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Motivation Model Calibration Results Appendix
Outline
Inequality in an OLG economywith heterogeneous cohorts and pension systems
(with Marcin Bielecki, Krzysztof Makarski and Marcin Waniek)
Joanna Tyrowicz
University of Warsaw and National Bank of Poland
MoPAct Workshop, ETLA, Helsinki, 20161 / 40
Motivation Model Calibration Results Appendix
Motivation
Wealth inequality increases due to:Demographic transitionPension reform: defined benefit → defined contribution
Effects for consumption inequality: unclear
Can policy instruments help?minimum pensions: ↑ pensions; ↓ labor supply incentivescontribution caps : obligatory savings replaced with private savings
Intuition insufficient
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Motivation Model Calibration Results Appendix
Motivation
Wealth inequality increases due to:Demographic transitionPension reform: defined benefit → defined contribution
Effects for consumption inequality: unclear
Can policy instruments help?minimum pensions: ↑ pensions; ↓ labor supply incentivescontribution caps : obligatory savings replaced with private savings
Intuition insufficient
2 / 40
Motivation Model Calibration Results Appendix
Literature review
Distributional effects of pension systems: OLG models with expost heterogeneity:
Castaneda et al. (2003, JPE); Fehr et al. (2008, RED); Song(2011, RED); Bucciol (2011, MD); Cremer and Pestieau (2011,EER); Kumru and Thanopoulos (2011, JPubE); Fehr and Uhde(2014, EM); St-Amant and Garon (2014, ITPF)
Ex ante + ex post heterogeneity: education affects mortality ratesHairault and Langot (2008, JEDC):McGrattan and Prescott (2014, NBER)Kindermann and Krueger (2014, NBER)
3 / 40
Motivation Model Calibration Results Appendix
Literature review
Distributional effects of pension systems: OLG models with expost heterogeneity:
Castaneda et al. (2003, JPE); Fehr et al. (2008, RED); Song(2011, RED); Bucciol (2011, MD); Cremer and Pestieau (2011,EER); Kumru and Thanopoulos (2011, JPubE); Fehr and Uhde(2014, EM); St-Amant and Garon (2014, ITPF)
Ex ante + ex post heterogeneity: education affects mortality ratesHairault and Langot (2008, JEDC):McGrattan and Prescott (2014, NBER)Kindermann and Krueger (2014, NBER)
3 / 40
Motivation Model Calibration Results Appendix
Our approach
Question 1: distributional effects of a pension system reform
Question 2: are standard instruments effective in reducing theincrease in inequality
Ex ante heterogeneous agents: age + within cohort
endowments + preferences ← not a standseparate endowments from preferencesmost countries: no data on mortality by education / income groups
4 / 40
Motivation Model Calibration Results Appendix
Our approach
Question 1: distributional effects of a pension system reform
Question 2: are standard instruments effective in reducing theincrease in inequality
Ex ante heterogeneous agents: age + within cohort
endowments + preferences ← not a standseparate endowments from preferencesmost countries: no data on mortality by education / income groups
4 / 40
Motivation Model Calibration Results Appendix
Results preview
DB → DC reform: both wealth and consumption inequalities ↑
Demographic transition ⇒ inequalities ↑, more than due to reform
Minimum pensions:reduce inequality from the reform by 40-50%work on the endowments margin, but not on preferences
Effects of the contribution cap: negligible
5 / 40
Motivation Model Calibration Results Appendix
Results preview
DB → DC reform: both wealth and consumption inequalities ↑Demographic transition ⇒ inequalities ↑, more than due to reform
Minimum pensions:reduce inequality from the reform by 40-50%work on the endowments margin, but not on preferences
Effects of the contribution cap: negligible
5 / 40
Motivation Model Calibration Results Appendix
Results preview
DB → DC reform: both wealth and consumption inequalities ↑Demographic transition ⇒ inequalities ↑, more than due to reform
Minimum pensions:reduce inequality from the reform by 40-50%work on the endowments margin, but not on preferences
Effects of the contribution cap: negligible
5 / 40
Motivation Model Calibration Results Appendix
Outline
1 Motivation
2 Model
3 Calibration
4 Results
5 Appendix
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Motivation Model Calibration Results Appendix
Method
ModelDeterministic (→we are working on it!)OLGex ante heterogeneity: endowments + preferences
Calibrate to Poland in 1999
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Motivation Model Calibration Results Appendix
Method
ModelDeterministic (→we are working on it!)OLGex ante heterogeneity: endowments + preferences
Calibrate to Poland in 1999
7 / 40
Motivation Model Calibration Results Appendix
Households I
“Born” at age 20 (j = 1) and live up to 100 years (J = 80)Subject to time and cohort dependent survival probability πBelong to a type k:
productivity level ωtime discounting δrelative leisure preference φ
Choose labor supply l endogenouslyMaximize remaining lifetime utility derived from consumption cand leisure 1− l:
Uj,k,t =J−j∑s=0
[δskπj+s,t+sπj,t
[cφkj+s,k,t+s (1− lj+s,k,t+s)1−φk
]]
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Motivation Model Calibration Results Appendix
Households II
Subject to the budget constraint
(1 + τ ct )cj,k,t + sj,k,t = (1− τ lt )(1− τ)wtωklj,k,t ← labor income+ (1 + (1− τkt )rt)sj−1,k,t−1 ← capital income+ (1− τ lt )bj,k,t ← pension income+ beqj,k,t ← bequests−Υt ← lump-sum tax
There exists a closed-form solution to this problem
9 / 40
Motivation Model Calibration Results Appendix
Producers
Perfectly competitive representative firmStandard Cobb-Douglas production function
Yt = Kαt (ztLt)1−α
Profit maximization implies
wt = zt(1− α)kαtrt = αkα−1
t − d
where d is the capital depreciation rateand k is capital per effective unit of labor
10 / 40
Motivation Model Calibration Results Appendix
Government
Spends a fixed share of GDP (g) on government consumptionCollects taxes TCloses the gap between pension system contributions and benefitsCan take on debt D
Tt +Dt = (1 + rt)Dt−1 + gYt + subsidyt
We fix debt at constant 45% debt to GDP ratio.Consumption tax varies to satisfy the government constraint.
11 / 40
Motivation Model Calibration Results Appendix
Pension System
Pay As You Go Defined Benefit (PAYG DB)
bJ ,k,t = ρ · gross wageJ−1,k,t−1
Pay As You Go Defined Contribution (PAYG DC)
bJ ,k,t =accumulated sum of contributionsJ ,k,t
expected remaining lifetimeJ ,t
Pensions indexed by the rate of annual payroll growth
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Motivation Model Calibration Results Appendix
Instrument 1: minimum pensions
Definitionbj,k,t ≥ ρmin · gross average waget
We set ρmin = 0.2 → 4% coverage (consistent with the data)
ExpectationsDirectly affects only the left tail of income distributionIncreases lifetime incomes of targeted group → consumptioninequality should decreaseLower incentives to work → possible reduction in hours workedLower incentives for private savings → possible increase inconsumption
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Motivation Model Calibration Results Appendix
Instrument 1: minimum pensions
Definitionbj,k,t ≥ ρmin · gross average waget
We set ρmin = 0.2 → 4% coverage (consistent with the data)
ExpectationsDirectly affects only the left tail of income distributionIncreases lifetime incomes of targeted group → consumptioninequality should decreaseLower incentives to work → possible reduction in hours workedLower incentives for private savings → possible increase inconsumption
13 / 40
Motivation Model Calibration Results Appendix
Instrument 1: contribution cap
Definition:
τ effj,k,t = min
τ,τcap · gross average waget
wtωklj,k,t
To replicate 2% coverage, τcap = 1.7 (lower than de iure 2.5)
ExpectationsAffects directly only the right tail of income distributionLower contributions of targeted group → higher voluntary savingrates → wealth inequalities ↑Matters because market interest rates and social security indexationdiffer
14 / 40
Motivation Model Calibration Results Appendix
Instrument 1: contribution cap
Definition:
τ effj,k,t = min
τ,τcap · gross average waget
wtωklj,k,t
To replicate 2% coverage, τcap = 1.7 (lower than de iure 2.5)
ExpectationsAffects directly only the right tail of income distributionLower contributions of targeted group → higher voluntary savingrates → wealth inequalities ↑Matters because market interest rates and social security indexationdiffer
14 / 40
Motivation Model Calibration Results Appendix
Solution procedure
Gauss-Seidel iterative algorithmSteady states (initial and final)
1 Guess an initial value for k2 Use it to compute the prices3 Have households of each type and age solve their problem given
prices4 Aggregate individual labor supply and savings to get new values
for L and K5 If the new value for k satisfies predefined norm, finish,
else update k and return to point (2)
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Motivation Model Calibration Results Appendix
1 Motivation
2 Model
3 Calibration
4 Results
5 Appendix
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Motivation Model Calibration Results Appendix
Exogenous assumptions
Projections for Poland provided by the European Commission
Population Size TFP Growth
Kept constant across scenarios, don’t affect results
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Motivation Model Calibration Results Appendix
Exogenous assumptions
Projections for Poland provided by the European Commission
Population Size TFP Growth
Kept constant across scenarios, don’t affect results
17 / 40
Motivation Model Calibration Results Appendix
Within cohort heterogeneity - endowments
Structure of Earnings Survey, 1998, Poland
Productivity ω
Resulting: 10 values for ω
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Motivation Model Calibration Results Appendix
Within cohort heterogeneity - leisure preference
Structure of Earnings Survey, 1998, Poland
Leisure Preference φ
Resulting: 4 values for φ
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Motivation Model Calibration Results Appendix
Within cohort heterogeneity - time preference
Again: no data on mortality rates or wealth by income oreducation groups
Calibrate the central value of δ to match the investment rateSplit population ad hoc to 3 groups:
discount factors are (0.98δ, δ, 1.02δ)
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Motivation Model Calibration Results Appendix
Within cohort heterogeneity - summary outcomes I
In total we have 120 types within each cohort
The resulting consumption Gini index in the initial steady stateis 25.5, consistent with Brzezinski (2011)
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Motivation Model Calibration Results Appendix
Within cohort heterogeneity - summary outcomes II0
24
6
0 20 40 60 80age
Lowest omega multiplierStandard omega multiplierHighest omega multiplier
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Motivation Model Calibration Results Appendix
Within cohort heterogeneity - summary outcomes III−
50
510
0 20 40 60 80age
Lowest delta multiplierHighest delta multiplierStandard multipliersLowest phi multiplierHighest phi multiplier
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Motivation Model Calibration Results Appendix
1 Motivation
2 Model
3 Calibration
4 Results
5 Appendix
24 / 40
Motivation Model Calibration Results Appendix
Minimum pensions coverage0
.2.4
.6.8
1
2000 2050 2100 2150 2200 2250year
Defined Benefit with minimum pensionsDefined Contribution with minimum pensions
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Motivation Model Calibration Results Appendix
Macroeconomic effects
No instrument Minimum pension Contribution capDB DC (to DB) DB DC DB DC
Capital 52.6% 60.4% 52.7% 60.3% 52.6% 60.5%Tax rate
initial 11.00 11.00 11.00 11.00 11.00 11.00final 15.44 10.95 15.43 11.99 15.46 10.95diff. (in pp) 4.44 - 0.05 4.43 0.99 4.46 - 0.05
Pension system deficitinitial 1.46 1.56 1.46final 3.95 0.00 4.02 0.87 3.97 0.00diff (in pp) 2.49 2.46 -0.69 2.51
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Motivation Model Calibration Results Appendix
Consumption Gini
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Motivation Model Calibration Results Appendix
Wealth Gini
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Motivation Model Calibration Results Appendix
Wealth Gini at retirement I
29 / 40
Motivation Model Calibration Results Appendix
Wealth Gini at retirement II
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Motivation Model Calibration Results Appendix
Inequality decomposition – endowments vs preferences
Instruments should reduce inequality stemming from endowments(luck) but not from preferencesTo isolate the effects of the two sources:
Shut down each channel separatelyKeep prices constant from the full model to avoid GE effectsSolve for decisions of households in partial equilibrium
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Motivation Model Calibration Results Appendix
Consumption inequality decomposition - minimum pensions
DB DC
32 / 40
Motivation Model Calibration Results Appendix
Wealth inequality decomposition - minimum pensions
DB DC
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Motivation Model Calibration Results Appendix
Wealth inequality decomposition - minimum pensions
DB DC
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Motivation Model Calibration Results Appendix
Wealth Gini at retirement
DB DC
35 / 40
Motivation Model Calibration Results Appendix
Welfare effects
Defined Benefit Defined Contribution
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Motivation Model Calibration Results Appendix
Conclusions
Consumption inequality increase due toaging processesDB→DC reform
Minimum pensionseffective in reducing consumption inequality resulting from theDB→DC reform by 40-50%with 80% coverage minimum pension costs 1 pp higherconsumption tax (transfer of about 0.9% GDP)wealth inequality increases
Contribution cap has virtually no effects
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Motivation Model Calibration Results Appendix
Thank you for your attention
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Motivation Model Calibration Results Appendix
Household Sector Closed Form Solution IFor j < J (working):
cj,t = Ωj,t + Γj,t(1 + τ ct )
[∑J−j−1s=0
((1 + φ) δs πj+s,t+s
πj,t
)+∑J−js=J−j
(δsπj+s,t+s
πj,t
)]lj,t = 1− φ(1 + τ ct )cj,t
(1− τ lt )(1− τ)wtsj,t = (1− τ lt )(1− τ)wtlj,t + (1 + (1− τkt )rt)sj−1,t−1 − (1 + τ ct )cj,t,
with
Ωj,t =J−j−1∑s=0
(1− τ lt+s)(1− τ)wt+s + beqj+s,t+s −Υt+s∏si=1(1 + (1− τkt+i)rt+i)
Γj,t =J−j∑s=J−j
(1− τ lt+s)bj+s,t+s + beqj+s,t+s −Υt+s∏si=1(1 + (1− τkt+i)rt+i)
.
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Motivation Model Calibration Results Appendix
Household Sector Closed Form Solution II
For j ≥ J (retired):
cj,t = Γj,t(1 + τ ct )
[∑J−js=J−j
(δsπj+s,t+s
πj,t
)]lj,t = 0sj,t = (1− τ lt )bιj,t + (1 + (1− τkt )rt)sj−1,t−1 − (1 + τ ct )cj,t,
with
Γj,t =J−j∑s=0
(1− τ lt+s)bj+s,t+s + beqj+s,t+s −Υt+s∏si=1(1 + (1− τkt+i)rt+i)
.
40 / 40