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‘The Child is Father of the Man:’ by Implications for the Demographic Transition * David de la Croix ** Omar Licandro DOCUMENTO DE TRABAJO 2008-04 Serie Innovación CÁTEDRA Fedea – Banco Sabadell

January 2008

* Department of Economics and CORE, Université catholique de Louvain ** European University Institute.

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David de la Croix Department of Economics and CORE, Université catholique de Louvain

Omar Licandro European University Institute and CEPR

September 2007

Abstract

We propose a new theory of the demographic transition based on the evidence that body development during childhood is an important predictor of adult life expectancy. Fertility, childhood development, longevity, education and income growth all result from individual decisions. Parents face a trade-of between the number of children they have and the spending they can aford on each of them in childhood. These childhood development spending will determine children longevity when adults. It is in this sense that we refer to Wordsworth’s aphorism that “The Child is Father of the Man.” Parents face a second trade-of in allocating their time between increasing their own human capital and rearing children. The model displays diferent regimes. In a Malthusian regime with no education fertility increases with adult life expectancy. In the modern growth regime, life expectancy and fertility move in opposite directions. The dynamics display the key features of the demographic transition, including the hump in both population growth and fertility, and replicate the observed rise in educational attainment, adult life expectancy and economic growth. Consistent with the empirical evidence, a distinctive implication of our theory is that improvements in childhood development precede the increase in education.

JEL Classification Numbers: J11, I12, N30, I20, J24. Keywords: Life Expectancy, Height, Human Capital, Fertility, Mortality.

¹De la Croix acknowledges the financial support of the Belgian French speaking community (Grant ARC 03/08-235 “New Macroeconomic Approaches to the Development Problem”) and the Belgian Federal Government (Grant PAI P6/07 “Economic Policy and Finance in the Global Economy: Equilibrium Analysis and Social Evaluation”). Licandro acknowledges the financial support of the Spanish Ministry of Sciences and Technology (SEJ2004-0459/ECON). We thank Joerg Baten and Richard Steckel for making their data available to us, and Juan Carlos Cordoba, Moshe Hazan, Omer Moav, David Weil and Hosny Zoabi for very useful comments on an earlier draft.

1 Introduction

The transition from a world of low economic growth with high mortality and high fertility to one with low mortality and fertility but sustained growth has been the subject of intensive research in recent years.¹ In this literature, the relation between growth and fertility results from the quantity/quality trade-of faced by parents between the number of children and their education. Longevity is generally taken as given or modeled through some aggregate externality.

In this paper, we analyze the demographic transition using a new theory of endogenous fertility and mortality, based on a fundamental parental concern: providing children with appropriate health care and nutrition and promoting good attitudes towards health during childhood ensures a longer life for future generations² Starting with Kermack, McKendrick, and McKinlay (1934), who showed that the first fifteen years of life were central in determining the longevity of the adult, the relationship between early development and late mortality within cohorts has been well-established. Another important contribution in the field is that of Barker and Osmond (1986) who related lower childhood health status to higher incidence of heart disease in later life. This idea also had an echo in the literary tradition as witnessed by the aphorism (with apologies to feminists) “The Child is Father of the Man” (Wordsworth 1802), meaning that the way a child is brought up determines what he or she will become in the future.

The main mechanism suggested in the literature to explain the link between childhood development and longevity is through improvements in nutrition and physiological status, as emphasized by Fogel (1994). Another mechanism stressed by epidemiologists links infections and related inflammations during childhood to the appearance of specific diseases in old age (Crimmins and Finch 2006). The contribution of this paper is to propose a new theory of the demographic transition based on the evidence that body development during childhood is an important predictor of life expectancy.

¹Rostow (1960) presents an early attempt to understand the transition from stagnation to growth. The first modern treatment of the issue is in the seminal paper by Galor and Weil (2000). Doepke (2006) contains a recent survey.

²A survey of the related epidemiology literature can be found in Harris (2001)).

The key and novel mechanism we propose is that parents face a trade-of between the quantity of children they have and the amount they can aford to spend on each of them during childhood. Parents like to have children, but they also care about their longevity. By ensuring an appropriate physical development for their children and protecting them from infections, parents provide them with greater health capital and a longer life.³ Such provision is costly though, and its cost is proportional to the number of children. As a consequence, having many children prevents parents of spending much on their health capital. The proposed quality/quantity trade-of makes longevity and fertility negatively related.

We are aware that longevity does not depend solely on childhood development. Adults’ investment in health and government spending on the elderly also contribute considerably to reductions in mortality. However, adding these mechanisms into our setup would not alter the trade-of we want to put forward, and, therefore, we abstract from these additional mechanisms in order to streamline the argument. We are also aware that there are at least two diferent types of health capital, as pointed out by Murphy and Topel (2006). One extends life expectancy so that individuals can enjoy consumption and leisure for longer; the other increases the quality of life, raising utility from a given quantity of consumption and leisure. In this paper, since we are mainly interested in longevity, we restrict the analysis to the first type of health capital.

³The state of the debate between nature and nurture may be synthesized by the following statement “The efects of genes depend on the environment,” (Pinker 2004) where genes are associated with nature, and the environment as a short-hand for the efects of human behavior with nurture. In this paper, we are mainly interested in understanding how changes in human behavior, for a given distribution of genes, afect childhood development. In other words, by restricting the analysis to a representative cohort member, a standard assumption in OLG models, we do not pay much attention to within cohort diferences in genes, but concentrate on how changes in nurture over time afect childhood development on average. The implicit assumption is that the complex interaction between nature and nurture shaping the distribution of childhood development across cohort members in society has no significant efect on the mean over a period of time covering one or two centuries.

In this paper, the key mechanism relating demographics and growth is the Ben-Porath hypothesis that longevity positively afects education and human capital accumulation, by extending an individual active life.⁴ Following Boucekkine, de la Croix, and Licandro (2002), we assume that adults decide about their own optimal amount of education, and we take basic education, even if provided by parents, as being exogenously given. In addition to the trade-of between the number and development of children stressed above, adults face a trade-of between having children and improving their own education, which makes the number of children and schooling negatively related. This is similar to the trade-of faced by parents in a Beckerian world, where they care about the quantity and quality (education) of their ofsprings. Letting parents to care also about the education of their children would certainty complicate the resolution of the model, obscuring the trade-of between the number of children and their childhood development, which is the key mechanism in our theory.

The dynamics of our model displays the key features of the demographic transition, including the hump in total net fertility rate and in population growth. In particular, it is able to replicate the observed rise in life expectancy and educational attainment, as well as the initial increase and then decline in fertility. If the mechanisms we describe predominate, the logic of the demographic transition could well be the reverse of that which is usually assumed: the key trade-of is not between fertility and education, with efects on longevity as a byproduct, but between fertility and healthier and longer living children, with a subsequent efect on education. This timing is evident in the data presented in the next section.

⁴Conditions for this mechanism to hold in the presence of endogenous fertility are derived in Hazan and Zoabi (2006). In a recent controversial paper, Hazan (2006) argues that the Ben-Porath hypothesis holds only if longevity afects lifetime labor supply positively, which he claims is not consistent with US data. The main argument in our paper would also hold if, as in Hazan and Zoabi, childhood development afects education directly because healthier children perform better.

In our model, fertility, childhood development, longevity, education and income growth all result from individual decisions. In this sense, this paper difers from the previous attempts in the literature to endogenize fertility and longevity simultaneously. Many papers have the standard education/fertility trade-of with exogenous longevity (for example, (Doepke 2004)). Other papers model health investment either by households (Chakraborty and Das (2005) and Sanso and Aisa (2006)) or by the government (Chakraborty (2004) and Aisa and Pueyo (2006)) but have exogenous fertility. A few treat both fertility and longevity as endogenous variables, but the mechanism leading to longer lives always relies on an externality: more aggregate human capital or more aggregate income leads to higher life expectancy (Blackburn and Cipriani (2002), Lagerloef (2003), Cervellati and Sunde (2005) and (2007) and Hazan and Zoabi (2006)).

Two recent papers have modeled the trade-of between the number and survival of children exclusively in the context of pre-modern societies. In Galor and Moav (2005)’s paper, there is an evolutionary trade-of (i.e. not faced by individuals but by nature), between the survival to adulthood of each ofspring and the number of ofspring that can be supported. Lagerloef (2007) suggested that agents chose how aggressively to behave, given that less aggressive agents stand a better chance of surviving long enough to have children, but gain less resources, so more of their children die early from starvation. In both cases, the trade-of is between the number and survival of children, while in our paper it is between the number of children and adult longevity.

This paper is organized as follows. Section 2 presents some evidence on the link between the improvement in childhood development and the demographic transition. In Section 3, we present the problem for individuals, and solve for the optimal allocation. Section 4 is devoted to the study of the dynamics of dynasties. Aggregate dynamics, including balanced growth path analysis, are studied in Section 5. A simulation of the demographic transition is proposed in Section 6. Section 7 presents our conclusions.

2 Data on Childhood Development and Fertility

Height is a frequently used indicator in microeconomic studies of the relationship between health and income. Weil (2007) finds that the efect on wages of an additional centimeter of height ranges between 3.3% and 9.4%, depending on the data set used. In a second step, he exploits the correlation between height and direct measures of health such as the adult survival rate to evaluate health’s role in accounting for income diferences among countries; he finds that eliminating health variations would reduce world income variance by a third.

Height is a simple measure of childhood development, since both better nutrition and lower exposure to infections leads to increased height.⁵ Height is constant after, say, the age of 18, but is still a good predictor of life expectancy and mortality in old age. According to Waaler (1984), the trend towards greater height found in the data means that younger cohorts, which have grown up with better nutrition, will have better health and live longer as adults.

The height of conscripts has been systematically recorded by the Swedish army since 1820, which provides time-series information on changes in height throughout the demographic transition. Figure 1 presents data for the cohorts born between 1760 and 1960. The left panel shows that the height of soldiers (measured at approximately age 20) is highly correlated with the life expectancy of the same generation.⁶ The right panel of Figure 1 shows that body height and years of schooling are positively correlated and, more importantly, that changes in height precede changes in education. Figure 2 completes the picture by reporting the net fertility rate for the same period. We observe that fertility and height are positively correlated over the period 1800-1830, uncorrelated between 1830 and 1900, and negatively correlated afterwards.

⁵According to Silventoinen (2003), height is a good indicator of childhood living conditions (mostly family background), not only in developing countries but also in modern Western societies. In poor societies, the proportion of cross-sectional variation in body height explained by living conditions is larger than in developed countries, with lower heritability of height as well as larger socioeconomic diferences in height.

⁶Notice that this strong correlation over time can also be established in a cross section of countries: Baten and Komlos (1998) regressed life expectancy at birth on adult height and explained 68% of the variance for a sample of 17 countries in 1860.

Figure 1: Height, Life Expectancy and Education in Sweden

Sources: Sandberg and Steckel (1997) for height data from 1820; Floud (1984) for height data before 1820 from Denmark); The “Human Mortality Database” for life expectancy data; and de la Croix, Lindh, and Malmberg (2007) for education data.

Figure 2: Height and Net Fertility in Sweden Net fertility is computed as the product of the fertility rate (Statistics Sweden) with the probability of survival until age 15 (Human Mortality Database).

Figure 3: Fertility Rates in Germany Sources: Baten (2003) for height data; Knodel (1974) for fertility data. Fertility is the ratio of total fertility rate to a benchmark.

Further insights into the links between childhood development and fertility during the demographic transition can be gained by combining two data sets. Baten (2003) classified the former provinces of the German empire into six categories according to conscripts’ height in 1906, i.e., for men born in the 1880s. In Figure 2 we retain the two extreme categories: provinces with the tallest (168.70 cm and more) and the shortest (166.50 cm and less) soldiers. The Princeton European Fertility Project provides information on fertility in these provinces for the years 1867-1933 (see Knodel (1974)). In the period 1870-1890, which is when the soldiers of 1906 were born, we can see that fertility rates were systematically higher in the provinces with shorter soldiers, which is consistent with the idea of a trade-of between the number of children and childhood development (as measured by adult height). Later on, fertility rates dropped and converged.

3 The Model: Individuals

Here we describe a continuous-time overlapping generations (OLG) model with endogenous fertility and mortality inspired by de la Croix and Licandro (1999) and Boucekkine, de la Croix, and Licandro (2002), who modeled the link between longevity and education in a framework where all the demographic variables are exogenous.

Let us denote by B the age of puberty, i.e., the age at which individuals acquire regular fertility. B assumed to be is constant. Individuals reaching puberty at time t are said to belong to cohort t, whose size is denoted by P (t). Life expectancy at age B is denoted by A(t), which is referred to below as life expectancy. We abstract from infant mortality, and assume that the survival law is rectangular, with mortality rates equal to zero for ages below B + A, and individuals dying with probability one at this age. Consequently, is life expectancy at birth. Choices are made by individuals reaching puberty.⁷ Preferences are represented by (we have dropped the index t to ease the exposition)

\[\int_ {0} ^ {A} c (z) \mathrm{d} z + \bar {H} (\beta \ln \hat {n} + \delta \ln \hat {A}).\tag{1}\]

We assume that individuals do not consume until they reach age B. c(t) represents consumption at age B + t. Preferences in consumption are linear for simplicity and the time preference parameter is assumed to be zero. Under this assumption, the equilibrium interest rate is zero and the marginal value of the intertemporal budget constraint, the associated Lagrange multiplier, is unity. In addition to their own consumption flow, individuals value the number of children, denoted by ˆn, as well as the life expectancy of their children, denoted by A. H is the average human capital per worker, which will be defined below. It multiplies the term associated with children to keep utility balanced in a growing economy, implying that the value of children and their health depends on the average human capital of the society. Parameter , weights the marginal utility of children relative to adult consumption. The marginal utility of the quality of children is weighted by δ,

⁷Modeling family behavior is not a simple issue. As children grow, parents take child preferences into account more and more, but parents still have something so say as long as they support children financially until they find a job, leave home and become fully independent. Since modeling this complex process is beyond the objective of this paper, we assume that children become fully independent at one stroke.

The technology producing human capital depends on the time allocated to education and the average human capital per worker:

\[h = \mu (\theta + T) ^ {\alpha} \bar {H}.\]

The productivity parameter and the parameter which relates to schooling before puberty, are strictly positive, and . θ ensures that human capital, and hence income, are positive even if individuals choose not to to school after age

The budget constraint takes the form

\[\int_ {0} ^ {A} c (z) \mathrm{d} z + \hat {n} \Psi (\hat {A}) = \mu (\theta + T) ^ {\alpha} \bar {H} (A - T - \phi \hat {n}).\tag{2}\]

The right hand side is the total flow of labor income. For simplicity, we assume that people have and raise their children immediately after finishing their studies and before becoming active in the labor market. This greatly simplifies the dynastic structure of the model. Raising a child takes a time interval of length , implying that individuals work for a period of length . The wage per unit of human capital is unity. Parental expenditure on each child’s development is

\[\Psi (\hat {A}) = \kappa \frac {1}{2} \frac {\bar {H}}{A} \hat {A} ^ {2},\tag{3}\]

which implies that the expenditure is quadratic in and inversely related to This formulation is inspired by the economic theory of capital adjustment costs, and is consistent with the complex interaction between nurture and nature observed by biologists and psychologists. It stresses the dificulty of raising life expectancy above that of the parents, reflecting how the genes interact with human behavior (the environment) in building up a child body. The parameter measures the costs of developing children and represents health technology in a broad sense. Finally, note that the integral in Equation (2) may be substituted in Equation (1). The resulting objective function, depending on and is concave.

Figure 4 summarizes the OLG structure of the model. Generation is born at becomes independent at goes to school until and enters the labor market at . Their children belong to generation , since is the age at which individuals have children, and children reach puberty after a period of length Individuals chose their own education , the number of children and the quality of children as measured by their life expectancy . Their choice depends on three types of parameters. First, those related to preferences, and Second, the parameters associated with child rearing and childhood development, , B and Finally, the educational technology parameters and The analysis of the demographic transition in Section 6 is built on letting some of these parameters change smoothly with time. To simplify the exercise, it will be assumed that all these parameters are cohort specific.

⁸An example of a technology of childhood development is given by Dalgaard and Strulik (2006), where the metabolic energy to create a new cell is an exponential function of the body mass the individua wants to reach.

The maximization of utility (1), subject to the budget constraint (2) and to the positivity constraints and , can be interior or corner. We make the following assumption about preferences:

Assumption 1 Preferences satisfy

Assumption 1 states that the preference weight attached to childhood development, cannot exceed twice the weight attached to the number of children, The quantity/quality trade-of depends on the ratio of marginal utilities to marginal costs, which crucially depends on the factor two because of the quadratic form of the adjustment costs. A similar condition can be found in Moav (2005) and de la Croix and Doepke (2006), when parents face the standard fertility/education trade-of.

We make the following assumptions about education technology

Assumption 2 The productivity of education technology satisfies:

\[\mu > \max \left[ \frac {(\beta - \delta / 2) \alpha^ {2}}{\theta^ {1 + \alpha} (1 + \alpha)}, \frac {\delta \alpha}{2 \theta^ {1 + \alpha}} \right] \equiv \underline {{\mu}}.\]

This assumption requires the productivity coeficient to be large enough. Let us establish the main proposition on individual behavior.

Proposition 1 Under Assumptions 1 and there exist two thresholds and , such that:

, there is a unique interior solution satisfying

\[\hat {A} ^ {2} = \frac {\delta}{\kappa \hat {n}} A,\tag{4}\]

\[T = \frac {\alpha}{1 + \alpha} (A - \phi \hat {n}) - \frac {\theta}{1 + \alpha},\tag{5}\]

\[\hat {n} = \frac {\beta - \delta / 2}{\mu \phi} (\theta + T) ^ {- \alpha}.\tag{6}\]

, there is a unique corner solution with positive consumption satisfying

\[\hat {A} ^ {2} = \frac {\delta}{\kappa \hat {n}} A,\tag{7}\]

\[T = 0,\tag{8}\]

\[\hat {n} = \frac {\beta - \delta / 2}{\mu \phi \theta^ {\alpha}}.\tag{9}\]

, there is a unique corner solution with zero consumption satisfying

\[\hat {A} ^ {2} = \frac {\delta}{\kappa} \frac {\mu \theta^ {\alpha} \phi}{\beta - \delta / 2} A,\tag{10}\]

\[T = 0,\tag{11}\]

\[\hat {n} = \frac {\beta - \delta / 2}{\beta \phi} A.\tag{12}\]

Proof. Using the Kuhn-Tucker conditions for constrained optimization, we can identify the two thresholds and and characterize the diferent regimes. See Appendix A. Restriction in Proposition 1 states that parental life expectancy has to be large enough for schooling to be positive. At the interior solution, Equation (4) shows the trade-of faced by parents between the number and the life expectancy of their children. The relation is negative, since the total cost of providing children with health cares increases as their number increases. Equation (5) is the standard Ben-Porath (1967)

result, as described by de la Croix and Licandro (1999), where life expectancy positively afects the time allocated to education since it allows people to work for a longer time. The term in Equation (5) shows an additional trade-of of having children: parents expecting to have many children will postpone their entry into the labor market, reducing the incentives to take additional education. This trade-of also shows up in Equation (6).

When , parental life expectancy A is not long enough to render optimal a positive investment in education. For lower levels of life expectancy, i.e. when both education and consumption are zero.⁹ Expected life time earnings are so low that parents use all their resources in bearing a limited number of children.

From now on, the interior solution, (4)-(6), and the corner solutions, (7)-(9) and (10)- (12), are referred to as . The efect of an increase in A is given by Corollary 1:

Corollary 1

for , for , and otherwise

and otherwise

Proof. See Appendix A.

In the interior solution, increased life expectancy raises optimal schooling and human capital levels via the Ben-Porath efect. This increases the opportunity cost (time cost) of raising children. Hence, the optimal number of children drops as life expectancy increases.

In the corner solutions, since , a change in parental life expectancy does not afect education, canceling the Ben-Porath efect. In the corner regime (7)-(9) the number of children remains constant whatever the life expectancy, but childhood development is still positively afected by life expectancy, since the eficiency of health activities depends positively on parental life expectancy. In the corner regime (10)-(12), when consumption C is zero, however, the efect of life expectancy on the quantity and quality of children reverses, since the number of children is directly determined by the constraint, which allows for more children as life expectancy increases. This is a Malthusian efect. Consequently, for the study of the transition from Malthusian to modern growth in Section it makes sense to initially put the economy into this last regime.

c ≥ c¯

A

⁹Imposing a strictly positive minimum consumption level with for the sake of realism would not change the results. The only diference is that would be larger and increasing in ¯c.

4 Dynasties

Since individual decisions do not depend on aggregate variables, we can study the dynamics of life expectancy, fertility and education within dynasties separately, before the analysis of the aggregates.

Let us consider the dynamics of life expectancy first. At any point t, individuals reaching puberty belong to a representative dynasty with life expectancy A (t). Let us denote as the life expectancy of the first generation of this dynasty. The operator defines a diference equation governing the evolution of the dynasty’s life expectancy since

\[A _ {i + 1} = \hat {A} _ {i} = f _ {A} (A _ {i}),\]

where the index , is associated with generations. From Proposition 1, for any initial value there exists a sequence of solutions and for i = 1, 2, 3,etc. where is the ith consecutive application of operator X. In the following, we characterize the dynamics of life expectancy. Once this has been done, the sequence determines through the operators and the date and size of the following generations.

Proposition 2 Under Assumptions 1 and the dynamics of life expectancy following , with given, are monotonic. A stationary solution exists, is unique and globally stable.

Proof. See Appendix A.

Proposition 2 states that life expectancy converges to a constant value in the long run. Consequently, from Proposition 1, fertility and education also converge to a constant value. Demographic variables are then stationary in a model of endogenous growth, meaning that the demographic transition only occurs, as the name itself indicates, as a transitional phenomenon.

The results obtained so far allow us to assess some theoretical characteristics of the demographic transition in our model. Consider Figure 5. The lower panel plots the function . It describes a situation where the globally stable steady state is in the interior regime. The top panel shows fertility as a function of life expectancy. Suppose now that initial life expectancy is very low, below The dynamics of life expectancy will be monotonic and converge to the steady state. A rise in life expectancy will first drive fertility up (as long as the economy is in the Malthusian regime , then fertility will peak in the zone where (i.e. where but , and then decrease (in the interior regime). Schooling will be zero until we reach the interior regime and will then increase monotonically. This sharp characterization is very much in line with the stylized facts of the demographic transition as reported in Figures 1 and 2.

In this description of the theoretical dynamics, we have assumed a steady state in the interior regime. A condition for such a situation to occur is given by Proposition 3.

Figure 5: A Steady State in the Interior Regime Proposition 3 The steady state is in the interior regime if

\[\kappa < \frac {4 \alpha \delta \theta^ {2 \alpha} \mu^ {2} \phi}{(2 \beta - \delta) (\alpha [ 2 \beta - \delta ] + 2 \theta^ {1 + \alpha} \mu)} \equiv \overline {{\kappa}}\tag{13}\]

Proof. See Appendix A.

Condition (13) states that if the childhood development technology is cheap enough, there is an interior steady state with positive education. Changing the value of κ is a natural way of generating a demographic transition: if κ is initially high, the economy is in one of the corner regimes. Once κ has increased, health becomes more afordable, the new steady state is in the interior regime, and life expectancy converges monotonically to this value.

Figure 6: Life cycle (from left to right) and living cohorts (from right to left)

5 Aggregates

Some definitions are useful to study the dynamics of the aggregate population, active population and human capital. In Figure 6, t and z represent time and cohort, respectively. Let us define as the age of the oldest cohort still alive at time t, which then represents the life expectancy at time t of cohort . By definition, is the life expectancy of cohort z. Then, given that generations z and are the same, has to be equal to . This is equivalent to introducing the variable change , implying that

\[\tilde {A} (t) = A \left(t - \tilde {A} (t)\right).\]

A similar argument applies to the functions and . Let us define and as the schooling time and the number of children of the youngest cohort entering the labor market at time t, i.e., cohort in Figure 6. Since and

\[\tilde {T} (t) = T (t - \tilde {T} (t) - \phi \tilde {n} (t)),\]

and

\[\tilde {n} (t) = \hat {n} (t - \tilde {T} (t) - \phi \tilde {n} (t)).\]

The total population is computed by integrating over all the living cohorts:

\[N (t) = \int_ {t - \tilde {A} (t)} ^ {t + B} P (z) \mathrm{d} z,\tag{14}\]

from the oldest to the youngest . The cohort size is given by

\[P (z + T (z) + B) = \hat {n} (z) P (z),\tag{15}\]

since members of cohort z have children at time , who belong to cohort . Aggregate human capital is defined by the human capital of active cohorts

\[H (t) = \int_ {t - \tilde {A} (t)} ^ {t - \tilde {T} (t) - \phi \tilde {n} (t)} P (z) \underbrace {\mu (\theta + T (z)) ^ {\alpha} \bar {H} (z)} _ {h (z)} \mathrm{d} z\tag{16}\]

where average human capital per worker is given by

\[\bar {H} (t) = \frac {H (t)}{E (t)},\]

and total employment is

\[E (t) = \int_ {t - \tilde {A} (t)} ^ {t - \tilde {T} (t) - \phi \tilde {n} (t)} P (z) \mathrm{d} z.\]

The technology producing the consumption good, the only final good in this economy, is linear in aggregate human capital with productivity one, implying that the real wage per unit of human capital is unity. Output per capita is then

The Balanced Growth Path

A balanced growth path is an equilibrium path where the population grows at rate η, human capital grows at rate γ, and, the demographic variable T, n and A are all constant, as defined in Section 5. From Equation (15), the grow rate of cohorts’ size is such that , i.e.

\[\eta = \frac {\ln (n)}{T + B},\]

with . The population growth rate depends on the fertility rate n and on the age at child’s birth . At a given fertility rate, the smaller the age at birth, the larger the frequency of births and thus the growth rate of the dynasty.

Total population, as defined in Equation (14), evolves along a balanced growth path following

\[N (t) = N ^ {\star} \mathrm{e} ^ {\eta t} = P ^ {\star} \frac {\mathrm{e} ^ {\eta B} - \mathrm{e} ^ {- \eta A}}{\eta} \mathrm{e} ^ {\eta t},\]

with . Population also grows at rate η and its size depends positively, as expected, on life expectancy. When η approaches zero, i.e., when population is constant, its size is given by , which is the product of the cohort size and life expectancy at birth. Along a balanced growth path, the active population is given by

\[E (t) = E ^ {\star} \mathrm{e} ^ {\eta t} = P ^ {\star} \frac {\mathrm{e} ^ {- \eta (T + \phi n)} - \mathrm{e} ^ {- \eta A}}{\eta} \mathrm{e} ^ {\eta t}.\]

Similarly as for total population, when η approaches zero converges to φn), where the term in brackets is the length of active life. Finally, the growth rate of

human capital γ satisfies

\[\gamma = \frac {P ^ {\star}}{E ^ {\star}} \mu (\theta + T) ^ {\alpha} \left(\mathrm{e} ^ {- \gamma (T + \phi n)} - \mathrm{e} ^ {- \gamma A}\right).\]

To understand this result better, let us diferentiate, at the balanced growth path, the definition of H(t) in Equation (16) with respect to time:

\[H ^ {\prime} (t) = P (t - T - \phi n) h (t - T - \phi n) - P (t - A) h (t - A).\]

The change in aggregate human capital is the diference between the human capital of the youngest workers and that of the oldest. From the human capital technology, and using the balanced growth path assumption

\[\gamma = \frac {H ^ {\prime} (t)}{H (t)} = \frac {P ^ {\star}}{E ^ {\star}} \mu (\theta + T) ^ {\alpha} \left(\mathrm{e} ^ {- \gamma (T + \phi n)} - \mathrm{e} ^ {- \gamma A}\right).\]

The first term on the r.h.s, , derives directly from the assumption that per worker human capital afects the human capital of the current cohort. If, instead of normalizing total human capital by E, we normalized it by P, this term would vanish. It basically corresponds to the length of active life. The second term reflects the fact that both the oldest and the youngest cohort share the same human capital technology, with a common length of education. For this reason, the term is common. Finally, the last term in brackets reflects the fact that aggregate human capital was not the same at the time the two cohorts were at school, the diference depending on the growth rate itself and the age diference between the cohorts. The vintage human capital nature of the model was pointed out by Boucekkine, de la Croix, and Licandro (2002): the human capital of new cohorts entering the labor market is larger than that of the retiring cohorts, because quality of schooling progressed with human capital accumulation.

The growth rate of per capita output is at the balanced growth path. No theorem is available to assess the asymptotic behavior of the solutions of our dynamic system directly, and in particular, whether income per capita converges to its balanced growth path.¹⁰ In the simulations in the next section though, the solution converges asymptotically to the balanced growth path.

6 Simulating the Demographic Transition

In this section, we investigate to what extent the trade-of between the number of children and their development can reproduce the key facts of the demographic transition. The transition is studied as the reaction to a change in the environment (the Industrial Revolution) occurring after 1820 and leading the economy to a new balanced growth path characterized by sustained growth and longer lives. For this purpose, we implement a change in the parameter κ and analyze how the economy adjusts to this change, using numerical simulations.¹¹

6.1 The Change in the Parameter κ

Before analyzing the demographic transition as a response to a change in the parameter κ, we need to discuss briefly the interpretation to be given to such a change. Indeed, many would think that the period 1800-1870 does not contain major technology changes in health that could have increased life expectancy at puberty so much.

¹⁰No direct stability theorem is available for delay diferential systems with more than one delay since the stability outcomes depend on the particular values of the delays. See Mahafy, Joiner, and Zak (1995).

¹¹A more sophisticated version of the model, in line with Galor and Weil (2000), would allow for an endogenous industrial revolution. This could be achieved by letting health technology κ depend upon population size or density.

It is known that ancient ideas persisted a long time in modern Europe and the confidence of consumers in medicine was low. As a consequence some authors claim that the rise of life expectancy in early modern Europe did not relied on medical advances. Johansson (1999) argues against this therapeutic nihilism that tends to deny that medicine had any efectiveness before the end of the nineteenth century. First, in the period 1500-1800, medicine showed an increasingly experimental attitude: no improvement was efected on the grounds of the disease theory (which was still mainly based on traditional ideas), but significant advances were made based on practice and empirical observations. For example, although the theoretical understanding of how drugs work only came progressively in the nineteenth century with the development of chemistry (Weatherall 1996), the efectiveness of the treatment of some important diseases was improved thanks to the practical use of new drugs coming from the New World. Second, the number of books containing lifestyle advice increasing significantly over the period 1750-1800, which provides some indirect evidence of the fact that lifestyle advice (concerning, for example, personal and domestic cleanliness) became popular. Third, Johansson (1999) reports that, as early as 1829, Dr.F.B. Hawkins wrote a book entitled Elements of Medical Statistics, in which he described what could be called an early modern epidemiological transition. He describes a set of diseases which were leading causes of death but can now (in 1829) be treated efectively: leprosy, plague, sweating sickness, ague, typhus, smallpox, syphilis and scurvy.

The cumulative efects of these improvements could have produced a net increase in the eficacy of medicine as early as in the eighteenth century (see de la Croix and Sommacal (2008) for further arguments).

6.2 The Demographic Transition over Time

We assume that the Industrial Revolution produced a change in the cost of childhood development κ. Setting the age of puberty, to 15, we first calibrate the model to reproduce a steady state having the following properties in the pre-1820 balanced growth path: economic stagnation , low life expectancy at age no education after puberty and a population growth rate of 0.5% per year. We also set the parameters to obtain the thresholds and , which ensure that the economy is initially in the Malthusian regime and ends in the modern growth regime (interior regime). There are too many parameters given the number of targets, and and α have been set arbitrarily , while δ and have been computed to match the properties given above. This leads to the following results: , and

Note that for these values Assumption 1 holds. We can also compute the threshold for required by Assumption 2. It is equal to 0.434, showing that Assumption 2 also holds in our example.

Next, we implement a drop in κ such that life expectancy is increased to 70 years at the new steady state. This requires us to divide κ by more than two, leading to at the new steady state. We assume that this change takes place smoothly, following a logistic curve:

\[\kappa (t) = \kappa_ {0} + \frac {\kappa_ {1} - \kappa_ {0}}{1 + e ^ {1 8 4 0 - t} / 5}.\]

With such a function, 99% of the change takes place between 1820 and We also assume that the parameter κ(t) is specific to generation t. Hence any change only afects new generations, leaving past decisions unafected.

¹²If, instead, the change were discrete, we would observe intervals of times with no births, corresponding to periods where everybody increases their length of schooling in a discrete way, giving rise to permanent replacement echoes which are typical of models with delays (Boucekkine, Germain, and Licandro 1997). In this case, the economy keeps fluctuating forever, moving from baby booms to baby busts. Non-monotonic convergence also occurs in the Galor and Weil model - see Lagerloef (2006).

Figure 7 depicts the simulation results.¹³ We first observe that, following the drop in the cost of childhood development, life expectancy increases monotonically over time and converges to the new steady state, in accordance with the prediction of Proposition 2. Cohorts’ education also increases monotonically, showing that the economy shifts from the Malthusian corner regime with no education to the interior regime with . Notice that the magnitude of the increase in T is about right, with schooling converging towards 5 years after puberty (15). Cohorts’ fertility (per individual, to be multiplied by 2 to get fertility per women) first increases as long as the economy is in the Malthusian state, then peaks in the corner regime with positive consumption, then drops monotonically as a consequence of the trade-of between education and number of children in the interior regime. The interaction of fertility and longevity over time leads to a hump-shaped population growth rate, which is one important characteristic of observed demographic transitions.

The robustness of the above results to alternative speeds of adjustment of κ has been investigated. Assuming a slower adjustment, but still achieved by the year 2000,¹⁴ leads to the economy spending a longer time in the intermediate corner regime, and education beginning to increase later, around 1900. The hump shaped population growth rate is still present, but arrives later. Lowering the threshold A by changing the calibration would restore the original timing.

In this simulation, the demographic transition is triggered by technological change, represented by a drop in the physical cost of childhood development. An alternative to this storyline could be the following. According to Galor and Moav (2002) a long period of struggle for survival during the Malthusian stagnation increased the weight attached to the quality of children in the population following an evolutionary process, which in turn fostered the take-of process. In our set-up this could be reflected by allowing changes in the preference parameters and δ. We could then replace the drop in κ in the above simulation by an increase in the weight of quality relative to quantity to match the rise in life expectancy between the initial and final steady states; the new relative weight of quality would be . The transitional dynamics following this change in parameters was computed using the same method and the results are similar to those displayed in Figure 7, indicating that the two stories – technical change and preference changes – lead to the same results.

¹³The simulation was performed using the method developed by Boucekkine, Licandro, and Paul (1997).

¹⁴It seems reasonable to assume that the majority of the possible gains in terms of childhood development have been exploited - height is no longer increasing much more any longer in developed countries, and further increases in longevity are more related to medical progress afecting adult and old-age health capital, which is not modeled here.

Figure 7: Example of dynamics - drop in κ

6.3 Regional Variations

We conclude from the above simulation exercise that our model is able to reproduce the main features of the take-of from stagnation to growth through the demographic transition. Another question is whether we can also shed some light on regional variations in the demographic transition. Considering the German data presented in Figure 3, we have seen that adult height (a proxy for childhood development) and fertility were negatively associated across provinces on the eve of the twentieth century. This is perfectly in line with the model when the economy is in the interior regime, i.e. in times of falling fertility. However, when fertility is rising, as it was the case in the 1860s in Germany, the economy is in the corner regime in which height and fertility are positively associated. It is thus not obvious a priori that the model can reproduce a world where shorter individuals have higher fertility at a time in when fertility is increasing.

One reason for diferent places exhibiting diferent fertility and height paths during the demographic transition is that the initial cost of health could vary in diferent places, and

Figure 8: Total Fertility Rates in Places A (dashed) and B (solid) the speed in the drop of the cost could be diferent. A pattern similar to that observed in Figure 3 can emerge if the place which initially had a higher cost κ benefited from a faster transition. Consider two economies with all parameters equal to those of the previous simulation except for κ. In place A, the initial κ is 10 % larger than in place B but it drops faster (the term ^−t is divided by 10 rather than by 5). The result of the simulation is shown in Figure 8. Fertility rates first rise then decline. There is initially a large gap between the two places. Place A, with the lower fertility in 1865- 1895 also has higher childhood development in that period, as witnessed by the gap in life expectancy computed for the cohort reaching puberty in 1900 (born in 1885 and being in the military in 1906): life expectancy is 47.4 in place A and 45.2 in place B. Hence, both the cross-sectional and time-series aspect of the demographic transition can be captured if diferential progress in health technology is allowed for.

7 Conclusion

The epidemiology literature stresses that life expectancy depends greatly on physical development during childhood. Both better nutrition and lower exposure to infections leads to increased body height and a longer life. We have proposed a theory of the demographic transition based on this fact. The novel mechanism of the model is that parents face a trade-of between the quantity of children they have and the amount they can aford to spend on childhood development of each of them. Parents like to have a lot of children, but they also care about their health and longevity as adults. Having many children prevents parents spending much on their health capital. If the cost of health decreases, parents will increase their investment in their children’s longevity. The number of children will first increase in the Malthusian regime as a consequence of higher lifetime income. As longevity rises, fertility starts falling as a result of the tradeof faced by parents between investing in their own human capital and spending time rearing children. Following the trade-of between the number of children and childhood development, adult longevity keeps increasing. The model we have developed reproduces the characteristics of the demographic transition well, displaying the appealing features that longer education delays birth and reduces fertility. Our theory can be seen as an alternative to the one based on a rise in the return to human capital investment induced by economic progress, leading parents to substitute quality for quantity. A distinctive implication of our theory is that improvements in childhood development should precede the increase in education. Taking height as a proxy for childhood development, we have observed just such a pattern in Swedish historical data. Our theory can also provide an explanation for the puzzling fact that height at age 18 is a strong predictor of education attained later in life (Magnusson, Rasmussen, and Gyllensten (2006) showed that Swedish men taller than 194 cm were two to three times more likely to obtain a higher education than men shorter than 165 cm), even after controlling for parental socioeconomic position, other shared family factors, and cognitive ability. A further test of our theory would consist of checking whether famil size is related to childhood development as measured by average height on historical micro-data.

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A Proofs of Propositions

Proof of Proposition 1

After substituting the integral in (2) into (1) and diving by , the objective becomes

\[\left(\beta \ln \hat {n} + \delta \ln \hat {A}\right) + \underbrace {\mu (\theta + T) ^ {\alpha} (A - T - \phi \hat {n}) - \hat {n} \left(\frac {\kappa}{2} \frac {\hat {A} ^ {2}}{A}\right)} _ {C}\]

which is maximized under the restrictions and

First order conditions to this problem are (omitting the Kuhn-Tucker conditions):

\[(1 + \eta) \hat {A} ^ {2} = \frac {\delta}{\kappa \hat {n}} A\tag{A.1}\]

\[(1 + \eta) \alpha \mu (\theta + T) ^ {\alpha - 1} (A - T - \phi \hat {n}) = (1 + \eta) \mu (\theta + T) ^ {\alpha} - \lambda\tag{A.2}\]

\[\frac {1}{\hat {n}} \left(\beta - \frac {\delta}{2}\right) = (1 + \eta) \mu (\theta + T) ^ {\alpha} \phi\tag{A.3}\]

where λ and η are the Kuhn-Tucker multipliers associated with the constraints and , respectively. The interior solution (4)-(6) is under The corner solution (7)-(9) results from the same system under , and finally, the corner solution (10)-(12) results from the first order conditions under Under Assumption 2, is not optimal.

Interior Regime. The solution to the first order conditions (4)-(6) exists and is unique if the loci in (5) and (6) cut once and only once for positive n and , and at the solution. The locus in (5) is a straight line with negative slope and cuts the axes at , see Figure A.1. The locus in (6) has a negative slope, is convex, and is such that ˆn goes to zero when T goes to infinity and cuts the ˆn axes at

Comparing these two points and imposing leads to the condition , where

\[\overline {{A}} = \frac {\beta - \frac {\delta}{2}}{\mu \theta^ {\alpha}} + \frac {\theta}{\alpha}.\]

Substituting (4) and (5) in the definition of C gives

\[C = \frac {\mu}{\alpha} (\theta + T) ^ {1 + \alpha} - \frac {\delta}{2},\]

which is positive under Assumption 2 for all

Corner regime . If , the straight line is above the convex curve at (see Figure A.1). A suficient condition for these two curves not to intersect in the positive plane is that the straight line is steeper than the convex curve at zero. This is guaranteed by Assumption 2. In that case, there is no interior solution, since negative values for T are not feasible. Consequently, the solution must be corner with

From equations (7)-(9), at this corner solution

\[C = \mu \theta^ {\alpha} \left(A - \frac {\beta - \delta / 2}{\mu \theta^ {\alpha}}\right) - \delta / 2,\]

which is positive for ,with

\[\underline {{A}} \equiv \frac {\beta}{\mu \theta^ {\alpha}}.\]

From Assumption 2, . It is easy to see that the solution is unique.

Corner regime . Finally, when , the optimal solution is (10)-(12), with both inequality constraints being binding. Uniqueness is trivial.

Proof of Corollary 1

For the interior solution, we apply the implicit function theorem to , which leads to

\[f _ {A} ^ {\prime} = \mathrm{d} \hat {A} / \mathrm{d} A = \sqrt {\frac {A \delta}{\hat {n} \kappa}} \frac {(T (\alpha + 1) + \theta + \alpha ((A - \hat {n} \phi) \alpha + \theta))}{2 A ((\alpha + 1) (T + \theta) - \hat {n} \alpha^ {2} \phi)}.\]

The numerator is positive. Under Assumption 2, the denominator is also positive. The results for and can be proved using the same arguments. For the corner solutions, the result is straightforward.

Proof of Proposition 2

Let us denote the function when when , and when . The dynamics of life expectancy following are monotonic because is continuous and non-decreasing.

Let us first prove the existence of a solution. From corollary 1, . It is easy to see that lim . To prove the existence it is enough to show

\[\lim _ {A \to \infty} \frac {f _ {A} (A)}{A} = \lim _ {A \to \infty} \frac {f _ {A 1} (A)}{A} = 0.\]

From (5) and (6)

\[\hat {n} = c t e (A - \phi \hat {n}) ^ {- \alpha}.\]

Substituting in (4), and dividing by gives

\[\left(\frac {\hat {A}}{A}\right) ^ {2} = c t e \frac {(A - \phi \hat {n}) ^ {\alpha}}{A}.\]

Since lim , it’s now easy to see that li

Let us now prove its unicity. For , the function is increasing and concave, with and , implying that if it crosses the diagonal on the interval , it crosses it only once.

Function is increasing and concave, with and , implying that if it crosses the diagonal on the interval , it crosses it only once.

Finally, let us prove that for any fixed point of in . From the implicit function theorem applied to (4)-(6),

\[\frac {\mathrm{d} \hat {A}}{\hat {A}} / \frac {\mathrm{d} A}{A} = \frac {1}{2} \frac {(1 + \alpha) (A - \phi \hat {n})}{A - (1 + \alpha) \hat {n}}.\]

At a fixed point of , since Corollary 1 shows that in this interval, the denominator must be strictly positive. It is then easy to see that if . Since, from Corollary 1, in this interval, for all if , which holds under Assumption 2.

Global stability is then trivial, since is above the diagonal before the unique steady state equilibrium and below it afterwards.

Proof of Proposition 3

A steady state for A in the interior regime exists if there is a solution to the system (4)-(6) evaluated at the steady state. Eliminating A and n from Equation (5) using Equations (4) and (6) we find that the steady state T should satisfy:

\[T (1 + \alpha) + \theta = \alpha \left(\frac {2 \delta \phi \mu (T + \theta) ^ {\alpha}}{\kappa (2 \beta - \delta)} - \frac {(2 \beta - \delta) (T + \theta) ^ {- \alpha}}{2 \mu}\right)\]

The left hand side is a linear increasing function of T. The right hand side is a concave function of T. A necessary and suficient condition for the existence and uniqueness of a solution is that the right hand side is larger than the left hand side at . This leads to Condition (13).