pareto efﬁcient income taxation - mitweb.mit.edu/iwerning/public/slidesnber-pe.pdf ·...

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


Introduction

Q: Good shape for tax schedule ?


Model

Main Results

Applications

Conclusions


Introduction

Q: Good shape for tax schedule ?Mirrlees (1971), Diamond (1998), Saez (2001)

positive: redistribution vs. efficiency

normative: Utilitarian social welfare function


Model

Main Results

Applications

Conclusions


Introduction

Q: Good shape for tax schedule ?Mirrlees (1971), Diamond (1998), Saez (2001)


normative: Utilitarian social welfare function

this paper: Pareto efficient taxation


normative: Utilitarian social welfare functionPareto Efficiency


Model

Main Results

Applications

Conclusions


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


Model

Main Results

Applications

Conclusions


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


Model

Main Results

Applications

Conclusions


Pareto Frontier

vH

vL

first best

constrained


Model

Main Results

Applications

Conclusions


Pareto Frontier

vH

vL


Model

Main Results

Applications

Conclusions


Pareto Frontier

vH

vL

VH+ VL


Model

Main Results

Applications

Conclusions


Pareto Frontier

vH

vL

VH+ VL

VH+(1- ) VL


Model

Main Results

Applications

Conclusions


Contribution

invert Mirrlees model...

...express in tractable way

...use it: some applications


Model

Main Results

Applications

Conclusions


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 )


Model

Main Results

Applications

Conclusions


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


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


Conditions

Main Results

Applications

Conclusions





U(c, Y, θ) = u(c) − θh(Y )


given T (Y )

v(θ) ≡ maxY

U(Y − T (Y ), Y, θ)

Introduction


Conditions

Main Results

Applications

Conclusions





U(c, Y, θ) = u(c) − θh(Y )


given T (Y )

v(θ) ≡ maxY

U(Y − T (Y ), Y, θ)

Government budget∫

T (Y (θ)) dF (θ) ≥ G

Introduction


Conditions

Main Results

Applications

Conclusions





U(c, Y, θ) = u(c) − θh(Y )


given T (Y )

v(θ) ≡ maxY

U(Y − T (Y ), Y, θ)

Resource feasible∫

(

Y (θ) − c(θ))

dF (θ) ≥ G

Introduction


Conditions

Main Results

Applications

Conclusions





U(c, Y, θ) = u(c) − θh(Y )


given T (Y )

v′(θ) = Uθ(Y (θ) − T (Y (θ)), Y (θ), θ)


(

Y (θ) − c(θ))

dF (θ) ≥ G

Introduction


Conditions

Main Results

Applications

Conclusions





U(c, Y, θ) = u(c) − θh(Y )


given T (Y )

v′(θ) = −h(Y (θ))


(

Y (θ) − c(θ))

dF (θ) ≥ G

Introduction


Conditions

Main Results

Applications

Conclusions





U(c, Y, θ) = u(c) − θh(Y )


given T (Y )

v′(θ) = −h(Y (θ))


(

Y (θ) − e(v(θ), Y (θ), θ))

dF (θ) ≥ G

Introduction


Conditions

Main Results

Applications

Conclusions


Planning Problem

Dual Pareto Problem

maximize net resources

subject to,v(θ) ≥ v(θ)

incentives

Introduction


Conditions

Main Results

Applications

Conclusions


Planning Problem

Dual Pareto Problem

maxY ,v

∫

(

Y (θ) − e(v(θ), Y (θ), θ))

dF (θ)


incentives

Introduction


Conditions

Main Results

Applications

Conclusions


Planning Problem

Dual Pareto Problem

maxY ,v

∫

(

Y (θ) − e(v(θ), Y (θ), θ))

dF (θ)


v′(θ) = −h(Y (θ))

Introduction


Conditions

Main Results

Applications

Conclusions


Planning Problem

Dual Pareto Problem

maxY ,v

∫

(

Y (θ) − e(v(θ), Y (θ), θ))

dF (θ)


v′(θ) = −h(Y (θ))

Y (θ) nonincreasing

Introduction


Conditions

Main Results

Applications

Conclusions


Efficiency Conditions

Lagrangian

L =

∫

(

Y (θ)−e(v(θ), Y (θ), θ))

dF (θ)

−

∫

(

v′(θ) + h(Y (θ)))

µ(θ) dθ

Introduction


Conditions

Main Results

Applications

Conclusions



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


Conditions

Main Results

Applications

Conclusions




L =

∫

(

Y (θ)−e(v(θ), Y (θ), θ))


+

∫


∫

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


Conditions

Main Results

Applications

Conclusions




L =

∫

(

Y (θ)−e(v(θ), Y (θ), θ))


+

∫


∫

h(Y (θ))µ(θ) dθ

First-order conditions

τ(θ)f(θ) = µ(θ)h′(Y (θ)) [Y (θ)]

Introduction


Conditions

Main Results

Applications

Conclusions




L =

∫

(

Y (θ)−e(v(θ), Y (θ), θ))


+

∫


∫

h(Y (θ))µ(θ) dθ


µ(θ) = τ(θ)f(θ)

h′(Y (θ))[Y (θ)]

Introduction


Conditions

Main Results

Applications

Conclusions




L =

∫

(

Y (θ)−e(v(θ), Y (θ), θ))


+

∫


∫

h(Y (θ))µ(θ) dθ



h′(Y (θ))[Y (θ)]

µ′(θ) ≤ ev

(

v(θ), Y (θ), θ)

f(θ) [v(θ)]

Introduction


Conditions

Main Results

Applications

Conclusions




L =

∫

(

Y (θ)−e(v(θ), Y (θ), θ))


+

∫


∫

h(Y (θ))µ(θ) dθ



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



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



Inefficiencies

Applications

Conclusions




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



Inefficiencies

Applications

Conclusions




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



Inefficiencies

Applications

Conclusions


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



Inefficiencies

Applications

Conclusions


Simple Tax Reform

Y

T

Introduction

Model



Inefficiencies

Applications

Conclusions


Simple Tax Reform

Y

T

g′(Y )g(Y )

small (f ′(θ)f(θ)

large) inefficiency

Introduction

Model



Inefficiencies

Applications

Conclusions


Laffer

lower taxes increase revenue

Pareto improvements “Laffer” effect

Proposition. T1(Y ) ≻ T0(Y ) T1(Y ) ≤ T0(Y )

Introduction

Model



Inefficiencies

Applications

Conclusions


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



Inefficiencies

Applications

Conclusions


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)




without empirical knowledgeanything goes

Introduction

Model



Inefficiencies

Applications

Conclusions


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)




without empirical knowledgeanything goes

need information on f(θ) to restrict T (Y )

Introduction

Model



Inefficiencies

Applications

Conclusions


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



Inefficiencies

Applications

Conclusions


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



Inefficiencies

Applications

Conclusions


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



Inefficiencies

Applications

Conclusions


Empirical Implementation

needed1. current tax function T (Y )

2. distribution of income g(Y )

3. utility function U(c, Y, θ)

Introduction

Model



Inefficiencies

Applications

Conclusions






in principle: #1 and #2 easy#3 usual deal

Introduction

Model



Inefficiencies

Applications

Conclusions







Diamond (1998) and Saez (2001)

Introduction

Model



Inefficiencies

Applications

Conclusions







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



Inefficiencies

Applications

Conclusions


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



Inefficiencies

Applications

Conclusions


Output Density



Y i = 1n

∑

Y it


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



Inefficiencies

Applications

Conclusions


Output Density



Y i = 1n

∑

Y it


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



Inefficiencies

Applications

Conclusions


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


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


Conclusions


Top Tax Rate

u(c) = c1−σ/(1 − σ) and h(Y ) = αY η


τ ≡ limθ→0


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


Conclusions


Top Tax Rate

u(c) = c1−σ/(1 − σ) and h(Y ) = αY η


τ ≡ limθ→0


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


Conclusions


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


Conclusions


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


Conclusions


Progressivity

Quasi-linear u(c) = c

result: can always increase progressivity

Introduction

Model

Main Results


Conclusions


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

Introduction

Model

Main Results

Applications

Conclusions❖ Conclusions


Conclusions

Pareto efficiency simple condition

generalizes zero-tax-at-the-top result

Pareto inefficient Laffer effects

flat taxes may be optimal...

...more progressivity always efficient

pareto efﬁcient income taxation - mitweb.mit.edu/iwerning/public/slidesnber-pe.pdf ·...

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