Vintage human capital, demographic trends and endogenous growth by Raouf Boucekkine* David de la Croix** Omar Licandro*** DOCUMENTO DE TRABAJO 2000-02

January, 2000

* IRES

** National Fund for Scientific Research

*** FEDEA

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Theorem 1.2. (Theorem of [10, Theorem 1.1]) Let be a finite field and let be the set of all elements of such that

Vintage human capital, demographic trends and endogenous growth*

Raouf Boucekkine David de la Croix Omar Licandro

December 24, 1999

Abstract

We study how economic growth is affected by demographics in an overlapping generations model with a realistic survival law. Individuals optimally chose the dates at which they leave school to enter the labor market and at which they retire. Endogenous growth arises thanks to the accumulation of generation-specific human capital. Favorable shifts in the survival probabilities always induce longer schooling and later retirement but have an ambiguous effect on growth. The relationship between the growth of population and per-capita growth is hump-shaped. Increases in longevity can be responsible for a switch from a no-growth regime to a sustained growth regime and for a positive relationship between fertility and growth to vanish.

JEL classification numbers:

Keywords: Human capital, longevity, fertility, growth, schooling.

*We are grateful to Henri Sneessens for a helpful discussion.

IRES, Université catholique de Louvain, Place Montesquieu 3, B-1348 Louvain-la-Neuve, Belgium. E-mail:boucekkine@ires.ucl.ac.be.

National Fund for Scientific Research and IRES, Université catholique de Louvain, Place Montesquieu 3, B-1348 Louvain-la-Neuve, Belgium. E-mail:delacroix@ires.ucl.ac.be. The financial support of the PAI programme P4/01 is gratefully acknowledged.

$FEDEA, c/ Jorge Juan 46, E-28001 Madrid, Spain. E-mail:licandro@fedea.es.

Introduction

The relationship between demographic trends and economics is a challenging area of research. Life expectancy at birth was below 50 years at the beginning of the century in Western Europe and it is it now reaching 80. The importance of the economic growth process in fostering such improvements has been stressed (Fogel, 1994), but the effect of past and future demographic trends on growth remain largely unexplored. One likely channel through which demographics affect growth is the size and quality of the work force. The aim of this paper is to study the effect of key demographic parameters on human capital accumulation and economic growth.

Figure 1: The decline in mortality – France – Survival laws

A first reason to study the effect of demographic trends on growth appears when we consider how big the demographic changes have been in the last centuries. Key statistics on mortality are presented in table 1 (built from Vallin (1991) and from Challier and Michel (1996)).

Crude death rate (deaths in % of population) started to decline in France and United-Kingdom during the eighteenth century. At the beginning of the twentieth century, the decline in crude death rate was slowed down by the progressive ageing of the population. Life expectancy at birth continued to increase (and there is no reason to believe that this increase will not continue in the future.) Much of the decline in mortality stems from a dramatic reduction in infant mortality during the early part of the period. In the twentieth century, the increasing distance between the survival law in 1899 and the one in 1969 for the ages 20–60 indicates that adults' mortality has been decreased substantially. Moreover, Erlich and Chuma (1990) report that the rate of increases of the life expectancies of the relatively older cohorts has been larger than that of the younger ones (for the USA).

Despite this huge drop in mortality, the growth rate of population, which is equal to the fertility rate minus the death rate plus net migrations, has not been that affected in

Western Europe, see Table 1 (data from Maddison (1995)). Indeed, empirical studies show that the fall in mortality rates is eventually followed by a steady and continuous decline in fertility (Erlich and Lui, 1997). In most parts of the European continent, fertility has now reached or even fallen below the replacement level. The future scenario of a zero population growth is now considered seriously.

Finally, one remarkable feature of the last two centuries is the continuous increase in the time spent at school.

Table 1: Long-run data

The need to model the vintage structure of the population provides the second motivation of the paper. Indeed, even if the empirical debate on demographics and growth is highly controversial, it stresses the importance of age-specific population characteristics (death rate, activity rate, education ...) to understand economic growth. Several authors have tried to disentangle empirically the effects of demographic changes on growth. In their empirical study of the determinants of growth, Barro and Sala-I-Martin (1995) stress the importance of life expectancy: a 13 year increase in life-expectancy is estimated to raise the annual growth rate by 1.4 percentage points. The authors think that life expectancy has such a strong, positive relation with growth as it proxies for features other than good health reflecting desirable performance of a society. Concerning the effect of population growth on per capita GDP growth, the initial view was that a high rate of population growth can not supported by a corresponding increase in investment, thus lowering growth per capita (Coale and Hoover, 1958). This negative relationship stems for population control policies. In the more recent literature, there is in general non significant correlation in cross-country studies and a slightly positive causality from population to growth in time-series analyzes (see e.g. Kapuria-Foreman (1995)). Considering both cross-section and time series data (Kelley and Schmidt, 1995), the impact of population growth has changed over time: it is not significant in the sixties and the seventies but becomes large and significantly negative in the eighties. Moreover, the empirical evidence in Kelley and Schmidt (1995) suggests that the effect of population growth varies with the level of economic development and can be positive for some developed countries. As far as mortality is concerned, Blanchet (1988) reports that reductions in crude death rate stimulate per capital economic growth. Kelley and Schmidt (1995) stand out the ambiguous effect of crude death rates. Indeed, growth is slowed by the deaths of the workers but can be enhanced by the deaths of dependents. They provide several elements showing the importance of age-specific mortality rates.

One central result of Kelley and Schmidt (1995) is that a decrease in the crude death rate increases economic growth, especially in the least developed countries, where mortality reduction is concentrated in the younger and working ages. This is less true when those gains occur in the retired cohort. The importance of the characteristics of the population by age to understand the effect of demographic shocks on growth is comforted by other recent studies.

One important channel through which demographic trends affect growth is obviously the size and quality of the labor force, which are both determined by schooling and retirement decisions of agents. As stressed by the study of Ram and Schultz (1979) for India, improvements in health and longevity conditions induce an increase in investment in schooling. As a consequence, population growth caused by an increase in longevity can be favorable to economic growth.

A key element is that different generations have different learning experiences and that the aggregate stock of human capital is built from the human capital of the different generations. The aforementioned various empirical studies stress the role of the age-structure in understanding the effect of demographic variables on growth. The most important characteristic of a growth theory designed to shed light on these issues is clearly to capture the vintage nature of human capital.

Accordingly, we develop an overlapping generations model à la Blanchard (1985), in which the aggregate human capital is built from a sequence of generation-specific human capitals. To stress the specific-role of the different cohorts, we assume that agents optimally choose the length of the three following activities: learning, working and being retired. Each individual has thus to decide on the length of time devoted to schooling before starting to work and on the retirement age.

Another desirable characteristic of our model is that it includes a relatively rich description of demographics and a realistic but still tractable survival law. This realistic survival law and the fact that the probability of death is increasing with age clearly represent an improvement with respect to previous papers on the subject (Galor and Stark (1992), Zhang, Wang and Lee (1998), Kalemli-Ozcan, Ryder and Weil (1998) and de la Croix and Licandro (1999)).

1 The model

Time is continuous and the equilibrium is evaluated from time 0 onward. At each point in time there is a continuum of generations indexed by the date at which they are born, t. There is a unique material good, the price of which is normalized to 1, that can be used for consumption. This good is produced from a technology using labor as the only input.

For instance, Crenshaw, Ameen and Christenson (1997) regress economic growth rates on age-specific population growth rates and conclude that “economies lie fallow during baby booms, but grow rapidly as boomers age and take up their economic roles in societies.”

Each vintage of human capital is related to a particular generation. This assumption is different from Chari and Hopenhayn (1991), where individuals of the same generation are associated to different vintages of human capital.

Demographic structure

The set of individuals born in t is an interval of measure , with and . Each individual has an uncertain lifetime. The unconditional probability for an individual born in t of being alive in z is given by the function :

\[m (z - t) = \frac {e ^ {- \beta (z - t)} - \alpha}{1 - \alpha}\tag{1}\]

with either or . This is a more general formulation than the one of Blanchard (1985) in which and . The case corresponds to a concave survival law which brings the model closer to the empirical mortality tables. We thus assume in the sequel.

Figure 2: Survival laws

Equation (1) implies that there is an upper bound on longevity. Indeed the maximum age A that an individual can reach is obtained by setting , leading to

\[A = \frac {- \log (\alpha)}{\beta}.\tag{2}\]

Moreover, the unconditional life expectancy is

\[\Lambda = \int_ {t} ^ {t + \mathrm{A}} (z - t) \frac {\beta e ^ {- \beta (z - t)}}{1 - \alpha} \mathrm{d} z = \frac {1}{\beta} + \frac {\alpha \log (\alpha)}{(1 - \alpha) \beta}\tag{3}\]

and we retrieve (Blanchard, 1985) when .

From equation (3) a rise in life expectancy can arise either through an increase in the parameter or through an increase in the parameter . As illustrated in figure 3, these two parameter shifts do not lead to the same changes in the survival probabilities. When increases, the improvement in life expectancy relies more on reducing death rates for young agents. When increases, the old agents benefit the most from the drop in death rates.

Denoting the set of individuals born in t still living in z, the measure of this set is

\[\mu (V _ {t, z}) = \zeta e ^ {n t} m (z - t) \text { for } z \in [ t, t + A ].\tag{4}\]

Although each agent is uncertain about the time of his death, the measure of each generation declines deterministically through time.

The size of the population at time t is given by

\[\int_ {t - \mathrm{A}} ^ {t} \zeta e ^ {n z} m (t - z) \mathrm{d} z = \zeta e ^ {n t} \kappa \quad \mathrm{with} \quad \kappa = \frac {n (1 - \alpha) - \alpha \beta (1 - \alpha^ {n / \beta})}{n (1 - \alpha) (n + \beta)}.\]

Computing the ratio of the new cohort to total population we find that the fertility rate is equal to . Hence, given the two parameters of the survival law , we can fix n and deduce the fertility rate , or alternatively, fix , from which we deduce the growth rate of population n. Notice that , which stems for the positive relationship between fertility and population growth.

Contrary to the literature in which a simpler demographic structure is made endogenous (Erlich and Chuma (1990), Strulik (1996), Galor and Weil (1998), Blackburn and Cipriani (1998), and Mateos (1998)), we consider demographics as exogenous and study the effects of these parameters on growth.

The households' problem

An individual born at time , derives the following expected utility:

\[\int_ {t} ^ {t + \mathrm{A}} c (t, z) m (z - t) \mathrm{d} z - \frac {\bar {H} (t)}{\phi} \int_ {t} ^ {t + \mathrm{P} (t)} (z - t) m (z - t) \mathrm{d} z,\tag{5}\]

is consumption of generation t member at time z. To get closed-form solutions of the model, the utility drawn from consumption is assumed linear. The pure time preference parameter is zero. The negative term on the right hand side is the disutility from studying and working until the retirement age . This disutility is proportional to the age of the agent, reflecting that the hardness of work increases with age. It is also proportional to the per capita stock of human capital , which captures the average level of knowledge of the society. is a parameter inversely related to the disutility of work.

By continuity, all the results derived below hold when we allow for a small subjective discount rate.

We assume the existence of complete markets. All lending and borrowing contracts between generations are insured by competitive life insurance companies. The intertemporal budget constraint of the agent born in t is:

\[\int_ {t} ^ {t + \mathrm{A}} c (t, z) R (t, z) \mathrm{d} z = \int_ {t + \mathrm{T} (t)} ^ {t + \mathrm{P} (t)} \omega (t, z) R (t, z) \mathrm{d} z.\tag{6}\]

is the contingent price paid by a member of generation t to receive one unit of the physical good at time z in case he is still alive. By definition, . The left-hand side is the actual cost of the contingent life-cycle consumptions. The right-hand-side is the actual value of contingent earnings. The agent goes to school until time (if he is still alive). After this education period, he earns a spot wage .

Spot wages depend on individual human capital, :

\[\omega (t, z) = h (t) w (z),\tag{7}\]

where is the wage per unit of human capital. The individual's human capital is a function of the time spent at school and of the average human capital of the society at birth:

\[h (t) = \mu \bar {H} (t) \mathrm{T} (t).\tag{8}\]

is a productivity parameter. The presence of introduces the typical externality (see Lucas (1988) and Azariadis and Drazen (1990)) which relates positively the future quality of the agent to the cultural ambiance of the society (through for instance the quality of the school).

The problem of the agent born in t is to select a consumption contingent plan, the duration of his education and the retirement age in order to maximize his expected utility subject to his inter-temporal budget constraint, to the constraint , and given the per capita human capital and the sequence of contingent wages and prices. We accordingly build the following Lagrangean:

\[\begin{array}{c} \int_ {t} ^ {t + \mathrm{A}} c (t, z) m (z - t) \mathrm{d} z - \frac {\bar {H} (t)}{\phi} \int_ {t} ^ {t + \mathrm{P} (t)} (z - t) m (z - t) \mathrm{d} z \\ - \lambda (t) \left[ \int_ {t} ^ {t + \mathrm{A}} c (t, z) R (t, z) \mathrm{d} z - \mu \mathrm{T} (t) \bar {H} (t) \int_ {t + \mathrm{T} (t)} ^ {t + \mathrm{P} (t)} w (z) R (t, z) \mathrm{d} z \right] - \nu (t) [ \mathrm{P} (t) - \mathrm{A} ], \end{array}\]

where is the Lagrange multiplier of the inter-temporal budget constraint and is the Kuhn-Tucker multiplier associated to the inequality constraint . The first order necessary conditions for a maximum are

\[m (z - t) - \lambda (t) R (t, z) = 0\tag{9}\]

\[\bar {H} (t) \left[ \frac {\mathrm{P} (t)}{\phi} m (\mathrm{P} (t)) - \mathrm{T} (t) \lambda (t) R (t, t + \mathrm{P} (t)) \mu w (t + \mathrm{P} (t)) \right] - \nu (t) = 0\tag{10}\]

We do not explicitly introduce obsolescence of , although this would not change the results as long as the individual's human capital never becomes fully depreciated.

\[\nu (t) \geqslant 0, \quad \mathrm{P} (t) \leqslant \mathrm{A}, \quad \nu (t) (\mathrm{P} (t) - \mathrm{A}) = 0\tag{11}\]

\[\int_ {t + \mathrm{T} (t)} ^ {t + \mathrm{P} (t)} w (z) R (t, z) \mathrm{d} z - \mathrm{T} (t) R (t, t + \mathrm{T} (t)) w (t + \mathrm{T} (t)) = 0.\tag{12}\]

Since and , we obtain from equation (9)

\[\lambda (t) = 1.\tag{13}\]

Using this, we may rewrite contingent prices as

\[R (t, z) = m (z - t).\tag{14}\]

Equation (14) reflects that, with a linear utility, contingent prices are just equal to the survival probabilities. The first order necessary condition for the retirement age is given by (10). The first term between brackets is the marginal utility cost of postponing retirement and the second term is the marginal utility of additional labor income. At an interior solution the two should be equal. Using (11), (13) and (14) and solving for yields

\[\mathrm{P} (t) = \min \left[ \mathrm{T} (t) \mu \phi w (t + \mathrm{P} (t)), \mathrm{A} \right].\tag{15}\]

The first order necessary condition for the schooling time is (12). The first term is the marginal gain of increasing the time spent at school and the second is the marginal cost, i.e. the loss in wage income if the entry on the job market is delayed.

The firms' problem

The production function is assumed to allow one unit of efficient labor to be transformed into one unit of good:

\[Y (t) = H (t).\tag{16}\]

Hence, firms employ the whole labor force to produce as long as the wage per unit of human capital is lower or equal to one. The equilibrium in the labor market thus implies that the wage per unit of human capital is constant through time and equal to one:

\[w (t) = 1.\tag{17}\]

Using this result in equations (12) and (15), it appears that, at equilibrium, and are constant through time and do not depend on agent's date of birth. For this reason, we drop the time argument for these two variables in the following.

2 Equilibrium schooling and retirement decisions

In this section we focus on the derivation of the schooling optimal solution. We first define

\[\eta = \mu \phi\]

which can be seen as the ratio of the productivity of schooling to its cost in terms of disutility. We prove that the existence of an interior solution depends crucially on the value of the parameter . Using (15) and (17), the interior retirement solution is

\[\mathrm{P} = \eta \mathrm{T}\]

which requires . Hence, the productivity of schooling should be greater than the corresponding disutility of time. Using (12) and (14) and the equilibrium relation (17) the interior solution for T should satisfy:

\[\mathrm{T} m (\mathrm{T}) = \int_ {\mathrm{T}} ^ {\eta^ {\mathrm{T}}} m (t) \mathrm{d} t,\tag{18}\]

under the restriction P < A, which is equivalent to

\[0 < \mathrm{T} < \frac {- \ln (\alpha)}{\beta \eta} \equiv \mathrm{T} _ {m a x} (\eta).\tag{19}\]

It should be noted that equation (18) is checked by the trivial root T = 0. The welfare comparison between this trivial solution and the interior solution is presented at the end of this section. Hereafter, we define . represents the discounted flow of wages for one unit of human capital as a function of the time spent at school.

Optimal schooling and the productivity of education

Proposition 1 The household problem has no interior solution if .

Proof: Since an interior solution should be lower than and the survival law is strictly positive and strictly decreasing on the open interval , we have:

\[(\eta - 1) x m (\eta x) < G (x) = \int_ {x} ^ {\eta x} m (t) \mathrm{d} t < (\eta - 1) x m (x),\]

for all . Consequently, if , we get the strict inequality: , , which means that the equation has no solution on the latter interval.

Proposition 1 implies that the optimal period of work must be greater than the optimal period of schooling. This can be explained using equation (18): since the life-cycle earnings profile is constant, the cost from increasing the schooling time is just equal to the time spent at school. The benefit is the sum of future wages ( ) discounted by their respective survival probabilities. Since the survival law is decreasing, we must work a period longer than the schooling time to meet this optimality condition. For this reason, we need to have an interior solution. This property does not depend on the particular survival law and should hold for any survival law, which must be decreasing. To assume is then natural. Notice also that, to produce one unit of good we first need to go to school and then to work. For this reason, the productivity of the education technology, , must be larger than the disutility of both schooling and working, say two times .

Does an interior solution exists if ? The three following lemmas provide the existence and uniqueness results in this case.

η < 2

However, the assumption of constant life-cycle earnings profile is in somewhat crucial. If the earning profile were increasing, with a steep slope, the discounted income could also be increasing, in which case could be consistent with an interior solution.

Lemma 1 There exists a unique number such that:

(i) and solves equation (18),

(ii) there exists at least one interior solution if .

Proof: see appendix.

Lemma 2 Equation (18) has a unique solution on when .

Proof: see appendix.

Lemma 3 An interior solution exists and is unique if and only if .

Proof: see appendix.

Lemma 1 has defined a threshold on below which the disutility of work is so strong and/or the productivity is so weak that it is optimal to never go retired. In order to have an interior retirement period, we have assumed that the disutility of labor is increasing in age, to take into account that the hardness of work is increasing when people approach their maximum age A. In particular, we have assumed that it is linearly increasing. However, linearity has the implication that a very high productivity of education can lead individuals to postpone retirement forever. Alternatively, we could assume that the disutility function is such that disutility converges to infinity when the age approaches A, in which case would be infinite and the solution for the retirement age should be interior for any . This assumption should be consistent with the deterioration of health for relatively old persons. To assume smaller than seems then natural.

We now turn to study the occurrence of corner solutions. What happens if or if ? The latter case is less trivial and we will study it in details. We should note here that when , solves (18). In this case, the retirement timing decision, evaluated according to the interior solution rule, yields . This makes clear that at , the “interior” regime reaches the corner solution regime T < P = A, which can be defined by:

\[\mathrm{T} m (\mathrm{T}) = \int_ {\mathrm{T}} ^ {\mathrm{A}} m (t) \mathrm{d} t,\tag{20}\]

subject to 0 < T < A. Note that equation (20) holds when T = A. Since this implies T = P = A, we disregard this solution. The following lemma establishes the appropriate existence and uniqueness properties for this corner regime.

Lemma 4 Equation (20) has a unique root strictly comprised between 0 and A.

Proof: see appendix.

Notice that, apart from the survival law, disutility is almost quadratic in P, which allows for linearity in the P-T relation, as represented by equation (15).

Corollary 1 The unique solution of equation (20) strictly comprised between 0 and A is exactly as defined in Lemma 1.

The proof is trivial. By construction, solves (20). Since , and since by Lemma 4, (20) admits a unique positive root strictly lower than A, it follows that is exactly this solution. We are now able to state a general proposition describing the conditions under which the different regimes occur.

Proposition 2 (i) There exists a unique interior solution to (18), and if and only if .

\[(i i) \text { If } \eta \geqslant \eta^ {\star}, \mathrm{T} ^ {\star} = \mathrm{T} _ {m a x} (\eta^ {\star}) \text { and } \mathrm{P} ^ {\star} = \mathrm{A}.\]

\[(i i i) I f 1 < \eta \leqslant 2, \mathrm{T} ^ {\star} = \mathrm{P} ^ {\star} = 0.\]

Proof: Given Proposition 1 and Lemmas 1 to 4, we should show that the solution T = P = 0 is always dominated by the interior solution when it exists for the proof to be completed. This is achieved by comparing the indirect utility function at the two solutions. For , the indirect utility is

\[\mathcal {V} (\mathrm{T}) \propto \eta \int_ {\mathrm{T}} ^ {\eta \mathrm{T}} \mathrm{T} \frac {e ^ {- \beta t} - \alpha}{1 - \alpha} \mathrm{d} t - \int_ {0} ^ {\eta \mathrm{T}} t \frac {e ^ {- \beta t} - \alpha}{1 - \alpha} \mathrm{d} t\]

Simple but tedious computations leads to

\[\begin{array}{r c l} \mathcal {V} ^ {\prime} (\mathrm{T}) & = & \frac {\eta \left(- e ^ {- \beta \mathrm{T}} + e ^ {- \beta \eta \mathrm{T}} + \mathrm{T} \beta (e ^ {- \beta \mathrm{T}} + \alpha (\eta - 2))\right)}{\beta (\alpha - 1)} \\ \mathcal {V} ^ {\prime \prime} (\mathrm{T}) & = & \frac {\eta \left((2 - \beta \mathrm{T}) e ^ {- \beta \mathrm{T}} - \eta e ^ {- \beta \eta \mathrm{T}} + \alpha (\eta - 2))\right)}{\alpha - 1} \\ \lim _ {\mathrm{T} \to 0} \mathcal {V} ^ {\prime} (\mathrm{T}) & = & 0 \\ \lim _ {\mathrm{T} \to 0} \mathcal {V} ^ {\prime \prime} (\mathrm{T}) & = & (\eta - 2) \eta > 0 \end{array}\]

from which we deduce that V has a horizontal slope and is concave at the right of T = 0. The solution T = 0 is thus dominated by a slightly greater T and is not the optimum.

Proposition 2 is illustrated in Figure 4 for specific values of the parameters. The values are: and which leads to a life expectancy of 73 years and a maximum age of 115 years. T, P and A are plotted for different values of starting from 1. The optimal schooling length (solid line), and the optimal retirement age (dotted line) are zero for and they increase with as long as . When the retirement age hits the maximum age (at ), T and P remains constant at and .

Life expectancy and optimal schooling

A key property of our model is that a decrease in the death rates, or equivalently, an increase in life expectancy induces individuals to study more. This prediction is consistent with the joint observation of a large increase in both life expectancy and years of schooling during the last century.

T,P,A

Figure 4: Optimal schooling and retirement as a function of Proposition 3 A rise in life expectancy increases the optimal length of schooling.

Proof:

- For the interior solution, i.e. when , we integrate equation (18), which leads to

\[0 = e ^ {- \beta \eta T} - e ^ {- \beta T} + T (e ^ {- \beta T} - 2 \alpha) \beta + \alpha \beta \eta T\]

Using the implicit function theorem, the partial derivatives are:

\[\frac {\mathrm{dT}}{\mathrm{d} \beta} = \frac {- \mathrm{T}}{\beta} > 0 \quad \mathrm{and} \quad \frac {\mathrm{dT}}{\mathrm{d} \alpha} = \frac {\mathrm{T} (\eta - 2)}{e ^ {- \eta \beta \mathrm{T}} - (2 - \beta \mathrm{T}) e ^ {- \beta \mathrm{T}} - \alpha (\eta - 2)} > 0\]

where the denominator of the last expression is shown to be positive in the proof of Lemma 2. (see the appendix)

- In the case where , and the derivatives are

\[\frac {\mathrm{dT}}{\mathrm{d} \beta} = \frac {\log (\alpha)}{\beta^ {2} \eta^ {\star}} > 0 \quad \mathrm{and} \quad \frac {\mathrm{dT}}{\mathrm{d} \alpha} = \frac {- 1}{\alpha \beta \eta^ {\star}} > 0\]

From these expressions we conclude that an increase in or in always lead the agents to study longer.

A rise in life expectancy makes it more profitable to study longer and to retire later, since it makes feasible to work for a longer period. The positive effect of longevity on schooling does not depend here on the assumptions made on fertility, as (or n) does not intervene in the determination of T. Finally, the following corollary establishes that the elasticity of T to is equal to 1.

In a more general set-up where the marginal productivity of labor is decreasing, changes in fertility could affect the expected wage profile via the labor market equilibrium, producing some indirect effects on schooling and retirement decisions.

Corollary 2 For any , there exists a constant independent of such that .

Proof: From the expressions of the derivatives terms , we can infer that the product is independent of unless the threshold value depends on . But is computed from equation (18) with as defined in Section 2, and one can easily check that the terms in vanish from such an equation so that does not depend on . In contrast, does depend on .

Using (3) we note that the share of expected life devoted to schooling is independent of :

\[\frac {\mathrm{T}}{\Lambda} = \frac {- A _ {1} (1 - \alpha)}{1 - \alpha + \alpha \log \alpha}.\tag{21}\]

Hence, although increases in lead the agents to study more, they will keep the share of schooling in their (expected) life unchanged. As a consequence, changes in the survival law caused by shifts in are useless to model the effect of changes in the ratio T/ on economic growth. However, shifts in do affect the ratio T/ (notice that depends also on ) making the proposed survival law much more interesting than the Poisson law generally used in the literature. Section “From Malthus to Solow” gives a nice example of it.

As far as retirement is concerned, we retrieve in our propositions the typical results of partial equilibrium analysis: “The comparative statics suggested that you should retire early to the extent that your circumstances involve high disutility of effort, low wages, low life expectancy ...” (Kingston, 1997).

3 The balanced growth path

Consistently with the technology (16), the productive aggregate human capital stock is computed from the capital stock of all generations currently at work:

\[H (t) = \int_ {t - \overline {{\mathrm{P}}} (t)} ^ {t - \overline {{\mathrm{T}}} (t)} \zeta e ^ {n z} m (t - z) h (z) \mathrm{d} z,\tag{22}\]

where is the last generation that entered the job market at t and is the last generation that retired at t. Function evaluated at birth gives the interval of time spent at school for any generation. Then, . Given that for all , then for all . For simplicity we assume that initial conditions are such that this also holds for all . The same reasoning applies to for all .

The average human capital at the root of the externality (8) is obtained by dividing the aggregate human capital by the size of the population:

\[\bar {H} (t) = \frac {H (t)}{\kappa e ^ {n t} \zeta}.\tag{23}\]

This assumption presents three advantages. First, it is the simplest way to eliminate scale effects. This allows to define a balanced growth path even in the presence of positive population growth. Second, is the average human capital of a worker times the activity rate; the strength of the externality depends thus on the density of workers in the population, without depending on the size of the economy. Indeed, for a given scale of the economy, the larger is the active population, the stronger is the externality. Third, this formulation amounts to link the externality to the output per capita (which would have been the adequate assumption in a learning-by-doing set-up).

The dynamics of human capital accumulation can be obtained by combining (8) with (22) and (23):

\[H (t) = \int_ {t - \mathrm{P}} ^ {t - \mathrm{T}} m (t - z) \frac {\mu \mathrm{T} H (z)}{\kappa} \mathrm{d} z\tag{24}\]

To evaluate , for , we need to know an entire span of initial conditions for , from -P to 0. Equation (24) is a delayed integral equation, with delays T and P. We analyze the balanced growth path in this section and postpone the study of the dynamics to section 4.

Existence and uniqueness of the balanced growth path

Using (24), there exists a steady state growth path , with H and two constants, H nonzero, if and only if the following integral equation holds:

\[\frac {\mu \mathrm{T}}{\kappa} \int_ {\mathrm{T}} ^ {\mathrm{P}} m (z) e ^ {- \gamma z} \mathrm{d} z = 1.\tag{25}\]

The following preliminary results are then easy to establish.

Lemma 5 From (25), we can conclude that

(i) if , is not a solution unless

\[\frac {\mu \mathrm{T}}{\kappa} \int_ {\mathrm{T}} ^ {\mathrm{P}} m (z) d z = 1,\]

(ii) if a solution exists, it should be unique.

The proof is trivial. Property (i) is obtained by setting in equation (25). The uniqueness result (ii) can be proved trivially by contradiction given that the survival law is continuous and strictly positive in the domain comprised between T and P.

Hereafter, we assume that , and we look for nonzero long run growth rates solutions. By performing the integration in (25), the existence of solutions turns out to be a fixed-point problem with:

\[Q (x) \equiv \frac {\mu \mathrm{T}}{\kappa (\alpha - 1)} \left(\frac {x}{x + \beta} (e ^ {- (x + \beta) \mathrm{P}} - e ^ {- (x + \beta) \mathrm{T}}) + \alpha (e ^ {- x \mathrm{T}} - e ^ {- x \mathrm{P}})\right).\]

Of course, x = 0 is a fixed point of , but it is a solution to (25) only under the condition stated in point (i) of Lemma 5. It is now possible to state the following existence-uniqueness proposition of steady state growth paths:

Proposition 4 Assume that . Denote by , with . Then:

(i) If , there exists a unique strictly negative growth rate.

(ii) If , we can define . More importantly, there exists a unique strictly positive (Resp. negative) long run growth rate if and only if (Resp. ). If , .

Proof: One can check easily that

\[\lim _ {x \to + \infty} Q (x) = 0, \quad \lim _ {x \to - \infty} Q (x) = - \infty \quad \mathrm{and} \quad \lim _ {x \to - \infty} {\frac {Q (x)}{x}} = + \infty .\]

Since , a strictly positive (Resp. negative) fixed-point exists if (Resp. ). Uniqueness follows from Lemma 5, property (ii). Note that if , zero is the unique solution for the growth rate. So to conclude, we have to compute . Since is the left-hand-side of (25) multiplied by , we can easily show that

\[Q ^ {\prime} (0) = \frac {\mu \mathrm{T}}{\kappa} \int_ {\mathrm{T}} ^ {\mathrm{P}} m (z) \mathrm{d} z\]

From (18) and , we get . The rest of the proposition is then quite trivially derived. Indeed, it is shown in appendix that function is strictly increasing on the interval so that is the maximum of on . We know that once excluded the corner solution , the optimal T values are equal or lower than for by Lemma 3 and Proposition 2. So optimal T always belongs to the interval . The conditions involving and are then obvious. Note that the case recovers exactly the case studied in Lemma 5, property i, in which: is a solution.

The first part of Proposition 4 states that, at given and , a too low fertility rate or a too low productivity of education lead inevitably to a negative growth rate. Low fertility is bad for growth as we reason here about absolute growth and not about per capita growth. A deeper analysis of the effect of fertility on per-capita growth is provided below. Low productivity of education is also bad for growth for obvious reasons. The second part of Proposition 4 defines a threshold on schooling above which growth is positive. This threshold depends on fertility. As the private decision on schooling does not depend on fertility and as the function is increasing, the threshold is less binding if fertility is high. Let us now go deeper into the study of the effect of demographic variables on growth.

Life expectancy and growth

We first consider the effect of life-expectancy on per-capita growth. There are two different ways to analyze this issue. A first one is to consider that the growth rate of population is constant, requiring that fertility adjusts to mortality changes. A second one is to reason at given fertility rate; we must then take into account that reducing the mortality rates generates an increase in the population growth rate. In the history of developed countries in the two last centuries, we do not observe that the huge increase in longevity has been followed by an increase in the population growth rate. The first case seems thus more adapted to developed countries, while the second can be suited to study developing countries during the twentieth century. We study both cases in this section.

A rise in life expectancy can come from an increase in or in . As explained in sections “Demographic structure” and “Life expectancy and schooling”, changes in these parameters have different implications on the way life expectancy improves. For simplicity, this section is devoted to a rise in life expectancy derived from an increase in , but some discussion is devoted to the specificities of changing . Moreover, even for changes in , it is very hard to provide a general treatment for the role of life expectancy on per-capita growth. In order to have some insight, we first present the results of some numerical computations and then we provide some general results.

Figures 5 and 6 show the relation between life expectancy and growth. In both cases, changes in life expectancy are derived from shifts in , taking as constant. Given that life expectancy, fertility and population growth are related by definition, a change in should be simultaneously followed by a movement in the fertility rate or in the growth rate of population. Figure 5 let the fertility rate to adjust in order to have a constant population rate and Figure 6 supposes that the fertility rate stays constant, implying that the population growth rate adjusts.

Both figures show a hump-shaped relation between life expectancy and the per-capita growth rate. Starting from a situation in which agents have a short horizon (low ), a rise in first leads to an increase in the growth rate. After some point, the sign of the effect changes and a rise in life expectancy leads to a drop in . Intuitively, the total effect of an increase in life expectancy results from combining three factors: (a) agents die later on average, thus the depreciation rate of aggregate human capital decreases; (b) agents tend to study more because the expected flow of future wages has risen, and the human capital per capita increases; (c) the economy consists more of old agents who did their schooling a long time ago. The two first effects have a positive influence on the growth rate but the third effect has a negative influence. When life expectancy is relatively short, the weight of old population is relatively small and the third factor is less important than the others. However, when life expectancy is high there is a large population of old individuals, making the third factor to prevail.

In order to discuss the general validity of these numerical results, Proposition 5 provides a mathematical proof for the case where the growth rate of population stays constant for different values of .

Proposition 5 A rise in life expectancy through at given population growth has a positive effect on economic growth for low levels of life expectancy and a negative effect on economic growth for high levels of life expectancy.

Proof: see appendix.

73

57

Our reference calibration for Figure 5 is , , n = 0, and which leads to a life expectancy of 73 years, a maximum age of 115 years, an optimal schooling time of 27 years, an optimal retirement age of 57 years and a growth rate of 2% per year. We then let vary between -.027 and -.07 and reports the corresponding growth rate . For Figure 6, the fertility rate is fixed at 1/73, varies again between -.027 and -.07, and we report the corresponding growth rate as n is now endogenous.

γ

1/73,

β

6,

-.07

Figure 5: Growth and life expectancy at given n

Figure 6: Growth and life expectancy at given

Observe that as mentioned in the proof of Corollary 2, does not depend on but well on . That is why we have chosen the parameter to study the relationship between growth and life expectancy. The same analysis through is analytically intractable. However, numerical simulations show that under reasonable parameterizations, the relationship between growth and life expectancy as measured by is qualitatively similar to the one theoretically established between the growth rate and .

From an empirical point of view, we should thus observe that the effect of life expectancy on growth is positive for countries with a relatively low life expectancy, but could be negative in more advanced countries. The positive effect is clearly the one stressed by Kelley and Schmidt (1995) for less developed countries. In the same direction, Ram and Schultz (1979) write for India: “In a society where life is short, labor earns a pittance; (...) A turn towards a better future comes when the span of life increases. Incentives become worthwhile to acquire schooling, and the time spent at work becomes more productive. The stock of human capital in the form of better health and more schooling becomes larger, and it enhances the quality of labor. These investments in human capital matter. In Marshall's perceptive words, capital consists in a great part of knowledge and knowledge in the most powerful engine of production."

Finally, we can explore for our numerical example the difference between a rise in and a rise in . Let us keep our benchmark case , , , and look for the growth-maximizing life expectancy, say, . Keeping fixed at its current value, -.01472, is equal to 98.2363 and is obtained with . The maximum growth rate is then 2.14 %. Keeping fixed at 5.4365, is equal to 89.9421 and is obtained with . The maximum growth rate is then 2.05 %. In this latter case, the maximum growth is lower and is attained with a lower life expectancy. Hence, starting from our numerical benchmark, improvements in longevity are more “growth promoting” if they arise through a shift in which increases the weight of young generations.

Population growth and economic growth

As stressed in the introduction, the empirical evidence on the effect of population growth on economic growth is mixed. To analyze the effect of population growth on the steady state growth path, one can derive the following relation by differentiating both sides of equation (25) with respect to

\[\frac {\partial \gamma}{\partial n} = \frac {\int_ {0} ^ {\mathrm{A}} z m (z) e ^ {- n z} d z}{\mu^ {\mathrm{T}} \Gamma \int_ {\mathrm{T}} ^ {\mathrm{P}} z m (z) e ^ {- \gamma z} d z},\]

since can be rewritten as . Note that as expected. We would like to investigate the much less trivial relationship between the population growth n and the per capita growth rate (i.e. the position of with respect to 1). We will show that our model is able to generate an interior n-maximizer for the per capita growth rate, more precisely that the functional relation between these two variables is hump-shaped. This feature relies on the vintage nature of our economy. Indeed, when n is relatively low, the share of retired workers in the population is relatively high. Increasing n thus increases the active population and the growth rate. However, when n is very high, the students are the main group in the population. Lowering n would then increase the active population. In the two extreme cases, n very low and n very high, the size of the active population compared to total population is small which depresses growth. There is thus a level of n which maximizes the activity rate. There is in fact a growth-maximizing size of the active population. To this size corresponds a “growth-maximizing” demographic growth rate.

To prove the latter claim, observe first that given the function in , , is strictly positive in the interval [T, P] except eventually when , we have the

Recall that optimal T and P are independent of n.

To this “activity” effect, one should add a “composition” effect bearing on the composition of the labor force: higher population growth implies a higher proportion of young workers in the active population; as young workers have more human capital than old workers because they have been educated more recently. With this effect alone, population growth has a positive influence on economy growth.

strict inequalities:

\[\mathrm{T} \int_ {\mathrm{T}} ^ {\mathrm{P}} m (z) e ^ {- \gamma z} d z < \int_ {\mathrm{T}} ^ {\mathrm{P}} z m (z) e ^ {- \gamma z} d z < P \int_ {\mathrm{T}} ^ {\mathrm{P}} m (z) e ^ {- \gamma z} d z.\]

Using (25) and the integral definition of given just above, it is then possible to find strict upper and lower bounds for involving only . Concretely, after some trivial algebra, we find

\[\frac {\int_ {0} ^ {A} z m (z) e ^ {- n z} d z}{P \int_ {0} ^ {A} m (z) e ^ {- n z} d z} < \frac {\partial \gamma}{\partial n} < \frac {\int_ {0} ^ {A} z m (z) e ^ {- n z} d z}{T \int_ {0} ^ {A} m (z) e ^ {- n z} d z}.\tag{26}\]

Call (G) these inequalities. We can then state the following proposition:

Proposition 6 Assume that . There exists a population growth rate finite value such that the long run per capita growth rate of the economy reaches its (interior) maximum at .

Proof: To prove this claim, we prove the strict inequalities

\[\lim _ {n \to + \infty} \frac {\partial \gamma}{\partial n} < 1 \quad \text {and} \quad \lim _ {n \to - \infty} \frac {\partial \gamma}{\partial n} > 1.\]

To this end, we use the inequalities (26).

A sufficient condition for to be strictly lower than 1 is

\[\frac {\int_ {0} ^ {A} z m (z) e ^ {- n z} d z}{T \int_ {0} ^ {A} m (z) e ^ {- n z} d z} \leqslant 1.\]

This is in turn equivalent to . Since , and since is trivially strictly decreasing in , it follows that there exists a threshold such that for every . Note that can be written explicitly in terms of . Simple but very tedious computations show that decreases from when tends to to zero when tends to .

Similarly, we can obtain a sufficient condition for to be strictly greater than 1. In particular, we can define a function , with as defined in Lemma 1 when P=A. Using the left side inequality in (26), we know that a sufficient condition for is . For the same reasons as before, there exists a threshold such that for every .

We can now conclude. When tends to , tends to zero. Since , we can conclude that . When tends to , tends to . Since , with equality only when , it follows that .

Proposition 6 is illustrated by the numerical example displayed in figure 7 with the same calibration as above. n varies between -.05 to .05 and the figures show the corresponding fertility rate and growth rate per capita . In this example, the growth maximizing n is slightly positive. Positive economic growth arises only for n inside a closed interval. We also notice the strength of the effect of n.

From Malthus to Solow

In a recent paper Galor and Weil (1999) argue that one of the most significant challenges facing economists interested in growth and development is to model the long transition process from thousands of years of Malthusian stagnation through the demographic transition to modern growth. Several authors (see Hansen and Prescott (1999) and Doepke (1999)) have modelled this transition by assuming an exogenous differential in the rate of technological progress between the agricultural sector and the industrial sector. At some point in time, industry becomes profitable and the transition starts.

As an alternative to this assumption of differential technological progress, we notice that demographic and health changes started in the eighteenth century can be at the root of the transition. One key element to this transition can be the following: an initial exogenous drop in mortality has risen the expected rate of return to human capital investments, has led to more schooling and eventually to a higher rate of per capita growth. In terms of our model, this corresponds to a shift from a regime with almost no schooling and no growth to a regime with positive schooling and growth.

It is interesting to notice that such a shift has an impact on the relationship between fertility and growth. In a Malthusian economy, land is in fixed supply and technology is constant. Exogenous drops in the population level (e.g. the Black Death) are reflected in higher real wages, faster population growth and faster economic growth per capita. For this reason, there is a positive Malthusian relationship between income per capita and population growth. In the modern regime of growth, this relationship is no longer in place.

To illustrate how the shift in regime and the relationship between fertility and growth is affected by a rise in life expectancy we compare two different balanced growth path corresponding to two different set of parameters.

In the first economy, Solow, we have kept the same calibration as before. Life expectancy is 73 years and there is a balanced growth path with a positive growth rate. The relationship between growth per capita and population is represented by the solid line in figure 8 in the range . We next considered a Malthusian economy. The only difference with the Solow economy is in the value of . All the other technological and preference parameters are kept the same. The lower generates an economy with a lower life expectancy and a very slow pace of growth. The relationship between growth per capita and population growth is represented by the dotted line in Figure 8. The last column of the table gives the share of expected life devoted to education. As it was already clear from equation (21), this share rises with . The share of education goes from 33 % to 37 %. This increase is consistent with the empirical evidence over the two last centuries, and justifies to model the rise in life expectancy as a shift in rather than in .

From the inspection of the two plots in figure 8, we observe that the relationship between fertility and steady state growth around n = 0 is increasing in Malthus and almost flat in Solow. We conclude that a rise in the parameter leading the economy from Malthus to Solow can be responsible (provided that the balanced growth path is stable, see next section) for (1) the switch from a no-growth regime to a sustained growth regime and (2) the vanishing of the positive relationship between fertility and growth. An intuitive explanation of this result is as follows: At the growth-maximizing population growth, an increase in n should increase the weight of the student population by the same amount it reduces the weight of retired population in order to let the activity rate unchanged. An increase in induces agents to study more in percentage of life and to retire later, which rises the stationary ratio of the student population to the retired population for a given . The growth-maximizing level of population growth is thus lower in the Solow regime, in order to balance the size of young and old population. If n increases above this threshold, the share of students in population would further increase, which is already large in the modern economy, and this is not good for growth.

4 The dynamics of human capital

We now turn our attention to the analysis of the stability of the balanced growth path. The dynamics of human capital accumulation can be derived from the integral equation (24). This integral equation defines a functional fixed-point problem which can be transformed into a functional differential equation as follows.

Differentiating (24) with respect to time, we find the following DDE,

\[H ^ {\prime} (t) = \frac {\mu \mathrm{T}}{\kappa} \left[ m (\mathrm{T}) H (t - \mathrm{T}) - m (\mathrm{P}) H (t - \mathrm{P}) - \int_ {t - \mathrm{P}} ^ {t - \mathrm{T}} \beta \frac {e ^ {- \beta (t - z)}}{1 - \alpha} H (z) \mathrm{d} z \right].\tag{27}\]

At time t, individuals of generation t - T enter the job market with human capital . individuals of generation t - P quit the labor market and retire. The third term represents the workers who die before the retirement age.

To study the dynamics of this economy, we define detrended human capital as

\[\hat {H} (t) = H (t) e ^ {- \gamma t},\]

in which is the growth rate of the economy along the balanced path. The DDE (27) becomes, ,

\[\begin{array}{r c l} \hat {H} ^ {\prime} (t) & = & \frac {\mu \mathrm{T}}{\kappa (1 - \alpha)} \left[ (e ^ {- (\beta + \gamma) \mathrm{T}} - \alpha e ^ {- \gamma \mathrm{T}}) \hat {H} (t - \mathrm{T}) - (e ^ {- (\beta + \gamma) \mathrm{P}} - \alpha e ^ {- \gamma \mathrm{P}}) \hat {H} (t - \mathrm{P}) \right. \\ & & \left. - \beta \int_ {t - \mathrm{P}} ^ {t - \mathrm{T}} e ^ {- (\beta + \gamma) (t - z)} \hat {H} (z) \mathrm{d} z \right] - \gamma \hat {H} (t). \end{array}\tag{28}\]

To obtain a usable DDE, we differentiate once more equation (28) with respect to time and replace the integral by its value from (28). This leads to

\[\hat {H} ^ {\prime \prime} (t) = - \gamma (\beta + \gamma) \hat {H} (t) - (\beta + 2 \gamma) \hat {H} ^ {\prime} (t)\]

β,

Note that this does not happen when longevity increases through , illustrating the interest of taking a realistic survival law.

\[\begin{array}{l l} + & \frac {\mu \mathrm{T}}{(1 - \alpha) \kappa} \left[ (\gamma e ^ {- (\beta + \gamma) \mathrm{T}} - \alpha (\beta + \gamma) e ^ {- \gamma \mathrm{T}}) \hat {H} (t - \mathrm{T}) \right. \\ & \left. - (\gamma e ^ {- (\beta + \gamma) \mathrm{P}} - \alpha (\beta + \gamma) e ^ {- \gamma \mathrm{P}}) \hat {H} (t - \mathrm{P}) \right] \\ + & \frac {\mu \mathrm{T}}{(1 - \alpha) \kappa} \left[ (e ^ {- (\beta + \gamma) \mathrm{T}} - \alpha e ^ {- \gamma \mathrm{T}}) \hat {H} ^ {\prime} (t - \mathrm{T}) - (e ^ {- (\beta + \gamma) \mathrm{P}} - \alpha e ^ {- \gamma \mathrm{P}}) \hat {H} ^ {\prime} (t - \mathrm{P}) \right]. \end{array} \tag {29}\]

Stability analysis

The obtained functional differential equation is a scalar second order differential-difference equation with delayed derivative terms. Using standard auxiliary concatenation rules, it can be written as a first order bi-dimensional delay differential equation. No theorem is available to assess directly the asymptotic behavior of the solutions of this kind of dynamic system. So we study the stability of our system for the different parameterizations given in the preceding section. To this end, we use the algorithm designed by Engelborghs and Roose (1999) for numerical stability assessment of delay differential equations. Figure 9 gives the distribution of the rightmost roots of the characteristic equation of our delay differential equation as computed by the latter algorithm. All roots have strictly negative real part except the trivial zero root. Hence, if the economy is not on its balanced growth path before t = 0, an oscillatory transition to the latter should take place.

It is possible to assess the convergence speed of the economy using the computed spectra. The closer to zero is the smallest real part (in absolute value) of the nonzero

Such a theorem, called Hayes theorem as stated in Bellman and Cooke (1963), is only available for scalar and autonomous delay differential equations with a single delay, as e.g. in Boucekkine, del Rio and Licandro (1999). In particular, no direct stability theorem is available for delay differential systems with more than one delay since in this case the stability outcomes depend on the particular values of the delays. See Mahaffy, Joiner and Zak (1995).

Figure 10: Dynamic simulation of a drop in fertility in Solow computed eigenvalues, the slower is convergence. From Figure 9, one can conclude that the model “Malthus” converges more quickly to the corresponding steady state than the model “Solow”. A rise in life expectancy tends to decrease the convergence speed as it takes more time to replace the existing stock of human capital.

Dynamic simulation

An example of the transitory oscillatory dynamics is provided in Figure 10. The initial conditions are in which is the stationary growth rate of the Solow economy with . These parameters imply that and that the optimal time spent at school is . We assume that there is a permanent unexpected change in fertility at . The size of new generations after time zero is instead of for . The growth rate of total population changes thus slowly from to . The new stationary growth rate is , which is very close to the previous one in per-capita terms.

Considering the transition from a balanced growth path to the other, we observe that the change in fertility is first followed by a substantial increase in . Per capita growth rises from 2 % to 2.4 % during 27 years (the schooling length). During this period, the activity rate increases systematically as the weight of students decreases. After this period the generations born after t=0, which are smaller, start entering the labor market. This has a negative effect on the externality and dampers growth. After t=P=57, the old generations born before 0 are progressively substituted by smaller cohorts in the retired population, which has a positive effect. We then observe the replacement echoes which are typical to models with delays (see Boucekkine, Germain and Licandro (1997)).

Hence, the effect of a baby burst is first to increase per capita growth, since it reduces the weight of children in total population. However, when these cohorts enter the labor market, the per capita growth rate starts decreasing until this population group attains retirement. This cycle takes sixty years and it is reproduced in the future due to human capital replacement echoes.

An important consequence of this exercise is that the short-run dynamics of population shocks takes at least sixty years. Since time series employed in most empirical studies are shorter than sixty years, we must be careful in taking this evidence as stationary or in using stationary models to account for it.

Conclusion

We have proposed a simple endogenous growth model in which, generations after generations, the economy accumulates human capital. The production function is close to a vintage capital technology where each vintage is related to a generation of agents endowed with a specific human capital. The key arbitrage in this economy lies in the choice by agents on how long they remain at school before entering the labor market, and when they retire. The survival law is a central determinant of their decision.

The interest of the model relies in its demographic structure. The human capital of the society is build from the specific human capital of each generation. Demographics is described by three free parameters: two parameters of the survival law and the fertility rate. From these three parameters one can infer the maximum lifespan for the agents, their life expectancy at birth and the growth rate of population.

The bottom line of the paper is to analyze the effect of exogenous demographic changes on growth. We obtained the following results: Favorable shifts in the survival probabilities always induce longer schooling and later retirement. First, this does not necessarily imply that growth per capita is enhanced. The effect of life expectancy on growth is positive for low levels of longevity and becomes negative after some threshold. The negative effect comes from the ageing of the work force. The way longevity increases is important: improvements in longevity have different effects depending on whether the reduction in the death rates affects young or old agents. Second, the effect of population growth on per capita growth should be interpreted in the light of the vintage structure of aggregate human capital. There is a “growth-maximizing” population growth (or fertility rate), implying an adequate percentage of students and pensioners.

Our model gives an explanation to the transition from a Malthusian economy to a modern-growth economy on the sole basis of demographic shifts. First, an exogenous rise in longevity leads people to study longer and this can be responsible for a shift from a no-growth path to a balanced path with positive growth. Second, a positive relationship between fertility and growth (for realistic fertility rate) can be observed in the Malthusian economy but can vanish in the modern regime. This is because rises in fertility further increase the share of students in population, which is already large in the modern economy.

Finally, as far as the dynamics are concerned, the change in aggregate production are described by a scalar second order differential-difference equation with delayed derivatives terms. For the numerical examples used in the article we have shown that, if the economy is not on its balanced growth path before t = 0, an oscillatory transition to the latter should take place.

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Appendix

Proof of Lemma 1

Denote by . is continuously differentiable. It is easy to prove that if and , is strictly negative on an interval for a sufficiently small . Indeed, , and . In order to prove that an interior solution exists, it is sufficient to prove that since function is obviously continuously differentiable for .

Clearly, we have since the inequality reported in the proof of Proposition 1, , is obviously true for and . On the other hand, we have . Note that function takes the value zero at and is an increasing function of when . This implies that since here we take and . Since , the latter result implies that Lemma 1 is valid if additionally we can prove that function is strictly decreasing for . To this end, we have to study in some details the latter function.

We have . Note that since if , . The latter inequality holds because the left hand side of the inequality takes the value 0 at and is a strictly increasing function of when . Now, observe that is equivalent to . By proving that , we have just established that , . However, , which is strictly positive for all positive and for all . Since , we conclude that , and . This means that , . Given that and that , it follows that there exists (a unique) such that properties (i) and (ii) of Lemma 1 hold. Indeed, there exists a unique such that for , , and for . It follows that for . Since is strictly negative in the neighborhood of , we conclude that there exists at least an interior solution . Note that when , is a solution of equation (18).

Proof of Lemma 2

Using the notations of the proof of the previous Lemma, we aim at solving the equation . Trivially, . Since we know that is strictly negative in the neighborhood of x = 0, we can deduce that equation (18) has at least a solution on . We need more information about to conclude for uniqueness. We compute the first and second order derivatives of this function:

\[M ^ {\prime} (x) = \eta e ^ {- \eta \beta x} - (2 - \beta x) e ^ {- \beta x} - \alpha (\eta - 2),\]

and

\[M ^ {\prime \prime} (x) = - \eta^ {2} \beta e ^ {- \eta \beta x} - (\beta^ {2} x - 3 \beta) e ^ {- \beta x}.\]

Note that and that . Since is strictly negative in the neighborhood of x = 0 and becomes infinitely large if x grows, there exists c > 0 such that and . Observe that if has no other root, has itself a unique non-zero root. Indeed has a unique root because it turns out that function is strictly convex. Given the analytical form of , it is sufficient to prove the latter claim for . In this case, is equivalent to the inequality (with ): . Given that the latter inequality is obvious since for all x > 0, function is indeed strictly convex for any . The function is represented in figure 11.

Proof of Lemma 3

Uniqueness is a direct consequence of the previous lemma. To prove the lemma, we only need to prove that no interior solution exists if . By the proof of Lemma 1, we know that if . Since as mentioned before , has a root greater than . Since by the previous lemma, function only admits a unique strictly positive root, there cannot be another root belonging to the interval An interior solution cannot arise in this case.

Proof of Lemma 4

Note that equation (20) is equivalent to the equation: . Note that since we can show that the function in , is zero at and is strictly decreasing for all . Also note that . Moreover, , which implies that and . The last inequality implies that function is strictly positive in the (left) neighborhood of A. Since , function has at least one root comprised between 0 and A. Now notice that since and , there exists d, 0 < d < A, such that and . As in the proof of Lemma 2, it is sufficient to prove that has a unique root to conclude for uniqueness of the roots of . And as in the proof of Lemma 2, this is achieved here by showing that is strictly monotonic. Indeed, for all positive x.

Proof that the function is strictly increasing on the interval .

Given that , on is equivalent to the inequality on this interval. Note that this inequality holds in the neighborhood of x=0 since . Observe also that function is strictly decreasing for all positive x since . Indeed, the derivative of this function with respect to x is simply . Therefore, the Lemma holds if the inequality is fulfilled at . It could be checked that the latter required property is equivalent to the inequality . This is indeed true since function takes the value zero at x=1 and is strictly increasing for .

Proof of Proposition 5

Proposition 4 is most useful to demonstrate the desired property. Note that by the latter proposition, the sign of the growth rate depends on the position of with respect to , and on the value of the equilibrium schooling time T. We can write as

\[\kappa_ {1} = \frac {1}{\mu} \left(\frac {1}{n + \beta} - \frac {\alpha \beta (1 - \alpha^ {\frac {n}{\beta}})}{n (1 - \alpha) (n + \beta)}\right).\]

As for , it is equal by definition to where . Finally, the value of optimal schooling time T can be expressed as an explicit function of by Corollary 2. For any fixed , there exists such that . Since , we have the restriction .

We shall prove that a rise in life expectancy has a positive (Resp. negative) effect on economic growth for low (Resp. high) levels of life expectancy. To this end, we will prove that when tends to , and that when tends to 0. We will use the same mathematical strategy in all cases. First, we show that the ratio admits a limit either when tends to or to 0. Indeed, the fixed point problem giving rise to the long run growth rate value, , can be rewritten in terms of z once T and P are replaced by their -functional expressions:

\[- \beta^ {2} \frac {\kappa_ {1} (\alpha - 1)}{A _ {1}} = F _ {\eta} (z) \equiv \frac {e ^ {\eta A _ {1} (1 + z)} - e ^ {A _ {1} (1 + z)}}{1 + z} + \alpha \frac {e ^ {A _ {1} z} - e ^ {\eta A _ {1} z}}{z}.\tag{30}\]

Since the left hand side of the equation above admits a limit when tends either to or 0, z admits a limit in the latter cases if the continuous function , as defined in (30), is monotonic. The derivative can be written after rearranging terms as the difference

\[F _ {\eta} ^ {\prime} (z) = G (z, \eta A _ {1}) - G (z, A _ {1}),\]

with

\[G (z, \theta) = e ^ {\theta z} \left(- \frac {e ^ {\theta}}{(1 + z) ^ {2}} + \frac {\theta e ^ {\theta}}{1 + z} + \frac {\alpha}{z ^ {2}} - \frac {\alpha \theta}{z}\right).\]

We shall restrict to be positive and to fulfill since . It is now easy to prove that is monotonic: Indeed, function is decreasing with respect to for any provided . In effect, .

Once the limit of the ratio z computed, one can recover the limit of the growth rate since from the fixed-point equation, , we can infer that

\[\gamma = G _ {\eta} (z) = \frac {- A _ {1}}{\beta \kappa_ {1} (\alpha - 1)} \left(z \frac {e ^ {\eta A _ {1} (1 + z)} - e ^ {A _ {1} (1 + z)}}{1 + z} + \alpha \left(e ^ {A _ {1} z} - e ^ {\eta A _ {1} z}\right)\right).\tag{31}\]

We will use exactly this strategy for each case study. Let us give some details of the achieved computations. Consider the limit case tends to . Observe that in this case the parameter behaves as a function with , for sufficiently big (in absolute value). Since , it follows that for sufficiently big . By Proposition 4, (i), we conclude that for sufficiently big (but finite) . What is the limit value of From (30), we can deduce that the ratio tends to when tends to . Indeed, the left hand side of (30) tends to , which is only consistent with z going to in the right hand side of the equality. This implies, even without using equation (31) that tends to when tends to . Consequently, when tends to , which is the demanded property in this limit case.

When tends to 0, the limit of the left hand side of (30) depends on the sign of the population growth rate, , since the limit behavior of in this case depends on this parameter value. Let us consider the most realistic case to illustrate the final outcome of the computations, which is indeed independent of the sign of . In this case, behaves as a function for sufficiently small with for . Since , it follows that for sufficiently small . Moreover since the threshold value , as defined in Proposition 4 ii), tends to when tends to 0, we get for sufficiently small . Hence by Proposition 4 ii), for sufficiently small . Let us compute the limit value of . The left hand side of equation (30) tends to 0, which is only consistent with going to in the right hand side of the equality. Now using this result and equation (31), the limit value of turns out to be 0. Since for sufficiently small , the zero limit value implies that when tends to 0.

COLECCION RESUMENES

98-01: “Negociación colectiva, rentabilidad bursátil y estructura de capital en España”, Alejandro Inurrieta.

TEXTOS EXPRESS

99-02: “Economic implications of the demographic change in Spain: Call for research”, Namkee Ahn.

99-01: “Efectos macroeconómicos de la finalización de las ayudas comunitarias”, Simón Sosvilla-Rivero y José A. Herce.

DOCUMENTOS DE TRABAJO

2000-02: “Vintage human capital, demographic trends and endogenous growth”, Raouf Boucekkine, David de la Croix y Omar Licandro.

2000-01: "Vintage capital and the dynamics of the AK model", Raouf Boucekkine, Omar Licandro, Luis A. Puch y Fernando del Río.

99-21: “Los efectos macroeconómicos de la Agenda 2000”, Simón Sosvilla-Rivero y José A. Herce.

99-20: “Unemployment duration and workers’ wage aspirations in Spain”, Namkee Ahn y J. Ignacio García-Pérez.

99-19: “El patrón inversor de los establecimientos industriales de la Comunidad de Madrid”, Ana Goicolea, Omar Licandro y Reyes Maroto.

99-18: “Crecimiento óptimo, depreciación endógena y subutilización del capital”, Omar Licandro, Luis A. Puch y J. Ramón Ruiz Tamarit.

99-17: “Panel data and tourism demand. The case of Tenerife”, F. J. Ledesma-Rodríguez, M. Navarro-Ibáñez y J. V. Pérez-Rodríguez.

99-16: “Redistribution in the Spanish pension system: An approach to its life time effects”, Joan Gil y Guillem López-Casasnovas.

99-15: “The plutocratic bias in the CPI: Evidence from Spain”, Javier Ruiz Castillo, Eduardo Ley y Mario Izquierdo.

99-14: “Local responses to a global monetary policy: The regional structure of financial systems”, Juan J. de Lucio y Mario Izquierdo.

99-13: “The importance of the embodied question revisited”, Raouf Boucekkine, Fernando del Río y Omar Licandro.

99-12: “Convergencia en tasas de inflación en la Unión Europea”, Irene Olloqui y Simón Sosvilla-Rivero.

99-11: "Provisión óptima de suelo urbano para usos no lucrativos. El valor de las zonas verdes". Pere Riera.

99-10: “Liberalización del mercado del suelo. Rigideces y competencia en el sistema urbanístico español”, Pere Riera.

99-09: “A note on the changing relationship between fertility and female employment rates in developed countries”, Namkee Ahn y Pedro Mira.

99-08: “The future of pension systems in Europe. A reappraisal”, Michele Boldrin, Juan José Dolado, Juan F. Jimeno y Franco Peracchi.

99-07: "On the profitability of technical trading rules based on artificial neural networks: Evidence from the Madrid stock market", Fernando Fernández-Rodríguez, Christian González-Martel y Simón Sosvilla-Rivero.