The existing view in the literature is that linking pensions to individual incomes rather than average wages reduces distortions, but removes insurance and hence cannot raise welfare in an economy with idiosyncratic income shocks. We study alternative channels of providing  insurance and efficiency gain from both strengthening the labor supply incentives and higher capital accumulation due to partial funding. We show labor tax progression can effectively substitute for the insurance implicit in redistributive social security. Thus, privatizing social security can deliver aggregate welfare gains even under uncertainty about future incomes.

1. Progressing into efficiency:
the role for labor tax progression in privatizing social security
Oliwia Komada (GRAPE and WSE)
Krzysztof Makarski (GRAPE and WSE)
Joanna Tyrowicz (GRAPE, UW, and IZA)
Public Sector Economics – Virtual Conference
Zagreb, 2020
1

2. Motivation

3. Motivation
Social security is essentially about insurance:
• annuity
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• redistribution
Cooley & Soares 1996, Tabellini 2000
2

4. Motivation
Social security is essentially about insurance:
• annuity
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• redistribution
Cooley & Soares 1996, Tabellini 2000
Prevailing consensus:
• redistribution is costly (inefficiency)
e.g. Diamond 1977 + large and diverse subsequent literature
• insurance motive dominates deadweight loss due to redistribution
Davidoff et al. 2005, Nishiyama & Smetters 2007, Fehr et al. 2008
2

5. Motivation
Social security is essentially about insurance:
• annuity
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• redistribution
Cooley & Soares 1996, Tabellini 2000
Prevailing consensus:
• redistribution is costly (inefficiency)
e.g. Diamond 1977 + large and diverse subsequent literature
• insurance motive dominates deadweight loss due to redistribution
Davidoff et al. 2005, Nishiyama & Smetters 2007, Fehr et al. 2008
Our approach: more instruments of redistribution than social security.
2

6. Our contribution: labor tax progressivity
• Equivalence of insurance during working period and retirement
• Social security can be less inefficient → welfare improvement
3

7. Our contribution: labor tax progressivity
• Equivalence of insurance during working period and retirement
• Social security can be less inefficient → welfare improvement
Quantitative model
• Idiosyncratic income shocks, abstract from value of annuity
• Calibrated to US
3

8. Table of contents
Motivation
Theoretical model
Theoretical results
Quantitative model
Calibration
Results
Conclusions
4

9. Theoretical model

10. (Stylized) theoretical model: partial equilibrium OLG model
Households:
• 2-periods, population is constant
• type θ ∈ H, L, and θ-specific productivity ω ∈ {ωL, ωH }, and ωH > ωL
with y(θ) = wt ωθlt (θ) (and ˜y(θ) = ˜wt ωθlt (θ))
5

11. (Stylized) theoretical model: partial equilibrium OLG model
Households:
• 2-periods, population is constant
• type θ ∈ H, L, and θ-specific productivity ω ∈ {ωL, ωH }, and ωH > ωL
with y(θ) = wt ωθlt (θ) (and ˜y(θ) = ˜wt ωθlt (θ))
• choose labor, consumption and assets
first period: c1,t (θ) + a1,t+1(θ) = (1 − τ)˜wt ωθ t (θ) − T(˜y(θ))
second period: c2,t+1(θ) = (1 + r)a1,t+1(θ) + b2,t+1(θ)
T(y(θ)) is the progressive income tax and τ is social security contribution
5

12. (Stylized) theoretical model: partial equilibrium OLG model
Households:
• 2-periods, population is constant
• type θ ∈ H, L, and θ-specific productivity ω ∈ {ωL, ωH }, and ωH > ωL
with y(θ) = wt ωθlt (θ) (and ˜y(θ) = ˜wt ωθlt (θ))
• choose labor, consumption and assets
first period: c1,t (θ) + a1,t+1(θ) = (1 − τ)˜wt ωθ t (θ) − T(˜y(θ))
second period: c2,t+1(θ) = (1 + r)a1,t+1(θ) + b2,t+1(θ)
T(y(θ)) is the progressive income tax and τ is social security contribution
• GHH preferences: Frisch elasticity + risk aversion
U(θ) =
1
1 − σ
(c1,t (θ) −
φ
1 + η
zt l1,t (θ)1+η
+ βc2,t+1(θ))1−σ
5

13. (Stylized) theoretical model: partial equilibrium OLG model
Government
• exogenously given level of revenue,
˜Rt =
θ∈{θL,θH }
T(ωθ ˜wt t (θ)), with ˜R = Rt /zt = constant
• spent on exogenously given government expenditure gt ,
• and lump-sum grants to all agents µt .
6

14. (Stylized) theoretical model: partial equilibrium OLG model
Government
• exogenously given level of revenue,
˜Rt =
θ∈{θL,θH }
T(ωθ ˜wt t (θ)), with ˜R = Rt /zt = constant
• spent on exogenously given government expenditure gt ,
• and lump-sum grants to all agents µt .
• Labor taxation is progressive:
T(˜y) = τl · ωθ ˜wt t (θ) − µt
6

15. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
θ∈{L,H}
ωθ 1,t+1(θ).
7

16. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
θ∈{L,H}
ωθ 1,t+1(θ).
Bismarck (no redistribution)
bBIS
2,t+1(θ) = τ wt (1 + γ) ωθ 1,t (θ)
7

17. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
θ∈{L,H}
ωθ 1,t+1(θ).
Bismarck (no redistribution)
bBIS
2,t+1(θ) = τ wt (1 + γ) ωθ 1,t (θ)
In stationary equilibrium:
˜bBEV
2 (θ) = τ ˜w
1
2
θ∈{L,H}
ωθ 1(θ) = ˜bBEV
2 (θL) = ˜bBEV
2 (θH )
˜bBIS
2 (θ) = τ ˜w ωθ 1(θ)
7

18. Theoretical results

19. Basic observations (1)
1. Labor supply is higher under Bismarckian than under Beveridgean social security
∀θ
BIS
t (θ) > BEV
t (θ)
8

20. Basic observations (1)
1. Labor supply is higher under Bismarckian than under Beveridgean social security
∀θ
BIS
t (θ) > BEV
t (θ)
2. θH workers work more in both BIS and BEV than θL,
8

21. Basic observations (1)
1. Labor supply is higher under Bismarckian than under Beveridgean social security
∀θ
BIS
t (θ) > BEV
t (θ)
2. θH workers work more in both BIS and BEV than θL,
8

22. Basic observations (1)
1. Labor supply is higher under Bismarckian than under Beveridgean social security
∀θ
BIS
t (θ) > BEV
t (θ)
2. θH workers work more in both BIS and BEV than θL, and ratio is constant
BEV
(θH )
BEV (θL)
=
BIS
(θH )
BIS (θL)
=
ωH
ωL
≡ 1/η
> 1
3. ∆ in labor supply between social security systems does not depend on type θ
BIS
(θ) − BEV
(θ)
BEV (θ)
=
(1 − τl (1 − τ))
(1 − τ − τl (1 − τ))
1/η
− 1 ≡ ξ1/η
− 1
8

23. Basic observations (1)
1. Labor supply is higher under Bismarckian than under Beveridgean social security
∀θ
BIS
t (θ) > BEV
t (θ)
2. θH workers work more in both BIS and BEV than θL, and ratio is constant
BEV
(θH )
BEV (θL)
=
BIS
(θH )
BIS (θL)
=
ωH
ωL
≡ 1/η
> 1
3. ∆ in labor supply between social security systems does not depend on type θ
BIS
(θ) − BEV
(θ)
BEV (θ)
=
(1 − τl (1 − τ))
(1 − τ − τl (1 − τ))
1/η
− 1 ≡ ξ1/η
− 1
4. ∆ in gov’nt revenue between social security systems does not depend on type θ
RBIS
− RBEV
RBEV
≡ ξ1/η
− 1
8

24. Basic observations (2)
Continuity in η
lim
η→0
ξ1/η
− 1 = ∞ and lim
η→∞
ξ1/η
− 1 = 0
1. The smaller η, the larger ∆ in labor supply between BIS and BEV
9

25. Basic observations (2)
Continuity in η
lim
η→0
ξ1/η
− 1 = ∞ and lim
η→∞
ξ1/η
− 1 = 0
1. The smaller η, the larger ∆ in labor supply between BIS and BEV
2. The smaller η, the larger ∆ in govn’t revenue between BIS and BEV
9

26. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
10

27. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
10

28. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
W (θH ) ↑ & W (θL) ↑
10

29. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
W (θH ) ↑ & W (θL) ↑
−
1
2
τwt (ωθ
BEV
1,t (θ) − ω−θ
BEV
1,t (−θ))
pension system redistribution
10

30. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
W (θH ) ↑ & W (θL) ↑
−
1
2
τwt (ωθ
BEV
1,t (θ) − ω−θ
BEV
1,t (−θ))
pension system redistribution
W (θH ) ↑ & W (θL) ↓
10

31. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
W (θH ) ↑ & W (θL) ↑
−
1
2
τwt (ωθ
BEV
1,t (θ) − ω−θ
BEV
1,t (−θ))
pension system redistribution
W (θH ) ↑ & W (θL) ↓
+ (µBIS
t (θ) − µBEV
t (θ)
tax system redistribution
10

32. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
W (θH ) ↑ & W (θL) ↑
−
1
2
τwt (ωθ
BEV
1,t (θ) − ω−θ
BEV
1,t (−θ))
pension system redistribution
W (θH ) ↑ & W (θL) ↓
+ (µBIS
t (θ) − µBEV
t (θ)
tax system redistribution
redistribution
10

33. Basic intuitions
• BIS (lower distortions), so BEV→BIS ⇒ efficiency gain ( (θ) ↑)
• BEV (more redistribution), so BEV→BIS ⇒ insurance loss
• In BEV social security transfers from θH to θL are strictly positive.
They are zero in BIS.
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = (1 − τ (1 − τ))ωθwt ( BIS
1 (θ) − BEV
1 (θ))
efficiency gain
W (θH ) ↑ & W (θL) ↑
−
1
2
τwt (ωθ
BEV
1,t (θ) − ω−θ
BEV
1,t (−θ))
pension system redistribution
W (θH ) ↑ & W (θL) ↓
+ (µBIS
t (θ) − µBEV
t (θ)
tax system redistribution
redistribution ⇐ NEW
10

34. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
11

35. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
11

36. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
11

37. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
−→ distribute extra government revenue as lump-sum grants µ
3 ∃ η > 1 such that W of both types ↑
11

38. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
−→ distribute extra government revenue as lump-sum grants µ
3 ∃ η > 1 such that W of both types ↑
11

39. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
−→ distribute extra government revenue as lump-sum grants µ
3 ∃ η > 1 such that W of both types ↑
∆µ from RBIS
− RBEV
> 0 compensates θL
11

40. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
−→ distribute extra government revenue as lump-sum grants µ
3 ∃ η > 1 such that W of both types ↑
∆µ from RBIS
− RBEV
> 0 compensates θL
for θH c(θH ) with higher consumption than under BEV
11

41. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
−→ distribute extra government revenue as lump-sum grants µ
3 ∃ η > 1 such that W of both types ↑
∆µ from RBIS
− RBEV
> 0 compensates θL
for θH c(θH ) with higher consumption than under BEV
(and ∃ η > η when it is no longer the case)
11

42. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ + redistribution ↑)
2 θL may have lower benefits under BIS than under BEV
(efficiency ↑ + redistribution ↓)
−→ distribute extra government revenue as lump-sum grants µ
3 ∃ η > 1 such that W of both types ↑
∆µ from RBIS
− RBEV
> 0 compensates θL
for θH c(θH ) with higher consumption than under BEV
(and ∃ η > η when it is no longer the case)
4 ∀ η > ˜η > η a policy bundle { BEV→BIS and ∆µ > 0} raises welfare in
Pareto sense
11

43. Quantitative model

44. Consumers
• uncertain lifetimes: live for 16 periods, with survival πj < 1
• uninsurable productivity risk: + endogenous labor supply
• CRRA utility function
• pay taxes (progressive on labor, linear on consumption and capital gains)
• contribute to social security, face natural borrowing constraint
12

45. Consumers
• uncertain lifetimes: live for 16 periods, with survival πj < 1
• uninsurable productivity risk: + endogenous labor supply
• CRRA utility function
• pay taxes (progressive on labor, linear on consumption and capital gains)
• contribute to social security, face natural borrowing constraint
Firms and markets
• Cobb-Douglas production function, capital depreciates at rate d
• no annuity, financial markets with (risk free) interest rate
12

46. Government
• Finances government spending Gt , constant as a share of GDP,
• Balances pension system: subsidyt
• Services debt: ∆Dt + rt Dt = Dt − Dt−1 + rt Dt
• Collects taxes on capital, consumption, labor
(progressive given by Benabou form)
Gt + subsidyt + ∆Dt + rt Dt = τk,t rt At + τc,t Ct + Taxl,t
13

47. Policy experiment
Status quo: current US social security
• redistribution through AIME
14

48. Policy experiment
Status quo: current US social security
• redistribution through AIME
• high distortion (no link between LS and future pension benefits)
aj+1,t+1 + ˜cj,t + Υt = yj,t − T (yj,t ) + (1 + ˜rt )aj,t + Γj,t
14

49. Policy experiment
Status quo: current US social security
• redistribution through AIME
• high distortion (no link between LS and future pension benefits)
aj+1,t+1 + ˜cj,t + Υt = yj,t − T (yj,t ) + (1 + ˜rt )aj,t + Γj,t
14

50. Policy experiment
Status quo: current US social security
• redistribution through AIME
• high distortion (no link between LS and future pension benefits)
aj+1,t+1 + ˜cj,t + Υt = yj,t − T (yj,t ) + (1 + ˜rt )aj,t + Γj,t
Alternative: fully individualized social security and lump-sum grants
• no redistribution through social security
14

51. Policy experiment
Status quo: current US social security
• redistribution through AIME
• high distortion (no link between LS and future pension benefits)
aj+1,t+1 + ˜cj,t + Υt = yj,t − T (yj,t ) + (1 + ˜rt )aj,t + Γj,t
Alternative: fully individualized social security and lump-sum grants
• no redistribution through social security
• no distortion
fj+1,t+1 + aj+1,t+1 + ˜cj,t + Υt = yj,t − T (yj,t ) + (1 + ˜rt )aj,t + Γj,t
+(1 + ˜rt )fj,t + τt wt ωj,t lj,t · µj,t
14

52. Calibration

53. Calibration to replicate US economy (2015)
Preferences: instantaneous utility function take CRRA form with
• Risk aversion in equal to 2
• Preference for leisure φ matches average hours 33%
• Discounting rate δ matches interest rate 4.5%
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 (¯j = 9)
Taxes {τc , τk , τl } match revenue as % of GDP {2.8%, 5.4%, 9.2%}
Depreciation rate d matches investment rate of 22%
15

54. Results

55. Labor supply
16

56. Welfare effects
17

57. What stands behind those results?
1. High productivity raise labor supply more
consistent with Hugget, 2010 (JPE)
“high productivity agents work too little and low productivity agents work too much under the U.S. system as
compared to the solution to the planning problem”
18

58. What stands behind those results?
1. High productivity raise labor supply more
consistent with Hugget, 2010 (JPE)
“high productivity agents work too little and low productivity agents work too much under the U.S. system as
compared to the solution to the planning problem”
2. Progression in taxes is more efficient than in pensions
Typical reforms (e.g. health systems) yield changes in the Kakwani index of 1-2 percentage points, we do about 0.4
of percentage point.
18

59. What stands behind those results?
1. High productivity raise labor supply more
consistent with Hugget, 2010 (JPE)
“high productivity agents work too little and low productivity agents work too much under the U.S. system as
compared to the solution to the planning problem”
2. Progression in taxes is more efficient than in pensions
Typical reforms (e.g. health systems) yield changes in the Kakwani index of 1-2 percentage points, we do about 0.4
of percentage point.
3. Gains depend on Frisch elasticity
consistent with Heathcote et al, 2008 (JME), 2017 (QJE)
18

60. Conclusions

61. Conclusions
1. Progression in tax system can effectively substitute for progression in social security ...
19

62. Conclusions
1. Progression in tax system can effectively substitute for progression in social security ...
2. ... generating welfare gains [potentially: Pareto improvement]
19

63. Conclusions
1. Progression in tax system can effectively substitute for progression in social security ...
2. ... generating welfare gains [potentially: Pareto improvement]
3. Important role for response of labor to the features of the pension system
19

64. Questions or suggestions?
Thank you!
w: grape.org.pl
t: grape org
f: grape.org
e: j.tyrowicz@grape.org.pl
20