Malthus on Population and Human Welfare

Based on Gregory Clark, A Farewell to Alms (Princeton, 2007)

A continuously rising standard of living would have been an alien concept to anyone living in Europe (or anywhere else) before 1500. People lived and worked much as their parents and grandparents had, and most of them produced food. In 1500 there were large parts of Europe — including modern-day England, France, Germany, Austria, Hungary and Poland — in which three-quarters of the population were agricultural workers. Belgium and the Netherlands were the most urbanized parts of the continent, and yet even there, more than half of the population worked in agriculture.1 For the continent as a whole, two out of three people produced food, and one out of three people did everything else.

Their world was a world without intensive growth. There was certainly accumulation of physical capital: draught animals, windmills and waterwheels increasingly replaced human labour. There was also significant technological progress. Clockwork and the flywheel both appeared before 1400, setting the stage for a machine age that would be a long time coming. But these things didn’t translate into an increased standard of living for the average person.

Thomas Malthus sought to explain this long stasis in his Essay on the Principle of Population (1798). Malthus argued that population growth responds to good (or bad) times in a way that brings those good (or bad) times to an end, causing the standard of living of the great mass of people to fluctuate around the subsistence level. Gregory Clark has developed a simple model of this mechanism, and argues that it is a fairly accurate depiction of human existence during all but the last few hundred years. It might also describe some parts of the world today.

Clark’s Model of the Malthusian World

Clark begins with three assumption:

  1. The birth rate, defined as the number of births per year among a group of 1000 people, rises as average income rises.
  2. The death rate, defined as the number of deaths per year among a group of 1000 people, falls as average income rises.
  3. Average income falls as population rises.

Raising a child requires significant resources. An increase in average income makes people better able and more willing to bear this cost, leading to an increase in the birth rate. Higher average incomes also lead to better nutrition and better sanitation, causing the death rate to fall. The last assumption follows from the relative fixity of a resource such as farmland. If the quantity of farmland is fixed, adding another farmer will raise the total crop, because there is always something useful that he can do. However, it reduces the crop produced by each farmer, because each farmer now has less farmland to work with. The same phenomenon — diminishing marginal productivity, in the parlance of economists — implies that for workers generally, average income falls as the number of workers rises.

The figure above represents these three assumptions graphically. The curve BR shows the way in which the birth rate (BR, measured vertically) varies with average income (Y, measured horizontally); and the curve DR shows the way in which the death rate (DR, also measured vertically) varies with average income. The curve in the bottom graph (let’s call it the productivity curve) shows the relationship between population (P, measured vertically) and average income.

Assumption 1 is satisfied because the BR curve is upward sloping, and assumption 2 is satisfied because the DR curve is downward sloping. Assumption 3 is satisfied because the productivity curve is downward sloping. Having used these assumptions to construct the figure, we have no further use for them and need not think of them again.

A country characterized by these three assumptions will fall into a state that will remain unchanged unless disturbed by some outside force. Economists call such a state an equilibrium. The equilibrium in Clark’s model is described by the values of five endogenous variables. Four of these variables appear in the figure: birth rate, death rate, population and average income. The fifth, life expectancy, is determined by the death rate. In a society with stable demographics, life expectancy is the reciprocal of the death rate. For example, if the death rate is 25 per thousand, the average person lives 1000/25, or 40, years. If the death rate is only 20 per thousand, the average person lives 1000/20, or 50, years.

So, the equilibrium — the unchanging state — consists of the values of five variables. To find them, first note that the population is constant only when the rate of population growth is zero. Since this rate is the difference between the birth rate and the death rate,2 equilibrium requires these two rates to be equal. The top half of the figure shows that they are equal only when the average income is Y’.3 The bottom graph shows that population is P’ when average income is Y’, and the top graph shows that the birth rate and the death rate are BR’ and DR’. Life expectancy is 1/DR’ years.

Famine, Plague, and War

The Europeans of the Middle Ages experienced repeated episodes of famine, repeated outbreaks of the plague, and wars too numerous to count. These events had a devastating impact on Europe’s population. The Black Death alone is conservatively estimated to have killed a quarter of the population in the middle of the fourteenth century — and then it came back again and again. Europe’s population fell from 73 million in 1300 to 45 million in 1400, and then recovered to 69 million in 1500.4 The implied rate of population growth over the entire fifteenth century is only 0.4% per year.

Clark’s model sheds light on this period in two ways. First, it tells us something about the lives of the people who survived these disasters. Second, it shows the resilience of the equilibrium described above. Although these events initially devastate the country, it still finds its way back to the equilibrium.

Look at the figure above, and imagine that we are initially in the equilibrium. Then the Black Death comes along, and kills a large part of the population. That means that the population falls from P’ to something a lot smaller than P’. The productivity curve shows that the decline in population corresponds to an increase in average income. These horrible events, perversely, raise the average person’s standard of living. (For example, if there are fewer farmers, each farmer has more land to work with, and is able to produce a bigger crop. The survivors eat better than they did before.) The top half of the diagram shows that at the higher average income, the birth rate is higher than the death rate. This imbalance will, slowly and over time, increase the population. As the population rises, the average income falls, shrinking the gap between the birth rate and the death rate and slowing the rate of population growth. When the population reaches P’, average income has fallen back to Y’ and the gap between the birth rate and the death rate has closed, so the population stops growing. The economy is right back where it started and it stays there.

This example shows Malthus’s mechanism in action. Famine and plague and war reduce the population, but they also create a period of prosperity that drives up the population. The increase in population brings the period of prosperity to an end. People are ultimately no better off than they were before, but also no worse off.

Makin’ Whoopee

Malthus was concerned with showing that things that look like they are good are not, in the end, all that good. Let’s imagine, for example, that cities are able to improve their sewage systems so that they become healthier places to live. And let’s imagine that this general improvement in health leads to a higher birth rate at every income. This event is represented in the model by an upward shift in the BR curve: the vertical distance that represents the birth rate is bigger at every income.

At the time of the shift, the population is P’ and average income is Y’. At this income, the upward shift in the BR curve opens up a gap between the birth rate and the death rate, so that the population starts to rise. The productivity curve shows that average income falls as the population rises. The fall in average income reduces the gap between the birth rate and death rate. When the population reaches P” and average income falls to Y”, this gap is zero, meaning that the population stops rising. The new equilibrium has population P” and average income Y”. The birth rate and the death rate are both higher than they were before, and the higher death rate implies a shorter life expectancy. People are both poorer and shorter lived in the new equilibrium. A supposedly good thing — better sanitation — has made people worse off.

Technological Change

Technological change might have been slow in the Middle Ages, but it did exist. Why didn’t it improve the material well-being of the average person? To find out why, let’s go back to the model. A single technological improvement raises the average income at every population, that is, it shifts the productivity curve to the right.

At the initial population P’, average income is read off the new productivity curve, and is substantially greater than Y’. But at every income greater than Y’, the birth rate exceeds the death rate, so the population is growing. The population keeps growing, and average income keeps falling, until population reaches P” and average income has fallen back to Y’. At this income, the gap between the birth rate and the death rate is closed, bringing the population growth to an end and halting the decline in average income . The new equilibrium has a higher population than the old, but the same average income, the same birth and death rates, and the same life expectancy.

A technological change nevertheless has temporary benefits. The rightward shift of the productivity curve produces a period of prosperity that is brought to an end, once again, by population growth. And since high incomes reduce the death rate, there is also a temporary increase in life expectancy. A technological change is a good thing, just not for long.

However, one can imagine a sequence of technological changes, with each change occurring before the economy has completely adjusted to the last one. Then the adjustment process is never completed, and the prosperity is never brought to an end. That is also the message of the Solow growth model. In that model, something called total factor productivity is a proxy for the state of technology. The model shows that, over longer horizons, per capita incomes grow at the same rate as total factor productivity. In other words, we owe our rising standard of living entirely to improving technology.

The Evidence

Clark argues that this model is a good description of the human experience everywhere until a few hundred years ago. The shift away from Malthusian stagnation and toward continuous growth occurred first in England and then spread to other parts of western Europe. Some parts of the rest of the world also shifted to growth around this time, while other parts remained Malthusian.

The figure above is part of Clark’s evidence for this conjecture.5 It graphs England’s per capita income against its population over a period of 600 years. For almost the first 400 years of data, there is a fairly strong negative correlation between per capita income and population. This finding conforms to the predictions of the Malthusian model (think of England is sliding up and down a fairly stable productivity curve). England breaks away from the Malthusian world in the middle of the seventeenth century: incomes climb steadily in the presence of a stable population. This new pattern continues until the middle of the eighteenth century, when the Industrial Revolution occurs. A third pattern is established in its aftermath: rapid population growth coupled with rapid income growth.

  1. These figures are from Robert Allen, The British Industrial Revolution in Global Perspective (Cambridge, 2009), p. 17.
  2. The population rises when people are born at a faster rate than they die, and shrinks when they die at a faster rate than they are born.
  3. Remember that the birth rate is the vertical distance to the BR curve, and the death rate is the vertical distance to the DR curve.
  4. These numbers are from Douglass North and Robert Thomas, The Rise of the Western World (Cambridge, 1973), p. 71.
  5. A version of this diagram appears on page 30 of Clark’s book. Clark uses it to argue that England was Malthusian until 1800, but to my mind, this claim requires a very liberal interpretation of his own model.