Society for Nonlinear Dynamics in Economics, Dallas, 2019

1. Efficiency versus insurance:
The role for fiscal policy in social security privatization
Joanna Tyrowicz (GRAPE, IAAEU, UW and IZA)
Oliwia Komada (GRAPE and WSE)
Krzysztof Makarski (GRAPE and WSE)
SNDE, Dallas, 2019
1 / 39

2. Motivation
• Longevity hazards DB viability: deficit ↑ 1.4% of GDP → reforms
Feldstein, BEA, SSA
2 / 39

3. Motivation
• Longevity hazards DB viability: deficit ↑ 1.4% of GDP → reforms
Feldstein, BEA, SSA
• (Any) Reform of social security redistributes
• between generations −→ timing of costs vs gains
• within generations −→ insurance inherent to pension systems features
2 / 39

4. Motivation
• Longevity hazards DB viability: deficit ↑ 1.4% of GDP → reforms
Feldstein, BEA, SSA
• (Any) Reform of social security redistributes
• between generations −→ timing of costs vs gains
• within generations −→ insurance inherent to pension systems features
• Deterministic setup: horse-race between
• efficiency (+)
• fiscal cost of the transition period (–)
2 / 39

5. Motivation
• Longevity hazards DB viability: deficit ↑ 1.4% of GDP → reforms
Feldstein, BEA, SSA
• (Any) Reform of social security redistributes
• between generations −→ timing of costs vs gains
• within generations −→ insurance inherent to pension systems features
• Deterministic setup: horse-race between
• efficiency (+)
• fiscal cost of the transition period (–)
• Stochastic setup: insurance (–)
Nishiyama & Smetters (2007, QJE) and subsequent literature
2 / 39

6. Motivation
• Longevity hazards DB viability: deficit ↑ 1.4% of GDP → reforms
Feldstein, BEA, SSA
• (Any) Reform of social security redistributes
• between generations −→ timing of costs vs gains
• within generations −→ insurance inherent to pension systems features
• Deterministic setup: horse-race between
• efficiency (+)
• fiscal cost of the transition period (–)
• Stochastic setup: insurance (–)
Nishiyama & Smetters (2007, QJE) and subsequent literature
2 / 39

7. Motivation → contribution
Fiscal policy around the pension reform matters:
• counteracts / reinforces redistribution
• affects efficiency (scope of distortions)
3 / 39

8. Motivation → contribution
Fiscal policy around the pension reform matters:
• counteracts / reinforces redistribution
• affects efficiency (scope of distortions)
Literature keeps “low profile” on accompanying fiscal policy
We contribute by
• introducing new fiscal closures: boost efficiency + substitute insurance
• quantifying the role of the insurance motive
3 / 39

9. Literature

10. Literature differs in terms of financing the reform
4 / 39

11. Literature differs in terms of financing the reform
• Financing within pension system
• contribution rates (20 papers)
Kumru & Thanopoulos (2011, JPE), Bruce & Turnovsky (2013, JPE)
• replacement rate (8 papers)
Boersch-Supan et al. (2014, AER), Kitao (2014, RED)
5 / 39

12. Literature differs in terms of financing the reform
• Financing within pension system
• contribution rates (20 papers)
Kumru & Thanopoulos (2011, JPE), Bruce & Turnovsky (2013, JPE)
• replacement rate (8 papers)
Boersch-Supan et al. (2014, AER), Kitao (2014, RED)
• Financing via fiscal policy
• labor tax (3 papers)
Bouzahzah et al. (2002, JEDC)
• consumption tax (10 papers)
Nishiyama & Smetters (2007, QJE), Diaz-Gimenez & Diaz-Saavedra (2009, RED)
• debt (5 papers )
Song, et al. (2015, AEJ) Lindbeck & Persson (2003, JEL)
5 / 39

13. Literature differs in terms of financing the reform
• Financing within pension system
• contribution rates (20 papers)
Kumru & Thanopoulos (2011, JPE), Bruce & Turnovsky (2013, JPE)
• replacement rate (8 papers)
Boersch-Supan et al. (2014, AER), Kitao (2014, RED)
• Financing via fiscal policy
• labor tax (3 papers)
Bouzahzah et al. (2002, JEDC)
• consumption tax (10 papers)
Nishiyama & Smetters (2007, QJE), Diaz-Gimenez & Diaz-Saavedra (2009, RED)
• debt (5 papers )
Song, et al. (2015, AEJ) Lindbeck & Persson (2003, JEL)
No papers with comparisons across fiscal closures
5 / 39

14. What we do: focus on fiscal policy for a given reform
• Scrutinize links between fiscal policy and pension system reform
• Propose new ways of financing the pensions system reform
• insurance → labor tax progression
• efficiency → capital income gains tax
• timing → smoothing reform cost with public debt
6 / 39

15. What we do: focus on fiscal policy for a given reform
• Scrutinize links between fiscal policy and pension system reform
• Propose new ways of financing the pensions system reform
• insurance → labor tax progression
• efficiency → capital income gains tax
• timing → smoothing reform cost with public debt
Preview of findings
• Nishiyama & Smetters is NOT universal → welfare ↑ in stochastic setup
• Labor tax progression mitigates insurance loss
• Capital income gains tax boost efficiency
6 / 39

16. What we do: focus on fiscal policy for a given reform
• Scrutinize links between fiscal policy and pension system reform
• Propose new ways of financing the pensions system reform
• insurance → labor tax progression
• efficiency → capital income gains tax
• timing → smoothing reform cost with public debt
Preview of findings
• Nishiyama & Smetters is NOT universal → welfare ↑ in stochastic setup
• Labor tax progression mitigates insurance loss
• Capital income gains tax boost efficiency
• Welfare gains and political support only sometimes overlap
• Public debt often “buys” political support
6 / 39

17. OLG model with income shocks, US
Procedure
Baseline: Redistributive PAYG DB
with aging → deficit
7 / 39

18. OLG model with income shocks, US
Procedure
Baseline: Redistributive PAYG DB
with aging → deficit
Reform: Individual DC, 50% funded
• 9 fiscal policies
• 2 pension system adjustments: contribution or benefits
• 7 fiscal closures: tax on labor income, consumption (with or without debt),
+ tax on capital income (w/ or wo/ debt) and progressive income tax
7 / 39

19. OLG model with income shocks, US
Procedure
Baseline: Redistributive PAYG DB
with aging → deficit
Reform: Individual DC, 50% funded
• 9 fiscal policies
• 2 pension system adjustments: contribution or benefits
• 7 fiscal closures: tax on labor income, consumption (with or without debt),
+ tax on capital income (w/ or wo/ debt) and progressive income tax
• government may behave differently in B and R → 81 combinations
7 / 39

20. OLG model with income shocks, US
Procedure
Baseline: Redistributive PAYG DB
with aging → deficit
Reform: Individual DC, 50% funded
• 9 fiscal policies
• 2 pension system adjustments: contribution or benefits
• 7 fiscal closures: tax on labor income, consumption (with or without debt),
+ tax on capital income (w/ or wo/ debt) and progressive income tax
• government may behave differently in B and R → 81 combinations
• compare welfare effect and political support
7 / 39

21. OLG model with income shocks, US
Procedure
Baseline: Redistributive PAYG DB
with aging → deficit
Reform: Individual DC, 50% funded
• 9 fiscal policies
• 2 pension system adjustments: contribution or benefits
• 7 fiscal closures: tax on labor income, consumption (with or without debt),
+ tax on capital income (w/ or wo/ debt) and progressive income tax
• government may behave differently in B and R → 81 combinations
• compare welfare effect and political support
• decompose welfare change into insurance and efficiency
7 / 39

22. Model
Consumers
• uncertain lifetimes: live for 16 periods, with survival rate πj,t < 1
unintended bequest redistributed within a cohort
8 / 39

23. Model
Consumers
• uncertain lifetimes: live for 16 periods, with survival rate πj,t < 1
unintended bequest redistributed within a cohort
• uninsurable earnings risk: endogenous labor supply + income shocks
process that follows AR(1) approximated by Markov chain
8 / 39

24. Model
Consumers
• uncertain lifetimes: live for 16 periods, with survival rate πj,t < 1
unintended bequest redistributed within a cohort
• uninsurable earnings risk: endogenous labor supply + income shocks
process that follows AR(1) approximated by Markov chain
• pay taxes (labor, consumption, capital gains) & contribute to pensions
8 / 39

25. Model
Consumers
• uncertain lifetimes: live for 16 periods, with survival rate πj,t < 1
unintended bequest redistributed within a cohort
• uninsurable earnings risk: endogenous labor supply + income shocks
process that follows AR(1) approximated by Markov chain
• pay taxes (labor, consumption, capital gains) & contribute to pensions
• no annuity financial markets with (risk free) interest rate
8 / 39

26. Model
Consumers
• uncertain lifetimes: live for 16 periods, with survival rate πj,t < 1
unintended bequest redistributed within a cohort
• uninsurable earnings risk: endogenous labor supply + income shocks
process that follows AR(1) approximated by Markov chain
• pay taxes (labor, consumption, capital gains) & contribute to pensions
• no annuity financial markets with (risk free) interest rate
Competitive producers
• Cobb-Douglas production function
• capital depreciation rate d
8 / 39

27. Pension system
Baseline scenario: PAYG DB
• equal benefit for whole cohort (provides insurance)
b ¯J,t = ρ · wavg,t
9 / 39

28. Pension system
Baseline scenario: PAYG DB
• equal benefit for whole cohort (provides insurance)
b ¯J,t = ρ · wavg,t
• indexed with payroll growth rate (GE labor ↑ ⇒ benefits ↑)
9 / 39

29. Pension system
Baseline scenario: PAYG DB
• equal benefit for whole cohort (provides insurance)
b ¯J,t = ρ · wavg,t
• indexed with payroll growth rate (GE labor ↑ ⇒ benefits ↑)
• longevity ↑ creates deficit −→ need for fiscal closure
9 / 39

30. Pension system
Baseline scenario: PAYG DB
• equal benefit for whole cohort (provides insurance)
b ¯J,t = ρ · wavg,t
• indexed with payroll growth rate (GE labor ↑ ⇒ benefits ↑)
• longevity ↑ creates deficit −→ need for fiscal closure
Reform scenario: partially funded DC
• contributions go into PAYG and funded pillar: τt = τP
t + τF
t
9 / 39

31. Pension system
Baseline scenario: PAYG DB
• equal benefit for whole cohort (provides insurance)
b ¯J,t = ρ · wavg,t
• indexed with payroll growth rate (GE labor ↑ ⇒ benefits ↑)
• longevity ↑ creates deficit −→ need for fiscal closure
Reform scenario: partially funded DC
• contributions go into PAYG and funded pillar: τt = τP
t + τF
t
• individual pension accounts indexed with payroll growth rate ⇒ insurance ↓
b ¯J,t =
accrued ‘savings’
life expectancyt
+
accrued savings
life expectancyt
9 / 39

32. Pension system
Baseline scenario: PAYG DB
• equal benefit for whole cohort (provides insurance)
b ¯J,t = ρ · wavg,t
• indexed with payroll growth rate (GE labor ↑ ⇒ benefits ↑)
• longevity ↑ creates deficit −→ need for fiscal closure
Reform scenario: partially funded DC
• contributions go into PAYG and funded pillar: τt = τP
t + τF
t
• individual pension accounts indexed with payroll growth rate ⇒ insurance ↓
b ¯J,t =
accrued ‘savings’
life expectancyt
+
accrued savings
life expectancyt
• funding generates deficit −→ need for fiscal closure.
9 / 39

33. Government
• Collects taxes
Tt = τl,t(1 − τt)wtLt + τk,trtAt + τc,tCt + Υt
J
j=1
Nj,t
• Finances government spending Gt = gzt
J
j=1 Nj,t,
• Balances pension system: subsidyt
• Services debt: ∆Dt = Dt − Dt−1
Gt + subsidyt + rtDt = Tt + ∆Dt
10 / 39

34. Fiscal policy accompanying the reform
• Three new closures details
• progressive labor tax → labor supply
• capital income gain tax (+ debt) → capital accumulation
11 / 39

35. Fiscal policy accompanying the reform
• Three new closures details
• progressive labor tax → labor supply
• capital income gain tax (+ debt) → capital accumulation
• Two closures within pension system details
• contributions → labor supply
• pensions → consumption (of retirees)
11 / 39

36. Fiscal policy accompanying the reform
• Three new closures details
• progressive labor tax → labor supply
• capital income gain tax (+ debt) → capital accumulation
• Two closures within pension system details
• contributions → labor supply
• pensions → consumption (of retirees)
• Four closures outside pension system details
• consumption tax (+ debt) → consumption
• labor tax (+ debt) → labor supply
11 / 39

37. Calibration

38. Calibration to replicate US economy (2015)
Preferences
• Preference for leisure φ matches average hours 33%
• Discounting rate δ matches interest rate 4%
12 / 39

39. Calibration to replicate US economy (2015)
Preferences
• Preference for leisure φ matches average hours 33%
• Discounting rate δ matches interest rate 4%
Idiosyncratic productivity shock based on Kruger and Ludwig (2013):
• Persistence η = 0.95
• Variance ση = 0.375
12 / 39

40. Calibration to replicate US economy (2015)
Preferences
• Preference for leisure φ matches average hours 33%
• Discounting rate δ matches interest rate 4%
Idiosyncratic productivity shock based on Kruger and Ludwig (2013):
• Persistence η = 0.95
• Variance ση = 0.375
Pension system
• Replacement rate ρ matches benefits as % of GDP 5.2%
• Contribution rate balances pension system in the initial steady state
• Pension eligibility age at 65
12 / 39

41. Calibration to replicate US economy (2015)
Preferences
• Preference for leisure φ matches average hours 33%
• Discounting rate δ matches interest rate 4%
Idiosyncratic productivity shock based on Kruger and Ludwig (2013):
• Persistence η = 0.95
• Variance ση = 0.375
Pension system
• Replacement rate ρ matches benefits as % of GDP 5.2%
• Contribution rate balances pension system in the initial steady state
• Pension eligibility age at 65
Taxes {τc, τl, τk} match revenue as % of GDP {9.2%, 3.8%, 3.6%}
Depreciation rate d matches investment rate of 25%
12 / 39

42. Calibration to replicate US economy (2015)
Demography is based on UN projections.
number of 20-year-olds mortality rates
13 / 39

43. Results

44. No reform: deficit ↑
Adjustment in pension parameters
contribution rate ↑ from 7.8% to 9%
tax on pensions ↑ from 0.0% to 17.3%
Adjustment in fiscal parameters
pension system deficit ↑
by 1pp of GDP
14 / 39

45. Funding the reform requires 2% of GDP (temporarily)
capital labor
Pension system deficit temporary ↑ from 0% to 2% of GDP
15 / 39

46. Major effects of the reforming
Pensions linked to contributions
1. reduced labor supply distortion (efficiency ↑)
2. income shocks carry over to retirement (insurance ↓ )
Intuition: fiscal policy can reinforce or attenuate these effects
16 / 39

47. Major effects of the reforming
Pensions linked to contributions
1. reduced labor supply distortion (efficiency ↑)
2. income shocks carry over to retirement (insurance ↓ )
Intuition: fiscal policy can reinforce or attenuate these effects
Eventually taxes decline
(relative to baseline scenario of permanent pension system deficit)
16 / 39

48. Result 1: insurance is small & efficiency is large
capital tax: the highest welfare gain due
to efficiency
progression: the smallest welfare loss
due to insurance
17 / 39

49. Result 2: loss of insurance important but not decisive
τk has larger gain than τc → positive overall welfare effect
18 / 39

50. Result 3: public debt helps gaining political support
Welfare effect – τk
19 / 39

51. Result 3: public debt helps gaining political support
Welfare effect - τk & debt + τk
20 / 39

52. Result 3: public debt helps gaining political support
Welfare effect - transition - τk & debt + τk
actual adjustments
21 / 39

53. Aggregate welfare effects and political support
Fiscal closure Baseline
τk d + τk prog. τ τb τc d + τc τl d + τl
Reform
τk 0.57 0.56 1.01 0.59 0.50 0.65 0.65 0.65 0.66
d + τk 0.54 0.54 0.99 0.56 0.47 0.63 0.63 0.63 0.64
prog. -0.45 -0.45 0.02 -0.13 -0.07 -0.35 -0.35 -0.36 -0.34
τ -0.13 -0.12 0.35 0.09 0.14 -0.03 -0.02 -0.03 -0.01
τb -0.15 -0.14 0.33 0.07 0.13 -0.05 -0.04 -0.05 -0.03
τc -0.14 -0.14 0.33 0.11 0.17 -0.04 -0.03 -0.05 -0.03
d + τc -0.16 -0.16 0.31 0.09 0.15 -0.07 -0.06 -0.07 -0.05
τl -0.46 -0.46 0.01 -0.11 -0.03 -0.36 -0.35 -0.37 -0.35
d + τl -0.45 -0.45 0.01 -0.1 -0.02 -0.36 -0.35 -0.36 -0.35
• τk is always a good idea
22 / 39

54. Aggregate welfare effects and political support
Fiscal closure Baseline
τk d + τk prog. τ τb τc d + τc τl d + τl
Reform
τk 0.57 0.56 1.01 0.59 0.50 0.65 0.65 0.65 0.66
d + τk 0.54 0.54 0.99 0.56 0.47 0.63 0.63 0.63 0.64
prog. -0.45 -0.45 0.02 -0.13 -0.07 -0.35 -0.35 -0.36 -0.34
τ -0.13 -0.12 0.35 0.09 0.14 -0.03 -0.02 -0.03 -0.01
τb -0.15 -0.14 0.33 0.07 0.13 -0.05 -0.04 -0.05 -0.03
τc -0.14 -0.14 0.33 0.11 0.17 -0.04 -0.03 -0.05 -0.03
d + τc -0.16 -0.16 0.31 0.09 0.15 -0.07 -0.06 -0.07 -0.05
τl -0.46 -0.46 0.01 -0.11 -0.03 -0.36 -0.35 -0.37 -0.35
d + τl -0.45 -0.45 0.01 -0.1 -0.02 -0.36 -0.35 -0.36 -0.35
• τk is always a good idea
• prog. (almost) always better then τl in the reform
22 / 39

55. Aggregate welfare effects and political support
Fiscal closure Baseline
τk d + τk prog. τ τb τc d + τc τl d + τl
Reform
τk 0.57 0.56 1.01 0.59 0.50 0.65 0.65 0.65 0.66
d + τk 0.54 0.54 0.99 0.56 0.47 0.63 0.63 0.63 0.64
prog. -0.45 -0.45 0.02 -0.13 -0.07 -0.35 -0.35 -0.36 -0.34
τ -0.13 -0.12 0.35 0.09 0.14 -0.03 -0.02 -0.03 -0.01
τb -0.15 -0.14 0.33 0.07 0.13 -0.05 -0.04 -0.05 -0.03
τc -0.14 -0.14 0.33 0.11 0.17 -0.04 -0.03 -0.05 -0.03
d + τc -0.16 -0.16 0.31 0.09 0.15 -0.07 -0.06 -0.07 -0.05
τl -0.46 -0.46 0.01 -0.11 -0.03 -0.36 -0.35 -0.37 -0.35
d + τl -0.45 -0.45 0.01 -0.1 -0.02 -0.36 -0.35 -0.36 -0.35
• τk is always a good idea
• prog. (almost) always better then τl in the reform
• little effect of debt on welfare
22 / 39

56. Aggregate welfare effects and political support
Fiscal closure Baseline
τk d + τk prog. τ τb τc d + τc τl d + τl
Reform
τk 0.57 0.56 1.01 0.59 0.50 0.65 0.65 0.65 0.66
d + τk 0.54 0.54 0.99 0.56 0.47 0.63 0.63 0.63 0.64
prog. -0.45 -0.45 0.02 -0.13 -0.07 -0.35 -0.35 -0.36 -0.34
τ -0.13 -0.12 0.35 0.09 0.14 -0.03 -0.02 -0.03 -0.01
τb -0.15 -0.14 0.33 0.07 0.13 -0.05 -0.04 -0.05 -0.03
τc -0.14 -0.14 0.33 0.11 0.17 -0.04 -0.03 -0.05 -0.03
d + τc -0.16 -0.16 0.31 0.09 0.15 -0.07 -0.06 -0.07 -0.05
τl -0.46 -0.46 0.01 -0.11 -0.03 -0.36 -0.35 -0.37 -0.35
d + τl -0.45 -0.45 0.01 -0.1 -0.02 -0.36 -0.35 -0.36 -0.35
• τk is always a good idea
• prog. (almost) always better then τl in the reform
• little effect of debt on welfare
• debt ’buys’ support (for good and bad reforms)
22 / 39

57. Summary
The same reform produces different effects depending on fiscal policy:
• efficiency effects of fiscal policy can be big
• insurance loss can effectively be substituted and is not very large
• smoothing by public debt buys political support
23 / 39

58. Summary
The same reform produces different effects depending on fiscal policy:
• efficiency effects of fiscal policy can be big
• insurance loss can effectively be substituted and is not very large
• smoothing by public debt buys political support
Features of this literature
• Labor has a roughly 10% response to reduced l.m. distortions
do people really “understand” the link between contributions and pensions
• Savings have a roughly 10% reaction to longevity
do people really “understand” the risk of old-age poverty
23 / 39

59. Where we get from here?
Work in progress:
• AIME + OADSI rather than fully redistributive DB
• Continuous income tax progression (e.g. Benabou, 2002)
• Analyze separately effects of longevity and reform
24 / 39

60. Where we get from here?
Work in progress:
• AIME + OADSI rather than fully redistributive DB
• Continuous income tax progression (e.g. Benabou, 2002)
• Analyze separately effects of longevity and reform
Why is capital tax so nice? −→ relatively less responsive to the tax hikes
• Longevity → capital accumulation ↑
• Funded pillar → capital accumulation ↑
24 / 39

61. Where we get from here?
Work in progress:
• AIME + OADSI rather than fully redistributive DB
• Continuous income tax progression (e.g. Benabou, 2002)
• Analyze separately effects of longevity and reform
Why is capital tax so nice? −→ relatively less responsive to the tax hikes
• Longevity → capital accumulation ↑
• Funded pillar → capital accumulation ↑
Preliminary results:
• Longevity is the main driver of changes in the economy
• Capital tax better than consumption tax under longevity
• Without longevity funding and DC reduce welfare regardless of fiscal closure
24 / 39

62. Thank you and
I am happy to take questions!
w: grape.org.pl
t: grape org
f: grape.org
e: j.tyrowicz@grape.org.pl
25 / 39

63. New fiscal closures: capital income gain tax
GO BACK
• capital income gain tax, τk,t
Tt = τl,t(1 − τt)wtLt + τk,trtAt + τc,tCt + Υt
J
j=1
Nj,t
Gt + subsidyt + rtDt = Tt + ∆Dt
• smoothing tax adjustments with public debt
• part of the costs of the reform shifted to the future generations
• fiscal rule
τk,t = (1 − )τfinal
k + τk,t−1 + D
Dt
Yt
−
D
Y
final
• debt in the final steady state the same as in the initial steady state

64. New fiscal closures: labor tax progression
GO BACK
• tr1 the lowest income threshold
• trn is the highest income threshold
• n is the number of income brackets
• m is a tax multiplier such that τi
l,t = τ0
l,t ∗ mi

65. New fiscal closures: labor tax progression
GO BACK
• tr1 the lowest income threshold
• trn is the highest income threshold
• n is the number of income brackets
• m is a tax multiplier such that τi
l,t = τ0
l,t ∗ mi
• Income threshold is multiple of average labor income, (1 − τt)wt
¯lt.
• In the initial steady state m = 1
• In the transition path m = 1.15 and n = 4

66. New fiscal closures: labor tax progression
Total gross labor income (1 − τt)wtLt is a sum of n + 1 components: earnings taxed
by one of n + 1 tax rate.
L
0
t =
¯J
j=1
Nj,t
Ω
min(ωj,t(sj,t)lj,t(sj,t), tr1)dPj,t
L
i
t =
¯J
j=1
Nj,t
Ω
max(min(ωj,t(sj,t)lj,t(sj,t − tr1), tri − tri−1), 0)dPj,t∀i = 1, ..., n
τ
0
l,t =
Gt + subsidyt + ∆Dt − Υ1
J
j=1 Nj,t − τc,1Ct − τk,1rtAt − n
i=0 Li
tτi
l
n
i=0 Li
t
τ
i
l,1 = m
i
∗ τ
0
l,1

67. Fiscal closures within pension system, subsidyt = 0
GO BACK
To keep pension system balanced government may adjust:
• contribution rate τ
• benefits bj (as a tax on benefits)
J
j= ¯Jt
Nj,t(1 − τb,t)bj,t = τt ¯wtLt and subsidyt = 0

68. Fiscal closures outside pension system, subsidyt = 0
GO BACK
• consumption tax, τc,t
• labor tax, τl,t
Tt = τl,t(1 − τt)wtLt + τk,trtAt + τc,tCt + Υt
J
j=1
Nj,t
Gt + subsidyt + rtDt = Tt + ∆Dt
• smoothing tax adjustments with public debt via a fiscal rule: ∀tax ∈ {l, c}
τtax,t = (1 − )τfinal
tax + τtax,t−1 + D (D/Y )t − (D/Y )final
• public debt in the final steady state = the initial steady state

69. Profile of average consumption for τk closure
other closures GO BACK
in line with Gourinchas & Parker (2002, Econometrica)

70. Profile of average savings for τk closure
other closures GO BACK

71. Profile of average labor for τk closure
other closures GO BACK

72. GO BACK
τk debt + τk progression
τ τl debt + τl
τb τc debt + τc

73. GO BACK
τk debt + τk progression
τ τl debt + τl
τb τc debt +τc

74. GO BACK
τk debt + τk progression
τ τl debt + τl
τb τc debt +τc

75. Capital
GO BACK

76. Labor
GO BACK

77. Model solving
GO BACK
• Gauss-Seidel iterative algorithm
• Guess an initial value for k = K/(zL) and compute prices
• Solve individual problem and aggregate it to find new K and L , thus k
• iterate until convergence

78. Model solving
GO BACK
• Gauss-Seidel iterative algorithm
• Guess an initial value for k = K/(zL) and compute prices
• Solve individual problem and aggregate it to find new K and L , thus k
• iterate until convergence
• Consumer problem (backward policy function iterations)

79. Model solving
GO BACK
• Gauss-Seidel iterative algorithm
• Guess an initial value for k = K/(zL) and compute prices
• Solve individual problem and aggregate it to find new K and L , thus k
• iterate until convergence
• Consumer problem (backward policy function iterations)
• implicit tax to reduce state space, Butler (2002)
• policy function iterations with picewise linear interpolation
• within period problem solved with Newton-Raphson
• given initial distribution at age j = 1, transition matrix for idiosyncratic
productivity and the policy functions compute the distribution in any successive
age j.
• aggregation done with Gaussian quadrature

80. Model solving
GO BACK
• Gauss-Seidel iterative algorithm
• Guess an initial value for k = K/(zL) and compute prices
• Solve individual problem and aggregate it to find new K and L , thus k
• iterate until convergence
• Consumer problem (backward policy function iterations)
• implicit tax to reduce state space, Butler (2002)
• policy function iterations with picewise linear interpolation
• within period problem solved with Newton-Raphson
• given initial distribution at age j = 1, transition matrix for idiosyncratic
productivity and the policy functions compute the distribution in any successive
age j.
• aggregation done with Gaussian quadrature
• Transition path, goes between the initial and final steady state