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Pareto Efficient Income Taxation - p. 1
Pareto Efficient Income Taxation
Iván WerningMIT
April 2007
NBER Public Economics meeting
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 2
Introduction
Q: Good shape for tax schedule ?
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 2
Introduction
Q: Good shape for tax schedule ?Mirrlees (1971), Diamond (1998), Saez (2001)
positive: redistribution vs. efficiency
normative: Utilitarian social welfare function
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 2
Introduction
Q: Good shape for tax schedule ?Mirrlees (1971), Diamond (1998), Saez (2001)
positive: redistribution vs. efficiency
normative: Utilitarian social welfare function
this paper: Pareto efficient taxation
positive: redistribution vs. efficiency
normative: Utilitarian social welfare functionPareto Efficiency
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 3
Old Motivation: “New New New...”
Why not Utilitarian? (∑
i Ui)
practical: cardinality U i → W (U i) (or even W i(U i))... which Utilitarian?conceptual: political process:social classes Coasian bargain...but max
∑
U i ?
philosophical: other notions of fairness and socialjustice
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 3
Old Motivation: “New New New...”
Why not Utilitarian? (∑
i Ui)
practical: cardinality U i → W (U i) (or even W i(U i))... which Utilitarian?conceptual: political process:social classes Coasian bargain...but max
∑
U i ?
philosophical: other notions of fairness and socialjustice
Pareto efficiency weaker criterion
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
first best
constrained
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
VH+ VL
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 4
Pareto Frontier
vH
vL
VH+ VL
VH+(1- ) VL
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 5
Contribution
invert Mirrlees model...
...express in tractable way
...use it: some applications
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 6
Results#0 restrictions generalize “zero-tax-at-the-top”#1 Any T (Y ). . .
efficient for many f(θ)
inefficient for many f(θ). . . anything goes
#2 Given T0(Y ) g(Y ) f(θ) (Saez, 2001)efficient set of T (Y ): large
inefficient set of T (Y ): large
#3 Simple test for efficiency of T0(Y )
Introduction❖ Introduction❖ Motivation❖ Contribution❖ Results
Model
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 7
Results
#4 Simple formulas...
bound on top tax rate
efficiency of a flat tax
#5 Increasing progressivitymaintains Pareto efficiency
#6 observable heterogeneitynot conditioning can be efficient
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
given T (Y )
v(θ) ≡ maxY
U(Y − T (Y ), Y, θ)
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
given T (Y )
v(θ) ≡ maxY
U(Y − T (Y ), Y, θ)
Government budget∫
T (Y (θ)) dF (θ) ≥ G
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
given T (Y )
v(θ) ≡ maxY
U(Y − T (Y ), Y, θ)
Resource feasible∫
(
Y (θ) − c(θ))
dF (θ) ≥ G
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
given T (Y )
v′(θ) = Uθ(Y (θ) − T (Y (θ)), Y (θ), θ)
Resource feasible∫
(
Y (θ) − c(θ))
dF (θ) ≥ G
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
given T (Y )
v′(θ) = −h(Y (θ))
Resource feasible∫
(
Y (θ) − c(θ))
dF (θ) ≥ G
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 8
SetupPositive side of Mirrlees (1971)
continuum of types θ ∼ F (θ)
additive preferences
U(c, Y, θ) = u(c) − θh(Y )
(e.g. Y = w · n and h(n) = αnη)
given T (Y )
v′(θ) = −h(Y (θ))
Resource feasible∫
(
Y (θ) − e(v(θ), Y (θ), θ))
dF (θ) ≥ G
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 9
Planning Problem
Dual Pareto Problem
maximize net resources
subject to,v(θ) ≥ v(θ)
incentives
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 9
Planning Problem
Dual Pareto Problem
maxY ,v
∫
(
Y (θ) − e(v(θ), Y (θ), θ))
dF (θ)
subject to,v(θ) ≥ v(θ)
incentives
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 9
Planning Problem
Dual Pareto Problem
maxY ,v
∫
(
Y (θ) − e(v(θ), Y (θ), θ))
dF (θ)
subject to,v(θ) ≥ v(θ)
v′(θ) = −h(Y (θ))
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 9
Planning Problem
Dual Pareto Problem
maxY ,v
∫
(
Y (θ) − e(v(θ), Y (θ), θ))
dF (θ)
subject to,v(θ) ≥ v(θ)
v′(θ) = −h(Y (θ))
Y (θ) nonincreasing
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)
−
∫
(
v′(θ) + h(Y (θ)))
µ(θ) dθ
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian (integrating by parts)
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)−v(θ)µ(θ) + µ(θ)v(θ)
+
∫
v(θ)µ′(θ)dθ −
∫
h(Y (θ))µ(θ) dθ
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian (integrating by parts)
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)−v(θ)µ(θ) + µ(θ)v(θ)
+
∫
v(θ)µ′(θ)dθ −
∫
h(Y (θ))µ(θ) dθ
First-order conditions(
1 − eY (v(θ), Y (θ), θ))
f(θ) = µ(θ)h′(Y (θ)) [Y (θ)]
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian (integrating by parts)
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)−v(θ)µ(θ) + µ(θ)v(θ)
+
∫
v(θ)µ′(θ)dθ −
∫
h(Y (θ))µ(θ) dθ
First-order conditions
τ(θ)f(θ) = µ(θ)h′(Y (θ)) [Y (θ)]
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian (integrating by parts)
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)−v(θ)µ(θ) + µ(θ)v(θ)
+
∫
v(θ)µ′(θ)dθ −
∫
h(Y (θ))µ(θ) dθ
First-order conditions
µ(θ) = τ(θ)f(θ)
h′(Y (θ))[Y (θ)]
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian (integrating by parts)
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)−v(θ)µ(θ) + µ(θ)v(θ)
+
∫
v(θ)µ′(θ)dθ −
∫
h(Y (θ))µ(θ) dθ
First-order conditions
µ(θ) = τ(θ)f(θ)
h′(Y (θ))[Y (θ)]
µ′(θ) ≤ ev
(
v(θ), Y (θ), θ)
f(θ) [v(θ)]
Introduction
Model❖ Setup❖ Planning Problem❖ Efficiency
Conditions
Main Results
Applications
Conclusions
Pareto Efficient Income Taxation - p. 10
Efficiency Conditions
Lagrangian (integrating by parts)
L =
∫
(
Y (θ)−e(v(θ), Y (θ), θ))
dF (θ)−v(θ)µ(θ) + µ(θ)v(θ)
+
∫
v(θ)µ′(θ)dθ −
∫
h(Y (θ))µ(θ) dθ
First-order conditions
µ(θ) = τ(θ)f(θ)
h′(Y (θ))[Y (θ)]
µ′(θ) ≤ ev
(
v(θ), Y (θ), θ)
f(θ) [v(θ)]
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 11
Efficiency Conditions
Proposition. T (Y ) is Pareto efficient if and only
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
τ(θ) ≥ 0 and τ(θ) ≤ 0.
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 11
Efficiency Conditions
Proposition. T (Y ) is Pareto efficient if and only
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
τ(θ) ≥ 0 and τ(θ) ≤ 0.
note: “zero-tax-at-top” special case
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 11
Efficiency Conditions
Proposition. T (Y ) is Pareto efficient if and only
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
τ(θ) ≥ 0 and τ(θ) ≤ 0.
note: “zero-tax-at-top” special case
more general condition:
τ(θ)f(θ)
h′(Y (θ))+
∫ θ
θ
1
u′(c(θ))f(θ) dθ is nonincreasing
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 12
Intuition
define
T (Y ) ≡
{
T (Y (θ)) − ε Y = Y (θ)
T (Y ) Y 6= Y (θ)
Proposition. T ≻ T
τ(θ)
(
θτ ′(θ)
τ(θ)+ 2
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 3(1 − τ(θ))
is violated at θ
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 13
Simple Tax Reform
Y
T
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 13
Simple Tax Reform
Y
T
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 13
Simple Tax Reform
Y
T
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 13
Simple Tax Reform
Y
T
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 13
Simple Tax Reform
Y
T
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 13
Simple Tax Reform
Y
T
g′(Y )g(Y )
small (f ′(θ)f(θ)
large) inefficiency
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 14
Laffer
lower taxes increase revenue
Pareto improvements “Laffer” effect
Proposition. T1(Y ) ≻ T0(Y ) T1(Y ) ≤ T0(Y )
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 15
Anything Goes
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
Proposition. For any T (Y )
exists set {f(θ)} Pareto efficient
exists set {f(θ)} Pareto inefficient
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 15
Anything Goes
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
Proposition. For any T (Y )
exists set {f(θ)} Pareto efficient
exists set {f(θ)} Pareto inefficient
without empirical knowledgeanything goes
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 15
Anything Goes
τ(θ)
(
θτ ′(θ)
τ(θ)+
d log f(θ)
d log θ−
d log h′(Y (θ))
d log θ
)
≤ 1 − τ(θ)
Proposition. For any T (Y )
exists set {f(θ)} Pareto efficient
exists set {f(θ)} Pareto inefficient
without empirical knowledgeanything goes
need information on f(θ) to restrict T (Y )
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 16
Identification and Test
observe g(Y ) identify (Saez, 2001)
θ(Y ) = (1 − T ′(Y ))u′(Y − T (Y ))
h′(Y )
f(θ(Y )) =g(Y )
θ′(Y )
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 16
Identification and Test
observe g(Y ) identify (Saez, 2001)
θ(Y ) = (1 − T ′(Y ))u′(Y − T (Y ))
h′(Y )
f(θ(Y )) =g(Y )
θ′(Y )
efficiency test...
d log g(Y )
d log Y≥ a(Y )
... for tax schedule in place
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 17
Graphical Test
define Rawlsian density:
α(Y ) =exp
(
∫ Y
0a(z) dz
)
∫
∞
0exp
(
∫ Y
0a(z) dz
)
graphical test:
g(Y )
α(Y )nondecreasing
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60
0.05
0.1
0.15
0.2
0.25
0.3
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 18
Empirical Implementation
needed1. current tax function T (Y )
2. distribution of income g(Y )
3. utility function U(c, Y, θ)
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 18
Empirical Implementation
needed1. current tax function T (Y )
2. distribution of income g(Y )
3. utility function U(c, Y, θ)
in principle: #1 and #2 easy#3 usual deal
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 18
Empirical Implementation
needed1. current tax function T (Y )
2. distribution of income g(Y )
3. utility function U(c, Y, θ)
in principle: #1 and #2 easy#3 usual deal
Diamond (1998) and Saez (2001)
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 18
Empirical Implementation
needed1. current tax function T (Y )
2. distribution of income g(Y )
3. utility function U(c, Y, θ)
in principle: #1 and #2 easy#3 usual deal
Diamond (1998) and Saez (2001)
some challenges...1. econometric: need to estimate g′(Y ) and g(Y )
2. conceptual: static modellifetime T (Y ) and g(Y ) (Fullerton and Rogers)
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 19
Output Density
IRS’s SOI Public Use Files for Individual tax returnslifetime g(Y )?
lifetime T (Y ) schedule?
Y i = 1n
∑
Y it
smooth density estimateassumed T (Y ) = .30 × Y
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 19
Output Density
IRS’s SOI Public Use Files for Individual tax returnslifetime g(Y )?
lifetime T (Y ) schedule?
Y i = 1n
∑
Y it
smooth density estimateassumed T (Y ) = .30 × Y
0 0.5 1 1.5 2 2.5
x 105
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
−5
Average Income (in 1990 dollars)
Kernel Density (bandwidth = 10,000) of Average Income Over Varying Time Periods in the United States
>= 10 years during 1979−19901982−19861987−1990
Figure 1: Density of incomeg(Y ).
0 2 4 6 8 10 12 14 16 18
x 104
−12
−10
−8
−6
−4
−2
0
2
Average Income (in 1990 dollars)
Elasticity of Kernel Density (bandwidth = 10,000) of Average Income Over Varying Time Periods in the United States
>= 10 years during 1979−19901982−19861987−1990
Figure 2: Implied elasticityY g′(Y )g(Y ) .
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 19
Output Density
IRS’s SOI Public Use Files for Individual tax returnslifetime g(Y )?
lifetime T (Y ) schedule?
Y i = 1n
∑
Y it
smooth density estimateassumed T (Y ) = .30 × Y
0 2 4 6 8 10
x 104
0
0.5
1
1.5
2
2.5
3
x 10−5Rawlsian Test against 1987−1990 Average Income Data (sigma = 0, eta = 2, T = .3Y)
0 1 2 3 4 5 6 7 8 9 10
x 104
0
1
2
3
4
5
x 10−5Rawlsian Test against 1987−1990 Average Income Data (sigma = 1, eta = 3, T = .3Y)
Introduction
Model
Main Results❖ Intuition❖ Anything Goes❖ Identification and
Test❖ Graphical Test❖ Empirical Strategy❖ Quantifying
Inefficiencies
Applications
Conclusions
Pareto Efficient Income Taxation - p. 20
Quantifying Inefficiencies
efficiency test qualitative
quantitative. . .
∆ ≡
∫
(
Y ∗(θ) − c∗(θ))
dF (θ) −
∫
(
Y (θ) − c(θ))
dF (θ)
does not count welfare improvements
v(θ) > v(θ)
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 21
Top Tax Rate
u(c) = c1−σ/(1 − σ) and h(Y ) = αY η
suppose top tax rate
τ ≡ limθ→0
τ(θ) = limY →∞
T ′(Y )
exists
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 21
Top Tax Rate
u(c) = c1−σ/(1 − σ) and h(Y ) = αY η
suppose top tax rate
τ ≡ limθ→0
τ(θ) = limY →∞
T ′(Y )
exists
efficiency condition bound
τ ≤σ + η − 1
ϕ + η − 2.
where ϕ = − limT→∞ d log g(Y )/d log Y .
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 21
Top Tax Rate
u(c) = c1−σ/(1 − σ) and h(Y ) = αY η
suppose top tax rate
τ ≡ limθ→0
τ(θ) = limY →∞
T ′(Y )
exists
efficiency condition bound
τ ≤σ + η − 1
ϕ + η − 2.
where ϕ = − limT→∞ d log g(Y )/d log Y .
Saez (2001): ϕ = 3
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 22
Top Tax Rate
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
elasticity 1/ η+1
up
pe
r b
ou
nd
on
τ
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 23
Flat Tax
linear tax necessary condition
τ ≤σ + η − 1
−d log g(Y )d log Y
+ η − 2
linear tax sufficient condition
τ ≤η − 1
−d log g(Y )d log Y
+ η − 1
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 24
Progressivity
Quasi-linear u(c) = c
result: can always increase progressivity
Introduction
Model
Main Results
Applications❖ Top Tax Rate❖ Flat Tax❖ Progressivity❖ Heterogeneity
Conclusions
Pareto Efficient Income Taxation - p. 25
Heterogeneity
groups = 1, . . . , N
f i(θ) and U i(c, Y, θ)
unobservable isingle T (Y )
average efficiency condition
observable imultiple T i(Y )
N efficiency conditions
observation:T i(Y ) = T (Y ) may be Pareto efficient
never optimal for Utilitarian