Optimal Redistributive Taxation in a Search Equilibrium Model.

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1 Optiml Redistributive Txtion in Serch Equilibrium Model. Mthis HUNGERBÜHLER Alexis PARMENTIER cdb November 16, 2005 Etienne LEHMANN bce Bruno VAN DER LINDEN cfeb Abstrct This pper chrcterizes optiml non-liner income txtion in n economy with continuum of unobservble productivity levels nd endogenous involuntry unemployment due to frictions in the lbor mrkets. Redistributive txtion distorts lbor demnd nd wges. Compred to their efficient vlues, gross wges, unemployment nd prticiption re lower. Averge tx rtes re incresing. Mrginl tx rtes re positive, even t the top. Finlly, numericl simultions suggest tht redistribution is much more importnt in our setting thn in comprble Mirrlees (1971) setting. Keywords: Optiml Income Txtion, Unemployment, Wge Brgining, Mtching JEL numbers: D82, H21, H24, J64. Deprtment of Economics, University of Nmur, Remprt de l Vierge 8, 5000 Nmur, Belgium. b ERMES, Université Pris 2, 12 Plce du Pnthéon, Pris Cedex 05, Frnce. c Deprtment of Economics - IRES, Université Ctholique de Louvin, Plce Montesquieu,3,B1348LouvinLNeuve,Belgium. d EUREQu, Université Pris 1, Boulevrd de l Hôpitl Pris Cedex 13, Frnce. e IZA, Bonn, Germny. f FNRS, Belgium. We thnk prticipnts in seminrs t EUREQu, the Dy for Belgin Lbour Economists, CREST, AFSE, Nmur, Evry, T2M 2004, ESEM 2004, EALE 2004, IDEI nd S t Etienne. We thnk Antoine d Autume, Robin Bodwy, Pierre Chuc, Brt Cockx, Per Engström, Jen Michel Grndmont, Jen Hindriks, Herwig Immervoll, Frncis Krmrz, Guy Lroque, Dvid Mrgolis, Dominique Meurs, Gwenël Piser, Thoms Piketty, Fbien Postel Viny, Jen Chrles Rochet, Emmnuel Sez, Etienne Wsmer, Fbrizio Zilibotti nd the two nonymous referees. The usul cvet pplies. The uthors cknowledge finncil support from the Belgin Federl Government (Grnt PAI P5/21, Equilibrium theory nd optimiztion for public policy nd industry regultion ). This reserch hs been done while Hungerbühler ws t IRES nd ERMES. Hungerbühler dditionlly thnks the Centre Cournot for Economic Studies for finncil support. 1

2 I Introduction This pper studies optiml non-liner income txtion in n economy with serch frictions nd privte informtion bout productivity levels. As in the seminl rticle of Mirrlees (1971), the government wnts to redistribute incomes from more to less productive gents, but fces n dverse selection problem becuse it only observes incomes nd not productivities. Hence, txtion cretes inefficiencies. Since Mirrlees (1971), the optiml tx literture hs focused on lbor supply s the source of such inefficiencies. This literture, however, does not cpture essentil spects of lbor mrkets. In prticulr, these re ssumed to be perfectly competitive nd unemployment only rises from preference for leisure. This rticle considers n lterntive, nd in our view importnt, source of inefficiency: Txes lter the outcomes of the wge brgin nd hence ffect lbor demnd. In turn, lbor demnd hs n impct on serch nd mtching frictions nd thus on unemployment. Our model is bsed on the serch nd mtching literture (e.g., Mortensen nd Pissrides (1999)). Firms post job vcncies nd individuls choose whether to prticipte or not in the lbor mrket. Those individuls who prticipte but fil to be mtched re clssified s unemployed, while those who do not prticipte re considered s inctive. Mtching frictions crete serch externlities nd generte rents tht re shred between workers nd firms through Nsh-brgining on wges. In order to focus shrply on the redistribution issue, we bstrct from stndrd inefficiencies rising from serch frictions by imposing the so-clled Hosios (1990) condition 1. In the presence of redistributive txtion, rise in the mrginl tx rte implies tht n increse in gross wges hs reduced impct on net wges. Therefore, it becomes less rewrding for workers to brgin ggressively nd gross wges fll (see, e.g., Lockwood nd Mnning (1993)). This in turn increses lbor demnd nd decreses unemployment. Such increse in lbor demnd is however not necessrily efficient, becuse rise in employment increses both gross output nd the resources bsorbed to crete dditionl vcncies. One cn show tht when employment is bove its efficient level, rise in lbor demnd decreses output net of vcncy costs. The government trdes off this efficiency loss ginst the equity gin of higher tx revenue induced by n increse in the mrginl tx rte. Hence, the optiml (i.e. second-best) wges nd thus the optiml unemployment rte re below their efficient (i.e. first-best) levels. A rise in the verge tx rte leds to wge pressure nd thus decreses lbor demnd. This cn however be compensted for by n equivlent increse in the mrginl tx rte. To obtin optiml wges below their efficient level, the government must set mrginl tx rtes bove verge tx rtes. This implies tht verge tx rtes re incresing through the whole distribution of productivities. Moreover, optiml mrginl tx rtes re positive t the top of the distribution. Since distorting employment t the top genertes no equity gins, the gross wge nd the employment rte of the most productive workers tke their efficient vlues. Hence, the wge pressure t the top generted by the positive level of txes hs to be compensted for by positive mrginl tx rte. Txtion lso ffects the individuls expected gin from prticipting in the lbor mrket. If the government decreses txes on low-productivity individuls to induce them to prticipte, dverse selection implies tht the government hs to give dditionl informtionl rents to ll individuls with higher productivity. This equity loss my well 1 An lterntive mechnism tht gurntees efficiency in the economy without txes is the competitive serch frmework proposed by Moen (1997). 2

3 outweigh the efficiency gin of incresed lbor mrket prticiption. As we show, this is the cse t the bottom of the productivity distribution. Hence, the government tkes the lowest-productivity individuls out of the lbor mrket by giving n ssistnce benefit to the inctive workers tht is necessrily higher thn the in-work benefit given to employed workers. As consequence, lbor mrket prticiption is below its efficient level. Some numericl nlysis illustrtes the properties of the optiml tx schedule. The optiml tx schedule is lmost liner. Depending on the chosen prmeters, optiml mrginl tx rtes lie between 42% nd 66%. To compre these numericl results with those obtined in Mirrlees setting, we clibrte the Mirrlees model to generte the sme distribution of ernings nd the sme elsticity of gross ernings with respect to the mrginl tx rte s our model. The results turn out to be considerbly different with our recommended mrginl tx rtes being t lest twice s high. The wy we model the impct of tx policy on lbor demnd is novel to the optiml tx literture. The stndrd Mirrlees model ssumes n ggregte production function with perfect substitution between the different types of lbor. Hence, hourly wges equl mrginl products of lbor nd re independent of txtion. In such model, txtion only ffects lbor supply. In prticulr, n increse in the mrginl tx rte mkes n dditionl working hour less rewrding in terms of dditionl net income. Hence, s in our model, gross income decreses. However, while in our model, the decrese in gross income emerges from the wge brgin nd induces rise in lbor demnd, in the Mirrlees model gross income decreses becuse of decline in lbor supply. Other uthors hve studied models where tx policy hs n impct on lbor demnd. Stiglitz (1982) considers two-skill model where the different types of lbor re not perfect substitutes in the production function. He shows tht in this cse, the government s tx policy ffects the hourly skill premium nd thus ernings. In contrst to the wge-brgining mechnism exposed in the present rticle, lbor demnd is ffected by the production function nd not by lbor mrket frictions. Engström (2002) extends the two-skill model in mtching frmework with exogenous hourly wges. As in the Mirrlees model, tx policy ffects lbor supply. This in turn chnges the rent to be shred between the worker nd the firm nd hence lbor demnd nd unemployment. The nlyticl frmework in our pper is different since we consider fixed working hours but endogenous wges. This pper is orgnized s follows. Section II presents the model nd derives the nlyticl results. Section III is devoted to numericl simultions. In section IV, we compre the optiml schedule in our setting to the optimum in Mirrlees-type setting. Section V concludes. II The model We consider sttic model 2 where jobs differ ccording to their exogenous productivity denoted by [ 0, 1 ] with 0 0 < 1 +. The intensive mrgin of the lbor supply is not tken into ccount here. Workers nd firms re ssumed to be risk neutrl. Directed serch is ssumed for simplicity 3.So,type- ctive workers serch for type- jobs. Firms open type-specific vcncies. Ech vcncy hs to be filled by single serching worker. Mtching workers nd vcncies is costly ctivity. Following Mortensen nd Pissrides 2 Our sttic model simplifies the dynmic version of the mtching model but still cptures its mjor mechnisms (see e.g. Boone nd Bovenberg (2002)). 3 Since we consider the whole distribution of productivity levels, directed serch seems to be less unrelistic ssumption thn undirected serch. 3

4 (1999) nd Pissrides (2000), we consider well-behved mtching function tht gives the number (mesure) of type- jobs formed s function of the number U of serching workers nd the number V of vcncies. Employment in segment is n incresing nd constnt-return-to-scle function H (U,V ). This mtching function implicitly cptures heterogeneities, frictions nd informtion imperfections on the lbor mrket. The size of the popultion is normlized to 1. Workers types re distributed ccording tocontinuousdensityf (.) nd c.d.f. F (.). These functions re common knowledge. Through costly screening, the productivity of worker is observed by the firm. We ssume the government does not hve this bility. The income tx system consists in continuously differentible non-liner tx function T(.) nd n untxed ssistnce benefit b. Since the government observes gross wges but not productivity levels, T (.) conditions only on the gross wge. We ssume tht the job serch cnnot be monitored by the government. The ssistnce benefit b is thus grnted to unemployed nd inctive individuls. Forechtype, w, L nd x denote respectively the gross wge (or equivlently the wge cost), the employment rte nd the workers ex-post surplus in cse of employment, with x w T (w ) b. Hence, type- employed workers receive w T(w )=x + b. Further, we define Σ x L s the workers expected surplus nd Y w L s the workers expected gross income. Let d>0 be the vlue of inctivity. Irrespective of their type, inctive (respectively unemployed) individuls receive b + d (resp. b). Posting type- vcncy costs κ. This prmeter cptures the cost of screening pplicnts nd the investment cost of creting worksttion. A type- filled (respectively unfilled) vcncy yields profit of w κ (resp. κ )tothefirm-owner. In the literture, the vcncy cost is either tken s fixed or s proportionl to productivity (see e.g. Pissrides (2000)). We therefore ssume tht 4 : 0 κ κ 1 (1) The timing of the model is: 1. The government commits to tx system defined s pir (T (.),b), witht (.) : R + 7 R nd b R. 2. Firms open vcncies nd workers decide whether or not to serch for job. 3. Mtching occurs. Once mtched, the firm nd the worker negotite the wge. 4. Trnsfers ccrue to the gents. This section is orgnized s follows. First, we del with the objective function nd the budget constrint of the government. Second, we introduce frictions nd the demnd side of the lbor mrket. Third, we chrcterize efficiency in the idelized cse where the government hs perfect informtion on productivities. Fourth, we del with the wge brgin. Fifth, prticiption nd incentive constrints re defined in the cse where the government observes wges but not productivities. Finlly, the properties of the secondbest optimum re derived. 4 A dot over vrible denotes the totl derivtive with respect to type (e.g κ = dκ /d). 4

5 II.1 The government We first present the government s objective nd its budget constrint. How productivity levels re llocted in the popultion is out of the scope of this rticle. People re simply not held responsible for their productivity. So, the government is redy to compenste for differences in productivity levels. We ssume the following objective for the government: Z 1 Ω = {p Φ [L (w T (w )) + (1 L ) b]+ (2) 0 (1 p ) Φ (b + d)} f () d where Φ 0 (.) > 0, Φ 00 (.) 0, p =1if type- workers prticipte in the lbor mrket nd p =0otherwise. This objective expresses the fct tht the government cres bout the distribution of expected utilities, nmely L (w T (w )) + (1 L ) b for those who re ctive nd b + d for inctive people. It encompsses s limiting cses the mximin criterion (mx b) nd the pure utilitrin criterion (whenever Φ 00 (.) =0). For expository resons, we neglect the issue of insurnce ginst the unemployment risk. As will soon be shown, over-employment is optiml t the second best for it contributes to the fulfillment of the incentive comptibility constrint. Under risk version nd when insurnce is incomplete, over-employment comes out for distinct reson, nmely becuse it enbles better shring of risks (see Hungerbühler et l. (2005)). Finlly, notice tht objective (2) cn be microfounded if we ssume tht the economy is mde of productivity-specific representtive households tht perfectly shre consumption between their employed nd unemployed members. From Σ = L (w T (w ) b), we rewrite this objective s: Z 1 {p Φ [Σ + b]+(1 p ) Φ (b + d)} f () d (3) Ω = 0 The government fces the budget constrint: Z 1 Z 1 0 p T (w ) L f () d = {p (1 L )+1 p } f () d b + E (4) 0 where E 0 is n exogenous mount of public expenditure. Since Y Σ = L [T (w )+b], we rewrite the government s budget constrint (4) s: Z 1 0 p (Y Σ ) f () d = b + E (5) II.2 The mtching process Following empiricl studies (see Blnchrd nd Dimond (1989) or Petrongolo nd Pissrides (2001)), we ssume Cobb-Dougls mtching function. The number of type- mtches is function of the number of type- vcncies V nd of the number of type- serching workers U ccording to: H = A (U ) γ (V ) 1 γ with γ (0, 1) All type- individuls either serch for job or sty inctive. If they serch, their number is U = f(). Their probbility of finding job (resp. the probbility of filling type- 5

6 vcncy) is L = H /U = A θ 1 γ (resp. H /V = A θ γ ) where θ V U denotes lbor mrket tightness in the type- segment of the lbor mrket. An increse in the number of vcnt jobs induces two types of serch externlities: It increses the probbility for n unemployed worker to find job, nd it diminishes the probbility for vcncy to mtch with worker. Similrly, n increse in the number of workers prticipting in the lbor mrket increses the probbility for vcnt jobs to be filled nd decreses the probbility forunemployedworkerstomtchwithfirm. ( w ) κ. The higher the gross wge w, the lower this return. Firms freely enter the mrket nd post vcncies s long s this return is positive. Therefore, in equilibrium, this return is nil (the so-clled free-entry condition ). One cn then derive the type- probbility of being employed (or the lbor demnd ): TheexpectedreturnofpostingvcncyisA θ γ µ L = A 1 γ w κ 1 γ γ The free-entry condition implies tht workers expected gross wge w L equls net output (i.e. totl output net of serch costs L θ κ ): Y = w L = L θ κ Tking (6) into ccount, we write net output under the free-entry condition s: (6) µ Y (w ) A 1 γ w κ 1 γ γ w (7) II.3 The first-best optimum In this subsection, we ssume tht the government perfectly observes productivities. The government chooses the ssistnce benefit b, thewgew, the prticiption indictor p nd workers expected surplus Σ to mximize the socil objective (3) subject to the budget constrint (5) nd the lbor demnd (6): s.t : mx w,p,σ,b Z 1 0 Z 1 0 {p Φ [Σ + b]+(1 p ) Φ (b + d)} f () d (8) p (Y (w ) Σ ) f () d = b + E As shown in ppendix A, the solution to this problem implies the following results. First, Σ = d for ll prticipting types. Hence, gents expected utility is equted cross prticiption/non-prticiption sttes nd productivity types. Second, the first-best gross wge w mximizes net output. Efficient (i.e. net output-mximizing) vlues of gross wges, employment, nd net output re: µ w = γ, L =(1 γ) 1 γ γ A 1 γ µ Y = Y (w)=γ (1 γ) 1 γ γ A 1 γ κ κ 1 γ γ 1 γ γ (9) To increse employment bove L, firms hve to open more vcncies. The resources spent on creting these vcncies re not offset by the increse in output. Mtching frictions 6

7 therefore imply tht full employment is not efficient 5. Equtions (1) nd (9) imply tht the efficient level of employment is non-decresing in. Finlly, let d be defined by: Y d = d if d > = 0 (10) The efficient prticiption condition implies tht every type bove (below) d prticiptes (stys inctive). II.4 Thewgebrgin We henceforth consider the cse where the government does not observe productivities. Therefore, the tx schedule T (.) is function of gross wges only. Once firm nd worker re mtched, they brgin over the wge. In the bsence of n greement, nothing is produced nd the worker gets the ssistnce benefit b. These outside options imply the existence of positive rent T (w ) b. As is stndrd in the literture (see Pissrides (2000)), this rent is shred by mximizing Nsh product. The wge w mximizes the Nsh product depending on the worker s nd the firm s surplus if they rech n greement: mx w [w T (w ) b] β [ w ] 1 β tking b nd T (.) s given nd where β (0, 1) denotes the worker s brgining power. For convenience, we redefine the Nsh product s: mx w [w T (w ) b] A 1 γ [ w ] 1 β β From Eqution (6), workers expected surpluses coincide with the (redefined) Nsh products if, s we henceforth ssume, the Hosios condition β = γ is fulfilled. This condition sttes tht the reltive weight of the firm s surplus in the Nsh product (1 β) /β is equl to the elsticity of lbor demnd with respect to the firm s surplus (1 γ) /γ. Underthe Hosios condition, efficiency is reched in the bsence of txes nd benefits. The results of this pper would therefore lso be obtined in the cse of ny other wge setting tht mximizes workers expected utility Σ + b, given the lbor demnd function (6) nd the tx system (T (.),b). This is in prticulr the cse with skill-specific monopoly unions. Henceforth, Σ will denote both the workers expected surplus evluted t brgined wges nd the mximised Nsh product: Σ =mx w [w T (w ) b] A 1 γ w κ 1 γ γ (11) The first-order condition leds to: w = γ (1 T 0 ) +(1 γ)(t + b) γ (1 T 0 )+1 γ (12) where T 0 T 0 (w ) denotes the mrginl tx rte nd T T (w ) denotes the level of txes for type- worker. Furthermore, the first-order condition of (11) implies: w w = γ 1 γ 5 A nd κ ressumedtobesuchthtl < 1 nd w >κ. 1 T 0 1 T+b w (13) 7

8 Since neither the firms nor the workers ex-post surplus cn be negtive, we conclude tht: T 0 1 Therefore, the worker s ex-post surplus x = w T (w ) b is necessrily n incresing function of the gross wge w. In the rest of this subsection, we consider the level of tx T nd the mrginl tx rte T 0 s prmeters. The tx schedule influences the lbor mrket equilibrium in two wys. First, higher levels of txes T (or benefits b) reduce the workers ex-post surplus. Therefore, workers clim higher wges. This positive effect on individul gross ernings is similr to wht occurs in lbor supply frmework when leisure is norml good. There, rise in the level of txes t given mrginl tx rtes increses lbor supply. Here this upwrd pressure on wges reduces employment. Second, keeping T constnt, the wge is decresing in the mrginl tx rte T 0 becuse unit rise in the gross wge increses net ernings t rte of one minus the mrginl tx rte. As the mrginl tx rte rises, workers ern less from ech increse in gross wges while the effect on firms profits remins unchnged. Therefore, workers hve fewer incentives to clim higher wges (see Mlcomson nd Srtor (1987), Lockwood nd Mnning (1993), Holmlund nd Kolm (1995), Pissrides (1998), Sørensen (1999) or Boone nd Bovenberg (2002) mong others). As in lbor supply frmework, higher mrginl tx rtes decrese individul gross ernings. However, in the lbor supply literture, the chnnel is different becuse there gross ernings re decresing due to lower working hours. Here, lbor demnd rises. Y T * T Figure 1: The impct of the mrginl tx rte T 0 on efficiency t given level of txes T. AriseinT 0 t given tx level T hs non-monotonic effect on net output Y.The gross wge w decreses ccording to Eqution (12). So, employment L nd therefore gross output L nd totl vcncy costs κ θ increse. When the wge level is higher (respectively lower) thn its efficient vlue w,theeffect on gross output (resp. on vcncy costs) domintes, so net output increses (resp. decreses). The reltionship between net output nd the mrginl tx rte is therefore hump shped (see Figure 1). Let the efficient mrginl tx rte T 0 be the one tht mximizes net output for given 8

9 level of tx. From Equtions (9) nd (13): T 0 = T + b γ = T + b w (14) This equlity estblishes n upwrd reltionship between the efficient mrginl tx rte ndtheleveloftxes. II.5 Prticiption nd incentive constrints A worker decides to serch s long s the expected utility when serching, Σ +b, ishigher thn the ssistnce benefit b plus the vlue of inctivity d. The prticiption constrint cnthenbewrittens: Σ d (15) Applying the envelope theorem to Eqution (11), we get: Σ = 1 γ µ 1 κ Σ (16) γ w κ Equtions (1) nd (16) imply tht the mximized Nsh product or equivlently the expected worker s surplus, Σ, is incresing in productivity. Therefore, there exists single threshold d 0 such tht workers endowed with < d sty inctive wheres workers with d serch for job. The strtegic interction between the firm nd the worker is modelled in reduced form wy. Given tx system (T (.),b), theworker-firm pir chooses the wge tht mximizes the Nsh product 6. Any tx system thus induces four tuple where [ d, 1 ]: w =rgmx wâ [wâ T (wâ) b] A 1 γ wâ x = w T (w ) b Σ =[w T (w ) b] A 1 γ w κ κ ³ d, {w,x, Σ } [d, 1 ] Σ d = d ³ The government cn be modeled s selecting tx system (T (.),b) nd tuple d, {w,x, Σ } [d, 1 ] subject ³ to these constrints. Equivlently, the government cn be modelled s choosing the tuple d, {w,x, Σ } [d, 1 ] directly subject to the incentive comptibility constrint: 1 γ γ 1 γ γ, Σ = x A 1 γ w κ 1 γ γ xâ A 1 γ wâ κ 1 γ γ, ã [ d, 1 ] (17) nd the prticiption constrint Σ d d. This equivlence comes from the fct tht ny lloction vilble to the government in the first problem is vilble to it in gme in 6 We ssume tht ernings re perfectly nd costlessly observed. Tx evsion is therefore neglected nd side-pyments re not llowed. 9

10 whichitselectsdirectmechnismndthenworker-firm pir selects report 7. This equivlence result is commonly known s the txtion principle (see Hmmond (1979) nd Guesnerie (1995)). By the reveltion principle, ttention cn then be restricted to equilibri tht entil truth telling. Stndrd principl-gent techniques then pply (see Slnié (1997) nd Lffont nd Mrtimort (2002)). The Nsh product cn be written s function N (, w, x) of the type, the gross wge w nd the worker s ex-post surplus x in the following wy: N (, w, x) A 1 γ µ w κ 1 γ γ The worker s surplus x hs to increse when the gross wge w increses to keep the Nsh product N (,.,.) unchnged. For ech pir (w, x) the mrginl rte of substitution: x = 1 γ x w N(,.,.) γ w is decresing function of the type. This single-crossing property is illustrted in Figure 2 tht displys the indifference curves in terms of the Nsh product N (,.,.) for workerfirm combintions with different productivity levels 0 >. The higher the productivity of mtch, the less elstic is the firm s surplus to the gross wge nd the less sensitive is the Nsh product to chnges in the gross wge. Given the single crossing property, the incentive comptibility constrint (17) is equivlent to the first order condition (16) nd the second-order condition which re (see ppendix B.1): [ d, 1 ] Σ = 1 γ γ µ 1 κ Σ nd ẇ > 0 w κ x x > N x=w-t(w)-b N x w w Figure 2: The single-crossing property. 7 The rewriting described bove shows the equivlence of the tx function to clss of direct mechnisms. The ltter mechnisms bsiclly tret the worker nd the firm s single gent with preferences described by the Nsh brgin objective. There my be other mechnisms vilble to the government tht exploit thefctthtthefirm nd the worker re different nd tht deliver better outcome (see e.g. Cremer nd McLen (1988)). However, they re not considered here. 10

11 II.6 The second-best optimum Under symmetric informtion, the government chooses the threshold d, the ssistnce benefit b, the wge nd the workers expected surplus functions (w, Σ ) to mximize the government s objective (3) subject to the budget constrint (5), the lbor demnd (6), the prticiption constrints (15) nd the incentive constrints (16) 8. The progrm of the government is therefore: Z 1 mx F ( d ) Φ (b + d)+ Φ (Σ + b) f () d (18) d,w,σ,b d s.t. : Z 1 d (Y (w ) Σ ) f () d = b + E Σ = 1 γ γ ( 1 w κ κ )Σ ½ Σd = d Σ d d if d > 0 d = 0 For ny vlue of d nd b, this is stndrd optiml control problem where workers expected surplus Σ is the stte vrible nd the gross wge w is the control vrible. The first-order conditions led to the following formultion of the equity-efficiency trde-off for ll d (see Appendix B.2): λ Y w f () = Z 1 γ 1 ª λ Φ 0 γ ( w ) 2 t Σt f (t) dt (19) where λ is the Lgrnge multiplier ssocited with the budget constrint nd Φ 0 Φ 0 (Σ + b). For< d,wedefine Φ 0 Φ 0 (b + d). Consider mrginl increse in the type- wge. The mximized Nsh product chieved by type- worker-firm combintions, Σ,isfixed by the incentive constrints for less productive pirs. The rise in the wge decreses the employment level L nd thereby gross output L but lso the resources spent on posting vcncies. The effect on net output Y is therefore mbiguous. If w <w (resp. >), this effect is positive (resp. negtive). Multiplying this by the number of type- gents f() nd the shdow cost of public funds λ, the left-hnd side of (19) mesures the socil vlue of the net mrginl chnge in output. This cptures the efficiency side of the trde-off. The right-hnd side of (19) represents the equity cost of higher gross wge for type- worker-firm combintions. As w rises t fixed Σ,moreproductiveworker-firm pirs find it more ttrctive to mimic type- combintions. To prevent this, the Nsh product ccruing to the former hs to grow. Looking t Eqution (16), the term in front of the integrl on the right-hnd side of (19) mesures by how much the rte of chnge of the mximized Nsh product Σ /Σ hs to grow when w mrginlly increses. The incentive comptibility constrints will remin stisfied if ll combintions with productivity higher thn benefit from n equivlent reltive increse in their Nsh product. For ny type t bove, this reltive increse times Σ t gives the rise in the Nsh product. Ech unit of the ltter genertes n increse in the socil welfre mesured by Φ 0 t nd implies budgetry cost equl to λ. The proof of the following normtive properties is set out in the ppendices. Proposition 1 The optiml levels of the gross wge nd employment re efficient t the top of the distribution. 8 Throughout the pper, we consider only the first-order incentive constrint nd verify ex-post in our simultions tht ẇ > 0. 11

12 The right-hnd side of (19) indictes tht chnging the gross wge t the top hs no distributionl effect. The government cn therefore set this gross wge t the level w 1 which mximizes net output. Proposition 2 For ll worker-firm pirs with productivity d < 1,theoptimlgross wges re below their efficient levels w nd the optiml employment levels re bove their efficient levels L. Consider tht the tx schedule hs been optimized for ll workers up to type. The mximized Nsh product of type- worker-firm combintions, Σ, is predetermined by the incentive comptibility constrints. This level of the Nsh product is depicted in Figure 3 by the curve denoted Σ.Letw decrese below its efficient vlue w. This only hs second-order effect on efficiency. But there is first-order effect on the Nsh products nd thus on workers expected surplus Σã for types ã>. By reducing w,thegovernment cn reduce the ltter (see Eqution (16) nd the downwrd shift of the curve lbeled Σã in Figure 3). The government cn extrct more tx revenues from these types bove. This gin in resources is vlued t the mrginl cost of public funds λ. The loss in the workers expected surplus for these types is vlued t the mrginl socil welfre Φ 0 ã, ã>. Following Eqution (19), these two effects re integrted over ll types bove. Sincethe ssistnce benefit is optimlly chosen, the property R 1 0 (Φ 0 t λ) f (t) dt =0holds (see Eqution (28) in ppendix B.2). As Σ is incresing in, Φ 0 is decresing in, ndthe right-hnd side of (19) is therefore lwys positive. In other words, the dditionl tx revenues re more vlued thn the loss in utility bove. Therefore the optiml vlue of w should be below its efficient level. x Σ ã > Σ ã w w * w Figure 3: The equity-efficiency trde-off. Another intuition for Proposition 2 is given by Figure 1. Keeping the level of txes unchnged up to T, rise in the mrginl tx T 0 cretes n equity gin since it enbles the government to tx richer workers more hevily. At the optimum, this equity gin hs to be compensted for by loss in efficiency. According to Figure 1, the mrginl tx rte is then necessrily higher thn the efficient one. Consequently, the wge rte is below its efficient level nd therefore there is over-employment. So, s in the stndrd optiml 12

13 income txtion model, incentive constrints led to decline in gross ernings. However, this reduction here implies rise in lbor demnd, wheres in the stndrd optiml income txtion literture it follows from reduction in lbor supply. Proposition 2 recommends over-employment for ll prticipting types (except t the top of the productivity distribution). Mny re convinced tht there ctully is underemployment, in prticulr t medium nd low productivity levels. Adopting this view, Proposition 2 hs to be considered in normtive wy. As the underlying model is highly stylized, this result should be considered with cre. The min messge is tht endogenizing wges nd lbor demnd leds to recommending fiscl stimultion of lbor demnds for prticipting types. The next proposition dels with the dverse effect of this stimultion on the optiml prticiption rte. Proposition 3 The optiml prticiption rte is lower thn or equl to its efficient vlue. Aggregte net output nd hence efficiency increse if individuls of types [ d, d) prticipte. But their prticiption lso gives to worker-firm pirs with productivity bove d the possibility of mimicking them. To void this mimicking, the government hs to give n dditionl informtionl rent to these more productive mtches. If this equity cost is higher thn the efficiency gin, the government prefers tht individuls of types [ d, d) do not prticipte. Combining Propositions 2 nd 3, the efficient nd the second-best optiml levels of totl employment L = R 1 0 L f () d cnnot be rnked. On the one hnd, fewer people prticipte in the lbor mrket. On the other hnd, more prticipnts re employed. Proposition 4 In-work benefits (if ny) re lower thn ssistnce benefits. An in-work benefit (i.e. T (w d )) tht is higher thn ssistnce benefits increses prticiption. However, proposition 3 shows tht the government chooses not to increse prticiption. This proposition implies tht n EITC (s defined by Sez (2002)) would not be optiml t the second best. Proposition 5 Optiml verge tx rtes re incresing with the wge level. mrginl tx rtes re positive everywhere. Optiml To decentrlize n optimum where wges re below their efficient vlues, optiml mrginl tx rtes re higher thn the verge tx rtes t ech point of the wge distribution (see Eqution (13)). This hs two striking consequences: optiml verge tx rtes re incresing through the whole distribution nd optiml mrginl tx rtes re positive, even t the top of the distribution. The first prt of proposition 5 should be stressed, since stndrd optiml income txtion models with endogenous lbor supply do not yield precise nlyticl results bout the shpe of verge tx rtes. With bounded distribution of productivity, this literture hs shown tht the optiml mrginl tx rte equls zero t the top. Therefore, one only knows tht the optiml verge tx rte is necessrily decresing close to the top of the distribution. Since the vlue of inctivity is unique, the second prt of Proposition 5 is in ccordnce with the stndrd optiml income txtion models, except t the top of the distribution. The reson why the optiml mrginl tx rte is positive t the top in our model is esily understood. As the government wnts to redistribute income in fvor of less productive 13

14 gents, the level of txes is positive t the top. This distorts the gross wge upwrds. To restore n efficient wge level (Proposition 1), positive mrginl tx rte is therefore needed t the top (see Eqution (14)). To produce positive mrginl tx rtes t the top of the distribution, the stndrd optiml income txtion literture needs specific ssumptions such s n unbounded Preto distribution of productivities (Dimond 1998 nd Sez 2001). There re no nlyticl results bout the profile of mrginl tx rtes. If the levels of wges nd of employment were efficient everywhere, the upwrd profile of verge tx rtes would require rising profile of mrginl tx rtes ccording to Eqution (14). However, Proposition 2 implies tht there is over-employment everywhere, except t the top. Mrginl txes re therefore higher thn their efficient vlues. From Propositions 1 nd 2, one knows tht the positive difference between optiml nd efficient employment levels hs to decline with productivity levels in the neighborhood of the top of the distribution. We do not know more. So, possible chnges in the intensity of the over-employment effect could led to non-monotonic reltionship between mrginl tx rtes nd wges. III Simultions III.1 Clibrtion The prmeters re chosen to roughly represent key figures for Frnce. For the productivity distribution, we use truncted log-norml density function 9 : " f () = K exp 1 µ # log log (µ 1 +(1 µ) 0 ) 2 2 ξ where K is scle prmeter. µ nd ξ re clibrted using the monthly wges of full time working individuls found in the French Lbor Force Survey 2002 (Enquête Emploi) 10, which gives ξ =0.4 nd µ =0.2. The lower bound of the productivity distribution, 0 is set t 0 nd the distribution is truncted t 1 =20000Euros. The distribution of in the benchmrk cse is displyed in solid lines in Figure 4. According to the empiricl literture (see Petrongolo nd Pissrides (2001)), the elsticity of the mtching function, γ, issett0.5. The Hosios condition then implies tht the worker nd the firm hve equl brgining power. There is no micro-evidence vilble for the vlues of the vcncy costs κ nd the scle prmeter of the mtching function, A. We clibrte these vlues to obtin unemployment rtes of 2% t the top nd 25% t the bottom in the economy without txes. Assuming n iso-elstic shpe so tht κ A 1 γ 1 = κ 0 l,wegetκ 0 =0.62 nd l =0.88. Similrly, there is no cler evidence on the vlue of leisure. We tke vlue of leisure d of 500 euros in the benchmrk cse nd conduct therefter sensivity nlyses with respect to this prmeter. To keep things simple, public expenditure E is set equl to zero. Finlly, the government s utility function is ssumed to be CES function of the expected utility, i.e. Φ (Ω) =Ω 1 σ / (1 σ). A 9 This form used to be typicl in the literture (Mirrlees (1971), Tuomol (1990) nd Bodwy et l. (2000); see Dimond (1998) nd Sez (2001) for critique).! " 10 To keep things simple, we pproximte the ctul tx system by considering tht (1 T) 0 / 1 T+b w is constnt, which is in line with the dt from the OECD tx dtbse. Unobserved productivity levels re then proportionl to gross wges w (see. Eqution (13)). Hence, µ nd ξ re the sme for both the wge nd productivity distributions. A sensitivity nlysis with respect to the productivity distribution will be conducted in subsection III

15 f Figure 4: Density Functions f(). The benchmrk cse is the solid line. A dotted line is used for (µ, ξ) =(0.2; 0.6) nd dshed line for (µ, ξ) =(0.15; 0.4). σ γ d µ ξ E Tble 1: Prmeter vlues in the benchmrk cse. government tht is more concerned bout redistribution will be chrcterized by higher vlue of σ. Iftype hs n expected utility N times tht of type 0,thesocilvlue of mrginl increse in expected utility is N σ higher for type 0 compred to. Inthe benchmrk cse, we tke σ equl to 1 (so Φ (Ω) =log(ω)). Tble 1 summrizes the vlues of the prmeters. III.2 The benchmrk As expected from Propositions 1 nd 2, employment is bove its efficient level for ll prticipting types d, except t the top of the distribution 11. As shown in Figure 5, over-employment is more pronounced for low productivity levels. This property is not surprising. As in the Mirrlees setting, our model predicts tht the government should highly distort the outcomes of the low productivity types becuse there re few of them. The corresponding mrginl tx rtes should therefore be high. However, introducing continuous extensive lbor supply mrgin might remove this property. In Sez (2002), mrginl tx rtes cn become negtive if prticiption elsticities re sufficiently high. Similrly, Boone nd Bovenberg (2004, 2005) show tht the introduction of serch costs might lso led to negtive mrginl tx rtes. At the bottom of the distribution, such decreses in mrginl tx rtes could ttenute the over-employment property of our optimum. In line with Proposition 3, the prticiption rte of 94% is below its level in the economy without txes (see Tble 2). Combining this result with the previously mentioned overemployment property for prticipting types, the net effect of these opposing forces is tht totl optiml employment L = R 1 0 L f () d is 1.8% bove its level in the economy without txes (see Tble 2). As depicted by the solid lines in Figure 6, the optiml tx function turns out to be 11 The simultion progrms re vilble on 15

16 L Figure 5: Employment levels in the benchmrk cse. Solid lines for the optimum. Dotted lines for the economy without txes. close to liner. Mrginl tx rtes re first decresing nd then slightly hump-shped. They lie between 52% nd 59%. The optiml level of ssistnce benefit is quite high t 809 Euros per month. III.3 III.3.1 Sensitivity nlysis The version to inequlity When the version to inequlity, σ, is doubled, the government puts more emphsis on redistribution in the equity-efficiency trde-off. Hence, wges re more downwrds distorted nd mrginl tx rtes shift upwrds by bout 5 percentge points (see Figure 6). The ssistnce benefit vries slightly. Since the benchmrk lredy leds to high degree of redistribution, these chnges in the tx system re rther smll. Tble 2 shows the effect of the government s preference for redistribution on totl employment. On the one hnd, n increse in σ decreses the prticiption rte. On the other hnd, the higher mount of redistribution increses employment distortions for the prticipting types. The net effect on totl employment is thus mbiguous. T 5000 Tm w w Figure 6: Dotted, solid nd dshed lines respectively for σ equl to 0.5, 1 nd 2. 16

17 Employment L b Prticiption WT SB WT SB Benchmrk % σ = % σ = % d = % d = % γ = % γ = % ξ = % µ = % E Y = % Tble 2: Numericl results. WT stnds for the economy without txes nd SB stnds for the second best. Y denotes ggregte efficient net output in n economy without txes, i.e. Y = R 1 Y f()d. d III.3.2 The vlue of inctivity Consider rise in the vlue of inctivity d. If the expected surplus Σ remined unchnged long the whole distribution, fewer individuls would prticipte in the lbor mrket. So the budget of the government would not be blnced nymore. Two types of djustments could restore the equlity between expenses nd receipts. First, the level of the ssistnce benefit b could decrese. Second, the decline in prticiption could be ttenuted. The ltter djustment tkes plce if the expected surplus Σ rises. This mens tht the government gives more informtionl rents to ctive workers. As illustrted by Figure 3, wges re then less distorted. Consequently, mrginl tx rtes decline. As cn be seen in Tble 2 nd Figure 7, s d increses, the optiml mix between efficiency nd equity requires tht both djustments tke plce: Mrginl tx rtes nd the ssistnce benefit re reduced. T 5000 Tm w w Figure 7: Dotted, solid nd dshed lines respectively for d equl to 300, 500 nd 700 Euros. 17

18 III.3.3 The lbor demnd elsticity nd the brgining power Since we ssume the Hosios condition to be stisfied, ny chnge in the elsticity of the mtching function γ implies n equivlent chnge in the workers brgining power. The higher the level of γ, the higher is the mtching effectiveness of serching workers. If γ 1, frictions on the lbor mrket dispper. Hence, s γ increses, lbor demnd becomes less elstic to the firm s expected surplus (see Eqution (6)). Thus, txtion becomes less distortive. Consequently, mrginl tx rtes shift upwrds (see Figure 8). Furthermore, rise in γ increses levels of efficient net output 12. This efficiency gin increses the expected surplus of ll prticipting types. Hence, prticiption increses. This improvement in efficiency is shred with the non-prticipting types through n increse in the ssistnce benefit b (see Tble 2). T 5000 Tm w w Figure 8: Dotted, solid nd dshed lines respectively for γ equl to 0.4, 0.5 nd 0.6. III.3.4 The form of the distribution In Figure 9 nd Tble 2, we report the effects of decrese in µ (dshed lines) nd of rise in ξ (dotted lines). Adecresein µ shifts the productivity distribution to the left (see Figure 4). The effects of decline in µ ndofriseind re therefore very similr. The impct on prticiption is negtive. To restore blnced budget, ssistnce benefits b decrese nd the decline in prticiption is ttenuted. For the ltter purpose, expected utility levels re incresed. Therefore, wges re less distorted nd mrginl tx rtes shift downwrds in Figure 9. Ariseinξ implies higher dispersion of productivities (see Figure 4). Therefore, there re more low productivity workers whose expected utility levels re highly vlued by the government. There re lso more high productivity workers whose expected utility levels hve lower socil vlue. Consequently, in the equity-efficiency trde-off, thegovernment puts more emphsis on redistribution. Therefore, mrginl tx rtes nd the level of ssistnce benefit increse, while prticiption decreses. These results re similr to those obtined by Knbur nd Tuoml (1994) in Mirrlees-type model of optiml income txtion. 12 The vcncy costs re reclibrted to keep the top nd bottom vlues of the unemployment rte in the economy without txes unchnged. Then, totl efficient net output Y increses from 1430 to 1794 nd to 2155 s γ increses from 0.4 to 0.5 nd

19 T 5000 Tm w w Figure 9: Dotted, solid nd dshed lines respectively for(µ; ξ) =(0.2; 0.6), =(0.2; 0.4) nd =(0.15; 0.4). III.3.5 Public expenditure When public expenditure increses to 359 (which mounts to 20% of totl efficient net output Y ) mrginl tx rtes re lmost unffected. The min difference is reduction in the ssistnce benefit (see Tble 2) nd n increse in the level of txes by bout the sme mount s the rise in public expenditure. Similr conclusions hold for different vlues of public expenditure. According to our simultion results, the optiml wy of finncing public expenditure is therefore minly through lump-sum txes. IV Compring the Mirrlees pproch to ours Our min contribution is methodologicl since we construct model where the efficiency distortions induced by income txtion re due to mtching frictions nd wge brgining insted of the stndrd consumption-leisure trde-off. Economists do observe the distribution of wges nd the elsticity of gross ernings with respect to the mrginl tx rte. Both the lbor supply-mirrlees pproch nd the brgining-mtching model re consitent with this observed elsticity. To wht extent the optiml tx schedule is sensitive to the micro-foundtion of this elsticity is the question we now ddress. To compre our model to frmework tht only incorportes the lbor supply choice of the individuls, we build model àlmirrlees (1971) tht genertes the sme distribution of wges nd the sme elsticity of gross ernings with respect to the mrginl tx rte t the equilibrium without txes. As e.g. Dimond (1998), we consider utility functions tht re qusi-liner in consumption. The worker s surplus is now x w T (w ) v(h ) b with v(h ), the disutility of work, v(h ) 0 > 0 nd v 00 (h ) > 0. Letη denote the elsticity of lbor supply, so 1/η = h v 00 (h) /v 0 (h). Employment rtes re ssumed equl to 1. Net output is given by Y = h v (h ) nd gross wges w re equl to h. Lbor supply h solves: v 0 (h )= 1 T 0 ( h ) (20) Σ now corresponds to the workers surplus for n optiml lbor supply: Σ = x = h v (h ) T ( h ) b 19

20 T w Tm w Figure 10: Benchmrk comprison of the two models. Dotted lines for the Mirrlees model. nd evolves ccording to: Σ = h 1 T 0 (w ) v 0 (h ) = h In this setting àlmirrlees, the optiml income txtion solves: (21) s.t. : mx d,h,σ,b Z 1 d Σ = h v 0 (h ) F ( d ) Φ (b + d)+ Z 1 d Φ (Σ + b) f () d { h v (h ) Σ } f () d = b + E ½ Σd = d Σ d d Optiml txtion verifies the following condition: R ³ T 0 1 (w ) 1 T 0 (w ) = 1 Φ0 t λ f () if f (t) dt d > 0 d = 0 µ 1+ 1 η which coincides with the clssic formul provided by the stndrd optiml income txtion literture when utility is qusi-liner in consumption (see e.g. Dimond (1998)). To compre it with the benchmrk model of Sections II nd III, we clibrte the Mirrlees model in such wy tht in the economy without txes, both models hve the sme distributions of wges 13 nd the sme elsticity of gross ernings with respect to one minus the mrginl tx rte. This elsticity equls 1 γ in our model 14, wheres it equls η in the Mirrlees model. Hence, we tke η =0.5. As Figure 10 nd Tble 3 show, the differences between the Mirrlees setting nd ours re quntittively very importnt. The optimum is much more redistributive when wges re brgined over nd the intensive mrgin of the lbor supply is neglected. Mrginl tx rtes re more thn twice s high. Assistnce benefits re lmost three times greter. Compred to the economy without txes, the gin in welfre Φ 1 (Ω) is considerbly higher nd the loss in net output Y is much lower. Furthermore, the profile of mrginl tx rtes is substntilly different. 13 This implies tht the distribution of bilities in the Mirrlees setting hs been ppropritely reprmetrized. 14 w From Eqution (12), we evlute 1 T 0 (1 T 0 ) w t T 0 = T = b =0. 20

21 Model Distribution Net output Y Welfre Φ 1 (Ω) b Prticiption WT SB WT SB SB Our Benchmrk % % % Mirrlees Benchmrk % % % Our With Preto % % % Mirrlees With Preto % % % Tble 3: Numericl results WT for the economy without txes nd SB for second best. Y denotes ggregte net output, i.e. Y = R 1 d Y f()d. We explin in two stges why our mrginl tx rtes re bove those found in the Mirrlees pproch. We first explin why this property would hold even if efficiency ws reched in the two settings. Next, we rgue tht this property is reinforced when tx rtes devite from their efficient vlues becuse they re optimlly chosen for redistributive purposes. In the Mirrlees model, efficient mrginl tx rtes re necessrily equl to zero since ny positive mrginl tx rte would distort lbor supply. In our model, since Hosios (1990), we know tht efficiency cn be reched in n economy without txes nd benefits. In n economy with redistributive txtion, efficiency cn, however, lso be chieved under the Hosios condition if mrginl tx rtes re efficient i.e. if they verify (14) (see Boone nd Bovenberg 2002). From this expression, efficient mrginl tx rtes re positive if T + b is positive. This condition sounds plusible if redistribution mtters. It is ctully verified by the second-best optimum (ccording to Propositions 4 nd 5). To sum up, if the solutions to the problem ddressed respectively by Mirrlees nd by us were both comptible with efficiency, the mrginl tx rtes would be higher in frictionl economy thn in competitive frictionless lbor mrket. Second, in both models, mrginl tx rtes re bove their efficient vlues, except t the top of the distribution. As we move to the left of the distribution, the frction of workers potentilly involved in mimicking others increses. This genertes greter nd greter upwrd pressure on mrginl tx rtes. In our model, the incentive comptibility constrint is expressed in terms of growth rtes of workers expected surplus (see Eqution 16). In the Mirrlees version, the incentive constrint is formulted in terms of bsolute chnges (see Eqution 21). Hence, in our model the upwrd pressure on mrginl tx rtes is stronger t the low end of the distribution. Figure 10 ssumes log-norml distribution. However, within the Mirrlees pproch, the tx schedule is highly sensitive to the skill distribution chosen. In prticulr, severl uthors (Dimond (1998) nd Sez (2001)) hve recently moved wy from the log-norml ssumption. They rgue tht the upper prt of the productivity distribution is better pproximted by n unbounded Preto function. Under this ssumption, they show tht optiml mrginl tx rtes re much higher t the top of the distribution. Therefore, we conduct sensitivity nlysis. We tke the sme log-norml distribution s before for the lower prt of the distribution but we ssume Preto distribution with density f pr () =K p / 1+π for the upper prt. Following Sez (2001), we tke π =2. The boundry between the two densities nd K p rechoseninsuchwythttheentire distribution is continuously differentible. This leds to boundry productivity level of lim =5510. For this distribution, Figure 11 nd the two bottom lines of Tble 3 compre 21