Pareto Efficient Income Taxation

Pareto Efficient Income Taxation Iván Werning MIT April 2007 NBER Public Economics meeting Pareto Efficient Income Taxation - p. 1

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

Motivation Contribution Results 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 function Pareto Efficiency Pareto Efficient Income Taxation - p. 2

Old Motivation: New New New... Motivation Contribution Results 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 social justice Pareto Efficient Income Taxation - p. 3

Pareto Frontier Motivation Contribution Results v L first best constrained v H Pareto Efficient Income Taxation - p. 4

Pareto Frontier Motivation Contribution Results v L v H Pareto Efficient Income Taxation - p. 4

Pareto Frontier Motivation Contribution Results v L V H + V L v H Pareto Efficient Income Taxation - p. 4

Pareto Frontier Motivation Contribution Results v L V H +(1- ) V L V H + V L v H Pareto Efficient Income Taxation - p. 4

Contribution Motivation Contribution Results invert Mirrlees model......express in tractable way...use it: some applications Pareto Efficient Income Taxation - p. 5

Results Motivation Contribution 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 T 0 (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 T 0 (Y ) Pareto Efficient Income Taxation - p. 6

Results Motivation Contribution Results #4 Simple formulas... bound on top tax rate efficiency of a flat tax #5 Increasing progressivity maintains Pareto efficiency #6 observable heterogeneity not conditioning can be efficient Pareto Efficient Income Taxation - p. 7

Setup Setup Planning Problem Efficiency Conditions Positive 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) = αn η ) Pareto Efficient Income Taxation - p. 8

Planning Problem Setup Planning Problem Efficiency Conditions Dual Pareto Problem maximize net resources subject to, ṽ(θ) v(θ) incentives Pareto Efficient Income Taxation - p. 9

Planning Problem Dual Pareto Problem Setup Planning Problem Efficiency Conditions max Ỹ,ṽ subject to, (Ỹ (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ(θ) v(θ) incentives Pareto Efficient Income Taxation - p. 9

Planning Problem Dual Pareto Problem Setup Planning Problem Efficiency Conditions max Ỹ,ṽ subject to, (Ỹ (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ(θ) v(θ) ṽ (θ) = h(ỹ (θ)) Pareto Efficient Income Taxation - p. 9

Planning Problem Dual Pareto Problem Setup Planning Problem Efficiency Conditions max Ỹ,ṽ subject to, (Ỹ (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ(θ) v(θ) ṽ (θ) = h(ỹ (θ)) Ỹ (θ) nonincreasing Pareto Efficient Income Taxation - p. 9

Efficiency Conditions Setup Planning Problem Efficiency Conditions Lagrangian (Ỹ L = (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ( ) ṽ (θ) + h(ỹ (θ)) µ(θ) dθ Pareto Efficient Income Taxation - p. 10

Efficiency Conditions Setup Planning Problem Efficiency Conditions Lagrangian (integrating by parts) (Ỹ L = (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ( θ)µ( θ) + µ(θ)ṽ(θ) + ṽ(θ)µ (θ)dθ h(ỹ (θ))µ(θ)dθ First-order conditions ( 1 ey (v(θ),y (θ),θ) ) f(θ) = µ(θ)h (Y (θ)) [Y (θ)] Pareto Efficient Income Taxation - p. 10

Efficiency Conditions Setup Planning Problem Efficiency Conditions Lagrangian (integrating by parts) (Ỹ L = (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ( θ)µ( θ) + µ(θ)ṽ(θ) + ṽ(θ)µ (θ)dθ h(ỹ (θ))µ(θ)dθ First-order conditions τ(θ)f(θ) = µ(θ)h (Y (θ)) [Y (θ)] Pareto Efficient Income Taxation - p. 10

Efficiency Conditions Setup Planning Problem Efficiency Conditions Lagrangian (integrating by parts) (Ỹ L = (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ( θ)µ( θ) + µ(θ)ṽ(θ) + ṽ(θ)µ (θ)dθ h(ỹ (θ))µ(θ)dθ First-order conditions f(θ) µ(θ) = τ(θ) h (Y (θ)) [Y (θ)] Pareto Efficient Income Taxation - p. 10

Efficiency Conditions Setup Planning Problem Efficiency Conditions Lagrangian (integrating by parts) (Ỹ L = (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ( θ)µ( θ) + µ(θ)ṽ(θ) + ṽ(θ)µ (θ)dθ h(ỹ (θ))µ(θ)dθ First-order conditions f(θ) µ(θ) = τ(θ) h (Y (θ)) [Y (θ)] µ (θ) e v ( v(θ),y (θ),θ ) f(θ) [v(θ)] Pareto Efficient Income Taxation - p. 10

Efficiency Conditions Setup Planning Problem Efficiency Conditions Lagrangian (integrating by parts) (Ỹ L = (θ) e(ṽ(θ), Ỹ (θ),θ) ) df(θ) ṽ( θ)µ( θ) + µ(θ)ṽ(θ) + ṽ(θ)µ (θ)dθ h(ỹ (θ))µ(θ)dθ First-order conditions f(θ) µ(θ) = τ(θ) h (Y (θ)) [Y (θ)] µ (θ) e v ( v(θ),y (θ),θ ) f(θ) [v(θ)] τ(θ) ( θ τ (θ) τ(θ) + d log f(θ) d log θ d log ) h (Y (θ)) 1 τ(θ) d log θ Pareto Efficient Income Taxation - p. 10

Efficiency Conditions Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies Proposition. T(Y ) is Pareto efficient if and only τ(θ) ( θ τ (θ) τ(θ) + d log f(θ) d log θ d log ) h (Y (θ)) 1 τ(θ) d log θ τ( θ) 0 and τ(θ) 0. Pareto Efficient Income Taxation - p. 11

Intuition Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies define ˆT(Y ) Proposition. ˆT T ( τ(θ) θ τ (θ) log f(θ) + 2d τ(θ) d log θ is violated at ˆθ { T(Y (ˆθ)) ε T(Y ) Y = Y (ˆθ) Y Y (ˆθ) d log ) h (Y (θ)) d log θ 3(1 τ(θ)) Pareto Efficient Income Taxation - p. 12

Simple Tax Reform T Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies Y Pareto Efficient Income Taxation - p. 13

Simple Tax Reform T Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies g (Y ) g(y ) small (f (θ) f(θ) large) inefficiency Y Pareto Efficient Income Taxation - p. 13

Laffer lower taxes increase revenue Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies Pareto improvements Laffer effect Proposition. T 1 (Y ) T 0 (Y ) T 1 (Y ) T 0 (Y ) Pareto Efficient Income Taxation - p. 14

Anything Goes Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies τ(θ) ( θ τ (θ) τ(θ) + d log f(θ) d log θ Proposition. For any T(Y ) exists set {f(θ)} exists set {f(θ)} d log ) h (Y (θ)) 1 τ(θ) d log θ Pareto efficient Pareto inefficient Pareto Efficient Income Taxation - p. 15

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

Identification and Test Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies observe g(y ) identify (Saez, 2001) efficiency test... θ(y ) = (1 T (Y )) u (Y T(Y )) h (Y ) f(θ(y )) = g(y ) θ (Y ) d log g(y ) d log Y... for tax schedule in place a(y ) Pareto Efficient Income Taxation - p. 16

Graphical Test Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies define Rawlsian density: ( ) Y exp a(z)dz 0 α(y ) = ( ) Y exp a(z)dz 0 0 graphical test: 0.3 0.25 g(y ) α(y ) nondecreasing 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 Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies needed 1. current tax function T(Y ) 2. distribution of income g(y ) 3. utility function U(c,Y,θ) in principle: #1 and #2 #3 usual deal easy Diamond (1998) and Saez (2001) Pareto Efficient Income Taxation - p. 18

Output Density Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies IRS s SOI Public Use Files for Individual tax returns lifetime g(y )? lifetime T(Y ) schedule? Y i = 1 n Y i t smooth density estimate assumed T(Y ) =.30 Y 2 x Elasticity of Kernel Density (bandwidth = 10,000) of Average Income Over Varying Time Periods in the United States Kernel Density 10 5 (bandwidth = 10,000) of Average Income Over Varying Time Periods in the United States 2 >= 10 years during 1979 1990 1982 1986 >= 10 years during 1979 1990 1.8 1987 1990 1982 1986 0 1987 1990 1.6 1.4 2 1.2 1 0.8 4 6 0.6 8 0.4 0.2 0 0 0.5 1 1.5 2 2.5 Average Income (in 1990 dollars) x 10 5 10 12 0 2 4 6 8 10 12 14 16 18 Average Income (in 1990 dollars) x 10 4 Figure 1: Density of income Figure 2: Implied elasticity Pareto Efficient Income Taxation - p. 19

Quantifying Inefficiencies efficiency test qualitative Intuition Anything Goes Identification and Test Graphical Test Empirical Strategy Quantifying Inefficiencies quantitative... (Ỹ (θ) c (θ) ) (Y ) df(θ) (θ) c(θ) df(θ) does not count welfare improvements ṽ(θ) > v(θ) Pareto Efficient Income Taxation - p. 20

Top Tax Rate Top Tax Rate Flat Tax Progressivity Heterogeneity u(c) = c 1 σ /(1 σ) and h(y ) = αy η suppose top tax rate exists τ lim θ 0 τ(θ) = lim Y T (Y ) Pareto Efficient Income Taxation - p. 21

Top Tax Rate Top Tax Rate Flat Tax Progressivity Heterogeneity u(c) = c 1 σ /(1 σ) and h(y ) = αy η suppose top tax rate exists efficiency condition τ lim θ 0 τ(θ) = lim Y T (Y ) bound τ σ + η 1 ϕ + η 2. where ϕ = lim T d log g(y )/d log Y. Pareto Efficient Income Taxation - p. 21

Top Tax Rate Top Tax Rate Flat Tax Progressivity Heterogeneity upper bound on τ 1 0.9 0.8 0.7 0.6 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 Top Tax Rate Flat Tax Progressivity Heterogeneity linear tax linear tax necessary condition τ σ + η 1 dlog g(y ) + η 2 d log Y sufficient condition η 1 τ dlog g(y ) + η 1 d log Y Pareto Efficient Income Taxation - p. 23

Progressivity Quasi-linear u(c) = c result: can always increase progressivity Top Tax Rate Flat Tax Progressivity Heterogeneity Pareto Efficient Income Taxation - p. 24

Heterogeneity Top Tax Rate Flat Tax Progressivity Heterogeneity groups = 1,...,N f i (θ) and U i (c,y,θ) unobservable i single T(Y ) average efficiency condition 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

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 - p. 26