## wpiea2020132-print-pdf

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---

### Demographic challenge and macro-fiscal implications
- Old-age dependency ratio (OADR) definition: number of people aged 65 or older as a percentage of the working-age population (people aged 15 to 64).
- Eurostat projection (euro area): OADR from 30% today to 52% in 2070, with two thirds of increase concentrated in the next two decades.
- By 2070: OADR highest in Portugal and Greece, lowest in Ireland and France.
- Life expectancy and fertility:
  - Remaining life expectancy at age 65 projected to average 23.6 years for men and 26.9 years for women by 2070 (around 5 years more than today).
  - Fertility rates well below natural recovery level and expected to remain low or decline in some countries.
  - Net inward migration will only partly mitigate ageing and is likely to fluctuate.
- Macroeconomic channels:
  - Shrinking labour force → lower growth rates of consumption, investment and GDP.
  - Life-cycle saving: workers accumulate, retirees dissave; rising life expectancy may raise precautionary savings.
  - Capital-labour ratio may increase → downward pressure on equilibrium real interest rate and dampened investment growth.
- Fiscal channels:
  - Public pension spending > half of total public expenditures in Europe today.
  - Ageing Report pension spending (euro area public pensions): 12.3 percentage points of GDP in 2016; 13.5 percentage points of GDP in 2040; 11.9 percentage points of GDP in 2070.
  - Workers per pensioner projected to shrink from three to two by 2070.
  - Ageing can compress tax bases and raise public debt-to-GDP ratios and long-term fiscal risks.

### The framework and the OGRE model
- Purpose:
  - Evaluate macroeconomic and fiscal impact of sudden, unanticipated reversals of pension reforms.
  - Replicate Ageing Report pension-expenditure projections and incorporate behavioural feedback via a dynamic general equilibrium OLG model.
- Model basics:
  - Name: OGRE (Overlapping Generations and Retirement).
  - Type: Gertler-type (Blanchard-Yaari-Weil-type) dynamic general equilibrium overlapping-generations model with demography, unemployment and wage bargaining.
  - Households: two generations (workers aged 20 to retirement age; retirees).
  - Labour market: hiring costs, wage bargaining; unemployment and wage bargaining modeled (Blanchard and Gali (2010) basis).
  - Fiscal sector: includes personal income tax, social security contributions, VAT, pension benefits, unemployment benefits, government consumption; PAYG pension system used in this paper.
  - Monetary policy: Taylor rule; closed-economy version used for analysis in the paper.
- Link to Ageing Report:
  - Baseline uses 2018 Ageing Report projections to capture already adopted but not fully implemented pension reforms.
  - OGRE introduces behavioural general-equilibrium feedbacks missing in the Ageing Report accounting framework.
- Calibration and country focus:
  - Applied to Germany and Slovakia as illustrative cases.
  - Calibrated to replicate Ageing Report pension-expenditure projections.

### Policy scenarios, reform reversals and scope
- Scenarios analyzed:
  - Baseline: Ageing Report projections integrated into OGRE.
  - Additional reforms: calibrated to stabilize public debt-to-GDP at initial 2016 level.
  - Reform reversals: mimic recent policy decisions (Germany cap on contribution rate and benefit ratio until 2040; Slovakia cap retirement age at 64 reached in 2045).
- Reform instruments considered:
  - Increase in retirement age.
  - Decline in benefit ratio.
  - Increase in contribution rate.
  - Combined reform mixes.
- Scope boundaries:
  - Drivers of reform reversals (political economy) not analyzed.
  - Switches to fully-funded regimes excluded.

### Key findings and contributions (summary)
- OGRE replicates Ageing Report projections for Germany and Slovakia closely.
- Population ageing has major macroeconomic and fiscal adverse implications if unaddressed.
- Accounting for already adopted pension reforms is important; omitting them overstates adverse ageing consequences.
- Increases in retirement age are most effective in alleviating ageing pressures.
- Reform packages that combine measures distribute adjustment burdens more equally across generations.
- Reform reversals are potentially very costly: they raise pension expenditure and public debt-to-GDP ratios and exacerbate adverse macroeconomic impacts; compensating reversals requires large alternative adjustments and may shift burdens across generations.
- Contribution: integrates Ageing Report pension-expenditure dynamics into an OLG general-equilibrium model to quantify macroeconomic and fiscal costs of reform reversals.

### Baseline scenario — headline quantitative results (until 2070)
- Common baseline outcome (no compensatory measures): public debt-to-GDP projected to increase by around 100 percentage points until 2070 compared to initial period for both Germany and Slovakia.
- Real GDP per capita projected deviations (until 2070):
  - Germany: decline by almost 14%.
  - Slovakia: decline by 9%.

### Germany — baseline long-run deviations (until 2070) and decomposition (values in % or percentage points as reported)
- GDP per capita (%): -13.9 (Ageing: -15.3; retirement age: 2.5; benefit ratio: 0.1; contribution rate: -1.2)
- Consumption: total: -1.5 (Ageing: -1.6; retirement age: 0.4; benefit ratio: 0.0; contribution rate: -0.2)
- Consumption: share of young (% of total): -4.6 (Ageing: -7.8; retirement age: 1.1; benefit ratio: 1.7; contribution rate: 0.4)
- Investment: -1.3 (Ageing: -1.4; retirement age: 0.1; benefit ratio: 0.0; contribution rate: 0.0)
- Unemployment rate (%): 0.8 (Ageing: -0.3; retirement age: 0.0; benefit ratio: 0.0; contribution rate: 1.1)
- Pension expenditure: 1.9 (Ageing: 3.8; retirement age: -0.6; benefit ratio: -1.3; contribution rate: 0.0)
- Primary balance: -2.9 (Ageing: -6.5; retirement age: 1.1; benefit ratio: 1.1; contribution rate: 1.4)
- Public debt: 100.3 (Ageing: 208.6; retirement age: -33.9; benefit ratio: -29.4; contribution rate: -44.9)
- Drivers and interpretation:
  - Decline in GDP per capita largely driven by ageing (decline by 15.3%).
  - Legislated increase in statutory retirement age to 65.5 years by 2070 is growth-enhancing.
  - Baseline pension expenditure increase of 1.9 percentage points of GDP by 2070 is almost halved thanks to adopted reforms.
  - Projected public debt increase of 100 percentage points of GDP reflects lower revenues and denominator effects.

### Slovakia — baseline long-run deviations (until 2070) and decomposition (values in % or percentage points as reported)
- GDP per capita (%): -9.0 (Ageing: -18.5; retirement age: 9.8; benefit ratio: 0.1; contribution rate: 0.0)
- Consumption: total: -0.4 (Ageing: -1.3; retirement age: 0.9; benefit ratio: 0.0; contribution rate: 0.0)
- Consumption: share of young (% of total): -4.2 (Ageing: -9.5; retirement age: 4.0; benefit ratio: 1.3; contribution rate: 0.0)
- Investment: -1.1 (Ageing: -2.0; retirement age: 0.8; benefit ratio: 0.0; contribution rate: 0.0)
- Unemployment rate (%): -1.0 (Ageing: -1.3; retirement age: 0.4; benefit ratio: 0.0; contribution rate: 0.0)
- Pension expenditure: 1.2 (Ageing: 4.1; retirement age: -2.2; benefit ratio: -0.7; contribution rate: 0.0)
- Primary balance: -2.4 (Ageing: -6.7; retirement age: 3.6; benefit ratio: 0.7; contribution rate: -0.1)
- Public debt: 103.0 (Ageing: 230.5; retirement age: -105.3; benefit ratio: -24.3; contribution rate: 2.1)
- Drivers and interpretation:
  - Ageing in Slovakia mainly driven by rising life expectancy.
  - Slovakia’s 2012 reform linking retirement age to life expectancy and indexation reduces ageing impact: without adopted reforms, ageing impact would be more than twice as strong.
  - Baseline pension expenditure increase of 1.2 percentage points of GDP matches Ageing Report; without reforms the increase would be three times as high.

### Policy scenarios with additional reform measures — Germany (long-run, deviations from baseline until 2070)
- Target: offset ageing-induced additional debt ≈ 100 percentage points of GDP.
- Single-instrument adjustments required to stabilize public debt at initial level:
  - Increase effective retirement age by 3.5 years (vs. baseline retirement age to 65.5 years by 2070).
  - Or reduce benefit ratio by 26 percentage points.
  - Or increase contribution rate by almost 11 percentage points.
- Table 4 selected long-run deviations (baseline vs. instrument; percentages as reported):
  - GDP per capita (%): Baseline -13.9; retirement age 6.1; benefit ratio 0.2; contribution rate -2.9; mix 1.1
  - Consumption: total: Baseline -1.5; retirement age 1.2; benefit ratio -0.1; contribution rate -0.6; mix 0.2
  - Pension expenditure (% of GDP): Baseline 1.9; retirement age -1.3; benefit ratio -4.5; contribution rate 0.1; mix -1.9
  - Public debt (% of GDP): Baseline 100.3; retirement age -108.6; benefit ratio -101.7; contribution rate -97.1; mix -102.3
- Interpretation:
  - Raising retirement age by 3.5 years yields GDP per capita 6.1% stronger relative to baseline.
  - Cutting benefit ratio leaves GDP per capita roughly unchanged relative to baseline.
  - Increasing contribution rates would worsen GDP per capita relative to baseline.

### Policy scenarios with additional reform measures — Slovakia (long-run, deviations from baseline until 2070)
- Single-instrument adjustments required to stabilize public debt at initial level:
  - Increase effective retirement age by 4.5 years (vs. baseline retirement age to 67.2 years by 2070).
  - Or reduce benefit ratio by 21 percentage points.
  - Or increase contribution rate by 7 percentage points.
- Table 5 selected long-run deviations (baseline vs. instrument; percentages as reported):
  - GDP per capita (%): Baseline -9.0; retirement age 8.4; benefit ratio 0.4; contribution rate 0.0; mix 2.9
  - Consumption: total: Baseline -0.4; retirement age 1.5; benefit ratio -0.1; contribution rate -0.1; mix 0.4
  - Pension expenditure (% of GDP): Baseline 1.2; retirement age -1.5; benefit ratio -2.9; contribution rate 0.0; mix -1.4
  - Public debt (% of GDP): Baseline 103.0; retirement age -97.5; benefit ratio -101.7; contribution rate -102.5; mix -100.9
- Interpretation:
  - Increasing retirement age can almost fully offset baseline adverse growth impact by enlarging working cohort and boosting consumption and investment.
  - Cutting entitlements has limited negative macro impact on GDP per capita relative to baseline.
  - Higher contribution rates leave results unchanged relative to baseline for Slovakia.

### Reform mixes and intergenerational considerations
- Single-instrument strategies concentrate burden on one generation:
  - Retirement-age increases affect workers (current and future).
  - Contribution-rate increases affect workers.
  - Benefit-ratio reductions affect pensioners.
- Reform mixes (each instrument ≈ one-third of adjustment) distribute burden more equally across generations and contain adverse macroeconomic impacts relative to single-instrument strategies.

### Costs of pension reform reversals — Germany (reversal: “double threshold” until 2040)
- Reversal scenario: freeze benefit ratio at 48% and cap contribution rate at 20% until 2040; thereafter both evolve roughly in line with Ageing Report.
- Baseline (replicated Ageing Report) by 2070: contribution rate ≈ 27% and benefit ratio <40% (as reference trajectories).
- Long-run deviations to 2070 (relative to initial steady state):
  - Baseline: GDP per capita (%) -13.9; Consumption: total -1.5; Consumption: share of young -4.6; Unemployment rate (%) 0.8; Pension expenditure 1.9; Public debt (% of GDP) 100.3; Retirement age (year) 65.5.
  - Reform reversal: GDP per capita (%) -13.3; Consumption: total -1.3; Consumption: share of young -6.5; Unemployment rate (%) 0.1; Pension expenditure 3.1; Public debt (% of GDP) 157.5; Retirement age (year) 65.5.
- Compensatory measures (assumed entirely via retirement-age increases) required to offset reversal:
  - GDP per capita (%): 10.1
  - Consumption: total: 2.0
  - Consumption: share of young (% of total): 4.2
  - Pension expenditure: -2.1
  - Public debt (% of GDP): -176.0
  - Retirement age (year): 65.5 +5.5 (effective retirement age would need to increase by 5.5 years to 71 years by 2070)
- Interpretation:
  - Reform reversal raises public debt-to-GDP by almost 60 percentage points more than baseline by 2070 (157.5 vs. 100.3).
  - Fully offsetting reversal fiscal cost would require a large retirement-age increase (5.5 years by 2070).

### Costs of pension reform reversals — Slovakia (reversal: cap retirement age at 64)
- Reversal assumed: automatic link between retirement age and life expectancy capped at age 64 in 2045.
- Long-run deviations to 2070 (relative to initial steady state):
  - Baseline: GDP per capita (%) -9.0; Consumption: total -0.4; Consumption: share of young -4.2; Unemployment rate (%) -1.0; Pension expenditure 1.2; Public debt (% of GDP) 103.0; Retirement age (year) 67.2.
  - Reform reversal: GDP per capita (%) -14.8; Consumption: total -0.9; Consumption: share of young -6.7; Unemployment rate (%) -1.3; Pension expenditure 2.6; Public debt (% of GDP) 156.1; Retirement age (year) 64.0.
- Compensatory measures (deviation from reversal scenario) required to stabilize debt:
  - Via benefit ratio reduction:
    - GDP per capita (%): 0.5
    - Consumption: share of young (% of total): 7.8
    - Pension expenditure: -4.2
    - Public debt (% of GDP): -145.4
    - Benefit ratio (%): -29.0
  - Via contribution rate increase:
    - GDP per capita (%): -0.1
    - Consumption: share of young (% of total): 2.2
    - Pension expenditure: 0.0
    - Public debt (% of GDP): -154.7
    - Contribution rate (%): 6.1
  - Via mix:
    - GDP per capita (%): 0.2
    - Consumption: share of young (% of total): 5.0
    - Pension expenditure: -2.1
    - Public debt (% of GDP): -150.5
    - Benefit ratio (%): -14.5; Contribution rate (%): 6.1
- Interpretation:
  - Reform reversal increases public debt-to-GDP by more than 50 percentage points beyond baseline (156.1 vs. 103.0).
  - Fully compensating the reversal would require either a benefit-ratio cut of more than 29 percentage points or a contribution-rate increase of more than 12 percentage points, or a combination.

### Policy implications (consolidated)
- Raising retirement age is the single most growth-friendly instrument to offset ageing fiscal pressures in the calibrated cases.
- Reform mixes spread adjustment burdens across generations and reduce reliance on a single instrument.
- Pension reform reversals have large fiscal costs; compensating them requires large and potentially painful adjustments that raise intergenerational fairness concerns.
- Integrating accounting-style Ageing Report projections into a behavioural OLG general-equilibrium framework is critical to capture feedbacks and quantify true macro-fiscal costs of reform reversals.

### Annex highlights — technical and calibration notes
- OGRE features: overlapping generations, CRRA preferences (parameter γ), discount factor β, capital depreciation δ, CES/Dixit–Stiglitz production, Rotemberg price adjustment, hiring costs, wage bargaining, Taylor-type monetary rule.
- Demography modeled via probabilities to be born (n_t), to retire (ω^Y_t), to pass away (ω^O_t); old/young ratios evolve endogenously.
- Fiscal block: PAYG pensions with replacement-rate ν_t; government budget constraint includes debt, distortionary taxes, lump-sum stabilizer.
- Steady-state calibration targets (selected reported values):
  - Germany calibrated vs targeted: Consumption (total) 60.7 / 54.6; Pension expenditure 7.2 / 7.8; Old-age dependency ratio 37.2 / 38.3; Public debt 75.4 / 75.4.
  - Slovakia calibrated vs targeted: Consumption (total) 60.8 / 60.4; Pension expenditure 6.9 / 6.7; Old-age dependency ratio 36.5 / 32.6; Public debt 53.3 / 53.3.
- Structural calibration (selected parameters):
  - Discount factor β: 0.999
  - Physical capital depreciation δ: 0.1
  - Physical capital income α: 0.385 / 0.32 (Germany / Slovakia)
  - Elasticity θ: 0.8 / 0.95
  - Price markup μ: 1.2
  - Gross pension-wage replacement rate ν: 0.43 / 0.52
  - Technology growth rate g_A (%): 1.53
  - Inverse intertemporal substitution γ: 1
  - Rotemberg price adjustment cost φ_P: 80
  - Taylor rule inflation response φ_π: 2.5
- Annex 2 accounting decomposition (Ageing Report): pension expenditure = (dependency ratio) × (coverage ratio) × (benefit ratio) × (labour market effect); OGRE reproduces and compares the four driving factors with the Ageing Report (Table 8) for Germany and Slovakia.

*Source: wpiea2020132-print-pdf — Introduction, Sections 3.1, 4.1, 4.2, Annex 1 and Annex 2; canonical URL: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020132-print-pdf.pdf*

### Introduction ______________________________________________________________5

### Introduction

### The demographic challenge
- Population ageing is underway in most advanced economies, including in Europe, and is expected to continue in the coming decades.
- The baby boom generation (those born between 1955 and 1970) will enter retirement in the next one and a half decades, concentrating much of the demographic shift.
- Old-age dependency ratio (OADR) definition and projections:
  - OADR = number of people aged 65 or older as a percentage of the working-age population (people aged 15 to 64).
  - Eurostat projects the euro area OADR to rise from 30% today to 52% in 2070, with two thirds of this increase concentrated in the next two decades.
  - By 2070, OADR will be highest in Portugal and Greece and lowest in Ireland and France.
- Driving demographic trends:
  - Remaining life expectancy at age 65 is projected to average 23.6 years for men and 26.9 years for women by 2070 (around 5 years more than today).
  - Fertility rates are well below the natural recovery level in most European countries and are expected to remain low or decline in some countries.
  - Net inward migration will only partly mitigate ageing and is likely to fluctuate over time.
- Macroeconomic implications of ageing:
  - Shrinking labour force and drop in growth rates of consumption, investment and GDP growth.
  - Changes in aggregate consumption and saving patterns due to life-cycle behavior: workers accumulate savings, retirees dissave; rising life expectancy may foster precautionary savings.
  - Capital-labour ratio may increase as labour supply shrinks, exerting downward pressure on the equilibrium real interest rate and dampening investment growth.
- Fiscal implications of ageing:
  - Public pension spending accounts for more than half of total public expenditures in Europe today and is projected to change over time.
  - Ageing-related projections from the 2018 Ageing Report for the euro area public pension spending:
    - 12.3 percentage points of GDP in 2016,
    - 13.5 percentage points of GDP in 2040,
    - 11.9 percentage points of GDP in 2070.
  - Number of workers per pensioner projected to shrink from three to two by 2070.
  - Ageing can compress tax bases (e.g., indirect tax revenues and social security contributions), pushing public debt-to-GDP ratios higher and raising long-term fiscal sustainability risks, especially for countries with already elevated public debt.

### The framework and model (OGRE)
- Purpose:
  - Provide a framework to evaluate macroeconomic and fiscal impact of sudden, unanticipated reversals of pension reforms.
  - Replicate Ageing Report pension expenditure projections and incorporate behavioral feedback via a dynamic general equilibrium OLG model.
- Model description:
  - Model name: OGRE (Overlapping Generations and Retirement).
  - OGRE is a Gertler-type (Blanchard-Weil-Yaari-type) dynamic general equilibrium model with overlapping-generations households, demography, unemployment and wage bargaining.
  - Households: two generations (workers and retirees).
    - Workers: either work (receive labour income and pay income taxes) or are unemployed (receive unemployment benefits).
    - Retirees: do not work and receive pension benefits.
  - Demography: population size and structure change over time via birth, retirement and death probabilities.
  - Fiscal sector: rich specification covering several public revenue and expenditure items, including pensions.
  - For this paper, the closed-economy version of OGRE is used.
- Link to Ageing Report:
  - The 2018 Ageing Report (European Commission) provides detailed, country-specific long-term projections and captures implementation delays of already adopted pension reforms; these projections are used to generate the baseline scenario.
  - The Ageing Report projections are based on a simple accounting framework and do not capture feedback effects between macroeconomic changes, age-related expenditures and pension parameters; OGRE is used to introduce behavioral feedbacks.
- Calibration and country focus:
  - The framework is applied to Germany and Slovakia as illustrative country cases.
  - OGRE can handle both PAYG and fully-funded regimes, though this paper focuses on PAYG parameter adjustments; systemic switches to fully-funded regimes are not analyzed here.
  - The model is calibrated to replicate the Ageing Report’s pension expenditure projections for the two countries.

### Policy scenarios and reform reversals
- Observed recent reform reversals and policy changes (not reflected in the 2018 Ageing Report projections because they were not adopted by then):
  - Germany:
    - Decided in 2018 to cap the expected decline in the benefit ratio and the expected increase in the contribution rate until 2025 at pre-fixed levels; contemplating an extension of the cap until 2040.
    - Government supported idea to introduce a basic pension as of 2021, conditional on 35 contributory years.
  - Slovakia:
    - Broke the automatic link between changes in life expectancy and retirement age by capping the retirement age at 64 years.
    - Legislated generous changes to minimum pensions and a Christmas bonus.
  - Spain:
    - Postponed implementation of the sustainability factor to 2023 (links replacement rate to changes in life expectancy).
  - Italy:
    - Partly backtracked previous reform achievements by temporarily facilitating early retirement.
  - Greece:
    - Faces risks related to court decisions reversing past pension cuts.
  - Netherlands:
    - Discussions on postponing the foreseen increase in the retirement age and loosening the link between life expectancy and retirement age.
- Framework capabilities:
  - Disentangles the future impact purely related to population ageing from the projected impact of already adopted but not fully implemented pension reforms.
  - Quantifies additional pension reform measures needed to compensate for ageing-induced increases in public debt-to-GDP.
  - Quantifies macroeconomic and fiscal effects of pension reform reversals and indicates policy measures needed to compensate for reversals.
- Scope boundaries:
  - The paper does not analyze drivers of reform reversals (e.g., political economy, moral hazard, median voter ageing), which are noted as important but beyond scope.
  - Focus is on pension reforms that adjust parameters of existing PAYG systems; switches to fully-funded regimes are excluded from the present analysis.

### Key findings and contributions
- Model fit and validation:
  - OGRE replicates the Ageing Report projections for Germany and Slovakia very well.
- Main empirical findings:
  - Population ageing has major macroeconomic and fiscal adverse implications if left unaddressed through reforms.
  - Accounting for already adopted pension reforms (as in the Ageing Report) is important; omitting them would overstate and bias ageing-related adverse consequences.
  - Pension reforms help contain adverse implications of ageing; the degree varies by measures and country specifics.
  - Increases in the retirement age appear most effective in alleviating ageing pressures.
  - Reform packages combining various measures spread adjustment burdens more equally across generations.
  - Reform reversals are potentially very costly:
    - They raise aggregate pension expenditure and public debt-to-GDP ratios.
    - They generally exacerbate the adverse macroeconomic impact of ageing.
    - Compensating reversed measures with alternative reforms can improve fiscal sustainability but may disproportionately impact one generation.
- Contribution to literature:
  - Provides a framework that combines detailed Ageing Report projections with an OLG general equilibrium model to capture behavioral feedbacks.
  - Offers a systematic quantification of macroeconomic and fiscal costs of pension reform reversals, an area with limited prior literature.

*Source: wpiea2020132-print-pdf — Introduction; canonical URL: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020132-print-pdf.pdf*

### 3.1    Key elements of the OGRE model

### 3.1    Key elements of the OGRE model

### Model structure and theoretical basis
- The model used is OGRE (Overlapping Generations and Retirement), based on Baksa and Munkacsi (2016a and 2016b).
- OGRE is a Gertler-type (Blanchard-Yaari-Weil-type) model.
- Central element: demographic block with two household types with perfect foresight:
  - Workers: people between the age of 20 and the retirement age.
  - Retirees: people who have reached the retirement age.
- Cohort sizes and total population evolve over time via three probabilities: the probability to be born, to retire, and to pass away. All three probabilities may change over time.
- Workers: either work (receive labour income, pay income taxes) or are unemployed (receive unemployment benefits); they consume and save in risk-free bonds in line with the Euler equation.
- Retirees: do not work, receive per capita pension payments set in real terms at the start of retirement and kept constant thereafter; retirees die with a certain probability and consume all savings by end of lifetime.
- Production side: firms hire workers and use physical capital; firms take account of price adjustment costs when setting prices.
- Labour market rigidities: hiring costs and wage bargaining affect unemployment; labour market and wage bargaining modeling based on Blanchard and Gali (2010).
- Monetary policy: characterized by a Taylor rule.
- Economy assumed closed.

### Fiscal sector and pension system
- Rich fiscal sector includes: personal income tax, social security contributions, VAT, pension benefits, unemployment benefits, government consumption.
- Public pension system: PAYG.
- Governments may issue government bonds to finance funding gaps.
- The model permits differentiation between PAYG, fully funded, and mixed pension schemes in principle, but this differentiation is not further considered in this paper.

### How the ageing shock enters the model and macroeconomic channels
- Ageing shock modeled via:
  - Declining or low probability to be born (equivalent to declining or very low fertility rate).
  - Falling probability to pass away (equivalent to increase in life expectancy).
- Result: lower share of workers relative to pensioners.
- Fertility rate determines persistence of labour force shrinkage over time.
- In the model:
  - Shrinking labour force → lower aggregate employment while unemployment rate will decline.
  - Scarcer labour input → adverse effect on production process.
  - Per capita labour productivity and labour force participation rate assumed not to change with age.
  - Compensation per employee tends to go up as labour force gets scarcer.
- General equilibrium effects:
  - Demand side: growth in aggregate consumption depressed by smaller number of workers and lower income growth; partly compensated by higher aggregate consumption from rising cohort of retirees.
  - Lower growth outlook → decline in aggregate investment → partly reduces projected rise in capital-labour ratio.
- Public finance channels:
  - Public pension spending grows with rising cohort of retirees → higher public expenditure absent built-in adjustments.
  - Lower economic activity and shrinking labour force → reduced government revenues via lower indirect taxes.
  - To finance deficits, government issues risk-free bonds; together with lower GDP growth this pushes up public debt-to-GDP ratio.

### Baseline scenario: integrating Ageing Report projections
- Baseline scenario uses Ageing Report projections to quantify impact of population ageing and already enacted pension reforms.
- Advantages of Ageing Report projections:
  - Based on detailed, country-specific data capturing pension system characteristics.
  - Account explicitly for the future impact of already enacted, but not yet fully implemented social security reforms.
  - Projections done under a no-policy change assumption (assumes no additional pension policy measures over projection horizon).
  - Based on common macroeconomic and demographic assumptions across countries facilitating cross-country comparison.
- Limitations of Ageing Report projections:
  - Based on simple accounting framework; ignores general equilibrium behavioural reactions and feedback effects.
  - Cannot assess how changes in retirement age affect employment or how consumption responds to lower pension benefit ratio.
- Integration approach:
  - Ageing shock defined by targeting the old-age dependency ratio from 2016 to 2070 as stated in the Ageing Report.
  - Translate changes in old-age dependency ratio into model probabilities to be born, to be retired, and to have passed away to derive projected cohort sizes.
  - Replicate Ageing Report’s projected path of public pension spending for 2016-70 within model structure.
  - Impact expressed as percentage point changes compared to the initial period, the year 2016.
  - Decompose macroeconomic and fiscal impact into demographic-driven part and part affected by implementation delay of previously adopted pension reforms.
  - Also reproduced the 4 underlying driving factors of the pension spending path used in the Ageing Report (results shown in Annex 2).

### Policy scenarios: additional reforms and reform reversals
- Two policy scenario types compared to the baseline: additional pension reforms and pension reform reversal.
- Country cases: Germany and Slovakia.
- Additional reform scenario:
  - Calibrate policy instruments such that they stabilize the public debt-to-GDP ratio at its initial level in 2016.
  - This requires full compensation of the public-debt impact of population ageing (used for presentation, not as prescriptive advice).
  - Focus on changes in public debt-to-GDP to account for changes in pension parameters (including contribution rate) and feedback via tax revenues.
  - Three types of pension reforms considered:
    - Increase in the retirement age.
    - Decline in the benefit ratio.
    - Increase in the contribution rate.
  - Assume additional reforms implemented at once; also analyze combined adjustment of all options.
- Reform reversal scenario:
  - Mimic current policy discussions as far as possible within model framework.
  - Germany reversal scenario:
    - Freeze the benefit ratio at its current level of 48% and assume the contribution rate would not exceed the threshold of 20% until 2040.
    - Assumes the agreed freeze of benefits ratio and contribution rate until 2025 is extended until 2040.
  - Slovakia reversal scenario:
    - Assume retirement age stops increasing from the year 2045 onwards, reflecting decision to cap retirement age at 64 years (to be reached in 2045).
  - Necessary compensatory measures under reform reversal scenario determined by what is needed to stabilize public debt-to-GDP ratio.

### Data and calibration (initial steady-state targets and demographic inputs)
- Calibration approach:
  - Macroeconomic variables in initial steady-state calibrated to multi-year averages (mostly 2009-2016) from Eurostat.
  - Targeted values include GDP growth and its main components, labour market variables, public finances, and country-specific pension variables from the 2018 Ageing Report (gross pension expenditures, (gross) replacement rate, benefit ratio).
  - Model reproduces Ageing Report’s change in public pension spending (including driving factors) over projection horizon until 2070.
- Demographic inputs:
  - Use Eurostat population data on absolute cohort sizes for Germany and Slovakia.
  - Translate data into initial-period probabilities (2016):
    - Probability to be born: number of people in the age cohort of the 19 year-olds.
    - Probability to pass away: size of the very old-age cohort (people aged 85 years and older).
    - Retirement probability: derived from the effective exit age from the labour market as reported by Eurostat.
  - Migration flows not explicitly modeled but indirectly captured via impacts on cohort sizes.
- Calibration matches (Table 1: calibrated vs targeted values, in % of GDP unless stated otherwise)
  - Germany:
    - Consumption (total): calibrated value 60.7, targeted value 54.6
    - Consumption (public): calibrated value 19.1, targeted value 19.1
    - Investment: calibrated value 20.1, targeted value 20.0
    - Compensation of employees: calibrated value 59.7, targeted value 56.3
    - Unemployment rate (%): calibrated value 5.7, targeted value 5.6
    - Unemployment benefit: calibrated value 1.5, targeted value 1.8
    - Pension expenditure: calibrated value 7.2, targeted value 7.8
    - Benefit ratio (%): calibrated value 34.1, targeted value 32.4
    - Old-age dependency ratio (%): calibrated value 37.2, targeted value 38.3
    - Public debt: calibrated value 75.4, targeted value 75.4
  - Slovakia:
    - Consumption (total): calibrated value 60.8, targeted value 60.4
    - Consumption (public): calibrated value 16.9, targeted value 16.9
    - Investment: calibrated value 22.3, targeted value 20.2
    - Compensation of employees: calibrated value 54.9, targeted value 46.0
    - Unemployment rate (%): calibrated value 12.8, targeted value 12.7
    - Unemployment benefit: calibrated value 0.7, targeted value 0.7
    - Pension expenditure: calibrated value 6.9, targeted value 6.7
    - Benefit ratio (%): calibrated value 34.4, targeted value 36.3
    - Old-age dependency ratio (%): calibrated value 36.5, targeted value 32.6
    - Public debt: calibrated value 53.3, targeted value 53.3
- Model performance and decompositions:
  - Model replicates Eurostat’s projected path of the old-age dependency ratio until 2070.
  - Germany: OADR projected to increase sharply until 2040 and more slowly thereafter; dynamics mostly driven by cohort effect (baby boom entering retirement) with strong contribution from probability to retire until 2040; life expectancy improvements matter but less so.
  - Slovakia: OADR projected to peak in 2060 before declining; change mostly due to strong improvement in life expectancy (probability of death); cohort effect smaller and declines after around 2060; low fertility rate also contributes to ageing.
  - Note: Although Slovakia’s fertility rate assumed to improve in the long run, long-run OADR impacted by today’s very low fertility rate.

*Source: wpiea2020132-print-pdf - 3.1    Key elements of the OGRE model*

### 4.1    The baseline scenario

### 4.1    The baseline scenario

### Baseline findings and decomposition
- Under the baseline scenario, population ageing has adverse macroeconomic and fiscal implications.
- Assuming no compensatory measures, public debt-to-GDP is projected to increase by around 100 percentage points until 2070 compared to the initial period for both Germany and Slovakia.
- Real GDP per capita is projected to decline by almost 14% in Germany and 9% in Slovakia compared to the initial period.
- Long-run model results are decomposed into:
  - Ageing impact (column “ageing” in Tables 2 and 3).
  - Impact of adopted pension reforms (columns under “impact of adopted pension reforms” in Tables 2 and 3).
- Previously adopted pension reforms can partly cushion or amplify macroeconomic impacts depending on the instrument (e.g., higher retirement age, higher contribution rates, lower benefit ratio).
- Results reported are from the closed-economy version of the model; results with the open-economy version are qualitatively similar but with a more pronounced adverse impact in the closed-economy version.

### Germany: baseline quantitative results and drivers
- Key long-run deviations (until 2070) for Germany (in % of GDP, unless stated otherwise):
  - GDP per capita (%): -13.9 (Ageing: -15.3; retirement age: 2.5; benefit ratio: 0.1; contribution rate: -1.2)
  - Consumption: total: -1.5 (Ageing: -1.6; retirement age: 0.4; benefit ratio: 0.0; contribution rate: -0.2)
  - Consumption: share of young (% of total): -4.6 (Ageing: -7.8; retirement age: 1.1; benefit ratio: 1.7; contribution rate: 0.4)
  - Investment: -1.3 (Ageing: -1.4; retirement age: 0.1; benefit ratio: 0.0; contribution rate: 0.0)
  - Unemployment rate (%): 0.8 (Ageing: -0.3; retirement age: 0.0; benefit ratio: 0.0; contribution rate: 1.1)
  - Pension expenditure: 1.9 (Ageing: 3.8; retirement age: -0.6; benefit ratio: -1.3; contribution rate: 0.0)
  - Primary balance: -2.9 (Ageing: -6.5; retirement age: 1.1; benefit ratio: 1.1; contribution rate: 1.4)
  - Public debt: 100.3 (Ageing: 208.6; retirement age: -33.9; benefit ratio: -29.4; contribution rate: -44.9)
- Drivers and interpretation:
  - The model-induced decline in GDP per capita by almost 14% by 2070 largely results from population ageing (decline by 15.3%).
  - The legislated gradual increase in the statutory retirement age until 2029 is expected to be growth-enhancing: effective retirement age increases from 64 years (in 2016) to 65.5 years by 2070.
  - Higher contribution rates can further worsen the impact on GDP per capita (by 1.2%), while a less generous benefit ratio has only a marginal positive growth impact.
  - Aggregate total consumption declines by 1.5 percentage points of GDP, largely driven by a fall in the workers’ share of total aggregate consumption (by 4.6 percentage points).
  - The decline in the young’s share of total aggregate consumption is driven by a shrinking young cohort relative to pensioners and forward-looking saving behavior by the young.
  - Baseline pension expenditure is projected to increase by 1.9 percentage points of GDP in 2070 compared to the initial period; ageing-induced increase in pension expenditures is almost halved thanks to previously adopted reforms (rise in retirement age and declining benefit ratio).
  - The projected public debt increase of 100 percentage points of GDP by 2070 for Germany reflects feedback effects via lower public revenues and a negative denominator effect; it abstracts from other factors such as national fiscal rules.

### Slovakia: baseline quantitative results and drivers
- Key long-run deviations (until 2070) for Slovakia (in % of GDP, unless stated otherwise):
  - GDP per capita (%): -9.0 (Ageing: -18.5; retirement age: 9.8; benefit ratio: 0.1; contribution rate: 0.0)
  - Consumption: total: -0.4 (Ageing: -1.3; retirement age: 0.9; benefit ratio: 0.0; contribution rate: 0.0)
  - Consumption: share of young (% of total): -4.2 (Ageing: -9.5; retirement age: 4.0; benefit ratio: 1.3; contribution rate: 0.0)
  - Investment: -1.1 (Ageing: -2.0; retirement age: 0.8; benefit ratio: 0.0; contribution rate: 0.0)
  - Unemployment rate (%): -1.0 (Ageing: -1.3; retirement age: 0.4; benefit ratio: 0.0; contribution rate: 0.0)
  - Pension expenditure: 1.2 (Ageing: 4.1; retirement age: -2.2; benefit ratio: -0.7; contribution rate: 0.0)
  - Primary balance: -2.4 (Ageing: -6.7; retirement age: 3.6; benefit ratio: 0.7; contribution rate: -0.1)
  - Public debt: 103.0 (Ageing: 230.5; retirement age: -105.3; benefit ratio: -24.3; contribution rate: 2.1)
- Drivers and interpretation:
  - Population ageing in Slovakia is mainly driven by rising life expectancy.
  - Slovakia’s 2012 pension reform automatically links the retirement age to changes in life expectancy and reduces the generosity of the pension system through changes in the indexation rule.
  - Baseline results predict a decline in GDP per capita by 9% by 2070 compared to the initial period; without the adopted pension reforms, the ageing impact would be more than twice as strong given projected strong increases in life expectancy.
  - The consumption share of the young is estimated to decline by 4.2 percentage points of total consumption; previously adopted pension reforms limit this decline by 4 and 1.3 percentage points through higher retirement age and reduced generosity, respectively.
  - Baseline pension expenditure is projected to increase by 1.2 percentage points of GDP by 2070 compared to the initial period; this is the same figure as projected in the Ageing Report. Without 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.

*Source: IMF working paper — Section 4.1 The baseline scenario (tables and text as provided).*

### 4.2    Policy scenarios with additional reform measures

### 4.2    Policy scenarios with additional reform measures

### Purpose and methodology
- Examine macroeconomic and fiscal impact of additional pension reforms beyond those already adopted.
- Calibrate additional reforms to fully compensate for the increase in the public-debt-to-GDP ratio due to ageing (i.e., keep public debt-to-GDP broadly constant at the initial level).
- Use the same dynamic general equilibrium overlapping-generations modelling framework as prior sections, with perfect foresight and immediate household reactions (including savings adjustments).
- Consider changes to three pension parameters: the retirement age, the benefit ratio, and the contribution rate, and evaluate single-instrument reforms and reform mixes.

### Key findings — Germany (long-run, deviations from baseline; long-run results until 2070)
- Ageing-induced additional debt to be offset: around 100 percentage points of GDP (context for adjustment magnitude).
- Required adjustments (to keep public debt roughly constant at initial level):
  - Increase in effective retirement age by 3.5 years (vs. baseline retirement age to 65.5 years by 2070).
  - Or reduce benefit ratio by 26 percentage points.
  - Or increase contribution rate by almost 11 percentage points.
- Table 4 long-run deviations from baseline (impact of additional pension reforms):
  - GDP per capita (%): Baseline -13.9; retirement age 6.1; benefit ratio 0.2; contribution rate -2.9; mix 1.1
  - Consumption: total: Baseline -1.5; retirement age 1.2; benefit ratio -0.1; contribution rate -0.6; mix 0.2
  - Consumption: share of young (% of total): Baseline -4.6; retirement age 2.5; benefit ratio 5.8; contribution rate 0.9; mix 3.1
  - Unemployment rate (%): Baseline 0.8; retirement age -0.1; benefit ratio -0.1; contribution rate 2.8; mix 0.8
  - Pension expenditure (% of GDP): Baseline 1.9; retirement age -1.3; benefit ratio -4.5; contribution rate 0.1; mix -1.9
  - Public debt (% of GDP): Baseline 100.3; retirement age -108.6; benefit ratio -101.7; contribution rate -97.1; mix -102.3
- Policy instruments (deviations from baseline):
  - Retirement age (year): Baseline 65.5; impact of additional reforms +3.5 (retirement-age column); mix +1.2
  - Benefit ratio (%): Baseline -8.0; benefit-ratio column -26.0; mix -8.7
  - Contribution rate (%): Baseline 4.4; contribution-rate column -10.6; mix 3.5
- Interpretation:
  - Raising retirement age by 3.5 years yields GDP per capita 6.1% stronger relative to baseline.
  - Cutting pension entitlements (benefit ratio reduction) leaves GDP per capita roughly unchanged relative to baseline.
  - Higher contribution rates in Germany would aggravate the adverse impact on GDP per capita relative to baseline.

### Key findings — Slovakia (long-run, deviations from baseline; long-run results until 2070)
- Required adjustments (to keep public debt roughly constant at initial level):
  - Increase in effective retirement age by 4.5 years (vs. baseline retirement age to more than 67 years by 2070).
  - Or reduce benefit ratio by 21 percentage points.
  - Or increase contribution rate by 7 percentage points.
- Table 5 long-run deviations from baseline (impact of additional pension reforms):
  - GDP per capita (%): Baseline -9.0; retirement age 8.4; benefit ratio 0.4; contribution rate 0.0; mix 2.9
  - Consumption: total: Baseline -0.4; retirement age 1.5; benefit ratio -0.1; contribution rate -0.1; mix 0.4
  - Consumption: share of young (% of total): Baseline -4.2; retirement age 2.8; benefit ratio 5.4; contribution rate 1.4; mix 3.2
  - Unemployment rate (%): Baseline -1.0; retirement age 0.0; benefit ratio -0.1; contribution rate 0.0; mix -0.1
  - Pension expenditure (% of GDP): Baseline 1.2; retirement age -1.5; benefit ratio -2.9; contribution rate 0.0; mix -1.4
  - Public debt (% of GDP): Baseline 103.0; retirement age -97.5; benefit ratio -101.7; contribution rate -102.5; mix -100.9
- Policy instruments (deviations from baseline):
  - Retirement age (year): Baseline 67.2; retirement-age column +4.5; mix +1.5
  - Benefit ratio (%): Baseline -5.0; benefit-ratio column -21.0; mix -7.0
  - Contribution rate (%): Baseline -0.2; contribution-rate column -6.8; mix 2.3
- Interpretation:
  - Increasing retirement age can almost fully offset the adverse growth impact in the baseline for Slovakia by enlarging the cohort of workers and boosting aggregate consumption and investment.
  - Cutting pension entitlements has limited negative macro impact on GDP per capita relative to baseline.
  - Higher contribution rates leave results unchanged relative to baseline for Slovakia.

### Reform mixes and intergenerational considerations
- Single-instrument reliance concentrates adjustment burden on one generation:
  - Lifting retirement age affects current and future workers (prolonged working lives).
  - Raising contribution rate affects cohorts of workers.
  - Lowering pension entitlements affects cohorts of pensioners.
- Reform mixes (each instrument contributing one-third of adjustment) distribute burden more equally across generations.
- Under reform mix scenarios for Germany and Slovakia (each instrument ~1/3 of total adjustment):
  - The necessary increase in retirement age, reductions in benefit ratio, and increases in contribution rate are each about one-third of the magnitudes required under single-instrument strategies.
  - Adverse ageing impact on GDP per capita is contained relative to baseline.

### Costs of pension reform reversals — Germany (reform reversal: “double threshold” until 2040)
- Reform reversal scenario defined: contribution rate capped at 20% and benefit ratio not falling below 48% until 2040; after 2040 both evolve roughly in line with Ageing Report projections.
- Baseline projections (Ageing Report replication): contribution rate ~27% and benefit ratio <40% by 2070 (solid lines in Figure 3).
- Fiscal and macro costs (Table 6, long-run deviations to 2070):
  - Baseline deviations from initial steady state until 2070:
    - GDP per capita (%): -13.9
    - Consumption: total: -1.5
    - Consumption: share of young (% of total): -4.6
    - Unemployment rate (%): 0.8
    - Pension expenditure: 1.9
    - Public debt (% of GDP): 100.3
    - Retirement age (year): 65.5
  - Reform reversal deviations (relative to initial steady state until 2070):
    - GDP per capita (%): -13.3
    - Consumption: total: -1.3
    - Consumption: share of young (% of total): -6.5
    - Unemployment rate (%): 0.1
    - Pension expenditure: 3.1
    - Public debt (% of GDP): 157.5
    - Retirement age (year): 65.5
  - Compensatory measures (deviation from the reform reversal scenario; assumed entirely via retirement-age increases):
    - GDP per capita (%): 10.1
    - Consumption: total: 2.0
    - Consumption: share of young (% of total): 4.2
    - Unemployment rate (%): -0.2
    - Pension expenditure: -2.1
    - Public debt (% of GDP): -176.0
    - Retirement age (year): 65.5 +5.5 (i.e., effective retirement age would need to increase by 5.5 years to 71 years by 2070)
- Interpretation:
  - Reform reversal raises public debt-to-GDP by almost 60 percentage points more than baseline by 2070 (public debt 157.5 vs. baseline 100.3).
  - GDP per capita would be only marginally better (decline by 13.3%) than under baseline.
  - To fully offset the reform reversal fiscal cost (almost 160 percentage points of GDP increase in public debt by 2070 relative to the initial period), Germany would need an effective retirement-age increase of 5.5 years by 2070; of this, 2 years are attributable to the reform reversal and 3.5 years to offset ageing under baseline.

### Costs of pension reform reversals — Slovakia (reform reversal: cap retirement age at 64)
- Reform reversal assumed: automatic link between retirement age and life expectancy capped at age 64 in 2045 (year retirement age would reach 64 under baseline).
- Fiscal and macro costs (Table 7, long-run deviations to 2070):
  - Baseline deviations from initial steady state until 2070:
    - GDP per capita (%): -9.0
    - Consumption: total: -0.4
    - Consumption: share of young (% of total): -4.2
    - Unemployment rate (%): -1.0
    - Pension expenditure: 1.2
    - Public debt (% of GDP): 103.0
    - Retirement age (year): 67.2
    - Benefit ratio (%): -5.0
    - Contribution rate (%): -0.2
  - Reform reversal deviations (relative to initial steady state until 2070):
    - GDP per capita (%): -14.8
    - Consumption: total: -0.9
    - Consumption: share of young (% of total): -6.7
    - Unemployment rate (%): -1.3
    - Pension expenditure: 2.6
    - Public debt (% of GDP): 156.1
    - Retirement age (year): 64.0
    - Benefit ratio (%): -5.0
    - Contribution rate (%): -0.2
  - Compensatory measures (deviation from the reform reversal scenario):
    - If compensatory via benefit ratio (column “benefit ratio”):
      - GDP per capita (%): 0.5
      - Consumption: total: -0.2
      - Consumption: share of young (% of total): 7.8
      - Unemployment rate (%): -0.1
      - Pension expenditure: -4.2
      - Public debt (% of GDP): -145.4
      - Benefit ratio (%): -29.0
    - If compensatory via contribution rate (column “contribution rate”):
      - GDP per capita (%): -0.1
      - Consumption: total: -0.1
      - Consumption: share of young (% of total): 2.2
      - Unemployment rate (%): 0.2
      - Pension expenditure: 0.0
      - Public debt (% of GDP): -154.7
      - Contribution rate (%): 6.1
    - If compensatory via mix (column “mix”):
      - GDP per capita (%): 0.2
      - Consumption: total: -0.2
      - Consumption: share of young (% of total): 5.0
      - Unemployment rate (%): 0.0
      - Pension expenditure: -2.1
      - Public debt (% of GDP): -150.5
      - Benefit ratio (%): -14.5
      - Contribution rate (%): 6.1
- Interpretation:
  - Reform reversal increases public debt-to-GDP by more than 50 percentage points beyond the baseline ageing-induced increase (156.1 vs. 103.0).
  - GDP per capita would shrink by almost 15% by 2070 relative to initial period under reform reversal.
  - To fully compensate if retirement age is capped at 64, Slovakia would need either:
    - Benefit ratio lowered by more than 29 percentage points; or
    - Contribution rate increased by more than 12 percentage points; or
    - A combination that halves the individual magnitudes but still leaves adverse GDP per capita effects.

### Conclusions and policy implications
- Mixed reform packages (retirement age increases, benefit ratio reductions, and contribution rate increases combined) help spread adjustment burdens more evenly across generations and contain adverse macroeconomic impacts relative to single-instrument reforms.
- Pension reform reversals can generate substantial fiscal costs (large additional public debt-to-GDP effects) and worsen macroeconomic outlooks; compensating these costs through alternative instruments would require large and potentially painful adjustments, with adverse implications for intergenerational fairness.
- The paper’s framework integrates Ageing Report pension-expenditure dynamics into a dynamic general equilibrium overlapping-generations model to capture feedback effects and to quantify macroeconomic and fiscal costs of reform reversals.

*Source: 4.2 Policy scenarios with additional reform measures (extract).*

### References

### References

### Demographic structure, ageing, and pensions
- Aksoy, Y., Basso, H. S., Smith, R. P., and Grasl, T. (2019): “Demographic structure and macroeconomic trends”. American Economic Journal: Macroeconomics, Vol. 11(1).
- B ̈orsch-Supan, A., A. Ludwig, and J. Winter (2006): “Ageing, pension reform and capital flows: a multi-country simulation model”. Economica, Vol 73(292).
- B ̈orsch-Supan, A. and J. Rausch (2018): “Die Kosten der doppelten Haltelinie”. MEA Discussion Papers, 3/2018.
- Carone, G., P. Eckefeldt, L. Giamboni, V. Laine and S. Pamies-Sumner (2016): “Pension reforms in the EU since the early 2000s: achievements and challenges ahead”. European Commission Discussion Paper, December 2016.
- Conesa, J. and D. Krueger (1999): “Social security reform with heterogeneous agents”. Review of Economic Dynamics, Vol 2.
- De la Croix, D., O. Pierrard and H. Sneessens (2013): “Ageing and pensions in general equilibrium: labor market imperfections matter”. Journal of Economic Dynamics and Control, Vol 37(1).
- European Central Bank (2018): “The economic impact of population ageing and pension reforms”. Issue 2/2018, March.
- European Commission (2017): “The 2018 Ageing Report: underlying assumptions and projection methodologies”. European Economy, October.
- European Commission (2018): “The 2018 Ageing Report: economic and budgetary projections for the EU Member States (2016-2070)”. European Economy, May.
- Fehr, H. (2000): “Pension reform during the demographic transition”. The Scandinavian Journal of Economics, Vol 102/3.
- Galasso, V. (2008): “Postponing retirement: the political effect of aging”. Journal of Public Economics, Vol. 92.
- Goraus, K., K. Makarski and J. Tyrowicz (2014): “Does social security reform reduce gains from increasing the retirement age?”. University of Warsaw Working Papers, No. 3/2014 (120).
- Heijdra, B. J., and W. E. Romp (2009): “Retirement, pensions, and ageing”. Journal of Public Economics, Vol. 93(3).
- Karam, P., D. Muir, J. Pereira and A. Tuladhar (2010): “Macroeconomic effects of public pension reforms”. IMF Working Paper, 10/297.
- Keuschnigg, C., T. Davoine and P. Schuster (2013): “Aging, informality and public finances in Poland and Slovakia: a general equilibrium approach”. Background Paper for the ECA Old-Age Insurance World Bank Report, Institute for Advanced Studies Research Report, May 2013.
- Kilponen, J., H. Kinnunen, and A. Ripatti (2006): “Population ageing in a small open economy - some policy experiments with a tractable general equilibrium model”. Bank of Finland Research Discussion Papers, 28.
- Weil, P. (1989): “Overlapping families of infinitely-lived agents”. Journal of Public Economics, 38(2).
- Yaari, M. E. (1965): “Uncertain lifetime, life insurance, and the theory of the consumer”. The Review of Economic Studies, Vol. 32(2).

### Overlapping-generations, life-cycle, and OLG modelling
- Baksa, D., and Z. Munkacsi (2016): “A detailed description of OGRE, the OLG model”. Bank of Lithuania Working Paper Series, No 31.
- Baksa, D., and Z. Munkacsi, (2016b), “Aging, (pension) reforms and the shadow economy in Southern Europe”. Bank of Lithuania Working Paper Series, No 32.
- Baksa, D., M. Constantinescu, and Z. Munkacsi (2016): “Aging, informality and public policies in a small open-economy”. Bank of Lithuania Discussion Paper Series, No 2.
- Gertler, M. (1999): “Government debt and social security in a life-cycle economy”. Carnegie-Rochester Conference Series on Public Policy, Vol. 50(1).
- Yaari, M. E. (1965): “Uncertain lifetime, life insurance, and the theory of the consumer”. The Review of Economic Studies, Vol. 32(2).
- Conesa, J. and D. Krueger (1999): “Social security reform with heterogeneous agents”. Review of Economic Dynamics, Vol 2.
- Weil, P. (1989): “Overlapping families of infinitely-lived agents”. Journal of Public Economics, 38(2).

### Labour supply, retirement decisions, and incentives
- Beetsma, R. W. Romp, and S. Wos (2013): “Intergenerational risk sharing, pensions, and endogenous labour supply in general equilibrium”. Scandinavian Journal of Economics, Vol. 115(1).
- Blanchard, O. J., and J. Gali. (2010): “Labor markets and monetary policy: a new Keynesian model with unemployment”. American Economic Journal: Macroeconomics, Vol. 2(2).
- Diaz-Gimenez, J., and J. Diaz-Saavedra (2009): “Delaying retirement in Spain”. Review of Economic Dynamics, Vol 12(1).
- Dolls, M. and C. Krolage (2019): “The effects of early retirement incentives on retirement decisions”. ifo Working Paper, No 291.
- Galasso, V. (2008): “Postponing retirement: the political effect of aging”. Journal of Public Economics, Vol. 92.
- Goraus, K., K. Makarski and J. Tyrowicz (2014): “Does social security reform reduce gains from increasing the retirement age?”. University of Warsaw Working Papers, No. 3/2014 (120).

### Macroeconomic effects, interest rates, and monetary policy
- Brand, C., M. Bielecki and A. Penalver (2018): “The natural rate of interest: estimates, drivers, and challenges to monetary policy”. ECB Occasional Paper, No 217.
- Kara, E. and L. von Thadden (2010): “Interest rate effects of demographic changes in a New-Keynesian life-cycle framework”. ECB Working Paper Series, No 1273.
- Rotemberg, J. J. (1982): “Sticky prices in the United States”. Journal of Political Economy, Vol. 90 (6).
- Schmitt-Grohe, S., and M. Uribe (2003): “Closing small open economy models”. Journal of International Economics, Vol. 61(1).
- Brand, C., M. Bielecki and A. Penalver (2018): “The natural rate of interest: estimates, drivers, and challenges to monetary policy”. ECB Occasional Paper, No 217.

### Methodology, simulations, and working papers
- Baksa, D., and Z. Munkacsi (2016): “A detailed description of OGRE, the OLG model”. Bank of Lithuania Working Paper Series, No 31.
- Baksa, D., and Z. Munkacsi, (2016b), “Aging, (pension) reforms and the shadow economy in Southern Europe”. Bank of Lithuania Working Paper Series, No 32.
- Baksa, D., M. Constantinescu, and Z. Munkacsi (2016): “Aging, informality and public policies in a small open-economy”. Bank of Lithuania Discussion Paper Series, No 2.
- Karam, P., D. Muir, J. Pereira and A. Tuladhar (2010): “Macroeconomic effects of public pension reforms”. IMF Working Paper, 10/297.
- Keuschnigg, C., T. Davoine and P. Schuster (2013): “Aging, informality and public finances in Poland and Slovakia: a general equilibrium approach”. Background Paper for the ECA Old-Age Insurance World Bank Report, Institute for Advanced Studies Research Report, May 2013.

*References list as provided in the source PDF.*

### Annex 1 Technical description of the OGRE model

### Annex 1 Technical description of the OGRE model

### Model overview
- OGRE is an overlapping-generations, small open-economy dynamic stochastic general equilibrium model with young (workers) and old (retired) cohorts, a pay-as-you-go pension system, nominal rigidities (Rotemberg price adjustment), hiring and investment adjustment costs, wage bargaining, and a Taylor-type monetary policy rule.
- Core preferences: CRRA utility with parameter γ and discount factor β. Firms have CES/Dixit–Stiglitz demand with elasticity parameter θ and production share α. Capital depreciates at rate δ. Adjustment cost functions and Rotemberg price-costs are quadratic.

### Demography and overlapping generations
- Population identities:
  - N_t = N^O_t + N^Y_t
  - N^Y_t = (1−ω^Y_{t−1}) N^Y_{t−1} + n_t N^Y_{t−1}
  - N^O_t = (1−ω^O_{t−1}) N^O_{t−1} + ω^Y_{t−1} N^Y_{t−1}
- Old/young ratio and young share:
  - s_t = N^O_t / N^Y_t
  - s_{Y,t} = N^Y_t / N_t = 1 / (1 + s_t)
- Cohort growth rates:
  - 1 + g_{N,Y,t} = N^Y_t / N^Y_{t−1} = 1−ω^Y_{t−1} + n_t
  - 1 + g_{N,O,t} = N^O_t / N^O_{t−1} = (1−ω^O_{t−1}) + ω^Y_{t−1} / s_{t−1}
- Aggregate population growth:
  - 1 + g_t = (1 + g_{N,Y,t}) (1 + s_t) / (1 + s_{t−1})

### Retired generation — optimization and aggregate consumption
- Retired individual Bellman problem (V^O) and budget constraint:
  - V^O(B^O_{a−1,t−1}(i)) = max { 1/(1−γ) C^{1−γ}_{O,a,t}(i) + β E_t (1−ω^O_t) V^O(B^O_{a,t}(i)) }
  - (1 + τ^C_t) p^O_t C_{O,a,t}(i) + (1−ω^O_t) B^O_{a,t}(i) = (1 + r_{t−1}) B^O_{a−1,t−1}(i) + TR^{YO}_{a,t}(i)
- Euler equation (retired):
  - β E_t [ (C_{O,a+1,t+1}(i) / C_{O,a,t}(i))^{−γ} (1 + r_t) ( (1+τ^C_t) p^O_t / (1+τ^C_{t+1}) p^O_{t+1} ) ] = 1
  - Rearranged: E_t C_{O,a+1,t+1}(i) = C_{O,a,t}(i) β^{1/γ} (1 + r_t)^{1/γ} Λ^O_{t,t+1}
  - Λ^O_{t,t+1} = E_t [ ((1+τ^C_t) p^O_t / (1+τ^C_{t+1}) p^O_{t+1})^{1/γ} ]
- Individual retired consumption closed-form as a function of present value of pension benefits and initial wealth; define aggregate MPC_O_t:
  - 1 / MPC_{O,t} = E_t ∑_{n=0}^∞ (1 + τ^C_{t+n}) p^O_{t+n} β^{n/γ} ∏_{k=1}^n (1−ω^O_{t+k−1}) (1 + r_{t+k−1})^{1/γ − 1} Λ^O_{t,t+k}
- Retired individual consumption function:
  - C_{O,a,t}(i) = MPC_{O,t} Ω^O_t TR^{YO}_{a,t}(i) + MPC_{O,t} (1 + r_{t−1}) B^O_{a−1,t−1}(i)
  - Ω^O_t = 1 + E_t [ (1−ω^O_t) / (1 + r_t) Ω^O_{t+1} ]
- Aggregate retired consumption:
  - C^O_t = MPC_{O,t} Ω^O_t TR_t + (1 + r_{t−1}) MPC_{O,t} (ω^Y_{t−1} B^Y_{t−1} + B^O_{t−1})

### Young generation — optimization and aggregate consumption
- Young individual Bellman problem (V^Y) and budget constraint:
  - V^Y_t(B^Y_{b−1,t−1}(i)) = max { 1/(1−γ) C^{1−γ}_{Y,b,t}(i) + E_t β [ (1−ω^Y_t) V^Y_{t+1}(B^Y_{b,t}(i)) + ω^Y_t V^O_{t+1}(B^{YO}_{b,t}(i)) ] }
  - (1+τ^C_t) p^Y_t C_{Y,b,t}(i) + (1−ω^Y_t) B^Y_{b,t}(i) + ω^Y_t B^{YO}_{b,t}(i) = (1 + r_{t−1}) B^Y_{b−1,t−1}(i) + (1−τ^L_t) w_t L_{b,t}(i) + w^U_t U_{b,t}(i) + Profit_{b,t}(i) − LTax_{b,t}(i)
- Euler equations (young) linking young consumption, future young consumption, and future old-age consumption via Λ terms:
  - E_t C_{Y,b+1,t+1}(i) = C_{Y,b,t}(i) β^{1/γ} (1 + r_t)^{1/γ} Λ^Y_{t,t+1}
  - E_t C^O_{0,t+1}(i) = C_{Y,b,t}(i) β^{1/γ} (1 + r_t)^{1/γ} Λ^{YO}_{t,t+1}
  - Λ^Y_{t,t+1} = E_t [ ((1+τ^C_t) p^Y_t / (1+τ^C_{t+1}) p^Y_{t+1})^{1/γ} ]
  - Λ^{YO}_{t,t+1} = E_t [ ((1+τ^C_t) p^Y_t / (1+τ^C_{t+1}) p^O_{t+1})^{1/γ} ]
- Lifetime income and expected pension benefit present value I^{YO}_{b,t}(i) recursive representation:
  - I^{YO}_{b,t}(i) = E_t [ ω^Y_t TR^{YO}_{0,t+1}(i) Ω^O_{t+1} + (1−ω^Y_t)(1 + r_{t+1})^{−1} I^{YO}_{b+1,t+1}(i) ]
- Define Inc^Y_{b,t}(i) = I^Y_{b,t}(i) + I^{YO}_{b,t}(i) and obtain individual consumption:
  - C_{Y,b,t}(i) = MPC_{Y,t} Inc^Y_{b,t}(i) + MPC_{Y,t} (1 + r_{t−1}) B^Y_{b−1,t−1}(i)
- Aggregate young consumption:
  - C^Y_t = MPC_{Y,t} Inc^Y_t + (1 + r_{t−1}) MPC_{Y,t} (1−ω^Y_{t−1}) B^Y_{t−1}

### Fiscal policy and pay-as-you-go pension plan
- Pension benefit at retirement for individual i:
  - TR^{YO}_{0,t}(i) = ν_t w_{t−1} L_{b−1,t−1}(i)  (replacement-rate ν_t)
- Aggregate pension flows:
  - TR^{YO}_t = ν_t ω^Y_{t−1} w_{t−1} L_{t−1}
  - TR_t = TR^{YO}_t + (1−ω^O_{t−1}) TR_{t−1}
- Government budget constraint:
  - Debt_t + LTax_t + Tax_t = Gov_t + TR_t + w^U_t U_t + (1 + r_{t−1}) Debt_{t−1}
  - Tax_t = τ^C_t (p^O_t C^O_t + p^Y_t C^Y_t) + (τ^{SSC}_t + τ^L_t) w_t L_t
- Lump-sum tax rule to stabilize debt-to-GDP:
  - LTax_t = Gov_t + TR_t + w^U_t U_t + (1 + r_{t−1}) Debt_{t−1} − {Debt/Y}^{Target}_t − Tax_t
- Bond market equilibrium:
  - Debt_t = B^Y_t + B^O_t

### Firms, production, price setting and labor
- Firms produce differentiated goods; Rotemberg price adjustment cost:
  - R(P_{Y,t}(i) / P_{Y,t−1}(i)) = φ_P / 2 [ P_{Y,t}(i) / (P_{Y,t−1}(i) (1 + π_{t−1}))^{θ−1} − 1 ]^2
- Total cost TC_t(i):
  - TC_t(i) = P_t Inv_t(i) + (1 + τ^{SSC}_t) W_t L_t(i) + P_t(i) Y_t R( P_t(i)/P_{t−1}(i) ) + HC_t H_t
- Employment law of motion with firing probability pr^F_t and hiring H_t(i) with convex adjustment S(·):
  - L_t(i) = (1−pr^F_t) L_{t−1}(i) + H_t(i) (1 − S(H_t(i)/H_{t−1}(i)))
- Capital accumulation with investment adjustment cost S(·):
  - K_t(i) = (1−δ) K_{t−1}(i) + Inv_t(i) (1 − S(Inv_t(i)/Inv_{t−1}(i)))
- Value-function Bellman includes expected continuation with survival (1−ω^Y_t), marginal cost MC_t, Tobin Q_t, and Lagrange multipliers for labor/hiring (Λ^H_t) and capital (Q_t).
- First-order conditions imply:
  - No-arbitrage modified by survival: q_t = (1−ω^Y_t) / (1 + r_t) ( r^K_{t+1} + q_{t+1} (1−δ) )
  - Effective real wage definition: w̄_t = (1 + τ^{SSC}_t) w_t + Λ^H_t − E_t [ (1−ω^Y_t)/(1 + r_t) (1−pr^F_{t+1}) Λ^H_{t+1} ]
  - Labor demand: w̄_t = mc_t (1−α)^{1/θ} Y_t^{1/θ} (A_t L_t)^{(1−θ)/θ} A_t
  - Marginal cost:
    - mc_t = [ α r^K_t^{1−θ} + (1−α) ( w̄_t / A_t )^{1−θ} ]^{1/(1−θ)}
- Price-setting condition (firm-level) (Rotemberg with expectations and survival):
  - (P_t(i)/P_t)^{−φ} − φ P_t(i) (P_t(i)/P_t)^{−φ−1} (1/P_t) Y_t − Y_t R(P_t(i)/P_{t−1}(i)) − P_t(i) Y_t R′(·) / ( P_{t−1}(i) (1 + π_{t−1})^{θ} ) + E_t (1−ω^Y_t)/(1 + i_t) P_{t+1}(i) Y_{t+1} R′(·) (...) + MC_t φ (P_t(i)/P_t)^{−φ−1} (1/P_t) Y_t = 0
- Aggregate simplified inflation relation (assuming symmetric firms and P_t(i)=P_t):
  - 1 + π_t = [ φ/(φ−1) ] mc_t + terms involving Rotemberg adjustment R(·) and expected future variables

### Retailers and consumption-smoothing
- Retailers set cohort-specific relative prices p_{j,t} to smooth short-run household consumption, with consumption retail adjustment cost G_{C,j}(·) convex:
  - p_{j,t} = 1 + G_{C,j}(C_{j,t}/C_{j,t−1}) + G'_{C,j}(C_{j,t}/C_{j,t−1}) (1/(1+g_t)) C_{j,t}/C_{j,t−1} − (1−ω^j_t)/(1 + r_t) G'_{C,j}(C_{j,t+1}/C_{j,t}) (1/(1+g_{t+1})) (C_{j,t+1}/C_{j,t})^2

### Wage bargaining
- Worker value functions (employed V^w_t and unemployed V^U_t) include survival, firing and hiring probabilities, and bargaining maximizes (V^w_t − V^U_t)^σ hc_t^{1−σ}.
- Nash-type first-order condition reduces to:
  - σ/(1−σ) hc_t^{1−τL_t} (1/(1 + τ^{SSC}_t)) = (1−τ^L_t) w_t − w^U_t + E_t [ (1/(1 + r_t)) (1−pr^F_{t+1})(1−pr^H_{t+1}) ... continuation term ]
- Leads to a recursive expression for bargaining surplus and relationship to hiring cost hc_t and matching probabilities pr^H_t, pr^F_t.

### Monetary policy
- Taylor-type rule with interest smoothing ρ_i, inflation response φ_π and monetary shock ε^i_t:
  - 1 + i_t = (1 + i_{t−1})^{ρ_i} [ (1 + r^n_t) (1 + π_t)^{φ_π} ]^{1−ρ_i} e^{ε^i_t}
- Fisher equation:
  - 1 + i_t = E_t (1 + r_t) (1 + π_{t+1})

### Normalized (detrended) model equations — notation x̃_t denotes variables normalized by population N_t
- Demography (normalized):
  - s_t = (1−ω^O_{t−1})/(1−ω^Y_{t−1} + n_t) s_{t−1} + ω^Y_{t−1}/(1−ω^Y_{t−1} + n_t)
  - s_{Y,t} = 1 / (1 + s_t)
  - 1 + g_{N,Y,t} = 1−ω^Y_{t−1} + n_t
  - 1 + g_{N,O,t} = (1−ω^O_{t−1}) + ω^Y_{t−1} / s_{t−1}
  - 1 + g_t = (1 + g_{N,Y,t}) (1 + s_t) / (1 + s_{t−1})
- Households (normalized):
  - C̃^O_t = MPC_{O,t} Ω^O_t T̃R_t + MPC_{O,t} (1 + r_{t−1})/(1 + g_t) [ ω^Y_{t−1} B̃^Y_{t−1} + B̃^O_{t−1} ]
  - 1 / MPC_{O,t} = (1 + τ^C_t) p^O_t + β^{1/γ} (1−ω^O_t) (1 + r_t)^{1/γ − 1} Λ^O_t E_t [ 1 / MPC_{O,t+1} ]
  - C̃^Y_t = MPC_{Y,t} Inc̃^Y_t + MPC_{Y,t} (1 + r_{t−1})/(1 + g_t) (1−ω^Y_{t−1}) B̃^Y_{t−1}
  - Inc̃^Y_t = (1−τ^L_t) w̃_t L̃_t + w̃^U_t Ũ_t + P̃rofit_t − L̃Tax_t + E_t [ (1 + g_{t+1}) T̃R^{YO}_{t+1} / (1 + r_t) Ω^O_{t+1} ] + E_t [ (1−ω^Y_t)/(1 + r_t) (1 + s_{t+1})/(1 + s_t) Inc̃^Y_{t+1} ]
  - 1 / MPC_{Y,t} = (1 + τ^C_t) p^Y_t + β^{1/γ} (1 + r_t)^{1/γ − 1} E_t [ (1−ω^Y_t) Λ^Y_t / MPC_{Y,t+1} + ω^Y_t Λ^{YO}_t / MPC_{O,t+1} ]
- Labor market (normalized) — wage bargaining:
  - σ/(1−σ) hc̃_t (1−τ^L_t)/(1 + τ^{SSC}_t) = (1−τ^L_t) w̃_t − w̃^U_t + E_t [ (1 + g_A_{t+1})/(1 + r_t) (1−pr^F_{t+1})(1−pr^H_{t+1}) ( σ/(1−σ) hc̃_{t+1} (1−τ^L_{t+1})/(1 + τ^{SSC}_{t+1}) ) ]
  - Ũ_t = s_{Y,t} − L̃_t
  - hc̃_t = κ (pr^H_t)^α HC
  - pr^H_t = (1 + g_{N,t}) H̃_t / ( Ũ_{t−1} + pr^F_t L̃_{t−1} )
- Producing firms (normalized):
  - (1/(1 + g_t)) K̃_{t−1} = α ( r^K_t / mc_t )^{−θ} Ỹ_t
  - q_t = (1−ω^Y_t)/(1 + r_t) ( r^K_{t+1} + q_{t+1} (1−δ) )
  - w̄̃_t = (1 + τ^{SSC}_t) w̃_t + λ̃^H_t − E_t [ (1−ω^Y_t)/(1 + r_t) (1−pr^F_{t+1}) (1 + g_{A,t+1}) λ̃^H_{t+1} ]
  - L̃_t = (1−α) ( w̄̃_t / (Ã_t mc_t) )^{−θ} Ỹ_t
  - mc_t = [ α (r^K_t)^{1−θ} + (1−α) ( w̄̃_t / Ã_t )^{1−θ} ]^{1/(1−θ)}
  - φ/(φ−1) mc_t = 1 + 1/(φ−1) R( (1 + π_t) (1 + π_{t−1})^γ ) + 1/(φ−1) R′(·) (1 + π_t (1 + π_{t−1}))^γ − E_t [ (1−ω^Y_t) φ^{−1} (1 + g_{t+1}) Ỹ_{t+1}/Ỹ_t R′(·) (...) ] / (1 + r_t)
- Retailers (normalized):
  - p^Y_t and p^O_t given by adjusted G_C,j(·) formulas in normalized variables (see original expressions)
- Fiscal policy and pension (normalized):
  - T̃R^{YO}_t = ν_t ω^Y_{t−1}/(1 + g_t) w̃_{t−1} L̃_{t−1}
  - T̃R_t = T̃R^{YO}_t + (1−ω^O_{t−1})/(1 + g_t) T̃R_{t−1}
  - Debt̃_t + L̃Tax_t + Tax̃_t = Goṽ_t + T̃R_t + (1 + r_{t−1})/(1 + g_t) Debt̃_{t−1}
  - Tax̃_t = τ^C_t ( p^O_t C̃^O_t + p^Y_t C̃^Y_t ) + (τ^{SSC}_t + τ^L_t) w̃_t L̃_t
  - Debt̃_t = B̃^Y_t + B̃^O_t
- Monetary policy (normalized):
  - 1 + i_t = (1 + i_{t−1})^{ρ_i} [ (1 + r^n_t) (1 + π_t)^{φ_π} ]^{1−ρ_i} e^{ε^i_t}
  - Fisher: 1 + i_t = E_t (1 + r_t) (1 + π_{t+1})
- Market clearing:
  - Ỹ_t = C̃^Y_t + C̃^O_t + Inṽ_t + Goṽ_t + R( (1 + π_t) (1 + π_{t−1})^γ ) Ỹ_t + hc̃_t H̃_t

### Steady-state solution procedure
- To compute steady state:
  - Solve demographic steady values:
    - s = ω^Y (1−ω^Y + n)^{-1} (1 − (1−ω^O)(1−ω^Y + n))^{−1} (expression as in text)
    - s_Y = 1/(1 + s)
    - 1 + g_{N,O} = 1−ω^O + ω^Y / s ; 1 + g = 1 + g_N ; 1 + g_N,Y = 1−ω^Y + n
  - Assume π = 0 implies i = r in steady state.
  - Tobin Q q = 1. No-arbitrage implies marginal product of capital:
    - r^K = (1 + r)/(1−ω^Y) − 1 + δ
  - Marginal cost mc = (φ−1)/φ (inverse markup).
  - From mc and r^K compute real wage w̄̃ and factor ratios:
    - K̃/Ỹ = (1 + g)^α ( r^K / mc )^{−θ}
    - L̃/Ỹ = (1−α) ( w̄̃ / (Ã mc) )^{−θ}
    - Inṽ/Ỹ = K̃/Ỹ [ 1 − (1−δ)/(1 + g) ]
    - H̃/Ỹ = L̃/Ỹ [ 1 − (1−pr^F)/(1 + g) ]
  - Wage-setting equilibrium uses hiring-cost-to-wage ratio WR = hc̃ / w̃ and unemployment benefit ratio WUR = w̃^U / w̃ to solve for pr^H and κ consistent with hc formula:
    - pr^H = 1 − [ σ/(1−σ) WR w̃ (1−τ^L) / (1 + τ^{SSC}) − (1−τ^L) w̃ + WUR w̃ ] / [ (1 + g_A)/(1 + r) (1−pr^F) (σ/(1−σ) WR w̃ / (1−τ^L) (1 + τ^{SSC})) ]
    - κ = hc̃ / ( pr^H^α HC )
  - Compute unemployment-to-GDP ratio and profit-to-GDP ratio:
    - Ũ/Ỹ = (1 + g_N) pr^H H̃/Ỹ − pr^F L̃/Ỹ
    - Profit̃/Ỹ = 1 − Inṽ/Ỹ − (1 + τ^{SSC}) w̃ L̃/Ỹ − hc̃ H̃/Ỹ
  - Use GDP identity to pin down levels and tax bases; compute steady pension TR̃/Ỹ using ν, ω^Y and wage/labor ratios and aggregate TR̃ via geometric survival factor:
    - TR̃^{YO}/Ỹ = ν ω^Y/(1 + g) w̃ L̃/Ỹ
    - TR̃/Ỹ = TR̃^{YO}/Ỹ (1 − (1−ω^O)/(1 + g))^{−1}
  - Given target Debt/Ỹ and Gov/Ỹ, compute lump-sum tax-to-GDP LTax̃/Ỹ to satisfy government budget (using Tax̃/Ỹ from distortionary taxes).
  - Compute MPC_O and MPC_Y in steady state (Λ terms equal one):
    - MPC_O = 1 / [ (1 + τ^C) p^O + β^{1/γ} (1−ω^O) (1 + r)^{1/γ − 1} Λ_O / MPC_O future ]
    - Closed-form expressions in steady state given in text:
      - MPC_O = [ 1 − (1−ω^O)(1 + r)^{1/γ − 1} β^{1/γ} ] / ( (1 + τ^C) p^O )
      - MPC_Y = [ 1 − β^{1/γ} (1 + r)^{1/γ − 1} (1−ω^Y) Λ_Y ] / [ (1 + τ^C) p^Y + β^{1/γ} (1 + r)^{1/γ − 1} ω^Y Λ^{YO} (1 / MPC_O) ]^{-1}
  - Pensioners’ discount factor steady state:
    - Ω_O = ( 1 − (1−ω^O)/(1 + r) )^{−1}
  - Compute young households’ expected lifetime income Inc̃^Y/Ỹ using initial guesses; compute B̃^Y/Ỹ from young budget constraint and MPC_Y; obtain B̃^O/Ỹ from bond market Debt̃/Ỹ − B̃^Y/Ỹ.
  - Compute C̃^O/Ỹ and C̃^Y/Ỹ from consumption functions; check aggregate consumption identity:
    - C̃/Ỹ ?= C̃^Y/Ỹ + C̃^O/Ỹ
  - Iterate initial guess for r until market-clearing condition holds (Newton–Raphson algorithm implied).

*Source: Annex 1 Technical description of the OGRE model (wpiea2020132-print-pdf) — IMF working paper chapter.*

### Annex 2 The Ageing Report’s accounting framework

### Annex 2 The Ageing Report’s accounting framework

### Accounting framework and decomposition of pension expenditure
- Pension expenditure (as ratio to GDP) is decomposed as:
  - population(65+) / population(20−64) × retirees / population(65+) × average pension income / GDP × hours worked × population(20−64) / hours worked
  - Summarized as: (dependency ratio) × (coverage ratio) × (benefit ratio) × (labour market effect)
- Changes in pension expenditures are explained by four effects:
  - Dependency ratio effect: quantifies the impact of changes in the old-age dependency ratio on pension expenditure.
  - Coverage ratio effect: looks at the number of pensioners relative to the population older than 64 years; captures developments of the effective exit age and the share of the population covered by the pension system.
  - Benefit ratio effect: indicates how average public pension spending develops relative to the average wage; assesses how changes to the legal framework of pension systems (concerning pension calculations and indexation rules) affect pension expenditure.
  - Labour market effect: describes the effect labour market behaviour/reforms have on pension expenditure.

### Comparison of driving factors of pension expenditures (OGRE vs Ageing Report)
- Table 8 summary (in % of GDP, unless stated otherwise):
  - Germany (OGRE vs AR):
    - Dependency Ratio Effect: 4.1 / 5.1
    - Coverage Ratio Effect: -0.4 / -1.0
    - Benefit Ratio Effect: -1.6 / -1.9
    - Labor Market Effect: -0.2 / -0.2
    - Residual: 0.0 / -0.2
    - Total: 1.9 / 1.8
  - Slovakia (OGRE vs AR):
    - Dependency Ratio Effect: 5.2 / 6.9
    - Coverage Ratio Effect: -1.0 / -3.2
    - Benefit Ratio Effect: -1.5 / -1.2
    - Labor Market Effect: -1.5 / -0.9
    - Residual: 0.0 / -0.6
    - Total: 1.2 / 1.0
- Note: The Ageing Report’s pension expenditures are rescaled to earnings-related pension expenditures.

### Calibration of structural parameters (Table 9)
- Name — Sign — Values (Germany / Slovakia where listed)
  - 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 α_HC: 0.5
  - Technology growth rate (%) g_A: 1.53
  - Inverse of intertemporal substitution γ: 1
  - Adjustment cost of physical capital investment φ_Inv: 2.5
  - Consumption habit φ_C: 4
  - Rotemberg price adjustment cost φ_P: 80
  - Price indexation γ_P: 0.5
  - Interest rate smoothing ρ_i: 0.5
  - Reaction to inflation in Taylor rule φ_π: 2.5

### Detailed calibration targets and calibrated values (Table 10) — selected series
- Germany (calibrated / targeted / source where given):
  - Consumption (total): 60.7 / 54.6 / Eurostat: National
  - Consumption (public): 19.1 / 19.1 / Accounts, 2011-2016
  - Investment: 20.1 / 20.0
  - Compensation of employees: 59.7 / 56.3
  - Unemployment rate (%): 5.7 / 5.6 / Eurostat
  - Unemployment benefit expenditure: 1.5 / 1.8 / 2009-2016
  - Firing probability (%): 13.9 / 13.9 / Eurostat: 2001-2015
  - Pension expenditure*: 7.2 / 7.8 / Ageing Report (2018): 2016
  - Benefit ratio (%)*: 34.1 / 32.4
  - Old-age dependency ratio** (%): 37.2 / 38.3
  - Fertility rate** (%): 1.9 / 1.9 / Eurostat
  - Probability of death** (%): 4.2 / 4.2 / 2006-2016
  - Probability of retirement** (%): 1.7 / 1.7
  - Value Added Tax Revenue: 7.0 / 7.0 / Eurostat: 1995-2016
  - Labor Income Tax Revenue: 8.7 / 8.7
  - Social Sec. Contr. Revenue: Employer: 6.5 / 6.5
  - Social Sec. Contr. Revenue: Employee: 6.2 / 6.2
  - Public debt: 75.4 / 75.4 / Eurostat: 2009-2016
- Slovakia (calibrated / targeted / source where given):
  - Consumption (total): 60.8 / 60.4 / Eurostat: National
  - Consumption (public): 16.9 / 16.9 / Accounts, 2011-2016
  - Investment: 22.3 / 20.2
  - Compensation of employees: 54.9 / 46.0
  - Unemployment rate (%): 12.8 / 12.7 / Eurostat
  - Unemployment benefit expenditure: 0.7 / 0.7 / 2011-2016
  - Firing probability (%): 11.1 / 11.1 / Eurostat: 2001-2015
  - Pension expenditure*: 6.9 / 6.7 / Ageing Report (2018): 2016
  - Benefit ratio (%)*: 34.4 / 36.3
  - Old-age dependency ratio** (%): 36.5 / 32.6
  - Fertility rate** (%): 2.6 / 2.6 / Eurostat
  - Probability of death** (%): 5.3 / 5.3 / 2002-2016
  - Probability of retirement** (%): 2.1 / 2.1
  - Value Added Tax Revenue: 6.5 / 6.5 / Eurostat: 2009-2016
  - Labor Income Tax Revenue: 2.9 / 2.9
  - Social Sec. Contr. Revenue: Employer: 7.3 / 7.3
  - Social Sec. Contr. Revenue: Employee: 3.0 / 3.0
  - Public debt: 53.3 / 53.3 / Eurostat: 2009-2016
- Notes:
  - *Earnings related pension expenditures
  - **Implied from population data and effective retirement rate

### Baseline scenario (figures and qualitative outcomes)
- Figures present deviations from initial steady-state for Germany and Slovakia across variables:
  - Germany (Figure 4): series shown for 2016, 2026, 2036, 2046, 2056, 2066 for:
    - GDP per capita (percentage) with contributions from: Contribution Rate, Benefit Ratio, Retirement Age, Aging, Total.
    - Share of Young Consumption (percentage of total consumption) with same contribution decomposition.
    - Unemployment Rate (percentage) with same contribution decomposition.
    - Public Debt (percentage of GDP) with same contribution decomposition.
  - Slovakia (Figure 5): series shown for 2016, 2026, 2036, 2046, 2056, 2066 for:
    - Public debt (percentage of GDP) with contributions from: Contribution Rate, Benefit Ratio, Retirement Age, Aging, Total.
    - GDP per capita (percentage) with same contribution decomposition.
    - Share of Young Consumption (percentage of total consumption) with same contribution decomposition.
    - Unemployment Rate (percentage) with same contribution decomposition.
- Visual emphasis: decomposition isolates the roles of Contribution Rate, Benefit Ratio, Retirement Age, and Aging in driving dynamics of GDP per capita, consumption composition, unemployment, and public debt over the projection horizon.

*Source: wpiea2020132-print-pdf - Annex 2 The Ageing Report’s accounting framework*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020132-print-pdf.pdf_