While the inequalities of endowments are widely recognized as areas of policy intervention, the dispersion in preferences may also imply inequalities of outcomes. In this paper, we analyze the inequalities in an OLG model with obligatory pension systems. We model both policy relevant pension systems (a defined benefit system — DB — and a transition from a DB to a defined contribution system, DC).

1. Motivation Model Calibration Results Appendix
Inequalities in an OLG economy
with heterogeneous cohorts and pension systems
(with Joanna Tyrowicz, Krzysztof Makarski and Marcin Waniek)
Marcin Bielecki
Faculty of Economics, University of Warsaw
4th NBP Summer Workshop
22-24 June 2015
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2. Motivation Model Calibration Results Appendix
Motivation
Many countries reform pension systems from defined benefit to
defined contribution
DC systems increase inequality
Demographic transition translates to higher inequality
Thus, it is important to know whether commonly used pension
system instruments – minimum pensions and contribution caps –
are effective in reducing inequality
The intuition is insufficient – we need quantitative answers
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3. Motivation Model Calibration Results Appendix
Literature Review
There exists a large body of literature analyzing the distributional
effects of pension systems with ex ante homogeneous agents:
Castaneda et al. (2003, JPE)
Nishiyama and Smetters (2007, QJE)
Fehr et al. (2008, RED)
Hairault and Langot (2008, JEDC)
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)
Kindermann and Krueger (2014, NBER)
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4. Motivation Model Calibration Results Appendix
Our Approach
ex ante homogeneous agents setup necessarily confounds effects
of preferences and endowments
ex ante heterogeneous agents setup allows to distinguish effects
from preferences and from endowments
We do not make a stand in the debate whether preferences differ –
we want to see a quantitative what-if effect
Heterogeneous cohorts (by age) and within cohorts (endowments
and preferences)
Thus, we are able to distinguish whether pension system
instruments – minimum pensions and contribution caps –
address inequalities due to endowments or due to preferences
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5. Motivation Model Calibration Results Appendix
Results Preview
We reform the pension system that relatively equalizes pension
outcomes (DB) to one that generates more inequality (DC)
Inequalities also increase due to the demographic transition –
effect stronger than the reform
Minimum pensions reduce inequality from the reform by 40-50%
Minimum pensions affect inequality due to endowments, but not
due to preferences
Contribution cap is ineffective
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6. Motivation Model Calibration Results Appendix
Method
We employ a deterministic heterogeneous agent general
equilibrium OLG model with within cohort heterogeneity
We calibrate the model to match closely the Polish economy
in 1999, as that was the year of the enacted pension system reform
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7. Motivation Model Calibration Results Appendix
Households I
Are “born” at age 20 (j = 1) and live up to 100 years (J = 80)
Belong to a type k:
time discounting δ
relative leisure preference φ
productivity level ω
Choose labor supply l endogenously
Maximize remaining lifetime utility derived from consumption c
and leisure 1 − l:
Uj,k,t =
J−j
s=0
δs
k
πj+s,t+s
πj,t
cφk
j+s,k,t+s (1 − lj+s,k,t+s)1−φk
Subject to the budget constraint (next slide)
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8. Motivation Model Calibration Results Appendix
Households II
Subject to the budget constraint
(1 + τc
t )cj,k,t + sj,k,t = (1 − τl
t )(1 − τ)wtωklj,k,t ← labor income
+ (1 + (1 − τk
t )rt)sj−1,k,t−1 ← capital income
+ (1 − τl
t )bj,k,t ← pension income
+ beqj,k,t ← bequests
− Υt ← lump-sum tax
There exists a closed-form solution to this problem
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9. Motivation Model Calibration Results Appendix
Producers
Perfectly competitive representative firm
Standard Cobb-Douglas production function
Yt = Kα
t (ztLt)1−α
,
where K is the aggregate capital stock, L is the aggregate labor
supply, and z denotes labor-augmenting exogenous technological
progress
Profit maximization implies
wt = zt(1 − α)ˆkα
t
rt = αˆkα−1
t − d
where d is the capital depreciation rate
and ˆk is capital per effective unit of labor
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10. Motivation Model Calibration Results Appendix
Government
Spends a fixed share of GDP g on government consumption
Collects taxes T
Closes the gap between pension system contributions and benefits
Can 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.
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11. Motivation Model Calibration Results Appendix
Pension System
Pay As You Go Defined Benefit (PAYG DB)
b ¯J,k,t = ρ · gross wage ¯J−1,k,t−1
Pay As You Go Defined Contribution (PAYG DC)
b ¯J,k,t =
accumulated sum of contributions ¯J,k,t
expected remaining lifetime ¯J,t
Pensions indexed by the rate of annual payroll growth
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12. Motivation Model Calibration Results Appendix
Minimum Pensions
Minimum pensions
bj,k,t ≥ ρmin · gross average waget
We set ρmin = 0.2 −→ 4% of population is affected by the
instrument, consistent with the data
Expected effects
Affects directly only the left tail of income distribution
Increases lifetime incomes of targeted group → consumption
inequality should decrease
Lower incentives to work → possible reduction in hours worked
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13. Motivation Model Calibration Results Appendix
Contribution Cap
Contribution cap
τeff
j,k,t = min τ,
τcap · gross average waget
wtωklj,k,t
We set τcap = 1.7 (lower than de iure 2.5) −→ 2% of population
is affected by the instrument, consistent with the data
Expected effects
Affects directly only the right tail of income distribution
Lower contributions of targeted group → higher saving rates
Matters for lifetime income only if market interest rates and social
security indexation differ
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14. Motivation Model Calibration Results Appendix
Solution Procedure
Gauss-Seidel iterative algorithm
Steady states (initial and final)
1 Guess an initial value for ˆk
2 Use it to compute the prices
3 Have households solve their problem given prices
4 Aggregate individual labor supply and savings to get new values
for L and K
5 If the new value for ˆk satisfies predefined norm, finish,
else update ˆk and return to point (2)
Transition path
1 Basing on the initial and final steady state values for ˆk guess an
initial path between the terminal points
...
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15. Motivation Model Calibration Results Appendix
Population Size and TFP Growth
Population Size TFP Growth
Projections for Poland provided by the European Commission
Kept constant across scenarios, don’t affect results
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16. Motivation Model Calibration Results Appendix
Within Cohort Heterogeneity I
Productivity Leisure Preference
ω φ
Based on Structure of Earnings Survey, 1998, Poland
Resulting 10 values for ω and 4 values for φ
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17. Motivation Model Calibration Results Appendix
Within Cohort Heterogeneity II
We calibrate the central value of δ to match the investment rate
We don’t have the data on stratified mortality rates or wealth
We split the model population ad hoc into 3 groups
Their discount factors are (0.98δ, δ, 1.02δ)
In total we have 120 types within each cohort
The resulting consumption Gini index in the initial steady state
is 25.5, consistent with Brzezinski (2011)
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18. Motivation Model Calibration Results Appendix
Results – lifecycle wealth profiles
0246
0 20 40 60 80
age
Lowest omega multiplier
Standard omega multiplier
Highest omega multiplier
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19. Motivation Model Calibration Results Appendix
Results – lifecycle wealth profiles
−50510
0 20 40 60 80
age
Lowest delta multiplier
Highest delta multiplier
Standard multipliers
Lowest phi multiplier
Highest phi multiplier
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20. Motivation Model Calibration Results Appendix
Minimum Pensions Coverage
0.2.4.6.81
2000 2050 2100 2150 2200 2250
year
Defined Benefit with minimum pensions
Defined Contribution with minimum pensions
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21. Motivation Model Calibration Results Appendix
Consumption Gini
.24.26.28.3
2000 2050 2100 2150 2200
year
DB: No instruments
DC: No instruments
DC: Minimum benefits
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22. Motivation Model Calibration Results Appendix
Wealth Gini
.85.9.9511.05
2000 2050 2100 2150 2200
year
DB: No instruments
DC: No instruments
DC: Minimum benefits
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23. Motivation Model Calibration Results Appendix
Inequality decomposition – endowments vs preferences
Instruments should reduce inequality stemming from endowments
(luck) but not from preferences
Need to isolate the effects of the two sources of differences
Shut down the differences in each dimension separately
Keep prices constant from the full model to avoid GE effects
Solve for decisions of households in partial equilibrium
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24. Motivation Model Calibration Results Appendix
Consumption inequality decomposition
Fixed endowments Fixed preferences
Differing preferences Differing endowments
.05.1.15.2.25.3
2000 2050 2100 2150 2200
year
DB: Fixed endowments, no instruments
DC: Fixed endowments, no instruments
DC: Fixed endowments, minimum benefits
.05.1.15.2.25.3
2000 2050 2100 2150 2200
year
DB: Fixed preferences, no instruments
DC: Fixed preferences, no instruments
DC: Fixed preferences, minimum benefits
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25. Motivation Model Calibration Results Appendix
Wealth inequality decomposition
Fixed endowments Fixed preferences
Differing preferences Differing endowments
0.2.4.6.81
2000 2050 2100 2150 2200
year
DB: Fixed endowments, no instruments
DC: Fixed endowments, no instruments
DC: Fixed endowments, minimum benefits
0.2.4.6
2000 2050 2100 2150 2200
year
DB: Fixed preferences, no instruments
DC: Fixed preferences, no instruments
DC: Fixed preferences, minimum benefits
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26. Motivation Model Calibration Results Appendix
Welfare Effects
Defined Benefit Defined Contribution
−.0004−.00020.0002
WeightedMeanCompensatingVariation
2000 2050 2100 2150 2200 2250
Year of birth
Minimum benefits
Contributions cap
−.003−.002−.0010
WeightedMeanCompensatingVariation
2000 2050 2100 2150 2200
Year of birth
Minimum benefits
Contributions cap
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27. Motivation Model Calibration Results Appendix
Macroeconomic Effects
No instrument Minimum pension Contribution cap
DB DC DB DC DB DC
Capital 52.6% 60.4% 52.7% 60.3% 52.6% 60.5%
Consumption tax rate (τc
)
initial 11.00 11.00 11.00 11.00 11.00 11.00
final 15.44 10.95 15.43 11.99 15.46 10.95
Pension system deficit
initial 1.46 1.56 1.46
final 3.95 0.00 4.02 0.87 3.97 0.00
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28. Motivation Model Calibration Results Appendix
Conclusions
Consumption inequalities increase due to
aging processes
DB→DC reform
Contribution cap has virtually no effects
Minimum pensions are effective in reducing consumption
inequalities resulting from the DB→DC reform by 40-50%
Provision of the minimum pension guarantee covering the majority
of the populace costs about 1 pp higher consumption tax
and requires a transfer of about 0.9% GDP
Minimum pensions increase wealth inequality
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29. Motivation Model Calibration Results Appendix
Thank you for your attention
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30. Motivation Model Calibration Results Appendix
Household Sector Closed Form Solution I
For j < ¯J (working):
cj,t =
Ωj,t + Γj,t
(1 + τc
t )
¯J−j−1
s=0 (1 + φ) δs πj+s,t+s
πj,t
+ J−j
s= ¯J−j
δs πj+s,t+s
πj,t
lj,t = 1 −
φ(1 + τc
t )cj,t
(1 − τl
t )(1 − τ)wt
sj,t = (1 − τl
t )(1 − τ)wtlj,t + (1 + (1 − τk
t )rt)sj−1,t−1 − (1 + τc
t )cj,t,
with
Ωj,t =
¯J−j−1
s=0
(1 − τl
t+s)(1 − τ)wt+s + beqj+s,t+s − Υt+s
s
i=1(1 + (1 − τk
t+i)rt+i)
Γj,t =
J−j
s= ¯J−j
(1 − τl
t+s)bj+s,t+s + beqj+s,t+s − Υt+s
s
i=1(1 + (1 − τk
t+i)rt+i)
.
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31. Motivation Model Calibration Results Appendix
Household Sector Closed Form Solution II
For j ≥ ¯J (retired):
cj,t =
Γj,t
(1 + τc
t ) J−j
s= ¯J−j
δs πj+s,t+s
πj,t
lj,t = 0
sj,t = (1 − τl
t )bι
j,t + (1 + (1 − τk
t )rt)sj−1,t−1 − (1 + τc
t )cj,t,
with
Γj,t =
J−j
s=0
(1 − τl
t+s)bj+s,t+s + beqj+s,t+s − Υt+s
s
i=1(1 + (1 − τk
t+i)rt+i)
.
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