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 España Tel.: +34 94 487 52 52 Fax: +34 94 424 46 21 Paseo de Recoletos, 10 28001 Madrid España Tel.: +34 91 374 54 00 Fax: +34 91 374 85 22 [email protected] www.fbbva.es

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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

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  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 el me responsabilidad responsabilidad alguna mismo de documentos facilitada por por los los mismo documentos o información información complementaria complementaria facilitada autores. autores. working paper does not not imThe BBVA BBVA Foundation’s Foundation’sdecision decisiontotopublish publishthis this working paper does imply anyresponsibility responsibilityfor foritsitscontents, contents,or or for for the the inclusion therein of ply any of any any supplementary supplementary documents documents or or information informationfacilitated facilitatedby bythe theauthors. authors. Documentos de de Trabajo tiene tiene como como objetivo objetivo la la rápida rápida difusión difusión de de La serie Documentos resultados del trabajo trabajo de investigación investigación entre entre los los especialistas especialistas de de esa esa área, área, los resultados para promover promover así así el intercambio intercambio de de ideas ideas yy el debate debate académico. académico. Cualquier Cualquier comentario sobre sobre sus sus contenidos contenidos será será bien bien recibido recibido yy debe debe hacerse hacerse llegar llegar comentario directamente aa los los autores, autores, cuyos cuyos datos datos de de contacto contacto aparecen aparecen en en la la Nota Nota directamente autores. sobre los autores. The Working Papers series series is is intended intended to to disseminate disseminate research research findings findings rapidly among among specialists specialists in in the the field field concerned, concerned, in in order order to to encourage encourage the the exchange exchange of ideas ideas and academic academic debate. debate. Comments Comments on on this this paper paper would would be be welcome and should should be be sent sent direct direct to to the the authors authors at at the the addresses addresses provided provided in the About About the the authors authorssection. section.

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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.

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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.

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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).

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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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Documento de Trabajo – Núm. 3/2012 Trabajo –– Núm. an overly optimistic view of the potential development of society in the Documento future. Ifde expectDocumento delife Trabajo Núm. 3/2012 3/2012

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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 Trabajo – Núm.of 3/2012 low birth rate, then sustainability quality life can be threatened in the long run. What we very simple indicator that integrates expectancyinattheany agerun. with thewe demographic structure sustainability and quality of life canlife be threatened long What propose in the sequelof is the a propose in the sequel is a very simple indicator that integrates life expectancy at any age with sustainability and quality of life can be threatened in the long run. What we propose in the sequel is a population, which allows to take life aging into account, it can sustainability very simple indicator thatusintegrates expectancy at anysince age with the affect demographic structurebeyond of the a the demographic structure of the population, which takethe aging into account, since of the very simple indicator that expectancy at any us agetowith demographic structure f life can be threatened in the long which run. What we integrates propose inlife the sequel a allows population, allows us to take aging intoisaccount, since it can affect sustainability beyond a certain point. it can affect sustainability beyond a structure certain population, which us to take aging point. into account, since it can affect sustainability beyond a ntegrates life expectancy at any age with allows the demographic of the certain point. us to take aging into certain account,point. since it can affect sustainability beyond a

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

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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

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  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.

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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.

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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.

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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. ———– (1964): Human Capital: A Theoretical and Empirical Analysis with Special Reference to Education. New York: Columbia University Press for the National Bureau of Economic Research (2nd edition, NBER, New York, 1975. 3rd edition, University of Chicago Press, Chicago, 1993).

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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. 9480, Cambridge, MA: National Bureau of Economic Research, January. 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.

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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.

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Documentos de Trabajo

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Documentos de Trabajo

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2012

Francisco J. Goerlich Gisbert Ángel Soler Guillén

Life Potential as a Basic Demographic Indicator

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