Inequalities in an OLG economy with heterogeneous cohorts and pension systems

Motivation Model Calibration Results Appendix

Inequalities in an OLG economywith heterogeneous cohorts and pension systems

(with Joanna Tyrowicz, Krzysztof Makarski and Marcin Waniek)

Marcin Bielecki

Faculty of Economics, University of Warsaw

4th NBP Summer Workshop22-24 June 2015

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Motivation Model Calibration Results Appendix

Motivation

Consumption inequality increases due to:Demographic transitionPension reform: defined benefit → defined contribution

Effects for wealth inequality: unclear

Can policy instruments help?minimum pensions: pensions ↑, labor supply incentives ↓contribution caps: mandatory savings replaced with private

Intuition insufficient – need quantitative answers

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Motivation Model Calibration Results Appendix

Motivation

Consumption inequality increases due to:Demographic transitionPension reform: defined benefit → defined contribution

Effects for wealth inequality: unclear

Can policy instruments help?minimum pensions: pensions ↑, labor supply incentives ↓contribution caps: mandatory savings replaced with private

Intuition insufficient – need quantitative answers

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Motivation Model Calibration Results Appendix

Literature review

Distributional effects of pension systems: OLG models withex 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 ratesHairault and Langot (2008, JEDC):McGrattan and Prescott (2014, NBER)Kindermann and Krueger (2014, NBER)

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Motivation Model Calibration Results Appendix

Literature review

Distributional effects of pension systems: OLG models withex 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 ratesHairault and Langot (2008, JEDC):McGrattan and Prescott (2014, NBER)Kindermann and Krueger (2014, NBER)

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Motivation Model Calibration Results Appendix

Our approach

Question 1: distributional effects of a pension system reform

Question 2: are standard instruments effective in reducingthe increase in inequality

Ex ante heterogeneous agents: age + within cohort

endowments + preferences ← not a standisolate the role of each separatelymost countries: no data on mortality by educationor income groups

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Motivation Model Calibration Results Appendix

Our approach

Question 1: distributional effects of a pension system reform

Question 2: are standard instruments effective in reducingthe increase in inequality

Ex ante heterogeneous agents: age + within cohort

endowments + preferences ← not a standisolate the role of each separatelymost countries: no data on mortality by educationor income groups

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Motivation Model Calibration Results Appendix

Results preview

DB→DC reform: consumption inequalities ↑, wealth inequalities ↓

Demographic transition: consumption inequalities ↑– effect larger than reform

Minimum pensions:reduce consumption inequality from the reform by approx. 40%work on the endowments margin, but not on preferences

Effects of the contribution cap: unnoticeable

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Motivation Model Calibration Results Appendix

Results preview

DB→DC reform: consumption inequalities ↑, wealth inequalities ↓Demographic transition: consumption inequalities ↑– effect larger than reform

Minimum pensions:reduce consumption inequality from the reform by approx. 40%work on the endowments margin, but not on preferences

Effects of the contribution cap: unnoticeable

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Motivation Model Calibration Results Appendix

Results preview

DB→DC reform: consumption inequalities ↑, wealth inequalities ↓Demographic transition: consumption inequalities ↑– effect larger than reform

Minimum pensions:reduce consumption inequality from the reform by approx. 40%work on the endowments margin, but not on preferences

Effects of the contribution cap: unnoticeable

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Motivation Model Calibration Results Appendix

Method

Modeldeterministic general equilibriumoverlapping generationsex ante heterogeneity: endowments + preferences

Calibrate to Poland in 1999

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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 endogenously until retirementMaximize 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

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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

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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.

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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)

Expected effectsAffects directly only the left tail of income distributionIncreases lifetime incomes of targeted group: consumptioninequality should decreaseLower incentives for private savings: possible increasein consumptionLower incentives to work: possible reduction in hours worked

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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)

Expected effectsAffects directly only the left tail of income distributionIncreases lifetime incomes of targeted group: consumptioninequality should decreaseLower incentives for private savings: possible increasein consumptionLower incentives to work: possible reduction in hours worked

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Motivation Model Calibration Results Appendix

Instrument 2: 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)

Expected effectsAffects directly only the right tail of income distributionLower contributions of targeted group: higher voluntary saving rates→ wealth inequalities ↑, capital accumulation ↑Matters because market interest rates and social security indexationdiffer

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Motivation Model Calibration Results Appendix

Instrument 2: 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)

Expected effectsAffects directly only the right tail of income distributionLower contributions of targeted group: higher voluntary saving rates→ wealth inequalities ↑, capital accumulation ↑Matters because market interest rates and social security indexationdiffer

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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 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)Transition path

1 Basing on the initial and final steady state values for k guess aninitial path between the terminal points...

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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

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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 incomeor education groups

Calibrate the central value of δ to match the interest 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

Productivity values and leisure preference parameters matchedto replicate dataDiscount factors differentiated ad hocIn 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

Minimum pensions coverage (demographic transition)0

.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

Consumption Gini.2

4.2

6.2

8.3

2000 2050 2100 2150 2200year

DB: No instrumentsDC: No instrumentsDC: Minimum benefits

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Motivation Model Calibration Results Appendix

Wealth Gini.8

5.9

.95

11.

05

2000 2050 2100 2150 2200year

DB: No instrumentsDC: No instrumentsDC: Minimum benefits

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Motivation Model Calibration Results Appendix

Inequality decomposition – endowments vs preferences

Clear arguments for reducing inequality stemmingfrom endowments (luck), not so much 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

Fixed endowments Fixed preferencesDiffering preferences Differing endowments

.05

.1.1

5.2

.25

.3

2000 2050 2100 2150 2200year

DB: Fixed endowments, no instrumentsDC: Fixed endowments, no instrumentsDC: Fixed endowments, minimum benefits

.05

.1.1

5.2

.25

.3

2000 2050 2100 2150 2200year

DB: Fixed preferences, no instrumentsDC: Fixed preferences, no instrumentsDC: Fixed preferences, minimum benefits

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Motivation Model Calibration Results Appendix

Wealth inequality decomposition

Fixed endowments Fixed preferencesDiffering preferences Differing endowments

0.2

.4.6

.81

2000 2050 2100 2150 2200year

DB: Fixed endowments, no instrumentsDC: Fixed endowments, no instrumentsDC: Fixed endowments, minimum benefits

0.2

.4.6

2000 2050 2100 2150 2200year

DB: Fixed preferences, no instrumentsDC: Fixed preferences, no instrumentsDC: Fixed preferences, minimum benefits

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Motivation Model Calibration Results Appendix

Macroeconomic effects

No instrument Minimum pension Contribution capDB DC DB DC DB DC

Capital 52.6% 60.4% 52.7% 60.3% 52.6% 60.5%Consumption tax rate (τ c)

initial 11.00 11.00 11.00 11.00 11.00 11.00final 15.44 10.95 15.43 11.99 15.46 10.95

Pension system deficitinitial 1.46 1.56 1.46final 3.95 0.00 4.02 0.87 3.97 0.00

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Motivation Model Calibration Results Appendix

Welfare effects

Defined Benefit Defined Contribution

−.0

004

−.0

002

0.0

002

Wei

ghte

d M

ean

Com

pens

atin

g V

aria

tion

2000 2050 2100 2150 2200 2250Year of birth

Minimum benefitsContributions cap

−.0

03−

.002

−.0

010

Wei

ghte

d M

ean

Com

pens

atin

g V

aria

tion

2000 2050 2100 2150 2200Year of birth

Minimum benefitsContributions cap

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Motivation Model Calibration Results Appendix

Conclusions

Consumption inequalities increase due todemographic transitionDB → DC reform

Minimum pensionseffective in reducing consumption inequality resultingfrom the DB → DC reform by approx. 40%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)

.

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