# Indicators
[](https://www.wifo.ac.at/jart/prj3/wifo/resources/person_dokument/person_dokument.jart?publikationsid=66484&mime_type=application/pdf)
This chapter is organized as a self-contained paper
> Spielauer, Martin, Thomas Horvath, Marian Fink, Gemma Abio, Guadalupe Souto, Ció
Patxot and Tanja Istenič (2020) **microWELT - Microsimulation Projection of Indicators of the Economic Effects of Population Ageing Based on Disaggregated National Transfer Accounts (NTAs)** [pdf](https://www.wifo.ac.at/jart/prj3/wifo/resources/person_dokument/person_dokument.jart?publikationsid=66484&mime_type=application/pdf)
## Abstract
This chapter studies how changes in the population composition by education and
family characteristics impact on indicators of the economic effects of
population ageing based on National Transfer Accounts (NTAs). NTAs constitute
cross-sectional per-capita age-profiles of the key variables of national
accounts consumption, income, saving, and public transfers, incorporating an
estimation of private transfers. A variety of indicators based on NTA data
combined with population projections was developed in the literature, of which
we have selected two for our analysis: The Support Ratio (SR) and the Impact
Index (IMP). We complement existing projections by using new disaggregated NTA
data by education and family type, contrasting the results to the same
indicators based on NTAs by age. Our projection analysis is performed using the
dynamic microsimulation model microWELT. The model provides the required
detailed socio-demographic projections and incorporates the NTA accounting
framework. Our results show that indicators based on disaggregated data can give
a very distinct picture of the economic effects of population ageing, as the
burden of ageing is alleviated by the education expansion. Our study compares
results for Austria and Spain.
## Introduction
Population ageing affects the proportion between the active and the economically
dependent population. Besides decreasing fertility and mortality, the post-war
baby boom and the subsequent baby bust created a kind of demographic cycle that
enlarged the otherwise gradual ageing process. The effect of baby-boomers
reaching retirement age is further aggravated by declining mortality. On the
other hand, education improvements and the increasing labour force integration
of women, as well as health improvements and policy reforms leading to an
increase in retirement age, can potentially mitigate this effect. In other
words, demographic measures, such as the purely age-based dependency ratio (the
population considered being in dependent age divided by the population in
working age where working age is set to, e.g. 18-64) might not adequately
capture the burden of population ageing on the economy. Specifically, the
dependency ratio is insensitive to the age-profiles of income, consumption, and
transfers. This effect is addressed by indicators based on National Transfer
Accounts (NTAs) which break down national accounts by age.
The most widely used indicator based on the NTA approach is the economic Support
Ratio (SR) measuring the changing relation between effective producers (or
effective labour) to effective consumers, the term effective referring to the
age shape of labour input and consumption used as weighting factors. It builds
on the older Support Ratio, which in its simplest form – as a purely demographic
measure such as the dependency ratio – relates the working-age population (e.g.
18-64) to the total population. Suggestions to apply weights to more adequately
represent producers and consumers – e.g. by accounting for labour force
participation – were first suggested by Cutler et al. (1990). For further
discussions on the definition of support ratios see Mason and Lee (2006), Patxot
et al. (2011) and Prskawetz & Sambt (2014). The economic Support Ratio (in the
following NTA SR) gives a more comprehensive measure in this line by replacing
the size of the working-age population by the population, weighted by the age
profile of the current average labour income. Similarly, the total population
size is replaced by the population, weighted by the age profile of current
consumption. Projections of the Support Ratio are available for many countries.
When the Support Ratio increases, economies experience the so-called
"demographic dividend" (a relative increase of the working-age population with
respect to the dependent population). The demographic dividend is still positive
in many developing countries, which are at early stages of their demographic
transition; It is exhausting in Europe as more baby boomers enter into
retirement. The evolution in Europe is estimated in Prskawetz & Sambt (2014)
using NTA data built in the AGENTA project.
As said above, education expansion is one of the sources that might mitigate the
impact of ageing. Mejía-Guevara et al. (2016) propose a decomposition of the
demographic dividend into age and education effects using NTA age profiles by
educational attainment. Their results confirm that the education component
partly offsets the future negative effect of ageing on the Support Ratio.
NTA data opens another window of opportunity to improve future projections by
offering a measure of family transfers and their complementary role in the
economy. Projections usually ignore the fact that the demographic transition is,
to a great extent, a gender transition, accompanied by a change in family
structures as females incorporate into the labour market. (Doepke, and Tertilt
(2016) stress the need to integrate changes in family structure in dynamic
macroeconomic models, summarising attempts made to do so both in the short run
and in the long run in growth models.) In other words, the Support Ratio, while
providing a less mechanic indicator of the effects of population ageing than the
demographic dependency ratio, ignores all improvements in human capital and
changes in female labour force participation and hence family structure.
In this paper, we study how accounting for the changing education composition of
the population and changes in family patterns affects the Support Ratio. This is
possible due to newly developed NTA data by education and family type (Abio et
al. 2020). While still being a cross-section measure ignoring all trends within
the distinguished population groups, incorporating the education and family
dimension adds some realism by accounting for critical composition effects.
Higher educated people on average obtain a higher income, stay in the labour
force longer, and – during active age – pay higher contributions; on the other
side higher educated people on average live longer lives and receive higher
pensions. We use the dynamic microWELT (Welfare Transfer) microsimulation model
(Spielauer et al. 2020a/b) to project the Support Ratio and for assessing the
individual contribution of the various processes such as education and familial
change, improvements in mortality, as well as mortality differentials by
education on this measure.
While the Support Ratio aims at providing an indicator of how the ratio of
labour input and (age-weighted) consumption would be different in a world which
besides the age distribution (and in our case education and family type) looked
exactly like the world today, it ignores that even under these simplifying
assumptions the economy would be different, as changes in labour relative to
capital input would alter wages and the interest rate. To overcome this
shortcoming, Lee & Mason (2017) introduced some elements of general equilibrium
into their projection capturing to some extent changes in factor prices (wages
and interest rates) resulting from changes in the relative size of capital and
labour. They also introduced another indicator (the Impact Index, IMP) which
complements the SR and allows capturing those changes in prices. The Impact
Index measures the changing relation between resources available for consumption
(total income, from labour and assets, net of savings, and effective consumers.
The calculation of these indexes is affected by wages and interest rates
resulting from a simple aggregate production function. Hence this index can be
computed both for a closed economy (changing wages and interest rates) and an
open economy (constant interest rate). As in the case of the Support Ratio, we
apply the microWELT model to calculate and project the Impact Index using
disaggregated NTA data.
This paper is organised in five parts. The first part gives an overview of the
microWELT model and its rational and application from the perspective of the
present research. The second part introduces the NTA approach and the
disaggregated NTA data used for our analysis. Both parts are kept very short as
extensive documentation of model and data are available in separate papers
(Spielauer et al. 2020a/b, Abio et al. 2020). The following two parts present
the calculation and projection of the Support Ratio and the Impact Index and
related analysis for the two studied countries. The final part discusses the
meaningfulness of the various indicators and the potential role of
microsimulation in adding realism to research based on the NTA approach.
Definitions of all the variables of the model are collected in the Appendix.
## The microWELT model
microWELT is a dynamic microsimulation platform developed for the study of the
interactions between welfare state regimes, private and public welfare
transfers, and population ageing accounting for educational change, life
expectancy differentials by education, and changing family patterns. microWELT
is developed at the Austrian Institute of Economic Research (WIFO) alongside an
international research program studying the distributional effects of four
welfare state regimes represented by Austria, Finland, UK, and Spain. The model
is part of the WELTRANSIM project, funded by the Joint Programming Initiative
"More Years, Better Lives". Project partners are the University of Barcelona,
the Finnish Institute for Economic Research, and the Finnish Centre for
Pensions.
microWELT is a portable continuous-time interacting population model based on
data readily available for many countries, most importantly the Euromod
database. While reproducing existing (Eurostat) demographic projections in
aggregate outcomes, microWELT also produces detailed family-demographic and
education projections integrated with a longitudinal accounting framework based
on the National Transfer Account (NTA) approach. Longitudinal accounting allows
the calculation of Full Generational Accounts as in Lee et al. (2017). microWELT
is fully documented and designed as a modular platform refinable and extendable
for a wide range of applications going beyond the WELTRANSIM project.
The starting population is generated from 2010 EUROMOD (Sutherland and Figari, 2013) input data and various parameters are estimated directly from this data
set. Other data sources are the harmonised European Labour Force Survey,
Eurostat projections, NTA variables by age and sex developed in the Agenta
Project as well as new disaggregated NTA data by age, sex, education and family
type developed within the WELTRANSIM project.
microWELT reproduces Eurostat population projections concerning fertility,
mortality rates by age and sex, as well as net migration numbers by age and sex.
While meeting these aggregate targets due to alignment, the model accounts for
the relative differences in rates by education. First birth rates (and thereby
childlessness) are modelled by education. Also, the model accounts for the
relative differences in mortality by education resulting in observed life
expectancy differentials, while overall results are aligned to mortality by age
and sex. microWELT models the intergenerational transmission of education which
improves the longitudinal consistency of economic accounting over the
life-course. For example, a person achieving higher education has a higher
likelihood to have grown up with higher educated parents, thus having received
higher family transfers as a child. Education (besides age, the presence of
children in the family, and the age of the youngest child) also impacts on the
likelihood of a woman to live in a partnership. In the simulation, partners are
matched by observed distributions by age and education. Persons are linked to
nuclear families. These links are maintained responding to events such as union
formation, new partnerships, leaving home, and death.
The model implements a parallel parameterisation of NTA variables by three
levels of disaggregation: by age, by age and sex, and by age, sex, education and
family type. NTAs can be used to calculate indicators. Also, microWELT
implements individual longitudinal accounts and mechanisms to balance national
accounts over time.
## Disaggregated NTA Data
NTA data are cross-sectional age profiles breaking down national accounting
variables on consumption, income, saving, and transfers (public and private).
Data distinguish between private and public consumption singling out education
and health (United Nations, 2013). Public transfers also distinguish pension
benefits. NTA data by sex are currently available for more than 50 countries
(www.ntaccounts.org), including most European countries (agenta-project.eu).
Alongside the WELTRANSIM Project, we further disaggregated NTA data by education
and family type. microWELT allows the user to choose the aggregation level,
allowing to compare simulation results based on aggregated profiles with
simulations accounting for composition effects along the education and family
dimensions. At their most disaggregated level, NTA variables in microWELT are
parameterised by the following population groups:
- Children age 0-16 and students age 17-25: by parent's education (the higher
of both parents if available; students 17-25 not living with parents form an
additional category)
- Non-students age 17-59: by sex, education, partnership status, and presence
of children in the household
- Persons 60+ by sex, education, partnership status, and childlessness versus
ever having had children
We distinguish three education levels, Low (ISCED 0-2), Medium (ISCED 3-4) and
High (ISCED 5+). Family types are constructed according to partnership status
and the presence of dependent children in the family (up to age 59) and
childlessness (60+). Dependent children are children up to 16 and students up to
25 if living with parents.
NTA data provide a detailed picture of how resources are re-distributed through
public and private transfers and asset re-allocation. microWELT implements a set
of 19 NTA variables.
- Private Consumption Education (CFE)
- Private Consumption Health (CFH)
- Private Consumption other than Education and Health (CFX)
- Public Consumption Education (CGE)
- Public Consumption Health (CGH)
- Public Consumption other than Education and Health (CGX)
- Public Transfers Pensions, Inflows (TGSOAI)
- Public Transfers Other Cash Inflows (TGXCI)
- Public Transfers Other In-Kind Inflows (TGXII)
- Public Transfers Education Inflows (TGEI)
- Public Transfers Health Inflows (TGHI)
- Public Transfers Outflows (TGO)
- Net Interhousehold Transfers (TFB)
- Net Intrahousehold Transfers (TFW)
- Private Saving (SF)
- Public Saving (SG)
- Labour Income (Yl)
- Private Asset Income (YAF)
- Public Asset Income (YAG)
Figure 1 illustrates NTA shapes for labour income and total consumption by
education. (For children and students up to 25, education refers to parents'
education.) Labour income is concentrated on work age, higher educated people
reaching higher incomes, especially later in their work career, and working
until a higher age. At the peak, the average labour income in the high education
group is much higher (up to 3 times in Austria) than in the low education group.
Consumption is much smoother; also, the differences between education groups are
smaller, adult people in the high education group consuming around 50% more than
in the lowest group.
Consumption profiles are flatter when using aggregate NTA data, whereas the
consumption profiles disaggregated by education increase clearly over age. This
is a consequence of the different educational compositions in the various age
groups and illustrates one of the problems in the application of the current
aggregated cross-sectional profiles in the future.
Figure 1: NTA age profiles of labour income and consumption
Source: Simulation output based on NTA data by education and family type
developed in the WELTRANSIM project.
The difference between labour income and consumption is a result of transfers –
both private and public – and asset re-allocation (asset income minus
dissaving). Figure 2 depicts the age profiles of private and public net
transfers by education.
Figure 2: NTA age profiles of public and private net transfers
Source: Simulation output based on NTA data by education and family type
developed in the WELTRANSIM project.
A comparison between the transfers in Spain and Austria reveals almost similar
patterns and absolute magnitudes (in €) for family transfers; given the lower
labour income in Spain, family transfers are thus higher in Spain in relative
terms. In contrast, public transfers are higher in Austria in absolute terms,
the difference being in the range of differences in labour income. Differences
by education are very pronounced – both in magnitude and shape – especially
regarding public net transfers.
## The Support Ratio
The Support Ratio, expressed as an index set to 1 in the base year, shows the
change in the relationship between available labour income to the current level
of consumption in the absence of relative changes in the age profiles of
consumption and labour.
Basing the calculation of the Support Ratio on disaggregated NTA data developed
in the WELTRASNIM Project, composition changes affect both the effective labour
and effective consumption. Figure 3 compares the changes in the two measures
calculated based on disaggregated NTA data to the measures based on NTA by age
only. Effective labour is simply the current (2010) age profile of labour
income, as depicted in Figure 1, re-weighted by projected population numbers,
set into relation to the initial 2010 value. Effective consumers are calculated
accordingly based on the age profiles of total consumption. The effects of (not)
ignoring composition effects are considerable: e.g. for Spain, changes in
effective consumers are double as high when basing the calculation on
disaggregated NTAs.
Figure 3: Effective Labour (L) and Effective Consumers (N)
Source: Simulation output based on NTA data by education and family type
(WELTRANSIM project) compared to outcomes based on aggregate NTAs (AGENTA
project).
The Support Ratio is calculated by dividing the index of Effective Labour by the
index of Effective Consumers. In Figure 4, we compare four measures:
- As a reference, we construct an index of the relation of the population
18-64 to the total population. This is a purely demographic index of the
sort NTA literature attempts to improve by applying age profiles instead of
arbitrary age brackets.
- The Support Ratio based on aggregated NTAs by age. Both for Spain and
Austria, the application of current age profiles of consumption and labour
leads to a higher decline of the index compared to the demographic reference
index.
- The Support Ratio based on disaggregated NTAs. Composition effects are
favourable in both Spain and Austria. This is especially the case in Spain,
where the decline in the index is now even less when compared to the purely
demographic index. In Austria, the effect is smaller; in the long run, the
positive effect of higher labour input is cancelled out by the relatively
high consumption of higher educated elderly.
- The fourth index relates the changes in effective labour based on
disaggregated NTA to the change in effective consumers based on aggregated
NTA. This indicator thus measures to which extent today's consumption by age
can be maintained accounting for the changing education composition of the
population, without maintaining the consumption standards within each
education group. Results show that the changing population composition by
education does not outweigh the ageing effect.
Figure 4: Support Ratios
Source: Simulation output based on NTA data by education and family type
(WELTRANSIM project) compared to outcomes based on aggregate NTAs (AGENTA
project).
Comparing the four indicators reveals some of the problems of an indicator based
on aggregate NTAs.
- NTA literature promotes the Support Ratio as a more realistic indicator of
the effect of population ageing compared to purely demographic measures
based on the relation of age groups. They find that for Europe, the Support
Ratio gives a more dramatic outlook than purely demographic measures
(Prskawetz & Sambt 2014). Our study shows that, when accounting for
composition effects, these results can be reversed. Also, our analysis
indicates similar trends, while the size of these results is very
country-specific.
- NTA measures based on aggregate NTAs not only ignore improvements in human
capital; the indicator is also based on average age-specific consumption. As
an indicator of the extent to which the living standard of a population can
be maintained, it refers to the average population by age and not to the
maintenance of the relative consumption standards as observed today for
given population groups by education and other characteristics. Our results
show that when using aggregate consumption by age as the standard reference,
a much higher level of consumption can be maintained due to composition
effects: for Spain, in the long run, 90% of the current age-specific
consumption can be maintained, compared to 75% indicated by the Support
Ratio based on aggregated NTAs.
The disaggregated NTAs developed in the WELTRANSIM Project do not only
distinguish by education, but also family type. The rationale is to provide
measures which allow for the distributional study not only by education but also
by family, especially between people with and without children. As a tool for
comparative studies, including a longitudinal perspective, microWELT projections
account for longevity differentials by education. In order to assess to which
extent such detail not only allows distributional studies but also impacts
summary measures such as the Support Ratio, we created two what-if scenarios:
- In Scenario A, we "switch off" the relative mortality differences by
education. This scenario does not affect overall age-specific mortality, but
who – at a given age and for given sex – dies. This scenario thus ignores
that people with higher education on average not only consume more but also
live longer lives.
- In Scenario B, we study the effect of the differences in the distribution of
family sizes by education (the concentration of reproduction; see Spielauer
2005 and Shkolnikov et al. 2004) on the Support Ratio. While microWELT
accounts for differences in timing and quantum of fertility by education, in
this scenario we assume a low concentration of reproduction setting
childlessness to a very low level (of 5%) regardless of education and
applying the same fertility patterns to all education groups. This scenario
does not affect overall age-specific fertility rates but distributes
children more evenly among women.
Figure 5 depicts the effect of mortality and fertility differentials by
education on the Support Ratio. As can be expected, ignoring longevity
differentials would lead to a smaller decline in the Support Ratio. The effect
of the concentration of reproduction is of comparable magnitude, but in the case
of Austria goes in the opposite direction.
Figure 5: The effect of mortality and fertility differentials on the Support
Ratio
Source: Simulation output based on NTA data by education and family type
developed in the WELTRANSIM project.
## The Impact Index
While the Support Ratio aims at providing an indicator of how the ratio of
labour input and (age-weighted) consumption would be different in a world which
besides the distribution by age (and in our case education and family type)
looked exactly like the world today, it ignores that even under these
simplifying assumptions the economy would be different, as changes in labour
relative to capital input would alter wages and the interest rate. To overcome
this shortcoming, Lee & Mason (2017) developed the Impact Index. In contrast to
the Support Ratio, the Impact Index is a broader measure which contains more
information and permits taking into account changes in wages and interest rates
resulting from the changing labour supply in relation to capital due to
demographic change. The Impact Index measures the change in the relationship
between resources available for consumption and the consumption level needed to
keep the initial standards. Compared to the Support Ratio, the Impact Index
replaces "effective producers" (a measure for labour input) with the consumption
which would be possible accounting for changes in wages and capital income
assuming a constant age-specific saving rate as observed today.
As in ageing societies, labour becomes relatively scarcer, wages increase,
softening the consequence of ageing on future consumption. Calculation of future
production as a function of capital and labour input requires the choice of a
production function. The Impact Index applies a Cobb Douglas assuming for
simplicity that the productivity growth is null. Besides the information
required to calculate the Support Ratio, the only additional information
required is an initial interest rate (as in Lee & Mason 2017, we use 4% for our
illustrations). NTAs do not contain information on capital stocks. As a proxy,
initial capital endowment by age is calculated based on the assumed interest
rate and kept constant in the simulation. In contrast, wages and interest rates
adapt to population change, assuming a closed economy, whereas only wages adapt
in the open economy version. While the saving rate is used for calculating
consumption, there is no longitudinal consistency between savings and the stocks
of capital, which are entirely based on today's estimated stocks by age. (See
Sánchez-Romero et al. 2013 for an alternative approach which better captures the
general equilibrium macroeconomic effects at the cost of less disaggregation.)
The equations behind the model calculations of the Impact Index are presented in
the Appendix.
In Figure 6, we depict the Impact Index comparing simulations based on
aggregated versus disaggregated NTA data and contrasting these indicators to
Support Ratios. The effects are considerable, both (1) accounting for the gap
between consumption needs and total available income net of savings, and changes
in prices compared to the SR (based on constant labour input), and (2) basing
the analysis on disaggregated data. For Spain, assuming an open economy, in the
long run, 90% of consumption could be maintained without changes in price
increases in labour force participation when using disaggregated data, compared
to 80% when using aggregate data. In contrast, the Support Ratio would decrease
to below 75%. The effects in Austria are similar in direction but smaller.
Figure 6: The Impact Index compared to the Support Ratio
Source: Simulation output based on NTA data by education and family type
(WELTRANSIM project) compared to outcomes based on aggregate NTAs (AGENTA
project).
Basing the Impact Index on disaggregated NTAs alleviates the burden of
population ageing by two effects. The first – as mentioned above – stems from an
increasing wage rate as labour becomes relatively scarce. Additionally, the
simple mechanics behind the calculation of the Impact Factor also leads to a
much higher increase in available capital, as the capital endowment by age and
education is assumed to be constant and observed endowments are considerably
higher for higher educated people driving up capital stocks by composition
effects. Consumption thus increases both due to higher wages and higher capital
income. The latter effect is particularly strong in Spain, leading to a
situation, where a closed economy (with lower interest rates) would result in a
higher decrease of the Impact Factor than an open economy.
Both the Support Ratio and the Impact Index take "effective consumers" (N) in
the denominator. As the number of effective consumers changes when applying
disaggregated NTAs, this thus also affects the Impact Index and its
interpretation: using aggregated NTAs, we express how future consumption is
related to today's consumption by age only. Using disaggregate NTAs, we relate
future consumption to consumption accounting for the differences in consumption
levels between education groups. In Figure 7 we compare three versions of the
Impact Index calculated by different combinations of aggregated and
disaggregated NTAs. Relating the future consumption based on disaggregated NTAs
to today's consumption based on aggregated NTAs, in the Spanish case, the Impact
Index stays above or close to 1 over the whole projection horizon. Changes in
the education composition thus compensate for the effect of ageing when
considering consumption by age. In the Austrian case, also this variant of the
Impact Index declines over time, whereas far less than other variants.
Figure 7: The Impact Index compared to the Support Ratio for different variants
of aggregation
Source: Simulation output based on NTA data by education and family (WELTRANSIM
project) compared to outcomes based on aggregate NTAs (AGENTA project).
## Discussion
This paper studied how changes in the population composition by education and
family characteristics impact on indicators of the economic effects of
population ageing based on National Transfer Accounts (NTAs). Results for two
indicators developed in the NTA literature were compared to results based on
NTAs by education and family type: The Support Ratio and the Impact Factor.
Our projection analysis was performed using the dynamic microsimulation model
microWELT, which provides the required detailed socio-demographic projections
and incorporates the NTA accounting framework. Our results show that indicators
based on disaggregated data give a very distinct picture of the economic effects
of population ageing, as the burden of ageing is considerably alleviated by the
education expansion
The Support Ratio was introduced in order to overcome the shortcomings of a pure
demographic measure of economic dependency, based entirely on age, claiming that
applying age profiles of economic profiles as observed today can give a more
realistic outlook of the economic burden of ageing which for European countries
seem to be more severe when measured by the Support Ratio. Accounting for the
composition effects introduced by education expansion, we show that these
results are reversed in the case of Spain, while for Austria, the Support ratio
stays below the purely demographic measure.
The Impact Index adds a simple economic model to the calculation of an indicator
of the burden of ageing, introducing wage and interest rate reactions which
increase the consumption potential in the future due to higher wages, and a
higher capital endowment of an aged population. Again, when basing the Impact
Index on disaggregated NTAs, results change considerably.
The calculation of the two indicators based on disaggregated NTAs constitutes a
first – and simple – application of the microWELT microsimulation model.
Detailed population projections are key for introducing realism to indicators of
the burden of population ageing. Microsimulation projections, by depicting
essential aspects of population heterogeneity, additionally allow the assessment
of distributional effects both cross-sectionally and longitudinally with
measures based on the whole life-courses of individuals. microWELT also provides
a platform for more realistic economic modelling, overcoming some problematic
assumptions, e.g. by explicitly modelling capital accumulation.
This paper is part of a series of related papers and other resources which
together build comprehensive documentation and presentation of the research
performed developing and using microWELT. All materials are available at the
project website www.microWELT.eu. One of the objectives of microWELT is the
provision of a modelling platform ready for applications beyond the WELTRANSIM
project. A collection of project descriptions and links to these projects is
provided on the project website. microWELT is an open-source project: the
application, the model code, a step-by-step implementation guide as well as
analysis scripts for parameter generation are available for download.
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## Appendix: Calculation of the Support Ratio and the Impact Index
**Variables**
> yl(x) Average labour income age x
> yk(x) Average capital income age x
> P(x) Population age x
> i(x) Average savings age x
> s(x) Saving rate age x - constant
> c(x) Average consumption age x (reference values for calculation of N)
> L Labour
> K Capital
> I Saving
> S Saving rate
> C Consumption
> r Interest rate
> w Wage rate
> Yl Labour Income
> Yk Capital Income
> Y Total Income Yl+Yk
> α Alpha – constant
> N Effective Consumers (population-weighted base-year consumption)
> l(x) Average labour age x – constant
> k(x) Average capital age x – constant
**Cobb Douglas Production Function**
> Y = L\^α K\^(1- α)
> Y = wL + rK
> w = αY / L
> r = (1- α) Y / K
> Yl = wL = αY
> Yk = rK = (1- α)Y
**Known**
> yl(x) NTA data of reference year
> yk(x) NTA data of reference year
> i(x) NTA data of reference year
> c(x) NTA data of reference year
> P(x) NTA data of reference year
> r parameter for reference year (used also to estimate the capital stock)
**Calculated for initial year**
> Yl = ∑ yl(x) \* P(x)
> Yk = ∑ yk(x) \* P(x)
> Y = Yl + Yk
> α = Yl / Y
> K = Yk / r
> L = ( Y / K\^(1- α))\^(1/α)
> w = Yl / L
> s(x) = i(x) / (yl(x) + ykx))
> l(x) = yl (x) / w
> k(x) = yk (x) / r
> N = ∑ cl(x) \* P(x)
**Simulation: calculate for an updated population by age P(x)**
**(a) Closed Economy**
> L = ∑ l(x)\* P(x)
> K = ∑ k(x)\* P(x)
> Y = L\^α K\^(1- α)
> Yl = αY
> YK = (1- α)Y
> w = αY / L
> r = (1- α) Y / K
> N = ∑ c(x) \* P(x)
> yl(x) = w \* l(x)
> yk(x) = r \* k(x)
> C = ∑ (1-s(x)) \* (yl(x) + yk(x) ) \* P(x)
**(b) Open Economy (difference to closed)**
> r is given exogenously (and assumed constant in the simulation)
> w = (Y – r \* K) / L (following directly from the identity Y = w \* L + r \*
> K)
> Yl = w \* L
> YK = r \* K
**Indices**
> SR = L / N Support Ratio (same for open and closed economy)
> IMP = C / N Impact Index (different for closed and open economy)