The New Palgrave Dictionary of Economics

Living Edition
| Editors: Palgrave Macmillan

Growth Take-Offs

  • Matthias Doepke
Living reference work entry


Following a phase of near-constant living standards lasting from Stone Age until the onset of the Industrial Revolution, a large number of countries have experienced growth takeoffs, in which stagnation gives way to sustained economic growth. What causes some countries to enter a growth takeoff while others remain poor? We discuss three mechanisms that can trigger a growth takeoff in a country previously trapped in poverty: fertility decline, structural change, and accelerating technological progress.


Child labour Demographic transition Economic growth Fertility Fixed factors Growth takeoffs Human capital Income–population feedback Industrial revolution Land Malthus’s theory of population Mortality Nutrition and development Population growth Productivity growth Skill-intensive technology Stagnation Structural change Technological progress Women’s work 

JEL Classification

D4 D10 

Viewed on a historical timescale, economic growth in the world economy is characterized by a long phase of stagnation in living standards, followed in many, but not all, countries by a growth take-off, that is, a transition to steady and sustained economic growth.

Figure 1 illustrates the basic facts. Before 1800, GDP per capita was low and near-constant in all world regions, with little cross-country variation in income levels. The first country to experience a growth take-off was Britain with the start of the Industrial Revolution, closely followed by other west European countries and the ‘Western Offshoots’ such as the United States. More recently, a number of Asian and Latin American countries have undergone a transition to rapid economic growth as well. In much of Africa, however, income per capita continues to stagnate. What causes some countries to enter a growth takeoff while others remain poor?
Fig. 1

The evolution of income per capita across world regions, years 1500–2001 (Note: The ‘Western Offshoots’ are defined as the United States, Canada, Australia, and New Zealand. ‘Asia’ excludes Japan. Source: Maddison (2003, Table 8c))

Explaining Stagnation

Before one can account for a growth takeoff after a phase of stagnation, it is essential to understand why economies stagnated in the first place. The explanation suggested by one of the earliest writers on the subject, British economist Thomas Malthus in his Essay on the Principle of Population of 1798, is widely accepted to the present day. The Malthusian model relies on two key ingredients: an agricultural production function that uses the fixed factor of land, and an income–population feedback where the population growth rate is an increasing function of income per capita.

Consider an aggregate production function of the form
$$ {Y}_t={A}_t{N}_t^{\alpha }{Z}^{1-\alpha }, $$
where Y t denotes output in period t, A t is productivity, N t is the size of the population, and Z is the fixed amount of land. (The results outlined below can be generalized to the case where physical capital also enters production.) In what follows, we use lower-case letters to denote per capita variables (y i  = Y t  = N t , and so on), and the growth rate of a variable x is written as γ(x). Output per capita is given by \( {y}_t=At{z}_t^{1-\alpha } \), so that its growth rate γ(y t ) satisfies:
$$ \gamma \left({y}_t\right)=\gamma \left({A}_t\right)+\left(1-\alpha \right)\gamma \left({z}_t\right). $$
Since land Z is constant, we have γ(z t ) = γ(Z)−γ(N t ) = –γ(N t ). Using this relationship, the growth equation can be rewritten as
$$ \gamma \left({y}_t\right)=\gamma \left({A}_t\right)-\left(1-\alpha \right)\gamma \left({z}_t\right). $$
Growth in income per capita is thus an increasing function of productivity growth and a decreasing function of population growth. The negative effect of population growth reflects the fact that land is a fixed factor: when the size of the population increases, there is less land for each person to work with, which lowers income per capita.
To turn the growth Eq. (2) into a theory of stagnation, one needs to specify how productivity A t and population N t evolve over time. Assume for now that productivity growth is constant, \( \gamma \left({A}_t\right)={\overline{\gamma}}_A \). The main assumption underlying the Malthusian theory of stagnation is that population growth is an increasing function of income per capita y t :
$$ \gamma \left({N}_t\right)=f\left({y}_t\right), $$
where f′ (y t ) > 0. A number of different justifications can be given for this relationship. One possibility is that children enter the utility function of parents as normal goods. A rise in income would then increase the demand for children, leading to higher population growth. Alternatively, the mechanism could also work through mortality. If higher income leads to better nutrition and, as a consequence, lower mortality rates, a positive relationship between income per capita and population growth follows. As an empirical matter, the assumption of a positive relationship appears to fit the experience of most pre-industrial economies rather well.
Using (3), the growth Eq. (2) reads:
$$ \gamma \left({y}_t\right)={\overline{\gamma}}_A-\left(1-\alpha \right)f\left({y}_t\right). $$
According to this equation, the growth rate of income per capita is a decreasing function of its level. If the detrimental effect of population growth is sufficiently strong, this mechanism leads to stagnation as the only possible long-run outcome. In a country where income per capita is initially rising, population growth will accelerate until it fully offsets productivity growth, \( \left(1-\alpha \right)f\left({y}_t\right)={\overline{y}}_A \), resulting in stagnation.

The Malthusian model is remarkably successful in terms of explaining economic growth (or the lack thereof) until Industrial Revolution. However, we now know that ultimately many countries managed to escape from the Malthusian trap. In these countries, living standards today are far superior to what almost any human alive before 1800 could have experienced. How can this drastic change in the economic fate of countries be explained?

Endogenous Population Growth

Given the growth Eq. (4), one scenario that could lead to a growth takeoff is a reversal of the income–population feedback. If the positive relationship described by the equation γ(N t ) = f (y t ) breaks down, and subsequent population growth is low, growth will ensue. Consider, for example, the case where population growth ceases altogether, γ(N t ) = 0. According to Eq. (2), growth in output per capita is then equal to productivity growth. Thus, as long as productivity keeps increasing, income per capita will grow indefinitely.

Historically, the Malthusian relationship between income and population growth did indeed break down in every single country that experienced a growth take-off. In a pattern known as the demographic transition, the high fertility and mortality rates of the pre-industrial era gave way to a new regime in which fertility, mortality, and population growth are low. In modern data, the relationship between income per capita and population growth is negative (both in a cross section of countries and in the time series for most rich countries), which is the opposite of what the Malthusian model assumes.

Figure 2 illustrates the demographic transition by comparing population growth in western Europe (the first region to experience a take-off) with Asia and Africa (the regions that stagnated the longest). In western Europe, population growth reached a peak at the end of the nineteenth century and has been declining since, despite rapid growth in income per capita. In Asia and Africa, in contrast, population growth has accelerated since the mid-nineteenth century, and is now much higher than in western Europe.
Fig. 2

The evolution of population growth across world regions, years 1500–2001 (Note: ‘Asia’ excludes Japan. Source: Maddison (2003, Table 8a))

A number of authors have developed theories that integrate models of economic growth and the demographic transition to explain growth take-offs. In this literature, fertility decline is usually interpreted as a substitution of child ‘quantity’ (a large number of children) by child ‘quality’ (fewer children in which parents invest in terms of education or human capital). As an example of a model capturing this tradeoff, consider the decision problem of a parent with preferences
$$ u\left(c,\;n,\;h\right)=\left(1{-} \beta \right) \log (c)+\beta \left[ \log (n)+\gamma \log (h)\right] $$
over consumption c, the number of children n, and the children’s human capital h, where β > 0 and 0 < γ < 1. The parent has to spend fraction φ of its time to raise each child, and can choose to spend an additional per-child fraction e on educating the children. The total child-rearing time is then given by (φ + e)n, and the budget constraint for the parent is c = (1−(φ + e)n)wH, where H is the parent’s human capital, w is the wage per unit of human capital, and the time endowment is normalized to one. A child’s human capital depends on the parent’s human capital H and education time e:
$$ h=1+\mu He, $$
where μ is the productivity of the education technology. Notice that a child receives at least one unit of human capital even if education e equals zero, which represents basic productive skills (such as physical strength) that do not rely on education. Lastly, the parent also has to observe a subsistence consumption constraint, \( c\ge \overline{c} \), where \( \overline{c} \) is the minimum amount of consumption required for survival.
In this model, the relationship between income and fertility depends on whether the optimal choices for education and consumption are at a corner. Assume that, initially, the wage w and the education productivity μ are so low that the subsistence constraint is binding and the parent chooses zero education (e = 0). The number of children is then constrained by the need to earn at least \( \overline{c} \) units of consumption:
$$ n=\frac{1}{\varphi}\left(1-\frac{\overline{c}}{wH}\right). $$
Under this regime, the relationship between income wH and fertility n is positive, as assumed by the Malthusian model.
The outcome changes substantially if, through an increase in the wage w and the education productivity μ, the economy enters a regime where the subsistence constraint is no longer binding, and education is positive: e > 0. Under this regime, parents spend a fixed fraction of their time on child rearing. The balance between child quality and quantity depends on parental human capital H. The optimal decision rules are:
$$ n=\frac{\beta }{\varphi +e}\mathrm{and} $$
$$ e=\frac{1}{1-\gamma}\left(\gamma \varphi -\frac{1}{\mu H}\right). $$
Equation (5) captures the trade-off between child quality and quantity: the number of children is a decreasing function of education e. Intuitively, investing a lot in each child renders children expensive, which reduces demand. Education e, in turn, depends positively on parental human capital H. An increase in income per capita (through a rise in H) therefore lowers fertility n, the opposite of the Malthusian assumption.

Given these results, an escape from the Malthusian trap is possible if some change in the economy generates increased investment in child quality. The literature proposes different candidates for the underlying cause of such an event. In Galor and Weil (2000), the take-off is ultimately a consequence of technological progress. Accelerating productivity growth increases the return to education (the parameter μ in the model outlined above), which eventually triggers the quantity–quality substitution and the growth take-off. Galor and Moav (2002), in contrast, suggest that evolving parental preferences (through an increase in the parameter γ) are the driving force behind fertility decline. Yet other authors have emphasized the role of declining mortality rates (Boucekkine et al. 2002; Cervellati and Sunde 2005; Doepke 2005; Kalemli-Ozcan 2002; Lagerlöf 2003a; Soares 2005), increasing female labour-force participation (Galor and Weil 1996; Lagerlöf 2003b), changes in the provision of old-age security (Boldrin and Jones 2002), changes in child-labour and education laws (Doepke and Zilibotti 2005), and the introduction of skill-intensive production technologies that raise the return to education (Doepke 2004).

Structural Change

Apart from endogenous population growth, the Malthusian model also relies on the presence of the fixed factor of land to generate stagnation. A second potential trigger for a growth take-off is therefore structural change that decreases the role of land. In pre-industrial economies, agriculture was the main mode of production. In contrast, in modern industrial economies the share of agriculture in output is small, and consequently land is less important. Translated into the growth Eq. (4), structural change amounts to a shift in the parameter α. In particular, an increase in α lowers the detrimental effect of population growth on income per capita. In the limit case of α = 1, income per capita is independent of the size or growth rate of the population, and is solely driven by productivity growth.

In Hansen and Prescott (2002), a decline of the role of land is generated endogenously in an environment where two competing technologies can be used for production. (Related contributions include Matsuyama 1992; Laitner 2000; Kögel and Prskawetz 2001; Gollin et al. 2002; Ngai 2004). In addition to the production function (1) above, an ‘industrial’, constant-returns technology is also available:
$$ {Y}_t^I={A}_t^I{N}_t^I, $$
where Y t I is industrial output, Y t I is productivity, and N t I is the amount of labour employed in the industrial sector. Productivity A t I is assumed to grow at a constant rate. The total amount of labour is allocated optimally between the traditional sector and the industrial sector. Given the linear production technology, output per worker in the modern sector is given by A t I . Early in development, when A t I is still low, it is optimal to allocate all workers to the traditional sector. During this phase the economy behaves just like a Malthusian economy where the modern technology does not exist at all.

Ultimately, however, the modern technology becomes sufficiently productive to be introduced. If w M is the (constant) marginal product of a worker in the Malthusian regime, the technology will be introduced once \( {A}_t^I>{w}_M \). From this point on, population growth no longer affects output per worker, since land is not used in the industrial sector. Output per worker therefore starts to grow at the rate of technological progress. Viewed through the lens of the Hansen–Prescott model, what initially appears as a structural break in economic history is merely the outcome of an optimal sectoral allocation decision in an otherwise stable economic environment.

Endogenous Technological Progress

Starting once again from the growth Eq. (4), a third potential trigger for a growth takeoff is a sustained increase in productivity growth that is large enough to ‘outrun’ population growth. Clearly, population growth cannot increase indefinitely, as there are physiological constraints on child bearing. Let \( {\overline{\gamma}}_N \) be an upper bound for population growth that cannot be exceeded for biological reasons. If now productivity growth satisfies
$$ {\gamma}_A>\left(1-\alpha \right){\overline{\gamma}}_N, $$
even at maximum population growth the detrimental effect of increasing population density does not suffice to negate productivity improvements, and improving living standards ensue.

A potential cause for accelerating productivity growth is scale effects in the production of ideas. An increase in world population implies that there are more people who might invent new, productive technologies. An increase in world population should therefore imply an acceleration of productivity growth. Scale effects of this kind underlie the takeoff models of Kremer (1993), Jones (2001), and Tamura (2002).


The three potential triggers for a growth take-off presented here should not be regarded as mutually exclusive alternatives, but rather as complementary explanations for a joint phenomenon. From an empirical perspective, there is little doubt that all three explanations are relevant: every country that underwent a growth takeoff also experienced a demographic transition, a sectoral shift from agriculture to industry and services, and an acceleration of productivity growth. Reflecting these observations, many papers in the literature already incorporate more than one of the mechanisms. For example, a number of authors propose models where accelerating endogenous productivity growth triggers a fertility transition. This is true, for example, of the seminal paper of Galor and Weil (2000) and, in a framework driven by human-capital externalities, for de la Croix and Doepke (2003). Similarly, Greenwood and Seshadri (2002) and Doepke (2004) integrate models of structural change with theories of fertility decline.

Building on the different mechanisms behind growth take-offs that have been proposed in recent years, a major challenge for future research is to understand why in many countries these mechanisms fail to work to the present day. Conceivably, a better understanding of the mechanisms that allowed some countries to overcome economic stagnation two centuries ago might help us learn how the same feat could be accomplished in poverty-stricken developing countries today.

See Also


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© The Author(s) 2008

Authors and Affiliations

  • Matthias Doepke
    • 1
  1. 1.