Endogenous Lifetime and
Economic Growth
¤
Shankha Chakraborty
Department of Economics, University of Oregon
Eugene, OR 97402-1285
Phone: (541) 346-4678, Fax: (541) 346-1243
shankhac@oregon.uoregon.edu
¤
For useful conversation and feedback, I would like to thank Costas Azariadis, Richard Barrett, Roger
Farmer, Gary Hansen, Dean Jamison, Amartya Lahiri, David Meltzer, Tomas Philipson, Jia Wang and sem-
inar participants at UCLA, UCSB, Harris School of Public Policy, and University of Oregon. All remaining
errors are mine.
Endogenous Lifetime and Economic Growth
Abstract
Conventional wisdom attributes the severity of mortality in poorer countries to
widespread poverty and inadequate living conditions. This paper considers the
possibility that persistent poverty may arise, in turn, from a high incidence of
mortality. Endogenous mortality risk is introduced in a two-period overlapping
generations model: probability of survival from the ¯rst period to the next
depends upon health capital that can be augmented through public investment.
Highmortalitysocieties donotgrowfastsinceshorter lifespansdiscouragesaving
and investment; multiple steady-states are possible. High mortality also reduces
returns on investments, like education, where risks are undiversi¯able. When
human capital drives economic growth, countries di®ering in only health capital
do not converge to similar living standards; `threshold e®ects' may also result.
Keywords: Health, Life expectancy, Mortality, Growth, Human capital.
JEL Classification: I1, I2, O1
1 Introduction
More than a billion people su®er from malnutrition in developing countries, infant mortality
rates there are three to four times higher than those in the richest nations and the burden
of disease twice as high [World Bank, 1993]. This paper studies the e®ect of such pervasive
ill-health, especially high mortality, on economic growth.
Health status and income levels are well known to systematically vary across nations,
particularly those in low- and middle-income categories. Figure 1 displays one such rela-
tionship, that between life expectancy at birth and log GDP per capita for 1996.
1
Life
expectancy at birth evidently shows a strong tendency to improve with per capita income,
ranging from as low as 37 years in Sierra Leone to as high as 77 in Costa Rica, more
than twelve times richer. A similar pattern is observed if we narrowed our focus to adult
life expectancy, a length of time more relevant for economically active individuals. Thus,
life expectancy at age 15 for the poorest nations fell 12 ¡ 13 years short of the 73 years
prevailing in the richest ones in 1990. Adult mortality risk, the probability that a typical
15-year old would die before reaching age 60, was thrice as high in Sub-Saharan Africa as
in the established market economies [World Bank, 1993, Table A.5]. Conventional wisdom
attributes the severity of mortality and morbidity in poor countries to widespread poverty
and inadequate living conditions. We argue here that there is as much possibility that
persistent poverty results from undesirable economic incentives that high mortality creates.
Ever since economists recognized the importance of human capital, considerable atten-
1
Data from World Bank [1998].
2
tion has been devoted to the various ways health improves economic outcomes.
2
But that
literature has usually focused on the implications of health and nutrition for labor market
outcomes. Yet health plays a role quite unlike any other human capital: by altering an
individual's lifespan and mortality risks, it fosters incentives to invest in assets, especially
those that realize high returns later in life. Given the central role capital accumulation
plays in growth theory, it is not surprising that life expectancy would nontrivially a®ect the
dynamics of per capita income.
With that in mind, this paper constructs a simple general equilibrium framework that
allows health capital and per capita income to be determined simultaneously. This is done
by introducing mortality risk in a two-period overlapping generations model. In particular,
the probability of surviving from the ¯rst period to the next is endogenously determined
through public investment in health. In low-income societies, when average lifespans are
short, individuals are less willing to postpone consumption. Consequently, they are also
less willing to save and invest. Due to pervasive poverty, neither can they a®ord substantial
private health investment.
During the early stages of development, a low income - low longevity society therefore
grows slower than what standard one-sector models predict. When the capital elasticity of
output is su±ciently large, increase in per capita income, over a certain range, draws a large
response in capital accumulation. Multiple steady-states result: societies with favorable
initial conditions (income or health) grow towards a high-income steady state, less fortunate
ones ¯nd high mortality a strong deterrent for saving and investment.
2
See Srauss and Thomas [1998] for an overview of this primarily empirical literature.
3
Besides this length-of-life e®ect, mortality rates a®ect investment decisions through rates
of return. Risks associated with investment in other forms of human capital, for instance
education, may not be fully diversi¯able. Higher mortality risks would then reduce returns
on such investment. Recent growth theories have persuasively argued that human capital,
by improving labor skills, inducing technological innovations and expanding the body of
economically useful knowledge, contributes to long-run growth [for instance, Romer, 1986
and Lucas, 1988]. When mortality considerations a®ect schooling decisions, countries that
di®er only in health capital do not converge to similar living standards. `Threshold e®ects'
[Azariadis and Drazen, 1990] may also result with the economy going through a phase of
slow growth and modest health improvements before taking-o® into sustained growth.
This paper complements several pieces of work in the literature. Among others, Ram and
Schultz [1979], Gersovitz [1983], and more recently, the World Bank [1993] have suggested
howlongevity improvements maypromote economic growth. Thecontribution of thecurrent
paper lies in systematically tying that intuition to a general equilibrium framework with
health investment.
In a model of endogenous growth, de la Croix and Licandro [1999] show that exogenous
life expectancy improvements initially raise economic growth, but may be growth depressing
once the average age of the workforce increases su±ciently. Kalemli-Ozcan et al. [1998]
examine how increased life expectancy promotes human capital investment in a Blanchard-
type [1985] model with constant probability of death.
The impact of prime-age and adult mortality on fertility choice and human capital
accumulation is analyzed by Meltzer (1995); mortality decline initially raises incentives to
4
have children and discourages human capital investment, but is followed by a phase of
sustained economic growth and declining population growth. Children are a source of old-
age support in Ehrlich and Lui [1991], where too, a demographic transition results. In both
these works, mortality declines are treated as exogenous to the development process.
Finally, several empirical works have motivated this paper to examine the health and
growth causality. Fogel [1997], for instance, estimates that better health and nutrition alone
may have contributed about 20-30% to British economic growth during 1780-1979. In their
analyses of the economic burden of malaria, both Gallup and Sachs [1998] and McCarthy et
al. [2000] ascribe the lackluster performance of several African nations to a severe incidence
of the disease. Using cross-country regressions, Barro and Sala-i-Martin [1995] and Knowles
andOwen[1995]showlifeexpectancytobeanimportantdeterminantofgrowth.
The rest of the paper is organized as follows. Section 2 sets up an overlapping gener-
ations economy with endogenous mortality risk. Since low-income societies cannot invest
much in health nor do they save a lot, convergence is shown to be slower during the early
stages of development. Depending upon the output share of capital, the model may also
deliver multiple steady-states. Section 3 studies the e®ect of mortality risk on schooling de-
cision. Educational investment is assumed to improve labor-skills that are passed on from
one generation to the next. Multiple equilibria result when such externalities are present.
In section 4, we brie°y turn to cross-country evidence that con¯rms the speci¯c linkages
stressed in the theory and indicates the growth e®ects to be signi¯cantly large. Section 5
concludes the discussion.
5
2 Length-of-life and Capital Accumulation
Longevity a®ects savings decisions through two channels: by increasing lifespans it promotes
investment in assets that pay o® later in life, and by reducing mortality risks, it increases
the return on such investment. This section considers the length-of-life e®ect,
3
postponing
until section 3 a discussion of the latter.
To keep the analysis simple, we shall ignore the e®ect of health on labor productivity
and labor market participation.
4
Moreover, we assume that preferences, technologies and
markets do not fundamentally di®er in a low-income country from those in a high-income
one.
Existing theoretical modelsthatincorporate mortalityusuallyassumeitto be exogenous.
5
To endogenize the mortality rate, we consider a two-period overlapping generations econ-
omy where the probability of surviving from the ¯rst period to the next depends upon the
health of individuals. Individuals can choose to improve this probability through appropri-
ate investments in health.
6
3
Theories of development often a higher savings propensity for the rich [see Lewis, 1954 or Leibenstein,
1957]. Lawrance [1991] ¯nds subjective rates of time preferences to be higher for poorer households, implying
a less willingness to save. The length-of-life e®ect is one way to explain both.
Deaton and Paxson [1994] attribute the higher savings rate in younger cohorts in Taiwan (1976-90) to life
expectancy improvements, among other factors.
4
See Gersovitz [1983] for a theoretical treatment, Fogel [1999] for a historical assessment.
5
Two exceptions are Gersovitz [1983] and Blackburn and Cipriani (1998). Gersovitz considers a partial
equilibrium model where mortality risk declines with consumption early in life. Our formalization di®ers by
assuming future survival depends upon (public) health expenditure, not past consumption.
Blackburn and Cipriani analyze the implications of infant mortality, which depends upon health-care
expenditure, on fertility and capital accumulation.
6
It is more plausible to think of health investment here as implementing technologies and programs already
available elsewhere (e.g., developed nations or through development agencies), rather than the direct cost of
inventing them.
6
In general, a person's health status depends upon her private health choices as well as on
public health facilities she can avail of. Public health intervention, particularly in developing
countries, is likely to play a critical role for an important reason. Table 1, from Dasgupta
[1993], divides mortality incidence in developing and developed countries according to causes
of death: 45% of all deaths in developing countries occurs due to infectious and parasitic
diseases, as opposed to only 5% in developed countries. Since these diseases, by their very
nature, create externalities, only public intervention can circumvent the underinvestment
that would result from private choices. Public health expenditure in new medical facilities,
sanitation improvements, disease control and inoculation programs augments private health
capital by reducing the economy-wide risk of contacting fatal diseases.
Private health inputs, on the other hand, may take various forms including nutrition
and preventative medicine early in life. A majority of people in developing countries may
be unable to a®ord these investments on a regular basis. Keeping this in mind, we ignore
all forms of private health investment and expand the scope of public health interventions
to include provision of subsidized nutritional supplements and clinical treatments.
7
2.1 The Environment
Consider now an overlapping generations economy where individuals may live for two pe-
riods: they live in youth for sure, but survive into old age with probability Á.
8
In any
period t, young agents of measure one are born, each with a time endowment of 1 unit.
7
Although this is the preferred interpretation here, note that the model can just as well be viewed as one
of private health investment by interpreting the health tax as a cost of purchasing private health inputs.
8
Since a new-born expects to live for 1+Á periods, the terms `life expectancy', `longevity', `length of life'
and `survival probability' are all used interchangably in the rest of the paper.
7
A generation-t individual supplies this labor time inelastically, earning a wage income w
t
.
Public health expenditure is ¯nanced through a proportional tax ¿
t
2 (0;1) on labor income.
Health investment, per generation-t person, therefore equals ¿
t
w
t
.
The survival probability of a generation-t person, Á
t
, depends upon her health stock,
h
t
. In particular, it is given by a non-decreasing concave function
Á
t
= Á(h
t
)(1)
that satis¯es Á(0) = 0; lim
h!1
Á(h)=¯ · 1andlim
h!0
Á
0
(h)=°<1.
9
Public health
investment augments private health capital through a constant returns technology:
h
t
= g(¿w
t
)=¿w
t
:
Each generation-t person gives birth to a new individual at the end of period t,beforeshe
realizes her `mortality shock'. This new individual becomes economically active only at the
beginning of period t+1; and in particular, does not inherit her parent's health stock.
10
To abstract from the risk associated with uncertain lifetimes and concentrate on the
length-of-life e®ect, we follow Yaari [1965] and Blanchard [1985] in assuming a perfect
annuities market whereby all savings are intermediated through mutual funds.
11
At the
end of her youth, each agent deposits her savings with a mutual fund. The mutual fund
9
One example that satis¯es these properties is Á(h)=h=(1 + h)with¯ = ° =1:
10
Incorporating health transfers through birth adds one more dimension to the dynamical system without
altering the basic insights.
11
If annuity markets were not perfect, a lower mortality rate would raise realized returns on investment
and encourage savings. In that case, the results that follow understate true magnitudes. But a perfect
annuities market assumption may not be far-fetched even for developing countries. When life-contingent
annuity markets are not well-developed, the family can self-insure against mortality risks through interfamily
transfers, substituting by more than 70% for perfect market annuities [Kotliko® and Spivak, 1981].
8
invests these savings in capital (the sole asset) and guarantees a gross return of
b
R
t+1
to
the surviving old.
12
If a fund earns a gross return R
t+1
on its investment, then perfect
competition would ensure that
b
R
t+1
= R
t+1
=Á
t
holds in equilibrium.
A generation-t agent maximizes her expected lifetime utility,
U
t
= u(c
t
t
)+Á
t
u(c
t
t+1
); (2)
subject to the period budget constraints:
c
t
t
· (1¡¿
t
)w
t
¡z
t
;
c
t
t+1
·
b
R
t+1
z
t
;
taking as given the vector of prices (w
t
;
b
R
t+1
). Here u denotes the period utility function
and z denotes savings of the youth. When u is homothetic, the savings function takes a
simple form,
z
t
=(1¡¿
t
)¾(
b
R
t+1
;Á
t
)w
t
; (3)
¾ being the savings propensity.
As in the standard neoclassical model, production is carried out using capital and la-
bor, through a constant returns-to-scale technology, F(K;L), that satis¯es the usual Inada
conditions. If k denotes the capital stock per worker, let f(k) ´ F(k;1) represent the
intensive-form production function. Perfect competition in the ¯nal goods market implies
12
An alternative assumption is one where the government takes over the assets of generation-t agents who
die prematurely and transfers them lump-sum to those alive. This gives qualititatively similar results as long
as the transfers are made to surviving members of the same cohort. If, however, these assets are transferred
to the progeny of the deceased (accidental bequests), asymptotic growth may result [Fuster, 1999].
9
that both labor and capital are paid their respective marginal products, so that prices are
given by:
w
t
= f(k
t
)¡k
t
f
0
(k
t
); (4)
R
t
=1+f
0
(k
t
)¡±; (5)
where ± is the depreciation rate of physical capital.
2.2 General Equilibrium
The asset market clears when the capital stock at t + 1 equals savings at t. When the
tax rate is exogenously given and time-invariant, ¿
t
= ¿ 8t, equilibria are characterized by
sequences of f(k
t
;h
t
)g that satisfy equations (1), (4), (5) and (6) ¡ (8) below:
k
t+1
=(1¡¿)¾(
b
R
t+1
;Á
t
)w(k
t
); (6)
h
t
= ¿w(k
t
); (7)
b
R
t+1
= R
t+1
=Á
t
: (8)
Substituting equations (1), (7) and (8) into (6) allows us to characterize the system by a
single ¯rst-order di®erence equation in the capital stock:
k
t+1
=(1¡¿)¾(k
t+1
;k
t
)w(k
t
); (9)
where ¾(k
t
;k
t+1
) ´ ¾(R(k
t+1
);Á(k
t
)) and Á(k
t
) ´ Á(¿w(k
t
)).
If youthful and old-age consumptions are gross substitutes, the propensity to save is
increasing in the interest factor and decreasing in the future capital stock, k
t+1
. An increase
in the current capital stock k
t
, on the other hand, allows society to invest more in health,
10
but has two opposing e®ects on the propensity to save. A higher capital stock allows for
increased health expenditure. This raises the chance of survival, Á
t
, and thus, the propensity
to save out of lifetime income. At the same time, since it lowers the equilibrium return on
savings,
b
R
t+1
, it tends to lower the propensity to save. As long as the direct length-of-life
e®ect dominates the indirect interest rate e®ect, the savings propensity is increasing in k
t
.
Assuming this is true,
13
equation (9) describes a monotonically increasing phase map in
(k
t
;k
t+1
)space.
It isnowrelativelystraightforwardto determinehowequilibriumsequencesoffk
t
gbehave.
Consider an economy that starts out with relatively low capital per worker. Since resources
are scarce, the economy is unable to invest much in the health of its population and mortal-
ity rates are high. High mortality rates make individuals e®ectively impatient, depressing
saving and investment. The economy's future stock of capital is lower than it would oth-
erwise be { the adverse e®ects of current high mortality spills into the future, constraining
health choices that future generations are able to make. As long as high mortality does
not severely depress savings, the economy grows by accumulating capital, only at a slower
pace. When saving and investment are particularly sensitive to mortality, however, history
matters: a low-income high-mortality society is unable to grow out of the `vicious cycle' of
poverty and ill-health.
To examine these ideas more closely, consider a version of the economy where the period
utility function is logarithmic and the aggregate technology Cobb-Douglas: u(c)=logc,
13
For instance, when the period utility function is CES, u(c)=c
1¡1=¾
=(1¡1=¾), the savings propensity
¾ = ÁR
¾¡1
=
¡
1+ÁR
¾¡1
¢
is increasing in Á:
11
f(k)=Ak
®
, ® 2 (0;1);A>0. The savings propensity is then simply Á=(1+Á), so that
¾(k
t
;k
t+1
) ´ ¾(k
t
)=
Á(¿w(k
t
))
1+Á(¿w(k
t
))
:
Equilibrium sequences of the capital stock now satisfy
k
t+1
=(1¡¿)¾(k
t
)w(k
t
) ´ J(k
t
); (10)
where,
J(k) ´ (1¡¿)(1¡®)
·
Á(Bk
®
t
)
1+Á(Bk
®
t
)
¸
Ak
®
t
;B´ ¿(1¡®)A:
The asymptotic behavior of this economy depends upon whether or not equation (10)
possesses multiple positive steady-states. When a unique positive steady-state exists, the
economy converges to it asymptotically as in the standard model. When multiple steady-
states exist, there are two of these. The lower one is asymptotically unstable, the higher one
asymptotically stable. The proposition below speci¯es conditions under which this happens
(refer to Appendix A for proof), while Figures 2 and 3 depict the dynamics of the economy.
Proposition 1 The dynamical system described by equation (10) possesses three steady-
states f0;k
1
;k
2
g,withk
2
> k
1
as long as ®>1=2. The two extreme steady-states, 0 and
k
2
, are asymptotically stable, while the intermediate one is not. When ®<1=2,twosteady
states f0;kg exist, only the positive one being asymptotically stable.
The economy converges monotonically to the positive steady-state k,inFigure2,ir-
respective of its initial position. Consider two countries that di®er only in their initial
health capital. In particular, both start with similar income levels but, for historical and
12
climatic reasons, individuals in one country enjoy the higher survival probability ¯.The
dynamic behavior of the society with better health is described by the dotted line in Figure
2, whereas the solid phase-line applies to the high-mortality society. In the long run, both
economies attain similar income levels. However, by virtue of having longer-lived citizens,
the low mortality society grows faster during the early stages by accumulating assets faster.
In the process, it is able to improve the living standards of all its transitional generations
in a manner that the less healthy society cannot.
Figure 3 illustrates the starker case of history dependence. Two countries now need not
converge to similar income levels even if they di®er only in their initial health capitals. Here,
unless a high mortality society starts out with a high enough capital stock (above k
1
), it is
unable to escape the twin traps of poverty and ill-health. Proposition 1 suggests that such
a trap exists when the output share of capital is greater than 1=2.
14
Intuitively, this results
from the dependence of the savings rate on the capital stock through health investment.
The higher the capital-elasticity of wage income ®, the more labor-income responds to the
availability of capital: with ®>1=2, accumulating capital allows for a relatively large
increase in wages that may be invested more extensively in mortality reduction. In other
words, small changes in the capital stock result in large gains in mortality reduction. In turn,
this increased longevity provides impetus to capital accumulation and it is this sensitive
14
Given existing estimates of the share of physical capital in the U.S., a value of ® greater than 0:5maybe
rationalized by broadening the concept of capital. By including human capital, we would expect the share
to be in the range (0:6;0:8) as in Mankiw et al. [1992]. Incorporating business capital, on the other hand,
gives an estimate of 0:71 as in Parente and Prescott [1994].
The exact restriction on ® in Proposition 1 is, however, sensitive to our annuities markets assumption. If
assets of the prematurely deceased were instead distributed lump-sum to existing members of that cohort,
the restriction weakens to ®>1=3:
13
two-way dependence that generates poverty traps.
Twoclari¯cationsareinorderhere. Wehaveassumedsofarthatthehealthtaxis
time invariant. Appendix B shows that individuals in low-income countries also prefer a
lower tax rate. When mortality rates are already high, the current utility cost of a higher
health tax hurts more than it helps by way of improving future consumption possibilities.
Besides being unable to invest enough resources in health, poorer economies also choose to
underinvest in health when the gains from that investment are not realized immediately.
15
Secondly, we have ignored the possibility that individuals may be altruistic. If house-
holds cared for their o®springs, they could substitute for their old-age consumption by
bequeathing assets to their progeny. In that case, even when mortality rates are high, cap-
ital accumulation may not su®er as much since individuals save to pass on those assets to
the next generation. Despite this, as long as individuals also cared for their old-age con-
sumption or were not fully altruistic, higher mortality rates would depress economic growth
[Ehrlich and Lui, 1991; Meltzer, 1995] and slow the process of convergence.
3 Mortality Risk and Investment in Education
Even when perfect annuities markets exist, mortality risks in certain kinds of investment
may be undiversi¯able. This is especially true of investments in \inalienable" human capital
[Hart and Moore, 1994], like education.
Based on age-speci¯c mortality rates and earnings pro¯les in developing countries,
15
This may partly explain why high-mortality low-income countries tend to spend a much smaller fraction
of GDP on public health [World Bank, 1993]. But as Table 1 suggests, developed countries also tend to
su®er from more `expensive' diseases.
14
Meltzer (1995) demonstrates that mortality declines can have large impact on school en-
rollment rates and signi¯cantly improve the social stock of human capital. The upshot is
that when human capital is the engine of economic growth, mortality rates magnify initial
cross-country di®erences in income and health into persistent di®erences in living standards.
This simple intuition is easy to capture with a minor modi¯cation of the previous anal-
ysis. Consider a similar setup except where individuals are endowed with 1 unit of labor
timeinyouthaswellasinoldage.Denotebyx
t
the average stock of skills of the period-t
workforce, now consisting of both young and old agents. Increments in x take the form
of labor-augmenting technological improvements that once invented are never lost. Con-
sequently, x
t
represents the skills that old members of generation t¡ 1acquiredthrough
schooling (see below), as also the skills that the youth in period-t inherit from their preced-
ing generation.
An individual born at t is e®ectively born with a labor e±ciency of x
t
. The individual
can, however, choose to improve her productivity in the second period of life. She can do so
by spending a fraction, s
t
2 [0;1], of her time in attending school when she is young. Her
future productivity depends upon the inherited stock of skills and investment in education
according to:
x
t+1
= x
t
¹(s
t
); (11)
where ¹ is an increasing and concave function, satisfying ¹(0) = 1.
Time spent by a young individual in school is chosen to maximize her lifetime income,
(1¡¿)(1¡s
t
)w
t
x
t
+
w
t+1
x
t+1
b
R
t+1
; (12)
15
where ¿ is the time-invariant tax rate on youthful wage income. The ¯rst-order condition
for an interior optimum equates returns on the two types of assets { physical and human
capital:
(1¡¿)w
t
w
t+1
=
b
R
t+1
= ¹
0
(s
t
): (13)
When ¹ is strictly concave, we can express optimal schooling choice as a function of prices:
s
¤
t
= s
µ
Á
t
w
t+1
w
t
R
t+1
¶
; (14)
where s is increasing in its argument.
Equation (14) captures the essence of the rate-of-return argument. Through perfect
market annuities, individuals can fully insure against mortality risks on physical capital
investment but are unable to do so on their educational investment. Hence, a mortality
decline raises the relative attractiveness of human capital. In equilibrium, another channel
reinforces this e®ect: mortality decline promotes capital accumulation through the length-
of-life e®ect, further raising the rewards to education in the form of higher future wages.
Interestingly, endogenous mortality risks may introduce threshold externalities of the
type that Azariadis and Drazen [1990] elucidate. There, private returns to education depend
uponthe social stockofhumancapital througha technological externality. Ifthis externality
sharply rises when human capital crosses a threshold value, it elicits rapid investment in
human capital and faster growth. Such externalities are one useful way to understand
why developing economies may often stagnate for a while before \taking-o®" into sustained
growth.
Our human capital technology (11) does not explicitly allow for such externalities. Nev-
16
ertheless, when mortality rates are endogenously determined by public health investment,
it alters private returns to education: one form of human capital injects an externality into
investments in another form of human capital.
An easy way to see this is to assume that
¹(s)=1+s
µ
;µ2 (0;1): (15)
Assume also that this is a small open economy that can borrow and lend abroad at the
¯xed interest factor R. This pins down the domestic capital stock at k,andthewagerate
at w ´ w(k). Using (14), we can then solve for the economy's growth rate as,
g
t
= ¹(s
¤
t
)¡1=[±Á(¿wx
t
)]
µ=(1¡µ)
; (16)
where ± ´ µ=[(1¡¿)R]. With constant mortality risks, say a survival probability equal to ¯,
this economy would have immediately jumped to the steady-state growth rate of (±¯)
µ=(1¡µ)
.
But when mortality is endogenous, the growth rate responds gradually to human cap-
ital accumulation since a rising stock of skills enables larger reductions in mortality rates.
When ¹ is moderately concave (µ>1=2), sharp changes in the growth rate are observed
for moderate changes in mortality rates (or, equivalently, knowledge capital).
16
Figure 4
illustrates this possibility of \take-o®": starting from a low stock of human capital, the
economy initially grows slowly and accelerates rapidly only when the social stock of skills
attains a critical mass (near bx).
17
16
This follows by considering the concavity of g(x).
17
This does not, however, allow multiple balanced growth paths with di®erent (positive) rates of skill-
investments to exist.
17
Another key implication of mortality is that small di®erences in health capital now imply
not only large di®erences in income levels, but possibly in growth rates too. To examine
this, let us return to the Cobb-Douglas technology and logarithmic preferences example.
Aggregate output now depends upon physical capital and e±ciency labor supply (N),
Y
t
= AK
®
t
N
t
1¡®
.Lettingk denote capital per e±ciency unit of labor, the intensive form
of the production is again y
t
= Ak
®
t
. To allow for multiple stable steady-states, we also
assume a linear skill accumulation technology:
¹(s)=1+µs; µ > 0: (17)
An individual born in period-t maximizes lifetime utility (2) subject to the lifetime
budget constraint (12) so that the savings function becomes
z
t
= ¾
t
[(1¡¿)(1¡s
t
)w
t
x
t
¡w
t+1
x
t+1
=R
t+1
]; (18)
where ¾ ´ Á=(1 +Á) denotes the savings propensity, as before.
Depending upon the initial stock of human capital, x
0
, the economy exhibits two types
of dynamic behavior: one, where no investment in schooling takes place, and another where
individuals invest in skills through schooling.
Consider ¯rst corner equilibria with no schooling. From (13), this occurs when skill
investment yields a return no greater than that obtainable with physical capital investment,
µ · (1¡¿)
w
t
R
t+1
Á
t
w
t+1
: (19)
As long as (19) is satis¯ed, the social stock of skills remains constant at its initial value,
x
t
= x
0
8t. E±ciency labor supply is then,
N
t
= x
t
+Á
t¡1
x
t
=(1+Á
t¡1
)x
0
:
18
Market clearing now requires that
N
t+1
k
t+1
= z
t
:
Using (18) and equilibrium prices, this may be expressed as
18
k
t+1
= G(k
t
); (20)
where
G(k) ´ (1¡¿)(1¡®)
·
1
1=[¾(k)(1¡¾(k))]+(1¡®)=®
¸
Ak
®
;
¾(k) ´
Á(Bx
0
k
®
)
1+Á(Bx
0
k
®
)
;B´ ¿(1¡®)A:
Whether or not poverty traps occur under no-schooling depends upon the steady-states
of equation (20). Not surprisingly, the dynamics here is similar to that obtained in the
previous section: povertytraps resultonlywhen®>1=2. Otherwise, theeconomyconverges
monotonically to its unique (positive) steady-state capital stock.
Despite this similarity, there is now an important distinction. With solely physical
capital investment, two economies that di®ered only in their initial survival probabilities
converged, in the long-run, to the same steady-state whenever ®<1=2. Better-health
societies grew faster, but only temporarily. In sharp contrast, initial di®erences in health
capital here may result in di®erent outcomes in the long-run.
Consider, for instance, a high-mortality society for which the no-schooling restriction
(19) is satis¯ed at initial levels of (x
0
;k
0
). A better-health society with the same initial
18
In general equilibrium, the no-schooling restriction becomes k
t+1
· ®(1¡¿)A[k
®
t
=Á(k
t
)] ´ H(k
t
): Since
H(k) ¸ G(k) 8k ¸ 0 (with strict equality at zero), an economy that starts at a no-schooling equilibrium
never switches to one with schooling.
19
resources, on the other hand, may have a su±ciently higher survival probability (say, ¯)
suchthatthisrestrictiondoesnothold. Asaresult,eventhoughbothcountriesshare
similar economic conditions, the society whose citizens live longer ¯nds it worthwhile to
invest in knowledge creation and experiences positive balanced growth. Contrast this to
the high-mortality society that remains stuck in a zero-growth equilibrium because its initial
conditions do not favor human capital investment.
When initial conditions admit interior schooling equilibria in both countries, initial
di®erences in mortality translate into widening income gaps, even though both converge to
the same growth path.
Dynamic equilibria, in this case, satisfy the following pair of di®erence equations in
(s
t
;k
t
):
k
t+1
=
®(1¡¿)A
µ
k
®
t
Á(k
t
)
; (21)
·
1¡s
t+1
+Á(k
t
)+
1¡®
®
¾(k
t
)
¸
(1+µs
t
)k
t+1
=(1¡¿)(1¡®)A(1¡s
t
)¾(k
t
)k
®
t
:(22)
Equation (21) follows from manipulating (13) using (17) and equilibrium prices, while sub-
stituting N
t
=[(1¡s
t
)+Á
t¡1
]x
t
into the market clearing condition gives equation (22).
SinceÁ
t
converges to¯ asymptotically, thesystem described bytheseequationspossesses
a positive steady-state (s
¤
;k
¤
) that is saddle-path stable; given (x
0
;k
0
); all sequences of
(s
t
;k
t
) converge monotonically to the steady-state growth rate of µs
¤
.
19
Obviously, two countries with di®erent initial mortality rates would now converge to the
same balanced growth path in the long-run. But despite this convergence in growth rates,
19
See Proposition 2 in Azariadis and Drazen (1990).
20
living standards in the two countries actually diverge. The low-mortality society always
invests more intensively in skills at a higher rate and thereby augments its health capital at
a faster pace. As a result it consistently enjoys a higher growth rate along its saddle-path
than the country with higher mortality-risks.
4 Some Empirical Evidence
The theory we have developed has emphasized the importance of health and longevity
in creating proper incentives for sustainable growth. But how empirically relevant is it?
Can this explain why a number of countries, especially those in Sub-Saharan Africa, have
languishedinill-healthandpovertyforsolong? Ismortalityindeedasigni¯cantdeterminant
of educational investment?
To answer these questions, we turn to cross-country regression analysis in this section.
The cross-country growth literature studies a gamut of issues ranging from the role of in-
stitutions, corruption, democracy, ethno-linguistic fractionalization, to the importance of
capital accumulation, public infrastructure, natural resources, or international trade for
economic growth. Only recently have researchers incorporated into their list of explanatory
variables measures of health status. Those that have, for instance Barro and Sala-i-Martin
[1995], Knowles and Owen [1995] and World Health Organization [1998], ¯nd health indi-
cators to be signi¯cant predictors of future growth rates. The regressions in this section are
in the spirit of those analyses, but are motivated by the speci¯c linkages that our theory
suggests.
The analysis is in terms of GDP per worker, rather than per capita, to control for
21
demographic shifts.
20
The time horizon considered is twenty years, covering 1970-90. The
data used are for 95 countries, though the actual number of countries used in each regression
di®ers depending upon data availability. Estimation is by ordinary least squares in all cases.
Appendix C provides details about data sources and de¯nitions.
Longevity was shown to have a level e®ect in Section 2. To test this we ¯rst regress the
end-of-period per worker GDP, i.e., GDP per worker in 1990 (LRGDP90), on the average
population growth rate (LPOP), the initial investment-to-GDP ratio (LINVSH), and the
initial enrollment rate in secondary education (ENROLS70). The secondary enrollment
rate is taken as a proxy for the initial stock of education capital. The result is reported
in column (1) of Table 2. As the human capital augmented neoclassical model suggests,
the investment rate and initial stock of human capital both signi¯cantly increase per capita
GDP, while population growth reduces it. Column (2) adds another explanatory variable,
life expectancy in 1970 (LIFEXP70) that the theory suggests has a signi¯cant e®ect on
capital accumulation. The results are striking | all variables except for life expectancy,
cease to be statistically signi¯cant. Indeed, in a regression of only life expectancy on the
end-of-period GDP per worker, life expectancy explains about 81% of the cross-country
variation in income levels (not reported in Table 2).
Our theory also suggests that longevity has a growth e®ect through human capital ac-
cumulation. The ¯rst step in testing this is to consider the relative e®ects of initial human
capital and initial life expectancy on the average growth rate for 1970-90. The ¯rst regres-
20
Demographic shifts are usually controlled for using fertilityrates inGDP per capita regressions. However,
since both mortality and fertility depend upon the underlying health stock, they are likely to be collinear.
22
sion, column (1) in Table 3, looks at the e®ect of the enrollment rate in 1970 (ENROLS70)
on the subsequent average growth rate (GR7090), controlling for initial income per worker
(LGDPEA70). Once again, as per the conditional convergence hypothesis, the coe±cient
on initial income is negative and highly signi¯cant. The initial stock of human capital
positively and signi¯cantly a®ects the future growth rate.
The second regression, in column (2), excludes human capital, and looks at the forces
of convergence versus initial life expectancy (LIFE70). The convergence prediction holds,
while the e®ect of life expectancy is high and signi¯cant. Moreover, initial per capita income
and life expectancy explain 42% of cross-country growth variation.
Column (3) of Table 3 includes both initial human capital and life expectancy as ex-
planatory variables. Observe how ENROLS70 ceases to be signi¯cant at the 5% level, while
LGDPEA70 and LIFE70 have the correct signs and continue to be statistically signi¯cant.
In other words, the theoretical predictions are consistent with the evidence.
Cross-country growth regressions often incorporate a dummy for Sub-Saharan Africa.
The coe±cient for this dummy is usually negative, suggesting geographical factors at work.
To see how much of that is region-speci¯c and how much simply the result of high mortality,
we next include a dummy for Sub-Saharan Africa (SSAFRICA). Also included is the average
shareofinvestmentinGDPbetween1965-69(INVSH). Columns (1) and (2) of Table 4
examine the result of this exercise with and without initial life expectancy. Inclusion of
life expectancy in the regression makes the initial stock of human capital insigni¯cant. It
also takes away the in°uence of the regional dummy variable, suggesting that Sub-Saharan
countries are poor not due to region-speci¯c characteristics, but because life expectancies
23
there are the lowest in the world. Moreover, the coe±cient on the investment-to-GDP ratio
has the incorrect sign, although statistically not di®erent from zero. Here too, a large part
of the e®ect of investment-to-GDP ratio on the growth rate seems to be explained by its
dependence on longevity.
Finally, as a crude sensitivity check, the regression model is extended to include several
other variables commonly used in cross-country regressions. These are the black-market
premium for exchange rates (BMP6590) as an indicator of corruption, terms of trade e®ects
(TOT6590), share of primary goods in exports (SXP) as an index of natural resources, the
number of telephones per worker as an index of public infrastructure (LTELPW), and the
initial total fertility rate (TFR70), to control for fertility induced demographic shifts. The
results appear in column (3) of Table 4. The only signi¯cant variables are initial GDP per
worker, initial life-expectancy and the share of primary goods in total exports.
21
All other
variables have the correct signs but none is highly signi¯cant.
Cross-country regressions are well known for their sensitivity to the list of variables
included [Levine and Renelt, 1992]. It is therefore encouraging to note that the only two
variables that strongly predict future growth rates in the regressions presented here are
initial income and longevity. In particular, human capital, in the form of education, public
infrastructure or other policy measures do not appear to be substantive factors contributing
to economic growth once a mortality measure is included. As Table 3 shows, most of
the variability in growth rates comes from initial incomes and initial di®erences in health.
21
The negative association between natural resources and economic growth was pointed out in Sachs and
Warner [1995]. Asea and Lahiri [1999] suggest that this results from the e®ect of natural resource abundance
on human capital investment.
24
The evidence complements what we already know from other micro- and disease-speci¯c
studies on the role of health. More rigorous econometric methods, however, are necessary
to ascertain the true magnitude of health's contribution to economic growth.
5 Conclusion
Widespread evidence, at both individual and aggregate levels, points to a strong correlation
between poverty and ill-health. This is often ascribed to the adverse impact of poverty
on health, but recent evidence suggests that health could equally well be an important
determinant of economic welfare. This paper has examined how better health, by improving
longevity and reducing mortality risks, may be conducive to growth and development.
The analysis points to health investment as a prerequisite for sustainable economic
growth. In particular, savings and investment rates are systematically low in high mortality
societies since low life expectancy raises the e®ective impatience of individuals and decreases
returns on human capital investment. The implication is that such societies would not grow
as fast as standard theory predicts even when technology is not the bottleneck. Mortality
risks may indeed be so growth-depressing as to result in `development traps'. These results
are broadly consistent with existing evidence on cross-country growth and regression results
indicate the growth e®ects of longevity improvements to be signi¯cantly large.
However, to better gauge the importance of health, one needs to quantify its contribution
better. There are two ways of doing this. One approach is use cross-country evidence and
more sophisticated econometric methods than have been used here. Since cross-country
analyses su®er from a variety of problems, a more promising approach may be the use
25
of quantitative methods, as have been used in McGrattan and Schmitz's [1998] analysis
of cross-country growth or Kalemli-Ozcan et al.'s [1998] study of the e®ects of (exogenous)
mortality decline. If the theory presented here is any indication, health capital could explain
a surprisingly large portion of cross-country income and growth di®erentials.
26
Appendix A: Proof of Proposition 1
To prove Proposition 1, we establish the following lemma.
Lemma 1
The function J(k) de¯ned in equation (10) satis¯es the following properties:
(i) J(0) = 0;
(ii) J
0
(k) ¸8k ¸ 0;
(iii) lim
k!1
J(k)=k < 1; and
(iv) lim
k!0
J
0
(k)=1 i® ®<1=2:
Proof. The ¯rst two properties are easy to check. To prove (iii), note that lim
h!1
¾ =
¯=(1 +¯). Therefore,
lim
k!1
J(k)
k
= C lim
k!1
¾(k)
k
1¡®
= C
µ
¯
1+¯
¶
lim
k!1
1
k
1¡®
=0;
where C ´ (1¡®)(1¡¿)A.
Now,
J
0
(k)=C(¾
k
k
®
+®¾k
®¡1
);
where,
¾
k
= ®¿(1¡®)A
Á
0
(1+Á)
2
k
®¡1
;
27
using the chain rule. Therefore, using lim
k!0
Á
0
(¿(1¡®)Ak
®
)=°,weobtain
lim
k!0
J
0
(k)=C lim
k!0
·
®¿(1¡®)A
Á
0
(1 +Á)
2
k
2®¡1
+®¾k
®¡1
¸
= C
·
®¿(1¡®)°Alim
k!0
k
2®¡1
+® lim
k!0
¾
k
(1¡®)k
¡®
¸
(using L'Hopital's Rule)
= ®C
·
¿(1¡®)°Alim
k!0
k
2®¡1
+¿®°Alim
k!0
k
2®¡1
¸
= 1 i® ®<1=2:
When ®>1=2; on the other hand, lim
k!0
J
0
(k)=0:
Proposition 1 follows immediately from these results. By (i), 0 is always a steady-state of
(10). By (ii) and (iii), the phase map is monotonically increasing and eventually falls below
the 45
o
line, so that at least one positive steady-state exists. Whether or not multiple such
steady-states exist depends upon the stability of 0. By (iv), 0 is a stable steady-state i®
®>1=2, intersecting the 45
o
line from below. In that case, J(k)intersectsthe45
o
line
from below at least once before falling below it. Hence, at least one positive steady-state
separates 0 from the asymptotically stable highest one.
Appendix B: The `Optimal' Tax Rate
Current public health expenditure bene¯ts only the currently young, who also bear the cost
of funding these investments. Therefore young agents born at t choose ¿
t
to maximize their
expected lifetime utility. Using the probability function (1), this maximization problem
becomes:
Max
fz
t
;¿
t
g
log[(1¡¿
t
)w
t
¡z
t
]+Á(¿
t
w
t
)log
h
b
R
t+1
z
t
i
:
28
The ¯rst-order condition for z
t
gives:
z
t
=(1¡¿
t
)
·
Á
t
1+Á
t
¸
w
t
; (B.1)
while that for ¿
t
equates the marginal utility cost to the (discounted) marginal utility gain
from a greater possibility of surviving:
w
t
c
t
t
= w
t
Á
0
(¿
t
w
t
)log
h
b
R
t+1
z
t
i
(B.2)
(B.1) can be used to simplify (B.2) and obtain an equation solely in the tax rate ¿
t
; given
the price vector (w
t
;
b
R
t+1
):
­(¿
t
) ´
1+Á(¿
t
w
t
)
1¡¿
t
= w
t
Á
0
(¿
t
w
t
)log
·
b
R
t+1
(1¡¿
t
)w
t
Á(¿
t
w
t
)
1+Á(¿
t
w
t
)
¸
´ ¤(¿
t
):
­ is monotonically increasing in ¿, while ¤ is inverted U-shaped satisfying lim
¿!0
¤=
¡1 = lim
¿!1
¤. When ­(¿)and¤(¿)intersect,theydosotwice;theoptimaltaxrate¿
¤
is obtained at the higher point of intersection. Moreover, ¿
¤
depends positively upon wage
income, w; as well as on the interest factor,
b
R.
29
Appendix C: Data Sources
The data used in Section 4 are from several sources. The list of variables, their de¯nitions
and sources are listed below. The Sachs-Warner [1995] dataset is available at
\http://www.nu®.ox.ac.uk/Economics/Growth/datasets.htm",
the Easterly-Levine [1997] dataset at
\http://www.worldbank.org/html/prdmg/grthweb/datasets.htm",
and the Barro-Lee [1993] dataset at both sites.
LRGDP90: Real GDP per capita in constant dollars (expressed in international prices, base
1985). Source: The Penn World 5.6 [see Summers and Heston, 1991]
ENROLS70: Total gross enrollment ratio for secondary education in 1970, from UNESCO.
Source: Barro-Lee [1993].
LPOP: Log of average population growth rate between 1970-90. Source: World Bank [1998].
INVSH: Average ratio of real domestic investment (private plus public) to real GDP between
1965 and 1969. Source: The Penn World Table 5.6.
LINVSH: Natural logarithm of INVSH.
LIFEXP70: Life expectancy at birth in 1970. Source: World Bank [1998]
LIFE70: Natural logarithm of LIFEXP70.
LGDPEA70: Natural logarithm of real (purchasing power parity adjusted) GDP per eco-
nomically active population in 1970. GDP data are from the Penn World Tables, Mark 5.6,
and are in constant 1985 international prices. The economically active population is de¯ned
30
as the population between the ages 15-64 and is taken from the World Data CD-ROM, 1995,
World Bank. Source: Sachs-Warner [1995].
GR7090: Average annual growth in real GDP per economically active population between
1970 and 1989. Based on the same original sources as LGDPEA70. Source: Sachs-Warner
[1995].
SSAFRICA: Dummy variable for Sub-Saharan Africa.
BMP6590: Average black market premium between 1965-69. BMP = (black market ex-
change rate/o±cial exchange rate) ¡1, where the exchange rate = local currency per dollar.
Source: Barro-Lee [1993].
TOT6569: Terms of trade shock (growth rate of export prices minus growth rate of import
prices) between 1965-69. Source: Barro-Lee [1993].
SXP: Share of exports of primary products in GNP in 1970. Source: Sachs-Warner [1995].
LTELPW: Log of telephones per 1000 workers. Source: Easterly-Levine [1997].
31
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5
6
7
8
9
10
30 40 50 60 70 80
Life Expectancy
Log per capita GDP
! = 0.76
Source: World Bank (1998). Per capita GNP is measured in PPP$. 
These 109 low- and middle-income countries are those 
in which 1996 per capita GNP was $9,635 or less.
Figure 1: Life Expectancy at Birth and Log Per Capita GDP
37
Cause of Death (%) Developing Developed
Countries Countries
Infectious and parasitic diseases 45 5
Diarrhoeal (13) -
Tuberculosis (8) -
Acute respiratory illness (17) 3
Perinatal diseases 81
Cancers 721
Circulatory and degenerative diseases 17 54
Injury 67
Other causes 17 12
Source: Dasgupta (1993), Table 4.3
Table1:CausesofDeathsinDevelopingandDevelopedCountries
Dependent Variable: LRGDP90
(t-statistic in brackets)
(1) (2)
CONSTANT 6.77 3.62
(13.23) (6.05)
LPOP -0.33 -0.09
(-2.87) (-0.87)
LINVSH 0.39 0.06
(3.19) (0.62)
ENROLS70 1.81 0.55
(4.31) (1.39)
LIFEXP70 0.07
(7.59)
R
2
0.71 0.83
R
2
0.70 0.82
OBS. 86 84
Table 2: Testing for the Length-of-Life E®ect
38
Dependent Variable: GR7090
(t-statistics in brackets)
(1) (2) (3)
CONSTANT 6.16 -25.85 -21.00
(2.72) (-7.44) (-4.23)
LGDPEA70 -0.73 -1.70 -1.80
(-2.36) (-5.53) (-5.63)
ENROLS70 4.45 1.36
(4.44) (1.35)
LIFE70 10.29 9.17
(7.90) (5.94)
R
2
0.19 0.42 0.42
R
2
0.17 0.41 0.41
OBS. 94 94 93
Table 3: Mortality Risk, Education and Growth Rates
39
Dependent Variable: GR7090
(t-statistic in brackets)
(1) (2) (3)
CONSTANT 9.14 -18.59 -9.69
(3.90) (-2.63) (-1.10)
LGDPEA70 -1.14 -1.79 -2.89
(-3.75) (-5.53) (-5.71)
ENROLS70 3.11 1.43 0.27
(3.07) (1.40) (0.24)
INVSH 1.65 -0.06 0.45
(1.85) (-0.07) (0.45)
LIFE70 8.58 8.34
(4.12) (3.71)
SSAFRICA -1.40 -0.33
(-2.93) (-0.63)
BMP6590 -0.52
(-0.78)
TOT6569 -5.02
(-1.52)
SXP -5.63
(-2.62)
LTELPW 0.55
(1.89)
R
2
0.31 0.42 0.55
R
2
0.28 0.39 0.48
OBS. 94 93 73
Table 4: E®ect of Longevity on Cross-country Growth
40
k
t"1
45
o
k
t
k
High Mortality
Low Mortality ()#$%
k
0
Figure 2: Convergence under the Length-of-Life E®ect
41
k
t
k
t"1
45
o
k
1
k
2
High Mortality
Low Mortality ()#$%
Figure 3: Non-convergence under the Length-of-Life E®ect
42
x
t
!
x
()
/( )
&$
''1(
g
t
')12/
Growth under high mortality with
Growth under low mortality ()#$%
Figure 4: Threshold E®ects and Mortality Risk
43