Pareto Efficient Income Taxation

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					                Pareto Efficient Income Taxation
                                        Iván Werning
                                               MIT



                                            April 2007
                                  NBER Public Economics meeting




Pareto Efficient Income Taxation                                   - p. 1
                                     Introduction
Introduction
y Introduction
y Motivation
                       Q:         Good shape for tax schedule   ?
y Contribution
y Results

Model

Main Results

Applications

Conclusions




Pareto Efficient Income Taxation                                     - p. 2
                                          Introduction
Introduction
y Introduction
y Motivation
                       Q:           Good shape for tax schedule             ?
y Contribution
y Results

Model
                           Mirrlees (1971), Diamond (1998), Saez (2001)
Main Results                      positive: redistribution vs. efficiency
Applications
                                  normative: Utilitarian social welfare function
Conclusions




Pareto Efficient Income Taxation                                                    - p. 2
                                          Introduction
Introduction
y Introduction
y Motivation
                       Q:           Good shape for tax schedule             ?
y Contribution
y Results

Model
                           Mirrlees (1971), Diamond (1998), Saez (2001)
Main Results                      positive: redistribution vs. efficiency
Applications
                                  normative: Utilitarian social welfare function
Conclusions



                           this paper: Pareto efficient taxation
                                  positive: redistribution vs. efficiency

                                  normative: Utilitarian social welfare function
                                             Pareto Efficiency



Pareto Efficient Income Taxation                                                    - p. 2
               Old Motivation: “New New New...”
Introduction
y Introduction             Why not Utilitarian? (         i U i)
y Motivation
y Contribution
y Results
                                  practical: cardinality U i → W (U i ) (or even W i (U i ))
                                  ... which Utilitarian?
Model

Main Results
                                  conceptual: political process:
                                  social classes        Coasian bargain
Applications
                                  ...but max U i ?
Conclusions
                                  philosophical: other notions of fairness and social
                                  justice




Pareto Efficient Income Taxation                                                                - p. 3
               Old Motivation: “New New New...”
Introduction
y Introduction             Why not Utilitarian? (         i U i)
y Motivation
y Contribution
y Results
                                  practical: cardinality U i → W (U i ) (or even W i (U i ))
                                  ... which Utilitarian?
Model

Main Results
                                  conceptual: political process:
                                  social classes        Coasian bargain
Applications
                                  ...but max U i ?
Conclusions
                                  philosophical: other notions of fairness and social
                                  justice

                           Pareto efficiency              weaker criterion




Pareto Efficient Income Taxation                                                                - p. 3
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results

Applications

Conclusions




                                                              first best


                                                     constrained

                                                                    vH

Pareto Efficient Income Taxation                                            - p. 4
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results

Applications

Conclusions




                                                     vH

Pareto Efficient Income Taxation                           - p. 4
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results

Applications

Conclusions




                                                     vH

Pareto Efficient Income Taxation                           - p. 4
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results

Applications

Conclusions




                                                     vH

Pareto Efficient Income Taxation                           - p. 4
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results

Applications

Conclusions




                                                     vH

Pareto Efficient Income Taxation                           - p. 4
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results

Applications

Conclusions




                                                     VH+ VL


                                                        vH

Pareto Efficient Income Taxation                               - p. 4
                                   Pareto Frontier
Introduction
y Introduction
y Motivation                  vL
y Contribution
y Results

Model

Main Results                                         VH+(1- ) VL
Applications

Conclusions




                                                         VH+ VL


                                                            vH

Pareto Efficient Income Taxation                                    - p. 4
                                     Contribution
Introduction               invert Mirrlees model...
y Introduction
y Motivation
y Contribution             ...express in tractable way
y Results

Model                      ...use it: some applications
Main Results

Applications

Conclusions




Pareto Efficient Income Taxation                           - p. 5
                                               Results
Introduction         #0 restrictions generalize “zero-tax-at-the-top”
y Introduction
y Motivation         #1 Any T (Y ). . .
y Contribution
y Results                 efficient for many f (θ)
Model
                               inefficient for many f (θ)
Main Results               . . . anything goes
Applications
                     #2 Given T0 (Y )      g(Y )       f (θ) (Saez, 2001)
Conclusions
                          efficient set of T (Y ): large
                                  inefficient set of T (Y ): large
                     #3 Simple test for efficiency of T0 (Y )




Pareto Efficient Income Taxation                                             - p. 6
                                              Results
Introduction
y Introduction
                     #4 Simple formulas...
y Motivation
y Contribution                    bound on top tax rate
y Results

Model
                                  efficiency of a flat tax
Main Results         #5 Increasing progressivity
Applications                 maintains Pareto efficiency
Conclusions
                     #6 observable heterogeneity
                            not conditioning can be efficient




Pareto Efficient Income Taxation                                - p. 7
                                          Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                           U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions




Pareto Efficient Income Taxation                                          - p. 8
                                            Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                            U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions


                           given T (Y )
                                          v(θ) ≡ max U (Y − T (Y ), Y, θ)
                                                  Y




Pareto Efficient Income Taxation                                             - p. 8
                                            Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                            U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions


                           given T (Y )
                                          v(θ) ≡ max U (Y − T (Y ), Y, θ)
                                                  Y


                           Government budget

                                                T (Y (θ)) dF (θ) ≥ G

Pareto Efficient Income Taxation                                             - p. 8
                                            Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                            U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions


                           given T (Y )
                                          v(θ) ≡ max U (Y − T (Y ), Y, θ)
                                                  Y


                           Resource feasible

                                               Y (θ) − c(θ) dF (θ) ≥ G

Pareto Efficient Income Taxation                                             - p. 8
                                           Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                            U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions


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

                           Resource feasible

                                               Y (θ) − c(θ) dF (θ) ≥ G

Pareto Efficient Income Taxation                                                    - p. 8
                                          Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                           U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions


                           given T (Y )
                                                v ′ (θ) = −h(Y (θ))


                           Resource feasible

                                               Y (θ) − c(θ) dF (θ) ≥ G

Pareto Efficient Income Taxation                                          - p. 8
                                            Setup
Introduction           Positive side of Mirrlees (1971)
Model
y Setup
                           continuum of types θ ∼ F (θ)
y Planning Problem
y Efficiency
  Conditions               additive preferences
Main Results
                                            U (c, Y, θ) = u(c) − θh(Y )
Applications
                           (e.g. Y = w · n and h(n) = αnη )
Conclusions


                           given T (Y )
                                                 v ′ (θ) = −h(Y (θ))


                           Resource feasible

                                          Y (θ) − e(v(θ), Y (θ), θ) dF (θ) ≥ G

Pareto Efficient Income Taxation                                                  - p. 8
                                  Planning Problem
Introduction
                       Dual Pareto Problem
Model
y Setup
y Planning Problem
y Efficiency                                maximize net resources
  Conditions

Main Results
                             subject to,
Applications
                                                ˜
                                                v (θ) ≥ v(θ)
Conclusions

                                                 incentives




Pareto Efficient Income Taxation                                     - p. 9
                                  Planning Problem
Introduction
                       Dual Pareto Problem
Model
y Setup
y Planning Problem
y Efficiency                          max     ˜         v     ˜
                                             Y (θ) − e(˜(θ), Y (θ), θ) dF (θ)
  Conditions                          ˜ v
                                      Y ,˜
Main Results
                             subject to,
Applications
                                                  ˜
                                                  v (θ) ≥ v(θ)
Conclusions

                                                  incentives




Pareto Efficient Income Taxation                                                 - p. 9
                                  Planning Problem
Introduction
                       Dual Pareto Problem
Model
y Setup
y Planning Problem
y Efficiency                          max     ˜         v     ˜
                                             Y (θ) − e(˜(θ), Y (θ), θ) dF (θ)
  Conditions                          ˜ v
                                      Y ,˜
Main Results
                             subject to,
Applications
                                                  ˜
                                                  v (θ) ≥ v(θ)
Conclusions
                                                            ˜
                                               v ′ (θ) = −h(Y (θ))
                                               ˜




Pareto Efficient Income Taxation                                                 - p. 9
                                  Planning Problem
Introduction
                       Dual Pareto Problem
Model
y Setup
y Planning Problem
y Efficiency                          max     ˜         v     ˜
                                             Y (θ) − e(˜(θ), Y (θ), θ) dF (θ)
  Conditions                          ˜ v
                                      Y ,˜
Main Results
                             subject to,
Applications
                                                  ˜
                                                  v (θ) ≥ v(θ)
Conclusions
                                                           ˜
                                              v ′ (θ) = −h(Y (θ))
                                              ˜
                                             ˜
                                             Y (θ) nonincreasing




Pareto Efficient Income Taxation                                                 - p. 9
                                  Efficiency Conditions
Introduction
                       Lagrangian
Model
y Setup
y Planning Problem
                          L=        ˜       v     ˜
                                    Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)
y Efficiency
  Conditions

Main Results
                                                      −                 ˜
                                                            v ′ (θ) + h(Y (θ)) µ(θ) dθ
                                                            ˜
Applications

Conclusions




Pareto Efficient Income Taxation                                                    - p. 10
                                  Efficiency Conditions
Introduction
                       Lagrangian (integrating by parts)
Model
y Setup
y Planning Problem
y Efficiency
                          L=        ˜       v     ˜                v ¯ ¯
                                    Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)−˜(θ)µ(θ) + µ(θ)˜(θ)
                                                                                  v
  Conditions

Main Results
                                               +    v (θ)µ′ (θ)dθ −
                                                    ˜                   ˜
                                                                      h(Y (θ))µ(θ) dθ
Applications

Conclusions




Pareto Efficient Income Taxation                                                   - p. 10
                                  Efficiency Conditions
Introduction
                       Lagrangian (integrating by parts)
Model
y Setup
y Planning Problem
y Efficiency
                          L=          ˜       v     ˜                v ¯ ¯
                                      Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)−˜(θ)µ(θ) + µ(θ)˜(θ)
                                                                                    v
  Conditions

Main Results
                                                   +    v (θ)µ′ (θ)dθ −
                                                        ˜                    ˜
                                                                           h(Y (θ))µ(θ) dθ
Applications
                       First-order conditions
Conclusions


                                   1 − eY (v(θ), Y (θ), θ) f (θ) = µ(θ)h′ (Y (θ))    [Y (θ)]




Pareto Efficient Income Taxation                                                        - p. 10
                                  Efficiency Conditions
Introduction
                       Lagrangian (integrating by parts)
Model
y Setup
y Planning Problem
y Efficiency
                          L=        ˜       v     ˜                v ¯ ¯
                                    Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)−˜(θ)µ(θ) + µ(θ)˜(θ)
                                                                                  v
  Conditions

Main Results
                                               +    v (θ)µ′ (θ)dθ −
                                                    ˜                      ˜
                                                                         h(Y (θ))µ(θ) dθ
Applications
                       First-order conditions
Conclusions


                                           τ (θ)f (θ) = µ(θ)h′ (Y (θ))             [Y (θ)]




Pareto Efficient Income Taxation                                                      - p. 10
                                  Efficiency Conditions
Introduction
                       Lagrangian (integrating by parts)
Model
y Setup
y Planning Problem
y Efficiency
                          L=        ˜       v     ˜                v ¯ ¯
                                    Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)−˜(θ)µ(θ) + µ(θ)˜(θ)
                                                                                  v
  Conditions

Main Results
                                               +    v (θ)µ′ (θ)dθ −
                                                    ˜                   ˜
                                                                      h(Y (θ))µ(θ) dθ
Applications
                       First-order conditions
Conclusions

                                                            f (θ)
                                             µ(θ) = τ (θ) ′                     [Y (θ)]
                                                         h (Y (θ))




Pareto Efficient Income Taxation                                                   - p. 10
                                  Efficiency Conditions
Introduction
                       Lagrangian (integrating by parts)
Model
y Setup
y Planning Problem
y Efficiency
                          L=        ˜       v     ˜                v ¯ ¯
                                    Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)−˜(θ)µ(θ) + µ(θ)˜(θ)
                                                                                  v
  Conditions

Main Results
                                               +     v (θ)µ′ (θ)dθ −
                                                     ˜                    ˜
                                                                        h(Y (θ))µ(θ) dθ
Applications
                       First-order conditions
Conclusions

                                                            f (θ)
                                             µ(θ) = τ (θ) ′                       [Y (θ)]
                                                         h (Y (θ))

                                         µ′ (θ) ≤ ev v(θ), Y (θ), θ f (θ)          [v(θ)]




Pareto Efficient Income Taxation                                                     - p. 10
                                  Efficiency Conditions
Introduction
                       Lagrangian (integrating by parts)
Model
y Setup
y Planning Problem
y Efficiency
                          L=        ˜       v     ˜                v ¯ ¯
                                    Y (θ)−e(˜(θ), Y (θ), θ) dF (θ)−˜(θ)µ(θ) + µ(θ)˜(θ)
                                                                                  v
  Conditions

Main Results
                                                +    v (θ)µ′ (θ)dθ −
                                                     ˜                    ˜
                                                                        h(Y (θ))µ(θ) dθ
Applications
                       First-order conditions
Conclusions

                                                             f (θ)
                                              µ(θ) = τ (θ) ′                      [Y (θ)]
                                                          h (Y (θ))

                                         µ′ (θ) ≤ ev v(θ), Y (θ), θ f (θ)          [v(θ)]

                                     τ ′ (θ) d log f (θ) d log h′ (Y (θ))
                             τ (θ) θ        +           −                 ≤ 1 − τ (θ)
                                     τ (θ)     d log θ       d log θ

Pareto Efficient Income Taxation                                                     - p. 10
                                  Efficiency Conditions
Introduction

Model                     Proposition. T (Y ) is Pareto efficient if and only
Main Results
y Intuition
y Anything Goes                      τ ′ (θ) d log f (θ) d log h′ (Y (θ))
y Identification and          τ (θ) θ        +           −                 ≤ 1 − τ (θ)
  Test                               τ (θ)     d log θ       d log θ
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies                               ¯
                                           τ (θ) ≥ 0 and τ (θ) ≤ 0.
Applications

Conclusions




Pareto Efficient Income Taxation                                                    - p. 11
                                  Efficiency Conditions
Introduction

Model                     Proposition. T (Y ) is Pareto efficient if and only
Main Results
y Intuition
y Anything Goes                      τ ′ (θ) d log f (θ) d log h′ (Y (θ))
y Identification and          τ (θ) θ        +           −                 ≤ 1 − τ (θ)
  Test                               τ (θ)     d log θ       d log θ
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies                               ¯
                                           τ (θ) ≥ 0 and τ (θ) ≤ 0.
Applications

Conclusions

                           note: “zero-tax-at-top”       special case




Pareto Efficient Income Taxation                                                    - p. 11
                                  Efficiency Conditions
Introduction

Model                     Proposition. T (Y ) is Pareto efficient if and only
Main Results
y Intuition
y Anything Goes                      τ ′ (θ) d log f (θ) d log h′ (Y (θ))
y Identification and          τ (θ) θ        +           −                 ≤ 1 − τ (θ)
  Test                               τ (θ)     d log θ       d log θ
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies                                  ¯
                                              τ (θ) ≥ 0 and τ (θ) ≤ 0.
Applications

Conclusions

                           note: “zero-tax-at-top”                 special case

                           more general condition:
                                                      ¯
                                                      θ
                                  τ (θ)f (θ)                  1        ˜ ˜
                                              +                     f (θ) dθ is nonincreasing
                                  h ′ (Y (θ))
                                                  θ
                                                                ˜
                                                          u′ (c(θ))
Pareto Efficient Income Taxation                                                                 - p. 11
                                            Intuition
Introduction               define
                                                         ˆ             ˆ
                                                  T (Y (θ)) − ε Y = Y (θ)
Model
                                      ˆ
                                      T (Y ) ≡
Main Results                                      T (Y )               ˆ
                                                                Y = Y (θ)
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test                      ˆ
                         Proposition. T ≻ T
y Empirical Strategy
y Quantifying
  Inefficiencies                 τ ′ (θ)    d log f (θ) d log h′ (Y (θ))
                        τ (θ) θ         +2            −                   ≤ 3(1 − τ (θ))
Applications                    τ (θ)        d log θ       d log θ
Conclusions
                                        ˆ
                         is violated at θ




Pareto Efficient Income Taxation                                                      - p. 12
                                  Simple Tax Reform
Introduction              T
Model

Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




                                                      Y

Pareto Efficient Income Taxation                           - p. 13
                                  Simple Tax Reform
Introduction              T
Model

Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




                                                      Y

Pareto Efficient Income Taxation                           - p. 13
                                  Simple Tax Reform
Introduction              T
Model

Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




                                                      Y

Pareto Efficient Income Taxation                           - p. 13
                                  Simple Tax Reform
Introduction              T
Model

Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




                                                      Y

Pareto Efficient Income Taxation                           - p. 13
                                  Simple Tax Reform
Introduction              T
Model

Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




                                                      Y

Pareto Efficient Income Taxation                           - p. 13
                                  Simple Tax Reform
Introduction              T
Model

Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




                                   g ′ (Y )            f ′ (θ)
                                                                                        Y
                                   g(Y )
                                              small   ( f (θ)    large)   inefficiency
Pareto Efficient Income Taxation                                                             - p. 13
                                             Laffer
Introduction               lower taxes        increase revenue
Model
                           Pareto improvements            “Laffer” effect
Main Results
y Intuition
y Anything Goes
y Identification and
  Test
y Graphical Test              Proposition. T1 (Y ) ≻ T0 (Y )     T1 (Y ) ≤ T0 (Y )
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




Pareto Efficient Income Taxation                                                      - p. 14
                                      Anything Goes
Introduction

Model
                                    τ ′ (θ) d log f (θ) d log h′ (Y (θ))
Main Results                τ (θ) θ        +           −                 ≤ 1 − τ (θ)
y Intuition
y Anything Goes
                                    τ (θ)     d log θ       d log θ
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
                                  Proposition. For any T (Y )
  Inefficiencies

Applications
                                    exists set {f (θ)}    Pareto efficient
Conclusions                         exists set {f (θ)}    Pareto inefficient




Pareto Efficient Income Taxation                                                   - p. 15
                                      Anything Goes
Introduction

Model
                                    τ ′ (θ) d log f (θ) d log h′ (Y (θ))
Main Results                τ (θ) θ        +           −                 ≤ 1 − τ (θ)
y Intuition
y Anything Goes
                                    τ (θ)     d log θ       d log θ
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
                                  Proposition. For any T (Y )
  Inefficiencies

Applications
                                    exists set {f (θ)}    Pareto efficient
Conclusions                         exists set {f (θ)}    Pareto inefficient

                           without empirical knowledge
                               anything goes




Pareto Efficient Income Taxation                                                   - p. 15
                                      Anything Goes
Introduction

Model
                                    τ ′ (θ) d log f (θ) d log h′ (Y (θ))
Main Results                τ (θ) θ        +           −                 ≤ 1 − τ (θ)
y Intuition
y Anything Goes
                                    τ (θ)     d log θ       d log θ
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
                                  Proposition. For any T (Y )
  Inefficiencies

Applications
                                    exists set {f (θ)}    Pareto efficient
Conclusions                         exists set {f (θ)}    Pareto inefficient

                           without empirical knowledge
                               anything goes

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


Pareto Efficient Income Taxation                                                   - p. 15
                                  Identification and Test
Introduction               observe g(Y ) identify (Saez, 2001)
Model
                                                                   u′ (Y − T (Y ))
Main Results                                θ(Y ) = (1 − T ′ (Y ))
y Intuition                                                             h′ (Y )
y Anything Goes
y Identification and                                  g(Y )
  Test                                  f (θ(Y )) = ′
y Graphical Test                                    θ (Y )
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




Pareto Efficient Income Taxation                                                      - p. 16
                                  Identification and Test
Introduction               observe g(Y ) identify (Saez, 2001)
Model
                                                                   u′ (Y − T (Y ))
Main Results                                θ(Y ) = (1 − T ′ (Y ))
y Intuition                                                             h′ (Y )
y Anything Goes
y Identification and                                  g(Y )
  Test                                  f (θ(Y )) = ′
y Graphical Test                                    θ (Y )
y Empirical Strategy
y Quantifying
  Inefficiencies
                           efficiency test...
Applications

Conclusions                                     d log g(Y )
                                                            ≥ a(Y )
                                                  d log Y
                           ... for tax schedule in place




Pareto Efficient Income Taxation                                                      - p. 16
                                            Graphical Test
Introduction               define Rawlsian density:
Model                                                                       Y
                                                                    exp    0
                                                                                 a(z) dz
Main Results
y Intuition                                      α(Y ) =         ∞               Y
y Anything Goes
y Identification and                                             0
                                                                     exp        0
                                                                                     a(z) dz
  Test                     graphical test:
y Graphical Test
y Empirical Strategy
y Quantifying
                                                      g(Y )
  Inefficiencies
                                                                     nondecreasing
                                                      α(Y )
Applications                  0.3

Conclusions                 0.25

                              0.2

                            0.15

                              0.1

                            0.05

                                  0
                                      0.5   1   1.5    2      2.5     3    3.5        4    4.5   5   5.5   6
Pareto Efficient Income Taxation                                                                            - p. 17
                         Empirical Implementation
Introduction               needed
Model                      1. current tax function T (Y )
Main Results               2. distribution of income g(Y )
y Intuition
y Anything Goes            3. utility function U (c, Y, θ)
y Identification and
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies

Applications

Conclusions




Pareto Efficient Income Taxation                              - p. 18
                         Empirical Implementation
Introduction               needed
Model                      1. current tax function T (Y )
Main Results               2. distribution of income g(Y )
y Intuition
y Anything Goes            3. utility function U (c, Y, θ)
y Identification and
  Test
y Graphical Test
y Empirical Strategy       in principle: #1 and #2      easy
y Quantifying
  Inefficiencies
                           #3 usual deal
Applications

Conclusions




Pareto Efficient Income Taxation                                - p. 18
                         Empirical Implementation
Introduction               needed
Model                      1. current tax function T (Y )
Main Results               2. distribution of income g(Y )
y Intuition
y Anything Goes            3. utility function U (c, Y, θ)
y Identification and
  Test
y Graphical Test
y Empirical Strategy       in principle: #1 and #2      easy
y Quantifying
  Inefficiencies
                           #3 usual deal
Applications
                           Diamond (1998) and Saez (2001)
Conclusions




Pareto Efficient Income Taxation                                - p. 18
                         Empirical Implementation
Introduction               needed
Model                      1. current tax function T (Y )
Main Results               2. distribution of income g(Y )
y Intuition
y Anything Goes            3. utility function U (c, Y, θ)
y Identification and
  Test
y Graphical Test
y Empirical Strategy       in principle: #1 and #2      easy
y Quantifying
  Inefficiencies
                           #3 usual deal
Applications
                           Diamond (1998) and Saez (2001)
Conclusions

                           some challenges...
                           1. econometric: need to estimate g ′ (Y ) and g(Y )
                           2. conceptual: static model
                                  lifetime T (Y ) and g(Y ) (Fullerton and Rogers)


Pareto Efficient Income Taxation                                                  - p. 18
                                         Output Density
Introduction               IRS’s SOI Public Use Files for Individual tax returns
Model                        lifetime g(Y )?
Main Results
y Intuition                       lifetime T (Y ) schedule?
y Anything Goes
y Identification and                  1
  Test                     Yi =      n
                                          Yti
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies
                           smooth density estimate
Applications
                           assumed T (Y ) = .30 × Y
Conclusions




Pareto Efficient Income Taxation                                                    - p. 19
                                                         Output Density
Introduction                   IRS’s SOI Public Use Files for Individual tax returns
Model                            lifetime g(Y )?
Main Results
y Intuition                            lifetime T (Y ) schedule?
y Anything Goes
y Identification and                               1
  Test                         Yi =               n
                                                                 Yti
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies
                               smooth density estimate
Applications
                               assumed T (Y ) = .30 × Y
                                                                                                            Elasticity of Kernel Density (bandwidth = 10,000) of Average Income Over Varying Time Periods in the United States
                       Kernel Density 10−5
                                    x (bandwidth = 10,000) of Average Income Over Varying Time Periods in the United States 2
                                  2
Conclusions                                                                        >= 10 years during 1979−1990
                                                                                   1982−1986                                                                                     >= 10 years during 1979−1990
                                1.8                                                1987−1990                                                                                     1982−1986
                                                                                                                              0                                                  1987−1990
                                1.6

                                                                                                                            −2
                                1.4


                                1.2                                                                                         −4

                                  1
                                                                                                                            −6
                                0.8


                                0.6                                                                                         −8


                                0.4
                                                                                                                           −10
                                0.2

                                                                                                                           −12
                                  0                                                                                           0       2        4       6       8       10       12      14       16       18
                                   0             0.5             1             1.5             2             2.5                                    Average Income (in 1990 dollars)                       4
                                                         Average Income (in 1990 dollars)                     5                                                                                       x 10
                                                                                                          x 10




Pareto Efficient Income Taxation
                                Figure 1:                         Density of income Figure 2:                                                                     Implied elasticity 19
                                                                                                                                                                                  - p.
                                                     Output Density
Introduction               IRS’s SOI Public Use Files for Individual tax returns
Model                        lifetime g(Y )?
Main Results
y Intuition                        lifetime T (Y ) schedule?
y Anything Goes
y Identification and                            1
  Test                     Yi =                n
                                                            Yti
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies
                           smooth density estimate
Applications
                           assumed T (Y ) = .30 × Y
                                    −5
                             Rawlsian Test against 1987−1990 Average Income Data (sigma = 0, eta = 2, T = .3Y)         −5
                                                                                                                 Rawlsian Test against 1987−1990 Average Income Data (sigma = 1, eta = 3, T = .3Y)
                               x 10                                                                               x 10

Conclusions                   3

                                                                                                                 5

                            2.5

                                                                                                                 4

                              2


                                                                                                                 3
                            1.5


                                                                                                                 2
                              1



                                                                                                                 1
                            0.5




                              0                                                                                  0
                               0           2            4            6           8            10                  0      1      2      3      4       5      6      7      8       9     10
                                                                                                           4                                                                                     4
                                                                                                       x 10                                                                                   x 10




Pareto Efficient Income Taxation                                                                                                                                                                      - p. 19
                         Quantifying Inefficiencies
Introduction               efficiency test       qualitative
Model
                           quantitative. . .
Main Results
y Intuition
y Anything Goes
y Identification and           ∆≡      ˜
                                      Y ∗ (θ) − c∗ (θ) dF (θ) −
                                                ˜                  Y (θ) − c(θ) dF (θ)
  Test
y Graphical Test
y Empirical Strategy
y Quantifying
  Inefficiencies            does not count welfare improvements
Applications
                                                    ˜
                                                    v (θ) > v(θ)
Conclusions




Pareto Efficient Income Taxation                                                    - p. 20
                                    Top Tax Rate
Introduction               u(c) = c1−σ /(1 − σ) and h(Y ) = αY η
Model
                           suppose top tax rate
Main Results

Applications                              τ ≡ lim τ (θ) = lim T ′ (Y )
                                          ¯
y Top Tax Rate                                 θ→0        Y →∞
y Flat Tax
y Progressivity
y Heterogeneity
                           exists
Conclusions




Pareto Efficient Income Taxation                                          - p. 21
                                     Top Tax Rate
Introduction               u(c) = c1−σ /(1 − σ) and h(Y ) = αY η
Model
                           suppose top tax rate
Main Results

Applications                               τ ≡ lim τ (θ) = lim T ′ (Y )
                                           ¯
y Top Tax Rate                                   θ→0       Y →∞
y Flat Tax
y Progressivity
y Heterogeneity
                           exists
Conclusions                efficiency condition         bound
                                                     σ+η−1
                                                  ¯
                                                  τ≤       .
                                                     ϕ+η−2
                           where ϕ = − limT →∞ d log g(Y )/d log Y .




Pareto Efficient Income Taxation                                           - p. 21
                                     Top Tax Rate
Introduction               u(c) = c1−σ /(1 − σ) and h(Y ) = αY η
Model
                           suppose top tax rate
Main Results

Applications                               τ ≡ lim τ (θ) = lim T ′ (Y )
                                           ¯
y Top Tax Rate                                   θ→0       Y →∞
y Flat Tax
y Progressivity
y Heterogeneity
                           exists
Conclusions                efficiency condition         bound
                                                     σ+η−1
                                                  ¯
                                                  τ≤       .
                                                     ϕ+η−2
                           where ϕ = − limT →∞ d log g(Y )/d log Y .

                           Saez (2001): ϕ = 3



Pareto Efficient Income Taxation                                           - p. 21
                                                       Top Tax Rate
                                              1
Introduction

Model                                       0.9

Main Results                                0.8

Applications
y Top Tax Rate                              0.7
y Flat Tax
                         upper bound on τ




y Progressivity
                                            0.6
y Heterogeneity

Conclusions                                 0.5


                                            0.4


                                            0.3


                                            0.2


                                            0.1


                                              0
                                               0   1    2   3   4       5      6   7   8   9     10
                                                                elasticity 1/η+1


Pareto Efficient Income Taxation                                                                - p. 22
                                        Flat Tax
Introduction               linear tax   necessary condition
Model
                                                   σ+η−1
                                          ¯
                                          τ≤
Main Results                                       log g(Y
                                               − d d log Y ) + η − 2
Applications
y Top Tax Rate
y Flat Tax
y Progressivity            linear tax   sufficient condition
y Heterogeneity
                                                      η−1
Conclusions                               ¯
                                          τ ≤ d log g(Y )
                                               − d log Y + η − 1




Pareto Efficient Income Taxation                                        - p. 23
                                    Progressivity
Introduction               Quasi-linear u(c) = c
Model
                           result: can always increase progressivity
Main Results

Applications
y Top Tax Rate
y Flat Tax
y Progressivity
y Heterogeneity

Conclusions




Pareto Efficient Income Taxation                                        - p. 24
                                         Heterogeneity
Introduction               groups = 1, . . . , N
                                               f i (θ)            U i (c, Y, θ)
Model
                                                         and
Main Results

Applications
y Top Tax Rate
                           unobservable i
y Flat Tax
y Progressivity
                           single T (Y )
y Heterogeneity                average efficiency condition
Conclusions
                           observable i
                           multiple T i (Y )
                               N efficiency conditions

                           observation:
                             T i (Y ) = T (Y ) may be Pareto efficient
                                  never optimal for Utilitarian

Pareto Efficient Income Taxation                                                   - p. 25
                                     Conclusions
Introduction               Pareto efficiency      simple condition
Model
                           generalizes zero-tax-at-the-top result
Main Results

Applications               Pareto inefficient      Laffer effects
Conclusions
y Conclusions              flat taxes may be optimal...

                           ...more progressivity always efficient




Pareto Efficient Income Taxation                                     - p. 26