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 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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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

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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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)

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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)

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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

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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

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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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Results preview

DB → DC reform: both wealth and consumption inequalities ↑Demographic transition ⇒ inequalities ↑, more than due to reform



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Results preview

DB → DC reform: both wealth and consumption inequalities ↑Demographic transition ⇒ inequalities ↑, more than due to reform



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Outline

1 Motivation

2 Model

3 Calibration

4 Results

5 Appendix

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Method

ModelDeterministicOLGex ante heterogeneity: endowments + preferences

Calibrate to Poland in 1999

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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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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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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

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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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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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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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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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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

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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

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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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1 Motivation

2 Model

3 Calibration

4 Results

5 Appendix

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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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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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Within cohort heterogeneity - endowments

Structure of Earnings Survey, 1998, Poland

Productivity ω

Resulting: 10 values for ω

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Within cohort heterogeneity - leisure preference

Structure of Earnings Survey, 1998, Poland

Leisure Preference φ

Resulting: 4 values for φ

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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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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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Within cohort heterogeneity - summary outcomes II0

24

6

0 20 40 60 80age

Lowest omega multiplierStandard omega multiplierHighest omega multiplier

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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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1 Motivation

2 Model

3 Calibration

4 Results

5 Appendix

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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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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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Wealth Gini.8

5.9

.95

11.

05

2000 2050 2100 2150 2200year

DB: No instrumentsDC: No instrumentsDC: Minimum benefits

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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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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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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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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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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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Conclusions

Consumption inequalities increase due toaging processesDB→DC reform

Minimum pensionseffective in reducing consumption inequalities 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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Thank you for your attention

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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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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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Economy & Finance

Inequalities in an OLG economy with heterogeneous cohorts and pension systems