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---

### Introduction and objective
- Investigates quantitative impacts of social security reforms on retirement and sectoral behavior, considering population aging.
- Develops a general equilibrium life-cycle model with heterogeneous agents and endogenous sectoral and retirement decisions.
- Model economy includes three working sectors: private formal, private informal and the government, each with its own retirement system.
- Calibrated to match key aspects of Brazil; reproduces distribution of individuals across sectors, application behavior for public jobs, retirement decisions, aggregate social security deficit and inequality.

### Reforms analyzed
- Unification of public and private pension systems (including the 2013 removal of the “integrality” provision for public retirees); operationalized as:
  - Applies a ceiling on public benefits equal to private sector y_max;
  - Contribution limited to 11% of min{earnings, y_max};
  - Eligibility and formulas otherwise unchanged (proxy for Law 12.618 (2013)).
- Raising the minimum retirement age:
  - Imposed minimum retirement age of 65 (mimics Constitutional Amendment 103, November 12th, 2019 in part).
  - Section 6.5 considers increasing minimum retirement age up to 72.
- Counterfactuals abstract from other external factors (e.g., real GDP growth outside the model, changes in educational composition).

### Model structure and key primitives
- Life-cycle general equilibrium with heterogeneous agents, endogenous sectoral choice, and endogenous retirement.
- Sectors: public (government), private formal, private informal; retirees modeled separately.
- Public sector:
  - Non-competitive wages, public sector wage premium θ.
  - Hiring via nationwide open exam; applicants pay time cost capp(j) and success probability Papp.
- Firms:
  - Formal sector: Cobb-Douglas technology Y_f = A_f K^α N_f^{1−α}.
  - Informal sector: linear technology Y_i = A_i N_i.
- Taxes and social security:
  - Capital income tax τk, consumption tax τc, labor income taxes τy(m), social security contributions τss(m) up to ymax.
  - Retirees pay τb on benefits; informal workers do not pay labor income taxes nor contribute to SS.
- Preferences and demographics:
  - Lifetime expected utility E0[ ∑J j=1 βj (∏k=1^j ψk) u( ̃cj , lj ) ] with period utility u( ̃cj , lj ) = [ ̃cγ j · l1−γ j ]^(1−σ) / (1−σ).
  - Population growth rate gn; survival probabilities ψj; accidental bequests ζ rebated lump-sum.

### Social security modalities and benefit formulas (Brazil, 2013 rules)
- Private sector (INSS) modalities:
  - Contribution modality: eligible after contributing for 35 years.
  - Age modality: eligible at normal retirement age 65 with ≥15 years of contribution.
  - Benefit formula b(j_r, x, mod, t_C) = Ψ(j_r, mod, t_C) · x.
  - Contribution-modality replacement rate f(j_r, t_C) = [0.31 t_C / E(j_r)] · [1 + (j_r + 0.31 t_C)/100].
  - Age-modality replacement rate Ψ(j_r, age, t_C) = max{ f(j_r, t_C), ˜Ψ(t_C) } with ˜Ψ(t_C) = min{ 0.70 + t_C/100, 1 }.
  - Private contributions (2013 rules):
    - τ_SS(P) = 8% if y ≤ R$14,972
    - τ_SS(P) = 9% if R$14,972 < y ≤ R$24,954
    - τ_SS(P) = 11% if R$24,954 < min{y, y_max}
  - Private social security ceiling y_max = R$50,000 per year (2013 rule).
  - Minimum-benefit (LOAS replication): calibrated minimum wage (annual, 2013) = R$ 8,136.
- Public sector modalities:
  - Wage base (pre-2013): average of 80% highest wages = model uses last wage; no upper limit until 2013.
  - Mandatory retirement at age 70; eligibility requires ≥10 years as public servant.
  - Contribution-modality for civil servants older than 60 with ≥35 years: b(y, Contribution) = y.
  - Age-modality for civil servants older than 65: b(y, t_C, Age) = min{ t_C/35, 1 } · y.
  - Public workers pay 11% of income to social security (2013 tax code).
  - Public retirees pay 11% tax over benefits in excess of private ceiling.

### Calibration, data, and equilibrium targets (selected)
- Benchmark year: 2013.
- Key calibrated parameters and targets:
  - β = 0.96 targets K/Y = 2.42; model: 2.5.
  - γ = 0.248 targets Participation Rate 73%; model: 72.3%.
  - Af = 2.194 targets GDP per capita R$ 27,468; model: R$ 26,520.
  - Ai = 3.03 targets Informal sector size 19.5%; model: 18.9%.
  - θ Public sector wage premium = 9.8% (PNADC).
  - τy(P) = 18%; τy(G) = 9%; τk = 15.5%; τSS(G) = 11%; τb = 11%.
  - αG Government consumption/GDP = 18.9%; ̄NG Size of government = 5%; ymax = R$ 50,000.
  - J Maximum age = 60 (agents live from 16–75).
  - {ηj} Age-efficiency profile estimated from PNADC; Π(z,z′) from PNADC.
  - Time endowment ̄H = 8,760 hours; H = 2,016 hours.
  - σ Risk aversion = 2.5.
- Equilibrium aggregates reported:
  - Interest rate = 8.47%.
  - Formal wage rate = 4.03 Reais per hour (1.5 times informal wage).
  - Consumption tax rate = 34.86%.

### Benchmark (2013) factual moments and model fit (selected)
- Social Security Deficit (Brazil, % GDP) — Table 1:
  - 2013: Private 0.94, Public 1.18, SS Deficit 2.12
  - 2014: Private 1.00, Public 1.18, SS Deficit 2.19
  - 2015: Private 1.44, Public 1.22, SS Deficit 2.66
  - 2016: Private 2.17, Public 1.18, SS Deficit 3.35
- Number of retirees:
  - Public sector retirees: around one million people.
  - Private sector retirees: around 29 million people.
- Calibration fit (Table 5 and 6 highlights):
  - Overall SS deficit (% GDP): Model 2.1% | Data 2.1%
  - Public sector SS deficit (%GDP): Model 1.3% | Data 1.2%
  - Fraction of contrib. modality claims: Model 59% | Data 61%
  - Average age at retirement: Model 52.2 | Data 54.8
  - Participation rate (20–55): Model 81.6 | Data 89.0
  - Formal share (20–55): Model 53.5 | Data 62.1
  - Informal share (20–55): Model 21.6 | Data 20.7
  - Public sector share (20–55): Model 6.5 | Data 6.2

### Results with 2013 demography (steady-state comparisons; Table 7)
- SS deficit/GDP:
  - Benchmark 2.1 | Unification 1.6 | Minimum age -1.0 | Both reforms -1.1
- Public SS deficit/GDP:
  - Benchmark 1.3 | Unification 1.0 | Minimum age -0.2 | Both reforms 0.1
- Private SS deficit/GDP:
  - Benchmark 0.8 | Unification 0.6 | Minimum age -0.7 | Both reforms -1.2
- Consumption tax:
  - Benchmark 34.9 | Unification 34.0 | Minimum age 30.8 | Both reforms 30.4
- Macro aggregates (normalized to benchmark = 100.0 where indicated):
  - Avg. Consumption (C): Benchmark 100.0 | Unification 102.3 | Minimum age 99.1 | Both reforms 100.7
  - Avg. Capital (K): Benchmark 100.0 | Unification 104.5 | Minimum age 100.1 | Both reforms 107.2
  - Avg. Output (Y): Benchmark 100.0 | Unification 108.9 | Minimum age 98.9 | Both reforms 102.3
- Labor and sectoral outcomes:
  - Participation rate (out of total population): Benchmark 73.0 | Unification 73.5 | Minimum age 71.5 | Both reforms 71.5
  - Fraction of retirees (out of total population): Benchmark 13.9 | Unification 15.1 | Minimum age 8.0 | Both reforms 8.0
  - Informality (out of total workforce): Benchmark 19.5 | Unification 14.5 | Minimum age 20.5 | Both reforms 16.1
  - Public sector applicants: Benchmark 0.72 | Unification 0.74 | Minimum age 0.67 | Both reforms 0.69
- Key takeaways:
  - Unifying pension systems reduces SS deficit by 0.5% of GDP; public deficit decreases by 0.3% of GDP.
  - Imposing minimum retirement age of 65 can generate a total pension surplus of 1% of GDP in the 2013-demography steady state.
  - Minimum-age reform lengthens contribution periods but may reduce participation and increase informality, lowering output in the minimum-age-only scenario.

### Long-run demography and 2060 scenarios (section 6.3; Table 8)
- Demographic shifts (2013 → 2060):
  - Share aged 65+ increases from 8% to 18.7%.
  - Fraction of retirees rises from 13.9% in 2013 to 31.5% in 2060.
  - Participation rate projected to decline (2060 w/o reforms participation rate 60.0).
  - Informal share of labor predicted to fall to 7.1% in 2060 (2060 w/o reforms).
- Fiscal outcomes (2013 → 2060 without reforms):
  - Overall SS deficit/GDP: 2.1 → 16.6.
  - Public SS deficit/GDP: 1.3 → 5.2.
  - Private SS deficit/GDP: 0.8 → 11.5.
  - Consumption tax required: 34.9% → 54.6%.
- 2060 steady-state comparisons (selected values from Table 8):
  - SS deficit/GDP: 2060 w/o reforms 16.6 | Unification 14.6 | Minimum age 10.8 | Both reforms 10.2.
  - Public SS deficit/GDP: 2060 w/o reforms 5.2 | Unification 3.2 | Minimum age 1.8 | Both reforms 1.4.
  - Private SS deficit/GDP: 2060 w/o reforms 11.5 | Unification 11.4 | Minimum age 9.0 | Both reforms 8.9.
  - Consumption tax: 2060 w/o reforms 54.6 | Unification 54.6 | Minimum age 51.7 | Both reforms 45.4.
  - Avg. Consumption (C) normalized: 2060 w/o reforms 106.7 | Unification 110.9 | Minimum age 107.0 | Both reforms 109.6.
  - Avg. Capital (K) normalized: 2060 w/o reforms 109.7 | Unification 120.4 | Minimum age 112.0 | Both reforms 119.7.
  - Avg. Output (Y) normalized: 2060 w/o reforms 99.3 | Unification 104.7 | Minimum age 99.4 | Both reforms 103.0.
  - K/Y normalized: 2060 w/o reforms 110.5 | Unification 115.0 | Minimum age 112.7 | Both reforms 116.2.
  - Participation rate: Benchmark 73.0 | 2060 w/o reforms 60.0 | Both reforms 58.7.
  - Fraction of retirees: Benchmark 13.9 | 2060 w/o reforms 31.5 | Both reforms 18.7.
  - Informality (out of total workforce): Benchmark 19.5 | 2060 w/o reforms 7.1 | Both reforms 6.3.
  - Public sector applicants: Benchmark 0.72 | 2060 w/o reforms 0.46 | Both reforms 0.39.
- Interpretation:
  - Implementing both reforms reduces SS deficit by almost 40% relative to 2060 no-reform case (16.6 → 10.2).
  - Minimum retirement age is the dominant reform for reducing retiree shares (31.5% → 18.7%).
  - Even with both reforms, a substantial residual deficit (~10.2% of GDP) remains in 2060.

### Role of informality and public sector responses
- Endogenous sectoral reallocation matters materially:
  - Omitting public sector yields much smaller estimated SS deficits: 2013 no-public-sector SS deficit 0.5% vs full-model 2.1%; 2060 no-public-sector SS deficit 11.7% vs full-model 16.6%.
  - Raising minimum retirement age has much smaller deficit-reduction effects without the public sector.
- Reforms trigger sectoral movements:
  - Unification can push high-skill individuals out of public sector, with productivity consequences.
  - Minimum-age reforms reduce incentives to take costly public exam, inducing reallocation toward informal sector or out of labor force for some groups.
- Table 9 and 10 decompositions show that fixing public-sector or informal worker behavior changes deficit and tax outcomes by multiple percentage points of GDP.

### Zero-deficit and required retirement ages (section 6.5)
- To eliminate SS deficit by 2060 under joint reform assumptions:
  - Minimum retirement age would need to be above 71 (i.e., at least 72 in scenarios explored).
- Effects of increasing minimum retirement age from 65 → 72 (Figure 12 interpretation):
  - Change in GDP increases range from 3.7% to 17% as minimum retirement age varies from 65 to 72.
  - Informality declines from 6.3% to 1.9% over the same range.
  - Mechanisms: longer work lives raise tax contributions and production, reduce retirement expenditures, lower equilibrium consumption tax, increase aggregate savings and capital, reduce interest rates, raise formal wages, and increase formal employment demand.

### Welfare and distributional impacts (section 6.6)
- Aggregate welfare (consumption-equivalent measure Δw) across retirement-age scenarios:
  - Long-run aggregate welfare increases between 2.9% with minimum retirement age of 65 and 10.5% with retirement age 72.
- Heterogeneous cohort effects:
  - Younger cohorts (16–30 and 31–45) uniformly gain from reforms due to higher GDP and lower consumption taxes.
  - Older cohorts (46–60 and 61–75) can be worse off under some reforms:
    - 46–60 group would fall by 7.9% under benchmark reforms (minimum retirement age 65) because many were retiring between 52 and 60.
    - 61–75 group prefers the old system if minimum retirement age exceeds 69; welfare improves for this group when minimum age is sufficiently high and macro gains offset stricter retirement conditions.
- Policy implication: aggregate welfare gains mask sizable distributional trade-offs; older cohorts face concentrated losses while younger cohorts capture gains.

### Policy implications and trade-offs (summary)
- Population aging alone implies a large fiscal challenge: SS deficit rising from 2.1% of GDP (2013) to 16.6% of GDP (2060) absent reform.
- Combining unification of pension systems and increasing minimum retirement age delivers the largest fiscal savings and reduces retiree shares substantially but does not eliminate the deficit in baseline joint-reform scenario (residual ~10.2% of GDP).
- Eliminating the deficit by 2060 plausibly requires minimum retirement ages above 71, which may be politically challenging given historical protests and distributional losses for older cohorts.
- Accounting for endogenous informality and public sector responses is critical for accurate assessment: ignoring these margins materially misestimates deficit and macro outcomes.
- Trade-offs:
  - Fiscal sustainability and higher long-run GDP, capital, and aggregate welfare versus concentrated welfare losses for older cohorts and potential short-run contractionary effects in some reform permutations.
  - Unification may reduce public-sector generosity but can induce high-skill exits from public employment, with productivity implications.

### Numerical solution and computational details (selected)
- Numerical implementation:
  - Solved in Fortran 90.
  - Discretized grids: assets = 52 points, average past earnings x = 10 points, income process z = 5 points, benefits b = 42 points.
  - Asset grid range: R$0 to R$3,790,202; x grid between 0 and ymax; b grid between R$0 and R$100,000.
- General equilibrium algorithm (iterative):
  1. Guess Θ ≡ (r, YG, ζ, τc, ̄q).
  2. Use formal firm FOCs to obtain wf.
  3. Solve agents’ problems backwards for policy functions.
  4. Forward induction to compute stationary distributions.
  5. Find ̃̄q to target ̄NG public sector size.
  6. Compute implied ̃Θ and check convergence ||̃Θ − Θ|| < ε; iterate as needed.

*Source: IMF Working Paper — Social Security Reforms, Retirement and Sectoral Decisions (calibrated to Brazil), excerpted from wpiea2025032-print-pdf.*

### 6.22013 Demography    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 

### Social Security Reforms, Retirement and Sectoral Decisions

### Introduction and Objective
- Investigates quantitative impacts of social security reforms on retirement and sectoral behavior, considering population aging.
- Develops a general equilibrium life-cycle model with heterogeneous agents and endogenous sectoral and retirement decisions.
- Model economy includes three working sectors: private formal, private informal and the government, each with its own retirement system.
- Calibrated to match key aspects of Brazil; reproduces distribution of individuals across sectors, application behavior for public jobs, retirement decisions, aggregate social security deficit and inequality.

### Demographic and Fiscal Motivation
- World life expectancy: increased from 65 years in 2000 to over 70 years nowadays (UN (2013) cited).
- Old age support ratio: expected to decrease from 3.8 nowadays to 2.1 in 2050 and 1.6 in 2100 for OECD countries.
- Example fiscal pressures:
  - 1998 Brazilian crisis: a fiscal deficit of more than 6% of GDP contributed to the crisis, with two-thirds of the deficit due to pension costs.
  - Lebanon: public retirees’ pensions are the third greatest expenditure item in the government’s budget, even though they account for less than 3% of the population.
- PAYG systems widespread; over half of OECD countries have different pension rules for workers in the public and private sectors.
- Majority of public sector pension schemes remain defined-benefit.

### Reforms Analyzed
- Unification of public and private pension systems (including the 2013 removal of the “integrality” provision for public retirees).
- Raising the minimum retirement age (including the 2019 reform that introduced a minimum retirement age of 65).
- Counterfactuals abstract from other external factors (e.g., real GDP growth outside the model, changes in educational composition) to isolate pension reform effects.

### Model and Literature Positioning
- Extends the literature by allowing endogenous retirement and endogenous sectoral choice across public, private formal, and private informal sectors.
- Addresses limitations of prior work that treated retirement as exogenous or omitted multiple working sectors and informality.
- Incorporates informality and an overpaid, stable public sector to assess how reforms affect sectoral composition and fiscal outcomes.

### Key Quantitative Findings (Brazil calibration)
- Demography-only (2060) effects without reforms:
  - Social security deficit: increases from 2.1% to 16.6% of GDP.
  - Participation rate: projected decline by 13 percentage points.
  - Proportion of retirees: rises from 13.9% in 2013 to 31.5% in 2060.
  - Public sector portion of deficit: increases from 1.3% to 5.2% of GDP.
  - Private sector portion of deficit: increases from 0.8% to 11.5% of GDP.
- Combined reforms (unification + minimum retirement age = 65):
  - Potential decrease in social security deficit of almost 40% relative to the demographic-only scenario.
  - Proportion of retired people falls from 31.5% to 18.7%.
  - Share of applicants to a government job falls by 15% as public sector attractiveness declines.
- Effectiveness of reforms:
  - Only raising the minimum retirement age can effectively decrease the number of retired people by forcing longer labor market participation.
  - Even with reforms, a remaining social security deficit of 10.2% of GDP is possible; to completely eliminate the deficit the minimum retirement age would need to increase to 72.
- Macroeconomic and welfare effects:
  - Positive impact on GDP as taxes decrease and labor market participation increases.
  - Long-run aggregate welfare changes are positive, but older cohorts would be worse off under reforms implemented in the past compared with no reforms.

### Role of Informality and Public Sector
- Incorporation of the public sector and informal economy materially affects results:
  - Omitting the public sector produces significantly smaller estimated social security deficits for 2013 and 2060 than the full model.
  - The effect of raising the minimum retirement age is considerably less pronounced without modeling the public sector and informality.
- Unification of pension systems can prompt high-skill individuals to exit the public sector, with potential productivity consequences.

### Policy Implications and Trade-offs
- Unifying public and private pension systems and increasing minimum retirement age together yield the largest fiscal savings and reduce retiree shares substantially.
- Political economy constraints: raising the minimum retirement age to levels (e.g., 72) that would eliminate the deficit may be politically challenging, as seen in protests in countries that raised retirement ages.
- Distributional trade-offs: aggregate welfare gains do not imply uniform gains across age groups; policymakers should weigh losses to older cohorts against long-run fiscal sustainability and GDP gains.

### Concluding Notes
- The paper provides a quantitative tool to analyze pension reforms that jointly considers informality, an overpaid and secure public sector, endogenous retirement, and sectoral choice.
- Results highlight that reforms targeting both pension system segregation and retirement age have the potential to materially reduce fiscal pressures induced by demographic change, but significant trade-offs and political constraints remain.

*Source: IMF Working Paper — Social Security Reforms, Retirement and Sectoral Decisions (calibrated to Brazil).*

### 2050. Sources: OECD (2017).

### 2050. Sources: OECD (2017).

### Retirement system heterogeneity and cross-country facts
- Several countries have retirement systems that vary according to the employment sector; public employees tend to have better retirement conditions relative to their private sector peers.
- On average, public sector pensions account for 20% of overall pensions expenditures.
- The general government accounts for, on average, 17% of the labor force, and their retirement benefits sum up to 1.5% of GDP.
- Unification of retirement schemes for private and public sector workers has been a focus of many proposed or implemented reforms (examples cited: France and Brazil).

### Why Brazil is used for the quantitative exercise
- Three reasons Brazil is considered representative of countries facing social security challenges:
  - Demographics: Life expectancy increased from 54.7 years in 1960 to 73.6 in 2012; IBGE projects the share of individuals aged 65 and over will grow from 4% in 1980 to 22% around 2050.
  - High pension spending relative to demographics: Figure 4 shows Brazil is as “young” as Chile and Mexico but spends a similar share of GDP with social security as Germany and the OECD average.
  - Large discrepancy between government employment share and government retirees’ pensions: government size is 11.4% of the labor force, while government retirees’ pensions represent 2% of GDP.
- Table 1: Social Security Deficit (Brazil, % GDP)
  - 2013: Private 0.94, Public 1.18, SS Deficit 2.12
  - 2014: Private 1.00, Public 1.18, SS Deficit 2.19
  - 2015: Private 1.44, Public 1.22, SS Deficit 2.66
  - 2016: Private 2.17, Public 1.18, SS Deficit 3.35
- Number of retirees:
  - Public sector retirees: around one million people.
  - Private sector retirees: around 29 million people.

### Key facts about Brazilian retirement system and labor markets
- Brazilian public pension system comprises three defined-benefit schemes:
  - mandatory private sector regime (Regime Geral de Previdência Social, RGPS);
  - mandatory public sector regime (Regimes Próprios de Previdência Social, RPPS);
  - noncontributory branch for rural workers, disabled people, and other low-income families.
- Lack of a minimum retirement age in the private sector until 2022 made early retirement prevalent.
- Table 2: Brazilian Early Retirement (ages and expected durations)
  - OECD: Retirement Age Men 64, Women 63; Expected Duration Men 16, Women 21
  - Latin America: Retirement Age Men 62, Women 60; Expected Duration Men 17, Women 21
  - World: Retirement Age Men 62, Women 60; Expected Duration Men 16, Women 21
  - Brazil (Private: Contribution): Retirement Age Men 56, Women 52; Expected Duration Men 23, Women 29
  - Brazil (Private: Age Modality Rural): Retirement Age Men 60, Women 55; Expected Duration Men 19, Women 26
  - Brazil (Private: Age Modality Urban): Retirement Age Men 65, Women 60; Expected Duration Men 16, Women 22
- Replacement rate: Brazil’s pension system has a replacement rate of 88%, substantially above the OECD average of 63.8% (OECD (2023)).
- Public sector job characteristics:
  - Public sector wage premium around 10% (Table 4).
  - Using 2013 PNAD data, average tenure: private sector job 5.3 years; public sector job 13.6 years.
  - Estimated public sector wage premium of 9.4% from regressions on 2012-2018 PNADC data (see Table 4).
- Education and public sector:
  - Statutory public servants account for 5% of the population but represent nearly 20% of individuals with more than 15 years of education.
  - Individuals taking the public exam represent 6% of the population but 23% of individuals with more than 15 years of education.
- Informality:
  - Large share of informal workers who do not pay most taxes and are not, in principle, under the protection of the Social Security Administration.
  - According to 2013 PNAD data, 18% of males aged 16 to 75 are in the informal private sector.
  - Only 17% of informal workers contributed to the social security system in 2013.
  - Informal sector absorbs on average less schooling and lower incomes.

### Implications for model design
- A model studying macroeconomic and sectoral consequences of social security reforms should include:
  - (i) a well-paid and stable public sector with costly access,
  - (ii) a public pension system that is relatively more attractive and expensive than the private pension system,
  - (iii) an informal sector with individuals that do not contribute to the social security system but may benefit from it.

### Model overview: structure and agents
- General equilibrium life-cycle model with heterogeneous agents, sectoral choices, and retirement behavior.
- Agents can be in the private sector, working for the government, or retired. All decisions are endogenous.
- Three working sectors: public, private formal, private informal.
- Government pays non-competitive wages to public workers, manages a PAYG retirement system for both public and private retirees, and taxes consumption, capital, and labor to balance its budget.
- Formal private firms use a Cobb-Douglas technology with capital and labor; informal firms produce with labor only. Goods are perfect substitutes. Firms act competitively.
- Uncertainty: idiosyncratic labor efficiency shocks and life span uncertainty. Agents save in a risk-free asset.

### Demography, preferences, and choices (model primitives)
- Population: continuum of mass one agents living at most J periods; time endowment ̄H hours per period.
- Survival probabilities: ψj probability of surviving to age j+1; population law of motion φj+1 = ψj / (1+gn) φj; ∑J j=1 φj = 1; population growth rate gn.
- Accidental bequests (ζ) equally distributed lump-sum to survivors; no private annuity markets.
- Lifetime expected utility:
  - E0[ ∑J j=1 βj (∏k=1^j ψk) u( ̃cj , lj ) ]
  - Period utility u( ̃cj , lj ) = [ ̃cγ j · l1−γ j ]^(1−σ) / (1−σ)
- Effective consumption ̃c = c + ε YG, where YG is public good consumption; parameters σ (risk aversion), γ (consumption share), ε (importance of public consumption).
- Labor choice discrete: {0, H < ̄H}; public servants mandatory to work; private workers and retirees choose.
- Agents can take a costly public sector entry exam with time cost capp(j); retirees cannot enter public sector; public workers cannot retake exam.
- Retirement and sectoral states are absorbing: informal private workers retire as private retirees; once in public sector, remain until retirement; retirees cannot reapply for public jobs.

### Labor income, productivity, and budget constraints
- Wages: wf (formal) and wi (informal) competitive wages.
- Output (pre-tax) if working for age j and sector m:
  - yj(ω,m) = ω e^{z+ηj} · hj, if m = Private or Retirement, with ω ∈ {wf, wi}
  - yj(ω,G) = (1+θ) wf e^{zG+ηj} · H, if m = Government
- Idiosyncratic productivity z follows first-order Markov process with transition matrix Π.
- Public sector has no uncertainty; zG is productivity at time of successful public exam.
- ηj deterministic age-specific labor efficiency; θ is public sector wage premium.
- Taxes and contributions:
  - Capital income tax τk and consumption tax τc apply to all.
  - Labor income tax τy(m); non-retired must additionally contribute τss(m) of labor income up to a maximum taxable income ymax to social security.
  - Revenue from τss(m) finances social security benefits; revenue from τy(m) finances other government expenditures.
  - Retirees pay tax rate τb on benefits. Informal workers do not pay labor income taxes nor contribute to SS.
- Borrowing constraint: assets nonnegative a_{j+1} ≥ 0; a_{J+1} = 0.
- Budget constraints:
  - Private formal: (1+τc) c + a' = [1+(1−τk) r] a + (1−τy(P)) yj(wf,P) − τss(P) min{ yj(wf,P), ymax } + ζ
  - Private informal: (1+τc) c + a' = [1+(1−τk) r] a + yj(wi,P) + ζ
  - Public sector: (1+τc) c + a' = [1+(1−τk) r] a + (1−τy(G) − τss(G)) yj(wf,G) + ζ
  - Retirees (formal): (1+τc) c + a' = [1+(1−τk) r] a + (1−τy(P)) yj(wf,R) + (1−τb) b + ζ
  - Retirees (informal): (1+τc) c + a' = [1+(1−τk) r] a + yj(wi,R) + (1−τb) b + ζ
- Retirement benefits b are calculated upon retirement with rules depending on worker’s state vector (detailed in Section 4 of source).

### Public sector recruitment mechanism (modeled on Brazilian practice)
- Hiring by public competition: nationwide open exam; best scores fill vacancies.
- In model: agents pay time cost capp(j) at age j to apply; score q_{jz} revealed next period:
  - log(q_{jz}) = log(e^{z+ηj}) + ν, where ν ∼ N(0,1).
- Test score decomposed into skill/age component and random component; random draws allow lower-skill individuals to pass.
- If q_{jz} ≥ q̄, agent becomes public servant from age j+1 onward; otherwise remains private.
- Threshold q̄ chosen by government in equilibrium to balance demand and supply of public servants.
- Probability of approval defined as Papp ≡ Pr(q_{jz} ≥ q̄).

*Source: Excerpt from IMF Working Paper (wpiea2025032-print-pdf).*

### 3.4    Social Security

### 3.4    Social Security

### Social security modalities and benefit determination
- Private sector workers face two retirement modalities: "contribution" and "age".
  - Contribution modality: worker may apply for retirement after contributing for a number of years (35, in the case of Brazil) even if they have not reached the normal retirement age.
  - Age modality: worker may apply for retirement only after reaching the normal retirement age (65 years old in the case of Brazil) and having contributed for a minimum number of years.
  - Agents in the model are free to choose the retirement modality that suits them best.
- Retirement benefits, b, are calculated as a replacement rate over average past earnings, x.
  - The replacement rate depends on the age of retirement, the modality, the number of years of contribution, among other institutional factors (detailed in Section 4).
- Public workers also have two modalities:
  - Contribution modality for public workers older than a certain age (60 years old in the case of Brazil) who contributed for a minimum number of years: replacement rate of 100% over their current labor income.
  - Age modality for civil servants older than 65: individuals are entitled only to a proportion of their last wage, depending on the number of years of contribution.

### Key model states related to social security
- Private sector agent state: s_P = (j,a,z,x,t_C) ∈ S_P ≡ {1,...,J} × R_+ × Z × X × {0,...,J}
  - j is age, a is asset holdings, z is idiosyncratic productivity, x is average past earnings, t_C is number of years contributed to SS.
- Public worker state: s_G = (j,a,z,t_C,t_G) ∈ S_G ≡ {1,...,J} × R_+ × Z × {0,...,J} × {0,...,J}
  - t_G is number of years worked in the government.
- Retiree state: s_R = (j,a,z,b) ∈ S_R ≡ {1,...,J} × R_+ × Z × B, where b denotes retirement benefits.

### 3.5    Value Functions

### Policy functions solved from recursive problems
- Policy functions for all m ∈ {P,G,R}:
  - working hours d_h(s_m) ∈ {0,H}
  - asset holdings d_a(s_m) ∈ R_+
  - consumption d_c(s_m) ∈ R_++
  - retirement d_ss(s_m) ∈ {0,1} for m ∈ {P,G}
  - informality d_inf(s_m) ∈ {0,1} for m ∈ {P,R}
  - public sector application d_app(s_P) ∈ {0,1}

### 3.5.1 Retired workers (value function)
- Retiree chooses between working formally, informally, or staying at home. Value function:
  - V(s_R) = max_{c,a',h,inf} u(ĉ, H̄ − h) + β ψ_j · E[V(s'_R)]
  - Budget constraint: (1+τ_c) c + a' =
    - [1+(1−τ_k) r] a + (1−τ_y(P)) y_j(w_f,P) + (1−τ_b(R)) b + ζ, if inf = 0
    - [1+(1−τ_k) r] a + y_j(w_i,P) + (1−τ_b(R)) b + ζ, if inf = 1
  - Constraints: (c,a') ≥ 0; h ∈ {0,H}; inf ∈ {0,1}
  - Next-period state: s'_R = (j+1,a',z',b)
  - Expectation: E[V(s'_R)] = ∑_{z'} Π(z,z') V(s'_R)

### 3.5.2 Public servants (value function)
- At each age j public sector worker decides retire or not (ret ∈ {0,1}).
  - V(s_G) = max_{c,a',ret} u(ĉ, H̄ − H) + β ψ_j · { (1−ret)·V(s'_G) + ret·V(s'_R) }
  - Budget constraint: (1+τ_c) c + a' = [1+(1−τ_k) r] a + (1−τ_ss(G) − τ_y(G)) y_j(w_f,G) + ζ
  - Constraints: (c,a') ≥ 0; ret ∈ {0,1}
  - Next-period states:
    - If not retire: s'_G = (j+1,a',z,t_C+1,t_G+1)
    - If retire: s'_R = (j+1,a',z,b' = b(s_G))
  - b(s_G) denotes retirement benefits as function of state variables.

### 3.5.3 Private workers (value function and decisions)
- Formal private worker decisions at each age j: consumption/saving, whether to retire (ret), whether to work formally/informally or stay home (h, inf), and whether to take the public exam (app).
  - V(s_P) = max_{c,a',h,ret,app,inf} u(ĉ, H̄ − h) + β ψ_j · {
      ret·E[V(s'_R)]
      + (1−ret)·app·P_app·V(s'_G)
      + (1−ret)·app·(1−P_app)·E[V(s'_P)]
      + (1−ret)·(1−app)·E[V(s'_P)]
    }
  - Budget constraint: (1+τ_c) c + a' =
    - [1+(1−τ_k) r] a + (1−τ_y(P)) y_j(w_f,P) − τ_ss(P) min{ y_j(w_f,P), y_max } + ζ, if inf = 0
    - [1+(1−τ_k) r] a + y_j(w_i,P) + ζ, if inf = 1
  - Evolution of (x',t'_C):
    - If inf = 0 and h = H:
      - x' = ((t_C − 1) x + min{ y_j(w_f,P), y_max }) / t_C
      - t'_C = t_C + 1
    - otherwise: (x,t_C) unchanged
  - Time/resource constraint: l + h + app · c_app(j) = H̄
  - Constraints: (c,a') ≥ 0; h ∈ {0,H}; ret, app, inf ∈ {0,1}
  - Next-period states:
    - If retire: s'_R = (j+1,a',z',b' = b(s_P))
    - If entering public career: s'_G = (j+1,a',z,t'_c,1)
    - If continue private: s'_P = (j+1,a',z',x',t'_c)

### 3.6    Agents’ Stationary Distribution
- Stationary distribution functions μ_m : S_m → [0,1] for m ∈ {P,G,R} with ∑_{(m,s_m)|j} μ_m(s_m) = φ_j for all j ∈ {1,...,J}.
- Stationary distribution constructed by forward induction using policy functions, given assumptions on entry (j = 1):
  - (i) every agent starts life cycle with zero initial assets, zero average past earnings and zero time of contribution;
  - (ii) everybody starts as a worker in the formal private sector;
  - (iii) initial distribution of idiosyncratic productivity is the invariant distribution of the Markov process for {z}.
- For remaining ages, distributions derived via forward induction considering policy functions, income transition matrix, survival probabilities, and probability of succeeding in public exam and entering public sector.

### 3.7    Technology (sector production)
- Two representative firms produce perfect substitute goods: formal and informal sector.
- Formal sector:
  - Production function: Y_f = F(K,N_f) = A_f K^α N_f^{1−α}
  - Capital depreciates at rate δ each period.
  - Formal firm problem: max_{K,N_f} K^α N_f^{1−α} − (1+τ_w) w_f N_f − (r+δ) K
  - τ_W is a tax rate charged by government to finance social security.
- Informal sector:
  - Linear technology in labor: Y_i = A_i N_i
  - Profit maximization: Π_i = max_{N_i} A_i N_i − w_i N_i
- Simplifying assumptions:
  - Abstract from firm heterogeneity within sectors; costs of informality are implicitly embedded in A_i.

### 3.8    The Government Sector
- Government taxes consumption, capital, income, wages, and social security benefits to finance:
  - social security coverage,
  - payroll of public servants and its own non-productive consumption.
- Government consumption: C_g = α_g Y_f (constant fraction of formal GDP).
- Government hires a share N̄_G ∈ [0,1] of the population as public servants to produce public good Y_G with production function Y_G = L_G (linear in effective labor supply).
- Government chooses N̄_G to balance demand and supply of public workers; empirical note: over 2005-2013 the Brazilian government employed consistently around 5% of the population (calculated using PNAD data from 2005-2013).

### 3.9    Equilibrium
- Recursive competitive equilibrium consists of allocations of households and firms, prices (wages and interest rate), government taxes and threshold score, a stationary distribution of agents, bequests and public goods such that:
  - (i) households and firms optimize;
  - (ii) individual and aggregate behaviors are consistent;
  - (iii) government sets threshold scores and consumption taxes to balance size of public sector in population and its budget constraint;
  - (iv) stationary distributions evolve according to policy functions;
  - (v) amount of public good and amount of bequests are consistent with individual behavior.
- Appendix B contains a complete equilibrium definition; Appendix C details the algorithm used to compute the equilibrium.

*Source: IMF Working Paper — excerpted section 3.4–4 (Social Security, Value Functions, Stationary Distribution, Technology, Government Sector, Equilibrium, Data and Calibration).*

### 4.4    Social Security System

### 4.4    Social Security System

### Overview
- The social security in Brazil is a pay-as-you-go system, which transfers income from workers to retirees.
- The system is financed with payroll taxes and has two different regimes: the private sector regime and the public sector regime.
- The benchmark year of calibration is 2013; retirement benefits structure is modeled to mimic retirement rules that prevailed in Brazil in 2013, before two major social security reforms.

### Age-efficiency and income process (estimation highlights)
- Age-efficiency profile plotted using age coefficients estimated from 2012-2018 PNADC.
- Table 4 (regression results, PNADC 2012q1-2018q4, men 16-75):
  - Public servant coefficients: 0.169*, 0.094*, 0.090* (standard errors: (0.004), (0.005), (0.006))
  - Informal worker coefficients: -0.188*, -0.056*, -0.053* (standard errors: (0.002), (0.002), (0.003))
  - Age and Age^2 coefficients vary by specification; e.g., Age 0.061*, 0.032*, 0.031*, 0.029*; Age^2 -0.00063*, -0.00028*, -0.00033*, -0.00031*.
  - Sample sizes: N = 1,832,912; 2,053,458; 1,773,520; 1,270,749 across columns.
  - R^2 reported as 0.196, 0.615, 0.899, 0.898.
- Controls include tenure, race (when no individual FE), occupation, sector, and education fixed effects. Robust standard errors reported. *p<0.001.

### 4.4.1 Private Benefits (INSS)
- Two retirement modalities under INSS: age modality and contribution modality.
- Eligibility:
  - Age modality: worker older than normal retirement age 65 and contributed more than 15 years.
  - Contribution modality: worker not at normal retirement age but contributed more than 35 years.
- Benefit formula:
  - b(j_r, x, mod, t_C) = Ψ(j_r, mod, t_C) · x
  - mod ∈ {Contrib, Age}; Ψ depends on retirement age j_r, modality mod, and years contributed t_C.
- Average lifetime earnings x computed as:
  - x_{j+1} = x_j (t_C−1) + min{y_j(w_f,P), y_max} over t_C, for j = 1,2,...,j_r
  - Only earnings from the formal sector are considered in x (informal workers assumed not to contribute).
- Contribution-modality replacement rate:
  - Ψ(j_r, Contrib, t_C) = f(j_r, t_C)
  - f(j_r, t_C) = [0.31 t_C / E(j_r)] · [1 + (j_r + 0.31 t_C)/100]
  - E(j_r) is life expectancy at retirement age j_r. Depending on j_r and t_C, f(j_r, t_C) can be greater than 1.
  - Example: a 58 years old worker with 35 years of contribution receives 80% of past earnings; same worker with 45 years receives 106%.
- Age-modality replacement rate:
  - Ψ(j_r, age, t_C) = max{ f(j_r, t_C), ˜Ψ(t_C) }
  - ˜Ψ(t_C) = min{ 0.70 + t_C/100, 1 }
  - 85% starting point for age modality comes from 70% in ˜Ψ plus 15% required for eligibility.
- Minimum-benefit provision:
  - Individuals older than 65 earning less than one fourth of the minimum wage entitled to minimum wage pension (LOAS replication).
  - Calibrated minimum wage (annual, 2013) = R$ 8,136.
- Private contributions (2013 rules):
  - τ_SS(P) = 8% if y ≤ R$14,972
  - τ_SS(P) = 9% if R$14,972 < y ≤ R$24,954
  - τ_SS(P) = 11% if R$24,954 < min{y, y_max}
- Private retirees have no tax on benefits.
- Private social security ceiling y_max set to R$50,000 per year (2013 rule).

### 4.4.2 Public Benefits
- Public servants have a different pension system with two modalities: contribution and age.
- Wage base (public sector, pre-2013): average of the 80% highest wages during public career; in model equals last wage (no upper limit until 2013).
- Mandatory retirement at age 70; eligibility requires at least 10 years working as a public servant.
- Contribution-modality (civil servants older than 60 with ≥35 years) benefits:
  - b(y, Contribution) = y (benefits match current labor income).
- Age-modality (civil servants older than 65) benefits:
  - b(y, t_C, Age) = min{ t_C/35, 1 } · y
- Public workers compulsorily pay 11% of income to social security (2013 tax code).
- Public retirees pay an 11% tax over benefits in excess of the private social security ceiling.

### Calibration moments and model fit (selected)
- Table 5: Non-target moments (model vs. data)
  - Overall SS deficit (% GDP): Model 2.1% | Data 2.1%
  - Public sector SS deficit (%GDP): Model 1.3% | Data 1.2%
  - Fraction of contrib. modality claims: Model 59% | Data 61%
  - Average age at retirement: Model 52.2 | Data 54.8
  - Average years contributed at retirement: Model 35.0 | Data 35.3
  - Mean effective earnings Formal/Informal: Model 1.4 | Data 1.5
  - Income GINI coefficient: Model 53.2 | Data 52.7
- Table 6: Sectoral composition of working age population (20-55), Model vs. Data
  - Participation rate: Model 81.6 | Data 89.0
  - Private sector: Model 75.1 | Data 82.8
  - Formal: Model 53.5 | Data 62.1
  - Informal: Model 21.6 | Data 20.7
  - Public sector: Model 6.5 | Data 6.2
  - Retirement (20-55): Model 2.2 | Data 3.1
- Equilibrium aggregates (section 5 highlights):
  - Interest rate = 8.47%.
  - Formal wage rate = 4.03 Reais per hour (1.5 times informal wage).
  - Consumption tax rate = 34.86%.
- Government sector calibration (selected):
  - Labor and capital tax rates: τ_y(P) = 18%, τ_k = 15.5%.
  - Public workers effective labor income tax τ_y(G) = 9%.
  - Government consumption share α_g = 19% of GDP.
  - Public sector size  ̄N_G = 5%.
  - Time cost to take public exam specified as quadratic: c_app(j) = α_app2 j^2 + α_app1 j + α_app0 with α_app2 = 6, α_app1 = −80, α_app0 = 266.
- Model replicates early retirement patterns, public servant age profile, and life-cycle profile of public sector applicants (Figure 8 comparisons).

### 6    Social Security Reforms — experiments overview
- Counterfactual demographics:
  - Using IBGE Tables of Population Projections 2000-2060 and model-implied age distribution, calibrated counterfactual population growth rate g_n = −0.013 to match Brazil’s 2060 profile.
- Two main reforms analyzed (steady-state comparisons):
  1. Unification of public and private retirement systems:
     - Applies a ceiling on public benefits equal to private sector y_max; contribution limited to 11% of min{earnings, y_max}; eligibility and formulas unchanged.
     - Proxy for effects of Law 12.618 (2013) which imposed a ceiling on benefits for new entrants in public sector.
  2. Increase in minimum retirement age:
     - Imposed minimum retirement age of 65 (mimics Constitutional Amendment 103, November 12th, 2019 for men; women 62 in reality, abstracted here).
     - Section 6.5 considers raising retirement age further up to 72 in alternative scenarios.
- Note: exercises compare steady states and abstract from transition dynamics and grandfathering rules.

### 6.2 Results with 2013 demography (Table 7 summary; values in %)
- Baseline (Benchmark) and policy counterfactuals:
  - SS deficit/GDP: Benchmark 2.1 | Unification 1.6 | Minimum age -1.0 | Both reforms -1.1
  - Public SS deficit/GDP: Benchmark 1.3 | Unification 1.0 | Minimum age -0.2 | Both reforms 0.1
  - Private SS deficit/GDP: Benchmark 0.8 | Unification 0.6 | Minimum age -0.7 | Both reforms -1.2
  - Consumption tax: Benchmark 34.9 | Unification 34.0 | Minimum age 30.8 | Both reforms 30.4
  - Avg. Consumption (C) normalized to benchmark 100.0: Unification 102.3 | Minimum age 99.1 | Both reforms 100.7
  - Avg. Capital (K) normalized 100.0: Unification 104.5 | Minimum age 100.1 | Both reforms 107.2
  - Avg. Output (Y) normalized 100.0: Unification 108.9 | Minimum age 98.9 | Both reforms 102.3
  - K/Y normalized 100.0: Unification 104.2 | Minimum age 101.2 | Both reforms 104.8
  - Participation rate (out of total population): Benchmark 73.0 | Unification 73.5 | Minimum age 71.5 | Both reforms 71.5
  - Fraction of retirees (out of total population): Benchmark 13.9 | Unification 15.1 | Minimum age 8.0 | Both reforms 8.0
  - Informality (out of total workforce): Benchmark 19.5 | Unification 14.5 | Minimum age 20.5 | Both reforms 16.1
  - Public sector applicants: Benchmark 0.72 | Unification 0.74 | Minimum age 0.67 | Both reforms 0.69
- Key findings from 2013-demography experiments:
  - Unifying pension systems reduces overall SS deficit by 0.5% of GDP; public pension deficit decreases by 0.3% of GDP (largest contributor to overall reduction).
  - Imposing minimum retirement age of 65 generates a total pension surplus of 1% of GDP (moving from deficit to surplus).
  - Minimum-age reform leads to longer contribution periods and later retirement for formal workers, increasing revenues and reducing private pension expenditures.
  - Minimum-age reform discourages labor force participation and increases informality, causing equilibrium output to fall by 1.1% in the minimum-age-only scenario.
  - In benchmark calibration nearly 60% of retirees opt for the contribution modality with average retirement age 52.2 years.

### Interpretation and implications (from model results)
- Reforms that reduce generosity or cap public benefits (unification) improve fiscal position, with the public sector accounting for a substantial share of SS deficits.
- Raising minimum retirement age can shift the system into surplus by extending contribution periods, but may have adverse labor-market effects (reduced participation, higher informality) and contractionary output effects in some scenarios.
- Reform impacts operate through agents’ sectoral and formality choices, timing of retirement, and career duration in the public sector; these margins affect tax revenues and SS spending.

*Source: wpiea2025032-print-pdf — 4.4 Social Security System (IMF Working Paper).*

### 6.3    Long-Run Demography

### 6.3    Long-Run Demography

### Macroeconomic and demographic shifts (2013 → 2060)
- Demographic composition changes:
  - Share of people aged 65 or older: from 8% to 18.7%.
  - Participation rate projected to decline substantially, with 40% of the population predicted to stay at home by 2060.
  - Fraction of retirees: from 13.9% in 2013 to 31.5% in 2060.
  - Informal share of labor: predicted to fall to 7.1% in 2060.
- Fiscal and tax impacts:
  - Overall social security (SS) deficit: increases from 2.1% to 16.6% of GDP.
  - Public SS deficit: rises from 1.3% to 5.2% of GDP.
  - Private SS deficit: increases from 0.8% to 11.5% of GDP.
  - Consumption tax required to finance deficits: from 34.9% to 54.6%.
- Aggregate production and capital:
  - Despite large tax increases, aggregate GDP shows only a slight fall while consumption and capital increase between 2013 and 2060 because the 2060 demography has relatively more people aged 35-55, increasing aggregate workforce productivity.

### Quantitative equilibrium outcomes with 2060 demography (Table 8 summary; values normalized to benchmark where specified)
- Baseline (Benchmark % column is 2013 benchmark; other columns calibrated with 2060 demography):
  - SS deficit/GDP: Benchmark 2.1; 2060 w/o reforms 16.6; Unification 14.6; Minimum age 10.8; Both reforms 10.2.
  - Public SS deficit/GDP: Benchmark 1.3; 2060 w/o reforms 5.2; Unification 3.2; Minimum age 1.8; Both reforms 1.4.
  - Private SS deficit/GDP: Benchmark 0.8; 2060 w/o reforms 11.5; Unification 11.4; Minimum age 9.0; Both reforms 8.9.
  - Consumption tax: Benchmark 34.9; 2060 w/o reforms 54.6; Unification 54.6; Minimum age 51.7; Both reforms 45.4.
  - Avg. Consumption (C) (normalized): Benchmark 100.0; 2060 w/o reforms 106.7; Unification 110.9; Minimum age 107.0; Both reforms 109.6.
  - Avg. Capital (K) (normalized): Benchmark 100.0; 2060 w/o reforms 109.7; Unification 120.4; Minimum age 112.0; Both reforms 119.7.
  - Avg. Output (Y) (normalized): Benchmark 100.0; 2060 w/o reforms 99.3; Unification 104.7; Minimum age 99.4; Both reforms 103.0.
  - K/Y (normalized): Benchmark 100.0; 2060 w/o reforms 110.5; Unification 115.0; Minimum age 112.7; Both reforms 116.2.
  - Participation rate (share of total population): Benchmark 73.0; 2060 w/o reforms 60.0; Unification 60.9; Minimum age 58.5; Both reforms 58.7.
  - Fraction of retirees (share of total population): Benchmark 13.9; 2060 w/o reforms 31.5; Unification 33.7; Minimum age 18.7; Both reforms 18.7.
  - Informality (out of total workforce): Benchmark 19.5; 2060 w/o reforms 7.1; Unification 3.8; Minimum age 8.6; Both reforms 6.3.
  - Public sector applicants: Benchmark 0.72; 2060 w/o reforms 0.46; Unification 0.47; Minimum age 0.38; Both reforms 0.39.
- Key interpretations:
  - Implementing both reforms (unification + minimum retirement age) reduces the social security deficit by almost 40% relative to the no-reform 2060 scenario (16.6 → 10.2).
  - The minimum retirement age is the reform that reduces the number of retirees (31.5% → 18.7%), because it forces people to stay longer in the labor market.
  - Aggregate SS deficit remains high even after reforms (around 10.2% of GDP in the joint-reform scenario).

### Sectoral reallocation and public sector effects
- Public sector ceilings and incentives:
  - Unification (imposing a social security ceiling) has limited effect on low-productivity public employees because the ceiling is relatively high for them.
  - Increasing minimum retirement age reduces incentives to take the costly public exam, inducing low-productivity individuals to reallocate away from the public sector into the informal sector or out of the labor force.
  - Example: Informal sector size increases from 7.1% to 8.6% under the minimum-age-only scenario, which contributes to lowering the public pension deficit in that scenario.
- Public sector application behavior:
  - From 2013 to 2060, proportion of applicants to public sector falls substantially; reforms reduce the average age of applicants and the mass of applicants is 14% lower under reforms.

### Life-cycle savings and assets (Figures 10 and 11 interpretation)
- Aggregate savings:
  - The 2060 economy has more workers in prime saving years (35–55), so aggregate savings expand.
  - With both reforms implemented, aggregate savings increase by 9.1% relative to the no-reform 2060 scenario.
  - Agents increase savings both because of incentives from lower taxation (in reform scenarios) and to smooth consumption in response to less generous pensions.
- Life-cycle asset profile:
  - An older population induces more savings for people older than 45; the 2060 profile is above the 2013 profile for ages >45.
  - With reforms, there is a notable jump in average assets from around age 33 to 62, reflecting precautionary saving to offset reduced retirement income; after age 62 assets fall below no-reform levels as retirees draw down assets.

### Labor market participation and retirement behavior (Figure 11 interpretation)
- Participation changes by age:
  - Most relevant reform effect: significant increase in participation for people aged 60–65 due to the new minimum retirement age.
  - In the 2060 economy without reforms, 12.8% of the population between 51 and 64 would be retired; with reforms this falls to zero for that group by legal force.
  - However, agents aged 35–50 have less incentive to work because they can no longer claim early retirement after 50; this effect dominates and overall participation rate mildly declines from 60.0% to 58.7% under the joint reforms.
- Informality over the life cycle:
  - Informality falls between 2013 and 2060 mainly because the 2013 informal workforce included many younger workers.
  - Comparing 2060 with and without reforms, informality changes little over the life cycle except near age 30 when agents shift towards the informal sector to avoid taxes and increase net income.

### Role of endogenous informality and public sector decisions (Section 6.4; Table 9 and 10 highlights)
- Table 9 decomposition (2060 demography):
  - Full optimal decisions with both reforms: SS deficit/GDP 10.2; Public SS deficit/GDP 1.4; Private SS deficit/GDP 8.9; Consumption tax 45.4; Avg. Consumption (C) 107; Avg. Capital (K) 120; Avg. Output (Y) 103; K/Y 116; Participation rate 58.7; Fraction of retirees 18.7.
  - If public and informal workers are kept fixed as in the benchmark (no sectoral reoptimization), SS deficit/GDP would be 11.4 (higher by 1.2 p.p.), public SS deficit/GDP 1.5, private SS deficit/GDP 9.8, consumption tax 47.5.
  - Allowing informal workers to reoptimize but keeping public worker decisions fixed reduces the deficit relative to the fully fixed case (illustrating the importance of endogenous sectoral moves).
- Economy without public sector (Table 10):
  - New Benchmark (no public sector): SS deficit/GDP 0.5 in baseline; with 2060 demography w/o reforms SS deficit/GDP 11.7; with minimum age SS deficit/GDP 9.2.
  - Public SS deficit/GDP is 0.0 in all columns (no public sector).
  - Consumption tax: New Benchmark 8.5; 2060 w/o reforms 24.6; minimum age 20.9.
  - Avg. Consumption (C) (normalized): 113.1; 120.5; 118.2 (columns respectively).
  - Participation rate: 71.6; 58.4; 56.5.
  - Fraction of retirees: 15.0; 33.6; 18.7.
  - Interpretation: Without a public sector, estimated SS deficits in 2013 and 2060 are considerably smaller than in the full model (0.5% and 11.7% versus 2.1% and 16.6%). Raising retirement age has a much smaller effect on the deficit when the public sector is absent. This underscores that ignoring endogenous informality and public sector responses can substantially misestimate long-run impacts.

### Zero-deficit policy and minimum retirement age (Section 6.5; Figure 12 interpretation)
- Required minimum retirement age to eliminate SS deficit by 2060:
  - A minimum retirement age above 71 would be necessary to eliminate the social security deficit by 2060 under the joint reform scenario.
- Effects of raising minimum retirement age from 65 → 72:
  - Change in GDP: increases range from 3.7% to 17% as minimum retirement age varies from 65 to 72.
  - Informality: size of informal sector declines from 6.3% to 1.9% as minimum retirement age increases (in the range studied).
  - Mechanisms: longer work lives raise tax contributions and production, reduce retirement expenditures, lower equilibrium consumption tax, increase aggregate savings and capital, reduce interest rates, raise formal wages, and increase formal employment demand.

### Welfare impacts across cohorts (Section 6.6; Figure 13 interpretation)
- Aggregate welfare measure: proportional change in consumption Δw that equalizes average utility between 2060 equilibria with and without reforms.
- Long-run aggregate welfare:
  - Welfare increases between 2.9% with the benchmark reforms (minimum retirement age of 65) and 10.5% with a retirement age of 72.
- Heterogeneous welfare effects by age group:
  - Younger cohorts (16–30 and 31–45) are better off under the reforms for all retirement ages considered, driven by higher GDP and lower consumption taxes.
  - Age groups 46–60 and 61–75 can be worse off under some reforms:
    - The 61–75 age group is directly affected by stricter retirement conditions and may prefer the old system if minimum retirement age exceeds 69.
    - Under benchmark reforms (minimum retirement age 65) the 46–60 age group welfare would fall by 7.9% because many were retiring between 52 and 60; when minimum retirement age is above 70, improved economic conditions can offset losses leading to welfare gains between 0.1% and 5% for this group.

### Summary implications and policy-relevant conclusions
- Population aging alone will drive a large increase in the SS deficit (from 2.1% of GDP in 2013 to 16.6% of GDP by 2060) if retirement rules remain unchanged.
- Reforms analyzed (unification of systems and raising minimum retirement age to 65) substantially reduce but do not eliminate the deficit; the minimum retirement age delivers the largest deficit reduction and reduces retiree shares.
- Accounting for endogenous sectoral responses (public sector and informality) is critical: keeping sectoral allocations fixed materially alters deficit and macroeconomic outcomes and leads to under- or over-estimation of reform impacts.
- Eliminating the deficit by 2060 likely requires a much higher minimum retirement age (above 71), with attendant distributional and political challenges because older cohorts may lose welfare while younger cohorts gain.
- Reforms have additional long-run benefits: increased savings, capital, output, and aggregate welfare, alongside reductions in informality—yet they entail transition costs and concentrated losses for older cohorts that complicate short-term feasibility.

*Source: IMF Working Paper — section 6.3, “Long-Run Demography,” as presented in the supplied content unit.*

### References

### wpiea2025032-print-pdf - References and Technical Appendices

### Major themes in the referenced literature
- Enforcement of labor regulation and informality; informality and wages in developing countries.
- Social security reform, retirement behavior, and macroeconomic implications of pension reform.
- Life-cycle models of consumption, wealth distribution, and earnings risk.
- Public sector pensions, early retirement incentives, and the effect of retirement age on labor supply and health.
- Methods for estimating income processes, risk aversion, and welfare/growth effects of fiscal rules.

### Appendix A — Agents’ stationary distribution (model structure)
- The stationary distribution is built from age-dependent measures μm(sm) for sectors m ∈ {P,G,R}.
- Initial entrants at j = 1: μP(sP) = φ ̄Γ(z) if sP = (1,0,z,0,0); μm(s m) = 0 for m = {R,G}.
- For j > 1, the measure of private agents (j,a′,z′,x′,t′C) equals the sum of:
  - agents who remained in private sector (term with factor (1−dapp(sP))(1−dss(sP))), and
  - private workers who applied to public sector and failed (term with factor dapp(sP)(1−Papp)).
- Transition kernel for idiosyncratic productivity is Π(z,z′).
- Updated average past earnings and contribution time ( ̃x′, ̃t′C) depend on optimal labor dh(sP), informality dinf(sP), and application decision dapp(sP):
  - ( ̃x′, ̃t′C) = ( [ x(tC−2)+min{y j−1(wf,P),ymax} ]/(tC−1), tC+1 ) if works formally; (x,tC) otherwise.
- Public sector distribution μG(j,a′,z′,t′C,t′G) comprises:
  - entrants who passed the public exam (dapp(sP)Papp), and
  - public workers who did not retire (1−dss(sG)).
- Retiree distribution μRP(j,a′,z′,b′) comprises:
  - existing retirees (μRP component with Π(z,z′)), and
  - recent private-sector retirees (αb·Π(z,z′) dss(sP) · μP(sP)), where αb is the interpolation coefficient on grid B.
- Retirees are split between private (RP) and public (RG) retirees in numerical implementation; stationary distributions are analogous.

### Appendix B — Definition of the stationary competitive equilibrium
- A recursive competitive equilibrium consists of:
  - value functions V : S m → R for m ∈ {P,G,R};
  - policy functions: dh:S m → {0,H}; da:S m → R+; dc:S m → R++; dss:S m → {0,1} for m∈{P,G};
    dinf:S m → {0,1} for m∈{P,R}; dapp:S P → {0,1};
  - competitive prices {r,wf,wi}, age-dependent time-invariant measures μm(sm), government transfers ζ, taxes, public good YG, and threshold score ̄q such that conditions (1)–(8) hold.
- Key equilibrium conditions (selected):
  - Aggregate capital law of motion: K′ = ∑m∈{P,G,R} ∑sm da(sm) μm(sm); K = K′/(1+gn).
  - Formal labor supply Nf = ∑m∈{P,R} ∑sm ez+ηj (1−dinf(sm)) dh(sm) μm(sm).
  - Government chooses ̄q to target ̄NG = ∑sG μG(sG).
  - Public goods consistency: YG = ∑sG μG(sG) ezG+ηj · H.
  - Final good market clearing: ∑m,s m dc(sm) μm(sm) + K′ + Cg = Yf + Yi + (1−δ)K.
  - Government budget balance condition detailed with τC, τK, τy(P), τw, τss(P), τss(G), τb, and Cg terms.
  - Bequests rebated to living: ζ = (1+r)/(1+gn) ∑m,s m da(sm)(1−ψj) μm(sm).

### Appendix C — Numerical solution and computational details
- Numerical implementation:
  - Solved in Fortran 90.
  - Discretized grids with points: assets = 52, average past earnings x = 10, income process z = 5, social security benefits b = 42.
  - Asset grid runs from R$0 to R$3,790,202, with points concentrated near the lower bound. The concentration follows a cubic polynomial.
  - x grid equally spaced between 0 and ymax.
  - b grid linearly spaced between R$0 and R$100,000. The upper bound chosen so more than 99% of benefits in equilibrium are below that value.
- General equilibrium algorithm (iterative):
  1. Guess initial Θ ≡ (r, YG, ζ, τc, ̄q).
  2. Use formal firm first order conditions to obtain wf.
  3. Solve agents’ problems backwards to obtain policy functions.
  4. Forward induction to compute stationary household distributions from policy functions.
  5. Find ̃̄q such that ̄NG of population are public servants.
  6. Compute implied ̃Θ ≡ ( ̃r, ̃YG, ̃ζ, ̃τc).
  7. Check convergence: ||̃Θ − Θ|| < ε; if not, update Θ and iterate.
- Notes on bounds:
  - Asset upper bound chosen as the smallest value such that no agent optimally chooses the upper bound.
  - Benefit upper bound chosen so that more than 99% of benefits distributed in equilibrium are lower than that value.

### Appendix D — Private agents’ decision tree (discrete choices)
- Within each period private agents choose, in sequence:
  - Whether to retire or not;
  - If not retired: whether to work formally, informally, or stay home;
  - For non-retirees, whether to take the public sector exam (incurs time cost, discounted from hours of work or leisure).
- Decision nodes include V(sP) with branches: Retire → Home; Not Retire → {Formal, Informal, Home} × {Test, No Test}.

### Appendix E — Data sources and sample definitions
- Aggregate sources:
  - 2013 Brazilian National Accounts for government consumption share of GDP.
  - 2013 Social Security Annual Report for social security deficit and public vs private retirement statistics.
  - 2013 IBGE mortality tables for survival probabilities.
- Micro data:
  - 2013 PNAD (cross-section) to calculate sectoral distribution (private, public, retirees), economic participation, test takers, and retiree statistics by age.
  - 2012–2018 PNADC (quarterly rotating panel) to estimate income process, age-efficiency profile, and public sector wage premium.
    - PNADC rotation: each household visited five times over five consecutive quarters; panel constructed over one-year intervals.
- Sample focus:
  - Men between 16 and 75 years old.
  - Informal worker defined as worker without a legal working contract (carteira de trabalho), not covered by social security and not paying income taxes.
  - Government agent defined as statutory government employee (subject to segregated retirement scheme).
  - Employees of publicly owned enterprises are not considered statutory public servants and are subject to private social security.

### Appendix F — Calibration summary (selected numerical targets and values)
- Internal calibration (Table F1):
  - β = 0.96 targets K/Y = 2.42; model: 2.5.
  - γ = 0.248 targets Participation Rate 73%; model: 72.3%.
  - Af = 2.194 targets GDP per capita R$ 27,468; model: R$ 26,520.
  - Ai = 3.03 targets Informal sector size 19.5%; model: 18.9%.
  - αapp0 = 266; αapp1 = −80; αapp2 = 6 target Test takers age profile (Figure 8).
  - τw = 0.085 targets Private SS deficit/GDP 0.8%; model: 0.9%.
- External calibration (Table F2), selected parameters and values:
  - J Maximum age = 60 (agents live from 16-75).
  - {ψj} Survival probabilities = Figure 6 (IBGE).
  - gn Population growth rate = 1.9% (IBGE).
  - ̄H Time endowment = 8,760 hours (24 hours/day x 365 days/year).
  - H Working hours = 2,016 hours (8 hours/day x 52 weeks/year).
  - σ Risk aversion = 2.5 (Attanasio (1999)).
  - ε Public good coef. = 0.5 (Ferreira and Nascimento (2005)).
  - α Capital share in output = 0.4.
  - δ Depreciation rate = 0.06 (Parente and Prescott (2002)).
  - Z Productivity space = Table 3 (PNADC); Π(z,z′) transition probabilities = Table 3 (PNADC).
  - {ηj} Age-efficiency profile = Figure 7 (PNADC).
  - θ Public sector wage premium = 9.8% (PNADC).
  - b(.) Benefits function = Section 4.4 (2013 rules).
  - τy(P) Private sector’s income tax = 18%.
  - τy(G) Public sector’s income tax = 9% (τy(P) − FGTS/union contributions).
  - τk Capital tax = 15.5%.
  - τSS(G) G worker SS contribution = 11% (2013 tax rates).
  - τSS(P) P worker SS contribution = Equation (6) (2013 tax rates).
  - τb G retiree benefit tax = 11% (2013 tax rates).
  - αG Govt. Consumption/GDP = 18.9% (National Accounts).
  - ̄NG Size of the government = 5% (2005-2013 PNAD).
  - ymax Private SS ceiling = R$ 50,000 (2013 rules).

*Italic: Content derived from "References" and Appendices A–F of Working Paper No. WP/2025/032.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025032-print-pdf.pdf_