Optimal Redistributive Taxation with both Labor Supply and Labor Demand Responses

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1 Optiml Redistributive Txtion with both Lbor Supply nd Lbor Demnd Responses Lurence JACQUET NHH Preliminry version Etienne LEHMANN y CREST Bruno VAN DER LINDEN z IRES - UCLouvin nd FNRS Mrch 28, 2011 Abstrct This pper chrcterizes the optiml redistributive tx schedule in mtching unemployment frmework with endogenous (voluntry) nonprticiption nd (involuntry) unemployment. The optiml employment tx rte is given by n inverse employment elsticity rule. This rule depends on the globl response of the employment rte, which depends not only on the prticiption (lbor supply) responses, but lso on the vcncy posting (lbor demnd) responses nd on the product of these two types of responses. For plusible prmeters, our mtching environment induces much lower employment tx rtes thn the usul competitive prticiption model. JEL Clssi ction: D82; H21; J64. Keywords: Optiml txtion, Lbor mrket frictions, Unemployment. I Introduction This pper nlyzes the optiml income tx schedule with endogenous voluntry nonprticiption nd involuntry unemployment. Individuls decide whether to prticipte or not to the lbor force (the extensive mrgin). Becuse of mtching frictions à l Mortensen nd Pissrides (1999), prticipting individul my be involuntry unemployed. The probbility for prticipnt to be recruited is endogenous nd depends on the number of vcncies rms nd pro tble to crete (the lbor demnd mrgin). Individuls di er both in their skills nd their costs of serching job. The skill heterogeneity implies Lurence Jcquet is lso reserch ssocite t IRES nd t BETA nd reserch network fellow t CESifo. Lurence.Jcquet@nhh.no y Etienne Lehmnn is lso reserch ssocite t IRES nd reserch network fellow t IZA, IDEP nd CESifo. etienne.lehmnn@ense.fr z Bruno Vn der Linden is lso reserch ssocite t ERMES- University Pris 2 nd reserch fellow t IZA. bruno.vnderlinden@uclouvin.be. 1

2 tht employed workers ern distinct wges. Costs of serching di er cross individuls of the sme skill level, which ccounts for the extensive mrgin s in Dimond (1980), Sez (2002) or Choné nd Lroque (2005, 2011). The government observes only ernings, so the redistribution is constrined to be second-best. This pper is interested in the employment tx, de ned s the tx the worker pys plus the unemployment bene t. 1 A higher level of employment tx reduces the returns to prticiption, thereby inducing some individuls to give up serch. In the stndrd cse with only the extensive mrgin, the optiml employment tx is inversely relted to the elsticity of the lbor supply, s in the extensive response model of Sez (2002). A lrger elsticity reduces the mgnitude of the optiml employment tx. When the lbor demnd is introduced, the optiml employment tx is inversely relted to the globl elsticity of employment. The ltter is the sum of three terms: the lbor supply elsticity, the lbor demnd elsticity with respect to the rm surplus nd the product of these two terms. 2 Intuitively, rise in the employment tx reduces the net (fter-tx) wge nd increses the gross (pre-tx) wge. In competitive setting, wges re exible nd equlize the lbor demnd nd supply. Hence, the gross wge response rises only when the lbor supply is elstic. In contrst, our model fetures unemployment nd wges re negotited below the productivity of the job. We mke the simplifying ssumption tht the workers nd their employers receive xed frction of the totl surplus. An increse in the employment tx reduces the totl surplus, thereby both the worker s nd the rm s surplus. Therefore, rise in the employment tx increses the gross wge, even in the bsence of prticiption responses. Firms thus nd less pro tble to crete vcncies nd the number of txpyers decreses through this lbor demnd response. Hence, the lrger the elsticity of the lbor demnd to the surplus of the rm, the lower is the optiml employment tx. The optiml employment tx is lso inversely relted to the product of the lbor demnd nd supply elsticities. The intuition behind this intercting term is the following. When tx increse reduces the rm s incentives to hire workers, the probbility for job seekers to nd job is reduced which reduces the return of prticiption. Therefore, the lrger the product of the lbor demnd nd lbor supply elsticities, the lower the employment tx. We numericlly investigte how introducing the lbor demnd responses ects the 1 In the literture, the employment tx is trditionlly clled prticiption tx in the bsence of (involuntry) unemployment. 2 In n ppendix vilble upon request, we show tht in the full informtion cse, the optiml employment tx is inversely relted to the lbor demnd elsticity only. Intuitively, in such cse, the government cn use the conditioning of txtion on the cost of serching to enforce individuls prticiption decisions without ny distortion of the lbor supply. The lbor supply elsticity does then not pper in the optiml tx formul. 2

3 optiml employment tx rtes nd show tht our mtching environment induces much lower employment tx rtes thn the usul competitive extensive response model. Severl ppers study the optiml income tx model under serch frictions on the lbor mrket. The optiml tx in Boone nd Bovenberg (2002) nd in Bodwy, Cu nd Mrceu (2003) cts s Pigouvin tx to correct the ine ciency tht rises from the serch-congestion externlities. Hungerbühler, Lehmnn, Prmentier nd Vn der Linden (2006) nd Lehmnn, Prmentier nd Vn der Linden (2011) consider insted n environment where these externlities re perfectly internlized by the wge setting process in the no-tx economy. The role of txtion is therefore to redistribute income nd not to restore e ciency. Hungerbühler nd Lehmnn (2009) consider both the redistributive spects nd congestion externlities. In ll of these ppers except Bodwy, Cu nd Mrceu (2003), rise in the mrginl tx rte increses the shre of the surplus tht the employer receives: higher mrginl tx rte discourges workers to clim for higher wges, thereby reducing the gross wge negotited nd boosting the lbor demnd. In contrst, we neglect this wge-cum-lbor demnd mrgin to stress the role of the lbor demnd responses in the optiml tx formul. This pper is orgnized s follows. Section II presents the model. Section III derives the optiml tx formul nd contrsts it with the cse of competitive lbor mrket nd lbor supply responses long the extensive mrgin. Section IV concludes. II The generl frmework Assume risk-neutrl individuls endowed with distinct skill levels denoted by. exogenous skill distribution is given by the continuous density function f(), de ned on the support [ 0 ; 1 ], with 0 < 0 < 1 1. The size of the popultion is normlized to 1. Jobs re skill-speci c. A worker of skill produces units of output if nd only if she is employed in type- job, 3 otherwise her production is nil. This ssumption of perfect segmenttion is mde for trctbility nd seems more relistic thn the polr one of unique lbor mrket for ll skill levels. At ech skill level, some people choose to sty out of the lbor force while some others do prticipte to the lbor mrket. We integrte this feture by ssuming tht individuls of given skill level di er in their cost of serching job. The distribution of conditionl on skill level is described by the conditionl density H 0 (:j ) over the support R +. The We ssume tht H (:j ) is twice continuously di erentible nd strictly positive for ll 2 R +. The chrcteristics nd my be distributed independently or 3 Allowing n gent to work in ny occuption which requires skill below her type opens the possibility of monotonicity constrints nd pooling tht re studied in Choné nd Lroque (2010). 3

4 my be correlted. Among individuls who prticipte to the lbor mrket, some fil to be recruited nd become unemployed. This involuntry unemployment is due to mtching frictions. The number of mtches between employers nd job seekers on the lbor mrket of skill is function of the stock of vcnt posts, V, nd the stock of job seekers, U, in the mrket (Mortensen nd Pissrides 1999). Therefore, M (V ; U ) denotes the mtching function on the lbor mrket of skill. If there were no frictions, the number of mtches would be determined by the short side of the mrket nd the mtching process would be e cient. But when job seekers nd employers hve to engge in costly nd time-consuming process of serch to nd ech other, the mtching function cptures the technology tht brings them together. The mtching process is ssumed not e cient hence M (V ; U ) < min(v ; U ). The mtching function M (V ; U ) is twice continuously di erentible on R 2 +, incresing nd concve in both rguments, veri es M (0; U ) = M (V ; 0) = 0 since mtches cnnot occur unless there re gents on both sides of the mrket nd exhibits constnt returns to scle. These ssumptions re lrgely empiriclly supported s discussed by Petrongolo nd Pissrides (2001). We ssume tht the government does neither observe individuls types (; ) nor the job-serch nd mtching processes. It only observes worker s gross wge w. Therefore, the tx T (:) : R + 7! R only depends on the gross wge w. Moreover, the government is unble to distinguish mong the non-employed individuls those who serched for job but filed to nd one (the involuntry unemployed) from the non prticipnts (the voluntry unemployed). Therefore, the government is constrined to give the sme level of welfre bene t b 2 R + to ll non-employed gents. The timing of the model is: 1. The government commits to tx system de ned s pir (T (:); b), with T (:) : R + 7! R which only depends on the gross wge w nd the welfre bene t b 2 R + for the non-employed. 2. For ech skill level, rms open vcncies. Creting vcncy of type costs > 0. Ech type -gent decides whether she prticiptes to the lbor mrket of type. 3. Mtching occurs. Once mtched, the rm nd the worker shre the rent hence set the wge. 4. Ech worker of skill produces units of goods, receives wge w = w nd pys txes or receive trnsfers. Txes nnce the ssistnce bene t b nd n exogenous mount of public expenditures R 0. Agents consume. 4

5 II.1 Prticiption decision An individul of type (; ) cn decide to sty out of the lbor force, in which cse her utility equls the welfre bene t b. Otherwise, she prticiptes. Then, she nds job with n endogenous probbility ` nd gets utility level equls to w T (w ) or she becomes unemployed with probbility 1 ` nd gets utility level equls to b. To prticipte, n gent of type (; ) should expect higher expected utility `(w T (w )) + (1 `) b thn in cse of non prticiption, b. Let = T (w ) + b denote the employment tx. We de ne the expected surplus of prticipnt of type s def ` (w T (w ) b) (1) i.e. the dditionl income she gets if she nds job rther thn being unemployed multiplied by the probbility of employment. Any individul of skill chooses to prticipte if her cost of serching job is lower thn the surplus she expects from nding job, i.e.. Let h denote the prticiption rte mong individuls of skill, i.e.: h = H ( j) Pr [ j ] (2) The mss of prticipnts of type equls U = h f(). We now de ne: P def H 0 ( j) H ( j) (3) s the elsticity of the prticiption rte mong individuls of skill with respect to the expected surplus of prticipnt, t =. Note tht P lso equls the elsticity of the prticiption rte mong individuls of skill with respect to di erence in income between employment nd unemployment w, controlling for chnges in the employment probbility `. The empiricl literture on the prticiption decisions is typiclly concerned with the ltter elsticity. II.2 Lbor demnd De ne mrket tightness s the rtio V =U : The probbility tht mtching is successful (i.e. the probbility of lling type- vcncy) equls m ( ) M (V ; U )=V = M (1; 1= ). Due to serch-mtching externlities, the mtching probbility decreses with the number of vcncies (V ) nd increse with the number of job-seekers (U ). Since M (V ; U ) exhibits constnt returns to scle, only tightness mtters nd m ( ) is decresing function of. Symmetriclly, the probbility tht job-seeker nds job is n incresing function of tightness m ( ) M (V ; U )=U = M ( ; 1) with the functions m ( ) nd m ( ) de ned from R + to (0; 1). Firms nd individuls being tomistic, they tke tightness s given. 5

6 When rm cretes vcncy of type, it lls it with probbility m ( ). The cretion of this vcncy costs > 0 to the rm. This cost includes the screening of pplicnts nd investment in equipment for the extr worker. The rm s expected pro t is m( ) ( w ). For given number of job-seekers, rise in the number of vcncies decreses this expected pro t becuse ech vcncy is lled with lower probbility. Firms crete vcncies until the free-entry condition m ( ) ( w ) = is met. This pins down the vlue of tightness s m 1 ( = ( w )). 4 In turn, it lso gives the probbility of nding job (or the lbor demnd) through m ( ) = L ( w ), where the lbor demnd function L (:) is de ned s: L ( w ) def At the equilibrium, one hs ` = L ( w ). m 1 w w (4) The L (:) function is reduced form tht cptures everything we need on the lbor demnd side. From the ssumptions mde on the mtching function, L (:) is twicecontinuously di erentible nd dmits vlues within (0; 1). As the wge w increses, rms get lower surplus ( w ) on ech lled vcncy, fewer vcncies re creted nd tightness decreses. This explins why the employment probbility ` decreses with the wge w. Moreover, due to the constnt-returns-to-scle ssumption, the probbility of being employed depends only on skill nd wge levels nd not on the number of prticipnts. If for given wge, there re twice more prticipnts, the free-entry condition leds to twice more vcncies, so the level of employment is twice higher nd the employment probbility is un ected. This property is in ccordnce with the empiricl evidence tht the size of the lbor force hs no lsting e ect on group-speci c unemployment rtes. Finlly, becuse lbor mrkets re perfectly segmented by skill, the probbility tht prticipnt of type nds job depends only on the wge level w nd not on wges in other segments of the lbor mrket. We then de ne the elsticity of the (type-) lbor demnd to the surplus of the rm w s: D def ( w ) L0 ( w ) L ( w ) = 1 ( ) > 0 (5) ( ) where (4) hs been used nd ( ) denotes the elsticity of the mtching function with respect to the mss of job-seekers U evluted t = m 1 ( = ( w )) (see Appendix A). The empiricl literture on lbor demnd is typiclly concerned with the elsticity of employment with respect to the level of wge. Controlling for prticiption decisions in our model, the ltter elsticity is negtive nd equls D (w = ( w )). 4 where m 1 (:) denotes the reciprocl of function 7! m (), holding constnt. 6

7 II.3 The wge setting: Exogenous shring of the rent Once rm nd worker re mtched, they shre the rent, i.e. the sum of the rm s surplus w nd of the worker s surplus w T (w ) b. In the bsence of n greement, nothing is produced nd the worker gets the welfre bene t b. The brgining process determines how the totl surplus S = T (w ) b is shred between the worker nd the rm. The result of the brgining cn be viewed s the outcome of the mximiztion of n objective (x; y) tht is incresing in the rm s x nd the employer s y surplus. For instnce, the generlized Nsh brgining frmework tkes the form (x; y) = x 1 y. However, di erent shpes cn be considered insted. In this pper, we consider Leontief h i speci ction (x; y) = min x 1 ; y to void n e ect of mrginl tx rtes on wges. This simpli ction enbles us to clerly identify the role of the lbor demnd responses in the optiml tx formul. The equilibrium wge solves: w w = rg mx min w 1 ; w T (w) b When the income tx function T (:) is di erentible with T 0 (:) 1 everywhere, the solution to this progrm is unique nd given by: w = + (1 ) (T (w ) + b) In this cse, it is equivlent 5 for the government to design n income tx function T (:), or to directly design the employment tx = T (w ) + b for ech skill level. Then w = + (1 ) (6) The gross wge w is incresing with the employment tx. An increse of the employment tx will reduce the employee s surplus hence the employee will o set her loss by lrger brgined wge w. Since w = (1 ) ( ) from (6), the employment probbility veri es: ` = L [(1 ) ( )] (7) 5 The government cn decentrlize n lloction chrcterized with di erentible 7! mpping by n income tx function w 7! T (w) when < 1 A given 7! leds to wge level given by (6). This wge is incresing in only when = (1 ) =@. Then, Eqution (6) cn be inverted to express skill s di erentible function = A (w) of the wge, with A 0 (w) = 1= [ + (1 =@]. The income tx function hs then to stisfy T (w) A(w), which implies: T 0 A0 (w) = This tx function veri es T 0 (w) 1 only =@ + 7

8 Combining (1) nd (6), the expected surplus from prticipting equls: nd the skill-speci c prticiption rte equls: = ( ) L [(1 ) ( )] (8) h = H [ ( ) L [(1 ) ( )] j] (9) Finlly, the skill speci c employment rte e equls the product of the prticiption rte h by the probbility ` for ech prticipnt to nd job: e = ` h = L [(1 ) ( )] H [ ( ) L [(1 ) ( )] j] (10) The employment rte responds to tx ccording to de = D + P + D P e d (11) where we used elsticities de ned in (3) nd (5). We henceforth refer to the term in brckets in (11) s the globl elsticity of employment. The product D P enters this formul becuse ny increse in the lbor demnd gives dditionl incentives for individuls to enter the lbor force, so it reinforces the lbor supply. This complementrity between lbor demnd nd lbor supply is key insight of the unemployment mtching theory. II.4 The government We ssume tht the government cres bout the distribution of expected utilities, nmely, ` (w T (w )) + (1 `) b = + b (from (8)) for those who prticipte nd b for nonprticipting individuls. More precisely, the government hs the following Bergson-Smuelson socil welfre function: Z 1 Z ( + b ) dh ( j) + (b) (1 H ( j)) f () d (12) 0 0 where 0 (:) > 0, 00 (:) 0. 6 The stronger the concvity of (:) the more verse to inequlity is the government. Assuming 00 (:) = 0 corresponds to the Benthmite Utilitrin criterion tht sums the individul expected utilities. The government fces the following budget constrint Z 1 0 e f() d b R = 0 (13) tht is written so tht the welfre bene t b is provided to ll gents in the economy but for ech dditionl worker of skill, the government sves the welfre bene t b nd collects 6 (:) is n incresing function of the expected utility nd does not vry with the heterogeneous individuls chrcteristics (; ). Another vlue judgment could give distinct welfre weights to identicl utility levels obtined by individuls with distinct chrcteristics. This is left for future reserch. 8

9 txes T (w ) (the sum of these being ). Tking (2) nd (7) into ccount, this budget constrint cn be rewritten s Z 1 0 L [(1 ) ( )] H ( ( ) L [(1 ) ( )] j) f() d(14) = b + R III The optiml tx policies The optiml tx problem consists in nding the optiml level of bene t b nd of employment tx t ech skill level to mximize the socil objective (12) subject to the budget constrint (14), tking (7) into ccount. This problem is solved in Appendix B. Let be the Lgrnge multiplier of the budget constrint. We interpret s the mrginl socil cost of the public funds R nd we let g denote the mrginl socil welfre weight given to workers of skill, expressed in terms of public funds, i.e. g def R 0 0 ( + b ) H 0 ( j) d h (15) Intuitively, the government is indi erent between giving one more euro to ech of the gent of skill nd giving g euros of public funds. Symmetriclly, we de ne g N def 0 (b) (16) s the mrginl socil welfre weight of non-prticipting individuls expressed in terms of public funds. The optiml tx policy is given in the following proposition, which is proved in the ppendix. Proposition 1 For ny skill level 2 A, the optiml tx schedule stis es: Z 1 g h + g N (1 h ) f () d = 1 (17) 0 = 1 g 1 + D w [ D + P + D P (17b) ] or = 1 g 1 + D 1 g (1 + D ) + [ D + P + D P ] w (18) Eqution (17) sttes tht the mrginl cost of public funds is weighted verge of the socil mrginl utilities of the workers (g ) nd of the unemployed (g N ). Eqution (17b) leds to (18). Our generl model encompsses two speci c cses. First, one cn retrieve the pure extensive mrgin model when the mtching function veri es M (V; U) = U nd = 1. 9

10 When M (V; U) = U, ny job-seeker becomes employed, s in Dimond (1980), Sez (2002) nd Choné nd Lroque (2005, 2011). If in ddition the workers hve ll the brgining power (i.e. = 1), eqution (6) leds to the equlity between the skill level nd the gross wge w. Under these two ssumptions, to which we henceforth refer to s the pure extensive response model, Eqution (17b) becomes identicl to the one obtined by Sez (2002) in the bsence of intensive response, i.e. w = 1 g P (19) Second, our model lso encompsses the polr pure lbor response model with xed prticiption decisions P = 0. Equtions (17)-(17b) then become: Z 1 g ` f () d = 1 (20) 0 = 1 g 1 + D w D (20b) To derive Eqution (17b), we consider in the spirit of Sez (2002) perturbtion of the optiml tx function tht consists in smll increse 7 dt (w ) > 0 in the tx libility t wge w. For constnt level of bene t b, this increse induces rise d = dt (w ) in the employment tx pid by workers of skill level, which implies mechnicl e ect, n employment response e ect nd socil welfre e ect tht we now describe. III.1 Mechnicl e ect Absent ny behviorl chnge, the government levies d dditionl txes on ech job of skill. Their mss is e f (). From (13), the mechnicl increse in tx revenue equls: M = e f () d (21) This e ect is identicl in our generl model, in the pure extensive response nd in the pure lbor demnd responses cses. III.2 Employment response e ect The increse in the employment tx d > 0 induces reduction in the employment rte e = ` h tht is given by (11). Using (6), employment chnges by: de = D + P + D P e This reduction integrtes direct chnge in prticiption, direct lbor demnd response nd the e ect of the lbor demnd response on the incentives for individuls to serch 7 The cse where the employment tx is decresed is symmetric s only rst-order e ects re considered. w d 10

11 job. As ech dditionl worker of skill increses the government s revenue by the employment tx, the employment e ect equls E = D + P + D P e f () d (22) w There re two di erences with the pure extensive response cse. First, the globl elsticity of employment D + P + D P is reduced to the sole lbor supply elsticity P in the pure extensive response cse where the lbor demnd elsticity D is nil. Second, in the generl model, the employment response e ect is multiplied by the frction of the surplus tht occurs to the worker. This ltter term would not pper if we hd expressed (22) s function of the the totl surplus of the job insted of the surplus w tht the workers receive. In the pure extensive response cse, the two terms re equls, while in the generl model, the workers surplus w is frction of the totl surplus. In the pure lbor demnd response cse, the globl elsticity of employment D + P + D P is reduced to the sole lbor demnd elsticity D. III.3 Socil welfre e ect We now describe how the reform ects the socil welfre function (12). Given our ssumption tht the government cres bout the distribution of expected utilities, one should determine how the reform modi es the expected surplus de ned in (1). On the one hnd, there is direct e ect on the surplus w extrcted by the worker. From (6), this chnge mounts to d (w ) = d On the other hnd, the lbor demnd response implies reduction in the probbility for job-seeker to nd job. From (5) nd (7), this term equls d` = D 1 ` d Combining these two e ects, the expected surplus is reduced by d = 1 + D ` d This reduction induces some individuls to stop prticipting. However, these pivotl individuls re indi erent between prticipting or not, so the chnge in their prticiption decisions hs no rst-order e ect on the socil objective. Recll tht g is the mrginl socil welfre weight given to workers of skill, expressed in terms of public funds (see (15)), the socil welfre e ect equls: W = g 1 + D e f () d (23) 11

12 In the pure extensive response model where = 1 nd D = 0, the welfre e ect (23) tkes simple form s 1 + D = 1. In the generl model, the workers py only frction of the tx. Moreover rise in txtion lso ects the probbility for jobseeker to nd job through the lbor demnd e ect. Hence, unit rise of the employment tx pid by workers of skill decreses their expected surplus by n mount 1 + D. The equlity 1 + D = 1 only holds in the generl model under speci c restriction, nmely: 1 = D (24) From (5), this restriction leds to the equlity between the worker s shre of the totl surplus nd the elsticity of the mtching function with respect to unemployment, the so-clled Hosios (1990) condition. The ltter gives the right blend of congestion externlities tht re inherent to the mtching process. Intuitively, this equlity indirectly sets up the wges so tht they e ciently coordinte the serch decisions of workers nd rms in the frictionl lbor mrket. There is no prticulr reson why it should be stis ed since it reltes prmeter of the resolution of brgining con ict to prmeter of the technology of mtching, s rgued by Pissrides (2000, p.198). The term tht corresponds to devition from the Hosios condition does not pper in our optiml tx formul for e ciency resons, but for tx incidence ones. As the frction of the surplus tht ech prty received fter the wge brgining re ssumed exogenous, txtion cnnot correct for congestion externlities. III.4 Optiml employment tx rtes A smll chnge in the employment tx must imply no rst-order e ect. Adding (21), (22) nd (23) nd rerrnging terms gives (17b). Rerrnging terms gin led to the optiml employment tx rtes given in (18). The sign of the employment tx rte is given by the di erence between the mechnicl (21) nd the socil welfre e ects (23). Redistribution therefore occurs from workers whose weights g re lower thn 1= 1 + D to workers with weights lrger thn this vlue nd to non prticipnts. R 1 0 In the pure lbor demnd response cse, the weighted verge of socil welfre weights g f () d equls 1 (see (20)). Under concve socil welfre function (:), the socil welfre weights g re decresing in the skill levels under the plusible ssumption tht expected surplus is incresing in the skill level. Therefore, if one lso ssumes tht the Hosios condition (24) holds, the employment tx on the lest skilled workers is negtive, cse tht Sez (2002) de nes s n Erned Income Tx Credit (EITC). In the pure extensive response cse nd in the generl model, the welfre of nonpr- 12

13 ticipnts hs to be tken into ccount. From (15) nd (16), one hs g N > g whenever the socil welfre function (:) is concve. In prticulr, when the socil welfre function is close to Mximin objective, one typiclly obtins g N > 1 > g. Assuming gin tht the Hosios condition holds, n EITC is then ruled out. III.5 Quntittive insights In this section, we numericlly investigte how introducing the lbor demnd responses ects the optiml employment tx rtes. For this purpose, we compute optiml employment tx rtes =w from (18) for di erent clibrted vlues of P, D nd g. P g 0 0:5 D 0 0: : =3 0:5 1 2=3 0: % 75% 66:7% 100% 60% 50% 0:25 80% 63; 2% 57:1% 66:7% 46:2% 40% 0:5 66:7% 54:5% 50% 50% 37:5% 33:3% Tble 1: Optiml employment txes =w under the Hosios condition We consider in Tble 1 cses where the Hosios condition (24) holds. the rst nd fourth columns of Tble 1 give =w in the pure extensive response model (where D = 0) while the rst row provides vlues of =w in the pure lbor demnd model (where P = 0). Incresing the lbor demnd elsticity then implies two e ects on the optiml employment tx rtes. First, the globl elsticity of employment D + P + D P increses, which tends to reduce the mgnitude of the employment tx rtes. Second, the reduction in tht tkes plce to keep the Hosios condition vlid does not chnge the rtio between the optiml employment tx nd the skill level. However, it reduces the rtio of wges w over skill, hence it tends to increse employment tx rte =w. Tble 1 suggests tht the former e ect domintes, whtever the elsticity of prticiption P nd the vlue of the welfre weight g. A lrger lbor demnd elsticity substntilly reduces the optiml employment tx. For instnce, when P = 0:25, =w shrinks by 23 percentge points (from 80% to 57%) under Mximin nd by 33 percentge points (from 67% to 40%) with mrginl socil welfre weight g equls to 0:5. The empiricl literture does not usully distinguish between the distinct components of the globl elsticity of employment with respect to the employment tx. However, depending on the reltive importnce of the lbor demnd nd lbor supply elsticities in explining the globl employment elsticity, the optiml employment tx rtes vry. To illustrte this, we ssume tht the Hosios condition (24) is vlid. Incresing D requires 13

14 g 0 0:5 D 0 0:25 0:5 0 0:25 0:5 1 0:8 2=3 1 0:8 2=3 D + P + D P 0:5 66:7% 71:4% 75% 50% 55:6% 60% 0:7 58:8% 64:1% 68:2% 41:7% 47:2% 51:7% 1 50% 55:6% 60% 33:3% 38:5% 42:9% Tble 2: Optiml employment txes =w under the Hosios condition to reduce to keep the Hosios condition stis ed. Therefore, from (18) with g < 1, we expect tht the employment tx rte will increse with D, the globl elsticity being constnt. This is con rmed in Tble 2. In ech row of Tble 2, the globl employment elsticity D + P + D D P is xed nd equls 0:5, 0:7 nd 1 respectively. Then, incresing from 0 to 0:5 increses the employment tx by bout 9 or 10 percentge points when g = 0. Employment tx rtes re lower when g = 0:5, nd decrese in D similr extent. P g 0:5 1 D 1 1 0:3 0:5 0:7 0:3 0:5 0:7 0 70% 50% 30% 57:1% 0 133:3% 0:25 60:9% 40% 22:2% 47:1% 0 61:5% 0:5 53:8% 33:3% 17:6% 40% 0 40% Tble 3: Otiml employment tx rtes =w when the Hosios condition is violted by Finlly, Tble 3 studies the impct of deviting from the Hosios condition (24), in prticulr when the worker s shre of the totl surplus increses. The elsticity D is ssumed constnt nd equls 1 nd the mrginl socil welfre weight g is equl to 0:5. Intuitively, given increse of the employment tx hs lrger impct on the welfre of the workers when is higher. Therefore, the optiml employment tx decreses with. Tble 3 con rms this intuition nd highlights tht the quntittive impct is substntil. For instnce, when P = 0:25 nd g = 0:5, incresing the worker s shre of the totl surplus from 0:3 to 0:7 divides the employment tx rte by nerly three (from 61% to 22%). The chnges re even lrger when g = 1. Under the Hosios condition, optiml employment tx rtes re nil since the numertor of (18) is equl to zero with these vlues of g, nd D. When P = 0:25, incresing the worker s shre of the totl surplus from 0:3 to 0:7 shifts the optiml employment tx rte from positive 47:1% to negtive 61:5%; n EITC then previls. 14

15 IV Conclusion The optiml tx schedule derived in the optiml tx model with lbor supply long the extensive mrgin is drsticlly modi ed when lbor demnd in frictionl economy is lso modeled. The employment tx is still n inverse elsticity rule however the elsticity term encpsultes not only lbor supply responses (s in the stndrd model) but lso lbor demnd responses nd the crossed e ects between lbor demnd nd lbor supply, both being neglected in the stndrd frmework. For plusible prmeters, mtching unemployment frictions induce much lower employment tx rtes thn the ones found in the usul competitive model. References [1] Bodwy R., Cu K. nd N. Mrceu (2003), Redistribution nd employment policies with endogenous unemployment, Journl of Public Economics, 87, [2] Choné, P. nd G. Lroque (2005), Optiml Incentives for Lbor Force Prticiption, Journl of Public Economics, 89 (2-3), [3] Choné, P. nd G. Lroque (2011), Optiml Txtion in the Extensive Model, Journl of Economic Theory, Forthcoming. [4] Dimond P. (1980), Income Txtion with Fixed Hours of Work, Journl of Public Economics, 13, [5] Dimond P. nd E. Sheshinski (1996), Economic Aspects of Optiml Disbility Bene ts, Journl of Public Economic Theory, 57, [6] Hosios, A. (1990), On the E ciency of Mtching nd Relted Models of Serch nd Unemployment, Review of Economic Studies, 57, [7] Hungerbühler, M., Lehmnn, E. (2009), On the Optimlity of Minimum Wge: New Insights from Optiml Tx Theory, Journl of Public Economics, 93 (3-4), [8] Hungerbühler, M., Lehmnn, E., Prmentier, A. nd B. Vn der Linden (2006), Optiml Redistributive Txtion in Serch Equilibrium Model, Review of Economic Studies, 2006, 73 (3), [9] Lehmnn, E., Prmentier, A. nd B. Vn der Linden (2011), Optiml income txtion with endogenous prticiption nd serch unemployment, CESifo Discussion Pper

16 [10] Meghir, C. nd D. Phillips (2008), Lbour supply nd txes, IZA Discusssion Pper Series, N [11] Mortensen, D. nd Pissrides, C. (1999), New developments in models of serch in the Lbor Mrket, in O. Ashenfelter nd D. Crd (eds.), Hndbook of Lbor Economics, vol 3, B, North-Hollnd, Amsterdm. [12] Petrongolo, B. nd Pissrides, C.A. (2001), Looking into the blck box: survey of the mtching function, Journl of Economic Literture, 39, [13] Pissrides, C. A (2000), Equilibrium Unemployment Theory, 2 nd Edition, Cmbridge: MIT Press. [14] Rubinstein, A. (1982), Perfect Equilibrium in Brgining Model, Econometric, 50(1): [15] Sez, E. (2002), Optiml Income Trnsfer Progrms:Intensive Versus Extensive Lbor Supply Responses, Qurterly Journl of Economics, 117, [16] Sez, E. (2003), The E ect of Mrginl Tx Rtes on Income: A Pnel Study of Brcket Creep, Journl of Public Economics, 87: [17] Thomson, W. (1994), Coopertive Models of Brgining, in Hndbook of Gme Theory, vol.2, R. umnn nd S. Hrt eds. Appendices A Link between the elsticity of the lbor demnd nd the elsticity of the mtching function Let (:) denote the elsticity of the mtching function M (:; :) with respect to the mss of job-seekers U. Becuse the mtching function is incresing in both rguments nd exhibits constnt returns to scle, depends only on the level of tightness nd one must hve () 2 (0; 1) for ll. From the de nition m () = M (1; 1=), the elsticity of the probbility of lling vcncy to the tightness level (i.e. ( =m ) (@m () =@ )) equls (). Hence the elsticity of the reciprocl m 1 (:) equls 1= m 1 (:). The log-di erentition of the L function (4) with respect to the rm s surplus w gives: dl L = ( ) d ( w ) w which leds to the second equlity in (5). The inequlity holds becuse () 2 (0; 1). 16

17 B Proof of Proposition 1 The Lgrngin of the optiml tx problem cn be written s Z 1 0 L ( ; b; ) f () d b R where L ( ; b; ) def Z 1 0 Z ( )L [(1 )( )] 0 ( ( ) L [(1 ) ( )] + b ) dh ( j) + (b) (1 H ( ( ) L [(1 ) ( )] j)) + L [(1 ) ( )] H ( ( ) L [(1 ) ( )] j) The rst-order condition with respect to b is written s ( Z ) ( )L [(1 )( )] 0 ( + b ) dh ( j) + 0 (b) (1 h ) f () d = 0 Using (15) nd (16) gives (17). The rst-order condition with respect to writes ( ; b; ), which gives, using (3) nd (5) Z 0 = 1 + D ` 0 ( + b D 1 + D ) dh ( j) P ` h Dividing both sides by h ` nd using (15) nd w = ( ) from (6) gives (17b). 17