Population Growth and Economic Progress: A Historical Analysis

1. Chapter 8 Population and the Origins of Sustained Economic Growth Charles I. Jones

2. The Role of Population Growth • In all of our models considered so far, population growth is playing a very important role. • In endogenous growth models, the number of ideas that drive economic progress is increasing with the number of people. • However, according to Thomas Malthus the growth in population inevitably results in resources drain. • In 1798, real wages in England stable for 200 years • In 1798, real wages are lower compared to 1500, at the level of 1200 • In this chapter, we make population growth endogenous by linking the number of children to the income level

3. Population and Living Standards Very low growth rates of population prior to 0 year CE: 0.00007% per year For one million (sic!) years prior to 0 CE real incomes were basically the same The period between one million years BCE and 0 CE is referred to as Malthusian era In 1 CE: average income per capita around $450 per year in today’s dollars, not growing between 1 CE and 1000 CE. 1000 CE to 1820 CE: average income grew to $670 a year, 0.05% growth rate per year

4. Malthusian Era The period between one million years BCE and 0 CE is referred to as Malthusian era In 1 CE: average income per capita around $450 per year in today’s dollars, not growing between 1 CE and 1000 CE. 1000 CE to 1820 CE: average income grew to $670 a year, 0.05% growth rate per year World population grew from 230 million in 0 CE to 261 million in 1000 CE, 0.02% growth per year; 438 million in 1500 CE, 1.04 billion in 1820 (less than today’s China) Little effort was spent during Malthusian era to accumulate human capital: universities only catered to a small class of individuals, education serving cultural and political purposes

5. Post-Malthusian Era • Around 1800, there occurs a rapid acceleration in both income per capita, and populatin growth rates • 1820-1870: population growth averages 0.4% per year, 1870-1913: 0.8%, 1919-1950: 0.9% per year • 1950-1973: 1.9% per year • Education not keeping pace: by 1841 (about the time of Industrial Revolution), only 5% of male workers and 2% of female workers worked in industries where literacy was required • The accelerated population growth rates did not lead declining living standards • 1820-1870: 0.5% growth in income per capita • 1870-1913: 1.3% growth

6. Modern Growth Era • Start of demographic transition • Shift toward smaller families: In the early 1990s population growth starts falling in Europe and North America • Decline in Total Fertility Rate (TFR), the average number of children per family • 1870: TFR=6 in the Netherlands and Germany, 5.5 in England, 4 in France • 1970: TFR=2 in Europe, implying near-zero population growth rates • Latin America followed in the middle of the 20th century, Asia in the late 20th century, Africa now has constant population growth rates • Fewer children started to acquire more education: US and the Netherlands had universal primary school education by the middle of the 19th century, by early 20th century secondary schooling widespread in the US

7. Production Function in the Malthusian Era Since vast majority of production was agricultural in the Malthusian era, we replace capital with land: X is stock of land, L is labor, Y is output, and B is TFP This production function is constant returns to scale In income per capita terms, The key property is, y is inversely related to population L: as L rises, more output is produced, but in per capita terms output will decrease due to decreasing returns. More people working on the same amount of land crowd out each other

8. Endogenizing the Population Growth Rate With this Malthusian production function, if we allow L to grow exogenously, eventually output per capita y will go down to zero, which is exactly what Malthus predicted about the future. We make population growth endogenous: the population growth is increasing with income per capita Improving living standards makes it easier to support more children in the family: where is a subsistence level of consumption, and is a parameter governing the response of population growth to income. The population growth can be negative in case per capita incomes fall below the subsistence level. If , population grows.

9. Population Growth and Population Size Combining the production function and endogenous population growth equations, we obtain the following: Population growth and size are negatively related to each other.

10. Malthusian Steady State At low levels of L, people are relatively rich, and population grows is positive For large L, income per capita is low, infant mortality is high, and population growth is negative For a specific population size L*, population growth rate is zero. This L* is Malthusian steady state. Whatever the initial size of L, it will converge to the Malthusian steady state.

11. Solving for the Malthusian Steady State We can solve for the value of the Malthusian steady state by setting in the equation to obtain The steady-state population size L* increases with more land X since more food can support larger population sizes. If the TFP (technology level) B increases, L* increases as well since higher technology makes land more productive (i.e. fertilizers). The greater the subsistence requirement , the lower the steady-state population L*.

12. Income per Capita and Subsistence Levels From endogenous population growth equation , we know that in the steady state the income per capita must equal the economy’s subistence level: Income per capita in steady state is only dictated by the subsistence level, no influence of technology B or land X. Increased living standards Suppose income per capita rose above y*. That would mean higher population growth rates. However, with fixed X and B the economy’s per capita output will start to decline: economy eats away at its own prosperity.

13. Decrease of Population and Wealth An exogenous decrease in the size of population temporarily rises living standards. 14th century Europe killed 30%-50% of the population. The survivors had greater access to resources per capita. Real wages in England doubled between 1350 and 1450. By the 1500s, real wages in England were back to their pre-disaster levels. England: population declined from 3.75 million to 2.5 million during Black Death, recovering by 1500 to 3.75 million

14. Subsistence Level and Wealth The level of subsistence level does not have to be low by biological standards. In case it is relatively high, the Malthusian steady state will correspond to a high living standard. Limited supply of land in the Malthusian model leads to a situation where any increases in income per capita are inevitably temporary.

15. Two Important Questions • We know that the size of the human population has increased exponentially over time • We also know that at the same time, income per capita has been increasing at a steady rate • The Malthusian model has to be modified to accommodate answers to these two questions in the following ways: • Allow for technological change • Relax the assumption of positive relationship between population growth rate and income per capita

16. Technological Change • An increase in the technology term B produces the following effects: • Everyone is more productive • Income per capita grows • Population growth rate grows because of With better technology, people have a higher income per capita, which increases their willingness to have children. Population level is permanently higher, but income per capita remains the same:

17. Technology in Malthusian Era Better technology allows to support more people on the same amount of land. Land is rivalrous, so gains in technology are offset by a greater number of people, so we have the same income per capita as before. Technology is not rivalrous, so a larger population does not decrease the technological level. Technological change then leads only to temporary gains in living standards along with a permanent increase in the size of population. Historically, this has been the case most of the time: stagnant living standards with increasing population numbers.

18. Continuous Technological Growth Take logs and differentiate the Malthusian production function: Since land X is in fixed supply, its log derivative is zero. Key message: growth in income per capita depends on how fast technology evolves compared to the growth of population. Let . If , population grows to fast, and per capita income is falling. In case , per capita income is rising.

19. Constant Technological Progress Suppose the growth rate of technology is fixed at a constant We can then combine and the Malthusian equation in the following diagram: is the Malthusian steady-state level of income per capita If , the income per capita level will converge to the Malthusian steady state. In other words, is a stable steady state.

20. Constant Technological Progress and Growth An increase in the growth rate of technology g will shift up the horizontal line so the Malthusian steady state will increase. However, with constant g, that is, if [read as “g” is not a function of time t], no sustained growth in income per capita is possible, only temporary increases due to a one-time increase in g. To have a growing population together with growth in income per capita, we must endogenize technological change.

21. Endogenous Technological Change From Chapter 5, the technological growth can be modeled as: [see equation 5.25] As population L grows, the rate of technological change increases as well. It can be shown (how?) that for sufficiently small values of L relative to B, as population grows, the growth rate will increase. Increases in will lead to a higher population growth, so we have a virtuous cycle where population and technological growth are reinforcing each other.

22. Kremer’s Model Michael Kremer (1993) uses the virtuous cycle reasoning to explain the historical relationship of population growth and population size. Assuming , and that all people work and create innovations so that the previous equation becomes Combining Kremer’s assumption with Malthusian equilibrium condition , we obtain the following: The population growth rate is increasing with the size of the population! This is a theoretical justification for the virtuous cycle reasoning.

23. Population Growth and Size: Empirical Test For all but the most recent past, there is a very strong positive relationship. Starting from the 1970s, the relationship breaks off. Standard Malthusian model without endogenous technological change cannot explain this graph. As the graph demonstrates, population growth rates start falling at one point. We thus need to alter our description of population dynamics.

24. Realistic Population Growth Rates Let us consider a more realistic relationship between population growth rate and y: For low levels of income per capita, we have a standard Malthusian (i.e. positive) relationship. However, now we have a turning point at which population growth rate starts falling with increases in income per capita. Since the dynamics of y are still governed by , the income per capita is behaving in the way illustrated by the figure.

25. Two Steady States The point is a stable steady state in the sense that, as long as , income per capita will end up at eventually. If , population growth rate is lower than g, and income per capita is increasing. Given our population growth function, richer families stop having more families at some level of wealth, so y grows indefinitely. The level of y equal to is an unstable steady state in the sense that a very small deviation from it will make y either grow indefinitely, or converge to .

26. Technology and a Single Steady State Theoretically, in the previous graph, once y jumps over the level of , sustained growth is guaranteed. However, the Black Death experience suggests this may never happen (income per capita doubled, but sustained growth was not realized). Imagine the technological growth rate g jumps to the level that it is above the maximum of the population growth rate function. In this case, the Malthusian steady state disappears, and the economy is on the sustained growth path, with y growing, and population growth rate leveled out at n*.

27. From Malthusian Doom to Sustained Growth In the beginning, the population is very small, and exists in a Malthusian steady state. Given , the growth rate of technology g is very small, too. For thousands of years, the population does not grow much. However, the virtuous cycle is at work: increasing speed of innovation leads to increasing population size, and vice versa: g is increasing, and Malthusian steady state is shifting to the right. This speeding up began in Europe around 1500 or 1600. The post-Malthusian period starts when the g line jumps over the hump of the population growth function, so the economy is on the sustained growth path.

28. Post-Malthusian Balanced Growth Path We can use the framework of chapters 2 through 5 to derive the balanced growth path equations: In case , the “stepping on toes” effect is not too strong compared to the “standing on shoulders effect”, and . In case the role of land in production is small, i.e. if is small, the growth is more likely to be positive. Historical growth pattern can be explained by expanding populationbringing about more innovators.

29. Fixed Production Factors and Growth Recall that, in our model, the magnitude of is crucial: if the value of g is large enough, there is no Malthusian steady state, and the economy grows continuously. Denoting , we can re-write That is, a sustained economic growth is achievable if: 1) The growth rate of technology is high, i.e. if is large 2) If the role of land is low, that is, if is low. More generally, an economy where the role of fixed factors of production is low, e.g. the role of land is insignificant, sustained growth is more likely.

30. Structural Changes and Sustained Growth • In Malthusian era, agricultural output made up the vast majority of total output. • Innovations resulted in reduced role of fixed factors of production like land. • Consider the valued added chain for a cotton shirt: • Use land to grow cotton • Spin thread from raw cotton • Weave thread into cloth • Sew raw cloth into a shirt according to a design developed by someone • Hire workers to transport shirts to the store • Hire clerks to show you to the shelf that has shirts on it • The role of land is rather limited in modern shirts’ production.

31. Declining Share of Land in Output 1750: farmland rents make up 20% of national income in England 1850: same share is 8% 2010: only 0.1% As time went by, increasing innovations reduced the drag on the economy produced by the fixed factors (land). In the end, g jumped over the hum of the population growth rate function, releasing us from the Malthusian era.

32. Why Not Do Without Fixed Factors? • Since the fixed factors of production are so pernicious, why not get rid of them from the start? • The innovations reducing the role of fixed factors may take time to realize • It is not easy to abandon agricultural production without starving to death • Food is probably the most important product for survival • A minimum amount of food is necessary to keep us alive • Labor must be allocated to food production until the minimum amount of food is safely provided • Limited, or even no labor is left for producing innovations

33. Engel’s Law As income per capita increases, the share of income spent on food never increases in the same proportion. Mathematically, the income elasticity of food is less than one. This empirical regularity is known as Engel’s Law. As a result, as economies grow richer, a smaller fraction of labor is channeled into agricultural production, and a smaller fraction of the nation’s output is comprised of agricultural products. England: in 1785, 40% of output is agricultural goods, in 1905 it’s only 5%. Germany: 50% in 1850, 25% in 1905 Sub-Saharan Africa: today, 50% of all output is agricultural goods U.S.: today, less than 1% Korea: today, 2.7% of GDP NK: 23.3%

34. Restructuring and Growth As economies grow and develop, they make a transition from predominantly agricultural structure towards the one mostly oriented on services: Malthusian Era: most output is agricultural Industrializing nations: agricultural output is replaced by manufacturing products until the latter dominate the economy Developed economies: Services dominate the economy, manufacturing is next, agriculture is marginal The key driver of this structural transformation is the population growth’s ability to produce increasingly more innovations.

35. Economics of Population Growth Recall that our analysis so far has crucially hinged on the assumption that population growth is related to per capita income in the following manner: Why do we have such a humped graph? Why is it that education does not rise until the population growth spikes?

36. Children and Their Education as Economic Goods In economic terms, children and their education are economic goods that families consume. Such framework allows us to apply the well-developed tools of utility functions and budget constraints to analyze families’ decisions on how many children to have, and how much education to provide for them. Gary Becker (1960): having children entails opportunity costs, so richer families choose to have fewer children because of higher opportunity costs (e.g. wages). However, there is a contradiction with Malthusian era evidence: wages were low, and the population did not grow fast.

37. Quantity-Quality Framework To resolve the contradiction with evidence on Malthusian era, let us adapt Becker’s (1960) framework by introducing the quality of children into the analysis. What’s a quality child? A quality child is an educated child. The basic tradeoff in this framework is between quantity of children, and their quality. For instance, large families will not be able to afford high educational standards for their children.

38. Family Budget Constraint • Consider an average family making a decision on how many children to have, and how much education to pay for. • Suppose an average family earns a per capita income y. • The family can spend its income in three ways: • Buying subsistence amount of consumption goods (food) • Spending money on children M (clothes, food, toys) • Spending money on educating children E (schooling fees) • Formally, the budget constraint is:

39. Number of Children and Education • We make the following assumptions regarding the number of children: • The more resources M we spend on children, the more children we can afford to have • The higher our income y is, the higher the opportunity cost of our time spent on children, so richer families have fewer children (remember this is Becker’s original idea!) where m is the number of children, and is a parameter. Assumptions regarding the amount of education: 1) Children get some basic education at home (mother tongue) 2) Education level increases with spending on education E Units of education of a child u are as follows:

40. Family’s Utility Function We assume families derive utility from having more children who are better educated. However, the returns in terms of utility to m and u are diminishing, which is a standard assumption. As a result, an average family’s utility function will look like this: Since logarithms tend to infinity as their arguments tend to zero, parents will always want at least one child. (Why is u never equal to zero?)

41. Family’s Optimal Choice It is important to realize that once the amount of education E is chosen by the parents, there is no other choice variable left! Indeed, suppose we set . Then from the budget constraint, it follows that . That is, the amount of money spent on children is uniquely identified because the income per capita y and basic consumption are constants. In the same way, E uniquely determines . We can thus rewrite our family utility function as and maximize it with respect to E.

42. First-Order Conditions Remember that in order to maximize a smooth function of one variable, it is enough to go over all levels of this variable where the first derivative at those levels is equal to zero. These special levels are called critical points, and the requirement that the first-order derivative be equal to zero is called the first-order condition. Luckily, we only have one such point. Indeed, taking derivatives with respect to E and equating the result to zero, we obtain the following First-Order Condition: Solving it, we obtain the level of education E an average family is going to choose:

43. Optimal Education Level We have just seen from the first-order conditions that the amount of education E an average family will choose will be given by: Expectedly, higher income per capita y will result in more education, while higher basic consumption needs and better home education will decrease the educational level chosen by the family. However, if y is very small, E can be negative. To avoid this, let us modify our optimal E a little bit: For low levels of y families will not provide education.

44. Optimal Number of Children We have already mentioned that we can use the budget constraint in order to infer the number of children m once the educational level E is known: Since the optimal educational level we just found is equal to we can plug it into the budget constraint above to obtain the optimal number of children an average family will choose to have:

45. Population Growth Function When , families are relatively poor, so they do not pay for any education for their children. As a result, if income y starts growing, they will use all of the additional resources (after paying for subsistence consumption ) on paying for the additional children. When , which is the peak of the population growth function, parents start paying for education, and increasing y leads to the choice to have fewer children. (We assume )

46. Population Growth in the Long-Run The long-run in this context corresponds to the levels of per capita income y so high that the population growth function flattens. Indeed, when y is high, the optimal number of children is given by , and the terms and both approach zero. The optimal number of children in the long-run becomes . The long-run population growth rate can be shown to be equal to , and it does not depend on the level of income y. For countries that have sufficiently high income levels y, the population growth rate is unchanged even as per capita incomes grow.

47. England and China Revisited Why is it England was the first to make a transition to sustained growth, and why did this not happen in China first? Remember the key to attaining sustained growth is to make sure that g rises above the hump of the population growth rate function. Higher levels of g occur either because of faster technological progress, or because of larger population size. Alternatively, the population growth function shifts down through differences in family fertility behavior (show that). • England • Borrowing ideas reduced stepping on toes effect • Institution of property rights increased incentives to innovate • China • Level of technology is higher compared to England • Rate of innovation not fast enough to outpace population growth

48. European Advantages: Other Reasons • The faster rate of technological innovation in Europe, even if China had much higher levels of technology, is likely to have allowed Europe to embark on the path to sustained growth rather early. • There are other explanations of why Europe advanced so early: • Protestantism favored children’s education and lowered fertility rates (but then, Confucianism favored education, too) • Late marriage age for European women and large number of unmarried women in Europe resulted in limited population growth (and, actually, limited innovation rate) • Higher mortality rates in Europe due to war, plague and urbanization lowered the hump of the population growth rate function, thus allowing Europe to escape the Malthusian steady state trap.

49. Geographical Dimension Vollrath (2011): different types of agriculture were practiced in different regions of the world. Arguably, Europe and Asia have more favorable conditions to foster agriculture compared to Africa and Americas. Diamond (1997): sizable advantages in terms of domesticable crops in Europe and southeast Asia, allowing these regions to sustain larger populations. Empirical evidence suggests that in general, the further away countries are from the equator, the better they perform economically.