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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, idiosyncraticproductivity 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, idiosyncraticproductivity 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, idiosyncraticproductivity 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, idiosyncraticproductivity 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 − λy1−τi
• The parameter τ measures the rate of progressivity:
◮ τ = 1 : full redistribution → yi = λ
◮ 0 < τ < 1: progressivity → T ′(y)T (y)/y > 1
◮ τ = 0 : no redistribution → flat tax 1− λ
◮ τ < 0 : regressivity → T ′(y)T (y)/y < 1
• Break-even income level: y0 = λ1
τ
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 4 /41
The tax/transfer function
T (yi) = yi − λy1−τi
• The parameter τ measures the rate of progressivity:
◮ τ = 1 : full redistribution → yi = λ
◮ 0 < τ < 1: progressivity → T ′(y)T (y)/y > 1
◮ τ = 0 : no redistribution → flat tax 1− λ
◮ τ < 0 : regressivity → T ′(y)T (y)/y < 1
• Break-even income level: y0 = λ1
τ
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: → τUS = 0.151
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 5 /41
Empirical fit to US micro data
• PSID 2000-06, age 20-60, N = 13, 721: R2 = 0.96: → τUS = 0.151
8.5
99.
510
10.5
1111
.512
12.5
13Lo
g of
pos
t−go
vern
men
t inc
ome
8.5 9 9.5 10 10.5 11 11.5 12 12.5 13Log of pre−government income
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 5 /41
Marginal and average tax rates
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x 105
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Household Income
Tax
Rat
e
Average Tax RateMarginal Tax Rate
• At the estimated value of τUS = 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-providedgoods (G), and skill-investment (s) effort:
Ui = vi(si) + E0
∞∑
t=0
(βδ)tui(cit, hit, G)
vi(si) = −1
κi
s2i2µ
ui (cit, hit, G) = log cit − exp(ϕi)h1+σit
1 + σ+ χ logG
κi ∼ Exp (η)
ϕi ∼ N(vϕ2, vϕ
)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 7 /41
Technology
• Output is CES aggregator over continuum of skill types:
Y =
[∫ ∞
0
N (s)θ−1
θ ds
] θ
θ−1
, θ ∈ (1,∞)
• Aggregate effective hours by skill type:
N(s) =
∫ 1
0
I{si=s} zihi di
• Aggregate resource constraint:
Y =
∫ 1
0
ci di+G
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 8 /41
Individual efficiency units of labor
log zit = αit + εit
• αit = αi,t−1 + ωit with ωit ∼ N(− vω
2 , vω)
αi0 = 0 ∀i
• εit i.i.d. over time with εit ∼ N(− vε
2 , vε)
• ϕ ⊥ κ ⊥ ω ⊥ ε 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 with ωit ∼ N(− vω
2 , vω)
αi0 = 0 ∀i
• εit i.i.d. over time with εit ∼ N(− vε
2 , vε)
• ϕ ⊥ κ ⊥ ω ⊥ ε cross-sectionally and longitudinally
• Pre-government earnings:
yit = p(si)︸ ︷︷ ︸skill price
× exp(αit + εit)︸ ︷︷ ︸efficiency
× hit︸︷︷︸hours
determined by skill, fortune, and diligence
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 9 /41
Government
• Disposable (post-government) earnings:
yi = λy1−τi
• Government budget constraint (no government debt):
G =
∫ 1
0
[yi − λy1−τ
i
]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 & θ = ∞, it is a RA economy
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 11 /41
A special case: the representative agent
maxC,H
U = logC −H1+σ
1 + σ+ χ log gY
s.t.
C = λY 1−τ and Y = H
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 12 /41
A special case: the representative agent
maxC,H
U = logC −H1+σ
1 + σ+ χ log gY
s.t.
C = 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
maxC,H
U = logC −H1+σ
1 + σ+ χ log gY
s.t.
C = λY 1−τ and Y = H
• Market clearing: C +G = Y
• Equilibrium allocations:
logHRA(τ) =1
1 + σlog(1− τ)
logCRA(g, τ) = log(1− g) + logHRA(τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 12 /41
Optimal policy in the RA economy
• Welfare function:
WRA(g, τ) = log(1− g) + χ log g + (1 + χ)log(1− τ)
1 + σ−
1− τ
1 + σ
• Welfare maximizing (g, τ) pair:
g∗ =χ
1 + χ
τ∗ = −χ
• 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) =
√ηµ (1− τ)
θ· κ (skill choice)
π1(τ) =
√η
θµ (1− τ)(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) =
√ηµ (1− τ)
θ· κ (skill choice)
π1(τ) =
√η
θµ (1− τ)(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) =
√ηµ (1− τ)
θ· κ (skill choice)
π1(τ) =
√η
θµ (1− τ)(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)) =1
θ2
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 15 /41
Equilibrium consumption allocation
log c∗(α, ϕ, s; g, τ) = logCRA(g, τ) + (1− τ) log p(s; τ)︸ ︷︷ ︸skill price
+(1− τ)α︸ ︷︷ ︸unins. shock
− (1− τ)ϕ︸ ︷︷ ︸pref. het.
+ M(vε)︸ ︷︷ ︸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) = logHRA(τ)− ϕ︸︷︷︸pref. het.
+1
σε
︸︷︷︸ins. shock
−1
σ(1− τ)M(vε)
︸ ︷︷ ︸level effect from ins. variation
• Response to ε mediated by tax-modified Frisch elasticity 1σ = 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 aliveagents, 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) = log(1 + g) + χ log g + (1 + χ)log(1− τ)
(1 + σ)(1− τ)−
1
(1 + σ)
+(1 + χ)
[−1
θ − 1log
(√ηθ
µ (1− τ)
)+
θ
θ − 1log
(θ
θ − 1
)]
−1
2θ(1− τ)−
[− log
(1−
(1− τ
θ
))−
(1− τ
θ
)]
− (1− τ)2vϕ2
−
(1− τ)
δ
1− δ
vω2
− log
1− δ exp
(−τ(1−τ)
2 vω
)
1− δ
−(1 + χ)σ1
σ2
vε2
+ (1 + χ)1
σvε
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 19 /41
Representative Agent component
W(g, τ) = log(1 + g) + χ log g + (1 + χ)log(1− τ)
(1 + σ)(1− τ)−
1
(1 + σ)︸ ︷︷ ︸Representative Agent Welfare = WRA(g, τ)
+(1 + χ)
[−1
θ − 1log
(√ηθ
µ (1− τ)
)+
θ
θ − 1log
(θ
θ − 1
)]
−1
2θ(1− τ)−
[− log
(1−
(1− τ
θ
))−
(1− τ
θ
)]
− (1− τ)2vϕ2
−
(1− τ)
δ
1− δ
vω2
− log
1− δ exp
(−τ(1−τ)
2 vω
)
1− δ
−(1 + χ)σ1
σ2
vε2
+ (1 + χ)1
σvε
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 20 /41
Exact expression for SWF
W(τ) = χ logχ− (1 + χ) log(1 + χ) + (1 + χ)log(1− τ)
(1 + σ)(1− τ)−
1
(1 + σ)
+(1 + χ)
[−1
θ − 1log
(√ηθ
µ (1− τ)
)+
θ
θ − 1log
(θ
θ − 1
)]
−1
2θ(1− τ)−
[− log
(1−
(1− τ
θ
))−
(1− τ
θ
)]
− (1− τ)2vϕ2
−
(1− τ)
δ
1− δ
vω2
− log
1− δ exp
(−τ(1−τ)
2 vω
)
1− δ
−(1 + χ)σ1
σ2
vε2
+ (1 + χ)1
σvε
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 21 /41
Skill investment component
W(τ) = χ logχ− (1 + χ) log(1 + χ) + (1 + χ)log(1− τ)
(1 + σ)(1− τ)−
1
(1 + σ)
+(1 + χ)
[−1
θ − 1log
(√ηθ
µ (1− τ)
)+
θ
θ − 1log
(θ
θ − 1
)]
︸ ︷︷ ︸productivity gain = logE [(p(s))] = log (Y/N)
−1
2θ(1− τ)
︸ ︷︷ ︸avg. education cost
−
[− log
(1−
(1− τ
θ
))−
(1− τ
θ
)]
︸ ︷︷ ︸consumption dispersion across skills
− (1− τ)2 vϕ
2
−
(1− τ)
δ
1− δ
vω2
− log
1− δ exp
(−τ(1−τ)
2 vω
)
1− δ
−(1 + χ)σ1
σ2
vε2
+ (1 + χ)1
σvε
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 22 /41
Skill investment component
5 10 15 20 250
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Elasticity of substitution in production (θ)
Opt
imal
deg
ree
of p
rogr
essi
vity
(τ)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 23 /41
Optimal marginal tax rate at the top
• Diamond-Saez formula (Mirlees approach):
t =1
1 + ζu + ζc(θ − 1)=
1 + σ
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 skillinvestment is large
◮ Key difference: exogenous vs endogenous skill distribution
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 24 /41
Uninsurable component
W(τ) = χ logχ− (1 + χ) log(1 + χ) + (1 + χ)log(1− τ)
(1 + σ)(1− τ)−
1
(1 + σ)
+(1 + χ)
[−1
θ − 1log
(√ηθ
µ (1− τ)
)+
θ
θ − 1log
(θ
θ − 1
)]
−1
2θ(1− τ)−
[− log
(1−
(1− τ
θ
))−
(1− τ
θ
)]
− (1− τ)2vϕ2︸ ︷︷ ︸
cons. disp. due to prefs
−
(1− τ)
δ
1− δ
vω2
− log
1− δ exp
(−τ(1−τ)
2 vω
)
1− δ
︸ ︷︷ ︸consumption dispersion due to uninsurable shocks ≈ (1 − τ)2 vα
2
−(1 + χ)σ1
σ2
vε2
+ (1 + χ)1
σvε
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 25 /41
Insurable component
W(τ) = χ logχ− (1 + χ) log(1 + χ) + (1 + χ)log(1− τ)
(1 + σ)(1− τ)−
1
(1 + σ)
+(1 + χ)
[−1
θ − 1log
(√ηθ
µ (1− τ)
)+
θ
θ − 1log
(θ
θ − 1
)]
−1
2θ(1− τ)−
[− log
(1−
(1− τ
θ
))−
(1− τ
θ
)]
− (1− τ)2 vϕ
2
−
(1− τ)
δ
1− δ
vω2
− log
1− δ exp
(−τ(1−τ)
2 vω
)
1− δ
−(1 + χ)σ1
σ2
vε2︸ ︷︷ ︸
hours dispersion
+ (1 + χ)1
σvε
︸︷︷︸prod. gain from ins. shock=log(N/H)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” 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: → χ = 0.23
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
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, logw) =1
σvε
var(log h) = vϕ +1
σ2vε
var0(log c) = (1− τ)2(vϕ +
1
θ2
)
var(logw) =1
θ2+
δ
1− δvω
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, logw) =1
σvε → vε = 0.17
var(log h) = vϕ +1
σ2vε → vϕ = 0.035
var0(log c) = (1− τ)2(vϕ +
1
θ2
)→ θ = 2.16
var(logw) =1
θ2+
δ
1− δvω → vω = 0.003
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 27 /41
Optimal progressivity
−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−8
−7
−6
−5
−4
−3
−2
−1
0
1
Progressivity rate (τ)
wel
fare
cha
nge
rel.
to o
ptim
um (
% o
f con
s.)
Social Welfare Function
Baseline Model
τUS= 0.151
τ*= 0.065
Welfare Gain = 0.5%
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 28 /41
Optimal progressivity: decomposition
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16
−14
−12
−10
−8
−6
−4
−2
0
2
4
Progressivity rate (τ)
wel
fare
cha
nge
rel.
to o
ptim
um (
% o
f con
s.)
Social Welfare Function
(1) Rep. Agentτ = −0.233
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 29 /41
Optimal progressivity: decomposition
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16
−14
−12
−10
−8
−6
−4
−2
0
2
4
Progressivity rate (τ)
wel
fare
cha
nge
rel.
to o
ptim
um (
% o
f con
s.)
Social Welfare Function
(2) + Skill Inv.τ = −0.062
(1) Rep. Agentτ = −0.233
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 29 /41
Optimal progressivity: decomposition
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16
−14
−12
−10
−8
−6
−4
−2
0
2
4
Progressivity rate (τ)
wel
fare
cha
nge
rel.
to o
ptim
um (
% o
f con
s.)
Social Welfare Function
(2) + Skill Inv.τ = −0.062
(1) Rep. Agentτ = −0.233
(3) + Pref. Het.τ = −0.020
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 29 /41
Optimal progressivity: decomposition
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16
−14
−12
−10
−8
−6
−4
−2
0
2
4
Progressivity rate (τ)
wel
fare
cha
nge
rel.
to o
ptim
um (
% o
f con
s.)
Social Welfare Function
(2) + Skill Inv.τ = −0.062
(1) Rep. Agentτ = −0.233
(3) + Pref. Het.τ = −0.020
(4) + Unins. Shocksτ = 0.076
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 29 /41
Optimal progressivity: decomposition
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16
−14
−12
−10
−8
−6
−4
−2
0
2
4
Progressivity rate (τ)
wel
fare
cha
nge
rel.
to o
ptim
um (
% o
f con
s.)
Social Welfare Function
(2) + Skill Inv.τ = −0.062
(1) Rep. Agentτ = −0.233
(3) + Pref. Het.τ = −0.020
(5) + Ins. Shocksτ* = 0.065
(4) + Unins. Shocksτ = 0.076
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 29 /41
Actual and optimal progressivity
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
−0.1
0
0.1
0.2
0.3
0.4
Income (1 = average income)
Ave
rage
tax
rate
Actual US τUS = 0.151
Utilitarian τ∗ = 0.065
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
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
−0.1
0
0.1
0.2
0.3
0.4
Income (1 = average income)
Ave
rage
tax
rate
Actual US τUS = 0.151
Utilitarian τ∗ = 0.065
Ins. Only τ∗ = −0.022
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 33 /41
Alternative assumptions on G
Case Utilitarian Insurance-onlyG
Yτ∗ Welf. gain τ∗ Welf. gain
g∗ endogenous χ = 0.233 0.189 0.065 0.49% −0.022 1.92%
g exogenous χ = 0 0.189 0.184 0.07% 0.092 0.20%
G 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 skillinvestment 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 skillinvestment 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 τmed = 0.083
• More progressivity because median voter poorer than average
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 36 /41
Progressive consumption taxation
c = λc1−τ
where c are expenditures and c 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 theefficient 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 =
∫
EQ(ε)B(ε)dε,
where Q(·) is the price of insurance and B(·) is the quantity
3. End of period: ε realized, consumption and hours chosen:
c+ δqb′ = λ [p(s) exp(α+ ε)h]1−τ +B(ε)
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” 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 = yt
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 = yt
• 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∗
ρ− r∗ = (1− τ) ((1− τ) + 1)vω2
• 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
4 5 6 7 8 9 100
1
2
x 10−4
Wage (average=1)
Den
sity
Top 1pct of the Wage Distribution
Model Wage DistributionLognormal Wage Distribution
Heathcote-Storesletten-Violante, ”Optimal Tax Progressivity” p. 41 /41