Redistributive Taxation in a Partial Insurance
Economy
Jonathan Heathcote
Federal Reserve Bank of Minneapolis
Kjetil Storesletten
Federal Reserve Bank of Minneapolis, and Oslo University
Gianluca Violante
New York University
Macroeconomic Dynamics with Heterogeneous Agents, June 10th, 2013
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Redistributive Taxation
• How progressive should earnings taxation be?
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Redistributive Taxation
• How progressive should earnings taxation be?
• Arguments in favor of progressivity:
1. Social insurance of privately-uninsurable shocks
2. Redistribution from high to low innate ability
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Redistributive Taxation
• How progressive should earnings taxation be?
• Arguments in favor of progressivity:
1. Social insurance of privately-uninsurable shocks
2. Redistribution from high to low innate ability
• Arguments against progressivity:
1. Discourages labor supply
2. Discourages human capital investment
3. Redistribution from low to high taste for leisure
4. Complicates financing of govt. spending
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Ramsey Approach
Planner takes policy instruments and market structure as given,
chooses the CE that maximizes welfare
• CE of an heterogeneous-agent, incomplete-market economy
• Nonlinear tax/transfer system
• Valued public expenditures also chosen by the government
Tractable equilibrium framework clarifies economic forces shaping
the optimal degree of progressivity
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Overview of the model
• Huggett (1994) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Overview of the model
• Huggett (1994) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
1. differential “innate” (learning) ability
2. endogenous skill investment + multiple-skill technology
3. endogenous labor supply
4. heterogeneity in preferences for leisure
5. valued government expenditures
6. additional partial private insurance (other assets, family, etc)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Overview of the model
• Huggett (1994) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
1. differential “innate” (learning) ability
2. endogenous skill investment + multiple-skill technology
3. endogenous labor supply
4. heterogeneity in preferences for leisure
5. valued government expenditures
6. additional partial private insurance (other assets, family, etc)
• Steady-state analysis
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Demographics and preferences
• Perpetual youth demographics with constant survival probability δ
• Preferences:
Ui = vi (si ) + E0
∞
X
(βδ)t ui (cit , hit , G)
t=0
vi (si )
ui (cit , hit , G)
=
1 s2i
−
κi 2µ
=
h1+σ
log cit − exp(ϕi ) it + χ log G
1+σ
κi
∼
ϕi
∼
Exp (η)
v
ϕ
N
, vϕ
2
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Technology
• Output is CES aggregator over continuum of skill types:
Y =
Z
∞
N (s)
0
θ−1
θ
ds
θ
θ−1
,
• Aggregate effective hours by skill type:
N (s) =
Z
1
I{si =s} zi hi di
0
• Aggregate resource constraint:
Y =
Z
1
0
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
ci di + G
θ ∈ (1, ∞)
Individual efficiency units of labor
log zit = αit + εit
• αit = αi,t−1 + ωit
• εit
with
i.i.d. over time
• ϕ⊥κ⊥ω⊥ε
with
− v2ω , vω
ωit ∼ N
αi0 = 0 ∀i
εit ∼ N
− v2ε , vε
cross-sectionally and longitudinally
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Individual efficiency units of labor
log zit = αit + εit
• αit = αi,t−1 + ωit
• εit
with
i.i.d. over time
• ϕ⊥κ⊥ω⊥ε
− v2ω , vω
ωit ∼ N
αi0 = 0 ∀i
with
εit ∼ N
− v2ε , vε
cross-sectionally and longitudinally
• Pre-government earnings:
yit = p(si ) × exp(αit + εit ) × hit
| {z }
|
{z
} |{z}
skill price
efficiency
determined by skill, fortune, and diligence
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
hours
Government
• Two-parameter tax/transfer function to redistribute and finance
publicly-provided goods G
• Disposable (post-government) earnings:
ỹi = λyi1−τ
• Government budget constraint (no government debt):
G=
Z
1
0
1−τ
yi − λyi
di
Government chooses (G, τ ), and λ balances the budget residually
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Our model of fiscal redistribution
T (yi ) = yi − λyi1−τ
• The parameter τ measures the rate of progressivity:
◮ τ =0:
no redistribution → flat tax 1 − λ
◮ τ =1:
full redistribution → y˜i = λ
◮ 0 < τ < 1:
◮ τ <0:
progressivity →
regressivity →
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
T ′ (y)
T (y)/y
T ′ (y)
T (y)/y
>1
<1
9
9.5
Log of Diisposable Income
10
10.5
11
11.5
12
Our model of fiscal redistribution
9
9.5
10
10.5
11
Log of Pre Government Income
11.5
• CPS 2005, N obs = 52, 539: R2 = 0.92 and τ = 0.18
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
12
Our model of fiscal redistribution
Marginal and average tax rates
0.5
0.4
0.3
0.2
0.1
US
US Marginal (τ
0
=0.18)
US
US Average (τ
=0.18)
−0.1
−0.2
−0.3
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Labor Income (1 = Average Earnings)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
5
Representative Agent Warm Up
max
C,H
U
=
C
s.t.
=
H 1+σ
log C −
+ χ log G
1+σ
λH 1−τ
Market clearing C + G = H
Define g = G/H
Equilibrium allocations:
log C
RA
(g, τ )
=
log H RA (g, τ )
=
1
log(1 − g) +
log(1 − τ )
(1 + σ)
1
log(1 − τ )
(1 + σ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Representative Agent Optimal Policy
• Welfare:
W RA (g, τ ) = log(1 − g) + χ log g + (1 + χ)
log(1 − τ )
1−τ
−
(1 + σ)
(1 + σ)
• Welfare maximizing (g, τ ) pair:
∗
=
τ∗
=
g
χ
1+χ
−χ
• Allocations are first best (same as with lump-sum taxes)
• Result for g ∗ will extend to heterogeneous agent setup
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Markets
• Competitive good and labor markets
• Competitive asset markets (all assets in zero net supply)
◮ Non-contingent bond
◮ Full set of insurance claims against ε shocks
◮ Perfect annuity against survival risk
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Markets
• Competitive good and labor markets
• Competitive asset markets (all assets in zero net supply)
◮ Non-contingent bond
◮ Full set of insurance claims against ε shocks
◮ Perfect annuity against survival risk
• Key result: There is an equilibrium with no bond trading
⇒ Can solve for equilibrium allocations in closed form
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
No bond-trade equilibrium
• Micro-foundations for Constantinides and Duffie (1996)
◮ CRRA, unit root shocks to log disposable income
◮ In equilibrium, no bond-trade ⇒ ct = ỹt
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
No bond-trade equilibrium
• Micro-foundations for Constantinides and Duffie (1996)
◮ CRRA, unit root shocks to log disposable income
◮ In equilibrium, no bond-trade ⇒ ct = ỹt
• Unit root disposable income micro-founded in our model:
1. Skill investment+shocks: → wages
2. Labor supply choice: wages → pre-tax earnings
3. Non-linear taxation: pre-tax earnings → after-tax earnings
4. Private risk sharing: after-tax earnings → disp. income
5. No bond trade: disposable income = consumption
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Equilibrium skill choice and skill price
• FOC →
s
κµ
(ϕ,s)
= (1 − βδ) ∂U0∂s
= (1 − τ ) ∂ log∂sp(s)
Equilibrium skill choice and skill price
• FOC →
s
κµ
(ϕ,s)
= (1 − βδ) ∂U0∂s
= (1 − τ ) ∂ log∂sp(s)
• Skill price has Mincerian shape: log p(s) = π0 + π1 s
s = (1 − τ )κµπ1
π1 =
r
η
θµ (1 − τ )
(return to skill)
Equilibrium skill choice and skill price
• FOC →
s
κµ
(ϕ,s)
= (1 − βδ) ∂U0∂s
= (1 − τ ) ∂ log∂sp(s)
• Skill price has Mincerian shape: log p(s) = π0 + π1 s
s = (1 − τ )κµπ1
π1 =
r
η
θµ (1 − τ )
(return to skill)
• Distribution of skill prices (in levels) is Pareto with parameter θ
1
var(log p(s)) = 2
θ
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Equilibrium consumption allocation
log c∗ (α, ϕ, s; g, τ ) =
log C RA (g, τ ) +
M(vε )
| {z }
level effect from ins. variation
+(1 − τ ) log p(s; τ ) − (1 − τ ) ϕ + (1 − τ ) α
{z
} | {z } | {z }
|
skill price
pref. het.
unins. shock
• Response to variation in (p(s), ϕ, α) mediated by progressivity
• Invariant to insurable shock ε
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Equilibrium hours allocation
∗
log h (ε, ϕ; g, τ )
=
log H
RA
(g, τ ) −
1
M(vε )
σ
b (1 − τ )
|
{z
}
level effect from ins. variation
1
ε
− ϕ +
|{z}
σ
b
|{z}
pref. het.
ins. shock
• Response to ε mediated by tax-modified Frisch elasticity
• Invariant to skill price p(s) and uninsurable shock α
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
1
σ̂
=
1−τ
σ+τ
Utilitarian Social Welfare Function
• Steady states with constant (g, τ )
W(g, τ ) ∝
∞
X
µk
k=−∞
• Government sets weights:
Z
1
Ui,k (·; g, τ )di
0
µk = β k × cohort size
◮ SWF becomes average period utility in the cross-section
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Exact expression for SWF
W(g, τ )
=
1
log(1 − τ )
−
log(1 − g) + χ log g + (1 + χ)
(1 + σ̂)(1 − τ ) (1 + σ̂)
s
"
!
#
ηθ
−1
θ
θ
+(1 + χ)
log
log
+
θ−1
µ (1 − τ )
θ−1
θ−1
1−τ
1−τ
1
−
− (1 − τ ) − − log 1 −
2θ
θ
θ
2 vϕ
− (1 − τ )
2



−τ (1−τ )
vω
1 − δ exp
2
δ
v
ω

− (1 − τ )
− log 
1−δ 2
1−δ
−(1 + χ)σ
1
1 vε
+
(1
+
χ)
vε
2
σ̂ 2
σ̂
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Representative Agent component
W(g, τ )
=
log(1 − τ )
1
log(1 − g) + χ log g + (1 + χ)
−
(1 + σ̂)(1 − τ ) (1 + σ̂)
|
{z
}
Representative Agent Welfare = W RA (g, τ )
"
s
!
θ
θ
ηθ
−1
log
log
+
θ−1
µ (1 − τ )
θ−1
θ−1
1−τ
1−τ
1
−
− (1 − τ ) − − log 1 −
2θ
θ
θ
vϕ
− (1 − τ )2
2



−τ (1−τ )
1 − δ exp
vω
2
v
δ
ω

− log 
− (1 − τ )
1−δ 2
1−δ
+(1 + χ)
1 vε
1
−(1 + χ)σ 2
+ (1 + χ) vε
σ̂ 2
σ̂
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
#
Skill investment component
W(τ )
=
W RA (τ )
+(1 + χ)
"
|
−1
log
θ−1
s
!
ηθ
θ
log
+
µ (1 − τ )
θ−1
{z
θ
θ−1
productivity = log E [(p(s))] = log (Y /N )
1−τ
1−τ
1
−
− (1 − τ ) − − log 1 −
2θ
θ
θ
|
{z
}
|
{z
}
avg. education cost
vϕ
− (1 − τ )
2

consumption dispersion across skills
2

−τ (1−τ )
vω
2
1 − δ exp
v
δ
ω
− log 
− (1 − τ )
1−δ 2
1−δ
1
1 vε
+ (1 + χ) vε
−(1 + χ)σ 2
σ̂ 2
σ̂
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”


#
}
Optimal τ as a function of θ
• Assume κ is the only source of heterogeneity
• Set σ = 2 and χ = 0.25
0.1
0.05
0
τ
∗
−0.05
−0.1
−0.15
−0.2
−0.25
1
2
3
4
5
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
6
θ
7
8
9
10
11
Uninsurable component
W(τ )
=
W RA (τ )
"
s
!
−1
θ
ηθ
θ
+(1 + χ)
log
log
+
θ−1
µ (1 − τ )
θ−1
θ−1
1−τ
1−τ
1
−
− (1 − τ ) − − log 1 −
2θ
θ
θ
vϕ
− (1 − τ )2
{z 2}
|
cons. disp. due to prefs


−τ (1−τ )
vω
2
1 − δ exp
v
δ
ω
− log 
−(1 − τ )
1−δ 2
1−δ
|
{z
consumption dispersion due to uninsurable shocks ≈
−(1 + χ)σ
1
1 vε
+
(1
+
χ)
vε
2
σ̂ 2
σ̂
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”


}
(1 − τ )2 v2α
#
Insurable component
W(τ )
=
W RA (τ )
s
"
!
−1
ηθ
θ
θ
log
+
log
θ−1
µ (1 − τ )
θ−1
θ−1
1−τ
1−τ
1
−
− (1 − τ ) − − log 1 −
2θ
θ
θ
2 vϕ
− (1 − τ )
2



−τ (1−τ )
vω
1 − δ exp
2
v
δ
ω

− log 
− (1 − τ )
1−δ 2
1−δ
+(1 + χ)
−(1 + χ)σ
1 vε
2 2
σ̂
| {z }
hours dispersion
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
+ (1 + χ)
1
vε
σ̂
|{z}
#
prod. gain from ins. shock=log(N/H)
Parameterization
• Parameter vector {χ, σ, δ, θ, vϕ , vω , vε , }
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Parameterization
• Parameter vector {χ, σ, δ, θ, vϕ , vω , vε , }
• To match G/Y = 0.20:
→ χ = 0.25
• Frisch elasticity (micro-evidence):
→σ=2
cov(log h, log w)
var(log h)
=
=
var 0 (log c) =
∆var(log w)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
=
1
vε
σ̂
1
vϕ + 2 vε
σ̂
1
(1 − τ )2 vϕ + 2
θ
vω
Parameterization
• Parameter vector {χ, σ, δ, θ, vϕ , vω , vε , }
• To match G/Y = 0.20:
→ χ = 0.25
• Frisch elasticity (micro-evidence):
→σ=2
cov(log h, log w)
=
var(log h)
=
var 0 (log c)
=
∆var(log w)
=
1
vε
σ̂
→ vε = 0.18
1
vϕ + 2 vε
→ vϕ = 0.06
σ̂
1
(1 − τ )2 vϕ + 2 → θ = 3
θ
vω
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
→ vω = 0.005, δ = 0.963
Optimal progressivity
Social Welfare Function
welf change rel. to optimum (% of cons.)
10
5
Welfare gain = 0.82 pct
0
−5
−10
−15
−20
−0.5
τUS = 0.18
τ* = 0.087
−0.4
−0.3
−0.2
−0.1
0
0.1
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
0.2
0.3
0.4
0.5
Optimal progressivity: decomposition
Social Welfare Function
welf change rel. to baseline optimum (% of cons.)
10
5
0
−5
−10
−15
−20
−0.5
(1) Rep. Agent τ = −0.25
−0.4
−0.3
−0.2
−0.1
0
0.1
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
0.2
0.3
0.4
0.5
Optimal progressivity: decomposition
Social Welfare Function
welf change rel. to baseline optimum (% of cons.)
10
(2) + Skill Inv. τ = −0.066
5
0
−5
−10
−15
−20
−0.5
(1) Rep. Agent τ = −0.25
−0.4
−0.3
−0.2
−0.1
0
0.1
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
0.2
0.3
0.4
0.5
Optimal progressivity: decomposition
Social Welfare Function
welf change rel. to baseline optimum (% of cons.)
10
(2) + Skill Inv. τ = −0.066
5
(3) + Pref. Het. τ = 0.00
0
−5
−10
−15
−20
−0.5
(1) Rep. Agent τ = −0.25
−0.4
−0.3
−0.2
−0.1
0
0.1
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
0.2
0.3
0.4
0.5
Optimal progressivity: decomposition
Social Welfare Function
welf change rel. to baseline optimum (% of cons.)
10
(2) + Skill Inv. τ = −0.066
5
(3) + Pref. Het. τ = 0.00
0
−5
(4) + Unins. Shocks τ = 0.102
−10
−15
−20
−0.5
(1) Rep. Agent τ = −0.25
−0.4
−0.3
−0.2
−0.1
0
0.1
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
0.2
0.3
0.4
0.5
Optimal progressivity: decomposition
Social Welfare Function
welf change rel. to baseline optimum (% of cons.)
10
(2) + Skill Inv. τ = −0.066
5
(3) + Pref. Het. τ = 0.00
0
(5) + Ins. Shocks τ = 0.087
−5
(4) + Unins. Shocks τ = 0.102
−10
−15
−20
−0.5
(1) Rep. Agent τ = −0.25
−0.4
−0.3
−0.2
−0.1
0
0.1
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
0.2
0.3
0.4
0.5
Actual and optimal progressivity
Average Tax Rate
0.4
Average tax rate
0.3
0.2
0.1
US
Actual US τ
0
=0.18
∗
Utilitarian τ =0.087
−0.1
−0.2
−0.3
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Labor Income (1 = Average Earnings)
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
5
Alternative SWF
Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ)
Turn off desire to redistribute
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Alternative SWF
Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ)
Turn off desire to redistribute
• Economy with heterogeneity in (κ, ϕ), and vω = τ = 0
• Compute CE allocations
• Compute Negishi weights s.t. planner’s allocation = CE
• Use these weights in the SWF
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Alternative SWF
Utilitarian
κ-neutral
ϕ-neutral
Insurance-only
Redist. wrt κ
Y
N
Y
N
Redist. wrt ϕ
Y
Y
N
N
Insurance wrt ω
Y
Y
Y
Y
τ∗
0.087
0.030
0.046
-0.012
Welf. gain (pct of c)
0.82
1.66
1.33
2.67
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Factors limiting progressivity
1. Discourages skill investment
2. Reduces labor supply
• especially important when G valued
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Factors limiting progressivity
1. Discourages skill investment
2. Reduces labor supply
• especially important when G valued
Welfare maxizing τ
var(log(λy 1−τ )) / var((log y))
Baseline
0.087
0.83
(1) Exog. skills
0.120
0.77
(2) σ = 20
0.219
0.61
(3) χ = 0
0.209
0.63
(1)+(2)
0.538
0.21
(1)+(2)+(3)
0.597
0.16
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Alternative assumptions on G
1. G endogenous and valued: χ = 0.25, G∗ = χ/(1 + χ) = 0.2
2. G endogenous but non valued: χ = 0, G∗ = 0
3. G exogenous and proportional to Y : G/Y = ḡ = 0.2
4. G exogenous and fixed in level: G = Ḡ = 0.2 × Y U S
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Alternative assumptions on G
1. G endogenous and valued: χ = 0.25, G∗ = χ/(1 + χ) = 0.2
2. G endogenous but non valued: χ = 0, G∗ = 0
3. G exogenous and proportional to Y : G/Y = ḡ = 0.2
4. G exogenous and fixed in level: G = Ḡ = 0.2 × Y U S
Utilitarian SWF
Insurance-only SWF
G
Y (τ ∗ )
τ∗
τ∗
G endogenous
χ = 0.25
0.200
0.087
-0.012
G endogenous
χ=0
0.000
0.209
0.103
g exogenous
ḡ = 0.2
0.200
0.209
0.103
G exogenous
Ḡ = 0.2 × Y (τ U S )
0.188
0.095
0.002
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”
Going forward
• Median voter choosing (g, τ ) once and for all
• Skill-biased technical change
• Rent-extraction by top earners? (Piketty-Saez view)
• Endogenous growth?
Heathcote-Storesletten-Violante, ”Redistributive Taxation in a Partial Insurance Economy”