Progressing towards efficiency: the role for labor tax progression in reforming social security

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)
EPCS, Hannover, 2023
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Motivation
Social security is essentially about insurance:
• old age (between cohorts) & mortality (annuitized)
Benartzi et al. 2011, Bruce & Turnovsky 2013, Reichling & Smetters 2015, Caliendo et al. 2017
• low income (within cohort redistribution)
Cooley & Soares 1996, Tabellini 2000
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Motivation
Prevailing consensus:
• privatization of social security brings efficiency gains,
• but reduces (within cohort) redistribution
• this insurance loss reduces overall welfare effect of such reforms
e.g. Nishiyama & Smetters (2007)
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Motivation
Our approach: replace redistribution in social security with tax progression
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Motivation
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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Table of contents
Motivation
Stylized theoretical model
Quantitative model
Results
Conclusions
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Stylized theoretical model
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Stylized theoretical model: partial equilibrium OLG model
Incomes:
• wage wt grows at the constant rate γ, wt = (1 + γ)t
, interest rate r = γ and 1 + r = δ−1
.
• two types θ ∈ {θL, θH } of measure one, income y(θ) = ωθwt ℓt (θ), ωθ ∈ {ωL, ωH }, ωH > ωL,
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Incomes:
.
Households: live for 2 periods, population is constant
• choose labor, consumption and assets
first period: c1,t (θ) + a1,t+1(θ) = (1 − τ)ωθwt ℓt (θ) − τℓ(1 − τ)ωθwt ℓt (θ) + µt
second period: c2,t+1(θ) = (1 + r)a1,t+1(θ) + b2,t+1(θ)
• lifetime budget constraint (allows to see the main result)
c1,t (θ) +
c2,t+1(θ)
1 + r
= RHSt (θ) = (1 − τ)ωθwt ℓ1,t (θ) − τℓ(1 − τ)wt ωθℓ1,t (θ) + µt +
b2,t+1(θ)
1 + r
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Incomes:
.
Households: live for 2 periods, population is constant
• choose labor, consumption and assets
first period: c1,t (θ) + a1,t+1(θ) = (1 − τ)ωθwt ℓt (θ) − τℓ(1 − τ)ωθwt ℓt (θ) + µt
second period: c2,t+1(θ) = (1 + r)a1,t+1(θ) + b2,t+1(θ)
• lifetime budget constraint (allows to see the main result)
c1,t (θ) +
c2,t+1(θ)
1 + r
= RHSt (θ) = (1 − τ)ωθwt ℓ1,t (θ) − τℓ(1 − τ)wt ωθℓ1,t (θ) + µt +
b2,t+1(θ)
1 + r
• GHH preferences: Frisch elasticity + risk aversion
U(θ) =
1
1 − σ
(c1,t (θ) −
ϕ(1 + γ)t
1 + 1
η
ℓ1,t (θ)
1+ 1
η )1−σ
+ β
1
1 − σ
c2,t+1(θ)1−σ
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Government:
• needs to finance exogenous level of expenditure g̃ = gt /(1 + γ)t
= constant,
• collects progressive income tax with fixed marginal rate and lump-sum grants
τℓ · (1 − τ)ωθwt ℓt (θ) − µt
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Government:
• needs to finance exogenous level of expenditure g̃ = gt /(1 + γ)t
= constant,
• collects progressive income tax with fixed marginal rate and lump-sum grants
τℓ · (1 − τ)ωθwt ℓt (θ) − µt
• The government budget constraint is
gt + 2 · µt =
X
θ∈{θL,θH }
τℓ · (1 − τ)ωθwt ℓt (θ),
whatever funds are left after covering government expenditures are spent on lump-sum grants µt .
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Social security
Beveridge (full redistribution): bBEV
2,t+1(θ) = τ
1
2
wt+1
X
θ∈{L,H}
ωθℓ1,t+1(θ)
Bismarck (no redistribution): bBIS
2,t+1(θ) = τ (1 + γ) wt ωθℓ1,t (θ)
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Social security
2,t+1(θ) = τ
1
2
wt+1
X
θ∈{L,H}
ωθℓ1,t+1(θ)
2,t+1(θ) = τ (1 + γ) wt ωθℓ1,t (θ)
Substituting into BC, recall wt+1 = wt (1 + γ):
RHSt (θ) = (1 − τ)ωθwt ℓ1,t (θ) − τℓ(1 − τ)wt ωθℓ1,t (θ) + µt +
b2,t+1(θ)
1 + r
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Social security
2,t+1(θ) = τ
1
2
wt+1
X
θ∈{L,H}
ωθℓ1,t+1(θ)
2,t+1(θ) = τ (1 + γ) wt ωθℓ1,t (θ)
Substituting into BC, recall wt+1 = wt (1 + γ):
RHSt (θ) = (1 − τ)ωθwt ℓ1,t (θ) − τℓ(1 − τ)wt ωθℓ1,t (θ) + µt +
b2,t+1(θ)
1 + r
RHS of Lifetime Budget
Beveridge: RHSt (θ) = (1 − τ)wt ωθℓ1,t (θ) − τℓ(1 − τ)wt ωθℓ1,t (θ) + µt + τ
1
2
wt
X
θ∈{L,H}
ωθℓ1,t+1(θ)
Bismarck: RHSt (θ) = (1 − τ)wt ωθℓ1,t (θ) − τℓ(1 − τ)wt ωθℓ1,t (θ) + µt + τwt ωθℓ1,t (θ)
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Basic intuitions
Reform and lifetime budget for type θ
RHSBIS
t (θ) − RHSBEV
t (θ) =
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Basic intuitions
RHSBIS
t (θ) − RHSBEV
t (θ) =
ωθwt (ℓBIS
1,t (θ) − ℓBEV
1,t (θ))
| {z }
efficiency gain
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Basic intuitions
RHSBIS
t (θ) − RHSBEV
t (θ) =
ωθwt (ℓBIS
1,t (θ) − ℓBEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
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Basic intuitions
RHSBIS
t (θ) − RHSBEV
t (θ) =
ωθwt (ℓBIS
1,t (θ) − ℓBEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωLℓBEV
1,t (θL) − ωH ℓBEV
1,t (θH ))
| {z }
social security redistribution
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Basic intuitions
RHSBIS
t (θ) − RHSBEV
t (θ) =
ωθwt (ℓBIS
1,t (θ) − ℓBEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωLℓBEV
1,t (θH ))
| {z }
W (θH ) ↑ W (θL) ↓
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Basic intuitions
RHSBIS
t (θ) − RHSBEV
t (θ) =
ωθwt (ℓBIS
1,t (θ) − ℓBEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωLℓBEV
1,t (θH ))
| {z }
W (θH ) ↑ W (θL) ↓
+ (µBIS
t − µBEV
t − τℓ(1 − τ)ωLwt (ℓBIS
1,t (θL) − ℓBEV
1,t (θL))
| {z }
⇑ NEW tax system redistribution
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Basic intuitions
RHSBIS
t (θ) − RHSBEV
t (θ) =
ωθwt (ℓBIS
1,t (θ) − ℓBEV
1,t (θ))
| {z }
efficiency gain
W (θH ) ↑ W (θL) ↑
−
1
2
τwt (ωLℓBEV
1,t (θH ))
| {z }
W (θH ) ↑ W (θL) ↓
+ (µBIS
t − µBEV
t − τℓ(1 − τ)ωLwt (ℓBIS
1,t (θL) − ℓBEV
1,t (θL))
| {z }
⇑ NEW tax system redistribution
W (θH ) ↓ W (θL) ↑
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Efficiency effect
Beveridge (full redistribution)
ℓBIS
1,t (θ) = [
1
ϕ
(1 − τℓ(1 − τ))ωθ]η
.
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Efficiency effect
ℓBIS
1,t (θ) = [
1
ϕ
(1 − τℓ(1 − τ))ωθ]η
.
Bismarck (no redistribution)
ℓBEV
1,t (θ) = [
1
ϕ
(1 − τℓ(1 − τ) − τ)ωθ]η
Reform = Beveridge → Bismarck, reduces distortions:
ℓBIS
1,t (θ) > ℓBEV
1,t (θ)
Welfare W (θ) ↑ =⇒ follows from the envelope theorem.
efficiency effect −→ labor wedge ↓, both types: ℓ(θ) ↑ and W (θ) ↑,
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Efficiency effect
ℓBIS
1,t (θ) = [
1
ϕ
(1 − τℓ(1 − τ))ωθ]η
.
Bismarck (no redistribution)
ℓBEV
1,t (θ) = [
1
ϕ
(1 − τℓ(1 − τ) − τ)ωθ]η
Reform = Beveridge → Bismarck, reduces distortions:
ℓBIS
1,t (θ) > ℓBEV
1,t (θ)
Welfare W (θ) ↑ =⇒ follows from the envelope theorem.
efficiency effect −→ labor wedge ↓, both types: ℓ(θ) ↑ and W (θ) ↑, what about redistribution?
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Redistribution through social security
Denote: PV Pen
t (θ) =
b2,t+1(θ)
1 + r
− τwt ωθℓ1,t (θ), then:
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Denote: PV Pen
t (θ) =
b2,t+1(θ)
1 + r
θH gains from less redistribution in social security
∆PV Pen
t (θH ) = PV Pen,BIS
t (θH ) − PV Pen,BEV
t (θH ) =
τwt
2
(ωH ℓBIS
(θH ) − ωLℓBEV
1 (θL)
| {z }
redistribution effect>0
]
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Denote: PV Pen
t (θ) =
b2,t+1(θ)
1 + r
∆PV Pen
t (θH ) =
τwt
2
(ωH ℓBIS
1 (θL)
| {z }
]
θL loses from less redistribution in social security
∆PV Pen
t (θL) = PV Pen,BIS
t (θL) − PV Pen,BEV
t (θL) =
τwt
2
(ωLℓBEV
1 (θL) − ωH ℓBIS
(θH )
| {z }
redistribution effect<0
]
11 / 34

Denote: PV Pen
t (θ) =
b2,t+1(θ)
1 + r
∆PV Pen
t (θH ) =
τwt
2
(ωH ℓBIS
1 (θL)
| {z }
]
∆PV Pen
t (θL) =
τwt
2
(ωLℓBEV
(θH )
| {z }
]
social security redistribution effect −→ benefits θH , harms: θL,
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Denote: PV Pen
t (θ) =
b2,t+1(θ)
1 + r
∆PV Pen
t (θH ) =
τwt
2
(ωH ℓBIS
1 (θL)
| {z }
]
∆PV Pen
t (θL) =
τwt
2
(ωLℓBEV
(θH )
| {z }
]
social security redistribution effect −→ benefits θH , harms: θL, can θL be compensated through taxes?
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Redistribution through tax system
1. % ∆ in labor supply is equal for both productivity types and increases with η
ℓBIS
(θ) − ℓBEV
(θ)
ℓBEV (θ)
=

(1 − τℓ(1 − τ))
(1 − τ − τℓ(1 − τ))
η
− 1 ≡ ξη
− 1
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ℓBIS
(θ) − ℓBEV
(θ)
ℓBEV (θ)
=

(1 − τℓ(1 − τ))
(1 − τ − τℓ(1 − τ))
η
− 1 ≡ ξη
− 1
2. % ∆ in government revenue increases with η (Frisch elasticity) ⇒ pool for µt ↑
RBIS
− RBEV
RBEV
≡ ξη
− 1
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ℓBIS
(θ) − ℓBEV
(θ)
ℓBEV (θ)
=

(1 − τℓ(1 − τ))
(1 − τ − τℓ(1 − τ))
η
− 1 ≡ ξη
− 1
RBIS
− RBEV
RBEV
≡ ξη
− 1
3. The change in net tax transfer for θL positive and for θH negative
∆µt − τℓ(1 − τ)ωLwt ∆ℓt (θL) 0 and ∆µt − τℓ(1 − τ)ωH wt ∆ℓt (θH ) 0
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ℓBIS
(θ) − ℓBEV
(θ)
ℓBEV (θ)
=

(1 − τℓ(1 − τ))
(1 − τ − τℓ(1 − τ))
η
− 1 ≡ ξη
− 1
RBIS
− RBEV
RBEV
≡ ξη
− 1
Reform + lump sum transfers bundle =⇒ positive (increasing in η) transfers from θH -type households to
the θL-type households through the tax system.
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ℓBIS
(θ) − ℓBEV
(θ)
ℓBEV (θ)
=

(1 − τℓ(1 − τ))
(1 − τ − τℓ(1 − τ))
η
− 1 ≡ ξη
− 1
RBIS
− RBEV
RBEV
≡ ξη
− 1
Tax system redistribution effect −→ benefits θL at the expense of θH ,
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ℓBIS
(θ) − ℓBEV
(θ)
ℓBEV (θ)
=

(1 − τℓ(1 − τ))
(1 − τ − τℓ(1 − τ))
η
− 1 ≡ ξη
− 1
RBIS
− RBEV
RBEV
≡ ξη
− 1
Tax system redistribution effect −→ benefits θL at the expense of θH , can it fully compensate θL for
the loss of redistribution in social security?
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Key results: Reform social security and extra government revenue → lump-sum grants µ
1 θH under BIS work more, have strictly higher pension benefits and pay higher taxes
(efficiency ↑ social security benefits ↑ tax liability ↑)
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2 θL under BIS work more, have (most likely) lower pension benefits and pay lower taxes
(efficiency ↑ social security benefits ↓ tax liability ↓)
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3 ∃ η̃ s.t. for θL HHs redistribution lost in pension system is fully compensated by tax system
∆PV Pen
(θL) = ∆Tax(θL)
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∆PV Pen
(θL) = ∆Tax(θL)
4 ∃ η ∈ (0, η̃) s.t. for η η reform with µ is a Pareto-improving (by continuity of the utility function)
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∆PV Pen
(θL) = ∆Tax(θL)
4 ∃ η ∈ (0, η̃) s.t. for η η reform with µ is a Pareto-improving (by continuity of the utility function)
5 ∃ η ∈ (0, η) s.t. for η η reform with µ is a Hicks-improving (by the same token)
13 / 34

Quantitative model
Consumers
• uncertain lifetimes: live for 16 periods, with survival πj 1
• ex ante heterogeneous productivity + uninsurable productivity risk
• consume, work and save based on CRRA instantaneous utility function 1
1−σ
c1−σ
− ϕ
1+1/η
ℓ1+1/η
• pay taxes (progressive on labor, linear on consumption and capital gains)
• contribute to social security, face natural borrowing constraint
15 / 34

Quantitative model
Consumers
• uncertain lifetimes: live for 16 periods, with survival πj 1
• ex ante heterogeneous productivity + uninsurable productivity risk
• consume, work and save based on CRRA instantaneous utility function 1
1−σ
c1−σ
− ϕ
1+1/η
ℓ1+1/η
• 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
15 / 34

Quantitative model
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 labor tax given by Benabou form)
Gt + subsidyt + rt Dt + Mt = τk,t rt At + τc,t Ct + Taxℓ,t + ∆Dt
where ∆Dt = Dt − Dt−1
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Policy experiment: comparative statics
Status quo: current US social security
• benefits redistributive, with high replacement rate for low income individuals
17 / 34

• distortion: no individual link between labor supply and pension benefits
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + (1 − τ)yj,t − Tt ((1 − τ)yj,t ) + Γj,t + 0 · τwt ωj,t ℓj,t + 0
17 / 34

Alternative: fully individualized social security and lump-sum grants
• benefits proportional to contribution, no redistribution through social security
17 / 34

• no distortion: direct individual link between labor supply and pension benefits
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + (1 − τ)yj,t − Tt ((1 − τ)yj,t ) + Γj,t + υR
j,t · τwt ωj,t ℓj,t + µt ,
17 / 34

• no distortion: direct individual link between labor supply and pension benefits
aj+1,t+1 + (1 + τc,t )cj,t = (1 + (1 − τk )rt )aj,t + (1 − τ)yj,t − Tt ((1 − τ)yj,t ) + Γj,t + υR
j,t · τwt ωj,t ℓj,t + µt ,
• additional tax revenue (from increased efficiency) goes into lump-sum grants
17 / 34

Calibration to replicate US economy (2015)
Preferences: instantaneous utility function take CRRA form with
• Risk aversion σ 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 , τℓ} 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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Distortion for η = 0.8
labor wedge formula
20 / 34

21 / 34

22 / 34

Labor supply reaction for η = 0.8
Average ∆ℓ ↑ 2.6% for HHs below median and 3.0% above median
Heathcote et al. (2008) argue for ↑ for high-productivity and ↓ for low-productivity.
23 / 34

Taxes and pensions changes
24 / 34

Distribution of welfare effects for η = 0.8
Under the veil of ignorance consumption equivalent increases by 0.3%
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Distribution of welfare effects for η = 0.8
Under the veil of ignorance consumption equivalent increases by 0.3%
Ex post almost universal gains (90%).
25 / 34

Welfare effect across η
26 / 34

Macroeconomic adjustment across η
27 / 34

Fiscal adjustment across η
28 / 34

Longevity makes the reform beneficial for even less responsive labor markets
29 / 34

Half-internalizing the reform is sufficient to deliver welfare gains (η = 0.8)
30 / 34

Conclusions
1. Progression in the tax system can effectively substitute for progression in social security ...
32 / 34

Conclusions
2. ... generating welfare gains
32 / 34

Conclusions
3. With rising longevity, the potential welfare gains are higher.
32 / 34

Conclusions
3. With rising longevity, the potential welfare gains are higher.
4. Important role for response of labor to the features of the pension system
32 / 34

Questions or suggestions?
Thank you!
w: grape.org.pl
t: grape org
f: grape.org
e: j.tyrowicz@grape.org.pl
33 / 34

Labor wedge as measure of distortion
With marginal labor income tax denoted as T ′
(yj,t (sj,t ))
ϕℓj,t (sj,t )
1
η =
cj,t (sj,t )−σ
1 + τc

1 − (1 − τ)T ′
(yj,t (sj,t )) − τ(1 − υj,t )

wt ωj,t (sj,t ),
Which gives the formula for wedge:
ϑj,t (sj,t ) =
(1 − τ)T ′
(yj,t (sj,t )) + τ(1 − υj,t ) − 1
1 + τc
+ 1.
Chari et al (2007), Berger et al (2019) and Boar and Midrigan (2020), Cociuba and Ueberfeldt (2020)
back to results
34 / 34

Progressing towards efficiency: the role for labor tax progression in reforming social security

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