LONGEVITY, SCHOOLING, AND GROWTH
Jacques Sadik1
[email protected]
tel 0097226718836 fax 0097226713385
The Hebrew University of Jerusalem
Abstract
This paper studies the transition from traditional learning to schooling and
explains the historical development of industrialized countries as well as why tropical
countries did not follow the same path. It also explains conditional convergence,
growth miracles, and leapfrogging. Since there is a positive feedback between growth
and longevity, development begins with slow growth but accelerates when longevity
increases. When longevity reaches the threshold where schooling is adopted, growth
jumps. In tropical climates this mechanism is blocked by the high prevalence of
diseases. Once this obstacle is removed by medical progress, a growth miracle can
occur.
Keywords: Growth, Convergence, Health, Longevity, Climate, Schooling
Jel classification O15, O33, O40
1
I am grateful to Oded Galor, Omer Moav, Chantal Sadik, Nathan Sussman, David
Weil, and Joseph Zeira, as well as to the participants in the EEA/ESM meeting in
Milan 2008 and those in the seminar at the Hebrew University of Jerusalem for their
numerous comments and suggestions.
1
1. Introduction
During most of human history, economic development was extremely slow;
people were poor by today’s standards and their life spans were short. Professions
were learned within the family or from a master. In the 18th century, during the first
Industrial Revolution, growth and longevity increased without a notable increase in
schooling, but had not yet reached their current levels. In the second half of the 19th
century, during a period often called the second Industrial Revolution, industrial
countries adopted and implemented technologies such as electricity and chemistry. In
addition, schooling became widely diffused in society and economic growth greatly
accelerated. By contrast, the growth of countries that did not make the transition to
schooling remained slow, whereas countries that experienced a delayed transition to
schooling caught up rapidly with the literate countries.
To explain these differences in development paths, we turn to empirical
studies that have shown that economic growth is positively correlated with the
distance from the equator2. As Weil [2005] stated, "There is good evidence that the
tropics constitute a bad health environment." Since these regions are also
characterized by a low level of human capital, this suggests that tropical climate may
be responsible for low longevity and, as a result, for the delayed transition to
schooling and thus a failure to make the related leap in economic growth. As climate
can be considered stable and exogenous, climatic factors may explain long-lasting
differences in economic development, and therefore the well-known positive
correlation between distance from the equator and economic growth.
This paper explains development through a positive feedback mechanism
between human capital accumulation and longevity, which in countries with a
temperate climate eventually triggers the transition to schooling. As the model
considers two ways of acquiring human capital, apprenticeship3 and schooling, it is
able to give a rationale for economic growth, without formal schooling which is an
important feature of the first Industrial Revolution. Once schooling is implemented,
the rates of human capital accumulation and economic growth leap as observed during
the second Industrial Revolution. In tropical countries, the presence of diseases that
2
See Sala-I-Martin [1997] and Hall and Jones [1999]
Apprenticeship encompasses a contract between the master and the apprentice, which is not
considered here. Nevertheless, this term is used to describe the way of accumulating human capital
without formal learning.
3
2
are not found in other parts of the world limits longevity and blocks this process.
When medical progress removes this obstacle, international spillovers cause the
transitory behavior of the system to exhibit conditional convergence, and economic
growth miracles occur.
Apprenticeship was universally used until the 19th century and is still
dominant in regions with low literacy. At low longevities, apprenticeship is chosen
since pupils participate in production during the learning time and earn a small wage.
By contrast the learning time spent at school does not contribute to production and
therefore pupils forego any income when studying. Nevertheless, in order to
accumulate a large human capital, school is preferable since, for example, literacy
obviates the need to learn everything by heart and mathematics enables the
application of a single formula to different problems. Therefore, it is assumed that
schooling is characterized by marginal returns that decrease less rapidly than those
linked to apprenticeship. When longevity is large enough, this advantage more than
compensates for the inconvenience of "losing time" to learn, and the economy shifts
to schooling, which is associated with a higher growth rate than apprenticeship. The
technology for human capital accumulation was discussed by Stokey [1991], who
concluded that the accumulation of human capital is affected by externalities passed
along by the previous generation. In the present model, these externalities may be
local and/or international. A high human capital in the previous generation improves
the quality of the learning system, which determines how much human capital is
acquired during the time devoted to learning. In the spirit of Basu and Weil [1998], it
is assumed that apprenticeship does not benefit from spillovers generated by
improvements in schooling. Technological knowledge is assumed to be available
everywhere. The worker's human capital determines which technology he or she will
be able to use and hence the respective wage rate. This builds on the work of Goldin
and Katz [1998], who documented the complementarities between technology and
human capital. A local amount of human capital inferior to that of the economically
leading country constitutes a barrier to the adoption of technology.4
Another externality of the accumulation of human capital by the previous
generation is its effect on longevity. High human capital means higher income, better
food, better hygiene, and hence longer life. However, longevity is bounded at a value
4
This differs from the models of Zeira [1998] and Sadik [2008], in which the appearance of barriers to technology adoption
during history was not considered.
3
that depends on disease prevalence and medical knowledge. If this maximum is lower
than the threshold for the transition to schooling, the development mechanism is
blocked at low growth rates. If longevity is bounded but surpasses this threshold,
international spillovers dictate that the steady-state growth rate is identical in all
countries using schooling, but levels of income per worker are determined by
longevities, such that countries plagued by diseases have the lowest income per
worker. Therefore, differences in learning systems explain the divergence between
industrial countries and those remaining in the apprenticeship regime. When
longevity in a follower country jumps, steady state income per workerincreases; the
country’s growth rate leaps and then converges again to its long-term value, as the
distance to the new steady state decreases.
The existence of a long term link between low longevity and low growth is
supported by historical evidences. Maddison [2001] separated the world in two
groups: group A (Australia, Canada, Japan, New Zealand, United States, and Western
Europe) and group B (all other countries). Between 1000 and 1820, the income per
capita of group A countries increased from 405$ to 1120$ and life expectancy
increased from 24 to 36 years; during the same period, the income per capita of group
B countries increased only from 440$ to 573$ and life expectancy remained
unchanged at 24 years. The model explains this early divergence by the presence in
group B of tropical diseases that never existed in group A. Thus, diseases constrained
life expectancy in group B for centuries and therefore growth in this group was
smaller then in group A, even before the adoption of schooling. Nevertheless, since
previous to the second Industrial Revolution all countries used apprenticeship, which
is associated with slow growth, the gap in income per capita between groups A and B
reached only a factor of two after 800 years. In the last two centuries, the gap in
income per capita between countries that adopted schooling and those for which the
population remained largely illiterate increased dramatically. Data on Sweden
reported by Cervellati and Sunde [2008] are also consistent with the mechanism
proposed in the model: adult life expectancy, primary school enrollment, and growth
all take off simultaneously at about the middle of the 19th century.
Empirical results obtained by various authors are in line with the predictions
of the model. Barro and Sala-I-Martin [1995], Bloom and Sachs [1998] and
Chakraborty [2004] found that low life expectancy at birth is highly predictive of
slow economic growth. Gallup et al. [1999] found that, during the period 1965 to
4
1990, severe malaria reduced growth by 1.2% per annum. Weil [2007] reported a
correlation of 0.767 between the fraction of 15-year-olds who will survive to age 60
and the log of GDP per capita. Chakraborty, Papageorgiu, and Pérez Sebastián [2005]
showed that in growth regression there is a threshold separating two regimes
characterized by different life expectancy.
The importance of international spillovers is well admitted. Grossman and
Helpman [1991], among others, explained the well-known positive correlation
between trade and growth by knowledge spillovers Parente and Prescott [1994]
showed that countries that became industrialized later grow faster than those that
became industrialized first. The power of international spillover once the burden of
diseases is removed is probably best illustrated by the eradication of malaria. In the
1950s and 1960s, a campaign to eradicate malaria was spearheaded by the World
Health Organization and the USA. This was an exogenous shock to longevity in those
countries in which the campaign succeeded, i.e., parts of southern Europe, Jamaica,
Hong-Kong, Malaysia, Mauritius, Singapore, and Taiwan, which, with the exception
of Jamaica, are typical examples of rapid economic catch-up.
The originality of the present model with respect to the wealth of literature on
the effects of life expectancy on growth5 is the introduction of two technologies for
human capital production as well as their interactions with the environment
characterized by climate and international knowledge spillovers. Adding a second
technology for human capital accumulation enables to differentiate the first from the
second Industrial Revolution. The disease burden plaguing tropical countries gives a
possible explanation for the concentration of countries characterized by low stocks of
human capital and low growth rates in tropical regions6. Finally, international
knowledge spillovers facilitate reproduction of growth miracles and conditional
convergence.7
5
See(Kalemi-Ozcan, Ryder, and Weil [2000], Jones [2001], Boucekkine, de la Croix, and Licandro
[2002], Lagerlof [2003], Cervellati and Sunde [2005], Soares [2005])
6
Cervellati and Sunde [2008] develop a model where high mortality delays the transition to the
industrial regime. The main differences between their model and the present one are that they consider
the demographic transition but not the different ways of acquiring human capital; therefore in their
model there is only one Industrial Revolution. Moreover, they do not model international spillovers but
suggest that they are important in determining the speed of transition for follower countries.
7
Galor Mountford [2008] study an other type of interaction between countries based on the influence
of trade on natality and human capital accumulation.
5
The approach developed in this paper differs also from that of a large part of
unified growth theory 8 since here the exit from the slow growth regime is triggered
by an increase in longevity and not in population size.
Another stream of literature9 has studied the role of institutions in the
provision of schooling by the state. In this paper, the mechanism presented for the
transition to schooling focuses on workers’ demand for schooling and is
complementary to that presented by the authors of those analyses.
The Industrial Revolution has also been explained through differences
between agriculture and industry.10 In these models, differences in the timing of
industrialization are a consequence of differences in agricultural productivity. A
previous study11 invoked a mechanism similar to the one used in the present model
and explained development differences in the light of variations in the endowment of
natural resources. This model presents a complementary explanation based on
climate.
The rest of this paper is organized as follows. Section 2 formalizes the
assumptions regarding production, longevity, and learning technologies, incorporating
them into an overlapping-generation model. Section 3 describes the path followed by
the first country to industrialize. Section 4 considers the equilibrium of follower
countries and shows how conditional convergence, growth miracles, and divergence
appear. Section 5 presents a brief discussion of the results. Section 6 concludes.
Proofs are provided in the Appendix.
2. The Model
The model is set in a small, open, overlapping generations’ economy that
produces a single homogeneous and perfectly tradable good. All markets are perfectly
competitive; knowledge flow freely. The world interest rate is r. At each period t, a
cohort of measure 1 is born. At the origin of time t=0 all countries are identical except
for their climates. Technological knowledge is developed in public institutions that
8
See for example Galor and Weil (2000), Galor (2006), (2010).
See Engermann and Sokoloff [1994], Galor and Moav [2006]), Galor, Moav, and Volltrath [2009].
10
See Kogel and Prskawetz [2001], Hansen and Prescott [2002], Tamura [2002]
11
See, Berdugo, Sadik, and Sussman [2003]
9
6
are exogenous to the economy and is always ahead of the level that workers and firms
are able to use12
2.1. Production
As one of the goals of this paper is to explain technological differences
between industrial and developing countries, it is assumed, following Acemoglu and
Zillibotti [2001], that human capital and technological levels are not independent of
each other and that a minimum level of human capital is needed in order to use a
given technology. Conversely, if a worker has more human capital than needed to
operate a machine, this extra human capital will not enhance his output. Therefore, at
equilibrium there is a one-to-one match between human capital and technology levels.
Hence, with the adequate choice of units, human capital and technology levels are
identical and human capital can alternatively be interpreted as the local technological
level. In particular, the level of human capital in the most developed or leader country
will be used to model the frontier of technology.
To capture these features in the simplest form, it is assumed that production
uses only human capital 13. Therefore, the production function is:
y tj  ht j
(1)
where y tj is the output per unit of time of a worker born in period t in country j who
has a human capital ht j after he completes his studying .
2.2. Individuals
The economy has overlapping generations of risk neutral individuals, who
may live for two periods each and maximize the expected present value of their
lifetime income... There is no utility of leisure. Since the economy is small and open,
the interest rate is exogenous and there is no need to solve for an individual's
allocation of resources.
All individuals are identical except for their longevity, which depends on their
countries' climate and on their income. There is no evolution in an individual's
12
Acemoglu and Zilibotti [1997] made a similar assumption and cited the works of
Hobsbawm [1968], who asserted that the technology of the British Industrial Revolution could have
been developed already during the 17th century. Mokyr [1990] also confirmed that gaps between
inventions and applications persisted for a long time.
13
It would be straightforward to introduce physical capital because in a small economy with free
capital flows the equilibrium amount of physical capital is determined by the technological level and
the international interest rate.
7
characteristics.14 Migration is not allowed. All individuals are certain to live for the
entirety of their first period and for a fraction  t j , which depends on their home
country j and on their date of birth, of a second period. Hence, the longevity of
inhabitants of country j born at time t is 1   t j . It is assumed that the only cost of
learning is foregone income. Individuals optimize the expected present value of their
life income, taking into account this cost and the possibility of earning a higher wage
by acquiring enough human capital to use an advanced technology. The effect of
current income on longevity is assumed to be an externality not taken into account by
individuals who expect to live the same time as their parents. Hence, people whose
parents had short longevities have less incentive to study.
2.3. Climate and longevity
A key variable of the model is the expected number of healthy days above
age 5, which is generally considered to be the minimum age to go to work or to
study. It is affected by diseases that shorten life and cause loss of learning or working
days. In order to keep the terminology simple, the number of healthy days above age
5 is designated by longevity or life. The main factors explaining an increase in
longevity are, according to Weil [2005], improvements in living standard, better
medical treatment, and improvements in public health measures.
Living standard is modeled by local income per worker. It is assumed that the
function relating longevity to income per worker is increasing, has decreasing returns
to scale, and that all effects of income on health via food intake, the quality of
medical treatment, and public health are taken into account by this function.
Nevertheless, an adequate income is a necessary but not sufficient condition to
achieve a long life. The kings living in the Middle Ages were certainly richer than the
representative individual of today but had a shorter life. Longevity is limited, among
other factors, by the possibility of contracting a lethal disease. Medical progress
enables the prevention or treatment of some of these diseases; therefore, longevity, if
not limited by income, increases each time a relevant cure is found.
Public health partly depends on scientific knowledge but climate also has a
large impact on the prevalence of diseases. There are privileged regions where, for a
given state of medical knowledge, life expectancy is longer than in other regions.
14
This aspect has been treated by Galor and Moav [2002], [2005]
8
15
To capture these features, the function expressing life expectancy 1   t j in region j
at time t can be written as:
(2)


1   t j  min 1  (ht j ) ,1   cj,hl ,
t
where   0 and   1 are constants.
The effect of climate on longevity before the inception of modern medicine is
well-accepted, and for that epoch maximal longevity  cj,h l can be considered to be
t
time invariant. Since then, medical progress lead to the treatment of a wide range of
diseases, increasing longevity in large parts of the world. It is assumed that new
treatments are created in the leader country and that a cure for a disease endemic in
country j shifts upward the maximum longevity (with the constraint 0   cj,hl  1 ) in
t
this country; hence, maximum longevity is an increasing step function whose levels
depend on the local climate in country j and on the state of the art technology htl . A
jump in this function is a spillover from the medical knowledge of the leader. This
function is illustrated in Figure 1.
2.4. Learning technologies
Individuals have two possibilities to acquire productive knowledge. The first is
by apprenticeship. The second is by attending school. In order to simplify the model,
combinations of formal and informal learning are not allowed. To reach the level of
human capital ht j , an individual born in period t in country j and using the learning
technique i must devote a time E itj to studying. It is assumed that the time devoted to
studies increases with the desired level of human capital and decreases with the
quality of the learning system, and that the learning processes have decreasing returns
to scale.
Two sources of externality determine the quality of the learning system. The
first one is the level of human capital of the local previous generation. A
15
The absence of factors such as mosquitoes transmitting parasites is probably a key
advantage. For example, Bloom and Sachs [1998] explained that the tropical climate presents the most
favorable conditions for malaria propagation because it is conducive both to the reproduction of the
parasite causing the disease and to that of the mosquito transmitting it. In a similar way, a tropical
climate favors the propagation of yellow fever, sleeping sickness, schistosomania, and dengue, to
name but a few.
9
straightforward example of this effect is the influence of parents’ education on their
offspring; the higher the parents' level of education, the lower the learning efforts
needed on the part of their children, both because parents act at the economy level by
demanding better schools and at the family level by helping their children. The
second source of externality stems from international spillovers. Followers can, to
some degree, imitate the leader and benefit from the best international learning
methods, which spill over through diverse channels such as books, computers, and
improvements in teaching programs; therefore, international spillovers substitute, in
part, for domestic spillovers. The magnitude of these international spillovers depends
on the degree of openness to foreign influences and on the gap between the follower
and the leader. Note that the leader has no one to imitate and depends solely on
spillovers from his own previous generation.
More formally, the time spent in acquiring the human capital ht j must be
independent of the units chosen to measure human capital; hence, it must be a function of
a human capital ratio. A simple expression satisfying these requirements is:
(3)
Ej
ht   it
 e
j
i

 (1   j )ht j1   j htl1 , i= I, S,




where e is a constant16,  i  1 is a constant characterizing the learning technology i, with i =
A for apprenticeship, i = S for schooling, and htl1 designating the level of the leader
country in the relevant learning system at time t-1. Since, in a given country, the
initial level of human capital is uniform and all workers are identical, they face the
same problem when optimizing their life income. Then by symmetry the optimal
learning time and the level of human capital in the next generation are unique17. The
j
parameter  represents the degree to which the jth economy is open to the beneficial
influence of the leader using the same learning technology.
An important difference between the two learning technologies is that
apprenticeship focuses on acquisition of the particular skills needed for the chosen
profession, whereas school provides all-purpose tools such as literacy and
mathematics. To accumulate a large amount of knowledge, school is preferable,
16
For the leader who does not receive any international spillover ( 
learning time corresponding to h  h
l
t
l
t 1
j
 0 ), we see that e is the
.
17
The level of human capital will be formally derived later on, as the result of the equilibrium of the
model.
10
since, for example, literacy dispenses with the necessity to learn all required
knowledge by heart and mathematics equips the student with the ability to apply a
single formula to different problems. Therefore, it is assumed that schooling is
characterized by marginal returns that decrease less rapidly than those associated
with apprenticeship.
Assumption 1
0   A  S 1
By contrast, an apprentice works and produces something during his studies.
Therefore, he earns a fraction 0  1  f A  1 of the wage rate he will earn at the end
of his studies18. If he goes to school, he does not earn anything and f S  1 .Hence the
opportunity cost of learning is f i E itj .
Finally, since apprenticeship does not involve the acquisition of literacy,
when schooling implies the use of books, it is assumed that international spillovers
between countries using apprenticeship and those using schooling are negligible. The
intuitive meaning of this assumption is that progress in teaching engineering in
America does not spillover to Africans teaching farming to their sons.
3. From the 17th to the 21st century
This section describes the path followed by the leader country. It is
characterized by the absence of both foreign spillovers (  j  0 ) and constraints on
longevity (  cl ,hl  1 ). Under these conditions, the human capital level of the country's
t
own previous generation determines entirely the quality of the learning system and the
expected longevity of the current generation.
In order to determine the equilibrium, the first step is to evaluate the learning
time. Individuals maximize the present value iitj of their expected life income.
Equations (1) and (3) lead to:
(4)
18

 j  E j
i  1  f i Eitj  t 1  it
1  r  e

j
it
i

 (1   j )ht j1   j htl1




j
Since the human capital of an apprentice is not ht until he completes, his studies, his marginal
product per unit of time is smaller then at the end of apprenticeship. It is modeled as (1  f A ) ht .
j
11
Where the first parenthesis corresponds to the duration of the first period 1
minus the opportunity cost of learning f i E itj plus the second period life expectation
 t j1 discounted by 1+r and the remaining expression is the human capital given by
equation (3).
The first-order condition determines the optimal learning time:19
i
E 
1   i  f i
j
it
(5)

j 
1  t 1 
 1 r 


Learning time is a proportion of life that depends on the method of learning.
We can now compute the growth rate lit of the product per unit of time worked
in the leading country. Taking Equation (3) with,  j  0 and replacing E itj by the
value given in Equation (5) leads to:
i
(6)
h  h
l
t
l l
it t 1


i
 l 
1  t 1  htl1



 ef i 1   i   1  r 
The expression for lit 20 has an intuitive meaning: the growth rate is equal to
the time devoted to study, normalized by e and elevated to the power  i . Therefore,
growth depends on longevity. This is a consequence of the assumption that the human
capital of an economy is that of its workers; therefore, in this model the trade-off
between present and future production is at the level of the individual, who chooses
how and how much to learn. The resource, which is divided between present and
future production, is the individual's working life. Moreover, as learning time is a
function of longevity, which itself depends on climate, growth will depend on
climate.21
19
The fact that this is a minimum is showed in the Appendix.
The growth rate considered here is that of income per unit of time of workers born at time t .All the
expressions for workers born at t-1can be deduced from those in the text by replacing t by t-.1 As GDP
per capita would be a cumbersome expression involving youg and old agents without adding any
insight, we will use income per unit of time of workers born at time t to conduct all the reasoning
thorough the paper.
21
This differs with respect to the endogenous growth models of Romer [1990], Grossman and
Helpman [1992], and Aghion and Howitt [1992], in which the trade-off between present and future
production is at the economy level and involves assigning people to research and development or to
production. The transition from a decision made at the economy level to one made at the individual
level leads to a growth rate devoid of the scale effect.
20
12
As we want the economy to make the transitions from apprenticeship to
schooling within the time frame of the model, the parameters must be restricted as
follows.
Assumption 2
a)
For the minimum longevity, the economy is stagnant but does not collapse:
lA  t  0   1 .
b)
For the minimum longevity, apprenticeship is preferred:
i Al (  t  0)  i Sl (  t  0) .
c)
For the maximum longevity, schooling is preferred:
i Al (  t  1)  iSl (  t  1) .
It is straightforward to see that Equation (6) and Assumption 2a imply:
(7)
A
e
f A 1   A 
.
Lemma 1: The set of 0  f A  1 such that Assumptions 2b and 2c are satisfied is not
empty.
Under these assumptions, the transition between the imitating and the
schooling regimes takes place at the value  AS .
Lemma 2. There is a value of longevity,
(8)
 AS
1
 l

 S  A




i


0

A
t

 1  r   l
 1 ,
 i   0  


 S t

such that:
a) o   AS  1 .
b) For  tl   AS , learning is done by apprenticeship and for  tl   AS
schooling is adopted.
If assumptions 1 and 2 are satisfied and the initial longevity is slightly
greater than  t  0 , the economy starts in the apprenticeship regime and grows
slowly (from equations (6) and (7) we see that in the apprenticeship
A
regime 
l
At

 tl1 

 ) . Then, the positive feedback between human capital,
 1 

1

r


quality of informal learning, and longevity leads to increasing growth accompanied
13
by a longer life. The epoch when growth is already sizable but schooling not yet
adopted, can be interpreted as the first Industrial Revolution. Eventually, longevity
reaches  AS , schooling is adopted, and growth jumps; this corresponds to the second
Industrial Revolution. Thereafter, the same mechanism continues at a faster rate
because the feedback between longevity and human capital is stronger. Growth
increases continuously until longevity reaches its maximum of two full periods.
The following proposition formalizes these results.
Proposition 1. Development path of the leader.
Under assumptions 1 and 2, the leader goes through the following stages:
a) The economy starts slightly above the minimum longevity and grows with
the use of apprenticeship. Growth of GDP per worker causes an increase in
longevity, which in turn augments the growth rate of income per worker.
b) When longevity reaches  AS , the economy shifts to schooling and the
1  S
 1.
1  A
growth rate is multiplied by a factor
c) Growth and longevity continue to increase until life duration reaches two
full periods. Then, the product per unit of time worked grows at the constant
rate of:
S
max
  1   A  
1 
  fA S
1 
 .
  A 1   S   1  r 
Figure 2 represents the leader's growth rate as a function of longevity.
4. Stagnation and growth miracles
This section considers the case of countries in which climate imposes an
effective limitation to longevity (  t j   cj,hl  1 ). The previous section demonstrated
t
that the type of learning technology, the learning time, and the growth rate all depend
on longevity, which implies that countries in which longevity is constrained at low
values grow at low rates. Those countries become followers but, if they are open to
foreign ideas, they can benefit from the leader's spillovers. Once the constraint on
longevity is removed, they may even experience growth miracles.
As we assumed that there are no learning spillovers between countries
employing apprenticeships and those using schooling, the two cases of intra-
14
technology spillovers are analogous. The dynamics of the follower are described by
lemma 3.
Lemma 3. The growth rate of the follower is given by:
ht j
(9)
ht j1

j
 1  t 1
  1 l r

 t 1
1
 1 r
i


 l 1  
 it 


j
 htl1

 j  1 .
h

 t 1

There are two parts in the right-hand side of Equation (9). The first one is the
quotient of longevities elevated to the power  i and multiplied by the growth rate of
the leader. It expresses the fact that, for the same quality of learning institutions
( ht j1  htl1 ), growth in countries in which longevity is low is inferior to that in
countries in which it is high, since people dedicate less time to learning. The second
part, corresponding to the spillover effect, increases with the opening of the follower
and with the widening of the gap between the leader and the follower; as the
ratio
htl1
can be much larger then one, this term can explain convergence and growth
ht j1
miracles.
Lemma 3 implies that divergence can take place if a country is closed to
spillovers from the learning system (  j  0 ) or because it has an unfavorable
climate, one that prevents longevity from surpassing the threshold  AS . For  j  0 ,
the follower does not benefit from spillovers, and if the life expectancy of its
inhabitants is shorter than that of the leader's, its growth rate will be smaller and there
will be divergence. Pomerantz ( ) cites China's isolation as one the cause of for its
divergence during the 18th and 19th centuries.
Countries in which longevity is constrained at a value inferior to  AS remain
in an apprenticeship regime and suffer from the fact that, in the absence of constraints
on longevity at  AS , as soon as  AS is reached the leader of the apprenticeship
technology will adopt schooling, so that followers see a changing leader with a
constant product per worker that grows at the low apprenticeship rate lAt . Those
follower countries are in a highly unfavorable situation compared with followers in
15
the schooling regime whose leader grows at  lSt . 22 Countries whose population are for
a large part illiterate indeed grow at a much slower rate then countries which adopted
schooling
To further study the influence of longevities on the system leader follower, it
will be assumed at this point that the leader has reached the maximal longevity
(  tl1  1 ) and that the follower’s longevity is constrained at a level 1   cj,hl .
t
Definition 1. A system of two countries is said to be in equilibrium if, for given
longevities, the ratio of their income per worker remains constant over time.
Lemma 4. Consider an open economy whose inhabitants’ longevity is
constrained at 1   chj l while the leader's inhabitants live two full periods; then, at
t
equilibrium, the gap in income per worker is given by:
(10)


1
*
1

 hil 
1 
 j   j   1 j r
h 
 

 i 
 1  c ,htl


 1 r
i







1
 1







Lemma 3 and 4 imply that opened economies converge to an equilibrium at
which they grow at the same rate as the leader while their income per worker is
determined by their inhabitants’ longevities. All opened countries whose inhabitants
have the same longevity as those of the leader converge to the same income per
worker. Countries in which longevity is limited to a smaller value converge to a lower
income per worker, the difference depending on their degree of openness.
Eliminating longevities between Equations (9) and (10) leads to the growth
rate of the follower as a function of its distance to its equilibrium point.
 htl1

1    j  1
 ht 1

 lit
*
l
 h 

1   j   ij   1
  hi 



j
(11)
ht j
ht j1
This expression, in which the follower's growth rate increases with its distance
to the equilibrium, is typical of conditional convergence.
22
An other plausible assumption is that in the apprenticeship regime, local skills are "schielded' from
international spillovers and hence that
 j ( A)  0 .
16
As climate is favorable to the propagation of different diseases in different
regions, medical progress can cause the longevity of a given follower country to
jump. A good example is the eradication of malaria already mentioned.23 This shocks
the system leader-follower which is thrown out of equilibrium; Equation (11) shows
that the follower will grow rapidly and converge to its new equilibrium. If this jump
is large, a growth miracle occurs. In this model, growth miracles are transitory and,
when the system approaches its equilibriom, the follower's growth rate converges to
that of the leader..The effect of a new cure is illustrated in Figure1, at point R the
longevity in country B jumps to 1.This sudden increase in longevity will cause a jump
in the growth rate and country B will grow at a faster rate than the leader [Equation
(9)]. Then, as the gap between the leader and follower decreases, the growth rate of
the follower converges to that of the leader. During this process, country B has
surpassed country C. This is an example of leapfrogging between two followers. If the
longevity of the follower surpasses that of the leader, the process will be similar until
catch-up. Then the country with the greatest longevity will grow faster and become
the leader [Equation 6)].24 Permanent shocks to longevity that may arise from new
treatments, such as vaccination, or new diseases, such as AIDS, have different effects
on economies that are opened or closed to learning spillovers. In an opened economy,
a shock has a transitory effect on growth and a permanent effect on the level of
income per worker. In a closed economy, it has a permanent effect on the growth rate.
The results of this section are summarized in the following proposition. Proposition
2. Divergence and convergence.
a) Countries whose inhabitants learn by apprenticeship diverge from those
countries whose inhabitants learn at school.
b) A closed follower (  j  0 ) in which the longevity of the previous
generation is 
23
j
t 1
ht j
grows at a rate j
ht 1

j
 1  t 1
  1 l r

 t 1
1
 1 r
i


 l . Therefore, the growth
 it


Conversely, a decrease in life expectancy, such as that in Africa, due to AIDS, will lower
the follower's equilibrium income and cause a transitory period of slow or negative growth
24
This explanation of leapfrogging based on comparative longevity gives a new point of view with
respect to explanations based on technological adoption such as that of Brezis, Krugman, and Tsiddon
[1993].
17
rate of the follower is determined by its life expectancy; if it is smaller than
that of the leader, it will diverge.
For a couple leader-follower using the same learning technology and for an
open follower (  j  0 ):
c) The ratio of the follower's growth rate to that of the leader's increases with
the longevity of the follower's inhabitants, with the openness of the follower
and with the gap leader-follower.
d) For given longevity, the ratio of the leader's income per worker to that of
the follower's decreases with the longevity of the follower's inhabitants and
with the openness of the follower. A country whose citizens have the same
longevity as those of the leader converges to the same income per worker.
e) In the long run, all countries in the same regime grow at the same rate.
f) Chocks to longevity such as medical progress can cause growth miracles
and leapfrogging.
5. Discussion
The present article emphasizes the role of geography as a prime determinant
of growth but does not mean to imply that institutions are unimportant. Our
interpretation is that geographical and institutional approaches are complementary, as
both of them establish the necessary conditions that must be satisfied for a country to
realize its growth potential. For example, the present model explains why HongKong and Singapore, which according to Acemoglu et al. [2001] benefited from good
institutions, had a GDP per capita in 1950 that was much lower than that of the
United States, but were able to experience a growth miracle after malaria
eradication25. Nevertheless, it does not explain why, after malaria eradication,
Jamaica did not follow the same convergence path as Hong Kong and Singapore.
However, the present model does show that medical progress can explain reversals of
fortune like those discussed by Acemoglu, Johnson, and Robinson [2002]. If medical
progress is biased toward treating diseases in the most developed countries, which
have temperate climates, this mechanism could be relevant in a significant number of
cases. In a recent work, Acemoglu and Johnson [2007] showed that life expectancy
25
Note that variations in results of the fight against malaria are attributed in the first place to diverse
climates and not to differences in income per capita or institutions (Gallup at al. 1999, Weil 2005).
18
had no significant effect on growth in a sample of countries, excluding Africa. This
suggests that health does not restrain growth in a significant number of the countries
comprising that sample. But longevity may still restrict growth in some countries,
particularly in Africa, where life expectancy is low. If growth depends on a set of
necessary conditions, a case per case approach in which economists try to determine
the binding condition in each slow-growth country may help in designing
development programs.
An important aspect not considered in this paper is demographic transition.
Intuitively, if part of the wage of an apprentice covers his food, as was often the case,
adoption of schooling, by suppressing income earned during apprenticeship, increases
the cost of children. Hence, an extension of the model can associate demographic
transition with the adoption of schooling. This mechanism has already been used in
the literature and therefore this aspect is not considered here.
Finally, as this model stresses the importance of schooling for growth, a
natural question arises regarding the role played by pedagogical progress. As innate
abilities differ among individuals, an important role of school is the orientation of its
pupils. Here this aspect is not considered; instead, the learning elasticity  S is given.
Nevertheless,  S may rise with pedagogical progress as today's schools may be better
in helping pupils choose subjects in which they have a high learning potential. This
could explain the results of Hazan [2009], who found that, during the last century, in
the United States, schooling increased while the number of hours worked during the
lifetime of Americans decreased due to increases in leisure and earlier retirement. A
formal development of models in which  S is endogenous is left for further research.
Concerning policy implications, the model stresses the importance of medical
spillovers to replace divergence in income per worker by convergence and, once this
result is obtained, to increase steady-state income. Of course, to benefit from
knowledge spillovers a country must be as open as possible.
6. Conclusion
This study sheds light on the importance of schooling in understanding
growth. It offers a model explaining the most important stylized facts characterizing
growth, such as the historical path followed by the Industrial World, the persistence
of poverty in some countries; conditional convergence, and recent growth miracles.
19
The modelisation of human capital accumulation when schooling is not
optimal offers a possible way to understand economic development before the
second Industrial Revolution as well as differences between growth with or without
schooling.
Disparities between countries are explained as a function of climate and lead
to predictions that benefit from strong empirical support. In particular the fact that
the most developed countries today, which benefit from a temperate climate, were
also the richest hundred years ago is intuitive in this model since climate is persistent
and exogenous.
The introduction of knowledge spillovers conditional to literacy lead to three
clubs of countries: First the less developed countries which did not yet make the
transition to schooling. These countries do not benefit from knowledge spillovers
from industrial countries and are still in the low growth regime. A second group is
constituted by countries which made recently the transition to schooling. These
countries benefit from strong spillovers and grow rapidly until they reach a new
steady state determined by the longevity of their inhabitants. The third group is
constituted by the old industrial countries which are the source of knowledge
spillovers. These countries grow at a constant rate.
c , a l
1
A
B
C
0
R
a
l
t
20
Fig. 1 Maximal longevity. In the leader, denoted by A,  cj,hl  1 , the constraint of
t
climate on longevity is not binding. In countries B and C, the climatic constraint is
binding. At point R, the maximal longevity of B surpasses that of C and reaches 1.
This represents the effect of a new drug treating an illness endemic in region B but
not in region C, which will generate a reversal of fortune between countries B and C
and convergence between B and the leader
21
lt
max
 SAS
 AAS
1
1   IS
2
longevity
Fig. 2 The leader's growth rate increases with longevity and jumps at the point of the
transition to schooling
22
Appendix
Proof that Ei ,jt corresponds to a minimum.
Taking the first derivative of ii j,t given by Equation (4) with respect to Eil,t and
equating it to zero yields:
 i
 j
 Ei ,t
 E j

j 
1  t 1   1   i  f i  it
 1 r 

 e


i

 (1   j )ht j1   j htl1  0 .




Equating the terms in the first set of brackets to zero leads to Equation (5).
The second derivative of ii j,t with respect to Eil,t is:


 Eitj  i
i 
 t j1   i   i 
 t j1 

1 
  j
1 
  1   I  f i 
 (1   j )ht j1   j htl1 

j 


j 2 
 Ei ,t   1  r  Ei ,t  Ei ,t  1  r 
 e 

 E j   i
 t j1 
I 
i
1 
1   i  it  (1   j )ht j1   j htl1  0
  j 1   i  f i 
2 
j
 Ei ,t
 e 
Ei,t   1  r 




■
Proof of lemma 1:
Introducing ht j1  htl1 , the expression for E itj and e given by Equations (5) and (7) in
Equation (4) yields:
1
i 
1 i
l
i ,t
  i 1   A  f A 





1


f
i
i 
 A
i

j 
1  t 1 
 1 r 


1 i
htl1 .
Introducing in this expression relevant values for longevity and f S  1 yields:
 1  S 

i Al   0  i Sl   0  f A  A 
 S  1   A 
 1  S 

i Al   1  iSl   1  f A  A 
 S  1   A 
 1  S 

so that A 
 S  1   A 
1
1
S
1 

1 

 1 r 
 A  S
S
1
1
1
S
1
S
1 

1 

 1 r 
 A  S
S
 1  S 

 f A  A 
 S  1   A 
1
,
1
S
.
As from Assumption 1 0   A   S  1 , it is clear that the left-hand side of this
inequality is smaller than the right-hand side. To justify our interpretation of f A as the
23
fraction of final income that is foregone during learning, it is enough to show that
f A  1 . First note that for  A   S the right-hand side of the inequality is equal to 1.
If
we

denote
Q  1   S

  1   S



1
1
S
A
1
S
and
Q
is
the
right-hand
side,
then,

1
 0 . Hence, Q is an increasing function of   A and is
1   S
S
equal to 1 when  =1. From Assumption 1,   1 and then Q<1, and therefore f A  1 .
■
Proof of Lemma 2
a) We have seen in the proof of Lemma 1 that:
1
i 
1 i
l
i ,t
  i 1   A  f A 


  A 1   i  f i 
i

j 
1  t 1 
 1 r 


1 i
htl1 ,
which implies that
 
1 
i  AS  
1  AS 
1  A  1 r 
l
A
1 A
l
t 1
h
 

 i   01  AS 
 1 r 
1 A
l
A
htl1
and
1
i  AS  
1  S
l
S
  S 1   A  f A 


  A 1   S  
S

j 
1  AS 
 1 r 


1 S
Equating these two expressions leads to  AS
l
t 1
h
j


 AS


 i   01 

1

r


l
S
1 S
htl1 .
1
 l

 S  A




i


0

A
t

 1  r   l
 1 , which is
 i   0  


 S t

Equation (8).
From Assumption 2b it is straightforward to see that  AS  0. Assumption 2c can be
1 

written as i   0 1 

 1 r 
1 A
l
A
1 

 i   01 

 1 r 
1 S
l
S
and therefore implies that
 AS  1 .
 
 
iSl  tl
iSl   0 
 tl 


b) From Assumption 1, l l  l
1
iA t
i A   0  1  r 
 S  A
is a function monotonically
increasing in  . Since the two incomes are equal at  t   AS by construction,
24
 t   AS
implies
that
 
 
iSl  tl
1
i Al  tl
and
 t   AS
implies
that
 
 
i Sl  tl
 1.
i Al  tl
■
Proof of proposition 1
Points a and c follow directly from lemmas 2, Equations (2), (6), and (7).
b), Lemma 2 demonstrates that the transition between imitating and schooling occurs
at  tl   AS . From Equations (6) and (7) at the transition, the growth rate is multiplied
by the following factor:
S


S


S
 
 S  AS   ef S 1   S   
 AS  S A   S 1   A   i Al  t  0  1   S

1
 
f A  l

■

I 
 A  AS  
  1 r 
  A 1   S   i S  t  0  1   A
A


 ef A 1   A  
Proof of Lemma 3
Dividing Equation (3) by ht j1 and using Equations (6) and (7) leads to:
ht j  Eitj 


ht j1  e 

j
 1  t 1
 1 r

 tl1
1
 1 r
i
l

 Eitj
j
j ht 1 
1    


ht j1   Eitl





i
 Eitl

 e





i
l

j
j ht 1 
1    


ht j1 

i


 l 1  
 it 


j
 htl1

 j  1
h

 t 1

■
Proof of Lemma 4
Equating the growth rate of the follower given by Equations (9) to that of the leader,
and solving leads to the conclusion.
■
Proof of proposition 2
Follows from lemma 3 and 4, and Equations (11)
25
■
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