Pareto Efficient Income Taxation

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

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

3 Motivation Contribution Results Q: Good shape for tax schedule? Mirrlees (1971), Diamond (1998), Saez (2001) positive: redistribution vs. efficiency normative: Utilitarian social welfare function Pareto Efficient Income Taxation - p. 2

4 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

5 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

6 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 efficiency weaker criterion Pareto Efficient Income Taxation - p. 3

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

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

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

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

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

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

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

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

15 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

16 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

17 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

18 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 ) given T(Y ) v(θ) max Y U(Y T(Y ),Y,θ) Pareto Efficient Income Taxation - p. 8

19 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 ) given T(Y ) v(θ) max Y U(Y T(Y ),Y,θ) Government budget T(Y (θ))df(θ) G Pareto Efficient Income Taxation - p. 8

20 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 ) given T(Y ) v(θ) max Y U(Y T(Y ),Y,θ) Resource feasible (Y (θ) c(θ) ) df(θ) G Pareto Efficient Income Taxation - p. 8

21 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 ) given T(Y ) v (θ) = U θ (Y (θ) T(Y (θ)),y (θ),θ) Resource feasible (Y (θ) c(θ) ) df(θ) G Pareto Efficient Income Taxation - p. 8

22 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 ) given T(Y ) v (θ) = h(y (θ)) Resource feasible (Y (θ) c(θ) ) df(θ) G Pareto Efficient Income Taxation - p. 8

23 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 ) 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

24 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

25 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

26 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

27 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

28 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

29 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 Pareto Efficient Income Taxation - p. 10

30 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

31 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

32 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

33 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

34 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

35 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

36 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. note: zero-tax-at-top special case Pareto Efficient Income Taxation - p. 11

37 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. note: zero-tax-at-top more general condition: special case τ(θ)f(θ) h (Y (θ)) + θ θ 1 u (c( θ)) f( θ)d θ is nonincreasing Pareto Efficient Income Taxation - p. 11

38 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

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

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

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

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

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

44 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

45 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

46 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

47 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 without empirical knowledge anything goes Pareto Efficient Income Taxation - p. 15

48 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 without empirical knowledge anything goes need information on f(θ) to restrict T(Y ) Pareto Efficient Income Taxation - p. 15

49 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

50 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

51 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: g(y ) α(y ) nondecreasing Pareto Efficient Income Taxation - p. 17

52 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) Pareto Efficient Income Taxation - p. 18

53 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 Pareto Efficient Income Taxation - p. 18

54 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

55 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) some challenges 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

56 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 Pareto Efficient Income Taxation - p. 19

57 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 >= 10 years during Average Income (in 1990 dollars) x Average Income (in 1990 dollars) x 10 4 Figure 1: Density of income Figure 2: Implied elasticity Pareto Efficient Income Taxation - p. 19

58 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 Rawlsian x 10 5 Test against Average Income Data (sigma = 0, eta = 2, T =.3Y) 3 Rawlsian x 10 5 Test against Average Income Data (sigma = 1, eta = 3, T =.3Y) x x 10 4 Pareto Efficient Income Taxation - p. 19

59 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

60 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

61 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

62 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. Saez (2001): ϕ = 3 Pareto Efficient Income Taxation - p. 21

63 Top Tax Rate Top Tax Rate Flat Tax Progressivity Heterogeneity upper bound on τ elasticity 1/η+1 Pareto Efficient Income Taxation - p. 22

64 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

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

66 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

67 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

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