return on capital multiplied by the number of years of income that the capital stock is worth. If income grows every year and the division stays the same, the return to capital should grow as fast as income, that is, the rate of return r should equal the growth rate of the economy g. Yet Piketty pointed out that for more than two hundred years r has stayed above g, and the data for sixteen rich countries since 1870 compiled by Jordà and his co-authors show the same.
Period averages, per cent a year. Returns are before tax and include changes in asset prices, covering housing, equities, bonds and deposits; country averages first, then an equal-weight mean over the sixteen countries; only 12 countries have return data for 1871–1913.
Calculated from the Jordà–Schularick–Taylor Macrohistory Database R6; see Jordà et al., "The Rate of Return on Everything, 1870–2015", Quarterly Journal of Economics, 2019.
The cause lies in how firms make decisions. In the order of the accounts, shareholders receive their share last; in actual management, their return comes first. Take an example. Suppose an electrical-goods company sells 100 million yuan of goods this year. It first pays 40 million for materials, freight and power, then 30 million in wages, 5 million in bank interest and 5 million in tax; the 20 million left over is the shareholders' profit. The next year the market is poor, customers cut their orders and revenue falls to 90 million. The company cannot decide how much its customers buy, and it cannot skip its interest or its tax; what management can adjust is wages and purchasing. It freezes pay, lays off some staff and pushes down its suppliers' prices. Wages fall to 25 million, purchasing to 35 million, and profit is still 20 million. The 10 million of lost revenue is borne entirely by employees and suppliers.
Two years of accounts for an electrical-goods company, in units of 10,000 yuan. Left to right is the order of the accounts, with the shareholders' profit last; when management decides, it fixes the right-hand segment first.
Hypothetical figures.
Listed companies work the same way. Executives' pay is tied to the share price, earnings per share and total shareholder return; the return to capital is their target, and wages are a cost they must hold down.
No one imposed this arrangement; it grew out of long-distance sea trade. A captain who sailed at the risk of dying at sea had to sell his cargo at a high enough price on his return to earn a reward that matched the risk. Later others paid to buy a share of the space in his hold, shared the risk and divided the proceeds in proportion to what they had put in; this was the first joint-stock company. The principle that greater risk earns an excess return has carried through to today's venture capital. There are millions of such decision-makers in the market, and every one of them asks for a return above the average, while the growth rate of the economy g is that average. Everyone cannot be above the average; that is arithmetically impossible. The part of the demand above the average can come only from new income created by technical progress, or from income that others already have. This has nothing to do with good or evil: when everyone makes the same choice, what they ask for in total exceeds what the whole society has to divide.
Total income can be increased in two ways: by investing in new technology, or by investing in foreign markets that are not yet saturated. A larger part of other people's income can also be obtained in two ways: by holding things in limited supply, such as land and houses, and collecting rent and interest; or by using debt, such as mortgages and government bonds, to claim in advance other people's future income. The first two add to the nation's income, and both capital and labour may gain; the last two do not, and the income capital gains equals the income workers lose. As capital accumulates, the same sum finds fewer opportunities for a high return, and capital shifts gradually from the first two ways to the last two. The data in the next two sections are consistent with this account.
We call capital used for production, such as factories, machinery and equipment, reproducible capital, to distinguish it from property such as land whose quantity does not grow. The chart below shows how many years of national income the world's reproducible capital is worth. In 1800 it was about 1.4 years; it rose steadily for a century and stood at around 2.8 years from 1904 to 1913, the high point of the first cycle; after the First World War it fell to about 1.8 years, and around the Second World War to about 1.5 years, its lowest point being in 1951; it then rose again, to about 3.1 years in 2020–2021 and about 2.8 years in 2023. The high points of the two cycles are close. But the values before 1980 are estimates, and the plausible range for 1913 runs from 1.8 to 3.5 years, so the closeness of the two peaks is only a hint, not evidence.
Before 1980, an estimate: the historical share of reproducible capital in domestic capital for France, Britain, the United States and Germany, joined by straight lines between benchmark years, taking the median each year, calibrating to the 1980 world figure and multiplying by the world ratio of domestic capital to national income; the shaded area is the plausible range of the estimate, not a statistical confidence interval. From 1980, direct world data. Grey marks 1914–1945.
Calculated from the country tables of the Piketty–Zucman (2014) data appendix and the World Inequality Database (WID).
We call 1815 to 1913 the first cycle; 1914 to 1945 was its clearing; 1950 to the present is the second cycle. Each cycle has two phases. The first phase, 1815–1870 in the first cycle and 1950–1980 in the second, came just after a clearing: capital was scarce, new technology to invest in was plentiful, and returns came mainly from the new income new technology created. These two periods are often called golden ages. In the second phase, 1870–1913 in the first cycle and 1980 to the present in the second, the easiest technical gains had been used up; part of capital went abroad in search of returns, and part of what stayed at home turned to land, property and lending. At some point foreign investment too failed to yield enough, while property at home kept accumulating; the ratio of capital to income reached its peak, and a clearing followed.
Choosing 1815 as the start does not mean the Industrial Revolution began that year. The spinning machine and the steam engine had appeared one after another in the eighteenth century, but as separate inventions; by around 1815 the steam engine was in general use in factories and mines, and ten years later on railways. 1815 was also the end of the Napoleonic Wars: for most European countries the previous clearing had just been completed, and post-war reconstruction began.
The world total alone does not show where capital was placed in each phase. Britain was the leading power of the first cycle and has the most complete records, so its wealth can be examined item by item: total wealth, land within it, reproducible capital within it, and government debt and foreign assets.
From 1815 to 1913 British national wealth was worth six to a little over eight years of national income, with highs in 1867–1868 and in the 1900s: the first came just after the London financial panic of 1866, and the second was followed by the First World War. After 1914 wealth fell to about three years of income, fluctuated between the wars, and only after 1950 began a slow recovery, reaching about four and a half years in 2025. The wars destroyed factories, equipment and cities, and also stripped bonds and shares of their value; in those thirty years capital was cleared on a large scale.
Market value, including land, housing, business assets and net foreign assets. Dashed: historical estimate, 1800–1979; solid: annual accounts from 1980.
World Inequality Database (WID), variable wnweali999.
Before the Industrial Revolution, lords holding large estates of farmland were the main collectors of rent. The value of British farmland was three years of national income in 1810 and only about a third of a year by 1913. The rise of industry freed people from rents and dues on the land; although industrial conflict between labour and capital soon followed, this was still a great advance. But from the 1990s land again came to make up a large part of wealth, this time residential land in cities: after 1996 the value of British residential land rose quickly relative to income.
Points on the dashed line are farmland values in benchmark years, joined by straight lines that do not show the years in between; farmland has annual data from 1980 and residential land from 1996.
Benchmark years: Piketty and Zucman (2014), UK tables; annual data: World Inequality Database, variables wnwagri999 and wnwlani999.
Reproducible capital shows how far capital was invested at home in technology, equipment and plant to expand production. British reproducible capital was about 2.6 times income in 1810 and about 3.1 times in 1855, then fluctuated, reaching a high of 3.25 times in 1913; in 1920 it fell to 0.61 times. In the second cycle it declined unevenly after 1980, fastest after 2020, and by 2024 was only 1.16 times. This corresponds to the move of manufacturing abroad after the oil crises and the hollowing-out of British domestic industry. Compare it with the value of land: as productive capital at home found less to invest in, capital turned to land and property. For young people in Britain after 1980 this meant fewer well-paid, secure jobs and an ever heavier burden of rent and mortgage payments.
Reproducible capital means domestic non-financial assets other than housing and farmland: machinery, equipment, plant and so on. Points on the dashed line are benchmark years, joined by straight lines that do not show the years in between; annual data from 1980.
Piketty and Zucman (2014), UK Table UK.6f; World Inequality Database, variable wnwodki999.
When the Napoleonic Wars ended in 1815, British government debt was worth about two and a half years of national income; in the next few years it rose to nearly three years by 1821, then fell gradually, to about a third of a year by 1913. Britain did not default; the debt came down gradually as income grew. Foreign assets moved the other way: after 1870 they rose faster, reaching about 1.7 times national income by the early 1890s and staying at roughly that level until 1913. This was the move abroad of the second phase: after the domestic financial bubble burst in 1866, from the 1870s British capital went abroad in growing amounts. The benefit of foreign markets was that productive capital at home could still find a respectable return, keep investing in technology and plant, and employ British workers to make goods for export; the cost fell on the people of the colonies. After the two world wars both trends reversed: by the 1960s foreign assets were almost entirely gone, and government debt rose again in 1946 to about two and a half years of income. The post-war Labour government nationalised part of industry and raised income tax; a capital gains tax followed in 1965.
Top: stocks, in years of national income. Bottom: annual flows, as a percentage of national income; the two cannot be added. Government debt is not counted in net national wealth.
Government debt, net foreign assets, capital income, net foreign capital income and government interest: World Inequality Database; farm rent 1855–1946: Piketty and Zucman (2014), UK tables (after Feinstein 1972).
How fast the gap accumulates when the whole return on capital is reinvested can be calculated. Suppose a country's income is 100 a year and its property is worth 500, five years of income; capital earns 5 per cent a year and the economy grows 2 per cent a year. Property then grows about 3 per cent a year faster than income, and the share of property income in national income rises from 25 per cent to 50 per cent in about 24 years and to 100 per cent in 48, when workers' income would be zero. The chart below lets you change the two growth rates and see how fast the share rises.
Start: property worth 5 years of national income. Move the two sliders to see how many years the share takes to reach 50% and 100%.
Each year the ratio of property to income is multiplied by (1 + r) ÷ (1 + g); the share of property income equals r times that ratio, computed year by year.
Of course this cannot actually happen: part of the return is always spent, taxed, or lost in war and inflation. By a calculation from the data compiled by Jordà and his co-authors, had the whole return on French wealth since 1870 been reinvested, with nothing spent, taxed or lost in war, French wealth in 2020 should have been more than 350 times national income; in fact it was about 6.9 times, close to its level in 1870.
We call the cutting back of the return that capital expects a clearing. There are two ways it happens. One is to cut the return a little each year over several decades: let prices rise moderately while holding interest rates down; tax high incomes and large estates more heavily; have creditors and debtors negotiate a reduction of debt. We call this a gradual clearing. The other is heavy loss in a short time: war, revolution, hyperinflation and waves of bank failures, in which what is lost is not only property on paper but factories, cities and human lives. We call this a violent clearing.
The first cycle ended in a violent clearing. The government bonds, foreign bonds, overseas railway shares, urban property and deposits accumulated in the century before 1914 all depended for their value on three conditions: peace, the gold standard and the free movement of capital. The two world wars removed all three at once. The wealth of Britain, France and Germany, worth six to seven years of national income in the 1910s, fell to two to four years in the 1920s to 1950s, lowest in the 1940s at about two and a half; the United States, untouched by fighting on its own soil, fell less. The share of wealth held by the richest 1 per cent in Britain fell from about 68 per cent in 1913 to about 45 per cent in 1950; in France over the same years from about 55 per cent to about 33 per cent. The cost of this clearing fell not only on property owners but on tens of millions of the dead and their families.
Decade averages; the last point on each line is 2024, from a different database with a different basis, and is not joined to the earlier points. Grey marks 1914–1945.
1810–2010: Piketty and Zucman (2014), data appendix Table A2; 2024: World Inequality Database.
The first phase of the second cycle saw the only gradual clearing on a large scale. British government debt fell from about 270 per cent of gross domestic product in 1946 to about 43 per cent in 1980, without default and with every interest payment made on time. It was done by capping interest rates on deposits and loans, requiring banks and pension funds to hold government bonds, and restricting the flow of money abroad. In sixteen rich countries the real return on Treasury bills and government bonds was on average about 6.3 percentage points a year below the growth rate of the economy between 1946 and 1980. In these years technology advanced quickly and total income grew fast, so cutting the old returns was correspondingly less painful.
Around 1980 the second cycle entered its second phase. The United States freed interest rates, foreign exchange and the financial industry from control, and firms put the return to shareholders first; in sixteen rich countries the real return on government debt rose above the growth rate again, by about 1.9 percentage points a year on average between 1981 and 2007. In every crisis central banks cut interest rates and bought assets, so that losses that would have occurred did not; the losses were postponed, and debt piled up for some more years. Today the ratio of the world's reproducible capital to income is back at its 1913 height. This does not prove that the second clearing has begun, but the ratio in the second cycle is now comparable to where it stood at the end of the first.
What grew by a fixed proportion was population; what grew more slowly was the harvest from the land; the accounting relation between them was that the food people eat can come only from the harvest. When the gap could widen no further, it was adjusted by famine, plague and war, or by marrying later and having fewer children.
What grows by a fixed proportion is the return capital expects; what grows more slowly is the income actually produced each year; the accounting relation between them is that the return can be paid only out of other people's income. When the gap can widen no further, it is adjusted by war, revolution and hyperinflation, or by cutting the return a little each year over decades.
What the two have in common is the same mathematical relation. Two quantities grow each at its own rate, and the ratio of the first to the second rises every year; for it to reach k times its starting value takes about 100 × ln k divided by the difference of the two growth rates in percentage points, in years. With a difference of 1 point, doubling takes about 70 years; with 3 points, about 23 to 24 years. What differs is where the limit lies. In the Malthusian age the limit was whether there was enough food; after the Industrial Revolution food is no longer the constraint, and the limit is how income is divided and how much a society will bear. So the adjustment in the first case is a fall in population, and in the second it is first of all a shrinking of property; only when the return cannot be cut year by year does the second end, like the first, in mass death.
On the agrarian side, we built a simulated farming world and ran it again and again from the same starting point, to test where the rise and fall of population comes from. So far we have made only one comparison: when the number of rulers was set by the number of commoners, population in three regions did not collapse once in fifteen hundred years; when the number of rulers was allowed to grow and shrink on its own, the fifty-year average population fell by more than three-tenths three to seven times. That comparison used a single random seed and has to be rerun with dozens of seeds. On the industrial side, the world's reproducible capital has twice risen to about three years of income: the first time ended in a clearing; the second peak came around 2020, and whether it will end in a clearing is not yet known. Our next tasks are to find out why both cycles rose to this height, whether the next adjustment will be gradual or violent, and whether this mathematical relation can be tested against the records of many countries. How far the simulated farming world has come is set out in What kind of mathematical system is human society?.