An Exploration in the Theory of Optimum Income Taxation

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1 A Exploratio i the Theory of Optimum Icome Taxatio J. A. Mirrlees, 1971 A Exploratio i the Theory of Optimum Icome Taxatio p. 1/12

2 Objectives ad Results Characterize the optimal icome tax schedule whe idividual productivities are ot observed by the plaer Is there a theoretical basis for progressive or regressive margial taxes? Evaluate optimal trade-off of labor icetives ad cosumptio isurace through use of a o-liear tax schedule Results Optimal margial tax rate is betwee 0 ad 1. Tax schedule is sesitive to distributio of skills ad preferece specificatio A Exploratio i the Theory of Optimum Icome Taxatio p. 2/12

3 Eviromet Demographics Static ecoomy with a cotiuum of cosumers, idexed by their labor productivity,. Prefereces u(x, y) where x is cosumptio ad y is labor supplied. Assume u is cot. diff. with u 1 > 0, u 2 < 0, u strictly cocave. Idividual Productivities F, with f() = F (). A aget who provides labor, y, produces output z = y Govermet Chooses a tax schedule such that a aget who produces z is left with c(z) = c(y) to cosume after tax. Restrict c to be upper semi-cotiuous. Timig Govermet aouces fuctio c(z) to which it ca commit, the aget observes ad choose y A Exploratio i the Theory of Optimum Icome Taxatio p. 3/12

4 Cosumer Problem (CP): Choose (x,y ) to maximize u(x,y) subject to x c(y) u 1 c (y) = u 2 (1) Let u = u(x,y ) be idirect utility as a fuctio of. The (1) gives: du d = u 1y c (y ) (2) = y u 2 (3) A Exploratio i the Theory of Optimum Icome Taxatio p. 4/12

5 Cosumer Problem (cot d) Propositio 1 There exists a umber 0 0 such that y = 0 if 0 ad y > 0 if > 0 Proof Let m < ad y m > 0. The u[c(my m ),y m ] < u [ c ( m y m), m y m] u So y m = 0 if y = 0. Propositio 2 Ay fuctio of, (x,y ) that solves (CP) for some USC fuctio, c(z), also solves (CP) for a o-decreasig USC futio, c(z) [defie c(z) = sup z z c(z )] margial tax rate eed ot be greater tha 100% A Exploratio i the Theory of Optimum Icome Taxatio p. 5/12

6 Govermet Problem Govermet chooses a fuctio, c(z) to maximise W = G(u )f()d (4) subject to x f()d H ( ) y f()d (5) du d = y u 2 (6) u = u(x,y ) (7) Take u as state, y as cotrol ad use Potriyagi Maximum Priciple. What fuctios, y, are admissible ad esure there exists c(z) such that (CP) is satisfied? A Exploratio i the Theory of Optimum Icome Taxatio p. 6/12

7 A Useful Theorem Assumptio B V (x,y) = yu 2 u 1 is icreasig i y for all x Theorem 1 Uder Assumptio B, z = y maximizes utility for every uder some cosumptio fuctio, c(z), iff 1. z is a o decreasig fuctio for > z < for all > 0 Ituitio: FOC from (CP) ca be writte as u 1 c (y) = u 2 (8) yc (y) = yu 2 u 1 (9) choose c(z) so thatc (z) provides correct wedge i MRS Implicatio: We kow which fuctios, y, i Govt problem are implemetable A Exploratio i the Theory of Optimum Icome Taxatio p. 7/12

8 Margial Tax Rates Defie the margial tax rate, θ, by θ = d d(wz) [wz c(z)] (10) So = ψ y 2 f() wθ = d dz [wz c(z)] = w + u 2 u 1 (11) 1 λg u 1 u 1 T m f(m)dm 0 (12) Propositio 3 Uder Assumptio B, the tax fuctio, wz c(z) is o-decreasig fuctio of z Implicatio Margial tax rates lies i [0, 1] A Exploratio i the Theory of Optimum Icome Taxatio p. 8/12

9 Additive Utility Let u 12 = 0. The V (x,y) is icreasig y. Let ψ y = y yu 2(x,y). Ca be roughly iterpreted as the icrease i the cost of providig the correct icetives as productivity icreases. ( ) Let v = w + u 2 u 1 /ψ y The solutio to the plaig problem - 2 D.E s: dv d = v (2 + f ) f du d = yu 2 subject to boudary coditios o v 0 ad v 1 2 u 1 + λg 2 (13) (14) A Exploratio i the Theory of Optimum Icome Taxatio p. 9/12

10 Additive Utility (cot d) Recall from Theorem 1 that we required dz d 0 to guaratee implemetability - eeds to be verified ex-post Mirrlees provides a algorithm to compute solutio A Example from Mirrlees u(x,y) = log x + log(1 y), G(u) = u logormal( 1,σ 2 ) σ = 0.39,1 A Exploratio i the Theory of Optimum Icome Taxatio p. 10/12

12 Aother Example u(x,y) = x1 γ 1 γ ψ y1+σ 1+σ logormal( 1 2 ν2,ν 2 ) γ 1.5 σ 2.5 ν Quatitative results are sesitive to fuctioal form ad parametrizatio A Exploratio i the Theory of Optimum Icome Taxatio p. 11/12

13 Aother Example - Results 2.5 Cosumptio 3.5 Output Mirrlees Full Ifo Autarky Productivity Mirrlees Full Ifo Autarky Productivity Effort Mirrlees Full Ifo Autarky Average Tax Rate Margial Tax Rate Productivity Productivity A Exploratio i the Theory of Optimum Icome Taxatio p. 12/12

Economics 2450A: Public Economics Section 3: Mirrlees Taxation

Economics 2450A: Public Economics Section 3: Mirrlees Taxation Ecoomics 2450A: Public Ecoomics Sectio 3: Mirrlees Taxatio Matteo Paradisi September 26, 2016 I today s sectio we will solve the Mirrlees tax problem. We will ad derive optimal taxes itroducig the cocept