A Theory of Income Taxation under Multidimensional Skill Heterogeneity

Transcription

1 A Theory of Income Taxation under Multidimensional Skill Heterogeneity Casey Rothschild Wellesley and Radcliffe Florian Scheuer Stanford and NBER November 2014 Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

2 Introduction Motivation Classic tradeoff for income taxes: equity vs. efficiency (Mirrlees, 1971) Recent debate Increased inequality shifts in sectoral composition of economy Wages vs. social marginal products This paper: income taxation in multi-sector economies with arbitrary externalities N activities, returns may deviate from marginal products naturally N-dimensional heterogeneity Driving forces for tax policy Income tax relative returns to activities across-activity shifts Pigouvian motives: fill in wedges between private and social returns Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

3 Model General Model N-dimensional skill space with support Θ and cdf F (θ) θ i skill for activity i Preferences U(c, e) =c h(m( e)) c h(l) (general in paper) Vector of activity-specific efforts e; convex h. Effort aggregator m increasing, quasiconvex, linear homogeneous Allocations c(θ), e(θ) l(θ) m(e(θ)), V (θ) c(θ) h(l(θ)) y(θ) i y i(θ), earned income (from each activity) Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

4 Model Model, ctd Aggregate output Y (E) = i Y i (E), where E is vector of E i Θ θ i e i (θ)df (θ) Each unit of effective effort in a given sector has the same private return For each i, there exists r i (E) such that y i (θ) =r i (E)θ i e i (θ) θ Y i (E) =r i (E)E i, r i may deviate from Y i Y / E i Examples Y CRS, r i = Y i no externalities (Rothschild/Scheuer 2013, for N =2) N = 2, rent-seeking r 1 Y 1, traditional r 2 = Y 2 (Rothschild/Scheuer 2014) Team production, Y = E 2, Y 1 = a(e 1 )E 2, Y 2 =(1 a(e 1 ))E 2 r 1 = a(e 1 )E 2 /E 1, r 2 =1 a(e 1 ), pos. and neg. externalities Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

5 Model Implementation Key: vector E returns r i (E) incomes shares q E (θ) and wages w E (θ) Direct implementation: announcement of unobservable θ observable c(θ), y(θ) (Income shares q i (θ) y i (θ)/y(θ) chosen to minimize cost of y.) m h.o.d. 1 q E (θ) andy(θ)/l(θ) w E (θ) independent of y Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

6 N Sectors (Endogenous) Wage Distributions E determines w E (θ) and q E (φ) for all θ Types θ (w, φ), distribution G E (w, φ) Wage distribution: w F E (w) df (θ) = dg E (z, φ) {θ w E (θ) w} w E Φ with density f E (w) = i f i E (w) and f i E (w) Φ qi E (φ)dg E (w, φ) Trace out Pareto frontier using weights Ψ(θ) and define analogously... Cumulative welfare weights Ψ E (w) Densities ψ E (w) = i ψi E (w) Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

7 Pareto Problem N Sectors Given E inner problem W (E) max V (w),l(w) V (w)dψ E (w) (1) s.t. V (w) = max u (c(w ), l(w )w ) w w w (2) r i (E)E i = wl(w)fe i (w)dw i (3) wl(w)f E (w)dw c(v (w), l(w))f E (w)dw. (4) Despite N-dimensional heterogeneity, standard problem except for (3) Outer problem: max E W (E) Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

8 N Sectors Marginal Tax Rate Formula Proposition In any Pareto optimum without bunching, ( 1 T (y(w)) = 1 ) ( fe i (w) f E (w) ξ i 1 + Ψ E (w) F E (w) wf E (w) i ( )) 1 ε(w) where ξ i are the multipliers on (3) and ε(w) is the wage elasticity of effort. Adjustment factor to otherwise standard formula (Diamond, 1998, Saez, 2001) Key object of interest: i f i E (w) f E (w) ξ i vs. where Pigouvian correction t i p solves r i (1 t i p) Y i i f i E (w) f E (w) ti p, Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

9 Two Sectors ξ i versus t i p from Outer Problem (for N=2) Let β j i (E) be the semi-elasticities of relative returns x j E (φ) = φ jr j (E)/r N (E) Lemma At any Pareto optimum, W (E) E i ( = r i ξi tp i ) β 1 i [I + R + (ξ 1 ξ 2 ) (C + S)], i = 1, 2 (5) S Sectoral shift effect I Incentive (Stiglitz, 1982) effect C Effort re-allocation effect on consistency constraints R Redistributive effect from utility re-allocation in objective If βi 1 = 0 ξ i = tp i Any deviation ξ i tp i is due to relative return effects Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

10 Applications No Externalities r i (E) = Y i (E) Y (E) must have CRS (Rothschild/Scheuer, 2013) Returns only depend on ρ E 1 /E 2 α(ρ) aggregate income share of sector 1, σ(ρ) substitution elasticity Proposition i f i E (w) f E (w) ξ i = ( α(ρ) f E 1(w) ) ξ with ξ f E (w) Vanishes as σ (linear technology) (I + R)/σ α(1 α)y + (C + S)/σ E.g. sector 1 high-wage, low redistributive preference: I, R > 0 Scale T up at low w, down at high w Less progressive tax schedule Indirectly redistribute from high to low w by manipulating relative returns Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

11 Applications No Externalities: Numerical Results tion Marginal Tax Rate Pareto Optimum SCPE wage Less progressive tax schedule than in SCPE Calibration Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

12 Applications Increasing/Decreasing Returns to Scale Homothetic Y (E) = h(ỹ (E)) where Ỹ (E) has CRS ε h elasticity of h: ε h > 1 IRS, ε h < 1 DRS σ substitution elasticity, α sector 1 income share under Ỹ Suppose Y is divided according to α: Y 1 = αy, Y 2 = (1 α)y Proposition i i fe i (w) ( f E (w) ξ i = α f E 1(w) ) ξ + (1 ε h ) f E (w) fe i (w) ( f E (w) ξ i = α f E 1(w) ) ξ f E (w) }{{} relative returns + (1 ε h ) }{{} Pigou Relative return adjustment: local versus aggregate income shares as before Pigouvian correction: global scaling of T, down if IRS, up if DRS Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

13 Team Production Applications Y (E) = E 2, Y 1 (E) = a(e 1 )Y (E), and Y 2 (E) = (1 a(e 1 ))E 2 : Activity 1 is always overpaid ( claiming credit ) t 1 p = 1 and t 2 p = a/(1 a) since r 1 = a(e 1 )E 2 /E 1 and r 2 = 1 a(e 1 ) Denote by ε a the E 1 elasticity of a Proposition i fe i (w) f E (w) ξ i = f E 1(w) ( 1 (1 a εa )ξ ) + f E 2(w) f E (w) f E (w) ( a ) 1 a + aξ Correction on activity 2 always less (in absolute value) than Pigouvian subsidy Subsidy E 2 r 1 /r 2 perverse shift to activity 1 Correction on activity 1 depends on ε a 1 a. Intuition: E 1 r 2, but r 1 depending on crowding If crowding dominates (ε a small), undercorrection as well Who bears externalities? Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

14 Conclusion Discussion Flexible toolkit for income tax design with general patterns of externalities Allows for variety of applications Quantitative work: requires both externalities and relative return elasticities E.g. income shares and their elasticities Key: Pigouvian wedge alone is not sufficient On whom are the externalities imposed? Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

15 Sectoral Shift Effect Extra Slides W (E) E i ( = r i ξi tp i ) β 1 i [I + R + (ξ 1 ξ 2 ) (C + S)] E i changes x 1 E θ1r1(e) (φ) = θ 2r 2(E) by β1 i, change in e 1 /e 2 effect on q 1 E = y 1/y E.g. β 1 i > 0 e 1 /e 2, Shift of income from 2 to 1 Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

16 Stiglitz (1982) Effect Extra Slides W (E) E i ( = r i ξi tp i ) β 1 i [I + R + (ξ 1 ξ 2 ) (C + S)] Suppose d ( f 1 E (w)/f E (w) ) /dw > 0 and E i x 1 Wage compression indirect redistribution to low wages Welfare improving if Ψ E (w) > F E (w) I is an integral of local Stiglitz effects Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

17 Re-allocation Effects Extra Slides W (E) E i ( = r i ξi tp i ) β 1 i [I + R + (ξ 1 ξ 2 ) (C + S)] 1 E i has different effects on workers with the same w but different q s High-q 1 workers see wage increase relative to low-q 1 workers at same w Move up along l(w)- and V (w)-schedule Effectively re-allocates effort (and utility) from activity 2 to 1 Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

18 Extra Slides A Simple Example Y 1 (E) = µ > 1, Y 2 (E) = E 2 Two types workers: θ 1 = 1 and θ 2 = 1 rent-seekers: θ 1 = θ R 1 > 1 and θ 2 = 0 Total rent-seeking effort is (with linear m) E 1 = θ R 1 e R + λ W e W, where λ W is fraction of workers who work in rent-seeking sector Workers may... be indifferent if E 1 = µ since then r 1 = µ/e 1 = 1 not work in rent-seeking sector if E 1 > µ all work in rent-seeking sector if E 1 < µ Optimum involves interior equilibrium, so E 1 = θ R 1 e R + λ W e W = µ λ W (e R, e W ) = µ θr 1 er e W Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

19 Extra Slides A Simple Example, ctd Output is sum of rents and production µ + (1 λ W (e R, e W ))e W = e W + θ1 R e R Utilitarian optimum maximizes output net of effort cost e W + θ1 R e R z(e W ) z(e R ) If gov t can control effort but not occupational choice, optimum involves z (e W ) = 1 and z (e R ) = θ R 1 Coincides with agents facing no distortionary taxes Zero tax on rent-seekers, even though completely unproductive (Pigouvian tax would be 100%) Taxing rent-seekers would attract workers into rent-seeking (since E 1 = µ) Rent-seekers are indirectly productive by crowding out workers Numerical Example General Income Shares Externalities Borne by One Activity Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

20 Extra Slides Income Tax Implementation Lemma Suppose the allocation {c(θ), y(θ), q(θ), E} is incentive compatible. Then (i) w E (θ) = w E (θ ) = w implies u(c(θ), y(θ)/w) = u(c(θ ), y(θ )/w), and (ii) {c(θ), y(θ), q(θ), E} can be implemented by offering the single non-linear income tax schedule T (y) corresponding to the retention function R (y) = y T (y) defined by R (y) max c { c (c(θ), u ) ( y(θ) u c, w E (θ) ) } y θ Θ w E (θ) and letting all agents choose one of their most preferred (c, y)-bundles from the resulting budget set B = {(c, y) c y T (y)}. Back Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

21 Extra Slides No Externalities: Calibration Isoelastic utility u(c, l) = c l 1+1/ε /(1 + 1/ε) with ε = 0.5 and linear m Cobb-Douglas production function Y = E α 1 E 1 α 2 (σ(ρ) = 1) Relative welfare weights Ψ(θ 1, θ 2 ) = 1 (1 F (θ 1, θ 2 )) r r = 1 no redistribution, r Rawlsian SWF Here: r = 1.3, moderately progressive Ψ(F( )) o Cumulative welfare weight Ψ F( ) Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

22 Extra Slides Identification of the Skill Distribution Remain agnostic about nature of the two latent sectors Basu/Ghosh (1978), Heckman/Honoré (1990) Suppose we observe a distribution of realized wages and it results from a Roy model with a bivariate lognormal skill distribution Then we can identify the (5 parameters of the) underlying skill distribution (up to a permutation of indices) 2011 CPS data on hourly wages in the US (as in Mankiw et al., 2009) MLE estimation of bivariate skill distribution (latent wages) Draw large sample (w 1, w 2 ), infer sectoral choice and w = max{w 1, w 2 } Infer effort from individual FOC l 1/ε = (1 τ)w (set τ = 25%) Infer sectoral incomes Y 1, Y 2 and income share α = Y 1 /(Y 1 + Y 2 ) Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

23 Calibration Results Extra Slides Empirical and Fitted Wage Distribution empirical fitted log(wage) Marginal Tax Rate µ 1 = 2.81 (.29), σ 1 = (.015) Pareto Optimum SCPE µ 2 = 1.74 (.714), σ 2 = (.369) ρ 12 =.030 (0.630) Mean wages: w 1 20, w 2 7 Sector 2 shares: 1 α 5.8% (income share) 12.3% of workers wage ρ 12 = 0 not rejected Likelihood ratio test rejects single sector model Back Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

24 Extra Slides Self-Confirming Policy Equilibrium Baseline that ignores general equilibrium effects and externalities What is the right baseline? Self-Confirming Policy Equilibrium Individual Choice T (y) Wage distribution Social Planner Choice Wage distribution (Optimal) tax schedule Mutual consistency: A tax schedule that is perceived as optimal given the wage distribution induced by that tax schedule T (y) confirmed as optimal with Saez (2001) methodology But tacit, exogenous wage, model used in confirming this is wrong Back Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

25 Extra Slides Rent Seeking: Numerical Example Y 1 (E) = µ = 1/10 Rent-seekers unproductive at the margin ε = 0.7 Pareto weights: Ψ(F ) = 1 (1 F ) r, r = 1.4 ρ = 1 utilitarian, ρ Rawlsian Skill distribution: independent F (θ 1, θ 2 ) = F 1 (θ 1 )F 2 (θ 2 ) F 1 (θ 1 ) truncated Pareto with α 1 = 3.4, Θ 1 = [.5, 5] F 2 (θ 2 ) truncated Pareto with α 2 = 7, Θ 2 = [.5, 3] If all top earners are rent-seekers, F 1 (θ 1 ) implies that the top tail of the income distribution is Pareto with k = 2 (Saez, 2001) Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

26 Extra Slides Rent Seeking: Numerical Example, ctd Marginal Tax Rate Optimal SCPE 3 2 Tax Schedule Average Tax Rate 1 Share of Rent Seekers wage wage Higher marginal tax rates in Pareto optimum Wage distribution extends further All agents with w 3 are completely unproductive rent-seekers Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November Back

27 Extra Slides Externalities from One Activity r i (E) = r i (E 1 ) for all i Rothschild/Scheuer (2014) with N = 2 and rent-seeking Proposition i fe i (w) f E (w) ξ i = f E 1(w) f E (w) ξ with ξ = t1 p + β1 1(I + R)/r 1 1 β1 1(C + S)/r 1 β1 1 = 0 just local income share weighted t1 p I, R > 0 if activity 1 high income, low redistributive preference β1 1 > 0 ξ > t1 p, overcorrection β1 1 < 0 ξ < t1 p, undercorrection Who bears the externalities? General N in paper, can allow for positive or mixed externalities Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November

28 Extra Slides Externalities Borne by One Activity r 1 (E) general, but r i (E) = r i fixed for i = 2,..., N Anything that affects r 1 induces relative return changes and externalities tp i = βi 1Y 1 /r i Example: resource transfer Y (E) = Y (E 1 ), Y 1 (E) = Y (E 1 ) E 2, Y 2 (E) = E 2 Proposition i f i E (w) f E (w) ξ i = i fe i (w) βi 1 Y 1 + I + R ξ with ξ = f E (w) r i 1 (β1 1/r 1 β2 1/r 2) (C + S) E.g. with positive externalities βi 1 > 0 Numerator: Pigouvian subsidy and direct redistributive effects of it If I, R > 0, increase in r 1 is in fact undesirable Denominator: multiplier from indirect sectoral shift Increase in r 1 always induces shift from activity 2 to 1 Reinforces/mitigates initial increase in r Back End 1 Casey Rothschild, Florian Scheuer Taxation and Multidimensional Heterogeneity November