This document discusses replacing redistribution in social security systems with tax progression to improve efficiency. It presents a theoretical model where households live for two periods and choose labor, consumption, and assets. There are two productivity types. Under the current Beveridge system, pensions are redistributed, while the proposed Bismarck system links pensions to individual contributions. The reform improves efficiency by increasing labor supply but reduces redistribution. It is shown that high-productivity households are strictly better off due to gains in both efficiency and redistribution. Low-productivity households may be worse off due to lower redistribution, but a tax progression and lump-sum grants can compensate and make the reform Pareto-improving under certain conditions. A quantitative

1. Progressing into efficiency:
the role for labor tax progression in privatizing social security
Oliwia Komada (FAME|GRAPE)
Krzysztof Makarski (FAME|GRAPE and Warsaw School of Economics)
Joanna Tyrowicz (FAME|GRAPE, University of Regensburg, and IZA)
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2. Motivation

3. Motivation
Social security is essentially about insurance:
• mortality (annuitized)
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• low income (redistribution)
Cooley & Soares 1996, Tabellini 2000
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4. Motivation
Social security is essentially about insurance:
• mortality (annuitized)
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• low income (redistribution)
Cooley & Soares 1996, Tabellini 2000
Prevailing consensus:
• redistribution is costly (distorts incentives)
e.g. Diamond 1977 + large and diverse subsequent literature
• rising longevity → linking pensions to individual contributions
McGrattan & Prescott (2017), Nishiyama & Smetters 2007
• reduces insurance against low income, so some is desirable
Davidoff et al. 2005, Fehr et al. 2008
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5. Motivation
Social security is essentially about insurance:
• mortality (annuitized)
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• low income (redistribution)
Cooley & Soares 1996, Tabellini 2000
Prevailing consensus:
• redistribution is costly (distorts incentives)
e.g. Diamond 1977 + large and diverse subsequent literature
• rising longevity → linking pensions to individual contributions
McGrattan & Prescott (2017), Nishiyama & Smetters 2007
• reduces insurance against low income, so some is desirable
Davidoff et al. 2005, Fehr et al. 2008
Our approach: replace redistribution in social security with tax progression
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6. Motivation
Social security is essentially about insurance:
• mortality (annuitized)
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• low income (redistribution)
Cooley & Soares 1996, Tabellini 2000
Prevailing consensus:
• redistribution is costly (distorts incentives)
e.g. Diamond 1977 + large and diverse subsequent literature
• rising longevity → linking pensions to individual contributions
McGrattan & Prescott (2017), Nishiyama & Smetters 2007
• reduces insurance against low income, so some is desirable
Davidoff et al. 2005, Fehr et al. 2008
Our approach: replace redistribution in social security with tax progression
Bottom line: Shift insurance from retirement to working period →
improve efficiency of social security → raise welfare.
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7. Table of contents
Motivation
Theoretical model
Quantitative model
Results
Conclusions
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8. Theoretical model

9. (Stylized) theoretical model: partial equilibrium OLG model
Incomes:
• wage wt grows at the constant rate γ, zt = (1 + γ)t
, interest rate r is constant
• two types θ ∈ {θH , θL}, with productivities ωθ ∈ {ωH , ωL}, and ωH > ωL
denote y(θ) = (1 − τ)wt ωθ`t (θ), and ỹ(θ) = (1 − τ)w̃ωθ`1,t (θ), w̃ = wt /zt
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10. (Stylized) theoretical model: partial equilibrium OLG model
Incomes:
• wage wt grows at the constant rate γ, zt = (1 + γ)t
, interest rate r is constant
• two types θ ∈ {θH , θL}, with productivities ωθ ∈ {ωH , ωL}, and ωH > ωL
denote y(θ) = (1 − τ)wt ωθ`t (θ), and ỹ(θ) = (1 − τ)w̃ωθ`1,t (θ), w̃ = wt /zt
Households:
• live for 2 periods, population is constant,
• choose labor, consumption and assets
first period: c1,t (θ) + a1,t+1(θ) = (1 − τ)wt ωθ`1,t (θ) − zt T(ỹ(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
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11. (Stylized) theoretical model: partial equilibrium OLG model
Incomes:
• wage wt grows at the constant rate γ, zt = (1 + γ)t
, interest rate r is constant
• two types θ ∈ {θH , θL}, with productivities ωθ ∈ {ωH , ωL}, and ωH > ωL
denote y(θ) = (1 − τ)wt ωθ`t (θ), and ỹ(θ) = (1 − τ)w̃ωθ`1,t (θ), w̃ = wt /zt
Households:
• live for 2 periods, population is constant,
• choose labor, consumption and assets
first period: c1,t (θ) + a1,t+1(θ) = (1 − τ)wt ωθ`1,t (θ) − zt T(ỹ(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 + 1
η
zt `1,t (θ)
1+ 1
η + +βc2,t+1(θ))1−σ
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12. (Stylized) theoretical model: partial equilibrium OLG model
Government:
• needs to finance exogenous level of expenditure g̃ = gt /zt = constant,
• collects progressive income tax with fixed marginal rate and lump-sum grants
T(ỹ(θ)) = τ` · ỹ(θ) − µ̃
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13. (Stylized) theoretical model: partial equilibrium OLG model
Government:
• needs to finance exogenous level of expenditure g̃ = gt /zt = constant,
• collects progressive income tax with fixed marginal rate and lump-sum grants
T(ỹ(θ)) = τ` · ỹ(θ) − µ̃
The implied government budget constraint is then
g̃ +
X
θ∈{θL,θH }
µt =
X
θ∈{θL,θH }
τ` · ỹ(θ),
whatever funds are left after covering government expenditures are spent on
lump-sum grants µt .
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14. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
X
θ∈{L,H}
ωθ`1,t+1(θ).
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15. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
X
θ∈{L,H}
ωθ`1,t+1(θ).
Bismarck (no redistribution)
bBIS
2,t+1(θ) = τ wt (1 + γ) ωθ`1,t (θ)
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16. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
X
θ∈{L,H}
ωθ`1,t+1(θ).
Bismarck (no redistribution)
bBIS
2,t+1(θ) = τ wt (1 + γ) ωθ`1,t (θ)
Reform reduces the distortions:
`BIS
1,t (θ) > `BEV
1,t (θ)
→ both types have efficiency gain,
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17. (Stylized) theoretical model: partial equilibrium OLG model
Social security
Beveridge (full redistribution)
bBEV
2,t+1(θ) = τ wt+1
1
2
X
θ∈{L,H}
ωθ`1,t+1(θ).
Bismarck (no redistribution)
bBIS
2,t+1(θ) = τ wt (1 + γ) ωθ`1,t (θ)
Reform reduces the distortions:
`BIS
1,t (θ) > `BEV
1,t (θ)
→ both types have efficiency gain, what about redistribution?
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18. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) =
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19. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
efficiency gain
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20. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
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21. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωθ`BEV
1,t (θ) − ω−θ`BEV
1,t (−θ))
| {z }
pension system redistribution
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22. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωθ`BEV
1,t (θ) − ω−θ`BEV
1,t (−θ))
| {z }
pension system redistribution
W (θH ) ↑ W (θL) ↓
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23. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωθ`BEV
1,t (θ) − ω−θ`BEV
1,t (−θ))
| {z }
pension system redistribution
W (θH ) ↑ W (θL) ↓
+ (µBIS
t − µBEV
t − τ`(1 − τ)ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
tax system redistribution
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24. Basic intuitions
With β = 1
1+r
, discounted lifetime consumption becomes
cBIS
t (θ) − cBEV
t (θ) = ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωθ`BEV
1,t (θ) − ω−θ`BEV
1,t (−θ))
| {z }
pension system redistribution
W (θH ) ↑ W (θL) ↓
+ (µBIS
t − µBEV
t − τ`(1 − τ)ωθwt (`BIS
1,t (θ) − `BEV
1,t (θ))
| {z }
tax system redistribution
⇑ NEW W (θH ) ↓ W (θL) ↑
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25. Effect on labor supply and government revenue
1. % ∆ in labor supply is equal for both productivity types and depends on η
(the larger η, the larger ∆ )
`BIS
(θ) − `BEV
(θ)
`BEV (θ)
=
(1 − τ`(1 − τ))
(1 − τ − τ`(1 − τ))
η
− 1 ≡ ξη
− 1
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26. Effect on labor supply and government revenue
1. % ∆ in labor supply is equal for both productivity types and depends on η
(the larger η, the larger ∆ )
`BIS
(θ) − `BEV
(θ)
`BEV (θ)
=
(1 − τ`(1 − τ))
(1 − τ − τ`(1 − τ))
η
− 1 ≡ ξη
− 1
2. % ∆ in gov’nt revenue increases with η (Frisch elasticity)
8

27. Effect on labor supply and government revenue
1. % ∆ in labor supply is equal for both productivity types and depends on η
(the larger η, the larger ∆ )
`BIS
(θ) − `BEV
(θ)
`BEV (θ)
=
(1 − τ`(1 − τ))
(1 − τ − τ`(1 − τ))
η
− 1 ≡ ξη
− 1
2. % ∆ in gov’nt revenue increases with η (Frisch elasticity)
3. for η ≥ η, where ξη
− 1 = τ
τ`(1−τ)
, tax revenue ↑ ⇒
labor tax progression ↑ ⇒ compensate social security redistribution ↓
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28. Key results
1 θH have strictly higher benefits under BIS than under BEV
(efficiency ↑ redistribution ↑)
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29. 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 ↑ but soc. sec. redistribution ↓)
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30. 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 ↑ but soc. sec. redistribution ↓)
9

31. 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 ↑ but soc. sec. redistribution ↓)
−→ reform social security and distribute extra government revenue as lump-sum
grants µ
3 for η η reform is a Pareto-improvement,
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32. 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 ↑ but soc. sec. redistribution ↓)
−→ reform social security and distribute extra government revenue as lump-sum
grants µ
3 for η η reform is a Pareto-improvement,
4 ∃ η ∈ (0, η) such that for η η reform is a Hicks-improvement
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33. Quantitative model

34. Consumers
• uncertain lifetimes: live for 16 periods, with survival πj 1
• ex ante heterogeneous productivity + uninsurable productivity risk
• chose between leisure, consumption and savings based on CRRA utility function
• pay taxes (progressive on labor, linear on consumption and capital gains)
• contribute to social security, face natural borrowing constraint
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35. Consumers
• uncertain lifetimes: live for 16 periods, with survival πj 1
• ex ante heterogeneous productivity + uninsurable productivity risk
• chose between leisure, consumption and savings based on 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
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36. Government
• Finances government spending Gt , constant between scenarios,
• Balances pension system: subsidyt
• Services debt: rt Dt ,
• Collects taxes on capital, consumption, labor, and covers lump-sum grant
(progressive given by Benabou form)
Gt + subsidyt + rt Dt + Mt = τk,t rt At + τc,t Ct + Tax`,t + ∆Dt
where ∆Dt = Dt − Dt−1
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37. Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
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38. Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
• distortion, households do not perceive labor supply and pension benefits link
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
+0 · τwt ωj,t `j,t + 0
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39. Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
• distortion, households do not perceive labor supply and pension benefits link
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
+0 · τwt ωj,t `j,t + 0
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40. Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
• distortion, households do not perceive labor supply and pension benefits link
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
+0 · τwt ωj,t `j,t + 0
Alternative: fully individualized social security and lump-sum grants
• benefits proportional to contribution, no redistribution through social security
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41. Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
• distortion, households do not perceive labor supply and pension benefits link
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
+0 · τwt ωj,t `j,t + 0
Alternative: fully individualized social security and lump-sum grants
• benefits proportional to contribution, no redistribution through social security
• no distortion, households perceive labor supply and pension benefit link,
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
υR
j,t · τwt ωj,t `j,t + µt ,
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42. Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
• distortion, households do not perceive labor supply and pension benefits link
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
+0 · τwt ωj,t `j,t + 0
Alternative: fully individualized social security and lump-sum grants
• benefits proportional to contribution, no redistribution through social security
• no distortion, households perceive labor supply and pension benefit link,
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + yj,t − T (yj,t ) + Γj,t
υR
j,t · τwt ωj,t `j,t + µt ,
• additional tax revenue translates into lump-sum grants
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43. Calibration to replicate US economy (2015)
Preferences: instantaneous utility function take CRRA form with
• Risk aversion is equal to 2
• Disutility form work φ matches average hours 33%
• Frisch elasticity η is equal to 0.8
• Discounting rate δ matches interest K/Y ratio 2.9
Productivity risk and age profiles shock based on Borella et. al (2018):
Pension system
• Replacement rate ρ matches benefits as % of GDP 5.0%
• Contribution rate balances pension system in the initial steady state
• Pension eligibility age at 65
Taxes {τc , τk , τ`} match revenue as % of GDP {2.8%, 5.4%, 9.2%}
Depreciation rate d based on Kehoe Ruhl (2010) equal to 0.06
Population survival probabilities based on UN forecast
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44. Results

45. Distortion for η = 0.8
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46. Distortion for η = 0.8
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47. Distortion for η = 0.8
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48. Labor supply reaction for η = 0.8
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49. Distribution of welfare effects for η = 0.8
Under the veil of ignorance
welfare measured as consumption equivalent increase by 0.3%
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50. Distribution of welfare effects for η = 0.8
Under the veil of ignorance
welfare measured as consumption equivalent increase by 0.3%
Ex post almost all (90%) gains.
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51. Welfare effect across η
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52. Macroeconomic adjustment across η
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53. Fiscal adjustment across η
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54. Longevity makes the reform beneficial for even less responsive labor markets
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55. Half-internalizing the reform is sufficient to deliver welfare gains (η = 0.8)
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56. Conclusions

57. Conclusions
1. Progression in the tax system can effectively substitute for progression in social security
...
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58. Conclusions
1. Progression in the tax system can effectively substitute for progression in social security
...
2. ... generating welfare gains [potentially: Pareto improvement]
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59. Conclusions
1. Progression in the tax system can effectively substitute for progression in social security
...
2. ... generating welfare gains [potentially: Pareto improvement]
3. With rising longevity, the potential welfare gains are higher.
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60. Conclusions
1. Progression in the tax system can effectively substitute for progression in social security
...
2. ... generating welfare gains [potentially: Pareto improvement]
3. With rising longevity, the potential welfare gains are higher.
4. Important role for response of labor to the features of the pension system
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61. Questions or suggestions?
Thank you!
w: grape.org.pl
t: grape org
f: grape.org
e: okomada@grape.org.pl
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