Contribution of demography to economic growth

Sánchez-Romero, Miguel; Abio, Gemma; Patxot, Concepció; Souto, Guadalupe Article Contribution of demography to economic growth SERIEs - Journal of the Spanish Economic Association Provided in Cooperation with: Spanish Economic Association Suggested Citation: Sánchez-Romero, Miguel; Abio, Gemma; Patxot, Concepció; Souto, Guadalupe (2018) : Contribution of demography to economic growth, SERIEs - Journal of the Spanish Economic Association, ISSN 1869-4195, Springer, Heidelberg, Vol. 9, Iss. 1, pp. 27-64, https://doi.org/10.1007/s13209-017-0164-y This Version is available at: https://hdl.handle.net/10419/195267 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ SERIEs (2018) 9:27–64 https://doi.org/10.1007/s13209-017-0164-y ORIGINAL ARTICLE Contribution of demography to economic growth Miguel Sánchez-Romero1·Gemma Abio2· Concepció Patxot2·Guadalupe Souto3 Received: 15 January 2017 / Accepted: 3 October 2017 / Published online: 12 October 2017 © The Author(s) 2017. This article is an open access publication Abstract From 1850 to 2000, in Western European countries life expectancy rose from 30–40 to 80years and the average number of children per woman fell from 4 to 5 children to slightly more than one. To gauge the economic consequences of these demographic trends, we implement an overlapping generations model with heterogeneity by level of education in which individuals optimally decide their consumption of market- and home-produced goods as well as the time spent on paid and unpaid work. We find that around 17% of the observed increase in per-capita income growth from 1850 to 2000 was due to the demographic transition. Around 50% of the demo- This project has received institutional support from the European Union’s Seventh Framework Programme for research, technological development and demonstration (AGENTA project, grant agreement No: 613247). Electronic supplementary material The online version of this article (doi:10.1007/s13209-017-0164-y) contains supplementary material, which is available to authorized users. BMiguel Sánchez-Romero [email protected] Gemma Abio [email protected] Concepció Patxot [email protected] Guadalupe Souto [email protected] 1Wittgenstein Centre for Demography and Global Human Capital (IIASA,VID/ÖAW, WU), Vienna, Austria 2Departament d’Economia, Universitat de Barcelona (UB), Barcelona, Spain 3Departament d’Economia Aplicada, Universitat Autónoma de Barcelona (UAB), Barcelona, Spain 123 28 SERIEs (2018) 9:27–64 graphic contribution is explained by the increase in the average productivity per worker (productivity component), which arises from the change in the population’s age structure and the rise in households’ saving rate. The remaining 50% is explained by the higher growth rate of workers relative to the total population (translation component). Keywords Demographic dividend ·Fertility, Mortality ·Per-capita income growth · Overlapping generations JEL Classification D58 ·E27 ·J11 ·N30 1 Introduction The importance of the demographic transition on per-capita income growth was neglected for a long time, mainly because of a myriad of inconsistent correlations between population and economic growth (Kelley 1988).1It was not until the 1990s, using empirical convergence models à la Barro (1991,1997), that several scholars were able to better isolate the effect of demography on economic growth (Kelley and Schmidt 1995;Bloom and Williamson 1997,1998). Their main finding was that demography has a strong and positive effect on economic growth when the workingagepopulationgrowsfasterthanthedependentpopulation,knownasfirst demographic dividend. Later on, Kelley and Schmidt (2005) added an important contribution by considering, in their convergence model, that changes in the age distribution of the population (known as the translation component) were likely to affect the productivity of workers (the productivity component). By doing so, they estimated demography to account for 20% of the per-capita income growth worldwide between 1965 and 1990, which was validated for the EU by several scholars (Prskawetz et al. 2007). Despitetheserecentfindingstherearestillmanyunanswered questions (Williamson 2013). For instance, to what extent does demography influence economic growth over a longer time span? What is the historical impact of demographic changes on economic growth? The demographic dividend literature has extensively used cross-country panel data for the period 1950–2010, which historically coincides with the period of most rapid population growth. However, as far as we know, there has been no study on the impact of demographic change on economic growth starting in the nineteenth century, exactly at the onset of the demographic transition in Europe (Livi-Bacci 2000;Lee 2003). The aim of this paper is to assess the impact of the demographic transition on percapita income growth along the period 1850–2000. We focus our analysis on Spain, since it is of great interest to economists, demographers, and historians due to the availability of historical data and the similarities with the East Asian “tiger economies” in the second half of the 20th century (Prados de la Escosura and Rosés 2010a). Spain started the demographic transition later than northern European countries (Livi-Bacci 2000). In 1850 the Spanish population size was around 15 million inhabitants, the 1We use per-capita income growth and economic growth interchangeably in this article. The demographic transition refers to the transition from high birth and death rates to low birth and death rates. 123 SERIEs (2018) 9:27–64 29 1850 1875 1900 1925 1950 1975 2000 Year 0 1 2 3 4 5 6 7 (a)(b) 1850 1875 1900 1925 1950 1975 2000 Year 0 10 20 30 40 50 60 70 80 90 Fig. 1 Spanish total fertility rate (TFR) and life expectancy at birth: Period 1850–2000. aTotal fertility rate (TFR), blife expectancy at birth. Source Authors’ estimates. Notes The TFR is the average number of children that would be born to a woman over her lifetime average woman expected to have between four and five children, and life expectancy at birth was close to 30years—due to an extremely high infant mortality—(Ramiro Fariñas and Sanz-Gimeno 2000). In 2000, the Spanish population was over 40 million people, the average woman expected to have 1.23 children, and life expectancy at birth was close to 80years (see Fig. 1). The Spanish population also witnessed an economic revolution during this period. According to Prados de la Escosura and Rosés (2009) the average labor income per worker rose from the equivalent of 3.000–3.500 euros per year in 1850 to more than 33.500 euros in 2000.2Moreover, the average number of hours worked declined by 36% points and the entrance into the labor market was delayed due to the educational expansion. Indeed, 64% of the cohort born in 1850 was illiterate and 34% had only primary education (Nuñez 2005). In 2000, by contrast, the average number of years of schooling was 8.4 for adults (Barro and Lee 2013). Thus, the increase is even more remarkable if we focus on the wage rate per hour worked which rose from the equivalent of 1.2 euros in 1850 to 19.2 in 2000 (Prados de la Escosura and Rosés 2009). The literature has frequently used convergence models to show the role of demographicchange on economicgrowth. However, the resultsof these econometricmodels usually suffer from endogeneity problems (Feyrer 2007). More importantly, it is not possible to extend this kind of analysis further in the past because of the lack of data. A new approach to answer this old question is to estimate the demographic dividend using overlapping generation models (OLG) since the accumulation of capital and labor are modeled endogenously. For instance, Sánchez-Romero (2013) followed this strategy to analyze the evolution of the demographic dividend in Taiwan. He inves- 2All figures are measured at constant prices of 2010. 123 30 SERIEs (2018) 9:27–64 tigates the economic impact of different demographic scenarios in order to assess the demographic dividend, obtaining similar results to those obtained using growth regression models. In this paper, we follow a similar strategy by implementing an OLG model. Nevertheless, our paper differs from Sánchez-Romero (2013) in two main aspects. First, the period of analysis increases from 40 to 150years. Second, the model is extended by introducing household production and non-homothetic preferences. We follow the works of Greenwood et al. (2005) and Ramey (2009) in order to account for the impact of technological progress on the value of time. In addition, we follow the work of Restuccia and Vandenbroucke (2013), that assume non-homothetic preferences, in order to assess the impact of the increase in longevity and technological progress on the simultaneous reduction in hours worked and the increase in schooling time during the last hundred and fifty years in the US. Thus, the combination of homeproduction with non-homothetic preferences allows us to account for the historical reduction in paid hours. The costs of rearing children to households are introduced in the utility function through the family size (Browning and Ejrnæs 2009) and in the time constraint. Given that we have detailed demographic information for the period 1850–2000, our family size not only changes over time but also by age of the household head. The units of equivalent adult consumption of market and home-produced goods rely on the AGENTA database (Vargha et al. 2015;Rentería et al. 2016). In addition, to control for the educational dividend caused by the educational transition (Crespo-Cuaresma et al. 2014), we introduce heterogeneous agents that differ in their educational attainment, using data from the Wittgenstein Data Explorer (Wittgenstein Centre for Demography and Global Human Capital 2015). Hence, the model takes as exogenously given inputs the evolution of vital rates (fertility and mortality) and human capital investments, ignoring their feedback effects. Comparingour baseline scenario to ahypothetical economy whose population faces fertility and mortality rates which were prevailing in Spain in 1800, we find that the changes in the age structure of the population accounts for 16.8% of per-capita income growth for the period 1850–2000. This result lies within the possible range of values (i.e., 16–44%) found in the literature for the period 1950–2010 (Kelley and Schmidt 2005). Additional counterfactual experiments show that fertility explains 14.5% of the impact of demographic changes in per-capita income growth, while mortality explains 6.4% of the impact of demographic changes in per-capita income growth. Moreover, we have further decomposed the contribution of demography to per-capita income growth in the translation component (the difference between the growth rate of workers and the total population) and the productivity component (the growth rate of output per worker). Our results suggest that over this period of one hundred and fifty years, the translation component accounted for 50% of the total income growth, while the productivity component accounted for 50%. The growth rate of output per worker is explained by two main factors. First, the transition from a young age structure to an aging population, since this demographic process leads to an increase in the average age of asset holders—older households own more assets than younger households— and in the average age of workers—older households have a higher income. Second, through a rise in the propensity to save due to the longer life expectancy. 123 SERIEs (2018) 9:27–64 31 The paper is organized as follows. Section 2details the theoretical model and its main theoretical implications. Section 3presents the Spanish demographic transition, the economic data, and the model calibration. The contribution of the demographic transition on the per-capita income growth rate is presented in Sect. 4and the impact of different model assumptions on our results are discussed in Sect. 5. Section 6 concludes. 2 The model We implement a large-scale OLG model à la Auerbach and Kotlikoff (1987) in which heterogeneous households by level of education endogenously choose consumption and the times spent in the market and in home production. Demographics, the educational attainment, and technological progress are exogenous. Firms are assumed to operate in perfectly competitive markets and produce under constant returns to scale. To account for the full effect of demography on the economy, we assume Spain to be a closed economy. Hence, changes in the population structure might have an impact on input prices. Moreover, to better capture the accumulation of capital over time, we consider the historical evolution of public pension expenditures. Thus, our individuals contribute a fraction of their labor income to the pension system and receive pension benefits when retired. 2.1 Household preferences and home production For expositional purposes, in this subsection we abstract from time subscripts. Households may belong to any of the three possible levels of education that we denote by e∈E={primary or less; secondary; tertiary}. Households derive utility from consumption of market-produced goods cm, home-produced goods ch, and leisure z.The period utility of a household, whose head has a level of education e∈E, is given by uh e(cm,ch,z)=φm elog cm 1+η(n)−¯cm+φh elog ch 1+η(n)−¯ch +φz e z1−1 σe−1 1−1 σe ,(1) where η(n)is a function that transforms the number of children in the household by age (n) to the number of equivalent adult consumers, ¯ci>0 is the subsistence level of consumption of type i∈{m,h},φi e>0 is the relative weight of good i∈{m,h,z} on the period utility, and σe>0 is the elasticity of substitution on leisure. The set of parameters {φm e,φh e,φz e,σ e}depends on the level of education in order to better account for differences in the labor supply. Home production requires intermediate goods and labor f(cx,h)=θcxρ−1 ρ+(1−θ)(h) ρ−1 ρρ ρ−1 ,with ρ≥0,(2) 123 32 SERIEs (2018) 9:27–64 where cxstands for goods purchased in the market and used as intermediate goods for home production, his the time spent on home production, θis assumed to be positive and between zero and one, and ρis the elasticity of substitution between the home input factors. According to Eq. (2) technological progress has an impact on home production through intermediate goods. Moreover, we implicitly assume that homeproduced goods are always consumed in the household and not sold in the market; i.e., ch=f(cx,h). 2.2 Household problem In each year thouseholds are heterogenous by their educational attainment (e), age (j), year of birth (t−j), assets accumulated (a), and the number of children raised at home (n). We denote by xe,j,t={ae,j,t,nj,t}the state variables of a household with a level of education eat age jin year t. Each household at age jin year tfaces a probability to surviving to the next age of πj+1,t+1, which is independent of other household characteristics.3Individuals can live to a maximum of 100years. Children are raised by the household between age 0 and 15 and do not work either in the market or at home.4At the age of 16 (J0) children start making decisions, leave their parents home, and establish their own households. After age 65 (JR) all individuals retire. Adults are endowed with one unit of time that they distribute between market work (), household chores (h), child rearing (hυ(n)), and leisure (z). Hours worked in the market are supplied in exchange of a (net of taxes) wage rate, whereas the time spent producing goods and rearing children is unpaid. Function υ(n)denotes the time spent rearing nchildren per unit of time devoted to home production. Hence, the functional form hυ(n)assumes, ceteris paribus, that individuals who either spend more hours producing goods at home or have more children also devote more time to childrearing and less time to labor. Annuity markets are absent and accidental bequests are distributed by the government to all households in the economy. Householdsoptimallychoosetheirconsumptionofmarketgoods, home production, intermediary goods, leisure time, and time spent on home production by maximizing their lifetime utility (V). Let the control variables of a household with a level of education eat age jin year tbe ce,j,t={cm e,j,t,ch e,j,t,cx e,j,t,he,j,t,ze,j,t}∈C. Thus, the household problem is equivalent to solving Vj,t(xe,j,t)=max ce,j,t∈Cuh e(cm e,j,t,ch e,j,t,ze,j,t)+πj+1,t+1Vj+1,t+1(xe,j+1,t+1)(3) 3Although the literature shows that there exists a positive correlation between educational attainment and longevity, we do not have information on death rates by educational attainment for the period analyzed (Lleras-Muney 2005). 4This assumption is necessary for reducing the complexity of the model. Existing studies for families working at the textile sector in Catalonia show that children above the age of 5 or 6 were progressively substituting the market work of their mothers as the number of offspring in the household increased (Camps- Cura 1998). Indeed, this pattern was common to many other countries (Bengtsson 2004). Moreover, the literature suggests that children between ages 0 and 17 in the US supplied on average 4h per week doing household chores, which represents one-sixth of the total average time devoted to such work by a prime-age adult (Ramey 2009). 123 SERIEs (2018) 9:27–64 33 subject to the budget constraint ae,j+1,t+1=Rtae,j,t+tr j,t+(1−τt)we,j,te,j,t−cx e,j,t−cm e,j,tfor J0≤j≤JR, Rtae,j,t+tr j,t+bt−cx e,j,t−cm e,j,tfor j>JR,(4) the time constraint e,j,t+he,j,t[1+υ(nj,t)]+ze,j,t=1,(5) and the standard boundary conditions ae,J0,·=0 and ae,100,·≥0.(6) Parameter Ris the capitalization factor, tr is the accidental bequests distributed by the government to the household, τis the social contribution rate to the pension system, we,j,t=rH te,jis the wage rate per hour worked of an individual with education eat age jin year t, which depends on the wage rate per efficient unit of labor (rH t) and the age-specific productivity by educational attainment (e,j), and btis the pension benefit received in year t. The optimal consumption path of market goods can be characterized by the Euler condition augmented by household size and subsistence level (see “Appendix A”): ˜ Cm e,j+1,t+1˜ Cm e,j,t=πj+1,t+1Rt+1with ˜ Cm e,j,t=cm e,j,t−¯cm[1+η(nj,t)].(7) Equation (7) indicates that households smooth the consumption above the subsistence level ¯cmfor all household members. The introduction of ¯cmand η(nj,t)are key for explaining the historical decline in the number of hours worked. This is because when labor productivity is low and the number of children is high, households need to work more hours in the market in order to finance the minimum consumption level of goods. Afterwards, as productivity rises, households need less hours in the market to finance the minimum consumption expenditure (Restuccia and Vandenbroucke 2013). Moreover, given that the equivalent adult consumers multiply ¯cmin (7), the same reasoning applies to η(nj,t). Thus, increases in the number of equivalent consumers force individuals to supply more hours to the market. However, this effect might be offset by the subsequent increase in the demand for childrearing within the household as we will explain below. The optimal hours worked in the market (or intensive labor supply) are given by the difference between the total available time and the sum of leisure and unpaid work, e,j,t=1−ze,j,t−he,j,t[1+υ(nj,t)],(8) where the optimal conditions of leisure and unpaid work are ze,j,t=φz e φm e ˜ Cm e,j,t cm e,j,t cm e,j,t (1−τt)we,j,tσe ,(9) 123 34 SERIEs (2018) 9:27–64 he,j,t=φh e φm e ch e,j,t ˜ Ch e,j,t ˜ Cm e,j,t cm e,j,t cm e,j,t yh e,j,t+θ 1−θyh e,j,tρ,(10) respectively. The term yh e,j,t=(1−τt)we,j,t[1+υ(nj,t)]is the opportunity cost of an hour devoted to home production and ˜ Ch e,j,tis equal to ch e,j,t−¯ch[1+η(nj,t)]. The non-homotheticity between market consumption and leisure, see Eqs. (1) and (9), implies that a rise in productivity leads to an increase in leisure and a decline in paid hours. This is reflected in the second ratio inside the parenthesis in Eq. (9). The strength of this positive effect on leisure is nonetheless liable to diminish as households become wealthier, since ˜ Cm e,j,twill converge toward cm e,j,t. Equation(10) makesexplicitthe importanceofthe minimum consumption ofhomeproduced goods (¯ch) and the elasticity of substitution between home input factors (ρ) for the evolution of home labor. For example, if we assume that ¯ch=0 (i.e. ch e,j,t=˜ Ch e,j,t) and that ρ=1, then an increase in productivity leads households to increase their time spent on unpaid work. However, for a sufficiently high value of ¯ch>0, an increase in productivity will make the marginal utility of home-produced goods to decline faster, which reduces the time spent on unpaid work. Similar to the effect of ¯cmon paid hours, a rise in productivity leads to a drop in home labor, since individuals substitute home labor for intermediate goods. Indeed, in the interior solution, the ratio of intermediate goods to labor in home production is cx e,j,t he,j,t =θ 1−θyh e,j,tρ .(11) Equation (11) shows that either a rise in wages, or the time spent rearing children per hour of home production, or a drop in the contribution rate, raises the ratio of intermediate goods to home labor for any ρ>0. The parameter ρis also crucial for explaining the impact of υ(n)on both the total unpaid work (i.e. h[1+υ(n)]) and paid work. We can distinguish three cases. If ρ<1, an increase in the time spent rearing children per unit of home labor will raise the number of total unpaid hours worked and reduce that of paid hours. This is because the increase in the marginal cost of home labor cannot be offset with the rise in intermediate goods, given that cxand hare close complements. Moreover, given that leisure does not depend on υ(n), the increase in υ(n)has the opposite effect on paid hours. If ρ=1, the increase in υ(n)will have no effect on the total time devoted to unpaid labor, since households offset the rise of υ(n)with a proportional increase in intermediate goods. And if ρ>1, a rise in υ(n)leads to a drop in the total number of unpaid hours because households substitute home labor for intermediate goods. Finally, households can, in addition to supplying labor to the market, specialize in home production when the marginal rate of substitution between leisure and consumption is greater than the effective wage rate per hour worked: ∂uh e ∂ze,j,t∂uh e ∂cm e,j,t >(1−τt)we,j,t.(12) 123 SERIEs (2018) 9:27–64 41 Birth cohort 1850 1875 1900 1925 1950 1975 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Primary or less Secondary Tertiary Age 20 30 40 50 60 70 0 1 2 3 4 Tertiary Secondary Primary or less Age 20 30 40 50 60 0 0,2 0,4 0,6 0,8 1 Tertiary Secondary Primary or less (a) (b) (c) Fig. 4 Decompositionof thestock of human capitalbyeducational attainment (bothsexescombined).aThe educational distribution, Ec(e),bthe endowment of efficient labor units, WP and clabor force participation rates, LF. Sources Educational distribution data is taken from the Wittgenstein Centre Data Explorer (Wittgenstein Centre for Demography and Global Human Capital 2015), the endowment of efficient labor units comes from MTAS (2010), and the average labor force participation rate between 1987 and 2013 is calculated using data from INE (2015a) 123 42 SERIEs (2018) 9:27–64 1850 1875 1900 1925 1950 1975 2000 Year 0 5 10 15 20 Authors' estimates OECD (2015) Nicolau (2005) EU Labor Force Survey EPA Censos de Poblacion Fig. 5 Labor force, Spain 1850–2000 (both sexes combined, in millions) 2013 for each educational group. The LF profile by educational attainment, shown in Fig. 4c, is assumed to remain fixed in Eq. (17), while the supply of labor will be endogenously chosen by households in the OLG model. We choose the period 1987–2013 because it includes two periods in which the unemployment rate declined (1987–1991 and 1995–2007) and another two periods in which it increased (1991– 1994 and 2008–2013). The educational distribution by birth cohort Ecin Spain is taken from the Wittgenstein Centre Data Explorer (Wittgenstein Centre for Demography and Global Human Capital 2015) for two reasons. First, this database offers information on historical reconstruction of educational attainment for the 20th century, and second, it provides harmonized projections until 2100 of the population by age and educational attainment (Lutz et al. 2014). We extract information for Spain on the shares of population by levels of education for the period 1970–2100, which allow us to calculate the educational distribution for each birth cohort born after year 1868.10 Since the available data is grouped by five-year age groups, we apply linear splines to interpolate the educational distribution for intermediate cohorts. Last, but not least, in order to guarantee an initial steady state, the educational attainment for cohorts born before 1868 are assumed to coincide with that of the cohort born in year 1868. To check the validity of our reconstruction of the stock of human capital ˆ Lt, we compare the labor force, which results from applying the formula 65 j=16 Nj,tELFe,jdEt−j(e), to existing estimates of the labor force from 1850 to 2000. Figure 5shows that our labor force estimates before 1950 are very similar to those reported by Nicolau (2005) and it fits well to the most recent estimates from widely used databases which validates our strategy. 10 Given the positive correlation between educational attainment and life expectancy, individuals born close to 1868, who reached tertiary education are likely to be overrepresented. Nevertheless, due to the late onset of the educational expansion in Spain, our main results are not affected. 123 SERIEs (2018) 9:27–64 43 Calendar year 1850 1875 1900 1925 1950 1975 2000 (in %) -20 -10 0 10 20 Data Data (Loess) Data (average growth) Fig. 6 Labor-augmenting technological progress, Spain 1850–2000. Source Authors’ calculations. Note The ‘Loess’ smoothing profile was computed using a bandwidth of 0.05 and rescaled to give an annual average growth rate of 1.40% 3.4 Labor-augmenting technology The labor-augmenting technology is calculated using reconstructed input factors in Sects. 3.2 and 3.3.WeuseOECD (2015c) data on the total number of workers from 1956 in order to account for the rise in unemployment. Our estimation of the laboraugmenting technological progress is calculated applying the formula Δln At=Δln(˜ Yt/Nt)−Δln(˜ Lt/Nt)−Δαt 1−αt ln(˜ Kt/˜ Yt), (18) where ˜ Ytis the value added, ˜ Ltis the stock of human capital, and ˜ Ktis the stock of physical capital. From (18) we obtain that the annual average labor-augmenting technological progress from 1850 to 2000 was 1.40%. We assume no productivity growth before 1850 and we also assume that our estimated average productivity growth for the period 1850-2000 from 2012 onwards. Figure 6shows the percentage change in the estimated labor-augmenting technological progress At. 3.5 Calibration To perform our quantitative experiment we need to find the parameters such that the model is capable of reproducing some key historical facts of the Spanish economy. We proceed as follows: In our baseline, we use the annual physical capital depreciation rate δtestimated in Sect. 3.2 and the labor share calculated by Prados de la Escosura and Rosés (2009). Hence, our modeled depreciation rate is, on average, close to 0.057 and the average labor share for the period 1850–2000 is 0.68. Since there is no information on the average pension benefit across cohorts, we calculate it indirectly. First, we decompose the ratio total public pension expenditures to compensation of employees using the identity 123 44 SERIEs (2018) 9:27–64 Pension expenditurest Comp. employeest =Avg. pension per indiv. 65+t Avg. salary per indiv.16−64t ×Pop. 65+t Pop.16−64t .(19) The first term on the right-hand side of (19) is the average replacement rate of the pension system in year t, which hereinafter we denote by ψt. The second term on the right-hand side is the old-age dependency ratio. Data on total public pension expenditures from 1850 to 2000 is taken from Comín and Díaz (2005). Compensation of employees is calculated by multiplying the labor share from Prados de la Escosura and Rosés (2009) to the value added along the period analyzed. Second, the model uses ψtto calculate the pension benefits received for all individuals above age 65 as follows bt=ψt j>JRNj,t rH tLt JR j=J0Nj,t for all t, where the last term on the right-hand side is the average labor income of the workingage population. The average replacement rate before 1850 and after 2000 is assumed to stay constant at the levels observed in 1850 and 2000, respectively. The model is comprised of sixteen parameters. Specifically, we have, for each educational group, the utility weights of each good on the household utility {φm e,φh e,φz e} and the intertemporal elasticity of substitution on leisure σe. Then, common to all educational groups, we have the minimum consumption level for each good {¯cm,¯ch} and the home-production technology {θ,ρ}. To reduce the dimension of the parameter set, we impose without loss of generality that φm e+φh e+φz e=1 and that work at home upon retirement accounts for 250min per day (Rentería et al. 2016). Thus, using the first-order conditions, we can indirectly calculate φz eas φz e=φh e(1−θ)(1−h)σe h, with h=.26 or, equivalently, 250min per day out of 60min ×(24–8) available hours. Moreover, we rely on the estimates of Ramey (2009) and set ρat 0.95. All remaining behavioral parameters are structurally estimated using the model. Let us denote by λ the 9 ×1 vector of parameters left to be determined: λ=[λe,¯cm,¯ch,θ]with λe={φh e,σ e}for e∈E. Further, we implement the following restrictions on the parameters, λ∈Λ.First, the intertemporal elasticities of substitution σemust be in the interval between zero and one. Second, the weight on the household utility of home-produced goods φh e must be non-negative. Third, the minimum consumption level of market- and homeproduced goods cannot be negative. And fourth, the share of intermediate goods on home production θmust be in the interval between zero and one. For a given λ, we solve the model in order to obtain the optimal labor supply of individuals over their lifecycle, the average hours worked, the output per capita, and the consumption per capita. Then, we compute the function F(λ)defined by 123 SERIEs (2018) 9:27–64 45 Table 1 Model parameters Parameter Symbol Values Firms technology Capital share αtPrados de la Escosura and Rosés (2010b) Capital depreciation rate δtPrados de la Escosura and Rosés (2009) Labor-augmenting technology AtAuthors’ estimates Home production Elasticity of substitution on labor ρ0.952 Ramey (2009) Factor share θ0.430 Household preferences Age at parental leave J016 Retirement age JR65 Subsistence level market goods ¯cm0.121 Subsistence level home goods ¯ch0.067 Level of education Primary or less Secondary Tertiary IES on leisure σe0.281 0.418 0.442 Weight of market goods φm e0.056 0.049 0.053 Weight of home goods φh e0.539 0.460 0.449 Weight of leisure φz e0.405 0.491 0.498 F(λ)= 65  j=16  e∈E Φ1 e,j(X;λ)2+ 2000  t=1851  i={l,y,c} Φi t(X;λ)2,(20) where Xdenotes the exogenous information set of the economic model. The first term corresponds to the difference between the labor supply by educational attainment, shown in Fig. 4c, and the individual labor supply by educational attainment obtained with the model, i.e., Φ1 e,j(X;λ)=γLFe,j−1 27 2013 t=1987 e,j,t. To transform the participation rates to actual hours worked, we set γto 0.32, which is equivalent to working 36h per week out of a total of 112h per week. The second term captures the difference between the observed average hours worked for the population between 16 and 65years, income per capita, and consumption per capita from 1850 to 2000, and those obtained with the model. Thus, we search for the value of λthat minimizes the function F(λ). Table 1reports the parameter values taken from the literature as well as those structurally estimated with the model. We can highlight in Table 1three key parameters: the elasticity of substitution between input factors in home production (ρ),the subsistence level of market-produced goods (¯cm), and the subsistence level of homeproduced goods (¯ch). The fact that the subsistence level of market- and home-produced goods are positive implies that the income effect dominated over the substitution effect when productivity was low. As a consequence, individuals had to work long hours in 123 46 SERIEs (2018) 9:27–64 order to finance the consumption of both goods in the nineteenth century. A similar result is obtained by Restuccia and Vandenbroucke (2013) analyzing the accumulation of human capital and the evolution of labor supply in the US for cohorts born between 1870 and 1970. It is also worth mentioning that the income effect varies over the lifecycle of the household, becoming stronger when the number of equivalent adult consumers in the household rises, and weaker when productivity increases. Another important remark is that ¯cmand ¯chdiffer in magnitude. Hence, the rate of change of market-produced goods relative to home-produced goods will differ over time as productivity increases. This result has also been found recently by Moro et al. (2017). The other key parameter is ρ, which we took from Ramey (2009) given that we did not have enough data to estimate it structurally. As we have commented in Sect. 2.2 a value of ρlower than one implies that home labor increases as productivity is on the rise. This effect is, however, offset by the existence of a positive minimum consumption of home-produced goods. The other effect of assuming a ρ<1 is that total unpaid hours marginally fall and paid hours marginally rise with the decline in υ(n), see Fig. 3b. Figure 7shows the in-sample performance of the baseline model with respect to the targeted time series. In Fig. 7a, we can see how well the model replicates the average per-capita hours worked by level of education between 1987 and 2013. Figure 7b compares the observed average fraction of hours worked by the working-age population to that obtained in the baseline. The discrepancy between both figures from 1976 to 2000 is explained by the fact that we do not consider the risk of unemployment in the model, whereas the unemployment rate rose to values over 20% during this period. In Fig. 7c, d we show how well the model replicates the evolution of the income per capita (or economic growth) and the consumption per capita from 1850 to 2000. The next section will apply the calibrated OLG model to the Spanish data to disentangle the contribution of demography to the observed economic growth. 4 Results In this section we quantify the Spanish demographic dividend or, equivalently, the contribution of demography to Spain’s economic growth from 1850 to 2000. In so doing, we first need to realize that assessing the demographic dividend over a long period of time by using the naïve demographic model (Y/N)gr =(Y/W)gr +(W)gr −(N)gr ,(21) where Wstandsforworkersand ‘gr’ denotestheaverage growthrate,givesanincorrect measure, since the growth rate of the support ratio (i.e. (W/N)gr) is zero in the long run. This is because in a stable population the growth rate of the population coincides with the growth rate of workers (Lotka 1939). Table 2shows the decomposition of the growth rate of per-capita output in Spain from 1850 to 2000. A naïve calculation using the first row of Table 2suggests that only 5% (i.e., = (Wgr −Ngr)/ (Y/N)gr =(.80–.72)/1.62) of the Spanish economic growth from 1850 to 2000 is explained by demographic changes. However, long-run demo- 123 SERIEs (2018) 9:27–64 47 20 25 30 35 40 45 50 55 60 Age 0 0.1 0.2 0.3 0.4 0.5 Fraction of market-hours worked Baseline, primary or less Data, primary or less Baseline, secondary Data, secondary Baseline, tertiary Data, tertiary 1850 1875 1900 1925 1950 1975 2000 Year 0 0.1 0.2 0.3 0.4 0.5 Fraction of hours worked Data Baseline (our model) 1850 1875 1900 1925 1950 1975 2000 Year 7 8 9 10 11 Data Baseline (our model) 1850 1875 1900 1925 1950 1975 2000 Year 7 8 9 10 11 Data Baseline (our model) (a) (b) (c) (d) Fig. 7 In-sample performance of the model, Spain 1850–2000. aFraction of per-capita hours worked by educational attainment, average of the 1987–2013 period, baverage hours worked by working-age population, cincome per capita (in logs) and dconsumption per capita (in logs). Source See text on Fig. 4 for panels 7a, b. National accounts data are taken from OECD (2015c)andPrados de la Escosura and Rosés (2009) graphic changes are translated into economic growth through productivity effects, known as the productivity component (Kelley and Schmidt 2005). For instance, some possible channels for demography to impact on productivity are: scale economies, population density, life-cycle savings, changes in the supply of labor, and changes in the human capital accumulation, among others. In order to control for some of the above mentioned channels, in this article, we follow the same strategy as in Sánchez-Romero (2013) to assess the Spanish demographic dividend. First, we show that our model is capable of reproducing the evolution of per-capita income along the period 1850–2000. In this regard, Fig. 7c shows that our 123 48 SERIEs (2018) 9:27–64 Table 2 Per-capitaoutputgrowthin Spain:1850–2000(annual average logarithmicratesin percent) Source Authors’ estimations and Prados de la Escosura and Rosés (2009) Period Output per capita (Y/N)gr Output per worker (Y/W)gr Workers Wgr Population Ngr 1850–2000 1.62 1.54 0.80 0.72 1850–1950 0.68 0.48 0.88 0.68 1951–1974 5.05 5.48 0.69 1.13 1975–2000 2.06 1.88 0.69 0.50 Bold values indicate the main results model mimics well the small growth of per-capita income for the period 1850–1925, theperiod of stagnation from 1925–1950,and the golden age of rapideconomic growth from 1950–1975. Second, based on the vital rates obtained for year 1800, we build a set of different demographic scenarios (from now on ‘experiments’) to disentangle the effect of demography for the period 1850 to 2000.11 We propose the following three experiments: •Experiment 1 In this experiment we cancel the effect of fertility and mortality. This experiment gives the structure of the population in year 1800 under stable conditions. Life expectancy at birth is fixed at 31.5years and the TFR is fixed at a value slightly above 5, which implies a young population structure (see the dotted line in Fig. 8a) and a constant annual population growth rate of 0.5% (see Fig. 8b). By comparing the economic outcomes of the baseline simulation to those of Experiment 1 we get the contribution of demography to economic growth. •Experiment 2 In this experiment we cancel the effect of the increase in longevity. This demographic scenario implies that the population would have increased until 1920 and it would have declined afterwards due to the fall in fertility below replacement level (see Fig. 8b). As a consequence, the age distribution of the population in year 2000 would have been older than under the baseline (see the triangle line in Fig. 8a). •Experiment 3 In this experiment we shut down the effect of the decline in fertility. The population growth rate would have continuously increased over the twentieth century until reaching a stable population growth rate of 3%. Thus, the age distribution of the population would have been even younger than under Experiment 1 (see circled line in Fig. 8a). Moreover, given the increasing population growth, during the period 1850–2000 the population growth rate would have been larger than the growth rate of the population between ages 16 and 65 (see Fig. 8b). Notice that setting Experiments 2 and 3 allows us to separate, as completely independent factors, the effect of fertility and mortality on economic growth. Given these three experiments, Sánchez-Romero (2013) shows that the impact of demography on per capita income growth can be easily estimated by calculating the relative contribution of each demographic factor to the observed economic growth, 11 Sánchez-Romero (2013) shows that fixing birth and death rates at the levels prevailing at the beginning of the period analyzed underestimates the demographic impact. Hence, a more correct approach is to fix birth and death rates at least a generation before the period analyzed. 123 SERIEs (2018) 9:27–64 49 0 102030405060708090100110 Age 0 1.0 2.0 3.0 4.0 Population distribution (in %) Baseline Experiment 1 Experiment 2 Experiment 3 Baseline Experiment 1 Experiment 2 Experiment 3 -0.5 0 0.5 1.0 1.5 2.0 Growth rate (in %) Total population growth, Ngr Population growth between age 16-65 1.66 1.56 -0.33 -0.15 0.50 0.50 0.85 0.72 (a) (b) Fig. 8 Population characteristics under different experiments, Spain 1850–2000. aPopulation distribution inyear 2000, Nj,2000/N2000 and bpopulation growth rates from 1850to 2000. Source Authors’calculation. Notes Experiment 1 assumes a fixed TFR around 5 and a life expectancy at birth of 31.5years. Experiment 2 assumes a fixed life expectancy at birth of 31.5years and a TFR as in the baseline. Experiment 3 assumes the TFR to be fixed around 5 and life expectancy evolving as in the baseline which is replicated by our baseline model. To gain some intuition about our counterfactual experiments, let us assume per capita income growth from time t0to time t,(Y/N)gr, is explained by an average exogenous increase in productivity (Agr), by demographic changes (Dem), and by other exogenous factors (I). Thus, per capita income growth from time t0to tis given by (Y/N)gr =Agr +Dem +I.(Baseline)(22) If we cancel—like in Experiment 1—the demographic changes (Dem), ceteris paribus other exogenous information (i.e. Agr +I), the new per capita income growth during 123 50 SERIEs (2018) 9:27–64 the same period ( Y/N)gr will be given by ( Y/N)gr =Agr +I.(Experiment)(23) Thus, from (22) and (23) the contribution of demography to the observed per capita income growth can be calculated as follows (Y/N)gr −( Y/N)gr (Y/N)gr =Dem Agr +Dem +I.(24) The same intuitive calculation can be done either for each demographic factor (i.e. experiments 2 and 3) or for any other factor affecting per capita income growth. Next we explain how demography affects per capita income and its relative contribution. Mortality and fertility effects. FollowingKelleyandSchmidt(2005)ourmodelcaptures the effect of demography on per capita income through two main channels: (i) the difference between the growth rate of workers and the growth rate of the population, or translation component, and (ii) through changes in the market labor supply and in household savings, caused by the rise in life expectancy and the fall in fertility, or productivity component. Notice the productivity component arises not only from changes in the behavior of individuals, but also from changes in the population’s age structure. Givenoureconomicsetup,thesetwocomponentsarewell-capturedbythefollowing decomposition of per capita income growth12 (Y/N)gr =α 1−αK/Ygr +(L/W)gr   Productivity component +(W)gr −(N)gr   Translation component +Agr.(25) By comparing per-capita income growth in the baseline to that in the experiments, the first term on the right-hand side of Eq. (25) mainly captures the change in household savings, the second term L/Wmainly reflects the change in the average hours worked, and the third term W/Nreflects the change in the support ratio.13 12 If we divide both sides of the Cobb–Douglas production function (13)byYαt t, solving for Yt,and dividing both sides by the total number of workers Wt,weget Yt Wt =Kt Ytαt 1−αtAt Lt Wt . The growth in output per worker comes from the growth in the capital-output ratio, the growth in human capital per worker, and the labor-augmenting technological progress. Thus, substituting the output per worker in (21)gives(25). 13 Given that the model does not distinguish at each age between intensive labor supply (i.e. hours worked) and extensive labor supply (i.e. labor participation), Wis the population between age 16 and 65 or workingage population. 123 SERIEs (2018) 9:27–64 57 sumption of a household, whose head is of age j, is obtained by maximizing (3) with respect to cm j,ch j,cx j,hj, and zj, subject to (4), the time constraint, and the boundary condition. The first-order conditions (FOC) are: cm j:∂uh e ∂cm j =πj+1 ∂Vj+1(xj+1) ∂aj+1 ,(26) ch j:∂uh e ∂ch j =∂uh e ∂cm j p∗ j+λ1,(27) where p∗ jis the shadow price of home-produced goods and services in a household whose head is age jand λ1is the Lagrange multiplier associated to home production. cx j:∂uh e ∂ch j ∂fh ∂cx j =∂uh e ∂cm j ,(28) hj,zj:∂uh e ∂ch j ∂fh ∂hj =∂uh e ∂zj (1+υ(nj)) for all j.(29) If households supply their labor in the market, the time spent on home production satisfies hj:∂uh e ∂ch j ∂fh ∂hj =∂uh e ∂cm j1−τjwj(1+υ(nj)). (30) The envelope condition (EC) is aj:∂Vj(xj) ∂aj =πj+1 ∂Vj+1(xj+1) ∂aj+1 Rj.(31) Combining (26) and (31)gives(7). Equation (11) is obtained dividing (30)by(28). B Market clearing conditions Let j∈J={0,...,100},t∈T={1500,...,2500}, and e∈E.Given initial values {¯cm,¯ch,φm e,φh e,φz e,σe,θ,ρ,αt,δt,gt,ψt,J0,JR}e∈E,j∈J,t∈T, demographics {Nj,t,nj,t,πj,t}j∈J,t∈T, the educational distribution Et(e)for cohorts born at time t∈T, and the age-specific productivity endowment by educational attainment {j,e}e∈E,t∈T, a recursive competitive equilibrium is a sequence of a set of household policy functions ce,j,t∈C, government policy functions {trj,t,τ t}j∈J,t∈T, and factor prices {rH t,rK t}t∈Tsuch that 1. Factor prices equal their marginal productivities. 123 58 SERIEs (2018) 9:27–64 2. The government’s budget constraint (14) is satisfied and all accidental bequests equal all transfers given 100  j=16 Nj,t(1−πj,t)E ae,j,tdEt−j(e)= 100−28  j=16 Nj,tπj,ttrj,t where trj,t=max{0,ξj,t}, Dt= 100  j=16 Nj,tmax{−ξj,t,0} with ξj,t=⎧ ⎨ ⎩ Nj+28,t(1−πj+28,t) Nj,tπj,tEae,j+28,tdEt−j−28(e)+1−πj,t πj,tEae,j,tdEt−j(e)if 16 ≤j≤44, Nj+28,t(1−πj+28,t) Nj,tπj,tEae,j+28,tdEt−j−28(e)if j>44. 3. Given the factor prices and government policy functions, household policy functions satisfy Eqs. (7)–(2), and the commodity of home production clears: 100  j=16 Nj,tE f(cx e,j,t,he,j,t)dEt−j(e)= 100  j=17 Nj,tE ch e,j,tdEt−j(e) 4. The stock of physical capital and the labor input are given by: Kt= 100  j=16 Nj,tE ae,j,tdEt−j(e) Lt= 100  j=16 Nj,tE e,je,j,tdEt−j(e) 5. The commodity market clears: Yt=Ct+St where the total consumption of market goods Ct=100 j=16 Nj,tEcm e,j,t+ cx e,j,tdEt−j(e)and Stis gross savings in year t. 123 SERIEs (2018) 9:27–64 59 C Population reconstruction Time is discrete. Individuals are assumed to live for a maximum of 100years. Let the survival probability to age jin year tbe Sj,t= j−1  x=0 πx,t−xwith S0,·=1,S100,·=0,(32) where πj,tis the conditional probability (of being alive at age jin year t) of surviving to age j+1 (with πj,·=0, for all j≥100). Let Nj,tbe the size of the population at age jin year t. We assume a closed population. Thus, the population at time t+1is given by the population in year tplus the total number of births in year t, denoted Bt, less the total number of deaths during the year Dt. The dynamics of the population can be written in matrix notation using a Leslie matrix (Leslie 1945;Preston et al. 2002) N(t+1)=(t)N(t), (33) with Γj,1(t)=L0,t 2S0,tfj,t+fj+1,t Lj+1,t Lj,tffab, Γj+1,j(t)=Lj+1,t Lj,t ,for j∈{1,...,99}at time t, (34) where Lj,t=Sj,t+Sj+1,t 2is the person years lived by the cohort between ages jand j+1 in period t,fj,tis the age-specific fertility rate at age jin year t,ffab is the fraction of females at birth (we assume ffab =0.4886, which is the standard value in the demographic literature). To reconstruct the population in Eqs. (33) and (34), we use a simplified version of a GIP model that matches the specific characteristics of our economic model: one gender without distinction between parity and region of birth, among others. The objective function used to solve the problem is: min {α1 t,α2 t,α3 t,μt,βt} t∈D1−ˆ Dt/Dt2+ t∈B1−ˆ Bt/Bt2+ t∈N1−ˆ Nt/Nt2 + t∈E1−ˆe0,t/e0,t2+ t∈T1−ˆ TFRt/TFRt2+ t∈C Ω−1  a=0(Na,t−ˆ Na,t)/Nt2 + T  t=t0 2  i=0αi t+1−αi t2+ T  t=t0 (μt+1−μt)2+ T  t=t0 (βt+1−βt)2,(35) subject to Eqs. (33)–(34) and to 3  k=1 αk tf(k) j=fj,twith 3  k=1 αk t=1,(36) 123 60 SERIEs (2018) 9:27–64 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 0 20 40 60 80 Year Life expectancy at birth Model e 0 Data e 0 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 0 1 2 3 4 5 6 7 8 9 Year Total fertility rate Model TFR Data TFR 1800 1820 1840 1860 1880 1900 1920 1940 1960 1980 2000 0 100 200 300 400 500 600 700 800 Year (thousands) Data births Model births Data deaths Model deaths 1775 1800 1825 1850 1875 1900 1925 1950 1975 2000 2025 5 10 15 20 25 30 35 40 45 Year Population size (in millions) GIP Population size Data Population size (Spain) (a) (b) (c) (d) Fig. 9 In-sampleperformance of the GIP modeltoexistingdemographic data. Spain:Selectedyear between 1787and 2000. aLife expectancy atbirth, btotal fertility rates,ctotal birthsand deathsand dtotal population j−1  x=0 πx,t=e2(μt+βtY1(x)+(1−βt)Y2(x)) 1+e2(μt+βtY1(x)+(1−βt)Y2(x)) with βt∈[0,1],(37) where {α1 t,α2 t,α3 t,μ t,β t}are the corresponding parameters for fertility and mortality, respectively; f(i) xand {Y1(·), Y2(·)}are actual age-specific fertility rates and two Brass logit model standards—where Y1(·), Y2(·)are associated to high mortality and low mortality rates, respectively–; and I≡{D,B,N,E,T,C} are the sets of deaths, births, total population, life expectancy, total fertility rates, and censuses used in the calculation. Crude migration rates are obtained using inverse population projection and are exogenous to the GIP model. Since GIP suffers from weak ergodicity, we use an initial population growth rate consistent with historical data prior to 1800 based on Livi-Bacci and Reher (1991) and Reher (1991). Figures 9and 10 show the in-sample performance of our population reconstruction with the existing demographic information. The demographic information used in Eq. (35) is depicted in these two figures. Specifically, in Fig. 9we plot the life expectancy, total fertility rate, total number of births, total number of deaths, and the total population; whereas in Fig. 10 we compare the population distribution for some selected census years to the associated censuses. 123 SERIEs (2018) 9:27–64 61 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1 0 20 40 60 80 100 Population size (in millions) Age Data GIP (a) (b) (c) (d) (e) (f) (g) (h) Fig. 10 In-sample performance of the GIP model to existing census data. 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