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). Our framework features within cohort heterogeneity of endowments (individual productivities) and heterogeneity of preferences (preference for leisure and time preference). We introduce two policy instruments, which are widely used: a contribution cap and a minimum pension. We show four main results. First, longevity increases aggregate consumption inequalities substantially in both pension systems, whereas the effect of a pension system reform works to reinforce the consumption inequalities and reduce the wealth inequalities. Second, the contribution cap has negligible effect on inequalities, but the role for minimum pension benefit guarantee is more pronounced. Third, the reduction in inequalities due to minimum pension benefit guarantee is achieved with virtually no effect on capital accumulation. The fourth result and the main policy implication of our study, is demonstrating that the minimum pension benefit guarantee addresses mostly the inequalities which stem from differentiated endowments and not those that stem from differentiated preferences.

1. Motivation Model Calibration Results Appendix
Inequalities in an OLG economy
with heterogeneous cohorts and pension systems
(with Marcin Bielecki, Krzysztof Makarski and Marcin Waniek)
Joanna Tyrowicz
University of Warsaw and National Bank of Poland
Netspar Workshop 2015, Utrecht
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2. Motivation Model Calibration Results Appendix
Motivation
Wealth inequality increases due to:
Demographic transition
Pension reform: defined benefit → defined contribution
Effects for consumption inequality: unclear
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3. Motivation Model Calibration Results Appendix
Motivation
Wealth inequality increases due to:
Demographic transition
Pension reform: defined benefit → defined contribution
Effects for consumption inequality: unclear
Can policy instruments help?
minimum pensions: ↑ pensions; ↓ labor supply incentives
contribution caps : obligatory savings replaced with private savings
Intuition insufficient
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4. Motivation Model Calibration Results Appendix
Literature review
Distributional effects of pension systems: OLG models with ex
post 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)
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5. Motivation Model Calibration Results Appendix
Literature review
Distributional effects of pension systems: OLG models with ex
post 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 rates
Hairault and Langot (2008, JEDC):
McGrattan and Prescott (2014, NBER)
Kindermann and Krueger (2014, NBER)
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6. Motivation Model Calibration Results Appendix
Our approach
Question 1: distributional effects of a pension system reform
Question 2: are standard instruments effective in reducing the
increase in inequality
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7. Motivation Model Calibration Results Appendix
Our approach
Question 1: distributional effects of a pension system reform
Question 2: are standard instruments effective in reducing the
increase in inequality
Ex ante heterogeneous agents: age + within cohort
endowments + preferences ← not a stand
separate endowments from preferences
most countries: no data on mortality by education / income groups
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8. Motivation Model Calibration Results Appendix
Results preview
DB → DC reform: both wealth and consumption inequalities ↑
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9. Motivation Model Calibration Results Appendix
Results preview
DB → DC reform: both wealth and consumption inequalities ↑
Demographic transition ⇒ inequalities ↑, more than due to reform
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10. 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: unnoticeable
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11. Motivation Model Calibration Results Appendix
Outline
1 Motivation
2 Model
3 Calibration
4 Results
5 Appendix
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12. Motivation Model Calibration Results Appendix
Method
Model
Deterministic
OLG
ex ante heterogeneity: endowments + preferences
Calibrate to Poland in 1999
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13. 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 π
Belong to a type k:
productivity level ω
time discounting δ
relative leisure preference φ
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
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14. 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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15. Motivation Model Calibration Results Appendix
Producers
Perfectly competitive representative firm
Standard Cobb-Douglas production function
Yt = Kα
t (ztLt)1−α
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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16. 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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17. 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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18. Motivation Model Calibration Results Appendix
Instrument 1: minimum pensions
Definition
bj,k,t ≥ ρmin · gross average waget
We set ρmin = 0.2 → 4% coverage (consistent with the data)
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19. Motivation Model Calibration Results Appendix
Instrument 1: minimum pensions
Definition
bj,k,t ≥ ρmin · gross average waget
We set ρmin = 0.2 → 4% coverage (consistent with the data)
Expectations
Directly affects 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
Lower incentives for private savings → possible increase in
consumption
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20. Motivation Model Calibration Results Appendix
Instrument 1: contribution cap
Definition:
τeff
j,k,t = min τ,
τcap · gross average waget
wtωklj,k,t
To replicate 2% coverage, τcap = 1.7 (lower than de iure 2.5)
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21. Motivation Model Calibration Results Appendix
Instrument 1: contribution cap
Definition:
τeff
j,k,t = min τ,
τcap · gross average waget
wtωklj,k,t
To replicate 2% coverage, τcap = 1.7 (lower than de iure 2.5)
Expectations
Affects directly only the right tail of income distribution
Lower contributions of targeted group → higher voluntary saving
rates → wealth inequalities ↑
Matters because market interest rates and social security indexation
differ
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22. 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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23. Motivation Model Calibration Results Appendix
1 Motivation
2 Model
3 Calibration
4 Results
5 Appendix
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24. Motivation Model Calibration Results Appendix
Exogenous assumptions
Projections for Poland provided by the European Commission
Population Size TFP Growth
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25. 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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26. 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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27. 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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28. Motivation Model Calibration Results Appendix
Within cohort heterogeneity - time preference
Again: no data on mortality rates or wealth by income or
education groups
Calibrate the central value of δ to match the investment rate
Split population ad hoc to 3 groups:
discount factors are (0.98δ, δ, 1.02δ)
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29. 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 state is 25.5,
consistent with Brzezinski (2011)
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30. Motivation Model Calibration Results Appendix
Within cohort heterogeneity - summary outcomes II
0246
0 20 40 60 80
age
Lowest omega multiplier
Standard omega multiplier
Highest omega multiplier
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31. Motivation Model Calibration Results Appendix
Within cohort heterogeneity - summary outcomes III
−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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32. Motivation Model Calibration Results Appendix
1 Motivation
2 Model
3 Calibration
4 Results
5 Appendix
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33. 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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34. 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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35. 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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36. Motivation Model Calibration Results Appendix
Inequality decomposition – endowments vs preferences
Instruments should reduce inequality stemming from endowments
(luck) but not from preferences
To isolate the effects of the two sources:
Shut down each channel separately
Keep prices constant from the full model to avoid GE effects
Solve for decisions of households in partial equilibrium
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37. 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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38. 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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39. 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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40. 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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41. Motivation Model Calibration Results Appendix
Conclusions
Consumption inequalities increase due to
aging processes
DB→DC reform
Minimum pensions
effective in reducing consumption inequalities resulting from the
DB→DC reform by 40-50%
with 80% coverage minimum pension costs 1 pp higher
consumption tax (transfer of about 0.9% GDP)
wealth inequality increases
Contribution cap has virtually no effects
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42. Motivation Model Calibration Results Appendix
Thank you for your attention
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43. 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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44. 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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