Economics 2450A: Public Economics Section 3-4: Mirrlees Taxation

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1 Ecoomics 2450A: Public Ecoomics Sectio 3-4: Mirrlees Taxatio Matteo Paradisi October 3, 206 I today s sectio we will solve the Mirrlees tax problem. We will ad derive optimal taxes itroducig the cocept of wedges ad study the model with ad without icome e ects. The Model Setup Suppose the aget has prefereces over cosumptio ad labor represeted by the utility fuctio u (c, l) that we assume separable ad quasi-liear such that u (c, l) c v (l). We assume that v 0 (l) > 0 ad v 00 (l) 0. Each aget ears icome z l whe supplyig l hours of labor ad cosumes c l T (l) after taxes. Idividuals are heterogeeous i the salary that represets their type ad we will iterpret as a measure of ability. Salaries are distributed accordig to f (), with 2 [, ]. Idividual welfare is aggregated through a social welfare fuctio G ( ), that we assume di eretiable ad cocave. Revelatio Priciple Throughout all of the tax problems that we study we will assume that the govermet caot observe the labor choice of the aget ad her type. Icome is the oly observed choice that the govermet ca target. We solve the model usig a revelatio mechaism. Our goal is to defie a optimal tax schedule that delivers a allocatio z (), c () to each aget. TheRevelatio Priciple claims that if a allocatio ca be implemeted through some mechaism, the it ca also be implemeted through a direct truthful mechaism where the aget reveals her iformatio about. We imagie that each aget reports to the govermet her type 0 ad that allocatios are a fuctio of 0 such that we ca write c ( 0 ), l ( 0 ), z ( 0 ) ad u ( 0 ). By revelatio priciple, the govermet caot do better tha defiig fuctios c (), z () such that the aget fids optimal to reveal her true productivity: z () z ( c () v c ( 0 0 ) ) v for every ad 0 where is the true type of the aget. Notice that sice is cotiuous we have a ifiity of costraits. I order to reduce the dimesioality of the problem, we assume that the margial rate of substitutio betwee cosumptio ad before-tax icome is decreasig i : MRS cz v0 (z () /) u 0 (c ()) decreases i This is the so called sigle-crossig coditio (or Spece-Mirrlees coditio). Sigle-crossig ad icetive compatibility imply the mootoicity of allocatios (i.e. c (), z () are icreasig i ). If mootoicity ad sigle-crossig are satisfied, we ca replace the icetive costrait with the first-order ecessary coditios of the aget that provide a local icetive coditio. Uder mootoicity ad siglecrossig the local coditios are also su ciet. While solvig these problems we will oly impose local icetive costraits ad igore the mootoicity of allocatios, which is the verified ex-post.

2 Icetive Compatibility We reduce the dimesioality of the problem by takig a first order approach that replaces the ifiity of costraits for each idividual with a local coditio relyig o the optimal revelatio choice. Whe reportig, the idividual of type solves the followig problem: z ( max c ( 0 0 ) ) v 0 the first order ecessary coditio for this problem is: c 0 ( 0 z 0 ( 0 ) z ( 0 ) ) v0 0 If the govermet wats the aget to reveal her true type, it must be: c 0 () z0 () z () v0 Uder the cocavity assumptio o the prefereces, this is a global icetive costrait coditio. Suppose we study local utility chages by totally di eretiatig the utility wrt, we get: du () d c 0 z 0 () z () () v0 + z () z () 2 v0 Notice that the term i the first bracket is the first order coditio of the aget. We ca thus write du () /d z () / 2 v 0 (z () /). This equatio pis dow the slope of the utility assiged to the aget at the optimum. By covexity of v ( ), the slope is always positive: the govermet assigs higher utility to higher types at the optimum. Higher types have a lower margial disutility of labor for a give level of hours worked ad they get iformatioal rets i the equilibrium. Labor Supply ad Labor Wedge The first order coditio is: The idividual solves the followig optimizatio problem: z max z T (z) v z T 0 (z) The secod term o the right-had-side of the equatio is the margial rate of substitutio betwee cosumptio ad icome ad we ca always write that T 0 () MRS(). Whe the aget is ot distorted, the MRS is equal to implyig T 0 (z) 0. We ca iterpret T 0 (z) as a wedge o the optimal labor supply: wheever it is di eret from zero, labor supply is distorted. Wedges are a cetral cocept i the optimal taxatio literature ad we will ecouter them throughout the class. From the optimality coditio, we ca derive the elasticity of labor wrt the et of tax wage. Rewrite the optimality coditio as: v 0 z ( T 0 (z)) Totally di eretiatig wrt ( Which implies the followig elasticity: " d ( T 0 (z)), wehave: v 0 (l) dl d ( T 0 (z)) v00 (l) dl T 0 (z)) ( T 0 (z)) l v0 (z) lv 00 (z) 2

3 Resource Costrait Suppose the govermet has a exogeous reveue requiremet E. Therev- eues collected through taxatio must be at least equal to E. Usig the aget s budget costrait we ca write the tax levied o a sigle idividual as T (z ()) z () c (). Summig over all the idividuals i the ecoomy we get: ˆ c () f () d ˆ z () f () d E This is the resource costrait for this ecoomy. Notice that ulike icetive costrait, this costrait is uique. 2 Optimal Icome Tax We ow solve the costraied maximizatio problem usig optimal cotrol theory. Istead of havig taxes as a choice variable, we assume that the govermet chooses a allocatio for each aget. Give the idividual s budget costrait, this is equivalet to choosig a tax level. The govermet problem is: ˆ max c(),u(),z() G (u ()) f () s.t. du () d z () 2 v0 z () ˆ c () f () d ˆ z () f () d E We solve the problem with a Hamiltoia where we iterpret as the cotiuous variable ad choose u () as state variable ad z () as cotrol. The icetive costrait becomes the law of motio of the state variable: it measures how utility chages across types i equilibrium. I order to setup the Hamiltoia, we eed to replace cosumptio i the resource costrait with a fuctio of state ad cotrol variables. Usig the defiitio of idirect utility, we ca write c () u ()+v (z () /). We replace this coditio ito the resource costrait ad setup the followig Hamiltoia: H G (u ()) + z () u () v z () f ()+µ() z () 2 v0 z () µ () deotes the multiplier o the icetive costrait of type ad is the multiplier o the resource costrait. The first order coditios of the optimal cotrol () v 0 (l ()) f ()+ µ () 2 The trasversality (boudary) coditios read: v 0 z () + z () v00 z () [G0 (u ()) ] f () µ 0 () (2) µ () µ ( ) 0 The Hamiltoia solutio requires µ ( ) u ( ) 0. However, if we wat to provide positive utility to type we must require µ ( ) 0. At the same time, sice at the optimum the icetive costraits will 3

4 be bidig dowwards, we require µ () 0. As it is stadard i this kid of problems the lowest type does ot wat to imitate ay other aget i equilibrium implyig that her icetive costrait is slack, while everyoe else is idi eret betwee her allocatio ad the allocatio provided to the immediately lower type. If we itegrate equatio (2) over the etire type space ad use trasversality coditios we fid: ˆ G 0 (u ()) f () d This is a expressio for the margial value of public fuds to the govermet. It states that the value of public fuds depeds o the margial social welfare gais across the etire type space ad it is equal to the welfare e ect of trasferrig $ to every idividual i the ecoomy. I other words, public fuds are more valuable the higher are the social welfare gais achievable i the ecoomy. We ca also itegrate equatio (2) to fid the value of µ (): µ () ˆ [ G 0 (u (m))] f (m) dm (3) Usig the defiitio of labor elasticity, we rearrage the followig: z () v 0 + z () z () z () v00 v 0 + Exploitig the defiitio of the tax wedge, we simplify equatio () to get: T 0 (z ()) µ () f () ( T 0 (z ())) + Usig the expressio for µ derived i equatio (3) we get: T 0 (z ()) T 0 (z ()) + [ g (m)] f (m) dm f () (4) The optimal tax is decreasig i the elasticity of labor supply. We defie g () G0 (u()) the relative social welfare weight of idividual such that g () f () d. Remember that aggregates the social welfare weights across the etire ecoomy. Thus, a higher g () meas that the govermet cares relatively more about idividual ad will tax her less. A Rawlsia govermet would have g () 0 for ay >ad the formula would reduce to: T 0 (z ()) T 0 (z ()) + F () f () The secod part of the expressio captures the ratio of the mass above type ad the desity at. It is a measure of thickess ad the lower it is the higher margial tax rate will be. 3 Diamod ABC Formula I this paragraph we derive a tax formula preseted i Diamod (998). We chage our assumptio about welfare weights ad assume that they are distributed accordig to a fuctio () withcdf (). The govermet objective fuctio becomes: ˆ u () () d 4

5 By assumptio We therefore have: () d implies. First order coditios ca be derived exactly as before. µ 0 () () f () ad after itegratio: µ () ˆ (f () ()) d () F () Usig the expressio above the tax formula reads: T 0 (z ()) + () F () T 0 (z ()) f () To write the ABC formula we divide ad multiply by F () to get: T 0 (z ()) + T 0 (z ()) () F () F () F () f () {z } {z } {z } A() B() C() A () captures the stadard elasticity ad e ciecy argumet. B () measures the desire for redistributio: if the sum of weights below is high relative to the mass above, the govermet will tax more. Fially, C () measures the thickess of the right tail of the distributio. A thicker tail will be associated to higher tax rates. Notice that i the Rawlsia case () for every >ad the formula coverges to the oe preseted i the previous paragraph. 4 Optimal Taxes With Icome E ects We ow relax the assumptio of o icome e ects. ũ (c, l) u (c) v (l) whereu 0 (c) > 0 ad u 00 (c) 0. Suppose the utility of the aget takes the form Elasticity of Labor Supply The optimality coditio for the labor supply choice becomes: v 0 (l) u 0 (c) ( T 0 (z)) The ucompesated respose of labor supply to the et of tax wage ( T 0 (z)) u0 (c)+l ( T 0 (z)) u 00 (c) v 00 (l) ( T 0 (z)) 2 2 u 00 (c) implyig the followig ucompesated elasticity: " u u0 (c) /l + v 0 (l) 2 u 00 (c) u 0 (c) 2 v 00 v (l) 0 (l) 2 u u 0 (c) 00 (c) 2 The respose of labor to icome chages is ( T 0 (z)) u 00 (c) v 00 (l) ( T 0 (z)) 2 2 u 00 (c) 5

6 Usig the Slutsky equatio (as we did i Sectio otes ( T 0 (z)) u0 (c)+l( T 0 (z)) u 00 (c) v 00 (l) ( T 0 (z)) 2 2 u 00 (c) u 0 (c) v 00 (l) ( T 0 (z)) 2 2 u 00 (c) l ( T 0 (z)) u 00 (c) v 00 (l) ( T 0 (z)) 2 2 u 00 (c) Therefore: " c v 0 (l) /l v 00 (l) ( T 0 (z)) 2 2 u 00 (c) Optimal Tax Everythig is similar to the previous case except for the fact that ow we caot replace the variable c () i the resource costrait usig the defiitio of idirect utility. We will defie cosumptio as a expediture fuctio c (ũ (),z(),) ad implicitly di eretiate it wrt to ũ () ad z (). Start from the defiitio of idirect utility: ũ () u ( c ()) v (z () /) It follows that the followig two coditios will hold at the optimum: dũ () u 0 ( c ()) d c () Rearragig: 0u 0 ( c ()) d c () v0 (z () /) dz () d c () dũ () u 0 ( c ()) The Hamiltoia for the problem is: d c () dz () v0 (z () /) u 0 ( c ()) H [G (u ()) + (z () c (ũ (),z(),))] f ()+µ() z () 2 v0 z () ad () v 0 (z () /) u 0 (c ()) f ()+ µ () 2 z () v 0 + z () v00 z G 0 (u ()) u 0 (c ()) f () µ 0 () I order to fid the equilibrium value of the multiplier, we ca itegrate the secod FOC: ˆ µ () G 0 (u (m)) u 0 f (m) dm (c (m)) We exploit the defiitio of the two elasticities to write: 6

7 z () v 0 + z () v00 z () v 0 z () v 0 z () 2 4+ z () +" u " c v 00 z() 3 5 v 0 z() The optimal tax formula will the become: T 0 (z ()) T 0 (z ()) +" u " c () f () (5) where () u0 (c())µ(). 5 Pareto E ciet Taxes We ow ask the questio of whether a tax system T 0 (z) i place is Pareto-optimal, meaig that there exists o feasible adjustmet i the tax schedule such that all idividuals i the ecoomy are weakly better o. We ca characterize the Pareto frotier of the previous problem by solvig the followig: s.t. max ˆ u () () d u (c ()) h (z () /) u (c ( 0 )) h (z ( 0 ) /) 8, 0 ˆ [z () c ()] f () d E By varyig the social margial welfare weights, we ca trace out every poit o the Pareto frotier. However, there might be poits o the Pareto frotier that ca be improved upo icreasig the utility of all the agets i the ecoomy. Werig (2007) develops a test for the Pareto optimality of a tax schedule. The first importat result of the paper is the followig: Propositio : A tax code fails to be costraied Pareto optimal if ad oly if there exists a feasible tax reform that (weakly) reduces taxes at all icomes. Proof: (if) suppose we weakly reduce taxes all over the etire ecoomy, the every idividual is at least as well o. (oly if) suppose there exists a Pareto improvig feasible tax reform T (z). The we have: U (z () T (z ()),z (),) U (z o () T 0 (z 0 ()),z 0 (),) U (z () T 0 (z ()),z (),) where the first iequality comes from the assumptio of Pareto-improvemet ad the secod from the assumptio that uder T 0 (z) the aget truthfully reveals her type ad chooses z 0 (). The chai of iequalities implies that T (z ()) T 0 (z ()) for every. Feasible meas that it satisfies the resource costrait. 7

8 Propositio implies that sice the resource costrait is satisfied ad both tax systems raise reveues at least equal to E, a Pareto improvemet ca oly occur through a tax reductio that does ot geerate a drop i reveues. This ca be iterpreted as a La er e ect: although the govermet lowers taxes, the behavioral respose (icrease i labor supply) is strog eough to more tha compesate the reveue loss. 6 A Test of the Pareto Optimality of the Tax Schedule I order to implemet the test, Werig takes a dual approach to the optimal taxatio problem that we studied i the previous paragraphs. We rewrite the problem such that istead of maximizig the social welfare fuctio, the govermet maximizes the resources to provide a miimum level v () of idirect utility to every aget i the ecoomy. We write the problem as follows: s.t. ˆ max u(),z() (z () c (v (),z(),)) f () d dv () d U ( c (v (),z(),),z(),) (6) v () v () 8 (7) Notice that c (v (),z(),) is the expediture fuctio that we itroduced to study optimal taxes with icome e ects. The problem would also have a mootoicity costrait that we relax for the momet, as we usually do. Notice that by chagig the levels of v () we ca characterize the etire Pareto frotier. 2 We solve the problem with a Lagragia by attachig multiplier () f () to (7) ad µ () tothe local icetive costrait: L ˆ ˆ + ˆ (z () c (v (),z(),)) f () d + µ () v 0 () d ˆ () v () f () d µ () U ( c (v (),z(),),z(),) d Notice that this is idetical to the Lagragia that we would obtai i a classical optimal tax problem with welfare weights () f (). 3 We itegrate µ () u0 () d by parts to obtai: 2 The first order coditios for this problem will be su ciet. We ca rewrite the problem i terms of ũ () u (c ()) ad h () h (z () /) sothattheobjectivefuctiobecomesh h () u (ũ ()) ad is cocave i ũ () ad h () that become the ew cotrol ad state variables. The problem would be: s.t. dv () d ˆ v () () f () d U ( c (v (),z(),),z(),) ˆ [z () c (v (),z(),)] f () d E 8

9 ˆ µ () v 0 () d µ ( ) v ( ) µ () v () ˆ µ 0 () v () d ad rewrite the Lagragia as follows: L ˆ ˆ (z () c (v (),z(),)) f () d + +µ ( ) v ( ) µ () v () ˆ µ 0 () v () d () v () f () d ˆ µ () U ( c (v (),z(),),z(),) d The FOC wrt z () is: d c (v (),z(),) dz () ad the FOC wrt v () is: f () d c (v (),z(),) µ () U c () + U z () 0 (8) dz () d c (v (),z(),) f () µ 0 d c (v (),z(),) () µ () U c () + () f () 0 (9) dv () dv () We kow from the paragraph about optimal taxatio with icome e ects that we ca write: d c (v (),z(),) dz () U z () U c () MRS() Also, sice T 0 () MRS() (see the discussio about wedges) we have that: d c (v (),z(),) dz () MRS() T 0 () The term i square brackets i equatio (8) ca be writte as follows: U c () d c (v (),z(),) dz () Therefore, euqatio (8) becomes: + U z () U c () U z () U c () + U z () U c () U z () U c () U c () U z () U c () 2 U c Uz U c () U c Usig MRS() T 0 () f () µ () U c () T 0 (), we ca rewrite the coditio T 0 () T 0 () f () µ () log MRS() c Now, we move to the secod FOC. We kow from before that: 9

10 d c (v (),z(),) dv () U c () We ca therefore rewrite (9) as: f () U c () µ 0 () µ () U c () U c () + () f () 0 Ay Pareto E ciet allocatio must satisfy (7) ad provide at least utility v ( ) to every aget. By the complemetarity-slackess coditio, this is equivalet to ask that () f () 0, which is that the multipliers associated to the costraits are ever egative. We rewrite the FOC imposig the followig iequality: If we chage variables ad defie ˆµ () U c () µ (), we have: U c µ 0 () µ () U c () f () (0) ˆµ 0 () U c () µ 0 ()+µ ()[U c ()+U cc () c 0 ()+U cz () z 0 ()] Substitutig ito (0) we fid: ˆµ 0 ()+ˆµ () U cc () c 0 ()+U cz () U c () f () The local icetive costrait of the aget (FOC for optimal reportig) implies that c 0 () /z 0 () U z () /U c (). It follows that: U cc () c 0 ()+U cz () U c () We fially establish two coditios for Pareto e U cc () c 0 () z 0 () + U cz () z 0 () U c () U cc () U z ()+U cz () U c () U c () 2 z 0 Uz () z 0 U c () z 0 ciecy: T 0 log MRS() +T 0 f () ˆµ () () ˆµ () () ˆµ () z 0 () f () Usig the tax schedule i place, the prefereces ad the skill distributio we ca derive the ˆµ from equatio (). We ca the use equatio (2) to test for the Pareto e ciecy of the tax schedule. 0

11 Applyig the Test ad Iterpretig the Coditios Suppose the aget has prefereces U (c, z, ) c (z/) with elasticity " / ( ). The FOCs of the dual problem read: µ 0 () () f () f () (3) T 0 () () +" T 0 f () µ () " The tax schedule is Pareto-optimal if ad oly if () f () 0, which implies µ 0 () f (). This iequality is the same as the oe derived i (2) sice U c () () /@c 0. Equatio (4) is the same as () whe we otice () /@ (+/") /. 4 Suppose that the margial tax rate is liear ad equal to, whe we put the two coditios together we get: f () µ0 () f () 8 Takig the derivative wrt the coditio becomes: " f 0 () +" f () (4) 8 (5) First, the coditio i (5) shows that for ay ad " there exists a set of f () such that is Pareto e ciet ad a set of f () such that it is ot Pareto e ciet. At the same time, for ay " ad f () we ca fid flat tax schedules that are e ciet ad set of s that are ie ciet. It follows that it is crucial to kow the distributio of skills. The test ca also be writte i terms of icome distributios that are easier to ifer from the data. Higher " makes the coditio harder to be satisfied: whe idividuals are reactig more to chages i taxes, a tax reductio is more likely to lead to a Pareto improvemet. Whe taxes are locally lowered at some, the idividuals below will ted to icrease their labor supply ad idividuals above will reduce the labor supply. The term f 0 () /f () measures the elasticity of the skill distributio ad captures how fast the skill distributio is decreasig at some. Highly egative elasticity of skill distributio at meas that the distributio decreases fast ad that the mass of idividuals below is sigificatly larger tha above, implyig a local La er e ect from the icrease i labor supply of idividuals below. I other words, by locally decreasig taxes at the govermet ca icrease reveues by icetivizig the labor supply of the large mass of idividuals below. For this reaso, whe the elasticity of the skill distributio is highly egative the test is harder to pass. 5 Refereces [] Diamod, Peter (998), Optimal Icome Taxatio: A Example with a U-Shaped Patter of Optimal Margial Tax Rates, America Ecoomic Review, 88, [2] Salaiè, Berard (2002), The Ecoomics of Taxatio, The MIT Press [3] Scheuer, Floria (206), Leture Notes (fid at pdf) [4] Werig, Iva (2007), Pareto E ciet Icome Taxatio, Workig Paper 4 Notice that MRS () z () log MRS () where +/". 5 I the secod part of the sequece you will study how to ru the same test i the iequality deflator framework usig the cocept of fiscal exterality.