Life Potential as a Basic Demographic Indicator

Documentos
de Trabajo
8
Documentos
de Trabajo
8
2012
Francisco J. Goerlich Gisbert
Ángel Soler Guillén
Life Potential
as a Basic
Demographic
Indicator
Plaza de San Nicolás, 4
48005 Bilbao
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dt_bbva_2012_8.indd 1
28/5/12 15:53:05
Life Potential as a Basic
Demographic Indicator
Francisco J. Goerlich Gisbert1,2
Ángel Soler Guillén1,2
U N I V E R S I T Y O F VA L E N C I A
I N S T I T U T O VA L E N C I A N O D E I N V E S T I G A C I O N E S E C O N Ó M I C A S ( I v i e )
1
2
Abstract
This working paper proposes an indicator that integrates
life expectancy with the demographic structure of the population for a given society, combining the simple indicators of mortality and aging. Life expectancy at birth is independent of the demographic structure of the population
and is, therefore, adequate for measuring overall mortality.
However, it neglects to take into account the fact that life
expectancy increases as society ages. We propose a simple
indicator that integrates life expectancy at different ages, not
only at birth, with the demographic structure of the population at a given point in time. The indicator has an intuitive
interpretation in terms of the life potential, or biological
capital, of society; and given that it is a weighted average,
its changes can be easily decomposed into reductions in
mortality (gains in life expectancy) and aging for different
age intervals.
Key words
Life expectancy, life table, aging, demography.
Resumen
Este documento de trabajo propone un indicador que integra la estructura demográfica de la población de una sociedad en la esperanza de vida, combinando la mortalidad y
el envejecimiento de la población. La esperanza de vida al
nacer es independiente de la estructura demográfica de la
población y, por tanto, es adecuada para medir la mortalidad. Sin embargo, no tiene en cuenta el hecho de que a medida que la esperanza de vida crece, la sociedad envejece.
Se propone un indicador simple que aglutina la esperanza
de vida a diferentes edades, no solo al nacer, con la estructura demográfica de la población en un momento dado del
tiempo. Dicho indicador tiene una interpretación intuitiva
en términos de potencial de vida, o capital biológico, de la
sociedad; y, dado que es una media ponderada, sus cambios
pueden ser fácilmente descompuestos en disminuciones de
la mortalidad (ganancias en esperanza de vida) y envejecimiento por intervalos de edad.
Palabras clave
Esperanza de vida, tablas de vida, envejecimiento, demografía.
publicar el
el presente
presente documento
documentode
de trabajo,
trabajo,lalaFundación
FundaciónBBVA
BBVAno
no asuasuAl publicar
alguna sobre
sobre su
su contenido
contenido ni
ni sobre
sobre la
la inclusión
inclusión en
en el
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me responsabilidad
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mismo de documentos
facilitada por
por los
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mismo
documentos o información
información complementaria
complementaria facilitada
autores.
autores.
working
paper
does
not not
imThe BBVA
BBVA Foundation’s
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decisiontotopublish
publishthis
this
working
paper
does
imply
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responsibilityfor
foritsitscontents,
contents,or
or for
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ply any
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supplementary documents
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theauthors.
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Versión: Abril
Junio 2012
2012
© los
autores,J.2012
Francisco
Goerlich Gisbert and Ángel Soler Guillén, 2012
© de esta
esta edición
edición // of
ofthis
thisedition:
edition:Fundación
FundaciónBBVA,
BBVA,2012
2012
edita // PUBLISHED
publishedBY
by
EDITA
Fundación BBVA,
BBVA, 2012
2012
Fundación
Plaza de
de San
San Nicolás,
Nicolás,4.4.48005
48005Bilbao
Bilbao
Documento de Trabajo – Núm. 8/2012
1.
Introduction
LIFE expectancy at birth summarizes in a single number the mortality conditions of a given
population, and it does so in a way that is independent of the age structure of the underlying
population. Essentially this means that the indicator is comparable, in time and across societies,
with populations having very different age structures. This feature has contributed in making
life expectancy one of the most widely used indicators in international comparisons on development. Additionally, life expectancy at birth is one of the simplest summary measures of
population health for a community (Murray et al. 2002), and as a consequence, of its degree of
development (Sen 1998, 1999).
For all these reasons, life expectancy becomes one essential dimension in the complex and
elusive concept of quality of life: without life there is no possibility to enjoy consumption opportunities as represented by per capita income, the other well-known development indicator widely
used in international comparisons. However, as has been recently recognized in the Stiglitz, Sen and
Fitoussi (2009) report, it is necessary to go beyond gross domestic product (GDP) in measuring the
progress of current societies. This was in fact the goal of the Human Development Index (HDI) of
the United Nations Development Program (http://hdr.undp.org), as well as many other proposals in
including life expectancy as part of synthetic quality of life indexes (Osberg and Sharpe 2002).
It is widely recognized that there is a high correlation between life expectancy at birth
and per capita income, in a given country and for a sufficiently long time span, as well as for a
cross-section of countries at different stages of development. However, this relationship is nonlinear, has no clear shape and moreover we may find countries with relatively low per capita
income that have a far superior life expectancy than countries with a higher per capita income
(Sen 1998). This relationship, known as the Preston (1975) curve, can be seen in figure 1, where
we can see that on average life expectancy is much lower for countries with lower per capita
income. The linear correlation coefficient between the variables represented in figure 1 is 0.62,
but clearly the relationship is non-linear. The curve drawn corresponds to the regression of life
expectancy at birth on the logarithm of GDP per capita, the correlation in this case rises to 0.80.
Taken at face value, we need a bit more than a 16% increase in per capita GDP for a one year
increase in life expectancy at birth, and so doubling per capita income represents an increment
of about six years in life expectancy at birth. The relationship drawn shows a decreasing elasticity, which for our sample values oscillates from about 0.13 to around 0.07.
3
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The Preston curve: Life expectancy at birth versus GDP per capita, 2009
FIGURE 1
1::
FIGURE
figure 1:
The Preston curve: Life expectancy at birth versus GDP per capita, 2009
85
85
Spain
Spain
Japan
Japan
Luxembourg
Luxembourg
80
80
United
United States
States
Qatar
Qatar
75
75
Life
Life expectancy
expectancy at
at birth
birth (years)
(years)
70
70
65
60
55
Equatorial Guinea
50
45
Afghanistan
40
0
le0 = 14.82 + 6.11 x log(GDP per capita) R2 = 0.64
10,000
10.000
20,000
20.000
30,000
30.000
40,000
50,000
60,000
40.000
50.000
60.000
GDP per capita, PPP (current international $)
70,000
70.000
80,000
80.000
90,000
90.000
100,000
100.000
Source: World Development Indicators World Bank (2011).
Source: World Development Indicators Word Bank. (2011).
An important conclusion from figure 1 is that, as income increases life expectancy at
birth has lower informational content regarding the development of a given country. In fact, we
An important conclusion from figure 1 is that, as income increases life expectancy at birth has
can see that the regression tends to over-fit the highest values of GDP per capita. At low level
lower informational content regarding the development of a given country. In fact, we can see that the
of income, the coefficient of variation of life expectancy for the countries shown in figure 1 is
regression tends to over-fit the highest values of GDP per capita. At low level of income, the
0.121, whereas for the high level income countries the coefficient of variation is just 0.0201,
coefficient of variation of life expectancy for the countries shown in figure 1 is 0.121, whereas for the
which signals the compression of life expectancy for the most developed countries2.
high level income countries the coefficient of variation is just 0.020,11 which signals the compression of
What is not evident from figure 1 is that as life expectancy increases society ages, a fact
life expectancy for the most developed countries.22
that results eventually from the increase in longevity. In the first stage of the demographic tranWhat is not evident from figure 1 is that as life expectancy increases society ages, a fact that
results eventually from the increase in longevity. In the first stage of the demographic transition
mortality falls at early childhood (Davis 1945; Vallin 2002), so the population pyramid widens at its
1
The
Worldfertility
Bank defines, for
2009,
low level income
countries
those with
GDP per
capita,incurbase,
but
conditions
andas mature
mature
societies
advance
the
base,
but as
as fertility adjusts
adjusts to
to the
the new
new mortality
mortality conditions
and
societies
advance
in the
rent PPP $ lower than 1,154.04 and high level income countries as those with a GDP per capita over
37,314.14 current PPP $. For lower and middle income countries, defined as those with a GDP per capita
lower than 4,449.04, current PPP $, the coefficient of variation is 0.138.
11
The
defines,
for
low level
income
countries as
as those
those with
with GDP
GDP per
per capita,
capita, current
current PPP
PPP $$ lower
lower than
than
2 World
The
WorldifBank
Bank
defines,
for 2009,
2009,
level limit,
incomelife
countries
Even
we do
not know
thelow
upper
at 37,314.14
birth should
bounded
fromandabove.
1,154.04
GDPexpectancy
per capita
capita over
over
currentbePPP
PPP
$. For
For lower
lower
middle
1,154.04 and
and high
high level
level income
income countries
countries as
as those
those with
with aa GDP
per
37,314.14 current
$.
and middle
This countries,
is not true
for per
capita
income,
however.
What
has
been true
historically
that theof
income
defined
than
4,449.04,
current
PPP $,
$, the
the is
coefficient
offorecasted
variation is
is
income countries,
defined as
as those
those with
with aa GDP
GDP per
per capita
capita lower
lower than
4,449.04,
current
PPP
coefficient
variation
limits
to
life
expectancy
have
been
broken
as
time
has
elapsed
(Oeppen
and
Vaupel
2002;
Willets
0.138.
0.138.
22 et al. 2004).
Even
at birth
birth should
should be
be bounded
bounded from
from above.
above. This
This is
is not
not true
true for
for per
per
Even if
if we
we do
do not
not know
know the
the upper
upper limit,
limit, life
life expectancy
expectancy at
capita
have been
been broken
broken as
as
capita income,
income, however.
however. What
What has
has been
been true
true historically
historically is
is that
that the
the forecasted
forecasted limits
limits to
to life
life expectancy
expectancy have
time
has
elapsed
(Oeppen
and
Vaupel
2002;
Willets
et
al.
2004).
time has elapsed (Oeppen and Vaupel 2002; Willets et al. 2004).
224
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sition mortality falls at early childhood (Davis 1945; Vallin 2002), so the population pyramid
widens at its base, but as fertility adjusts to the new mortality conditions and mature societies
subsequent stages of the epidemiological transition (Olshansky and Ault 1986) the base of the
advance in the subsequent stages of the epidemiological transition (Olshansky and Ault 1986)
population pyramid begins to shrink, and society grows older.
the base of the population pyramid begins to shrink, and society grows older.
Eventually, the reduction of mortality at all ages, as summarized by a continuous increase in
Eventually, the reduction of mortality at all ages, as summarized by a continuous increase
life expectancy, goes hand in hand with a reduction in fertility. Lower numbers of births are observed
in life expectancy, goes hand in hand with a reduction in fertility. Lower numbers of births are
in highly developed countries, and this contributes to the aging of the population.
observed in highly developed countries, and this contributes to the aging of the population.
If we substitute the logarithm of per capita GDP in the x-axis of figure 1, for the logarithm of
If we substitute the logarithm of per capita GDP in the x-axis of figure 1, for the logathe share of people who are 65 years old and over (a very simple index of aging) we get a very similar
rithm of the share of people who are 65 years old and over (a very simple index of aging) we
picture. This is shown in figure 2, where again a semi-logarithmic equation is drawn. Taken at face
get a very similar picture. This is shown in figure 2, where again a semi-logarithmic equation is
value, an additional year of life expectancy at birth is associated with an almost 10% increase in the
drawn. Taken at face value, an additional year of life expectancy at birth is associated with an
share of older people, so we get a high correlation between development, as measured by per capita
almost 10% increase in the share of older people, so we get a high correlation between developincome, and aging via life expectancy at birth.
ment, as measured by per capita income, and aging via life expectancy at birth.
FIGURE 2:
figure 2:
Life
at at
birth
versus
aging
of the
population,
20092009
Lifeexpectancy
expectancy
birth
versus
aging
of the
population,
85
Japan
80
75
Life expectancy at birth (years)
70
65
60
55
50
45
le0 = 50.60 + 10.27 x log(Pop of 65 years old and above) R2 = 0.48
Afghanistan
40
0
5
10
15
Population of 65 years old and above (%)
20
25
Source:
World
Development
Indicators
World
Bank (2011).
Source:
World
Development
Indicators
Word Bank.
(2011).
Acknowledging this correlation, however, does not provide any evidence of causality.
Acknowledging
this correlation,
however,
doesincome
not provide
evidence alone
of causality.
What figures
1 and 2 imply
is that either
per capita
or lifeany
expectancy
can giveWhat
us
figures 1 and 2 imply is that either per capita income or life expectancy alone can give us an overly
optimistic view of the potential development of society
in the future. If life expectancy increases only
5
because longevity increases, as is the case in advanced societies with a very low birth rate, then
3
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Trabajo
–– Núm.
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future. Ifde
expectDocumento
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Trabajo
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Documento de Trabajo – Núm. 3/2012
ancy increases
because
longevity
increases, asinisthe
thelong
case run.
in advanced
a very
sustainability
andonly
quality
of life
can be threatened
What wesocieties
proposewith
in the
sequel is a
Documento deand
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– Núm.of
3/2012
low birth rate, then sustainability
quality
life can be threatened in the long run. What we
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agerun.
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propose
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at
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2. Life potential: A basic demographic indicator
2.LifeLife
Potential:
A Basic
Demographic
Indicator
2.
potential:
A basic
demographic
indicator
2.define
Lifelife
potential:
basic
demographic
indicator
WE
potential forA
a given
individual
at age x as their
(uncertain) life expectancy given their
WE define
define life
life potential
potential for
WE
for aa given
given individual
individual at
at age
age xx as
as their
their(uncertain)
(uncertain)life
lifeexpectancy
expectancygiven
given their
A basic demographic
age, and theindicator
life potential for a society, L, as the aggregate over individual life potential. Hence,
WE
life
potential
for
asover
theirindividual
(uncertain)
expectancy
given their
theirdefine
age,theand
life potential
for aindividual
society,
asage
thex aggregate
over individual
life potential.
age,
and
lifethe
potential
foraa given
society,
L, as theL,at
aggregate
lifelife
potential.
Hence,
and
potential life
for aexpectancy
society, L,given
as
aggregate over individual life potential. Hence,
Hence,
r a given individual atage,
age
x as the
theirlife
(uncertain)
∞ thetheir
(1)
L = ∞∞P( x)e( x)dx
or a society, L, as the aggregate over individual life potential. Hence,
(1)
L = 0∞ P( x)e( x)dx
(1)
L = 00 P( x)e( x)dx (1)
∞
where P(x) is the population at age x, and e(x)
0 is the corresponding life expectancy. From (1) it is clear
L = ³ P( x)ewhere
( x)dxP(x) is the population at age x, and e(x) is (1)
the corresponding life expectancy. From (1) it is clear
0
that L is a weighted sum of life expectancies at different ages, and in the tradition of capital theory can
where
is the population
at age
x, and e(x)atisdifferent
the corresponding
expectancy.
(1) it is clear
that
L P(x)
isP(x)
a weighted
sum of life
ages, and inlife
the
of From
capital
where
is the population
atexpectancies
age x, and e(x)
is the corresponding
lifetradition
expectancy.
Fromtheory
(1) can
be
understood
as
the
biological
capital
of
a
society,
since
it
is
an
estimate
of
the
physical
support of
n at age x, and e(x) is the
corresponding
life
expectancy.
From (1)
ita is
that
L is a weighted
sum
of life expectancies
at clear
different
ages,
and
in
the tradition
of
capitalsupport
theory can
be
understood
as
the
biological
capital
of
society,
since
it
is
an
estimate
of
the
physical
of
it is clear that L is a weighted sum of life expectancies at different ages, and in the tradition of
any
other
form
of
human
capital,
such
as
education,
job
training
or
health
capital
(Shultz
1962;
Becker
life expectancies at different
ages,
and
thebiological
tradition
capital
theory
can since
be
understood
as
a society,
it is or
an health
estimate
of the
physical
support
any
other
form
ofinthe
human
capital,ofcapital
such
asofeducation,
training
capital
1962;
Beckerof
capital
theory
can
be understood
as the
biological job
capital
of a society,
since
it(Shultz
is an estimate
1962,
1964,itform
2007;
Because
L is difficult
to compare
among
societies
of 1962;
different
size,
gical capital of a society,
since
is2007;
anofGrosman
estimate
of1972).
the physical
support
of job training
any
other
human
capital,
such
as education,
or health
capital
(Shultz
Becker
1962,
Grosman
1972).
Because
is
to
societies
of
1962,
1964,
2007;
Grosman
1972).
Because
L human
is difficult
difficult
to compare
compare
among
societies
of different
different
size,
of the1964,
physical
support
of any
other
form ofL
capital,
such asamong
education,
job training
or size,
∞
apital, such as education,
job 1964,
training
or health
capital
(Shultz
1962; LBecker
∞
1962,
2007;
Grosman
1972).
Because
is difficult to compare among
societies of different size,
∞
health
capital
(Shultz 1962;
Beckerl.1962,
2007;
Grosman
1972).
L is, we
difficult
to
wewe
may
use
life
potential
per
capita,
Let
PP1964,
be
the
total
population,
P ==Because
P((xx))dx
dx
then define
life
may
use
life
potential
per
capita,
l.
Let
be
the
total
population,
P
P
,, we
we maytouse
life potential
capita,ofl. different
Let P be size,
the total population,
we then
then define
define life
life
n 1972). Because L is difficult
compare
amongper
societies
∞
0
0
compare
among
societiesper
of capita,
different
size,Pwe
use population,
life potentialPper
be the
we
may use
life potential
l. Let
be may
the total
= 0 capita,
P( x)dxl. ,Let
we Pthen
define life
∞
potential
per
capita
as
0
per
total
define life
life potential per capita as
er capita, l. Let P be thepotential
totalpopulation,
population,
potential
per capita
capitaPas
as= ³ P ( x)dx , we then define
³
³
³
³³
³
potential per capita as
l=
0
∞
L ∞∞
l l== L == ³ ω
ω((xx))ee((xx))dx
dx PPL 0³0∞
0
l = = ³ ω( x)e( x)dx
P 0 (2)
∞∞
(2)
(2)
(2)
(2)
(2)
L
= ³ ω( x)e( x)dx
∞
P 0
ω
(
x
)
=
P
(
x
)
P
with
the
property
that
So,
weighted
average
where
ω
where
,
with
the
that
average
ofof
ω
(
x
)
=
P
(
x
)
P
,
with
the
property
that
So, llllisis
weighted
average
life
where
So,
isisaaaaweighted
weighted
average
ofoflife
life
where ω( x ) = P ( x ) P , with the property that³³∞ω
ω((xx))dx
dx == 11 ... So,
∞
000
³
∞life expectancies,
are given
population
life expectancy
ω( x ) = P( xwhere
) P , the
withweights
the property
thatby ω
So, l Because
is a weighted
average of life
where
( x)dx = 1 .shares.
with the property that
.
So,
l
is
a
weighted
average
of
life
ω
(
x
)
dx
=
1
expectancies,
where
the
weights
are
given
by
population
shares.
Because
life
expectancy
0
expectancies,
where
shares. Because
Because life
life expectancy
expectancydecreases
decreases
wherethe
theweights
weightsare
aregiven
given by
by population
population
decreases
³expectancies,
decreases with age (at least beyond a certain point), l is increasing in life expectancy at any
0
with
least
beyond
aacertain
certain
point),
increasing
in life
expectancy
at
in
expectancies,
where
theaweights
are
given
population
shares.
Because
life
expectancy
decreases
with
age
(at
least
beyond
certainpoint),
point),
isby
increasing
expectancy
atatany
any
age
and
decreasing
inin
with
ageage
(at(at
least
beyond
lllisis
increasing
in
life
expectancy
anyage
ageand
anddecreasing
decreasing
age
and
decreasing
in
population
aging.
From
the
definition,
it
follows
that
l
can
be
interpreted
eights are given by population shares. Because life expectancy decreases
population
aging.
From
the
definition,
that
can
be
interpreted
as
the
life
expectancy
of
with
age (at
leastFrom
beyond
a certain
point),
lfollows
is increasing
in life
decreasing
population
aging.
Fromthe
the
definition,
follows
that lll can
interpreted
as
the
life
expectancy
ofofaaina
population
aging.
definition,
itititfollows
that
beexpectancy
interpretedat
asany
theage
lifeand
expectancy
certain point), l is increasing in life expectancy at any age and decreasing in
population aging. From the definition, it follows that l can be interpreted as the life expectancy of a
e definition, it follows that l can be interpreted as the life expectancy of a
6
Documento de Trabajo – Núm. 8/2012
as the life expectancy of a given population, as opposed to the life expectancy of a cohort at a
given age, which is the usual interpretation in demography3.
Given that life potential per capita has the nature of an average; it can also be interpret
as the expected remaining life of a citizen picked up randomly within the population. This is
not the case for the life expectancy at birth, except for a newborn. So when integrating life
expectancy with other economic variables that incorporate the demographic structure of the
society, such as GDP per capita or the unemployment rate, it may be a good reason to choose
life potential instead of life expectancy at birth4.
3.
Life Potential in Practice
TO build an operational measure for (1) we only need population classified by age and their
corresponding life expectancies. In the absence of individual (subjective) survival curves (Gan,
Hurd and McFadden 2003) individual data are not available, and therefore should rely on life
expectancy from standard life tables.
Published life tables are usually of the age-period type, so age-specific mortality rates
for a given period (usually a calendar year) are used to construct the life experience of a fictitious generation that is followed until it is extinguished. Life expectancy at different ages is
estimated by redistributing equally all future life years lived by the survivors of the generation
at a given age. In this way, period life tables represent the current mortality conditions, without
taking into account future improvements in mortality. Life expectancy at birth thus represents
the average time that an individual born at a given time can expect to live on average, with the
current mortality conditions. Figure 3 represents this set-up.
Fortunately, the Human Mortality Database (see http://www.mortality.org/) builds complete life tables for a great number of countries based on a common methodology with an open
3
If we partition the population into exhaustive and mutually exclusive groups, such as by region, gender
or ethnic groups, then L can be calculated as the sum of life potential over the different groups, and l is a
weighted sum of life potential per capita, where the weights are given by the relative importance of each
group in the population.
4
This nice interpretation of l was suggested by an anonymous referee.
7
Documento de Trabajo – Núm. 8/2012
Documento de Trabajo – Núm. 3/2012
ended age interval of 110 years old and above (Wilmoth et al. 2007). They also offer population data by one year-age intervals covering long periods of time, and dated 1st January. All the
Documento de Trabajo – Núm. 3/2012
FIGURE 3:
Life tables: Age-period
calculations in this paper use life tables and population data from this database.
g−3
Age (x)
g−2
g−1
g−2
g−1
Cohort (g)
Life
Life tables:
tables: Age-period
FIGURE
figure3:
3:
g−3
xAge
+ 3 (x)
Cohort
g = t −(g)
x
g=t−x
x+3
x+2
g+1
x+2
g+1
x+1
g+2
x+1
g+2
x
t
t+1
t+2
t+3
t+2
t+3
Time (t)
x
t
t+1
Time (t)
Period life tables estimate life expectancy at an exact age, x, ex; this is at the beginning of the
Period life tables estimate life expectancy at an exact age, x, ex; this is at the beginning
ageofinterval,
[x, xlife
+ 1)
in the
case
oflife
single
age years.
On
the
other
hand,
population
stock
is dated
Period
tables
estimate
expectancy
at an
exact
ex; this
is population
at the beginning
of theat a
the age
interval,
[x, x + 1)
in the
case
of single
age
years.
Onage,
the x,
other
hand,
stock
given
pointatina[x,
time,
but
recorded
ageOn
interval,
[x,hand,
x + interval,
1).
The empirical
counterpart
is dated
given
point
iniscase
time,
butforit aage
isgiven
recorded
for
a given
age
[x, x + 1).
The
age
interval,
x +t,1)
in it
the
oft, single
years.
the
other
population
stock is
dated at a of
empirical
counterpart
of (1)
discrete
with
structure
(1)given
frompoint
discrete
datat,with
structure
is adata
in time,
but
itthis
isfrom
recorded
for
given
agethis
interval,
[x, xis+ 1). The empirical counterpart of
(1) from discrete data with this structure is
110 +
L = ¦ Px ex 110 +
x =0
L = ¦ Px ex
(3)
(3)
(3)
x =0
e = ½.(e + e ) , P is the population in the age interval [x, x + 1) at a given point in time and
where
wherex e = ½.(x e +xe+1 ) , xP is the population in the age interval [x, x + 1) at a given point in
x
x
x +1
x
where ex = ½.(ex + ex +1 ) , Px is the population in the age interval [x, x + 1) at a given point in time and
and for
the open
ended age
usee110
e110. = ½.e110 .
fortime
the open
ended
age interval
weinterval
use e110we
= ½.
for the open ended age interval we use e110 = ½.e110 .
P
Using the weights ωx = x , where P = Σ x ≥ 0 Px , the empirical counterpart of life potential per
P
Using the weights ωx = Px , where P = Σ x ≥ 0 Px , the empirical counterpart of life potential per
P
capita is
capita is
8
110 +
+ ω e
l = 110
¦
x x
l=¦
x = 0 ωx ex
x =0
(4)
(4)
110
110++
110 +
LL == ¦
PPx eex
L =x¦
Pxx exx
=¦
0
(3)
(3)
(3)
xx==00
½.(
the population
in the
age interval
[x, x + 1) atataagiven
time
where eex =
½.(eex ++ eex +1 )) ,, PPxx is
givenpoint
pointin
timeand
and
where
thepopulation
populationin
in the
the age
age interval
interval [x,Documento
x + 1) atdea Trabajo
given
point
inintime
and
where xe=
xisisthe
x = ½.(xex + xe+x1+1 ) , P
– Núm. 8/2012
forfor
thethe
open
ended
age
interval
we
use
e e ===½.
..
for
the
open
½.
openended
endedage
ageinterval
intervalwe
weuse
use e110
½.ee110
e110
110
110
110 .
PPxP
Using
the
weights
x x, where P = Σ x ≥ 0 Px , the empirical counterpart of life potential per
x =
Using
the
weights
theempirical
empiricalcounterpart
counterpart
po- per
where PP==ΣΣxx≥≥00P
Pxx ,,, the
the
of
life
potential
Using
the
weightsω
=
counterpart
ofof
lifelife
potential
per
Using
the
weights
ωω
=
x x P , ,where
PP
tential
per
capita
is iscapita is
capita
capita
is
110 +
110++
110
l l==¦ ωωx eex l = x¦
¦ωxxexx
=0
xx==00
(4) (4)
(4)
(4)
which is simply a population weighted average of life expectancies5. Figure 4 shows life expectancy at birth and life potential per capita for a selection of developed countries: Spain,
66
6
Japan, the UK, the US, France and Sweden, over the same period of time; and table 1 shows the
numerical values for selected years. Overall, long run tendencies in life expectancy are clear
and well-known, and despite short periods related to wars or epidemics, life expectancy shows
an up-ward and steady trend. In 2007, the last common year available to all the countries considered, life expectancy at birth was 82.87 years in Japan, the highest observed value, followed
by France, 81.16 years, Sweden, 81.08 years, and Spain, 80.92 years. The lowest value is found
in the US with 78.32 years.
Tendencies for life potential per capita are less clear cut. For some periods and most
countries, life potential follows life expectancy closely. In fact, with the exception of Japan and
Spain, the correlation between both series is very high, in excess of 0.88. However, as we will
see in the sequel, this correlation changes abruptly with time, and life potential per capita appears to slow down, or even to fall in recent years in most countries with the exception of the
UK. In fact in this country life potential per capita falls at the beginning of the XX century, even
this is not shown in figure 1 and table 1. This particular evolution would be worth exploring.
It is interesting to note the particular evolution of life potential per capita in Japan and
Spain. Both countries show very high life expectancy at birth, but in both cases life potential
is currently falling, since the end of the 70s in the case of Japan, and since the beginning of the
80s in the case of Spain. This puts a precautionary note in the optimistic signal shown by the observation of life expectancy at birth alone, given that interpreting life potential as the biological
capital of society, both countries are, in fact, destroying this kind of capital. From this point of
5
It is worth noting that the indicator (4) was used by Usher (1973) in his imputation of the value of life
in the national accounts, but keeping constant the population structure and fixed to the base year. Maintaining the population structure or life expectancies constant in (4) we can construct counter-factual life
potential per capita, and thus be able to examine the evolution of l with one of its components taken as
given.
9
Documento de Trabajo – Núm. 8/2012
view, the country with the highest biological capital is the US, which given the observed lower
life expectancies signals a younger population than the other countries considered.
Given that l is a weighted average we can split the changes between two points in time,
or even the differences between two countries at a given point in time, into the contributions
due to changes in the demographic structure, ωx, and the contributions due to the changes in life
expectancies, ex. This is the goal of the so called shift-share analysis widely
usedde in
regional
Documento
Trabajo
– Núm. 3/2012
economics. These types of decompositions are never unique (Kitagawa 1955), but the decomposition that is easiest to interpret is the following
TABLE 1:
Life expectancy at birth and life potential per capita. Selected years from developed countries
Life expectancy at
birth and life potential per capita.
Spain
Japan Selected years from developed
UK
table 1:
Year
countries
Life expectancy Life potential Life expectancy Life potential Life expectancy Life potential
Spain
Japan
UK
1947
59,33
40,21
51,75
38,54
66,44
37,86
Year1955Life expectancy
Life
potential
Life
expectancy
Life
potential
Life
expectancy
Life
potential
66,78
42,74
65,77
44,41
70,21
38,88
1947
59,33
40,21
51,75
38,54
66,44
37,86
1977
74,39
44,40
75,38
44,96
73,25
40,17
1955
66,78
42,74
65,77
44,41
70,21
38,88
2007
80,92
42,47
82,87
41,37
79,72
42,46
1977
74,39
44,40
75,38
44,96
73,25
40,17
2007
80,92
42,47
82,87
41,37
79,72
42,46
US
France
Sweden
Life expectancy
Life potential Life expectancy
Life potential Life expectancy
Life potential
Year
US
France
Sweden
66,73 Life potential
40,27 Life expectancy
63,98 Life potential
37,87Life expectancy
69,47 Life potential
39,66
Year1947Life expectancy
19471955
66,73 69,63
40,27 42,35
63,98 68,47
37,87 39,37
69,47 72,60
39,66 40,84
19551977
69,63 73,38
42,35 43,64
68,47 73,83
39,37 42,18
72,60 75,44
40,84 40,54
19772007
73,38 78,32
43,64 43,99
73,83 81,16
42,18 43,77
75,44 81,08
40,54 42,22
2007
78,32
43,99
81,16
43,77
81,08
42,22
Source: Human Mortality Database and own elaboration.
Source: Human Mortality Database and own elaboration.
110 +
§ ωsx + ωtx · s
§ exs + ext ·
t
s
t
l −l = ¦¨
¸ .(ex − ex ) + ¦ (ωx − ωx ). ¨
¸ 2 ¹
2
x =0 ©
x =0
©
¹
s
t
110 +
(5)
(5)
where the first term can be interpreted as the contribution of changes in life expectancies, whereas the
where the first term can be interpreted as the contribution of changes in life expectancies,
second term can be interpreted as the contribution of changes in the demographic structure of society.
whereas the second term can be interpreted as the contribution of changes in the demographic
Table 2 shows, for selected time periods, the changes in life expectancy at birth and life
structure of society.
potential per capita, as well as the correlation between the two variables for the period. It also
illustrates the decomposition (5), showing the contribution of life expectancies and demographics to
the change in life potential per capita.
Several facts are worth mentioning. (i) For the period considered, changes in life expectancy at
birth are always positive, and they show no symptoms of exhaustion; a well-known fact. (ii) On the
other hand, changes in life potential per capita are more irregular. Spain and Japan show a negative
change in recent decades, which translates into a high negative correlation for these years. (iii)
Correlation between both variables is quite sensitive to the time period considered. The almost absent
10
correlation for Japan for the whole period, 1947–2007, is the result of a high positive correlation in the
early decades and a high negative correlation in recent decades. No clear pattern emerges in this
respect when we consider shorter sub-periods for the different countries. (iv) With the exception of the
Documento de Trabajo – Núm. 8/2012
Life expectancy at birth and life potential per capita. Historical international comparisons
a) Spain
b) Japan
85
50
48
80
50
48
80
46
42
70
40
65
38
36
60
Life expextancy at birth
44
Life potential per capita
75
44
42
70
40
65
38
36
60
34
34
55
32
Correlation coefficient: 0.37
50
1947 1953 1959 1965 1971 1977 1983 1989 1995 2001 2007
Life expextancy at birth
55
Correlation coefficient: 0.17
50
30
1947 1953 1959 1965 1971 1977 1983 1989 1995 2001 2007
Life expextancy at birth
Life potential per capita
c) UK
50
Life potential per capita
85
48
80
50
48
80
42
70
40
65
38
36
60
46
Life expextancy at birth
44
Life potential per capita
Life expextancy at birth
46
75
75
44
42
70
40
65
38
36
60
34
55
Correlation coefficient: 0.98
1947 1953 1959 1965 1971 1977 1983 1989 1995 2001 2007
Life expextancy at birth
34
55
32
Correlation coefficient: 0.86
30
50
1947 1953 1959 1965 1971 1977 1983 1989 1995 2001 2007
Life potential per capita
Life expextancy at birth
e) France
50
30
Life potential per capita
85
48
80
50
48
80
44
42
70
40
65
38
36
60
46
Life expextancy at birth
75
Life potential per capita
46
Life expextancy at birth
32
f) Sweden
85
75
44
42
70
40
65
38
36
60
34
55
Correlation coefficient: 0.97
50
30
d) US
85
50
32
Life potential per capita
Life expextancy at birth
46
75
Life potential per capita
85
1947 1953 1959 1965 1971 1977 1983 1989 1995 2001 2007
Life expextancy at birth
34
55
32
Correlation coefficient: 0.83
30
32
50
30
1947 1953 1959 1965 1971 1977 1983 1989 1995 2001 2007
Life potential per capita
Life expextancy at birth
Source: Human Mortality Database and own elaboration.
11
Life potential per capita
Life potential per capita
figure 4:
Documento de Trabajo – Núm. 8/2012
Table 2 shows, for selected time periods, the changes in life expectancy at birth and life
potential per capita, as well as the correlation between the two variables for the period. It also
illustrates the decomposition (5), showing the contribution of life expectancies and demographics to the change in life potential per capita.
Several facts are worth mentioning. (i) For the period considered, changes in life expectancy at birth are always positive, and they show no symptoms of exhaustion; a well-known
fact. (ii) On the other hand, changes in life potential per capita are more irregular. Spain and
Japan show a negative change in recent decades, which translates into a high negative correlation for these years. (iii) Correlation between both variables is quite sensitive to the time period
considered. The almost absent correlation for Japan for the whole period, 1947–2007, is the
result of a high positive correlation in the early decades and a high negative correlation in recent decades. No clear pattern emerges in this respect when we consider shorter sub-periods for
the different countries. (iv) With the exception of the firsts years considered for United States
and France, 1947–1955, where the demographics contribute slightly positive, the contribution
of demographics to changes in life potential per capita is invariably negative. Moreover, this
negative contribution is increasing in magnitude with time. In the two cases mentioned, Japan
and Spain, the negative contribution of demographics out-weighs the positive contribution of
improvements in life expectancies, resulting in the negative variation of life potential per capita
mentioned earlier. This is also the case for Sweden for the period 1955–1977.
Eventually, figure 5 shows a scatter plot between life expectancy at birth and life potential per capita for 2007 and 23 European countries; these are the EU-27 with the exception
of Greece, Cyprus, Malta and Romania, for which there is no data in the Human Mortality Database. This cross-section comparison at a point in time shows a relatively high association between both variables, with a correlation coefficient of 0.87, which is however far from perfect,
the rank correlation coefficient being 0.81; and also a high heterogeneity between countries for
the two variables considered. The previous time series analysis for individual countries warns
us against a simplistic interpretation of this relationship, on the contrary, it suggest that future
increments in life expectancy at birth will probably be associated with lower increments, or
even decrements, in life potential per capita.
12
Documento de Trabajo – Núm. 8/2012
table 2:
Changes in life expectancy at birth and life potential per capita. Selected periods from developed countries. Shift-share decomposition for life
potential per capita
a) Spain
Period
b) Japan
Changes in
Life expectancy at birth Life potential
Correlation
Decomposition
Changes in
Life expectancies Demographics
Correlation
Life expectancy at birth Life potential
Decomposition
Life expectancies Demographics
1947-1955
7,45
2,53
0,959
3,43
-0,90
14,02
5,87
0,996
6,42
-0,55
1955-1977
7,61
1,66
0,869
3,08
-1,42
9,61
0,56
0,598
5,32
-4,77
1977-2007
6,53
-1,93
-0,894
4,78
-6,71
7,49
-3,60
-0,964
6,21
-9,80
1947-2007
21,59
2,25
0,371
10,84
-8,59
31,12
2,83
0,171
16,64
-13,81
c) UK
Period
d) US
Changes in
Life expectancy at birth Life potential
Correlation
Decomposition
Changes in
Life expectancies Demographics
Correlation
Life expectancy at birth Life potential
Decomposition
Life expectancies Demographics
1947-1955
3,77
1,02
0,855
1,13
-0,11
2,90
2,08
0,999
1,71
0,37
1955-1977
3,04
1,29
0,943
1,80
-0,51
3,75
1,29
0,845
2,45
-1,16
1977-2007
6,47
2,29
0,991
5,01
-2,72
4,94
0,35
0,497
3,71
-3,36
1947-2007
13,28
4,61
0,984
7,93
-3,33
11,59
3,72
0,855
7,82
-4,10
e) France
Period
f) Sweden
Changes in
Life expectancy at birth Life potential
Correlation
Decomposition
Changes in
Life expectancies Demographics
Correlation
Life expectancy at birth Life potential
Decomposition
Life expectancies Demographics
1947-1955
4,49
1,49
0,929
0,97
0,53
3,13
1,18
0,931
1,77
-0,60
1955-1977
5,36
2,81
0,990
2,89
-0,08
2,84
-0,30
-0,134
1,71
-2,01
1977-2007
7,33
1,59
0,926
5,59
-4,00
5,64
1,68
0,958
4,30
-2,62
1947-2007
17,18
5,90
0,973
9,35
-3,46
11,61
2,56
0,826
7,79
-5,23
Note: Decomposition shows the formula (5) of the text, so it shows the contribution of the changes in life expectancies and demographics to the change in life potential per capita in the given period.
Source: Human Mortality Database and own elaboration.
13
Documento de Trabajo – Núm. 3/2012
Documento de Trabajo – Núm. 8/2012
Life expectancy
expectancy at
at birth
birth versus
versuslife
lifepotential
potentialper
percapita.
capita.EU-23.
EU-23.Year
Year 2007.
2007.
Life
figure5:
5:
FIGURE
84
LE(0)=34+1.08 LP
(8.2)
R²=0,76
82
ITA
Life expectancy at birth
80
DEU
SVN
78
FRA
SWE
ESP
NLD
AUT
BEL GBR
FIN
LUX
PRT
IRL
DNK
CZE
76
POL
SVK
74
HUN
EST
BGR
72
LTU
LVA
70
34
36
38
40
42
44
46
Life potential per capita
Note: For Greece, Cyprus, Malta and Romania there is no data available.
Note:
For Human
Greece, Cyprus,
Malta
and Romania
there elaboration.
is no data available.
Source:
Mortality
Database
and own
Source: Human Mortality Database and own elaboration.
contribution of improvements in life expectancies, resulting in the negative variation of life
4.
Final Comments
potential per capita mentioned earlier. This is also the case for Sweden for the period 1955–
1977.
THIS short paper has introduced a simple demographic indicator that integrates life expectancy
Eventually,
figure
5 shows a scatter
plot of
between
life expectancy
at itbirth
life
at different
ages with
the demographic
structure
population.
In this way,
triesand
to balance
potential
per capita
for in
2007
23 European
countries;
are the EU-27
with the
the observed
increment
lifeand
expectancy
with the
aging ofthese
the population
that characterizes
exception
Greece,Aging
Cyprus,
Maltatoand
for consequence
which there isofno
data in the Human
advanced of
societies.
appears
be Romania,
an inevitable
development,
and should
Mortality
This cross-section
comparison
at atopoint
in time
shows
a relatively high
therefore Database.
be incorporated
in social indicators
related
quality
of life
and sustainability.
association
both variables,
with aand
correlation
coefficient
of 0.87, which
howevercapWebetween
call the indicator
life potential,
it has an intuitive
interpretation
as theisbiological
far
coefficient
being
andhow
alsoaging
a high
heterogeneity
italfrom
of theperfect,
society the
at a rank
givencorrelation
point in time.
In this way,
we0.81;
can see
societies
could suffer
between
countries
for the
two variables
considered.
The previous
time
series
analysis
forThis
from a loss
in biological
capital,
thus affecting
sustainability
and quality
of life
in the
long run.
individual
countries
us of
against
a simplistic
interpretation
of who,
this relationship,
on the of
is the idea behind
thewarns
proposal
Herrero,
Martinez and
Villar (2010)
in their reformulation
contrary,
suggest that Index
future(HDI),
increments
in life
lifeexpectancy
expectancyat birth
at birth
willpotential
probably
the Humanit Development
substitute
for life
perbe
capita
associated
with lower
increments,
or even
decrements,
in in
life
of a given country,
in addition
to other
important
changes
thepotential
way theper
HDIcapita.
is calculated.
11
14
Documento de Trabajo – Núm. 8/2012
It is well known that life expectancy is independent of the population structure of society and that there are good reasons for this in measuring the incidence of mortality, essentially
to avoid the composition effect when comparing countries with different population pyramids.
But the same virtue becomes an inconvenient when we use demographic indicators to assess
other aspects related to future development. By taking into account the prevailing population
structure life potential provides a better estimate of future possibilities.
From a practical and computational point of view, life potential per capita is simply a
population weighted life expectancy of the society. Thus, the continuous increment in life expectancy at all ages is balanced with an increase in the share of old people with shorter life expectancy. Life potential is simple to calculate and has low data requirements, not going beyond
the information needed to calculate life tables. It also has an interesting interpretation in terms
of the average life expectancy of the population at a given date, or as the expected remaining
life of a citizen picked up at random within the population, and changes of the indicator can be
easily decomposed in its two components.
A practical application for some developed countries in a historical context shows the
clear and well-known tendency of increasing life expectancy, but a less clear cut tendency
for life potential per capita. In general, we observe stagnation of this last variable, and in two
particular cases, Japan and Spain, there is an important fall in life potential per capita in recent
decades, signaling an accelerated aging of the population in these two countries. Clearly, this
could be cause for concern, beyond the optimistic view that can be reached by looking at life
expectancy at birth alone. Aging is an important factor in developed societies, and this should
be incorporated in social indicators in a more satisfactory manner than simply looking at percentages of young or old people in society.
5.
References
Becker, G. S. (1962): “Investment in human capital: A theoretical analysis”. The Journal of Political
Economy 70 (5), Part 2: Investment in Human Beings, October, 9-49.
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edition, NBER, New York, 1975. 3rd edition, University of Chicago Press, Chicago, 1993).
15
Documento de Trabajo – Núm. 8/2012
Becker, G. S. (2007): “Health as human capital: Synthesis and extensions”, Oxford Economic Papers
59, 3, (July), 379-410.
Davis, K. (1945): “The world demography transition”. Academy of Political and Social Science, 273,
1-11.
Gan, L., H. Hurd y D. McFadden (2003): “Individual subjective survival curves”. Working Paper No.
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Goerlich, F. J. y R. Pinilla (2005): “Esperanza de Vida y Potencial de Vida a lo largo del siglo xx en
España”, Revista de Demografía Histórica, XXIII (II), 79-109.
Grossman, M. (1972): The Demand for Health – A Theoretical and Empirical Investigation. New York:
Columbia University Press for the National Bureau of Economic Research.
Herrero, C., R. Martínez y A. Villar (2010): “Improving the Measurement of Human Development”,
Research Paper No. 2010/12, United Nations Development Programme, Human Development
Reports, July. Available at:
http://hdr.undp.org/en/reports/global/hdr2010/papers/HDRP_2010_12.pdf.
Kitagawa, E. M. (1955): “Components of a difference between two rates”. Journal of the American Statistical Association 50 (272), 1168-1194.
Murray, J. L., J. A. Salomon, C. D. Mathers y A. D. López (2002): Summary Measures of Population
Health. Geneva: World Health Organization.
Oeppen, J. y J. W. Vaupel (2002): “Broken limits to life expectancy”. Science 296 (5570), May, 10291031.
Olshansky, S. y A. Ault (1986): “The fourth stage of the epidemiologic transition: The age of delayed
degenerative diseases”, The Milbank Quarterly, 64 (3), 355-391.
Osberg L. y A. Sharpe (2002): “An index of economic well-being for selected OECD countries”, Review
of Income y Wealth 48 (3), September, 291-316.
Preston, S.H. (1975): “The changing relation between mortality and level of economic development”
Population Studies 29 (2), 231-248.
Sen, A. (1998): “Mortality as an indicator of economic success y failure”. The Economic Journal 108,
(January), 1-25.
———– (1999): Development as Freedom. Nueva York: Alfred A. Knopf Inc. (Spanish edition: Desarrollo y libertad. Barcelona: Planeta, 2000).
Schultz, T. W. (1962): Investment in Human Beings. Chicago: University of Chicago Press.
Usher, D. (1973): “An imputation to the measure of economic growth for changes in life expectancy”,
with comments by Robert J. Willis; in M. Moss, Ed. The Measurement of Economic and Social
Performance. New York: National Bureau of Economic Research, Columbia University Press,
193-232.
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Vallín, J. (2002): “The end of the demographic transition: Relief or concern?”, Population and Development Review 28 (1), 105-120.
Willets, R. C., A. P. Gallop, P. A. Leandro, J. L. C. Lu, A. S. Macdonald, K. A. Miller, S. J. Richards,
N. Robjohns, J.P. Ryan y H.R. Waters (2004): “Longevity in the 21st century”, Institute of Actuaries and Faculty of Actuaries, March/April.
Wilmoth, J. R., K. Andreev, D. Jdanov y D. A. Glei, (2007): “Methods protocol for the Human Mortality
Database”. Mimeo, May 31, Version 5. Available at: http://www.mortality.org.
17
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NOTA SOBRE LOS AUTORES - About the authorS*
francisco j. goerlich gisbert is a graduate and doctor in economics from the
University of Valencia, and MSc in economics from the London School of Economics, University of London. He is currently professor in the Department of Economic Analysis of the University of Valencia and Senior Researcher at the Instituto
Valenciano de Investigaciones Económicas (Ivie). His research fields are applied
econometrics, regional economics and income distribution. He has published more
than forty articles in both national and international specialized journals and has
collaborated in more than ten books.
E-mail: [email protected]
is a graduate in economics and business studies from
the University of Valencia (1996) majoring in general economic situations and
specialising in the public sector. He was granted scholarships during his studies,
among them a collaboration scholarship from the Department of Economic Analysis in 1995. He completed his postgraduate studies at the University of Valencia
(1996-1998). He joined the Ivie in 1996. He is also assistant professor in the Economics Department of the University of Valencia. His specialization fields are human development, university education, labour market, immigration and human
capital. He is co-author of seventeen books and is responsible for the elaboration
of the Human Capital Series. He has also participated in the EU KLEMS project
and in building the data series of Human Development.
ángel soler guillén
E-mail: [email protected]
____________________________
Any comments on the contents of this paper can be addressed to: Francisco J. Goerlich Gisbert,
Universidad de Valencia, Departamento de Análisis Económico, Campus de Tarongers, Av de
Tarongers s/n, 46022-Valencia. E-mail: [email protected].
* The authors acknowledge financial support by the Spanish Ministry of Science and
Technology, ECO2011-23248 project, and the BBVA Foundation-Ivie research programme.
Results mentioned, but not shown, are available from the authors upon request. Comments
from an anonymous referee and Carmen Herrero are greatly appreciated, without implicating
them in our errors. Important note to readers of the cellulose versions of this paper: Many of
the figures in this study are best read in the electronic color version.
18
Documento de Trabajo – Núm. 8/2012
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19
Documentos
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2012
Francisco J. Goerlich Gisbert
Ángel Soler Guillén
Life Potential
as a Basic
Demographic
Indicator
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