Optimal Tax Progressivity: An Analytical Framework
Jonathan Heathcote
Federal Reserve Bank of Minneapolis
Kjetil Storesletten
Oslo University and Federal Reserve Bank of Minneapolis
Gianluca Violante
New York University
Federal Reserve Bank of Atlanta – January 14th, 2013
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 1 /41
Question
• How progressive should labor income taxation be?
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 2 /41
Question
• How progressive should labor income taxation be?
• Argument in favor of progressivity: missing markets
1. Social insurance of privately-uninsurable lifecycle shocks
2. Redistribution with respect to initial conditions
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 2 /41
Question
• How progressive should labor income taxation be?
• Argument in favor of progressivity: missing markets
1. Social insurance of privately-uninsurable lifecycle shocks
2. Redistribution with respect to initial conditions
• Arguments against progressivity: distortions
1. Labor supply
2. Human capital investment
3. Public provision of good and services
4. Redistribution from high to low “diligence” types
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 2 /41
Overview of the framework
• Huggett (1993) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
1. additional private insurance (other assets, family, etc)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 3 /41
Overview of the framework
• Huggett (1993) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
1. additional private insurance (other assets, family, etc)
2. differential “innate” diligence & (learning) ability
3. endogenous skill investment + multiple-skill technology
4. endogenous labor supply
5. government expenditures valued by households
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 3 /41
Overview of the framework
• Huggett (1993) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
1. additional private insurance (other assets, family, etc)
2. differential “innate” diligence & (learning) ability
3. endogenous skill investment + multiple-skill technology
4. endogenous labor supply
5. government expenditures valued by households
• Tractable equilibrium framework → closed-form SWF
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 3 /41
Overview of the framework
• Huggett (1993) economy: ∞-lived agents, idiosyncratic
productivity risk, and a risk-free bond in zero net-supply, plus:
1. additional private insurance (other assets, family, etc)
2. differential “innate” diligence & (learning) ability
3. endogenous skill investment + multiple-skill technology
4. endogenous labor supply
5. government expenditures valued by households
• Tractable equilibrium framework → closed-form SWF
• Ramsey approach: tax instruments & mkt structure taken as given
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 3 /41
The tax/transfer function
T (yi ) = yi − λyi1−τ
• The parameter τ measures the rate of progressivity:
◮ τ =1:
full redistribution → y˜i = λ
T ′ (y)
T (y)/y
◮ 0 < τ < 1:
progressivity →
◮ τ =0:
no redistribution → flat tax 1 − λ
◮ τ <0:
regressivity →
T ′ (y)
T (y)/y
>1
<1
• Break-even income level: y 0 = λ τ1
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 4 /41
The tax/transfer function
T (yi ) = yi − λyi1−τ
• The parameter τ measures the rate of progressivity:
◮ τ =1:
full redistribution → y˜i = λ
T ′ (y)
T (y)/y
◮ 0 < τ < 1:
progressivity →
◮ τ =0:
no redistribution → flat tax 1 − λ
◮ τ <0:
regressivity →
T ′ (y)
T (y)/y
>1
<1
• Break-even income level: y 0 = λ τ1
Restrictions: (i) no lump-sum transfer & (ii) T ′ (y) monotone
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 4 /41
Empirical fit to US micro data
• PSID 2000-06, age 20-60, N = 13, 721: R2 = 0.96: → τ U S = 0.151
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 5 /41
Empirical fit to US micro data
10 10.5 11 11.5 12 12.5 13
9.5
8.5
9
Log of post−government income
• PSID 2000-06, age 20-60, N = 13, 721: R2 = 0.96: → τ U S = 0.151
8.5
9
9.5
10
10.5
11
11.5
12
12.5
13
Log of pre−government income
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 5 /41
Marginal and average tax rates
0.5
0.4
0.3
Tax Rate
0.2
0.1
Average Tax Rate
Marginal Tax Rate
0
−0.1
−0.2
−0.3
−0.4
0.5
1
1.5
2
2.5
3
Household Income
3.5
4
4.5
5
5
x 10
• At the estimated value of τ U S = 0.151
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 6 /41
Demographics and preferences
• Perpetual youth demographics with constant survival probability δ
• Preferences over consumption (c), hours (h), publicly-provided
goods (G), and skill-investment (s) effort:
Ui = vi (si ) + E0
∞
X
(βδ)t ui (cit , hit , G)
t=0
vi (si )
ui (cit , hit , G)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
=
1 s2i
−
κi 2µ
=
h1+σ
log cit − exp(ϕi ) it + χ log G
1+σ
κi
∼
ϕi
∼
Exp (η)
v
ϕ
, vϕ
N
2
p. 7 /41
Technology
• Output is CES aggregator over continuum of skill types:
Y =
Z
∞
N (s)
0
θ−1
θ
ds
θ
θ−1
,
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
ci di + G
0
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 8 /41
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, ”Optimal Tax Progressivity”
p. 9 /41
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
hours
determined by skill, fortune, and diligence
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 9 /41
Government
• 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
• WLOG: G ≡ g · Y
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 10 /41
Markets
• Competitive good and labor markets
• Perfect annuity markets against survival risk
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 11 /41
Markets
• Competitive good and labor markets
• Perfect annuity markets against survival risk
• Competitive asset markets (all assets in zero net supply)
◮ Non state-contingent bond
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 11 /41
Markets
• Competitive good and labor markets
• Perfect annuity markets against survival risk
• Competitive asset markets (all assets in zero net supply)
◮ Non state-contingent bond
◮ Full set of insurance claims against ε shocks
If vε = 0, it is a bond economy
If vω = 0, it is a full insurance economy
If vω = vε = vϕ = 0 &
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
θ = ∞, it is a RA economy
p. 11 /41
A special case: the representative agent
max
C,H
U
=
C
s.t.
=
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
H 1+σ
log C −
+ χ log gY
1+σ
λY 1−τ and Y = H
p. 12 /41
A special case: the representative agent
max
C,H
U
=
C
s.t.
=
H 1+σ
log C −
+ χ log gY
1+σ
λY 1−τ and Y = H
• Market clearing: C + G = Y
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 12 /41
A special case: the representative agent
max
C,H
U
=
C
s.t.
=
H 1+σ
log C −
+ χ log gY
1+σ
λY 1−τ and Y = H
• Market clearing: C + G = Y
• Equilibrium allocations:
(τ )
=
1
log(1 − τ )
1+σ
log C RA (g, τ )
=
log(1 − g) + log H RA (τ )
log H
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
RA
p. 12 /41
Optimal policy in the RA economy
• Welfare function:
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:
∗
=
χ
1+χ
τ∗
=
−χ
g
• Solution for g ∗ will extend to heterogeneous-agent setup
• Allocations are first best (same as with lump-sum taxes)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 13 /41
Recursive stationary equilibrium
• Given (G, τ ), a stationary RCE is a value λ∗ , asset prices
{Q(·), q}, skill prices p(s), decision rules s(ϕ, κ, 0), c(α, ε, ϕ, s, b),
h(α, ε, ϕ, s, b), and aggregate quantities N (s) such that:
◮ households optimize
◮ markets clear
◮ the government budget constraint is balanced
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 14 /41
Recursive stationary equilibrium
• Given (G, τ ), a stationary RCE is a value λ∗ , asset prices
{Q(·), q}, skill prices p(s), decision rules s(ϕ, κ, 0), c(α, ε, ϕ, s, b),
h(α, ε, ϕ, s, b), and aggregate quantities N (s) such that:
◮ households optimize
◮ markets clear
◮ the government budget constraint is balanced
• In equilibrium, bonds are not traded
◮ b = 0 → allocations depend only on exogenous states
◮ α shocks remain uninsured, ε shocks fully insured
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 14 /41
Equilibrium skill choice and skill price
Equilibrium skill choice and skill price
• Skill price has Mincerian shape: log p(s; τ ) = π0 (τ ) + π1 (τ )s(κ; τ )
s(κ; τ )
π1 (τ )
=
r
ηµ (1 − τ )
·κ
θ
=
r
η
θµ (1 − τ )
(skill choice)
(marginal return to skill)
Equilibrium skill choice and skill price
• Skill price has Mincerian shape: log p(s; τ ) = π0 (τ ) + π1 (τ )s(κ; τ )
s(κ; τ )
π1 (τ )
=
r
ηµ (1 − τ )
·κ
θ
=
r
η
θµ (1 − τ )
(skill choice)
(marginal return to skill)
• Distribution of skill prices (in levels) is Pareto with parameter θ
Equilibrium skill choice and skill price
• Skill price has Mincerian shape: log p(s; τ ) = π0 (τ ) + π1 (τ )s(κ; τ )
s(κ; τ )
π1 (τ )
=
r
ηµ (1 − τ )
·κ
θ
=
r
η
θµ (1 − τ )
(skill choice)
(marginal return to skill)
• Distribution of skill prices (in levels) is Pareto with parameter θ
• Offsetting effects of τ on s and p on pre-tax wage inequality
var(log p(s; τ )) =
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
1
θ2
p. 15 /41
Equilibrium consumption allocation
log c∗ (α, ϕ, s; g, τ ) =
log C RA (g, τ ) + (1 − τ ) log p(s; τ )
|
{z
}
skill price
+(1 − τ ) α − (1 − τ ) ϕ +
| {z } | {z }
unins. shock
pref. het.
M(vε )
| {z }
level effect from ins. variation
• Response to variation in (p, ϕ, α) mediated by progressivity
• Invariant to insurable shock ε
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 16 /41
Equilibrium hours allocation
∗
log h (ε, ϕ; τ )
=
log H
RA
1
ε
(τ ) − ϕ +
|{z}
σ
b
|{z}
pref. het.
−
ins. shock
1
M(vε )
σ
b (1 − τ )
|
{z
}
level effect from ins. variation
• Response to ε mediated by tax-modified Frisch elasticity
1
σ̂
=
1−τ
σ+τ
• Invariant to skill price p, uninsurable shock α, and size of g
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 17 /41
Social Welfare Function
Planner chooses constant (g, τ )
Make two assumptions:
1. Planner puts equal weight on period utility of all currently alive
agents, discounts future at rate β
2. Skill investments are fully reversible
Then, SWF becomes average period utility in the cross-section
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 18 /41
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
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
1
1 vε
+
(1
+
χ)
vε
2
σ̂ 2
σ̂
p. 19 /41
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, ”Optimal Tax Progressivity”
p. 20 /41
Exact expression for SWF
W(τ )
=
1
log(1 − τ )
−
χ log χ − (1 + χ) log(1 + χ) + (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, ”Optimal Tax Progressivity”
p. 21 /41
Skill investment component
W(τ )
=
1
log(1 − τ )
−
(1 + σ̂)(1 − τ ) (1 + σ̂)
s
"
!
#
ηθ
−1
θ
θ
+(1 + χ)
log
log
+
θ−1
µ (1 − τ )
θ−1
θ−1
|
{z
}
χ log χ − (1 + χ) log(1 + χ) + (1 + χ)
productivity gain = 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
− (1 − τ )
−(1 + χ)σ

δ vω
− log 
1−δ 2
1 − δ exp
1 vε
1
+
(1
+
χ)
vε
2
σ̂ 2
σ̂
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
−τ (1−τ )
vω
2
1−δ


p. 22 /41
Skill investment component
0.18
Optimal degree of progressivity (τ)
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
5
10
15
20
25
Elasticity of substitution in production (θ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 23 /41
Optimal marginal tax rate at the top
• Diamond-Saez formula (Mirlees approach):
t̄ =
1
1+σ
=
1 + ζ u + ζ c (θ − 1)
θ+σ
◮ Decreasing in θ (increasing in thickness of Pareto tail)
• Our model: there is a range where τ is increasing in θ
◮ When complementarity is high, cost of distorting skill
investment is large
◮ Key difference: exogenous vs endogenous skill distribution
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 24 /41
Uninsurable component
W(τ )
=
1
log(1 − τ )
−
χ log χ − (1 + χ) log(1 + χ) + (1 + χ)
(1 + σ̂)(1 − τ ) (1 + σ̂)
s
!
"
#
θ
ηθ
θ
−1
log
log
+
+(1 + χ)
θ−1
µ (1 − τ )
θ−1
θ−1
1
1−τ
1−τ
− (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
δ
v
ω
−(1 − τ )
− log 
1−δ 2
1−δ
|
{z
consumption dispersion due to uninsurable shocks ≈
1
1 vε
+ (1 + χ) vε
−(1 + χ)σ 2
σ̂ 2
σ̂
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”


}
(1 − τ )2 v2α
p. 25 /41
Insurable component
W(τ )
=
1
log(1 − τ )
−
(1 + σ̂)(1 − τ ) (1 + σ̂)
s
!
"
#
θ
ηθ
θ
−1
log
+
log
+(1 + χ)
θ−1
µ (1 − τ )
θ−1
θ−1
1
1−τ
1−τ
− (1 − τ ) − − log 1 −
−
2θ
θ
θ
2 vϕ
− (1 − τ )
2



−τ (1−τ )
1
−
δ
exp
vω
2
v
δ
ω

− log 
− (1 − τ )
1−δ 2
1−δ
χ log χ − (1 + χ) log(1 + χ) + (1 + χ)
−(1 + χ)σ
1 vε
2 2
σ̂
| {z }
hours dispersion
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
+ (1 + χ)
1
vε
σ̂
|{z}
prod. gain from ins. shock=log(N/H)
p. 26 /41
Parameterization
• Parameter vector {χ, σ, θ, vϕ , vω , vε }
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 27 /41
Parameterization
• Parameter vector {χ, σ, θ, vϕ , vω , vε }
• To match G/Y = 0.19:
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
→ χ = 0.23
p. 27 /41
Parameterization
• Parameter vector {χ, σ, θ, vϕ , vω , vε }
• To match G/Y = 0.19:
→ χ = 0.23
• Frisch elasticity (micro-evidence):
→σ=2
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 27 /41
Parameterization
• Parameter vector {χ, σ, θ, vϕ , vω , vε }
• To match G/Y = 0.19:
→ χ = 0.23
• Frisch elasticity (micro-evidence):
→σ=2
cov(log h, log w)
var(log h)
=
=
var 0 (log c) =
var(log w)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
=
1
vε
σ̂
1
vϕ + 2 vε
σ̂
1
(1 − τ )2 vϕ + 2
θ
1
δ
vω
+
2
θ
1−δ
p. 27 /41
Parameterization
• Parameter vector {χ, σ, θ, vϕ , vω , vε }
• To match G/Y = 0.19:
→ χ = 0.23
• Frisch elasticity (micro-evidence):
→σ=2
cov(log h, log w)
=
var(log h)
=
var 0 (log c)
=
var(log w)
=
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
1
vε
σ̂
→ vε = 0.17
1
vϕ + 2 vε
→ vϕ = 0.035
σ̂
1
(1 − τ )2 vϕ + 2 → θ = 2.16
θ
1
δ
vω
+
2
θ
1−δ
→ vω = 0.003
p. 27 /41
Optimal progressivity
Social Welfare Function
welfare change rel. to optimum (% of cons.)
1
0
Welfare Gain = 0.5%
−1
−2
−3
−4
*
τ = 0.065
−5
US
τ
= 0.151
−6
−7
Baseline Model
−8
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 28 /41
Optimal progressivity: decomposition
welfare change rel. to optimum (% of cons.)
Social Welfare Function
4
2
0
−2
−4
−6
−8
−10
−12
(1) Rep. Agent
τ = −0.233
−14
−16
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 29 /41
Optimal progressivity: decomposition
welfare change rel. to optimum (% of cons.)
Social Welfare Function
4
(2) + Skill Inv.
τ = −0.062
2
0
−2
−4
−6
−8
−10
−12
(1) Rep. Agent
τ = −0.233
−14
−16
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 29 /41
Optimal progressivity: decomposition
welfare change rel. to optimum (% of cons.)
Social Welfare Function
4
(2) + Skill Inv.
τ = −0.062
2
(3) + Pref. Het.
τ = −0.020
0
−2
−4
−6
−8
−10
−12
(1) Rep. Agent
τ = −0.233
−14
−16
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 29 /41
Optimal progressivity: decomposition
welfare change rel. to optimum (% of cons.)
Social Welfare Function
4
(2) + Skill Inv.
τ = −0.062
2
(3) + Pref. Het.
τ = −0.020
0
−2
−4
−6
(4) + Unins. Shocks
τ = 0.076
−8
−10
−12
(1) Rep. Agent
τ = −0.233
−14
−16
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 29 /41
Optimal progressivity: decomposition
welfare change rel. to optimum (% of cons.)
Social Welfare Function
4
(2) + Skill Inv.
τ = −0.062
2
(3) + Pref. Het.
τ = −0.020
0
−2
(5) + Ins. Shocks
τ* = 0.065
−4
−6
(4) + Unins. Shocks
τ = 0.076
−8
−10
−12
(1) Rep. Agent
τ = −0.233
−14
−16
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Progressivity rate (τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 29 /41
Actual and optimal progressivity
0.4
Average tax rate
0.3
0.2
0.1
US
Actual US τ
0
= 0.151
∗
Utilitarian τ = 0.065
−0.1
−0.2
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Income (1 = average income)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 30 /41
Alternative SWF
Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ)
Switch off desire to redistribute
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 31 /41
Alternative SWF
Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ)
Switch off desire to redistribute
• Economy with heterogeneity in (κ, ϕ), and χ = vω = τ = 0
• Compute CE allocations
• Compute Negishi weights s.t. planner’s allocation = CE
• Use these weights in the SWF
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 31 /41
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∗
0.065
0.006
0.037
-0.022
Welf. gain (pct of c)
0.49
1.35
0.84
1.92
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 32 /41
Optimal progressivity: alternative SWF
0.4
Average tax rate
0.3
0.2
0.1
0
Actual US τUS = 0.151
Utilitarian τ∗ = 0.065
∗
Ins. Only τ = −0.022
−0.1
−0.2
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Income (1 = average income)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 33 /41
Alternative assumptions on G
Case
Utilitarian
G
Y
Insurance-only
τ∗
Welf. gain
τ∗
Welf. gain
g ∗ endogenous
χ = 0.233
0.189
0.065
0.49%
−0.022
1.92%
ḡ exogenous
χ=0
0.189
0.184
0.07%
0.092
0.20%
Ḡ exogenous
χ=0
0.189
0.071
0.45%
−0.011
1.78%
• With χ = 0, no externality and no push towards regressivity
• If G fixed in level, lower progressivity translates 1-1 into higher c
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 34 /41
Irreversible skill investment
• Assume skills are fixed after being chosen
• Planner has an additional incentive to increase progressivity:
◮ Can compress consumption inequality, while discouraging skill
investment only for new generations (not for current ones)
• Needed to preserve tractability: production is segregated by age
• Planner chooses τ ∗ = 0.137
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 35 /41
Irreversible skill investment
• Assume skills are fixed after being chosen
• Planner has an additional incentive to increase progressivity:
◮ Can compress consumption inequality, while discouraging skill
investment only for new generations (not for current ones)
• Needed to preserve tractability: production is segregated by age
• Planner chooses τ ∗ = 0.137
• Hybrid model: optimal progressivity is, again, τ ∗ = 0.065
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 35 /41
Political Economy
• Think of g and τ as outcomes of a majority rule voting process
• Median voter theorem applies in our model!
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 36 /41
Political Economy
• Think of g and τ as outcomes of a majority rule voting process
• Median voter theorem applies in our model!
◮ Agents agree over spending and choose g ∗ = χ/(1 + χ)
◮ ... but they disagree over τ
◮ In spite of multi-dimensional heterogeneity over (ϕ, κ, α),
preferences are single-peaked in τ
• Median voter chooses τ med = 0.083
• More progressivity because median voter poorer than average
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 36 /41
Progressive consumption taxation
c = λc̃1−τ
where c are expenditures and c̃ are units of final good
• Implement as a tax on total (labor plus asset) income less saving
• Consumption depends on α but not on ε
• Can redistribute wrt. uninsurable shocks without distorting the
efficient response of hours to insurable shocks
• Higher progressivity and higher welfare
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 37 /41
Budget constraints
1. Beginning of period: innovation ω to α shock is realized
2. Middle of period: buy insurance against ε:
b =
Z
Q(ε)B(ε)dε,
E
where Q(·) is the price of insurance and B(·) is the quantity
3. End of period: ε realized, consumption and hours chosen:
c + δqb′
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
=
λ [p(s) exp(α + ε)h]1−τ + B(ε)
p. 38 /41
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, ”Optimal Tax Progressivity”
p. 39 /41
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, ”Optimal Tax Progressivity”
p. 39 /41
Equilibrium risk-free rate r ∗
vω
ρ − r = (1 − τ ) ((1 − τ ) + 1)
2
∗
• Intertemporal dis-saving motive = precautionary saving motive
• Key: precautionary saving motive common across all agents
•
∂r ∗
∂τ
> 0: more progressivity ⇒ less precautionary saving ⇒
higher risk-free rate
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 40 /41
Upper tail of wage distribution
Top 1pct of the Wage Distribution
−4
x 10
Model Wage Distribution
Lognormal Wage Distribution
Density
2
1
0
4
5
6
7
8
9
10
Wage (average=1)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity”
p. 41 /41