A framework for assessing the costs of pension reform reversals

Daniel Baksa, Zsuzsa Munkacsi, Carolin Nerlich

Working Paper Series

A framework for assessing the costs of pension reform reversals

No 2396 / April 2020

Disclaimer: This paper should not be reported as representing the views of the European Central Bank (ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB.

Abstract

Several European countries are currently considering reversing parts of their pension reforms that were adopted previously to improve sustainability. In this paper we present a framework that allows us to quantify the macroeconomic and scal costs of such reversals. We thereby integrate the country-specic information from the latest Ageing Report into a dynamic general equilibrium model with overlapping generations. Focusing on Germany and Slovakia as country cases, our model replicates the Ageing Report's pension expenditure projections very well. We calculate the macroeconomic impact of rst the additional pension reforms needed to contain the public debt pressures arising from population ageing and second the costs of reform reversals. Our model results show that undoing past pension reforms would generate substantial adverse macroeconomic costs and could pose challenges for scal sustainability.

JEL Classication: H55, J11, J26

Keywords: public pension, reform reversals, population ageing, overlapping generations model,

Ageing Report

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Non-technical summary

Population is ageing in most advanced economies, including in Europe, and this trend is expected to continue in the coming decades. In view of the adverse macroeconomic and scal implications of ageing, many European countries have implemented signicant pension reforms in the past two decades to safeguard their public pension systems. More recently, however, the reform progress has stalled, and despite an unchanged demographic outlook several European countries reversed, or plan to do so, parts of their previously adopted pension reforms.

In this paper we oer a framework that allows us to evaluate the macroeconomic and scal costs of pension reform reversals. We thereby use a dynamic general equilibrium model with overlapping generations and combine it with the 2018 Ageing Report projections to exploit the country-specic information contained in it. The Ageing Report projections have the advantage that they are detailed, country-specic and account for implementation delays, i.e. they include the future impact of already adopted pension reforms. Not accounting for the latter would instead overstate and bias the adverse ageing-related consequences. Yet, the Ageing Report is based on a simple accounting framework which ignores general equilibrium behavioural reactions. By using a general equilibrium model with overlapping generations we can account for feedback eects between changes in pension parameters, pension expenditures and macroeconomic vari- ables. Specically, for our analysis we use the model by Baksa and Munkacsi (2016), which is a Gertler-type(Blanchard-Weil-Yaari-type) dynamic general equilibrium model with demogra- phy, overlapping generations, unemployment and wage bargaining. The model is calibrated for Germany and Slovakia. While both countries are confronted with population ageing and have partly reversed previously adopted pension reforms, they dier in terms of demographic driving forces and projected pension expenditures.

In line with the literature we nd that population ageing has major adverse implications if left unaddressed through reforms. However, dierent from most other studies, we do not only look at the demographic transition, but also account for already adopted pension reforms. Moreover, similar to other studies we nd evidence that pension reforms help to contain the adverse implications of ageing, although by a varying degree, depending on the concrete measures adopted and the country-specic circumstances. In particular, increases in the retirement age appear

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to help to alleviate ageing pressures most. The analysis also shows that reform packages that consist of various pension reform measures help to spread the adjustment burden more equally across generations which is proxied by the share of the young generation's consumption in total consumption. Finally, we nd strong evidence for the presumption that reversals of pension reforms are potentially very costly. In fact, reform reversals would not only result in higher aggregate pension expenditure and public debt-to-GDP ratios, but would in most cases also exacerbate the adverse macroeconomic impact of ageing. If the reversed reform elements were to be compensated by other reform measures, this would improve scal sustainability, but might disproportionally burden one generation at a time.

Our main contribution to the literature is twofold: First, our paper oers a framework that allows us to combine the long-term dynamics of pension expenditures, as projected in the Ageing Report, with the behavioural and feedback eects of our overlapping generations model. We use this framework to evaluate the macroeconomic and scal impact of population ageing and pension reforms. Second, our framework enables us to quantify the possible cost of pension reform reversals. To our knowledge, the cost of reform reversals have not been systematically analysed in the literature yet. Our framework allows us to look at the macroeconomic and scal cost of reform reversals and can be easily extended to other European countries than the two analysed in this paper.

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• Introduction

Population is ageing in most advanced economies, including in Europe, and this trend is expected to continue in the coming decades. Population ageing is widely seen to pose adverse scal and macroeconomic challenges that render policy action.3 Against this background, many countries in Europe have implemented signicant pension reforms in the past two decades to contain the adverse consequences of ageing. More recently, however, reform progress has stalled in Europe and despite an unchanged demographic outlook several countries have reversed, or plan to do so, parts of previously adopted pension reforms. For example, Germany recently decided to cap the decline in the benet ratio and the increase in the contribution rate until 2025 at certain levels, and is considering whether to extend this cap even until 2040. Slovakia decided to break the automatic link between changes in life expectancy and retirement age, by capping the retirement age at 64 years. In Spain, the government inter alia postponed the implementation of the sustainability factor to 2023, which links the replacement rate to changes in life expectancy. In Italy it was decided to partly backtrack previous reform achievements by inter alia temporarily facilitating early retirement. Greece is facing signicant risks related to court decisions reversing past pension cuts. In the Netherlands discussions are ongoing whether to postpone the foreseen increase in the retirement age, and thereafter to loosen the agreed link between changes in life expectancy and retirement age. All these reform reversals cause potentially substantial scal and macroeconomic costs in the long run, thereby putting additional burden on future generations. Moreover, this may lead to debt level pressures which are an area of concern particularly for countries with already elevated public debt-to-GDP ratios.

In this paper, we oer a framework that allows us to evaluate the adverse macroeconomic and scal impact of a sudden and not foreseen reversal of pension reforms. We thereby use a dynamic general equilibrium model with overlapping generations (OLG) and combine it with the 2018 Ageing Report to exploit the country-specic information contained in it.4 We focus on Germany and Slovakia as country cases. Concretely, on the basis of our model we try to replicate the Ageing Report's pension expenditure projections for these two countries. Looking at

• For an overview of the macroeconomic and scal implications of ageing see for example ECB (2018).

• See European Commission (2018). The Ageing Report's long-term projections are published every three years. They are jointly prepared by the Ageing Working Group and the European Commission. While the Ageing Report's pension expenditures are a central element of the projection exercise, total ageing cost projections also cover other public expenditure items, such as health and long-term care costs.

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a variety of scal and macroeconomic variables, we can disentangle the future impact that is purely related to population ageing from the projected impact of pension reforms that have been adopted in the past, but are not yet fully implemented. Within our framework we are also able to quantify the size of additional pension reform measures needed to compensate for the expected ageing-induced increase in the public debt-to-GDP ratio. Even more importantly with respect to our main policy question, the framework allows us to quantify the possible macroeconomic and scal eects of pension reform reversals. We also oer an indication of what would be needed in terms of policy measures to compensate for these reform reversals.

The model used in our framework is called OGRE (Baksa and Munkacsi 2016a, and 2016b), which is an acronym for Overlapping Generations and Retirement. It is a Gertler-type (Blanchard- Weil-Yaari-type) dynamic general equilibrium model with OLG households, demography, unemployment and wage bargaining.5 The model assumes two generations: the workers and the retirees. Workers either work, in which case they receive labour income and pay income taxes, or are unemployed, in which case they receive unemployment benets. The retirees do not work, but they receive pension benets. The size and structure of the population is changing over time, and changes are induced by the probabilities to be born, to retire and to pass away. The model has a rich scal sector which covers several public revenue and expenditure items, including pensions. For simplicity reasons we use the closed-economy version of the model.

The general t of our model with the Ageing Report projections is very good for the two countries analysed. While both countries are facing challenges due to population ageing, they dier in terms of demographic driving forces and projected pension expenditures. We do not expand the set of countries further as this would go beyond the scope of this paper. Yet, the framework can be easily applied to other countries, including those in which fully-funded schemes play a more prominent role, as OGRE can handle both pay-as-you-go and fully-funded regimes.6

Our results can be summarised as follows: in line with the literature we nd that population

• See Gertler (1999), Blanchard (1985), Weil (1989), and Yaari (1965). The labour market rigidities and wage
  bargaining are based on Blanchard and Gali (2010).
  6In this paper we only focus on pension reforms that adjust the parameters of existing pay-as-you-go systems, while we disregard any reforms that would imply a switch from a pay-as-you-go to a fully-funded regime. Baksa and Munkacsi (2016b) and Baksa et al. (2016) examined the latter, for instance. While not reported in the paper, we also calibrated the model for Spain and Portugal, the results of which could be shared upon request.

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ageing has major macroeconomic and scal adverse implications if left unaddressed through reforms.7 However, dierent from most other studies, we do not only look at the demographic transition, but also account for already adopted pension reforms, as captured in the 2018 Age- ing Report. This is an important aspect when calculating the additional pension reforms that would be needed to fully contain the adverse public debt impact of ageing. In line with the liter- ature, we nd evidence that pension reforms help to contain the adverse implications, although by a varying degree, depending on the concrete measures adopted and the country-specic cir- cumstances. In particular, increases in the retirement age appear to help to alleviate ageing pressures most. The analysis also shows, though, that reform packages that consist of various pension reform measures help to spread the adjustment burden more equally across generations. Finally, we nd strong evidence for the presumption that reversals of pension reforms are potentially very costly. In fact, reform reversals would not only result in higher aggregate pension expenditure and public debt-to-GDP ratios, but would in most cases also exacerbate the adverse macroeconomic impact of ageing. If the reversed reform elements were to be compensated by other reform measures, this would improve scal sustainability, but might disproportionally impact one generation.

Our main contribution to the literature is twofold: First, our paper oers a framework that allows us to replicate the long-term dynamics of pension expenditures, as projected in the Age- ing Report, very well, which we combine with the behavioural and feedback eects of our OLG model. We use this framework to evaluate the macroeconomic and scal impact of population ageing and pension reforms. The Ageing Report projections have the advantage that they are detailed, country-specic and account for implementation delays, i.e., the future impact of already adopted pension reforms. In particular the latter is important in view of the numerous pension reforms adopted in past years, as not accounting for them would overstate and bias the adverse ageing-related consequences.8 Yet, the Ageing Report projections are based on a simple accounting framework and thereby do not capture feedback eects between changes in

7On the macroeconomic impact of population ageing and pension reforms, although far from exhaustive, see for example Fehr (2000), Borsch -Supan et al. (2006), Kilponen et al. (2006), Diaz-Gimenez and Diaz-Saavedra (2009), Kara, E. and L. von Thadden (2010), Karam, et al. (2010), Keuschnigg et al. (2013), de la Croix et al. (2013), Goraus et al. (2014) and Baksa and Munkacsi (2016b). See also Conesa and Krueger (1999), Galasso (2008), Heijdra and Romp (2009), and Beetsma et al., (2013) who discussed the political feasibility of pension reforms and optimal pension policies.

8On importance of implementation delays of pension reforms for their macroeconomic impact, see Bi and Zubairy (2019).

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macroeconomic variables, age-related expenditures and pension parameters.

Second, our framework enables us to quantify the cost of reform reversals. There is a wide range of studies that analyse the economic and scal impact of population ageing and di erent kinds of pension reforms, including systemic reforms. To our knowledge, however, the macroeconomic costs of reform reversals have not been systematically analysed in the literature yet. The few exceptions include the papers by Borsch -Supan and Rausch (2018) and Dolls and Krolage (2019) which focus on the scal cost of pension reform reversals in Germany.

We do not look at the drivers of pension reform reversals in this paper. In fact, reform reversals can be, among other things, explained by moral hazard behaviours. For example, the median voter is ageing, which calls for swift policy action rather sooner than later. In addition, the political pressure to adopt reforms is distinctly vanishing in good economic times (Beetsma et al. (2013)). Yet, these aspects go well beyond the scope of this paper and would deserve a paper on its own.

The paper is organised as follows: In section 2 we briey outline the ageing challenge. In section 3 we present our framework and explain how we link the OLG model to the 2018 Ageing Report projections for our two country cases, Germany and Slovakia, to generate the baseline scenario. Moreover, we present the motivation for our policy scenarios, including the one on reform rever- sals. In section 4 we rst present the results for our baseline scenario. We then show the impact in case additional pension reforms were adopted to neutralize the public debt pressures arising from population ageing. In section 5, we assess the adverse impact of reform reversals by using Germany and Slovakia as illustrative examples. In section 6 we conclude.

• The demographic challenge

Europe's population is rapidly ageing and the demographic challenges are expected to become even more pressing over time. In particular in the next one and a half decades, the baby boom generation, i.e., the sizeable cohort of those born between 1955 and 1970, will enter retirement. The change in the relative size of the age cohorts is well captured by the old-age dependency

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ratio, which sets the elderly population (of age 65 and older) in relation to the working-age population (of age 15-64). According to Eurostat, the old-age dependency ratio of the euro area will almost double in the next half century, from 30% today to 52% in 2070, with two thirds of this increase concentrated in the next two decades (Figure 1, left-hand side).9 The demographic transition di ers, however, across countries. By 2070, the old-age dependency ratio is projected to be highest in Portugal and Greece, while it will be lowest in Ireland and France (Figure 1, right-hand side).

Population ageing is being driven by a number of demographic trends. Life expectancy will continue to rise, as people tend to live longer. According to Eurostat, remaining life expectancy at the age of 65 will average at 23.6 years for men and 26.9 years for women by 2070, which corresponds to around 5 years more than today. Moreover, fertility rates are already today well below the natural recovery level in most European countries. Looking ahead, they are expected to either remain low or even in some countries decline further. Finally, these negative demographic trends are expected to be only partly mitigated by net inward migration ows of workers, in particular as these ows are likely to uctuate over time.

Population ageing has considerable adverse macroeconomic implications. These materialise inter alia in form of a shrinking labour force as well as a drop in the growth rates of consumption, investment and GDP growth.10 The precise impact is strongly determined by the starting po- sition, and whether ageing mainly stems from higher life expectancy, lower fertility rates, or a combination of both. With respect to the labour force, the pool of workers will gradually age and, in particular, if fertility rates are very low, eventually shrink. A gradual contraction in the labour force and aggregate employment will, in turn, drag down GDP growth.11 Moreover, age- ing will a ect aggregate consumption and saving rates, as workers and pensioners have di erent marginal propensities to consume and save. Following the life-cycle-hypothesis, workers tend to accumulate savings and consume less than the elderly, but once in retirement people are likely

9The old-age dependency ratio is, however, only a rough proxy for the eective old-age dependency ratio, as the later takes on top of the statutory retirement age also incentives for earlier or later retirement into account. Moreover, the indicator does not acknowledge that the countries' statutory retirement age might be dierent from the 65-year-threshold.

1. For an overview of the main channels through which ageing can aect the economy, see e.g. Borsch -Supan and Ludwig (2006), Keuschnigg et al. (2013), de la Croix et al. (2013), ECB (2018), and Brand et al. (2018).
2. The adverse growth impact is expected to be even larger if one believes that productivity and innovation is age-dependent and hump-shaped, as shown e.g. in Aksoy et al. (2019). In our model, however, we do not assume productivity to be endogenous.

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									Figure 1: Old-age dependency ratio
(%, 2000-70)													(in %, 2016 and 2070)
75															IE
															FR
															BE
65															ES
															NL
															LU
55															SI
														FI
															EA
45															EE
														LT
															LV
35															AT
														MT
															DE
25															SK
														IT
															CY
15															GR
														PT
2000	2005	2010	2015	2020	2025	2030	2035	2040	2045	2050	2055	2060	2065	2070
															15	20	25	30	35	40	45	50	55	60	65	70

Sources: Eurostat, and authors' calculations

Note: The old-age dependency ratio (OADR) is dened as the number of people aged 65 or older as a percentage of the working-age population (i.e., people aged 15 to 64). In the chart on the left-hand side, the blue solid line shows the ratio for the euro area aggregate and the grey shaded area is determined by the largest and lowest values of the euro area countries over time. The chart on the right-hand side shows the OADR for 2016 (yellow dots) and 2070 (red dots) in all euro area countries

to dissave and increase consumption. At the same time, the prospect of rising life expectancy is likely to cause households to change their saving behaviour by fostering precautionary savings. At aggregate level, the impact can be expected to vary over time in line with changes in the underlying demographic structure and behavioural responses. The capital-labour ratio can be expected to increase with labour supply shrinking. As this will exert downward pressure on the equilibrium real interest rate, also investment growth is likely to be dampened.

Ageing poses considerable scal challenges. Public pension spending in Europe is elevated already today, accounting for more than half of total public expenditures. In several countries pension spending can be expected to further increase following a rising number of pensioners and given that most European countries have a pay-as-you-go (PAYG) system in place. This would aggravate the intergenerational burden sharing. Indeed, the 2018 Ageing Report projects public pension spending in the euro area to increase on average from 12.3 percentage points of GDP in 2016 to 13.5 percentage points of GDP in 2040, before falling back to 11.9 percentage points of GDP in 2070, notwithstanding large cross-country dierences. Moreover, rising pension expenditures can pose challenges for the nancing of pension systems, in particular for countries with a mandatory PAYG regime. The outlook of the old-age dependency ratio indicates that the number of workers potentially available to nance one pensioner will shrink from

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three to two by 2070. Assuming no additional changes to the main three pension parameters - i.e., the retirement age, the bene t ratio or the contribution provisions - population ageing can be expected to result in a widening of the nancing gap. Yet, challenges vary across countries, reecting di erences in the set-up of their social security systems, the underlying change in the age pro le and national preferences. In addition, population ageing may also a ect tax revenues. Most prominently, a shrinking labour force will ceteris paribus limit indirect tax revenues and social security contributions. Taking all these aspects together, ageing can be expected to push public debt-to-GDP ratios too. This, in turn, is likely to pose long-term scal sustainability risks, particularly in those countries with already elevated levels of public debt today.

To address the adverse implications of population ageing, many countries have adopted pension reforms.12 Especially countries that underwent an economic adjustment programme, such as Greece, Spain, Cyprus and Portugal, have adopted fundamental changes to their pension sys- tems. These included a wide range of measures, a ecting pension system rules as well as pension parameters. In contrast, systemic pension reforms that foresee a full or partial shift from PAYG schemes to fully-funded schemes have been limited to a few countries, mainly in Eastern Europe.

More recently, however, the reform progress has stalled and several countries are planning or have already reversed parts of their previously adopted pension reforms. For example, Germany decided in 2018 to cap both the expected decline in the bene t ratio and the expected increase in the contribution rate until 2025 at pre- xed levels. It is contemplating whether to extend this cap until 2040, in which case the strong cohort of baby boomer would bene t from this cap. More recently, the government supported the idea to introduce a basic pension as of 2021, conditional on 35 contributory years. Slovakia decided to implicitly break the automatic link between changes in life expectancy and the statutory retirement age, by capping the retirement age at 64 years. Moreover, the government legislated generous changes to minimum pensions and the Christmas bonus. In Spain, the government inter alia postponed the implementation of the sustainability factor to 2023, which should link the replacement rate to changes in life expectancy. In Italy previous reform achievements were partly abandoned by inter alia temporarily facilitating early retirement. Greece is facing signi cant risks related to court decisions

12For a detailed overview of past pension reforms in various EU countries see Carone et al. (2016) and European Commission (2017).

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that aim at undoing past pension cuts. In the Netherlands discussions are ongoing whether to postpone the foreseen increase in the retirement age, which thereby would loosen the agreed link between changes in life expectancy and retirement age. None of these reform reversals mentioned above were reected in the 2018 Ageing Report projections, as they were not adopted by the time.

• The framework

3.1 Key elements of the OGRE model

We estimate the macroeconomic and scal impact of population ageing and changes of the pension parameters on the basis of a dynamic general equilibrium OLG model. We use the model by Baksa and Munkacsi (2016a and 2016b), called OGRE (Overlapping Generations and Retire- ment), which is summarised in Annex 1.13

OGRE is a Gertler-type(Blanchard-Yaari-Weil-type) model.14 Its central element is the demographic block which considers two types of households, both with perfect foresight: workers (i.e., people between the age of 20 and the retirement age) and retirees (i.e., people who have reached the retirement age). The absolute and relative size of these two cohorts changes over time, and so does their sum, i.e., total population. The size of the cohorts is determined by three probabilities: the probability to be born, to retire and to pass away. All three probabilities may change over time.

According to the model, workers either work, in which case they receive labour income and pay income taxes, or are unemployed, in which case they receive unemployment bene ts. Workers use their disposable income or unemployment bene ts for consumption and savings in risk-free bonds, in line with the Euler equation. They retire, once they have reached retirement age. This is determined by the probability to retire.15 Retirees do not work, but receive pension bene ts. Per capita pension payments are set in real terms at the start of retirement, and are

1. For a more detailed description of the closed-economy version of the model see Baksa and Munkacsi (2016a). Compared to the original version of the model, we disregard the informal sector in this paper and the ageing shock is dierently modelled. In fact, in the original version of OGRE, the ageing shock was dened as a 10 percentage point increase in the old-age dependency ratio.
2. See Gertler (1999), Blanchard (1985), Yaari (1965) and Weil (1989).
3. In the model we use the eective retirement instead of the statutory retirement age.

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kept constant thereafter. Retirees die with a certain probability. By the end of their lifetime, retirees have consumed all their savings.

On the production side, rms hire workers and use physical capital. Firms take account of price adjustment costs when setting prices. Moreover, the model assumes labour market rigidities in form of hiring costs and wage bargaining, which have an inuence on the level of unemploy- ment.16 Moreover, the model has a rich scal sector with di erent kinds of public revenues (personal income tax, social security contributions, VAT) and public expenditure items (pen- sion bene ts, unemployment bene ts, government consumption).17 The public pension system is a PAYG system. Governments may issue government bonds to nance any funding gap, for example in the pension system. We assume the economy to be closed. Monetary policy is characterised by a Taylor rule.

The population ageing shock enters the model via a combination of a declining or low probability to be born (equivalent to a declining or very low fertility rate) and a falling probability to pass away (equivalent to an increase in life expectancy). Thereby, an ageing shock results in a lower share of workers relative to pensioners. In particular the fertility rate determines how persistently the labour force will shrink over time. In our model, a shrinking labour force results in lower aggregate employment while the unemployment rate will decline. As labour input becomes scarcer, this would adversely a ect the production process. We assume per capita labour productivity and the labour force participation rate not to change with age. Compensation per employee tends to go up as the labour force gets scarcer.

Moreover, the model accounts for other general equilibrium e ects of population ageing. On the demand side, growth in aggregate consumption can be expected to be depressed by a smaller number of workers and lower income growth. This e ect is partly compensated by higher aggregate consumption resulting from a rising cohort of retirees. Moreover, a more subdued growth outlook triggers a decline in aggregate investment, which would partly reduce the projected rise in the capital-labour ratio.

16The modelling of the labour market and the wage bargaining process is based on Blanchard and Gali (2010).

17In principle, the model allows to dierentiate between PAYG, fully funded, and mixed pension schemes. Yet, this dierentiation is not further considered in this paper.

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In our model, population ageing will a ect public nances through several channels: rst, public pension spending will grow with a rising cohort of retirees. Subsequently, this would result in higher public expenditure, particularly if there are no built-in adjustments on the side of social security contributions or pension bene t payments. Second, lower economic activity and a shrinking labour force reduces government revenues via lower indirect taxes by households and rms. To nance a budget de cit, the government can issue risk-free bonds. This together with lower GDP growth rates will push up the public debt-to-GDP ratio.

3.2 Ageing Report projections: generating the baseline scenario

In the baseline scenario we quantify the impact of population ageing and already enacted pension reforms by taking the Ageing Report projections. These projections o er many advantages. First, the projections are based on a very detailed, country-speci c set of data that allows us to capture the main characteristics of a country's pension system. Second, the projections account explicitly for the future impact of already enacted, but not yet fully implemented social security reforms on age-related spending. This is important in view of the numerous pension reforms adopted in past years, as their impact is not yet fully reected in actual data and will only become fully visible after some time. Instead, not accounting for their impact would overstate the adverse ageing-related consequences. Third, the projections are done under a no-policy change assumption. Concretely, in the baseline scenario it is assumed that no additional pension policy measures will be adopted over the projection horizon. This provides a more accurate picture of a country's actual adjustment needs. Finally, the projections are based on a set of common macroeconomic and demographic assumptions across countries. This allows for a cross-country comparison of the demographic transition and the adjustment needs.

However, the Ageing Report projections as such are not suitable for evaluating the macroeconomic implications of population ageing, pension reforms or their reversals. In fact, the projections are based on a simple accounting framework that ignores general equilibrium behavioural reactions. Thus, feedback e ects between changes in macroeconomic variables, age-related ex- penditures, and pension parameters cannot be captured. For example, it cannot be assessed how changes in the retirement age will a ect employment, or how consumption growth will react to a lower pension bene t ratio. By integrating the Ageing Report projections in our model allows us

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to benet from the information inherent in the Ageing Report while overcoming the limitations on the feedback eects.

The baseline scenario relies on two sets of information: the ageing shock and the pension expenditure dynamics projected in the Ageing Report. First, we dene the ageing shock by targeting the old-age dependency ratio from 2016 to 2070, as stated in the Ageing Report. Thereby, we translate the ageing shock into the model structure by allocating the level and change in the old-age dependency ratio to the probabilities to be born, to be retired and to have passed away.18 Thus, the projected size of the two age cohorts (i.e., the workers and the retirees) in a country over time is derived from the probabilities in the initial period and the changes over time. Thereby, we can explain to what extent changes in the old-age dependency ratio are due to changes in fertility, mortality and retirement. This is important for our analysis, as the impact of population ageing depends on the prevailing driving forces that trigger the demographic transition, as explained above. Moreover, with this approach we are able to capture changes in the demographic structure such as the sizeable cohort of the baby boom generation entering retirement, rising longevity and low fertility, as well as changes to the eective retirement age due to already enacted pension reforms.

Second, to sketch the macroeconomic and scal eect of population ageing in the baseline scenario we try to replicate the Ageing Report's projected path of public pension spending for the period 2016-70 in our model structure.19 Our estimates under the baseline scenario show the long-term impact of population ageing while accounting for already enacted pension reforms. The impact is expressed as percentage point changes compared to the initial period, which is the year 2016, in line with the latest Ageing Report. Moreover, the modelling approach allows us to decompose the macroeconomic and scal impact into the part that is purely driven by the demographic process and the part that is aected by the implementation delay of previously adopted pension reforms.

1. It is important to note that the way the ageing shock is modelled in this paper is dierent from the approach used in the earlier versions of OGRE. See footnote 13.
2. We also reproduced the 4 underlying driving factors of the pension spending path used in the Ageing Report. The results can be interpreted as a sensitivity analysis for the tness of the model and are shown in the Annex 2.

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3.3 The policy scenarios: additional pension reforms and reform reversals

A central part of the paper relates to the policy scenario analysis. We focus on two kinds of policy scenarios, which are compared to the baseline: a scenario with additional pension reforms and one assuming a pension reform reversal.20 The analysis is done for our two country cases: Germany and Slovakia.

In doing so, we rst determine the required size of the additional pension policy instruments. Thus, under the additional policy reform scenario we calibrate the policy instruments such that they stabilise the public debt-to-GDP ratio at its initial level, i.e., in 2016. Assuming a full compensation of the public-debt impact of population ageing is obviously a radical choice. How- ever, we use this approach for presentational purposes rather than to prescribe concrete policy advices. The main reason why we focus on changes in the public debt-to-GDP ratio rather than on changes in public pension spending is that this allows us to account for all inherent changes to the pension parameters, including the contribution rate, and for feedback e ects though changes in tax revenues. We distinguish three di erent types of pension reforms: increase in the retirement age, decline in the bene t ratio and increase in the contribution rate. We assume additional reforms to be implemented at once. We also analyse the impact if all policy options were adjusted at the same time in comparison to the baseline scenario.

Thereafter, we de ne the reform reversal scenario and gauge its potential macroeconomic im- pact. We try to mimic, as much as possible within our model framework, the current policy discussions on reform reversals in the two country cases. Concretely, in the reversal scenario for Germany we freeze the bene t ratio at its current level of 48% and assume that the contribution rate would not exceed the threshold of 20% until 2040. With this reform reversal scenario we assume that the agreed freeze of the bene ts ratio and contribution rate until 2025 will be extended until 2040.21 In Slovakia, we assume the retirement age to stop increasing from the year 2045 onwards. This reects the decision by the Slovakian government to freeze the retirement

1. It should be recalled that neither the policy scenarios nor the reform reversal scenarios are based on the Ageing Report.
2. In Germany, the so-called double threshold measure was adopted in mid-2018 to be e ective until 2025. Members of the government suggested extending this measure until 2040, which is broadly in line with the recommendation by the Deutsche Rentenkommission issued in March 2020. As, this measure was adopted after the publication of the 2018 Ageing Report, it is not reected in the Ageing Report projections.

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age at 64 years.22 Similar to the approach taken in the additional reform scenario, we determine the necessary compensatory measures under the reform reversal scenario by what is needed to stabilize the public debt-to-GDP ratio.

3.4 Data and calibration

The calibrated models for our two country cases match well the input data (see Table 1 and Annex 3). For the macroeconomic variables in the initial steady-state period, the calibrations are based on multi-year averages of national accounts data from Eurostat, in most cases covering the period 2009-2016, i.e., after the outbreak of the nancial crisis. The targeted values include GDP growth and its main components, as well as variables related to the labour market and public nances. Moreover, we replicate all important country-specic pension system variables, as reported in the 2018 Ageing Report. These include gross pension expenditures, the (gross) replacement rate and the benet ratio.23 With the OGRE model we are able to reproduce the Ageing Report's change in public pension spending (including its driving factors) over the projection horizon until 2070 (see Annex 2).

The main advantage of our approach is that we are able to capture the country-specic dynamics of public pension expenditures, as projected in the Ageing Report, and combine it with the behavioural and feedback eects of our OLG model. At the same time, it should be noted that it is not possible to fully replicate all elements of the Ageing Report projections. In fact, some of the variables used in the Ageing Report cannot be easily translated into our model structure. For example, with respect to pension expenditures, the Ageing Report projects total pension expenditure, which includes other components such as early pension payments, minimum pensions or disability pensions. In our model, we only look at earnings-related pensions. To make the model results comparable with the Ageing Report, the values are therefore rescaled. More- over, in the model we look at the eective retirement age, while the Ageing Report projects the eective exit rate from the labour market. We assume these two variables to be identical, while this is not necessarily the case as old-age workers can get unemployed.

1. In Slovakia, the government decided in end-2018 to implicitly break the automatic link between changes in life expectancy and retirement age, by capping the retirement age at 64 years. This will be reached in 2045.
2. When replicating the key pension system variables, we focus on the 2016 data provided in the 2018 Ageing Report (as 2016 is the starting year of the projections), to ensure comparability of the results.

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To model the demographic structure, we use Eurostat's population data on the absolute size of the respective cohorts for Germany and Slovakia.24 We translate the data such to obtain values for the three probabilities in the initial period for which the model is calibrated, i.e., for 2016. Concretely, in the initial period the probability to be born results from the number of people in the age cohort of the 19 year-olds, while the probability to pass away is based on the size of the very old-age cohort (i.e. of the people aged 85 years and older).25 The retirement probability is derived from the e ective exit age from the labour market, as reported by Eurostat.

Table 1: Calibrating the initial period

in % of GDP, unless stated otherwise

	Germany	Slovakia
	calibrated	targeted	calibrated	targeted
	value	value	value	value
Consumption (total)	60.7	54.6	60.8	60.4
Consumption (public)	19.1	19.1	16.9	16.9
Investment	20.1	20.0	22.3	20.2
Compensation of employees	59.7	56.3	54.9	46.0
Unemployment rate (%)	5.7	5.6	12.8	12.7
Unemployment bene t	1.5	1.8	0.7	0.7
Pension expenditure	7.2	7.8	6.9	6.7
Bene t ratio (%)	34.1	32.4	34.4	36.3
Old-age dependency ratio (%)	37.2	38.3	36.5	32.6
Public debt	75.4	75.4	53.3	53.3

Note: The targeted values are based on the 2018 Ageing Report and Eurostat data.

The model structure allows us to replicate Eurostat's projected path of the old-age dependency ratio until 2070 (see Figure 2, left-hand side panels). In fact, for Germany the old-age dependency ratio is projected to increase sharply until 2040 and more slowly thereafter, while for Slovakia the ratio will peak in 2060 before starting to decline thereafter. Moreover, the development of the old-age dependency ratio can be decomposed into the three driving forces, i.e., the probabilities to be born, to retire and to pass away. As shown in Figure 2 (upper right-hand side panel), for Germany, the dynamics of the old-age dependency ratio is mostly driven by the cohort e ect, as the sizeable baby boom generation will enter retirement within the next one and a half decades. This is reected in the strong contribution of the probability to retire until

24We use Eurostat data, in line with the Ageing Report. As a sensitivity analysis (available by the authors upon request) we replicated the analysis with UN population projections.

25While migration ows are not explicitly shown in the model, they are indirectly captured as they impact the absolute size of the cohorts.

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2040. While improvements in life expectancy are also projected to matter, their impact on the dependency ratio will be much smaller.

In Slovakia, the expected change in the old-age dependency ratio is mostly due to the strong improvement in life expectancy, as shown by the probability of death (see Figure 2, lower right-hand side panel). The projected impact of the cohort eect, which is smaller compared to Germany, will gradually decline as of around 2060. To a lower extent, population ageing in Slovakia is also driven by its very low fertility rate.26

Figure 2: Old-age dependency ratio and driving forces

(retirees/workers, in %)
		Germany - OADR			Germany - Driving Forces of OADR
56						56
51						51
46						46
41						41
36						36
2016Y	2026Y	2036Y	2046Y	2056Y	2066Y	2016Y	2026Y	2036Y	2046Y	2056Y	2066Y
	OGRE			Ageing Report 2018		Fertility Rate	Probability of Retirement		Probability of Death	Total
		Slovakia - OADR			Slovakia - Driving Forces of OADR
60						60
55						55
50						50
45						45
40						40
35						35
30						30
2016Y	2026Y	2036Y	2046Y	2056Y	2066Y	2016Y	2026Y	2036Y	2046Y	2056Y	2066Y
	OGRE			Ageing Report 2018		Fertility Rate	Probability of Retirement		Probability of Death	Total

Sources: Eurostat and authors' calculations.

26Although Slovakia's fertility rate is assumed to improve in the long-run, the old-age dependency ratio in the long-run is expected to be impacted by today's very low fertility rate.

• Results

4.1 The baseline scenario

Under the baseline scenario, we nd strong support for the general view that population ageing has adverse macroeconomic and scal implications. Assuming no compensatory measures, the results show an increase in the public debt-to-GDP ratio by around 100 percentage points until 2070, compared to the initial period, for both Germany and Slovakia.27 Moreover, real GDP per capita is projected to decline by almost 14% in Germany and 9% in Slovakia, compared to the initial period (see Tables 2 and 3).28 The long-run model results can be decomposed into the impact driven by the demographic transition (see column ageing" in Tables 2 and 3) and the impact that results from past pension reform measures (see the columns under impact of adopted pension reforms" in Tables 2 and 3).

In the case of Germany, the model-induced decline in GDP per capita by almost 14% by 2070 largely results from population ageing (decline by 15.3%). Yet, previously adopted pension reforms can be expected to partly cushion or amplify the macroeconomic impact, depending on the respective instruments.29 Specically, the legislated gradual increase in the statutory retirement age until 2029 is expected to have a potentially growth-enhancing impact, and will thereby partly cushion the adverse growth impact of population ageing. In line with our model, the eective retirement age will increase from 64 years (in 2016) to 65.5 years by 2070. The baseline scenario captures also other legislative settings: in case of rising pension spending, this requires compensatory adjustments in form of higher contribution rates or a lower benet ratio. Yet, higher contribution rates can be expected to further worsen the impact on GDP per capita (by 1.2%), while a less generous benet ratio will only have a marginal positive growth impact.

27The results are qualitatively very similar to the results with the open-economy version of the model. Yet, for simplicity reasons we only report the results with the closed economy version here, while the results with the open-economy model can be shared upon request. Compared to the open-economy version, the results show an overall more pronounced adverse impact of population ageing in the closed-economy version, as in particular variables, such as consumption, seem to react more strongly to changes in the degree of openness.

1. While the tables show the long-run results, compared to the initial period, the projected long-run develop- ments over time are shown in the charts in Annex 4 (for Germany) and Annex 5 (for Slovakia).
2. In Germany the pension system is prohibited to generate public debt. Instead, increases in pension spending need to be compensated by higher contribution rates and lower benet ratios. Moreover, in 2007 Germany legislated a reform that foresees a gradual increase in the statutory retirement age from 65 to 67 years by 2029. Back then, it was also decided to reduce the pathways to early retirement, inter alia through higher penalty deductions, and to harmonise the parameters for calculating pension benets in West and East Germany.

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Table 2: Baseline scenario: long-run results for Germany

in % of GDP, unless stated otherwise

	Baseline1	Ageing	Impact of adopted pension reforms:
Germany			retirement	benet	contribution
			age	ratio	rate
	(1+2+3+4)	(1)	(2)	(3)	(4)
GDP per capita (%)	-13.9	-15.3	2.5	0.1	-1.2
Consumption: total	-1.5	-1.6	0.4	0.0	-0.2
Consumption: share of young (% of total)	-4.6	-7.8	1.1	1.7	0.4
Investment	-1.3	-1.4	0.1	0.0	0.0
Unemployment rate (%)	0.8	-0.3	0.0	0.0	1.1
Pension expenditure	1.9	3.8	-0.6	-1.3	0.0
Primary balance	-2.9	-6.5	1.1	1.1	1.4
Public debt	100.3	208.6	-33.9	-29.4	-44.9

1Deviation from the respective values in the initial steady state (see Table 1) until 2070

The baseline results for Germany show a decline of aggregate total consumption by 1.5 percentage points of GDP, compared to the initial period. This is largely driven by the steep fall in the workers' share of total aggregate consumption (by 4.6 percentage points), which can be explained by two factors. First, the cohort of the young will shrink relative to the cohort of the pensioners. Second, as agents are forward-looking, the young cohort tends to consume less and save more to prepare for increased longevity. By building up a higher stock of savings during working lives, future pensioners will be better prepared for consumption smoothing. The aging-induced decline in the young's share of total aggregate consumption is partly reduced by the reforms implying a higher retirement age (cohort e ect) and a lower bene t ratio (lower disposable income of the pensioners).

Finally, the baseline results for Germany assume pension expenditure to increase by 1.9 percentage points of GDP in 2070, compared to the initial period. This gure is broadly comparable with the Ageing Report projections.30 The ageing impact on pension expenditures will be almost halfed thanks to previously adopted reforms, namely the rise in the retirement age and the declining bene t ratio. As the German pension system is prohibited to accumulate debt, the projected public debt increase of 100 percentage points of GDP by 2070 only reects the feedback e ects via lower public revenues and a negative denominator e ect. It abstracts from any others factors, like e.g. national scal rules, that could imply a di erent debt trajectory

30The 2018 Ageing Report projects pension expenditures in Germany to increase by 2.4 percentage points of GDP until 2070. See Table 1 in European Commission (2018).

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besides the ageing impact.

While the results for Slovakia are qualitatively very similar to those for Germany, they reveal important dierences between the two countries. In particular, dierences relate to the main driving forces of population ageing, the structure of the economy as well as the impact of past pension reforms. In fact, as shown in Figure 2, population ageing in Slovakia is mainly driven by rising life expectancy. To address the ageing challenge, Slovakia adopted in 2012 an important pension reform that rst automatically links the retirement age to changes in life expectancy and second reduces the generosity of the pension system through changes in the indexation rule.

The baseline results for Slovakia predict a decline in GDP per capita by 9% by 2070, compared to the initial period (see Table 3). Without the adopted pension reforms, namely the one adjusting the retirement in line with changes in life expectancy, the ageing impact would be more than twice as strong, given the projected strong increase in life expectancy.

Table 3: Baseline scenario: long-run results for Slovakia

in % of GDP, unless stated otherwise

	Baseline1	Ageing	Impact of adopted pension reforms:
Slovakia			retirement	benet	contribution
			age	ratio	rate
	(1+2+3+4)	(1)	(2)	(3)	(4)
GDP per capita (%)	-9.0	-18.5	9.8	0.1	0.0
Consumption: total	-0.4	-1.3	0.9	0.0	0.0
Consumption: share of young (% of total)	-4.2	-9.5	4.0	1.3	0.0
Investment	-1.1	-2.0	0.8	0.0	0.0
Unemployment rate (%)	-1.0	-1.3	0.4	0.0	0.0
Pension expenditure	1.2	4.1	-2.2	-0.7	0.0
Primary balance	-2.4	-6.7	3.6	0.7	-0.1
Public debt	103.0	230.5	-105.3	-24.3	2.1

1Deviation from the respective values in the initial steady state (see Table 1) until 2070

The consumption share of the young in Slovakia is overall estimated to decline by 4.2 percentage points of total consumption, compared to the initial period. The previously adopted pension reforms can be seen as improving the intergenerational burden sharing as they help to cushion the adverse ageing impact for the young relative to those for the pensioners. In fact, the reforms leading to a higher retirement age and reducing the generosity of the pension system limit the

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decline in the young's consumption share by 4 and 1.3 percentage points, respectively.

The baseline results for Slovakia show an increase in pension expenditure by 1.2 percentage points of GDP by 2070, compared to the initial period. This is the same gure as projected in the Ageing Report. Without the impact of the adopted reforms the increase in pension spending would be three times as high. The public debt-to-GDP ratio is estimated to increase by 103 percentage points of GDP by 2070, but would be considerably stronger without the adopted reforms.

4.2 Policy scenarios with additional reform measures

The spending pressures arising from population ageing call for additional pension reforms which would need to go well beyond those already adopted. Therefore, as a next step, we examine the macroeconomic and scal impact of additional pension reforms. We report the macroeconomic impact of the additional pension policies in comparison to the baseline scenario, i.e. by how much would the additional policy measures change the results obtained under the baseline scenario.

We calibrate the additional pension reforms by the eort that would be required to fully compensate for the increase in the public-debt-to-ratio due to ageing. In other words, the additional pension reforms need to ensure that the public debt-to-GDP ratio is kept broadly constant at the initial level. We use the same modelling framework as in the previous section, as this allows us to explicitly account for the economic feedback eects of pension reforms. The assumption of perfect foresight continues to hold. Thus, households are expected to react immediately to policy changes by inter alia adjusting their savings behaviour. We consider the same types of pension reforms as discussed above. Concretely, we analyse by how much the three pension parameters, i.e., the retirement age, the benet ratio and the contribution rate, would need to be changed in order to alleviate the public debt impact of population ageing, and assess their impact.

Taking again Germany and Slovakia as country cases, we nd the following results in the additional reform scenarios: in order to fully compensate for the ageing-induced debt increase of around 100 percentage points of GDP, Germany would need to increase the eective retirement

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age by 3.5 years until 2070, compared to an increase in the eective retirement age to 65.5 years by 2070 under the baseline scenario (see Table 4). Similarly, Slovakia would need to increase its retirement age by 4.5 years, compared to an increase to more than 67 years by 2070 under the baseline scenario (see Table 5). Alternatively, if the compensatory measures were only to rely on reforming pension entitlements, the benet ratio would need to be reduced by 26 percentage points in Germany and 21 percentage points in Slovakia, in order to keep the public-debt-to-ratio roughly constant at the initial level. For reforms targeting the nancing side, pension contributions would need to be increased by almost 11 percentage points in Germany and 7 percentage points in Slovakia.

Table 4: Additional pension reforms: long-run results for Germany

in % of GDP, unless stated otherwise

	Baseline1	Impact of additional pension reforms2
Germany		retirement	benet	contribution	mix
		age	ratio	rate
		(1)	(2)	(3)	(1)+(2)+(3)
GDP per capita (%)	-13.9	6.1	0.2	-2.9	1.1
Consumption: total	-1.5	1.2	-0.1	-0.6	0.2
Consumption: share of young (% of total)	-4.6	2.5	5.8	0.9	3.1
Unemployment rate (%)	0.8	-0.1	-0.1	2.8	0.8
Pension expenditure	1.9	-1.3	-4.5	0.1	-1.9
Public debt	100.3	-108.6	-101.7	-97.1	-102.3
Policy instruments
Retirement age (year)	65.5	3.5	-	-	1.2
Benet ratio (%)	-8.0	-	-26.0	-	-8.7
Contribution rate (%)	4.4	-	-	10.6	3.5

1Deviation from the respective values in the initial steady state (see Table 1) until 2070, long-run results for the retirement age

2Deviation from the baseline

Furthermore, in line with the literature, we nd evidence that the macroeconomic impact strongly diers depending on which reform measures have been adopted. For example, in the case of Germany an increase in the retirement age by 3.5 years, which would be necessary to keep the public-debt-to-ratio constant at the level in the initial period, would imply that GDP per capita would be 6.1% stronger, compared to the baseline (see Table 4). In the case of Slovakia, adjustments in the retirement age would allow to almost fully oset the adverse growth impact estimated in the baseline scenario. In fact, prolonging people's working lives helps to enlarge the

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cohort of workers compared to the baseline scenario.31 This can be expected to boost aggregate consumption and investment, while public pension expenditure could be scaled down consider- ably. Cutting pension entitlements, in turn, would leave GDP per capita roughly unchanged, compared to the baseline scenario (see Table 4, the column benet ratio" for Germany and Slovakia). Furthermore, higher contribution rates would in the case of Germany aggravate the adverse impact on GDP per capita, compared to the baseline scenario, while for Slovakia the results would be unchanged compared to the baseline scenario.

Table 5: Additional pension reforms: long-run results for Slovakia

in % of GDP, unless stated otherwise

	Baseline1	Impact of additional pension reforms2
Slovakia		retirement	benet	contribution	mix
		age	ratio	rate
		(1)	(2)	(3)	(1)+(2)+(3)
GDP per capita (%)	-9.0	8.4	0.4	0.0	2.9
Consumption: total	-0.4	1.5	-0.1	-0.1	0.4
Consumption: share of young (% of total)	-4.2	2.8	5.4	1.4	3.2
Unemployment rate (%)	-1.0	0.0	-0.1	0.0	-0.1
Pension expenditure	1.2	-1.5	-2.9	0.0	-1.4
Public debt	103.0	-97.5	-101.7	-102.5	-100.9
Policy instruments
Retirement age (year)	67.2	4.5	-	-	1.5
Benet ratio (%)	-5.0	-	-21.0	-	-7.0
Contribution rate (%)	-0.2	-	-	6.8	2.3

1Deviation from the respective values in the initial steady state (see Table 1) until 2070, long-run results for the retirement age

2Deviation from the baseline

Relying at one reform measure at a time could, however, turn out to be politically challenging as the additional adjustment burden would be placed disproportionally on one generation. For example, by lifting the retirement age the adjustment costs would be fully born by the cohort of workers, as their working lives would be prolonged. Raising the contribution rate would only aect the cohort of workers, while lower pension entitlement would only aect the cohort of pen- sioners. Instead, reform mixes composed of various measures can be expected to help spreading

31In this paper we abstract from the discussion how plausible it is to assume healthy ageing. Yet, this is a relevant question with respect to the eectiveness of the policy instrument of postponing the retirement age. We also abstract from age-dependent productivity rates.

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the adjustment burden more equally across generations. Therefore, we analyse the impact of a mixture of reforms. We assume that each reform measure would contribute by one-third to the adjustment e ort that is needed to keep public debt constant at its initial level.

The estimation results under the reform mix scenario are encouraging in terms of intergener- ational fairness and macroeconomic impact (see Tables 4 and 5, last column). Looking again at the examples of Germany and Slovakia, the necessary increase in the retirement age would be one-third of the adjustment coming only through changes in the retirement age. Likewise, pension entitlements and contribution rates would need to be adjusted by around one-third compared to a situation with only one single instrument. Overall, under the reform mix scenario the adverse ageing impact on GDP per capita would be contained, compared to the baseline scenario.

• Quantifying the costs of pension reform reversals

Several countries are currently discussing whether to reverse parts of their previously adopted pension reforms. Our framework allows us to evaluate the macroeconomic and scal costs of pension reform reversals. In the following, we study the impact of reform reversals by looking more closely at the cases of Germany and Slovakia. In both countries the reform reversals only emerged after the 2018 Ageing Report and therefore are not reected in our baseline scenario.

In Germany, previously adopted reform measures have been lately reversed and further revisions are under discussion. Concretely, in August 2018 the German government agreed on a reform reversal, by setting the so-called double threshold" until 2025. This double threshold" foresees that (i) the pension contribution rate should not rise above 20% (compared to 18.9% in the initial period), and (ii) the bene t ratio should not fall below 48% until 2025.32 On top of this, although not yet decided, it was proposed to prolong the double threshold" until 2040. This latter proposal, however, can be expected to be considerably more costly, as the large cohort of baby boomers will retire between 2025 and 2040, and would therefore bene t from the additional

32It should be noted that the benet ratio used in the model is not fully comparable to the national denition. The latter looks at the level of pensions in retirement relative to earnings for a standard pensioner who earns average income and pays contributions for 45 years. Therefore we re-scaled the benet ratio in Figure 3 to align it with the policy discussion.

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generosity foreseen under the double threshold" scenario extended to 2040.

Figure 3: Pension reform reversal in Germany: double threshold" until 2040

baseline - contribution rate (lhs)	reversal scenario - contribution rate (lhs)
baseline - benefit ratio (rhs)	reversal scenario - benefit ratio (rhs)

	30%
	28%
rate	25%
Contribution	23%
	20%
	18%
	15%
	2015	2017	2019	2021	2023	2025	2027	2029	2031	2033	2035	2037	2039	2041	2043	2045	2047	2049	2051	2053	2055	2057	2059	2061	2063	2065	2067	2069
	Sources: Borsch	-Supan and Rausch (2018), and authors' calculations.

50%

49%

48%

47%

46%

45%

44%

43%

42%

41%

40%

39%

Benefit ratio

We de ne the double threshold" until 2040 as our reform reversal scenario for Germany.33 Thus, in the reform reversal scenario the pension contribution rate should not rise above 20% and the bene t ratio should not fall below 48% until 2040. After 2040, both the bene t ratio and the contribution rate are assumed to develop roughly in line with the Ageing Report projections.34 This is shown in Figure 3, with the solid lines representing the baseline scenario and the dashed lines reecting the reform reversal scenario. Like in the previous sections, the baseline scenario replicates the 2018 Ageing Report projections, including the already adopted reforms, according to which Germany's contribution rate will increase to around 27% and its bene t ratio will decline to below 40% by 2070 (see Figure 3, solid lines). Under the baseline scenario, both ratios are projected to start worsening rapidly from the mid-2020s onwards.35

33The reform reversal scenario for Germany includes the double threshold" agreement until 2025, as this was only decided in mid-2018 and therefore is not included in the 2018 Ageing Report projections.

1. Yet, the paths of the dotted and solid lines do not fully match after the year 2040 given that the two scenarios imply dierent feedback eects until 2040 which will have a lagged impact. This is particularly relevant for the benet ratio.
2. In the baseline scenario, the contribution rate is projected to increase to 19.5% and the benet ratio to

In the reform reversal scenario, we quantify the costs of reform reversals by the adverse impact on the public debt ratio, in comparison to the debt impact under the baseline scenario. For Germany, we nd that the reform reversal would imply sizeable costs (see Table 6, columnreform reversal"). Specically, by 2070, the increase in the public debt-to-GDP ratio can be expected to be ceteris paribus almost 60 percentage points higher than under the baseline scenario, as a result of higher pension expenditures, adverse feedback eects and lower contribution rates. At the same time, the impact on GDP per capita would be only marginally better (decline by 13.3%) than under the baseline scenario.36

Table 6: Reform reversal scenario: long-run costs for Germany

in % of GDP, unless stated otherwise

	Baseline1
	Reform	Compensatory
Germany		reversal1	measures2,3
GDP per capita (%)	-13.9	-13.3	10.1
Consumption: total	-1.5	-1.3	2.0
Consumption: share of young (% of total)	-4.6	-6.5	4.2
Unemployment rate (%)	0.8	0.1	-0.2
Pension expenditure	1.9	3.1	-2.1
Public debt	100.3	157.5	-176.0
Policy instruments
Retirement age (year)	65.5	65.5	5.5

1Deviation from the initial steady-state until 2070; long-run result for the retirement age 2Deviation from the reform reversal scenario

3Change in retirement age to compensate for the debt impact of the reform reversal

Moreover, we calculate the compensatory measures for Germany that would be needed in order to oset the adverse debt impact under the reform reversal scenario of almost 160 percentage points of GDP by 2070, compared to the initial period (see Table 6, column compensatory mea- sures").37 We assume that the adjustment will be solely placed on increases in the retirement age. We nd that the compensatory measure needed to oset the eects of the reform reversal would be very painful for future generations. In fact, the results suggest that Germany would

decrease to 47.6% by 2025. This implies that the double threshold" will not be a binding constraint until 2025, except for a small gap arising for the bene t ratio towards the end of this period, while it will become considerably more constraining thereafter.

1. It should be noted, however, that the results might reect a potential upward bias, as the model does not explicitly account for behavioural changes of the labour supply in response to changes in the pension system. In fact, the labour force participation rate is kept constant.
2. Thus, the estimated increase in the public debt-to GDP ratio by almost 160 percentage points under the reform reversal scenario reects both the ageing-related debt impact, as derived under the baseline scenario, and the debt impact due to the reform reversal.

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have to increase its e ective retirement age by 5.5 years to 71 years by 2070 to be able to fully o set the increase in the public debt-to-GDP ratio.38 Out of this suggested total increase in the retirement age, 2 years would be on account of the reform reversal, while the remaining 3.5 years reect the compensatory measures needed to o set the ageing cost increase under the baseline scenario. Moreover, the compensatory measures in form of higher retirement age would help to largely alleviate the adverse macroeconomic and scal e ects of the pension reform reversal.

Slovakia is another interesting example to study the impact of reform reversals. In end-2018, Slovakia decided to reverse one of its previous pension reforms, namely by capping the automatic link between the retirement age and changes in life expectancy at the age of 64 years. In the baseline scenario, the e ective retirement age is expected to increase from its initial level of 61 years (2016) to above 67 years. In the reform reversal scenario we assume for Slovakia that the automatic link between the retirement age and increases in life expectancy will be capped in 2045, i.e., the year when the retirement age is expected to reach 64 years, according to the baseline scenario.

Like for Germany, we quantify the scal costs of the reform reversal in Slovakia by comparing the debt impact under the reform reversal scenario with that under the baseline scenario. Our results show that such a reform reversal would be very costly. In fact, the increase in the public debt-to-GDP ratio would be more than 50 percentage points higher than the estimated increase of around 100 percentage points of GDP under the baseline scenario (see Table 7). Moreover, the results under the reform reversal scenario suggest an even more adverse economic outlook than under the baseline scenario: GDP per capita can be expected to shrink by almost 15% by 2070, compared to the initial period. This is in line with earlier ndings for Slovakia, namely that, as discussed in section 3, the increase in life expectancy is the most dominant factor for the expected rise in the old-age dependency ratio. Moreover, capping the retirement age at 64 years would lower further the young's total consumption share and increase pension expenditure even more, compared to the baseline.

38Other simulations obtain similar results. Borsch -Supan and Rausch (2018) calculate the scal costs of the double threshold" assuming that it would be kept until 2060. These authors consider an alternative policy scenario, and ask by how much the VAT rate would need to be increased to nance the double threshold" via tax subsidies to the pension system. They nd that the VAT rate would need to increase by more than 6 percentage points by 2040 to oset the scal cost of the double threshold".

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Table 7: Reform reversal scenario: long-run costs for Slovakia

in % of GDP, unless stated otherwise

	Baseline1	Reform	Compensatory measures:2
		reversal1
Slovakia		benet	contribution	mix
			ratio	rate
			(1)	(2)	(1)+(2)
GDP per capita (%)	-9.0	-14.8	0.5	-0.1	0.2
Consumption: total	-0.4	-0.9	-0.2	-0.1	-0.2
Consumption: share of young (% of total)	-4.2	-6.7	7.8	2.2	5.0
Unemployment rate (%)	-1.0	-1.3	-0.1	0.2	0.0
Pension expenditure	1.2	2.6	-4.2	0.0	-2.1
Public debt	103.0	156.1	-145.4	-154.7	-150.5
Policy instruments
Retirement age (year)	67.2	64.0	-	-	-
Benet ratio (%)	-5.0	-5.0	-29.0	-	-14.5
Contribution rate (%)	-0.2	-0.2	-	12.2	6.1

1Deviation from the initial steady-state until 2070; long-run result for the retirement age 2Deviation from the reform reversal scenario

In addition, we analyse the hypothetical case that compensatory measures were adopted that would aim to oset the debt impact under the reform reversal in Slovakia, compared to the initial period. Concretely, we consider addressing the reform reversal costs through either lower benet ratios, higher contribution rates, or a combination of the two (see Table 7, last three columns). We nd that, if the retirement age were to be capped at 64 years, the benet ratio would need to be lowered massively, by more than 29 percentage points, in order to fully compensate for the debt increase until 2070. Alternatively, the contribution rate would need to be increased by more than 12 percentage points to fully compensate for the debt impact of the reform reversal. Although the size of the compensatory measures could be halved when combining the two compensatory measures (see Table 7, column mix"), they would not help to alleviate the adverse impact on GDP per capita.

• Conclusions

The paper oers a framework to examine the measures needed to cope with the macroeconomic and scal eects of population ageing, as well as to quantify the costs of pension reform re- versals. Concretely, we exploit the country-specic information contained in the 2018 Ageing Report which we integrate into a dynamic general equilibrium model with overlapping generations to capture feedback eects. In the baseline scenario we thereby account for the impact

ECB Working Paper Series No 2396 / April 2020

29

of already adopted pension reforms, as captured in the Ageing Report. We also examine the additional pension reforms that are needed to contain the public debt impact arising from the demographic transition, and assess their macroeconomic impact. We nd evidence that mixed reforms which consist of di erent pension reform measures help to spread the adjustment burden more equally across generations. Intergenerational equity of public pension systems is an important consideration, also in view of raising political pressures to revert past pension reforms. We study the costs of pension reform reversals in two countries: Germany and Slovakia. We nd that pension reform reversals in these two countries could generate substantial scal costs as measured by the additional public debt-to-GDP impact compared to the baseline scenario and potentially worsen the macroeconomic outlook. Addressing these costs through other policy instruments could be expected to require painful adjustments, thereby potentially worsening intergenerational fairness further.

Our paper contributes to the literature in two ways: rst, to our knowledge this is the rst paper which integrates the long-term dynamics of pension expenditures, as projected in the Ageing Re- port, into a dynamic general equilibrium model with overlapping generations. By this we are able to capture feedback e ects in a general equilibrium framework, and to systematically account for important country-speci c aspects and already adopted pension reforms which will become e ective over time. Second, our framework allows us to quantify the expected macroeconomic and scal costs of reform reversals. While most European countries have adopted important pension reforms in the last two decades, more recently pension reform reversals are being discussed or have already been decided. So far pension reform reversals have hardly been reected in the literature. We try to ll this gap with our framework that allows us to systematically quantify the costs of pension reform reversals.

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30

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Annex 1 Technical description of the OGRE model

In this technical appendix, rst, we focus on solving the optimizing problems of the young and old generations, we describe the pay-as-you-go pension system and the price and wage setting equations of the rms. At the end, we provide the normalized - detrended by population growth

• equations and the steady-state calculation of the model. Regarding any other technical detail, further information is available from the authors upon request.

A.1.1 Demography and overlapping generations

Demography

Total population (Nt) is equal to the sum of the number of old (retired) (NtO) and young (worker) people (NtY ):

Nt

NtY

NtO

• NtO + NtY
• (1 − ωtY−1)NtY−1 + ntNtY−1
• (1 − ωtO−1)NtO−1 + ωtY−1NtY−1

st denotes the ratio of the number of old and young people, while sYt denotes the share of young people in the whole population:

NtO

(1 − ωtO−1)NtO−1

+ ωtY−1NtY−1

O NtO−1 NtY−1

Y NtY−1

st =

=

= (1

− ωt−1)

+ ωt−1

NtY

NtY

NtY−1

NtY

NtY

=

(1 − ωtO−1)

s

t−1

+

ωtY−1

(1 − ωtY−1 + nt)

(1 − ωtY−1 + nt)

NY

NY

1

1

sY

=

t

=

t

=

=

NtO

t

Nt

NY + NO

1 + st

t

t

1 + NY

t

Then, we can express the growth rate of each cohort:

N,Y		NtY		(1 − ωtY−1)NtY−1	+ ntNtY−1		Y
1 + gt	=		=			= 1 − ωt−1	+ nt
NtY−1	NtY−1
N,O		NtO		(1 − ωtO−1)NtO−1	+ ωtY−1NtY−1		O	ωtY−1
1 + gt	=		=			= (1 − ωt−1) +
NtO−1	NtO−1		st−1

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34

Finally, population (and the BGP) growth follows as:

1 + gt = 1 + gN = NtY + NtO

=

NY

NO

=

NtY−1

+ NtY−1

t

t

t

NY

1

+ NO

1

NtY

1

NtO 1

t

−

t

−

−

−

NtY−1

+ NtY−1

N,Y		NtO
1 + gt	+
Y
		Nt−1	=

1 + st−1

	N,Y		NO		NY
1 + gt	+	t		t		N,Y	N,Y	)		1 + st
Y		Y
=			Nt−1		Nt	=	1 + gt	+ st(1 + gt	= (1 + gtN,Y )
				1 + st−1		1 + st−1
	1 + st−1

Retired generation

First-order conditions of a retired agent

'Retired' agent i of retired cohort a is one individual who retired a years ago. Each agent maximises the following Bellman-equation:

V O(BaO−1,t−1(i)) = max 1 −1 γ Ca,tO (i)1−γ + βEt(1 − ωtO)V O(Ba,tO (i))

subject to this budget constraint:

(1 + τtC )pOt Ca,tO (i) + (1 − ωtO)Ba,tO (i) = (1 + rt−1)BaO−1,t−1(i) + T Ra,tY O(i)

where O denotes the retired cohort, the T Ra,tY O(i) is the pension benet that was set by the

government a years ago. Here, we assume that T Rn,tY O+n(i) = T R0Y,tO(i) ∀n > 0, i.e. the gov-

ernment in the pay-as-you-go pension system sets the real value of the benet at the time of

retirement, and provides the same real amount until the pensioner passes away. Ca,tO (i) is the

level of individual consumption, Ba,tO (i) is the individual risk-free bond, pOt is the retailer price of the pensioners' consumption basket, and τtC is the consumption tax rate.

The rst-order conditions are as follows:

Ca,tO (i)	:	Ca,tO (i)−γ +	λa,tO (1 + τtC )ptO = 0
O	(i)	:	O	O	O	O
Ba,t	βEt(1 − ωt	)VBa,tO (i)	+ Et(1 − ωt	)λa,t = 0

The one-period-aheadEnvelope-theorem is:

EtV O = −EtλO (1 + rt)

Ba,t(i) a+1,t+1

ECB Working Paper Series No 2396 / April 2020

35

The rst-order conditions imply the Euler equation:

	Ca,tO (i)γ	(1 + τtC )ptO
βEt		(1 + rt)		= 1
CaO+1,t+1(i)γ	(1 + τtC+1)ptO+1
which can be rearranged:
			1	1

EtCaO+1,t+1(i) = Ca,tO (i)β γ (1 + rt)γ ΛOt

where

1

• #

Λt,tO	+1 = Et	(1 + τtC )ptO	γ
(1 + τtC+1)ptO+1

Based on the Euler-equation, all future individual retired consumptions follow:

n	n	1
Y
EtCaO+n,t+n(i) = Ca,tO (i)β γ Et	(1 + rt+k−1)γ Λt,tO +k
k=1

Individual consumption of a retired agent

First, we derive the intertemporal budget constraint from the one-period budget constraint:

∞	Q	kn=1(1 − ωtO+k−1)(1 + τtC+n)ptO+nCaO+n,t+n(i)	∞	Q	kn=1(1 − ωtO+k−1)T RaY+On,t+n	(i)
Et n=0	kn=1(1 + rt+k	−	1)	= Et n=0	kn=1(1 + rt+k	−	1)	+
X		Q		X		Q

+(1 + rt−1)BaO−1,t−1(i)

if k > n and rt+k = 0.

We can use the Euler equation for future consumptions:

∞

(1

)(1 + τC

)pO

n

(1 + rt k

1

n

ωO

CO

(i)β γ

n

)γ ΛO

Q

k=1

− t+k−1

kn=1(1 + rt+k 1)Q

+ −1

Et

Q(i)

−

t,t+k

=

X∞

(1 ωO

)nT RY O

n=0

= E

X

− t+k−1

a+n,t+n

+ +(1 + r

t−1

)BO

(i)

−

t

kn=1(1 + rt+k

1)

a−1,t−1

n=0

Q

If we rearrange, we get consumption of agent i of cohort a at time t as a function of the present value of pension benets, other income and initial wealth:

CO i

Et

n∞=0 Q

n

k=1(1+rt+k−1)

(i)

k=1

P

(1−ωtO+k

1)T RaY+On,t+n

n

−

a,t( ) =

Et

∞

(1 + τC

n

n

Q(1

ωO

)(1 + r

1

−

1

+

)pO

β γ

t+k−1

)γ

ΛO

Pn=0

t+n

t+n

Qk=1

−

t+k−1

t,t+k

ECB Working Paper Series No 2396 / April 2020

36

(1 + r

t−1

)BO

(i)

+

n

a−1,t−1

1

−1Λt,tO +k

Et Pn∞=0(1 + τtC+n)ptO+nβ γ

Qkn=1(1 − ωtO+k−1)(1 + rt+k−1)γ

Finally, using the assumption that T Rn,tY O+n(i) = T R0Y,tO(i) ∀n > 0:

CO (i) = T RY O(i)

Et

nn∞=0

n

+

Qk=1(1+rt+k−1)

P

k=1(1−ωtO+k

1)

n

−

a,t

a,t

n

Q(1

)(1 + rt+k

1

1

ΛO

Et

n∞=0(1 + τtC+n)ptO+nβ γ

k

ωO

1)γ

−

P

(1 + r

t−1

)BO

(i)Q

=1

− t+k−1

−

t,t+k

+

n

a−1,t−1

1

Et Pn∞=0(1 + τtC+n)ptO+nβ γ

Qkn=1(1 − ωtO+k−1)(1 + rt+k−1)γ −1Λt,tO +k

The denominators are the function of non-individual variables, then we can denote it as an

aggregate variable:

1		∞		nn						1
			X		Y	(1 − ωtO+k−1)(1 + rt+k−1) γ −1Λt,tO +k
	MP CO	= Et	(1 + τtC+n)ptO+nβ γ
	t	n=0		k=1
			∞		n	n		1
			X			Y
= (1 + τtC )ptO + Et	(1 + τtC+n)ptO+nβ γ	(1 − ωtO+k−1)(1 + rt+k−1) γ −1Λt,tO +k
			n=1			k=1
= (1 + τtC )ptO +
1			1	∞				n	n	1
+β γ (1 − ωtO)(1 + rt) γ −1Λt,tO +1Et	X					Y	(1 − ωtO+k)(1 + rt+k) γ −1ΛtO+1,t+k
(1 + τtC+n+1)ptO+n+1β γ
					n=0		1		k=1
			1		1
= (1 + τtC )ptO	+ β γ (1	− ωtO)(1 + rt) γ −1Λt,tO +1Et
MP CO
								t+1

Using the same recursive substitution for the future expected pension benet, the consumption function of agent i of cohort a at time t is:

CO	(i) = MP COΩOT RY O(i) + MP CO(1 + r	t−1	)BO	(i)
a,t	t t a,t	t	a−1,t−1

where ΩO is the discount factor of the retired households:

ΩO	= 1 + Et	1 − ωtO	ΩO
t			t+1
		1 + rt

Aggregate consumption of the retired cohort

Aggregate consumption is equal to the sum of pension benets, other income and initial wealth:

ECB Working Paper Series No 2396 / April 2020

37

∞			∞
X	(i)CO	(i) = MPCOΩO	X	(i)T RY O(i) +
NO	NO
a,t	a,t	t t	a,t	a,t
a=0			a=0
			∞

• MP CtO(1 + rt−1) X Na,tO (i)BaO−1,t−1(i)
  a=0

First, the number of old people declines over time:

NaO+1,t	=	(1	− ωtO−1)Na,tO −1
NaO+2,t	=	(1	− ωtO−1)(1 − ωtO−2)Na,tO −2

...

and

∞

NtO = X Na,tO

a=0

Now, we can express aggregate pension income in period t of those who get retired at period t, one period before, etc.:

NO T RY O(i) = T RY O
0,t	0,t	t
N1O,tT R1Y,tO(i) = (1 − ωtO−1)N0O,t−1T R0Y,tO−1(i) = (1 − ωtO−1)T RtY−O1
N2O,tT R2Y,tO(i) =	(1 − ωtO−1)(1 − ωtO−2)N0O,t−2T R0Y,tO−2(i) = (1 − ωtO−1)(1 − ωtO−2)T RtY−O2

...

using T Rn,tY O+n(i) = T R0Y,tO(i) ∀n > 0 again. Then, adding up all pensions implies:

	∞
T Rt ≡	X
Na,tO (i)T Ra,tY O(i) = T RtY O + (1 − ωtO−1)T RtY−O1 + ...
	a=0

• T RtY O + (1 − ωtO−1)T Rt−1

Now, aggregate consumption of the retired cohort cohort is dened as:

ECB Working Paper Series No 2396 / April 2020

38

∞

X(1 − ωtO−1)NaO−1,t−1Ba,tO −1(i) a=1

∞

CtO = X Na,tO Ca,tO (i)

a=0

while total savings of the retired is:

∞			∞
X	(i) = NO BO	(i) +	X	(i)
NO BO	NO BO
a,t a,t−1	0,t 0,t−1		a,t a,t−1
a=0			a=1

Here, we need to be careful with the just-retired agents: they were young one period before without knowing about their next period retirement. We can use the law of large numbers to

get the following expression: NO	= ωY	NY		:
0,t	t−1	t−1
O O		O	∞	Y,last	Y Y	BY
	X	t−1
N0,tB0,t−1(i) = N0,t	Bb,t−1 (i) ' ωt−1Nt−1	−
b=1	NtY 1

where the last refers to the fact that those who get retired today spent their last year in the young cohort in the previous year.

Then, from t − 1 to t it is easy to see that: P∞ NO = P∞ (1 − ωO )NO which implies

a=1 a,t a=1 t−1 a−1,t−1

that

∞

X Na,tO Ba,tO −1(i) = ωtY−1BtY−1 + a=0

Here, the second term means that only those retired agents cumulate savings who expect to survive the next period. Hence, the amount of aggregate old-age savings from the previous

period is BO = P∞ (1 −ωO )NO BO (i). Then, overall savings of the retired cohort in

t−1 a=1 t−1 a−1,t−1 a,t−1

period t can be expressed easily by adding just-retired savings from the previous period's young cohorts:

∞

X Na,tO Ba,tO −1(i) = ωtY−1BtY−1 + BtO−1

a=0

As a last step, we put together all parts of the equation, so, aggregate consumption of formal goods of the retired cohort is:

CtO = MP CtOΩOt T Rt + (1 + rt−1)MP CtO(ωtY−1BtY−1 + BtO−1).

ECB Working Paper Series No 2396 / April 2020

39

Young generation

First-order conditions of a young agent

'Young' agent i of young cohort b is one individual of its cohort who started to work (was born) b years ago. The Bellman-equation of a young individual is:

VtY (BbY−1,t−1(i)) = max	1	1	γ Cb,tY	(i)1−γ + Etβ	(1 − ωtY )VtY+1(Bb,tY (i)) + ωtY VtO+1(Bb,tY O(i))
		−

while the budget constraint is:

(1 + τtC )pYt Cb,tY (i) + (1 − ωtY )Bb,tY (i) + ωtY Bb,tY O(i) =

• (1 + rt−1)BbY−1,t−1(i) + (1 − τtL)wtLb,t(i) + wtU Ub,t(i) + P rofitb,t(i) − LT axb,t(i)

where Cb,tY (i) denotes the young agent's individual consumption, pYt is the retailer price of the young consumption basket, Bb,tY (i) is young households risk-free bond, Bb,tY O(i) is young house- holds' state-contigentrisk-free bond, Lb,t(i) is individuals' working hours, wt is the real wage, P rofitb,t(i) denotes the prots from the rms, and LT axb,t(i) is the other lump-sum taxes. If the given household is unemployed (Ut), she receives unemployment benet (wtU ) from the government, but this benet will not be taken into account at the time of retirement.

The rst-order conditions are as follows:

Cb,tY (i) : Cb,tY (i)−γ + λYb,t(1 + τtC )pYt = 0

Bb,tY (i) : βEt(1 − ωtY )VBYY + Et(1 − ωtY )λYb,t = 0 b,t

Bb,tY O(i) : βEtωtY VBYY O + EtωtY λYb,t = 0 b,t

The one-period-aheadEnvelope-theorem is:

EtVBb,tY = −EtλYb+1,t+1(1 + rt)

Also, from the retired agent's optimization we know that:

EtVBb,tY O = −EtλO0,t+1(1 + rt) = −EtλOb+1,t+1(1 + rt)

ECB Working Paper Series No 2396 / April 2020

40

where EtλO0,t+1 = EtλOb+1,t+1 because someone who was young in t gets retired in t + 1. Thus, the Euler equations of the young individual are:

CY

(i)−γ

(1 + τC )pY

βEt

b+1,t+1

(1 + rt)

t

t

= 1

Cb,tY (i)−γ

(1 + τtC+1)ptY+1

O

−γ

(1 + τC )ptY

Cb+1,t+1(i)

βEt

(1 + rt)

t

= 1

Y

γ

C

O

Cb,t(i)−

(1 + τt+1)pt+1

Rearranging results in the following:

EtCbY+1,t+1(i) =

Cb,tY

1

1

(i)β γ (1 + rt)γ Λt,tY

+1

EtC0O,t+1(i) =

Cb,tY

1

1

(i)β γ (1 + rt)γ Λt,tY O+1

where

"

1

(1 + τtC )ptY

#

Λt,tY

+1 = Et

(1 + τtC+1)ptY+1

1

• #

Λt,tY O+1 = Et	(1 + τtC )ptY	γ
(1 + τtC+1)ptO+1

Also, we can express each period's consumption as a function of period-t consumption and the discount rate:

n	n	1
Y
EtCbY+n,t+n(i) = Cb,tY (i)β γ Et	(1 + rt+k−1)γ Λt,tY +k
k=1
Individual consumption of a young agent

First of all, we would like to stress that one needs to be careful when deriving the young agent's individual consumption because old-age incomes and expenditures must be taken into account, too. Moreover, the young agents also consider the probability of retirement, for instance, in period t the probability that a young agent becomes retired in period t + 1 is ωtY , while the probability that the same agent becomes retired in period t + 2 is (1 − ωtY )ωtY+1. So, the rst term of the left-hand side of this equation shows the stream of lifetime consumption if the agent stays young, then, from the second term onwards she retires with some probability in each period:

ECB Working Paper Series No 2396 / April 2020

41

∞	Q	kn=1(1 − ωtY+k−1)(1 + τtC+n)ptY+nCbY+n,t+n(i)	+
Et n=0	kn=1(1 + rt+k−1)
X		Q

Y	∞	Q	kn=2(1 − ωtO+k−1)(1 + τtC+n)ptO+nCnO−1,t+n(i)
+Etωt	n=1	kn=1(1 + rt+k−1)
	X		Q

!

+

Y	Y	∞	Q	kn=3(1 − ωtO+k−1)(1 + τtC+n)ptO+nCnO−2,t+n(i)
+Et(1 − ωt	)ωt+1	n=2	kn=1(1 + rt+k−1)
		X		Q

!

+ ...

∞

kn=1(1 − ωtY+k−1) (1 − τtL+n)wt+nLb+n,t+n(i) + wtU+nUb+n,t+n(i) + P rofitb+n,t+n(i) − LT axb+n,t+n(i)

+

= Et n=0 Q

kn=1(1 + rt+k−1)

X

+(1 + rt−1)BbY−1,t−1(i) +

Q

Y

∞

Y O

kn=2(1 − ωtO+k−1)

+Etωt

n=1 T Rn−1,t+n

(i)

Qkn=1(1 + rt+k−1)

+

X

∞

Q

n

(1

ωO

)

+Et(1 − ωt

)ωt+1 n=2 T Rn−2,t+n(i)

Qkn=1(1 + rt+k−1)

+

)

X

∞

Q

−

n

(1 ωO

Y Y

Y O

k=3

t+k−1

+Et(1 − ωt

)(1 − ωt+1)ωt+2 n=3 T Rn−3,t+n(i) Qkn=1(1 + rt+k−1) + ...

X

Q

− t+k−1

Y

Y

Y

Y O

k=4

Based on the Euler-equations, we can express expected future consumptions. Let's consider an agent who is young in period t, then her consumption functions in the next periods after getting retired are:

n	n	1
EtCn,tO +n+1(i) = EtC0O,t+1(i)β γ	Y	(1 + rt+k−1)γ ΛtO+1,t+k
	k=2

On the other hand, if the agent stays young in period t + 1 and gets retired after that, her future old-age consumptions look like:

						nn	1
			EtCn,tO +n+2(i) = EtC0O,t+2(i)β γ	(1 + rt+k−1)γ ΛtO+2,t+k
						k=3
						Y
Now, we plug them in the intertemporal budget constraint:
∞	Q	n	Y	C Y Y	(i)
n=0		k=1	(1 + rt+k−1)
X		k=1(1 − ωt+k−1)(1 + τt+n)pt+nCb+n,t+n
		Q		+
Et			n

∞

n

Q

n

O

C

O O

1

O

t

n=1

β

k=1

(1 + rt+k−1)

Λt+1,t+k

+Etω

Y

X

k=2(1 − ωt+k−1)(1 + τt+n)pt+nC0,t+1

(i)(1 + rt+k−1)

Q

n

!

+

∞

n

Q

n

O

C

O O

1

O

− t

t+1

n=2

β

k=1

(1 + rt+k−1)

Λt+2,t+k

Y

Y

X

k=3(1 − ωt+k−1)(1 + τt+n)pt+nC0,t+2

(i)(1 + rt+k−1)

+Et(1 ω

)ω

Q

n

!

• ... =

ECB Working Paper Series No 2396 / April 2020

42

∞	Q	n	Y	L	U
n=0			k=1	(1 + rt+k−1)
X		k=1(1 − ωt+k−1) (1 − τt+n)wb+n,t+nLb+n,t+n(i) + wt+nUb+n,t+n(i) + P rofitb+n,t+n(i) − LT axb+n,t+n(i)
			Q		+
= Et				n

+(1 + rt−1)BbY−1,t−1(i) +

∞

X

+EtωtY

n=1

+Et(1 − ωtY

n−1,t+n

n

(1 + rt+k−1)

Qk=1

T R

Y O

(i)

k=2

(1

− ωtO+k

−

1)

+

n

∞

Q

n

(1

ωO

)

t+1 n=2

n−2,t+n

Qk=1(1 + rt+k−1)

)ω

X

T R

(i)

Q

n

−

t+k

−

1

+ ...

Y

Y O

k=3

After that, we use the other Euler equation (the one that shows the substitution between period-t young and period-t + 1 old-age consumption):

∞	Q	n	Y	C Y Y	(i)
n=0		k=1	(1 + rt+k−1)
X		k=1(1 − ωt+k−1)(1 + τt+n)pt+nCb+n,t+n
		Q		+
Et			n

∞

n

Q

n

O

C O Y

1

Y O

1

O

t

n=1

β

k=1(1 + rt+k−1)

Λt,t+1(1 + rt+k−1)

Λt+1,t+k

+Etω

Y

X

k=2(1 − ωt+k−1)(1 + τt+n)pt+nCb,t

(i)(1 + rt)

Q

n

!

+

− t

t+1

n=2

n

Q

O

k=1(1 + rt+k−1)

1

Y O

1

O

!

γ

γ

γ

Y

Y

∞

β

n

C O Y

(i)(1 + rt+1)

+Et(1

ω

)ω

X

k=3

(1 − ωt+k−1)(1 + τt+n)pt+nCb+1,t+1

Λt+1,t+2(1 + rt+k−1)

Λt+2,t+k

+ ...

)Q

n

(1

) (1

)

b+n,t+n

b+n,t+n(

b+n,t+n

b+n,t+n

b+n,t+n

∞

Q

n

ωY

τL

w

L

i

+ wU U

(i) + P rofit

(i)

LT ax

(i)

n=0

k=1(1 + rt+k−1)

X

k=1

− t+k−1

−

t+n

Q

t+n

−

+

= Et

n

+(1 + rt−1)BbY−1,t−1(i) +

∞

X

+EtωtY

n=1

+Et(1 − ωtY

n−1,t+n

n

(1 + rt+k−1)

Qk=1

T R

Y O

(i)

k=2

(1

− ωtO+k

−

1)

+

n

∞

Q

n

(1

ωO

)

t+1 n=2

n−2,t+n

Qk=1(1 + rt+k−1)

)ω

X

T R

(i)

Q

n

−

t+k

−

1

+ ...

Y

Y O

k=3

Concentrating on consumptions:

∞	Q	kn=1(1 − ωtY+k−1)(1 + τtC+n)ptY+nCbY+n,t+n(i)	+
Et n=0	kn=1(1 + rt+k−1)
X		Q

		n	Q	kn=2(1 − ωtO+k−1)(1 + τtC+n)ptO+nCb,tY	1	1
Y	∞	β γ	(i)(1 + rt) γ Λt,tY O+1(1 + rt+k−1) γ ΛtO+1,t+k
+Etωt	n=1		kn=1(1 + rt+k−1)
	X			Q

!

+

+Et(1 − ωt )ωt+1	n=2	n	Q	kn=1	(1 + rt+k−1)	1	!	+ ...
				1
Y Y	∞	β γ		kn=3(1 − ωtO+k−1)(1 + τtC+n)ptO+nCbY+1,t+1(i)(1 + rt+1) γ ΛtY+1O	,t+2(1 + rt+k−1) γ ΛtO+2,t+k
	X			Q

ECB Working Paper Series No 2396 / April 2020

43

We rearrange and get the following:

(1 + τtC )pYt Cb,tY (i) +

∞

X

+Cb,tY (i)EtωtY

n=1

n	1	1
(1 + τtC+n)ptO+nβ γ	Qkn=2(1 − ωtO+k−1)(1 + rt) γ	Λt,tY O+1(1 + rt+k−1) γ ΛtO+1,t+k

Qn

k=1(1 + rt+k−1)

!

+

+(1 + τtC+1)ptY+1CbY+1,t+1(i)	(1 − ωtY )	+
+EtCb+1,t+1		(1 + rt)		Q	kn=1	(1 + rt+k−1)		!	+ ...
(i)(1 − ωt )ωt+1	n=2	n	1
						1
Y	Y Y		∞ (1 + τtC+n)ptO+nβ γ		kn=3(1 − ωtO+k−1)(1 + rt+1) γ ΛtY+1O	,t+2(1 + rt+k−1) γ ΛtO+2,t+k
			X			Q

Simplifying before recursive substitution gives us:

(1 + τtC )pYt Cb,tY (i) +

b,t

Y

1

1

n=1

C O

n

Q

n

k=2

(1 + rt+k−1)

1

O

!

1 + rt

γ

γ

Y

ωt

(1 + rt)

Λt,t+1

β

X

(1 + τt+n)pt+n

β

k=2(1

−

ωt+k−1)(1 + rt+k−1)

Λt+1,t+k

+C

(i)Et

Q

n

+

+Et(1 + τC	)pY	CY			(i)	(1 − ωtY )	+
t+1	t+1	b+1,t+1		(1 + rt)
					1	1
Y		Y		ωtY+1(1 + rt+1) γ ΛtY+1O	,t+2β γ
+EtCb+1,t+1	(i)(1 − ωt	)
	(1 + rt)(1 + rt+1)

n=2	C O		n	Q	k=3	(1 + rt+k−1)	1	O	!
	γ	γ
∞	β		n	−	O
X	(1 + τt+n)pt+n			k=3(1	ωt+k−1)(1 + rt+k−1)		Λt+2,t+k	+ ...
				Q
					n

Now, we can use	1	from the retired agents' optimization:
MP CtO+1
Cb,tY (i) "(1 + τtC )ptY + Etβ γ	ωtY (1 + rt)γ	−1Λt,tY O+1 MP CO	#	+
		1	1	1
				t+1

+EtCY	(i)	(1 − ωtY )
b+1,t+1		(1 + rt)

"		1	1	−1		1	#
(1 + τC	)pY	+ β γ ωY	(1 + rt+1)γ	ΛY O		+ ...
t+1	t+1	t+1			t+1,t+2 MP CO
						t+2

And, using the Euler-equation again (to have period-t consumption only):

EtCbY+n,t+n(i) = Cb,tY

n

n

1

(i)β γ Et

k=1

(1 + rt+k−1)γ Λt,tY +k

Y

Finally, we get:

1

= Cb,tY (i) (1 + τtC )ptY

1

−1

1

1

+

Cb,tY

(i)

+ EtωtY (1 + rt) γ

β γ Λt,tY O+1

MP CtY

MP CtO+1

1

1

Y

1

1

(1 − ωt )

1

Cb,tY

(i)Et(1 + rt) γ β γ Λt,tY +1

(1 + τtC+1)ptY+1

+ ωtY+1(1 + rt+1) γ −1β γ ΛtY+1O,t+2

+ ...

MP CtO+2

(1 + rt)

ECB Working Paper Series No 2396 / April 2020

44

where

MP CY	= (1 + τtC )ptY + Etβ γ	(1 + rt)γ −1	"(1 − ωtY )Λt,tY	+1 MP CY	+ ωtY Λt,tY O+1 MP CO	#
1	1	1		1	1
t				t+1	t+1

Similarly, the young agent's budget constraint contains old-age income items, i.e., expected revenues from the pension fund.

Y O	Y O Y O		(1 − ωtY )ωtY+1	O Y O
Ib,t	(i) = Etωt Ωt+1T R0,t+1(i) + Et		Ωt+2T R0,t+2(i) +
(1 + rt+1)
+Et	(1 − ωtY )(1 − ωtY+1)ωtY+2	ΩO	T RP G,Y O(i) + ...
	(1 + rt+1)(1 + rt+2)	t+3		0,t+3

Again, we use that T Rn,tP G,Y+nO(i) = T R0P,tG,Y O(i) ∀n > 0. In a recursive way it looks as:

Y O

i

E

ωY T RY O

i

E

(1 − ωtY )

Y O

i

)

Ib,t

( ) =

t

t

0,t+1( )Ωt+1 +

t (1 + rt+1)Ib+1,t+1(

Furthermore, young-age income is:

Ib,t

∞

∞

Y

n

L

U

n=0

Q

k=1(1 + rt+k−1)

Y

X

k=1(1 − ωt+k−1)

(1 − τt+n)wt+nLb+n,t+n(i) + wt+nUb+n,t+n(i) + P rofitb+n,t+n(i) − T axb+n,t+n(i)

(i)

= Et

n

=

L

U

Q

1 − ωtY

Y

= (1 − τt

)wtLb,t(i) + wt Ub,t(i) + P rofitb,t(i) − LT axb,t(i) + Et

Ib+1,t+1

(i)

1 + rt

If we add the present value of young income and expected pension benets, we can introduce a

new variable:

IY O(i)

IncYb,t(i) = Ib,tY (i) + b,t

1 + rt

= (1	−	τL)wtLb,t(i) + wU Ub,t(i) + P rofitb,t(i)	−	LT axb,t(i) + Et	ωtY	T RY O	(i)Ωt+1	+ Et	1 − ωtY	IncY	(i)
1 + rt
	t	t		0,t+1			1 + rt	b+1,t+1

Thus, the individual consumption function of agent i of cohort b in period t is

Cb,tY (i) = MP CtY IncYb,t(i) + (1 + rt−1)MP CtY BbY−1,t−1(i)

Introducing a new variable for life-time income, and using marginal propensity to consume:

ECB Working Paper Series No 2396 / April 2020

45

CY

(i)

=

MPCY

IncY (i) + MP CY

(1 + rt

−1

)BY

(i)

b,t

t

b,t

t

b−1,t−1

ωtY

Y

L

U

Y O

Incb,t

(i)

=

(1 − τt

)wtLb,t(i) + wt

Ub,t(i) + P rofitb,t(i)

− LT axb,t(i) + Et

T R0,t+1

(i)Ωt+1

+

1 + rt

+Et

1 − ωtY

IncbY+1,t+1(i)

MP CtY

=

1 + rt

+ ωtY Λt,tY O+1 MP CtO+1

(1 + τtC )ptY + Et(1 + rt) γ −1β γ (1 − ωtY )Λt,tY +1 MP CtY+1

1

1

1

1

1

Aggregate consumption of the young cohort

As a rst step we need to express the total number of young people. If Nb,tY is the number of b-year old workers, the total number of workers is

∞

NtY = X Nb,tY

b=0

Following the previous idea, we sum up all consumptions, incomes and savings:

∞	∞	∞
X	X	X
Nb,tY Cb,tY (i) = MP CtY	Nb,tY Incb,tY (i) + (1 + rt−1)MP CtY	Nb,tY BbY−1,t−1(i)
b=0	b=0	b=0

where we note that the new young workers in time t have zero savings from the previous period.

∞

∞

∞

NY

NY

CY (i) = MP CY

NY

IncY

(i) + (1 + r

)MP CY

X

NY

b−1,t−1

BY

(i)

X

b,t

t

X

b,t

t−1

t

b,t

−

−

b−1,t−1

b,t

b,t

NY

b=0

b=0

b=1

b

1,t 1

Rearranging gives us:

∞

∞

∞

X

X

X

NbY−1,t−1BbY−1,t−1(i)

Nb,tY Cb,tY (i) = MP CtY

Nb,tY Incb,tY (i) + (1 + rt−1)MP CtY (1 − ωtY−1)

b=0

b=0

b=1

Aggregate values are dened as:

∞

X

CtY

≡

Nb,tY Cb,tY (i)

b=0

∞

X

BtY−1 ≡

NbY−1,t−1BbY−1,t−1(i)

b=1

∞

X

InctY

≡

Nb,tY Incb,tY (i)

b=0

ECB Working Paper Series No 2396 / April 2020

46

P∞ NY BY (i)

b=1 b−1,t−1 b−1,t−1

It is important to note that in each period, independently from the survival probabilities, each young agent saves for the next period, hence, the overall savings BtY−1 =

are divided among those who remain young and get retired. As a result, the aggregate consumption functions are:

CtY = MP CtY IncYt + (1 + rt−1)MP CtY (1 − ωtY−1)BtY−1

Now we need to aggregate the supporting variables as well. First of all, we rename individual contemporary income as follows:

IncYb,t(i) = Ib,tY (i) + Ib,tY O(i)

Aggregating gives us:

IncYt = ItY + ItY O

After aggregating and rearranging we get:

∞	∞	(1 − τtL)wtLb,t(i) + P rofitb,t(i) − T axb+n,t+n(i) +
b=0 Nb,tY Ib,tY (i) = b=0 Nb,tY
X	X
E (1 − ωtY )	∞ NY Y	i
			X
+ t	1 + rt		b,tIb+1,t+1( )
			b=0
∞				1 ∞
= b=0 Nb,tY (1 − τtL)wtLb,t(i) + P rofitb,t(i) − LT axb,t(i)	+ Et 1 + rt b=0 NbY+1,t+1IbY+1,t+1(i)
X					X

Because ItY+1 contains the income of the new-born people as well, the last term can be rearranged, using the law of large numbers as follows:

∞

Et X NbY+1,t+1IbY+1,t+1(i) = EtItY+1 − EtNb,tY +1Ib,tY +1(i) =

b=0	1 − NtY+1	! = EtIt+1	1 − NtY+1 !
= EtIt+1
Y		Nb,tY +1	Y		ntNtY

Then, total young income is:

Y

= (1 −

τL

w

L

t +

P rofit

LT ax

t +

E

1 − ωtY

Y

It

t −

t (1 + rt)(1 + gtN,Y+1 )It+1

t )

t

ECB Working Paper Series No 2396 / April 2020

47

A similar exercise can be done for pension benets. First, we dene ItY O which can be rearranged as:

∞∞

ItY O = X Nb,tY Ib,tY O(i) = EtωtY X Nb,tY EtT R0Y,tO+1(i)ΩOt+1 +

		b=0				b=0
	(1 − ωtY )	∞	NY Y O	i E NO	T RY O i O
			X
+	(1 + rt+1)			b,tIb+1,t+1( ) = t 0,t+1	0,t+1( )Ωt+1 +
1	b=0
		∞
					X
+Et	(1 + rt+1)		NbY+1,t+1IbY+1O,t+1(i)
					b=0

Now, similarly to total young income, the last term can be expressed as:

E	∞	NY	Y O i E 1 − ωtY Y O
	X	b+1,t+1Ib+1,t+1( ) = t		It+1
	t	1 + gN,Y
	b=0			t+1

We also know that the following expression holds:

EtN0O,t+1T R0Y,tO+1(i)ΩOt+1 = EtT RtY+1O ΩOt+1

Finally, the expected income of the young after getting retired is

Y O		E	T RY O O	+	E	1 − ωtY	Y O
It		t (1 + rt+1)(1 + gtN,Y+1 )It+1
=	t	t+1Ωt+1

Based on the derivation above, we can express the aggregate version of the young household's income as:

IncY

τL

w

L

wU U

P rofit

LT ax

E

T RtY+1O

O

E

1 − ωtY

IncY

= (1 −

t +

t +

t 1 + rt

Ωt+1

t (1 + rt)(1 + gtN,Y+1 )

t

t )

t

t

t −

t +

+

t+1

where aggregate unemployment is as follows:

Ut = NtY − Lt

Aggregating the young households' budget constraints

The individual budget constraint of a young agent is as follows:

(1 + τtC )pYt Cb,tY (i) + (1 − ωtY )Bb,tY (i) + ωtY Bb,tY ∗(i) =

ECB Working Paper Series No 2396 / April 2020

48

• (1 − τtL)wtLb,t(i) + wtU Ub,t(i) + P rofitb,t(i) − LT axb,t(i) + (1 + rt−1)BbY−1,t−1(i)

Aggregating implies the following:

∞	NY	τC pY CY i	∞	NY	ωY	BY i	∞	NY	ωY BY ∗ i
X	b,t(1 +	t ) t b,t( ) +	X	b,t(1 − t )	b,t( ) +	X	b,t	t b,t ( ) =
b=0			b=0				b=0

∞	+ (1 + rt−1)
= b=0 Nb,tY (1 − τtL)wtLb,t(i) + wtU Ub,t(i) + P rofitb,t(i) − LT axb,t(i)
X

where the denition of aggregate savings is:

∞
X	BY	(i)
NY
b,t	b−1,t−1
b=1

∞∞

X Nb,tY BbY−1,t−1(i) = X(1 − ωtY−1)NbY−1,t−1BbY−1,t−1(i)

b=1b=1

After aggregation, there is no dierence between the BtY and BtY ∗. So, we can easily express aggregate budget constraint:

(1 + τtC )pYt CtY + BtY = (1 − τtL)wtLt + wtU Ut + P rofitt − LT axt + (1 + rt−1)(1 − ωtY−1)BtY−1

A.1.2: Fiscal policy and pay-as-you-go pension plan

Pension system

To account for the overall expenditure of the pension system, we need to count the number of just-retired and retired agents. The number of just-retired agents, (those who were young one period before) is:

		∞
NO	=	X	NY
ωY
0,t		t−1	b−1,t−1
		b=1

and the total number of retired agents in period t is the just-retired agents plus those who survived the previous periods:

NtO = N0O,t + (1 − ωtO)N1O,t−1 + (1 − ωtO)(1 − ωtO−1)N2O,t−2 + ...

Individual (i)'s pension in the year of retirement t is based on replacement rate νt and the previous period income:

ECB Working Paper Series No 2396 / April 2020

49

• R0Y,tO(i) = νtwt−1Lb−1,t−1(i)

We need to use the following expressions to aggregate:

∞

NO T RY O(i) = ν

NO w

(i) = ν

ωY

X

(i)

0,t

0,t

t

0,t

t−1

b−1,t−1

t

t−1

b−1,t−1

t−1

b−1,t−1

b=1

T RY O

= ν ωY

w

t−1

L

t−1

t

t

t−1

Furthermore, total pension expenditure of all retired people is as follows:

T Rt = T RtY O + (1 − ωtO−1)T RtY−O1 + (1 − ωtO−1)(1 − ωtO−2)T RtY−O2 + ...

which can be rewritten as:

T Rt = T RtY O + (1 − ωtO−1)T Rt−1

Rest of the scal sector

The government budget constraint is as follows:

Debt	+ LT ax + T ax	t	=	Gov + T R	+ wU U	+ (1 + r	t−1	)Debt
t	t			t	t	t t		t−1
		T axt	=	τtC (ptOCtO + ptY CtY ) + (τtSSC + τtL)wtLt

where Gov denotes the other - exogenous - current expenditures, Debt is the level of public debt. The government wants to stabilizes the public debt-to-GDP ratio by adjusting the level of lump-sum taxes:

LT axt = Govt + T Rt + wtU Ut + (1 + rt−1)Debtt−1 −	Debt		T arget
		Yt − T axt
Y

The households nance government debt and the bond market equilibrium is the following:

Debtt = BtY + BtO

ECB Working Paper Series No 2396 / April 2020

50

A.1.3: Firms' optimization

The young households own the rms and the labor union, hence, in their optimization we take

into account their survival probability.

Production and nominal price setting

The rms produce di erentiated products and due to their monopolistic power they are able to set optimal prices. However, nominal price setting is costly. Hence their optimal price settings conditional on the current a future cost of the price settings (Rotemberg, 1982). The cost function can be given by a quadratic convex function:

	PtY (i)	! =	φP		PtY (i)	− 1!	2
R
PtY−1(i)		2		PtY−1(i) (1 + πt−1)ϑ

where Pt(i) is the optimal individual price in period t, π denotes the ination. If the rms adjust their prices by the previous period ination is cost-less, debends on the size of ϑ.

Total cost of production follows as:

T Ct(i) = PtInvt(i) + (1 + τtSSC )WtLt(i) + Pt(i)YtR PP−t(i()i) + HCtHt

t 1

Law motion of employment can be described as the function of previous level of employment, ring probability and the actual hiring (that also imposed by convex adjustment cost):

Lt(i) = (1 − prF )Lt−1(i) + Ht(i) 1 − S Ht(i)

tHt−1(i)

Hiring cost is the positive function of the hiring-probability:

HCt = κprtH αHC

Ht

prH =

t Ut−1 + prtF Lt−1

The rms are responsible for capital accumulation:

Invt(i)

Kt(i) = (1 − δ)Kt−1(i) + Invt(i) 1 − S Invt−1(i)

where S (·) function denotes the investment adjustment cost, any changes in investment which

ECB Working Paper Series No 2396 / April 2020

51

dier from the balanced growth path is costly. The cost functions are the following:

S

Invt(i)

=

φInv

1

Invt(i)

− 1

2

Invt−1(i)

2

1 + gt

Invt−1(i)

S Ht 1(i)

= 2

1 + gN Ht 1(i) − 1!

2

Ht(i)

φH

1

Ht(i)

−

t

−

We can write up the Bellman-equation:

V (Pt−1(i), Lt−1(i), Ht−1(i), Kt−1(i), Invt−1(i)) = Pt(i)Yt(i) − T Ct(i) +

+Et(1 − ωtY )	V (Pt(i), Lt(i), Ht(i), Kt(i), Invt(i))
1 + it

+MCt	hαΘ Kt−1(i)	−θ
	1		θ 1

1

+ (1 − α)θ (AtLt(i))

θ−θ	1iθ−1	− Yt(i)! +
		θ

+λtH (1 − prtF )Lt−1(i) + Ht(i) 1 − S	Ht(i)	− Lt(i)
Ht−1(i)
+Qt (1 − δ)Kt−1(i) + Invt(i) 1 − S		Inv (i)		− Kt(i)
t
Invt 1(i)
		−

where Q is the nominal Tobin-Q.

We can plug the demand for individual product, that can be derived from a Dixit-Stiglitz aggregator function:

V (...) = Pt(i)

Pt(i)

−ϕ

Yt − T Ct(i) + Et(1 − ωtY )

V (...)

Pt

1 + it

−

− Pt

Yt!

+

+ MCt

hα θ Kt−1(i) θ

+ (1 − α) θ (AtLt(i)) θ i

1

θ−1

1

θ−1

θ

Pt(i)

−ϕ

θ

1

+ λtH (1 − prtF )Lt−1(i) + Ht(i) 1 − S

Ht−1(i)

− Lt(i)

Ht(i)

+ Qt (1 − δ)Kt−1

(i) + Invt(i) 1 − S Invt−t 1(i) − Kt(i)

Inv (i)

Substitute out total cost:

V (...) =	Pt(i)	Pt(i)		−ϕ	Yt − PtInvt(i) − (1 + τtSSC )WtLt(i) −
Pt
−	Pt(i)YtR		Pt(i)		− HCtHt(i) + Et(1 − ωtY )	V (...)
Pt−1(i)	1 + it	Pt(i)
		h 1				θ−1	1	θ−1i	θ
					θ−1

−ϕ

!

+ MCt α θ Kt−1(i) θ + (1 − α) θ (AtLt(i)) θ	−
Pt

Yt +

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52

+	ΛtH (1 − prtF )Lt−1(i) + Ht(i) 1 − S	Ht−1(i)	− Lt(i)
					Ht(i)
+	Qt (1 − δ)Kt−1(i) + Invt(i) 1 − S	Invt−t 1(i)	− Kt(i)
				Inv (i)

The rst-order conditions are as follows:

∂Vt

Pt(i)

−ϕ

Pt(i)

−ϕ−1 1

Pt(i)

=

Yt − ϕPt(i)

Yt − YtR

−

∂Pt(i)

Pt

Pt

Pt

Pt−1(i)

− Pt(i)YtR0

Pt−t 1(i) Pt−1(i) (1 + πt−1)ϑ +

P (i)

1

VPt(i)

Pt(i)

−ϕ−1

1

+ Et(1 −

ωtY )

+ MCtϕ

Yt = 0

1 + it

Pt

Pt

∂Vt

=

E

1 − ωtY V

Kt(i) −

Q

t = 0

∂Kt(i)

t

1 + it

1 −

−

= 0

∂Vt

HC

E

1 − ωtY

V

H

S

Ht(i)

S0

Ht(i)

1 Ht(i)

∂Ht(i)

= −

+

+ Λt

1 + gt Ht−1(i)

t

t 1 + it

Ht(i)

Ht−1(i)

Ht−1(i)

∂Vt

Yt(i)

1

Y

=

(1 + τSSC )Wt + MCt(1

α)

θ

Θ

1At

ΛH + Et

1 − ωt VL

i = 0

−

−

1

− t

t( )

∂Lt(i)

t

1 − ωtY V

(AtLt(i)) θ

1 + it

∂Vt

P

E

Q

S

S0

1

Invt(i)

=

−

t +

t

Invt(i) +

t

1 −

(·) −

(·)

∂Invt(i)

1 + it

1 + gt

Invt−1(i)

Using the Envelope-theorem we get:

VPt−1(i)

VLt−1(i)

VHt−1(i)

VKt−1(i)

VInvt−1(i)

P (i)

P (i)

= Pt(i)YtR0

t

t

Pt 1(i)

Pt

−

1(i)2 (1 + πt

−

1)ϑ

=

(1 − prtF )ΛtH

−

=

Ht(i)

1

Ht(i)

2

Ht−1(i)

1 + gtN

Ht−1(i)

ΛtH S0

= αθ MC

Yt(i)

1

1+ Q (1 δ)

1

θ

−

t

t

K

t−1

(i)θ

2

=

QtS0(·)

1

Inv (i)

t

1 + gt

Invt

1(i)

−

Stepping one period ahead gives us:
VPt(i) = Pt+1(i)Yt+1R0	Pt+1(i)		Pt+1(i)
Pt(i)		Pt(i)2 (1 + πt)ϑ

ECB Working Paper Series No 2396 / April 2020

53

VLt(i)

VHt(i)

VKt(i)

VInvt(i)

=

(1 − prtF+1)ΛtH+1

2

= ΛH S0

Ht+1(i)

1

Ht+1(i)

Ht(i)

1 + gtN+1

Ht(i)

t+1

= αθ MC

Yt+1(i)1

1

+ Q (1 δ)

1

t+1

θ

t+1 −

Kt(i)θ

=

Qt+1S0

(·)

1

Invt+1(i)

2

1 + gt+1

Invt(i)

As a simplication, we can introduce a new variable for the marginal product of capital which

modies the no-arbitrage condition:

rtK = αθ mct		Yt(i)	1
	1
			1			θ
			1 − ωtY	Kt−1(i)θ					)
q		=		rK	+	q	t+1(1	−	δ
	1 + rt
	t		t+1

The eective real wage can be given as: labor demand can be expressed as:

1 − ωY

w¯t = (1 + τtSSC )wt + λHt − Et 1 + it (1 − prtF+1)λHt+1 t

And the labor demand has the following form:

			1
		1	Yt(i)
		Θ
w¯t = mct(1	−	α)θ				At
		1
		(AtLt(i))θ

If we substitute out the capital and labor from the production function we can get marginal cost

function:

mct	=	"αrtK	−		+ (1 − α)	At	1
	#
		1		θ			w¯t		1−θ	1−θ

The rms' hiring decision can be given as:

hct =

Et

1 − ωtY

λH

1

S0

Ht+1(i)

Ht+1(i)

2

+

Ht(i)

+

1 + rt t+1 1 + gtN+1

Ht(i)

(i)!

λtH 1 − S Ht 1

(i)

− S0

Ht 1

(i)

1 + gN

Ht 1

Ht(i)

Ht(i)

1

Ht(i)

−

−

t

−

ECB Working Paper Series No 2396 / April 2020

54

The investment decision is as follows:

E

1 − ωtY

qt+1 S

0(·)

Invt+1(i)

2

q

1

S

S

1 Invt(i)

1 =

t

+

t

−

(·) −

0(·)

1 + rt

1 + gt+1

Invt(i)

1 + gt

Invt−1(i)

The price-setting curve can be expressed as:

• Pt(i)−ϕ Yt − ϕPt(i) Pt(i)−ϕ−1 1 Yt − YtR Pt(i) −
  PtPt PtPt−1(i)

P (i)

1

−

Pt(i)YtR0

t

+

Pt 1(i)

Pt

−

1

(i) (1 + πt

−

1)ϑ

(1 − ωtY )P

−

−ϕ−1 1 Y

E

i Y

R

Pt+1(i)

Pt+1(i)

MC

ϕ

Pt(i)

+

t

t+1( )

t+1

0

+

t

t = 0

1 + it

Pt(i)

Pt(i)2 (1 + πt−1)ϑ

Pt

Pt

We can assume that all rms follows the same price setting behaviour, it means the individual

and the aggregate prices are the identical, and we can introduce the ination 1 + πt =

Pt

Pt−1

1

1 + π

1

1 + π

1 + π

1 +

R

t

+

R0

t

t

−

ϕ

−

1

(1 + πt 1)γ

ϕ

−

1

(1 + πt

1)γ

(1 + πt 1)γ

−

−

(1+πt)γ

−

=

ϕ mct

Et 1 − ωtY Yt+1 R0

(1+πt)γ

1+πt+1

1+πt+1

−

ϕ − 1

Yt

1 + rt

ϕ − 1

Retailers

We can assume that in short run the households can not adjust their consumptions immediately. There is a retailer sector that sets the cohort speci c relative prices for the consumption baskets in a way to smooth out the short run cyclical adjustment of the households consumption.

∞ n	1 − ωtj+k	1	j j	j		C,j Ctj+n
X Y	1 + rt+k	−1	(pt+nCt+n − Ct+n	1 + G					!!)
	Cj	−
	−					t+n	1
n=0 k=1

Taking the rst-order condition of consumption:

t

= 1 +

Ctj−1 !

+

Ctj−1 !

1 + gt Ctj−1 −

1 + rt

Ctj

! 1 + gt+1

Ctj

!

2

pj

GC,j

Ctj

GC,j0

Ctj

1

Ctj

1 − ωtj

GC,j

0

Ctj+1

1

Ctj+1

ECB Working Paper Series No 2396 / April 2020

55

where the adjustment cost is a usual convex function:

Cj

φC,j

1 Cj

2

! =

− 1!

GC,j

t

t

Cj

2

1 + gt

Cj

t−1

t−1

Wage bargaining

The value function of the workers is as follows:

1

h(1

− prtF+1 + prtF+1prtH+1)Vtw+1 + prtF+1(1 − prtH+1)VtU+1i

Vtw = (1 − τtL)wt + Et

1 + rt

The value function of the unemployed is:

VtU = wtU + Et

1

h(1 − prtH+1)VtU+1 + prtH+1Vtw+1i

1 + rt

Wage-bargaining is as follows:

max(Vtw − VtU )σhct1−σ wt

The rst-order conditions are:

σ(Vtw − VtU )σ−1hct1−σ(1 − τtL) − (1 − σ)(Vtw − VtU )σhct−σ(1 + τtSSC ) = 0

σ(Vtw − VtU )−1hct(1 − τtL) = (1 − σ)(1 + τtSSC )
	σ	hc	1 − τtL	V w		V U
	1 − σ	t 1 + τtSSC =	−
		t	t
We can express the dierences between the two value functions:
Vtw − VtY,U = (1 − τtL)wt	− wtU + Et 1 + rt "(1 − prtF+1 + prtF+1prtH+1)Vtw+1 +
						1
+prtF+1(1 − prtH+1)VtY,U+1 # − Et 1 + rt "(1 − prtH+1)VtU+1 + prtH+1Vtw+1#	=
						1

"#

= (1 − τL)wt − wU + Et 1 (1 − prF,F )(1 − prH ) V w − V U

t t 1 + rt t+1 t+1 t+1 t+1

ECB Working Paper Series No 2396 / April 2020

56

Finally, we can plug in the rst-order conditions:

σ

hc

1 − τtL

τL

w

wU

1 − σ

t 1 + τtSSC =

(1 −

t −

+

t )

t

1 − σ hct+1

1 + τtSSC+1

!

+

Et 1 + rt

(1 − prt+1)(1 − prt+1)

1

F

H

σ

1 − τtL+1

A.1.4: Monetary policy

The behaviour of the central bank can be discribed by the Taylor-type monetary policy rule:

1 + it = (1 + it−1)ρi (1 + rtn) (1 + πt)φπ 1−ρi eεit

where ρi is interest smoothing, φπ denotes the inationary reaction, and εit assigns the monetary policy shock. Once the ination is reaches the target again, the central should set the interest rate at its exible price equilibrium level.

The real interest rate is de ned by the Fisher-identity:

1 + it = Et(1 + rt)(1 + πt+1)

A.1.5: Normalized model equations

Each variable must be detrended: individual variables are normalized by population (Nt) because there is only population growth in the model. This section lists all the nal equations of the model: detrended variables are denoted by x˜t.

Demography:

s =		(1 − ωtO−1)	s	+	ωtY−1
t			(1 − ωtY−1 + nt) t−1		(1	− ωtY−1 + nt)
sY	=		1
	1 + st
t
1 + gtY	= 1 − ωtY−1 + nt	ωY
1 + gtO	= (1 − ωtO−1) +	st−1
					t−1		1 + st
1 + gt	=	1 + gtN = (1 + gtN,Y )
1 + st−1

ECB Working Paper Series No 2396 / April 2020

57

Households:

˜O Ct

1

MP CtO

ΩOt

˜Y Ct

˜ Y

Inct

1

MP CtY

ΛOt

ΛYt

ΛYt O

˜Y Bt

O ˜

O

O 1 + rt−1

Y ˜Y

˜O

=

MP C

t

T R

tΩt

1

t

1

t−1 t−1 +1 t−1

1 + gt

+ MP C

ω B

B

=

(1 + τtC )ptO + β γ (1 − ωtO)(1 + rt) γ −1ΛtOEt

MP CO

t+1

= 1 + Et

1 − ωtO

ΩO

1 + rt

t+1

Y 1 + rt−1

Y ˜

Y

Y

˜Y

=

MPCt

Inct

+ MP Ct

(1

− ωt−1)Bt−1

1 + gt

τL

w

L˜

wU U˜

P rof˜ it

LT˜ax

E

g

T˜RtY+1O

O

E

1 − ωtY

1 + st+1

Inc˜ Y

= (1 −

+

−

+

t(1 +

t+1) 1 + rt

Ωt+1

+

t 1 + rt

t

) ˜t

t

+ ˜t

t

t

t

1 + st t+1

t

t

t

t

− t

t MP CtY+1

t t

MP CtO+1

1

1

1

1

= (1 + τC )pY + β γ (1 + r ) γ −1E (1 ωY )ΛY

+ ωY ΛY O

(1 + τtC )ptO

1

γ

= Et

(1 + τtC+1)ptO+1

(1 + τtC )ptY

1

γ

= Et

(1 + τtC+1)ptY+1

(1 + τtC )ptY

1

γ

= Et

(1 + τtC+1)ptO+1

(1 + rt−1)(1 − ωtY−1)

L

˜

U ˜

˜

˜

˜Y

C

Y ˜Y

=

(1 −

τt

)w˜tLt

+ w˜t

Ut

+ P rofitt

− LT axt

+

Bt−1 − (1 + τt )pt Ct

1 + gt

Labor market:

σ	hc˜ t	1 − τtL
1 − σ	1 + τtSSC

˜

Ut

˜

hct

prtH

Producing rms:

= (1 − τtL)w˜t − w˜tU +

		1 + gtA+1		F		H
+	Et			(1	− prt+1)(1	− prt+1)
1 + rt
=	Y	˜
st	− Lt
=	κprtH αHC		˜
= (1 + gtN )			Ht
˜		F ˜
			Ut−1	+ prt	Lt−1

!

• 1 − τL
• • t+1 1 − σ hct+1 1 + τtSSC+1

• ˜

1 + gt Kt−1 qt

˜

w¯t

=

α

rK

−θ

mct

Y˜t

t

1

− ωtY

rK

q

δ

) Y

=

1 + rt

t+1

+

t+1(1 −

=

(1 +

τSSC w

λ˜H

−

E

1 − ωt

(1 −

prF

gA

λ˜H

1 + rt

t

) ˜t +

t

t

t+1)(1 +

t+1)

t+1

ECB Working Paper Series No 2396 / April 2020

58

˜

Lt

mct

˜

hct

= (1 − α)

˜

−θ

˜

A˜tmct

A˜t

w¯t

Yt

=

"αrtK − + (1 − α) A˜t

1

1−θ

#

1

˜

1−θ

θ

w¯t

=

t(1 +

Y

t+1

˜

˜

2

t+1) 1 + rt

H˜t

H˜t

+

E

g

1

− ωt

λ˜H

S0

Ht+1

Ht+1

+ λ˜tH 1 − S

˜

− S0

˜

˜

H˜t−1

H˜t−1 H˜t−1

Ht

Ht

Ht

1

=

t

(1 +

Y

t+1

˜

2

t 1 −

(·) −

˜

t+1) 1 + rt

(·) Inv˜ t

+

(·) Inv˜ t−1

E

g

1

− ωt

q

S0

Invt+1

q

S

S0

Invt

ϕ − 1 mct

= 1 + ϕ − 1 R

(1 + πt−t1)γ + ϕ − 1 R0 (1 + πt−t1)γ (1 + πt−t1)γ −

ϕ

1

1 + π

1

1 + π

1 + π

Et

1 − ωtY (1 + gt+1) Y˜t˜+1 R0

(1+πt)γ

(1+πt)γ

1+πt+1

1+πt+1

−

ϕ − 1

Yt

˜

1 + rt

L˜t

=

F

L˜t−1 + H˜t 1 − S

1

11−+ gtt

H˜t−t

pr

H

t

=

1 + gt

t−1

+

t

1 −

˜

t−1

Inv˜

K˜

1 − δ

K˜

Inv˜

S

Invt

− R (1 + πt−t1)γ Y˜t − hc˜ tH˜t

Prof˜ itt

= Y˜t − Inv˜ t − (1 + τtSSC )w˜tL˜t

1 + π

Retailers:

t

= 1 +

˜Y

!

+

˜Y

˜Y

Y

˜Y

!

(1 +

t+1)

˜Y

!

2

C˜Y

C˜Y

! C˜Y −

1 + rt

C˜Y

C˜Y

pY

GC,Y

Ct

GC,Y 0

Ct

Ct

1 −

ωt

GC,Y 0

Ct+1

g

Ct+1

t−1

!

t−1

t−1

t

!

t

!

2

t

= 1 +

˜O

+

˜O

˜O

Y

˜O

(1 +

t+1)

˜O

C˜O

C˜O

! C˜O −

1 + rt

C˜Y

C˜O

pO

GC,O

Ct

GC,O0

Ct

Ct

1 −

ωt

GC,O0

Ct+1

g

Ct+1

t−1

t−1

t−1

t

t

Fiscal policy and the pension system:

T˜RY O

= ν

ωtY−1

w˜

t−1

L˜

t

t 1 + gt

t−1

T˜R

= T˜RY O

+

1 − ωtO−1

T˜R

t−1

t

t

1 + gt

˜

˜

˜

˜

˜

1 + rt−1

˜

Debtt + LT axt + T axt

= Govt + T Rt +

1 + gt

Debtt−1

˜

˜

C

O

˜O

Y ˜Y

SSC

L

T axt

= τt

(pt

Ct

+ pt

Ct

) + (τt

+ τt )w˜tLt

ECB Working Paper Series No 2396 / April 2020

59

LT˜axt

˜

Debtt

• • ˜ U ˜
• Govt + T Rt + w˜t Ut
• • Y ˜O
• Bt + Bt

	+ r			t−1 − T˜axt −	Debt		T arget
+	11 + tg−t	1	Debt˜			Y˜t
Y

Monetary policy:

• + it = (1 + it−1)ρi (1 + rtn) (1 + πt)φπ 1−ρi eεit
• + it = Et(1 + rt)(1 + πt+1)

Market clearing:

Y˜t = C˜tY + C˜tO + Inv˜ t + Gov˜	t + R	1 + π	t	Y˜t + hc˜ tH˜t
(1 + πt		1)γ
			−

A.1.6: Steady state of the model

To be able to calculate the steady-state solution we need to specify initial guess for r and we calibrate the hiring cost to gross wage ratio. Then the rest of the variables and equations can be solved numerically. As a function of the initial guess, we can determine the variables of production, labor market and those of the government and pension system. Finally, we turn to the consumption and savings functions. At the end, using the market clearing equations we can check whether our initial guesses are correct.

First, the demographic equations are:

s

sY

• + gN,O

1 + g

=			ωY			1 −	(1 − ωO)	!	−1
	(1 − ωY + n)	(1 − ωY + n)
=			1
	1 + s
=	1	− ωO +	ωY
s
=	1	+ gN = 1 + gN,Y = 1 − ωY + n

Then, we need to guess an initial value for r which is veried by the Newton-Raphson algorithm. Assuming π = 0 in the steady state implies:

i = r

ECB Working Paper Series No 2396 / April 2020

60

The Tobin-Q (q) is one in the steady-state equilibrium, so the initial assumption for r and the no-arbitrage condition imply the steady-state value of the marginal product of capital:

rK =	1 + r	− 1 + δ
1 − ωY

The rms' supply curve in the steady state gives us the marginal cost as the inverse of the markup:

mc =	ϕ − 1
ϕ

Based on the marginal cost function we can calculate the real wage:

1

˜

˜ (mc1−θ − αrK 1−θ )1−θ

w¯ = A

1 − α

We can calculate the capital and labor per production ratios from the input demand functions of the rms:

˜

K

˜

Y

˜

L

˜

Y

˜	also implies	˜	:
K˜	Inv˜
Y		Y

=	(1 + g)α		rK	!	−θ
mc
=	(1 − α)		˜		−θ	A˜
Amc˜	!
			w¯			1

˜		˜	1 −	1 − δ
Inv	=	K
Y˜	Y˜	1 + g

˜	also implies	˜	:
L˜	H˜
Y		Y

Y˜	=	Y˜	1 −	11−+ g	!
˜		˜		pr	F
H		L

In the steady state the wage setting equations imply the real wage of the households:

˜H	˜
λ	= hc

ECB Working Paper Series No 2396 / April 2020

61

And we can use our assumption about the hiring cost gross wage ratio:

˜

W R = hc w˜

We can express the real wage from the rm equation:

w˜

τSSC w

hc˜

1 − ωY

prF

gA

hc˜

= (1 +

−

1 + r (1 −

)(1 +

¯

t

) ˜ +

)

˜

w˜

=

w¯

(1 + τSSC ) + W R

−

1−ωY

(1

−

prF )(1 + gA)WR

t

1+r

The unemployment benet is exogenous, can be also calibrated in terms of real wage:

w˜U

W UR =

w˜

These assumption can be used for the wage bargaining equation and we can express the prH :

		σ	hc˜		1 − τtL
	1 − σ		+ τtSSC
	1
σ	W Rw˜		1 − τtL
		SSC
1 − σ	1
+ τt

prH

= (1 − τtL)w˜ − w˜U +

1 − σ hc˜

1 +−τSSC !

+

1 + r

(1 − prF )(1 − prH )

1 + gA

σ

1

τL

= (1 − τtL)w˜ − W URw˜ +

1 − σ W Rw˜ 1 +−τSSC !

+

1 + r

(1 − prF )(1 − prH )

1 + gA

σ

1

τL

σ

1−τtL

L

W Rw˜

− (1 − τt )w˜ + W URw˜

=

1

1−σ

1+τtSSC

−

1+r (1 − prF )

1−σ W Rw˜ 1+−τSSC

1+gA

σ

1

τL

Based on prH we can calculate κ that consistent with our wage cost ratio:

							˜
		κ	=				hc
		prH αHC
And we can express the unemployment-to-GDP ratio:
	˜		1 + g	N		˜			˜
	U	=			H	− prF	L
	Y˜	prH		Y˜		Y˜

ECB Working Paper Series No 2396 / April 2020

62

We can also add the prot-to-GDP ratio:

˜		˜	SSC		˜	˜	˜
P rofit		Inv		L	H
		= 1 −		− (1 + τt	)w˜		− hc
Y˜	Y˜	Y˜	Y˜

We can use the unemployment-to-GDP ratio to express the level of GDP from the unemployment denition since we now the labor-to-GDP ratio and the share of young population in the steady state:

• sY
  Y =
• • ˜
    U˜ + L˜
    Y Y

Based on the GDP identity we can express the total consumption:

	˜		˜		˜		˜
	C	= 1 −	Inv		Gov	˜	H
	Y˜	Y˜	−	Y˜	− hc	Y˜

The retailers relative prices are one in the steady state, since we know all tax bases we can also express the sum of distortionary tax revenue of the government:

˜		˜		˜
T ax	= τtC	C	+ (τSSC + τL)w˜	L
˜		˜	˜
Y		Y		Y

Using the assumption of replacement ratio and labor market variables we can calculate the steady-state pension expenditures. Using the pension expenditures, the assumptions for public debt-to-GDP ratio, and government expenditure to GDP ratios we can calculate the equilibrium level of the tax burden:

˜ Y O				ω	Y		˜
T R	=	ν		w˜	L
˜	1 + g	˜
Y					Y	1 − ωO	−1
T˜R		T˜RY O
		=						1 −		!
	Y˜			Y˜			1 + g

The level of the public debt-to-GDP ratio is given, based on the expenditures and distortionary tax revenues we can calculate the lump-sumtax-to-GDP ratio:

˜

˜

˜

˜

+

1 + r

− 1

˜

˜

LT ax

=

Gov

+

T R

+ w˜U

U

Debt

−

T ax

Y˜

Y˜

Y˜

Y˜

1 + g

Y˜

Y˜

ΛY , ΛO, and ΛY O are one in the

steady-state, we can use them to calculate the MP CO and

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63

MP CY :
MP CO =		1 − (1 − ωO)(1 + r)γ1−1β γ1	ΛO
			(1 + τC )pY + β γ				−	1
MP CY =	1 − β γ	(1 + τC )pO			(1 + r)γ	−1ωY ΛY O MP CO
(1 + r)γ −1(1 − ωY )ΛY
	1	1			1	1		1

The pensioners' discount factor in the steady state is the following:

ΩO = 1 − 1 − ωO !−1 1 + r

We can express the young households' expected lifetime-income-to-GDP ratio by using the initial

˜

guess of Y :

˜ Y

= 1 −

Y

!

−1

˜

+

˜

−

˜

+ (1 + g)

˜ Y O

O

!

Y˜

11−+ r

(1 − τL)w˜ Y˜

Y˜

Y˜

Y˜

1 + r

Inc

ω

L

P rofit

LT ax

TR

Ω

Based on the young consumption function one can substitute out the young consumption-to- GDP ratio in the budget constraint, and express the young bond-to-GDP ratio:

˜Y

L

˜

˜

˜

C

C

˜

Y

L

P rofit

LT ax

Y Inc

B

=

(1 − τ

)w˜ Y˜

+

Y˜

−

Y˜

− (1 + τ

)p

MP C

Y˜

˜

C

Y

1 + (MP C

Y

(1 + τ

)p

C

−

1)

1+r

(1 − ω

)

Y

1+g

Now, we can express the old households' bond-to-GDP ratio from the bond market equilibrium:

˜O		˜		˜Y
B	=	Debt	−	B
Y˜	Y˜		Y˜

And based on the consumption functions we can calculate the consumption-to-GDP ratios:

Y˜

= MP CO Y˜

ΩO + MP CO 1 + g "ωY

Y˜

+ Y˜

#

˜O

˜

1 + r

˜Y

˜O

C

TR

B

B

˜Y

˜

Y

1 + r

˜Y

C

Inc

B

= MP CY

+ MP CY

(1

− ωY )

Y˜

Y˜

1 + g

Y˜

Finally we need to check if the initial assumptions for r is correct. It means that we need to check if the total consumption is equal with the sum of the young and retired consumption.

ECB Working Paper Series No 2396 / April 2020

64

Otherwise the algorithm should choose another initial value until the condition is satised.

˜	?	˜Y		˜O
C	C		C
	=		+
˜	˜	˜
Y		Y		Y

If we have the right initial values, we can calcuate the levels of all the normalized variables, and run the simulations.

ECB Working Paper Series No 2396 / April 2020

65

Annex 2 The Ageing Report's accounting framework39 and the

calibration results with OGRE

pension expenditure

=

population(65+)

×

retirees

×

average pension income

×

population(20 − 64)

GDP

population

(20

− 64)

population

(65+)

GDP

hours worked

hours worked

• (dependency ratio) × (coverage ratio) × (benet ratio) × (labour market eect)

Changes in pension expenditures can be explained by the following factors:

• Dependency ratio eect: quanti es the impact of changes in the old-age dependcy ratio on pension expenditure;
• Coverage ratio eect: looks at the number of pensioners relative to the population older than 64 years. The ratio can capture how developments of the e ective exit age and the share of the population covered by the pension system inuence pension expenditure;
• Benet ratio eect: indicates how average public pension spending develops relative to the average wage. The ratio assesses how changes to the legal framework of pension systems (concerning pension calculations and indexation rules) a ect pension expenditure; and
• Labour market eect: describes the e ect labour market behaviour/reforms have on pension expenditure.

Table 8: Comparison of the driving factors of pension expenditures

in OGRE and the Ageing Report

in % of GDP, unless stated otherwise

	Germany	Slovakia
	OGRE	AR1	OGRE	AR1
Dependency Ratio Eect	4.1	5.1	5.2	6.9
Coverage Ratio Eect	-0.4	-1.0	-1.0	-3.2
Benet Ratio Eect	-1.6	-1.9	-1.5	-1.2
Labor Market Eect	-0.2	-0.2	-1.5	-0.9
Residual	0.0	-0.2	0.0	-0.6
Total	1.9	1.8	1.2	1.0

1The Ageing Report's pension expenditures are rescaled to earnings-related pension expenditures.

39For further information see European Commission (2018).

ECB Working Paper Series No 2396 / April 2020

66

Annex 3 Calibration of the parameters

Table 9: Structural parameters

Name	Sign	Values
Germany	Slovakia
Discount factor	β	0.999
Physical capital depreciation rate	δ	0.1
Physical capital income	α	0.385	0.32
Elasticity of production technology	θ	0.8	0.95
Price markup	µ	1.2
Gross pension-wage replacement rate	ν	0.43	0.52
Ratio of hiring cost to wage	WR	0.1675	0.575
Bargaining power	σ	0.75
Elasticity of hiring cost	αHC	0.5
Technology growth rate (%)	gA	1.5	3
Inverse of intertemporal substitution	γ	1
Adjustment cost of physical capital investment	φInv	2.5
Consumption habit	φC	4
Rotember price adjustment cost	φP	80
Price indexation	γP	0.5
Interest rate smoothing	ρi	0.5
Reaction to ination in Taylor rule	φπ	2.5

ECB Working Paper Series No 2396 / April 2020

67

2020 April / 2396 No Series Paper Working ECB

Table 10: Detailed calibration table

			Germany				Slovakia
	calibrated	targeted		source		calibrated	targeted	source
	value	value				value	value
Consumption (total)	60.7	54.6		Eurostat: National		60.8	60.4	Eurostat: National
Consumption (public)	19.1	19.1		Accounts, 2011-2016		16.9	16.9	Accounts, 2011-2016
Investment	20.1	20.0				22.3	20.2
Compensation of employees	59.7	56.3				54.9	46.0
Unemployment rate (%)	5.7	5.6		Eurostat:		12.8	12.7	Eurostat:
Unemployment tbene expenditure	1.5	1.8	2009-2016		0.7	0.7	2011-2016
Firing probability (%)	13.9	13.9		Eurostat: 2001-2015		11.1	11.1	Eurostat: 2001-2015
Pension expenditure*	7.2	7.8		Ageing Report (2018): 2016		6.9	6.7	Ageing Report (2018): 2016
tBene ratio (%)*	34.1	32.4			34.4	36.3
Old-age dependency ratio** (%)	37.2	38.3				36.5	32.6
Fertility rate** (%)	1.9	1.9		Eurostat:		2.6	2.6	Eurostat:
Probability of death** (%)	4.2	4.2		2006-2016		5.3	5.3	2002-2016
Probability of retirement** (%)	1.7	1.7				2.1	2.1
Value Added Tax Revenue	7.0	7.0				6.5	6.5
Labor Income Tax Revenue	8.7	8.7		Eurostat: 1995-2016		2.9	2.9	Eurostat: 2009-2016
Social Sec. Contr. Revenue: Employer	6.5	6.5			7.3	7.3
Social Sec. Contr. Revenue: Employee	6.2	6.2				3.0	3.0
Public debt	75.4	75.4		Eurostat: 2009-2016		53.3	53.3	Eurostat: 2009-2016

*Earnings related pension expenditures

**Implied from population data and ectivee retirement rate

68

Annex 4 Baseline scenario - Germany

(deviation from initial steady-state)

GDP per capita

Share of Young Consumption

(percentage)

(percentage of total consumption)

5

4

0

2

0

-5

-2

-10

-4

-15

-6

-8

-20

2026Y

2036Y

2046Y

2056Y

2066Y

-10

2026Y

2036Y

2046Y

2056Y

2066Y

2016Y

2016Y

Contribution Rate

Benefit Ratio

Retirement Age

Aging

Total

Contribution Rate

Benefit Ratio

Retirement Age

Aging

Total

Unemployment Rate

Public Debt

(percentage)

(percentage of GDP)

1.5

250

1

200

150

0.5

100

0

50

-0.5

0

-50

-1

-100

-1.5

-150

2016Y

2026Y

2036Y

2046Y

2056Y

2066Y

2016Y

2026Y

2036Y

2046Y

2056Y

2066Y

Contribution Rate

Benefit Ratio

Retirement Age

Aging

Total

Contribution Rate

Benefit Ratio

Retirement Age

Aging

Total

Annex 5 Baseline scenario - Slovakia

(deviation from initial steady-state)

		GDP per capita					Share of Young Consumption
		(percentage)						(percentage of total consumption)
15							8
10							6
						4
5
						2
0							0
						-2
-5
						-4
-10							-6
-15							-8
						-10
-20							-12
2016Y	2026Y	2036Y	2046Y	2056Y		2066Y	2016Y	2026Y	2036Y	2046Y	2056Y		2066Y
Contribution Rate	Benefit Ratio	Retirement Age		Aging	Total	Contribution Rate	Benefit Ratio	Retirement Age		Aging	Total
		Unemployment Rate						Public debt
		(percentage)						(percentage of GDP)
1.5							300
1							250
0.5							200
0							150
-0.5							100
-1							50
-1.5							0
-2							-50
-2.5							-100
-3	2026Y	2036Y	2046Y	2056Y		2066Y	-150	2026	2036	2046	2056		2066
2016Y		2016
Contribution Rate	Benefit Ratio	Retirement Age		Aging	Total	Contribution Rate	Benefit Ratio	Retirement Age		Aging	Total

Acknowledgements

The views expressed here are those of the authors and do not necessarily represent the views of the European Central Bank, the Eurosystem or the International Monetary Fund, its Executive Board or IMF Management. We thank, in alphabetical order, Ruo Chen, Johannes Clemens, Werner Ebert, Ahmed El Ashram, Csaba Feher, Christophe Kamps, Daehaeng Kim, Nir Klein, Zuzana Mucka, Ludovit Odor, Jenni Paakkonen, Alex Pienkowski, Andre Santos, Andrea Schaechter, an anonymous referee, and participants at the 2019 International Institute for Public Finance conference, the 2019 European Economic Association conference, an IMF seminar and ECB Fiscal Policy Forum for their valuable comments.

Daniel Baksa

International Monetary Fund, Washington, D.C., United States; email: [e-mail]

Zsuzsa Munkacsi

International Monetary Fund, Washington, D.C., United States; email: [e-mail]

Carolin Nerlich

European Central Bank, Frankfurt am Main, Germany; email: [e-mail]

© European Central Bank, 2020

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Telephone [phone]

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This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library or from RePEc: Research Papers in Economics. Information on all of the papers published in the ECB Working Paper Series can be found on the ECB's website.

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ISBN 978-92-899-4039-9

ISSN 1725-2806

doi:10.2866/475743

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