Optimal Taxation with On-the-Job Search

Transcription

1 Optmal Taxaton wth On-the-Job Search Jesper Bagger Espen R. Moen Rune M. Vejln Royal Holloway BI Norwegan Busness School Aarhus Unversty February 5, 28 Abstract We provde a theoretcal and emprcal study of the optmal taxaton of labor ncome n the presence of search frctons and unobservable amentes usng comprehensve Dansh matched employer-employee data. Heterogeneous workers undertake costly search off- and on-the-job n order to locate more productve jobs that pay hgher wages. More productve workers search harder, resultng n equlbrum sortng where low-type workers are overrepresented n low-wage jobs whle hgh-type workers are overrepresented n hgh-wage jobs. Absent taxes, worker search effort s effcent, because the socal and prvate gans from search concde. The optmal tax system balance effcency and equty concerns at the margn. Equty concerns make t desrable to levy low taxes on (or ndeed, subsdze) low-wage jobs ncludng unemployment, and levy hgh taxes on hgh-wage jobs. Effcency concerns lmt how much taxes an optmal tax system levy on hgh-pad jobs, as hgh taxes dstort the workers ncentves to search. Usng detaled mcro data on wages, labor market transtons, and ncome tax flngs we estmate the structural model and compare the the actual Dansh tax regme wth the optmal one. Prelmnary results suggest the optmal tax schedule exhbts less progressvty than the actual tax system n place. The model allows us to quantfy the welfare gans from adoptng an optmal ncome tax schedule. Key words: Optmal taxaton, On-the-Job Search, Sortng, Job ladder, Matched Employer-Employee Data. JEL codes: H2, H3 J63, J64. Introducton The equlbrum allocaton of resources does not materalze costlessly n a market wth search frctons, as t does n a perfectly compettve market. That s, mprovng the We would lke to thank Vdar Chrstansen for comments on an early draft.

2 allocaton of resources n a frctonal market requres resources n tself, and s therefore value creaton. Ths value creaton may be dstorted by taxaton. In the context of a frctonal labor market, workers and frms undertake costly search n order to dentfy the jobs n whch ther productvty (and therefore, wage) s the hghest. Taxes on labor ncome wll have an allocatve effect, as t reduces the ncentves of workers to search for more hgh-productve, well-pad jobs. We study the optmal taxaton of labor ncome n a frctonal labor market where workers search to fnd a better job both whle employed and unemployed.. Workers are ex ante heterogeneous n ther productvty, and hence face dfferent job prospects. A benevolent planner has a preference for ncome equalty, where ncome s measured as (expected) lfetme dscounted ncome. On the one hand, n order to redstrbute from hgh- to low-productve workers, the planner wants to levy hgh taxes on ndvduals wth a hgh current ncome (whch tend to be hgh-productve workers) and redstrbute the tax revenues towards agents wth a low current ncome (whch tend to be lowproductve workers). On the other hand, workers of all types expend resources to fnd more productve jobs that pay better. All workers enter as unemployed, and through job search, frst as unemployed and then as employed, gradually clmb the wage/productvty ladder. Hence, search effort yelds both prvate and socal returns, but s costly and crucally not deductble (hence on-the job search may well be undertaken durng leasure tme). Increasng taxes on wages at a gven rung of the wage/productvty ladder reduces the ncentves to search at lower rungs, but does not affect the ncentves to search on hgher rungs. Hence, n order to protect the creaton of value, the planner wants to levy lower taxes on hgher wage levels. The optmal tax system trades off the equty and effcency concerns of the planner. We frst construct a model n whch both unemployed and employed workers search (on-the-job search). In ths model, search s random, however n the absence of taxes the equlbrum s constraned effcent. Workers of dfferent types face dfferent wage dstrbutons when they search. The role of frms are very much played down, they do not contrbute actvely to the search process, and wages are equal to productvty. We frst derve analytcally condtons for an optmal tax system. We then extend the model by allowng for amentes, whch are not observable by the planner and hence We assume that taxaton does not dstort other margns, lke the ntensve and the extensve margn of labor supply. Hence our focus s solely on the workers search decsons. In a rcher model wth endogenous labor supply, the dstortonary effects of taxes on the workers search decsons wll come n addton to (the well-studed) dstortons n labor supply. 2

3 cannot be taxed. When choosng between jobs, a worker takes both wages net of taxes and amentes nto account, and ths creates a new margn that s dstorted by taxes. Equty concerns mply that the planner levy taxes that ncrease n gross wages. After successful search, a worker therefore has a tendency to accept too few jobs that offer a hgher wage and too many jobs that offer a lower wage than the wage n the current job, relatve to what the planner would lke the worker to do. Ths s because workers, n contrast to the planner, consder wages net of taxes when choosng between jobs. In solaton ths effect tends to ncrease the effcency loss of taxes. Agan we derve analytcal condtons for an optmal tax system. We solve the basc model (wthout amentes) numercally, and gve examples of optmal tax systems. Fnally, the model s estmated on Dansh matched employer-employee data set wth detaled nformaton on ndvdual level ncome tax rates. The data cover the entre Dansh populaton of workers and frms durng , a perod where the ncome tax regme n Denmark s stable, and contan nformaton on daly ndvdual labor market hstores, hourly wages, and crucally detaled nformaton on ndvdual tax flngs. Ths part of the data allow us to smulate ndvdual-level margnal tax rates (as n Kleven and Schultz, 24). Hence, we estmate the structural model usng the actual tax functon faced by ndvdual workers, ncludng ndvdual crcumstances such as martal status, non-labor ncome, and ndvdual level deductons. 2 We construct an Indrect Inference where we ft a number of moments related to workers labor market behavor. Identfcaton of the structural parameters s dscussed. Wth the structural parameter estmates n hand we proceed to compute the optmal tax regme for the estmated economy, and compare ths to the actual Dansh tax regme. Prelmnary results suggest the optmal tax schedule exhbts less progressvty than the actual tax system n place. The model allows us to quantfy the welfare gans from adoptng an optmal ncome tax schedule. The exstng lterature on optmal taxaton and search s mostly concerned wth the search decsons of unemployed workers. Hungerbuhler, Lehmann, Parmenter, and van der Lnden (26) analyse optmal taxaton n a one-shot unemployment search model. In ther model, frms use resources to open vacances and wages are determned by wage barganng. They assume (lke us) that workers are rsk neutral, whle the planner has preferences over the (expected) ncome dstrbuton over dfferent worker types. They show how a revelaton mechansm can be appled at the barganng stage, so that the worker and the frm bargan over whch type to reveal to the planner. As 2 We are grateful to Henrk Kleven and Esben Schultz for makng ther tax smulator avalable to us. 3

4 a result, the revelaton prncple can be used to derve the optmal mechansm. Under the optmal taxaton scheme, the employment level s optmal for the most productve worker-frm pars, whle there s over-employment for the lower types that do search. Golosov, Mazero, and Menzo (23) study optmal taxaton n a one-shot compettve search equlbrum model wth dentcal, rsk averse workers and heterogeneous frms. There s a fxed cost for workers from sendng an applcaton. The equlbrum wthout taxaton s neffcent, as optmal rsk sharng requres that workers are compensated for applyng to jobs they do not get. In the constraned effcent equlbrum, the unemployment nsurance s set so that workers are ndfferent between searchng for any job and not searchng, as ths gves maxmum nsurance gven workers ncentve compatblty constrant. There s no transfers between workers searchng for dfferent frm types; the frms n effect fnance the unemployment beneft of the workers they attract but do not hre. As a result, taxes are regressve. Sh and Wen (999) analyze the effect of taxes n a model of random unemployment search, n whch workers accumulate captal. Hgher labor taxes dscourage workng, and leads to lower nvestments by frms and lower wages. Captal taxaton on the other hand ncreases labor supply, as workers get a lower return on ther captal. Hence captal taxaton may mprove the allocaton of resources. Domej (25) analyzes optmal taxaton wthn the same modelng framework, and fnd that the optmal captal tax s zero f and only f the Hoso s condton s satsfed. Jang (22) uses a smlar setup to analyse the welfare effects of a UK tax reform. Arseneau and Chugh (22) studes taxaton n a calbrated DSGE model wth search frctons, and argues that cyclcal varatons n the search-based labor wedge call for taxes that vary over the busness cycle. Wlemme (27) studes taxaton n a model of msmatch, and shows that taxes should be regressve to correct for workers not beng suffcently selectve. Geromchalos (25) study optmal taxaton wth rsk averse workers n a one-shot urn ball model of the labor market. A couple of recent papers analyze taxaton and on-the-job search. Sleet and Yazc (27) studes optmal taxaton n a Burdett and Mortensen (998) model of on-the job search. A tax on labor ncome reduces net wages for workers, and hence ncreases ther pre-tax reservaton wage. As a result, the entre wage dstrbuton shfts, and ths nfluences the dvson of rents between workers and employers. Bagger, Hejlesen, Sumya, and Vejln (27) evaluates equlbrum effects on labor allocaton of a seres of tax reforms n Denmark and also analyze optmal tax reforms usng an equlbrum on-the-job search model wth Burdett and Mortensen (998) wage settng. 4

5 On a conceptual level, our paper s also related to papers outsde the search lterature. Saez (22) analyses a model of taxaton n whch taxes nfluence partcpaton (the extensve margn) as well as whch frm type (level) to work for (the ntensve margn). Workng for a frm at a hgher level gves hgher ncome, but ths may come at a cost. If the extensve margn s suffcently mportant, taxes for low-ncome employed workers may be lower than for unemployed workers. Ths s studed n more detal n Chrstansen (25). Although the Saez (22) model s very dfferent from ours, there are nterestng smlartes between the two: In Saez model, reducng taxes at a gven level nduces some workers who prevously were choosng an occupaton one level above or below swtches to that level. In our model, by contrast, reducng taxes at a gven job type reduces the ncentves to search for workers n that job type, ncreases the ncentves to search for all workers further down the n the job ladder, and leaves the ncentves unchanged for all workers hgher up n the job ladder. Another related paper s Best and Kleven (23). They study an envronment wth learnng by dong, so that the wage of an old worker depends on hs labor supply as young. A hgher margnal ncome tax on old workers thereby reduces the labor supply of young workers. The model s calbrated to wage data for young and old workers from the PSID. Estmates of the effect of labor supply on future wages are obtaned from exstng studes of the effect of experence on wages. The authors fnd that wth agedependent taxes, the presence of on-the-job learnng mples that an optmal tax system prescrbes lower taxes for old workers than for young workers. Wth age-ndependent taxes, t mples that the optmal tax system prescrbes a lower margnal tax rates across the board. 2 Basc Model There are J worker types n the economy. The fracton of workers of type j s denoted by τ j,.e. J j= τ j =. Workers dscount the future at a pure dscount rate r and ext the market at rate λ to be replaced by new unemployed workers. The effectve dscount rate s thus r r + λ. Workers search equally effcent off- and on-the-job. The number of frms s exogenously gven. The job ladder has n + rungs wth rungs ndexed by {,..., n}. Each rung s assocated wth a productvty level y > y, wth the lowest rung = beng the value of home producton (.e. productvty durng unemployment). After successful search, the probablty that a worker of type j draws a job of productvty y s denoted f j, and 5

6 defne F j k= f j k. Our assumpton that productvtes are dscretely dstrbuted s made for tractablty only, extendng the analyss to allow for a contnuum of job productvtes s straght-forward. The offer dstrbuton s ndependent of the search ntensty and acceptance decsons of workers, and s a prmtve of the model. Our modelng of the productvty dstrbuton of workers and frms s general, and may capture dfferent assumptons regardng the productvty dstrbuton of frms, complementartes between worker and frm types, and of match-specfc productvty components. Let us gve an example of an underlyng structure that gves rse to a job dstrbuton of our type: Suppose there are L frm-types n the economy. The fracton of frms of type l s ξ l. All frms exhbt constant returns to scale n producton. The productvty of a type-j worker and a type-l frm s stochastc and revealed at the pont when they meet. Denote by f(j, l) the probablty that the realzed productvty s y. It follows that f j = L l= ξ l f(j, l). Ths producton structure, wthout any restrctons on the probabltes, are suffcently rch to capture the producton functons of most on-the-job search models n the lterature. Worker types are ndexed so that a hgher j means a hgher type. More precsely we assume that f F j+ stochastcally domnates F j for all j. In addton we assume that f j+ /f j s strctly ncreasng n whenever properly defned (f j from zero. ) and dfferent An employed worker may be ht by a negatve employment shock and enter unemployment. The rate at whch ths happens s s. In prncple, s may depend on both worker- and job type, n whch case we wrte s = s j. The arrval rate of job offers, p, to a worker s proportonal to the number of effcency unts of search, e, provded by the worker. That s, p = Ae, where A s an exogenous constant. We may thnk of A as proportonal to the number of frms K n the economy, A = ak. The aggregate number of matches s NAē, where N s the aggregate number of workers n the economy (whch we wll normalze to ), and ē ther average search ntensty. Hence our search technology dffers from the standard search technology n that a worker s search ntensty does not create congeston effects and thereby lower job fndng rates for other workers. Ths seems a natural assumpton f workers search for frms, and frms have a constant-return-to-scale producton technology and gve a job offer to all workers that approach them. The cost of search effort s κ j c(e), where the coeffcent κ j may be worker-type dependent. If not, we normalze κ to. The wage a worker receves gross of taxes s equal to hs productvty y. Ths s 6

7 consstent wth a model of wage barganng n whch the worker has all the barganng power. It follows that absent taxes, there are no externaltes assocated wth the workers choce of search ntensty. Hgher search ntensty of a worker does not create congeston externaltes for other workers, and although t ncreases the hrng probablty of frms, ths does not create postve externaltes as the worker s wage s equal to hs productvty. 3 We assume that the government cannot observe worker types drectly and restrct the tax system to be contngent on current wages. Hence the government, through taxaton, determnes the net-of-tax wage w that a worker (rrespectve of hs or het type) receves n a job of productvty y. Note that w s the net-of-tax unemployment ncome. We requre that w s nondecreasng n, so that ncome taxaton leaves the rankng of job unaltered. It follows that the probablty dstrbuton of w s the same as that of y. 4 An mportant ssue s the preferences of the agents. As n Golosov, Mazero, and Menzo (23), Best and Kleven (23) and others we assume that agents n the economy are rsk neutral, whle the planner s welfare weghts on the dfferent groups of workers depend on ther expected net present ncome. 5 Let W j be the net present ncome of an unemployed worker of type j less of search costs. Note that W j s also the lfetme ncome of a worker that enters the market. An enterng type-j worker s objectve s to maxmze W j. A planner s objectve s to maxmze the welfare functon Ω = J κ j Φ(W j ) () j= where Φ(W j ) s strctly ncreasng and concave. A drvng assumpton n our analyss s that search effort s unobservable to the planner, and cannot be contracted upon. Furthermore, we assume that search cost s assocated wth dstress when searchng for a job and reduced lesure, not reduced ncome. Hence the search cost s born fully by the worker, whle the gan n terms of 3 The search-and wage determnaton processes are specal cases of those developed by Damond (982), Mortensen (982), and Pssardes (985), wth an exogenous number of vacances and wth the workers barganng power equal to. See also Pssardes (994) for a model wth on-the-job search. 4 A smpler way to proceed s to mpose a partcular functonal form of the tax functon, for nstance by followng Heathcote, Storesletten, and Volante (27) and assume that the tax functon s gven by z λz τ. It follows that the after-tax-ncome of a worker s λz τ and the elastcty of the after-tax-ncome s τ. 5 One way of justfyng ths assumpton s that dosyncratc rsk s shared n famles of workers, where all the famly members are of the same type. 7

8 hgher ncome s partly taxed away. We also assume that the search cost s ndependent of the worker s ncome. Hence the level of wages n the dfferent jobs do not nfluence search ntensty, only the wage dfferentals. At ths pont we follow Saez (22) when he assumes that the choce of sectors do not depend on the ncome levels n the dfferent sectors, only the dfferences n ncome between them. 3 Optmal taxaton In ths secton we wll frst solve the model for a gven set of taxes, and then derve the optmal tax system. 3. Asset values For any varable X j, defne EXj = n k=+ f j k Xj k and the operator as Xj = n k=+ f j k (Xj k Xj ) for any varable Xj. Let W j denote the expected dscounted ncome of a type-j worker n a type- job. Analogously, let M j denote the expected dscounted future ncome gross of taxes for a type-j worker n a type- job. It follows that (r + s)w j = w + Ae j W j + sw j c(ej ) (2) c (e j ) = A W j (3) (r + s)m j = y + Ae j j M j + sm j c(ej ) (4) For a gven vector (w,...w n ), and a gven W j, the model can be solved recursvely for each type separately:. At the top: W j n = w n + sw j r + s e j n = M j n = y n + sm j r + s 8

9 2. Recursvely further down W j = w + Ae j EW j + sw j c(ej ) r + s + Ae j ( F j c (e j ) = A W j ) M j = y + Ae j EM j + sm j c(ej ) r + s + Ae j ( F j ) Lemma For any vector (w, w,...w n ) the vector W j,...w j n exsts and s unque. Proof. For a gven Ŵ j j, equaton (2) -(4) unquely defne W = W j (Ŵ j ) for all. In partcular, W j = W j (Ŵ j ). Equlbrum s a fxed-pont to ths mappng. As the mappng s contnuous and defned on a compact and convex set, Brouwer s fxedpont theorem ensures exstence. Furthermore, Maxwell s suffcency condton s satsfed (W j (Ŵ j + ) < W j (Ŵ j ) + k ) for some k < ), whch mples that the fxed-pont s unque. Consder now a change n w. For a worker n a job k >, ths only nfluences the npv ncome W j, and snce the transton rate to unemployment s s ndependently of the job type, t follows that dw j k = s dw j k >. (5) r + s From the envelope theorem t follows that the effect of an ncrease n w on W j dw j = + s dw j r + s + Ae j ( F j (6) )). s The effect of a change n w on W l s recursvely defned as dw l = Ae j l + s dw j r + s + Ae j l ( F l) k= l+ f j dw j k (7) An ncrease n w reduces search effort at w, and ncreases search effort at lower wages. From (3) t follows that de j l = A d W l /c (e j l ) (8) 9

10 Remark d W l > for all l. Proof. We know that ths s true for l =. Suppose t s true for all l < l. Suppose t s not true for l. Then we know from (2) that W j l decreases, but that can only be true f W j l ncreases, a contradcton. Remark 2 The hgher s the worker type, the hgher s the search ntensty at a gven current wage level w. Proof. Consder worker types j and j +. Snce F j+ stochastcally domnates F j. Hence for all, W j < W j+. From equaton (2) t follows that W j+ > W j+. by (2), but wth sub- Remark 3 Note that W j sttuted n for W j j. It follows that W s ndependent of W j = W j +. Defne W j s r+s W j. Ths smplfes the dervatves, j as t s suffcent to calculate the dervatve of W and then add the dervatve of W j at the end. In partcular t follows that we can wrte rw j = w + Ae j ( W j + s r+s W j ). Ths smplfes the calculaton of the dervatve of W j. Fnally, consder M j. Note that a change n w does not affect M j drectly, only through ts effect on e j.6 From (4) we have that (r + s)m j = y + e j A M j + sm j c(ej ) = y + e j A W j c(ej ) + ej A (M j W j ) + sm j (9) From the envelope theorem t follows that dm j s gven by For k >, t follows that (r + s) dm j = dej A (M W ) + s dm j () dm k = s dm () r + s 6 Recall that, due to the envelope theorem, margnal changes n e j has no effects on W j. Note the smlarty wth the lterature on suffcent statstcs for welfare analyss, see Chetty (29).

11 Further down the job ladder, t follows that (r + s) dm l = A l e j dw l k= l+ dm k f j dw + dej l (M j dw l W j j l ) + sdm (2) A change n w does not drectly nfluence M j, only ndrectly through ej. A change n nfluences bot the return to search and the cost of search. Takng the dervatve of e j (9), usng the frst order condton for the worker s choce of e j of unemployment) gves (). (and nclude the effect The effects of a change n w on M l are frst that effects of changes n the values M j l+,...m j ths s reflected n the frst term. Second, a change n w also nfluences the search effort, whch s captured by the second term. The last term captures the effects through M j. Fnally, note that the equvalent of Remark 3 apples for M j. Remark 4. Suppose taxes (y w ) are constant (n dollar) at and above w and that s =. Then (M j W j ) =, and hence dm j dm <. 3.2 The planner s problem =. If taxes are proportonal, then Suppose the planner n steady state needs to rase an amount M n net present value (NPV) ncome. 7 The planner takes gross ncomes (y,..., y n ) as gven, and chooses a vector (w,..., w n ) of net wages so as to maxmze welfare. therefore reads The planner s problem max w,...,w n n τ j M j M j= S.T. n τ j Φ(W j ) j= n τ j W j j= c (e j ) = A W h w w + 7 Equvalently, suppose the planner needs to rase a steady state flow revenue of r M.

12 The frst constrant says that the planner cannot use more than the total value of resources avalable. Note that as c(e) enters lnearly n both W and M, t cancels out. Hence the constrant requres that the ncome avalable must be equal to the total ncome generated. In what follows we assume that wages are ncreasng n types, and then check that they actually do n optmum afterwards. The Lagrangan reads (gnorng the constrant w w + ) n n n L = τ j Φ(W j ) + λ τ j M j τ j W j M j= j= j=,j µ j (A W j ej ), (3) where λ and µ j are the Lagrangan parameters assocated wth the constrants. The frst order condton for w reads (where dm j effects of changes n effort levels) s gven by 2, and thus ncludes the n j= τ j (Φ (W j ) λ)dw j = λ n j= τ j dm j. (4) The left-hand sde s the welfare effect of ncreasng w over and above the shadow value of ncome. The rght-hand sde s the cost n terms of reduced search effort. Intutvely, the value of dw j depends on how large fracton of the tme (dscount rate weghted) that a worker of type j spends n a job of type. The lower s the worker type, the more tme the worker spends n unemployment, as nflow to unemployment typcally s ndependent of or decreasng n a worker type j type whle outflow s hgher for hgh-type workers wth a hgher search effort. Ths bas may be even stronger n low-type jobs. Low-type workers have a hgher probablty of meetng a low-type job, and hence may have a hgher nflow rate to these jobs than hgh-type workers. At the same tme the outflow rate s stll hgher for the hgh-type workers. Let g j = Φ (W j ), Ẇ = n j= τ j dw j, and Ṁ = n j= τ j dm j, and fnally let ω j = W j /Ẇ. Then we can rewrte (4) as 2

13 n τ j (g j λ)ω j = λṁ (5) Ẇ j= From the planner s perspectve, ncreasng w has two effects: a dstrbutonal effect and an ncentve effect. The left-hand sde of (5) reflects the dstrbutonal effect, t shows how an NPV-dollar used on ncreasng w s dstrbuted on the dfferent worker types. A hgh correlaton between the welfare weghts g and the dstrbuton weghts ω j mples a postve dstrbutonal effect of ncreasng w. The rght-hand sde shows the effcency loss of ncreasng w,.e., the effect on the total amount of avalable resources per npv dollar spent on ncreasng w j. As we have seen, the effcency loss tends to be postve for low values of and negatve for hgh values of. The effcency concerns thus puts a lmt on how much the planner wants to redstrbute. At the lower end of the wage herarchy, low-type workers are overrepresented. Hence for lower wages, ω j tend to be decreasng n worker type. Hence the correlaton between g j and ω j tends to be postve, and ths calls for low (or negatve) taxes at low wage levels. On the other hand, hgh wages /low taxes at the lower end of the wage dstrbuton tends to reduce search effort, whle hgher wages at hgh wage levels tend to ncrease search effort. Effcency consderatons therefore set a lmt on how much the planner wants to redstrbute. The unemployment state s somewhat specal, as all agents start as unemployed, and all restart as unemployed after a negatve employment shock. Hence t may well be that the fracton of hgh-types among the unemployed s hgher than the fracton among workers n the low-pad jobs. In addton, ncreasng the ncome of unemployed workers clearly has a negatve ncentve effects for unemployed workers. Hence t s not a pror clear that the planner wll set the unemployment beneft partcularly hgh, but rather obtan redstrbuton by subsdzng low-wage jobs. Ths wll be explored numercally below. Remark 5 Suppose the planner has lnear preferences, so that the welfare weghts ω j are equal for all j ndependently of W j. Then the planner wll set y j w j = M/r for all j. Wth such a tax polcy, the planner wll not dstort the workers search effort, and effcency s obtaned. Ths s equal to a poll tax. 3

14 4 Includng amentes Suppose now that a job comes wth two attrbutes, productvty y and other qualtes, whch we denote by z and refer to as amentes. Amentes cannot be observed by the planner, and hence cannot be taxed. The utlty flow of a job s thus the sum of wages net of taxes and amentes. The jont dstrbuton of amentes and productvty types for a person of type j can be wrtten as F j (, z). Let F j denote the margnal probablty that a worker of type j draws a job of productvty type (or just type) or lower. Let f j denote the probablty that y = y. Let F j (z ) denote the condtonal dstrbuton of z gven. In order to smplfy the analyss we assume that the draws of productvtes and amentes are ndependent. Ths means that we can wrte the dstrbuton of z as G(z). We wll modfy ths assumpton n future work. Above, we requred that that taxes wll not nfluence the workers rankng of jobs. Ths s consstent wth the assumpton that margnal taxes are less than percent. However, wth amentes the tax system wll nfluence the rankng of jobs, and lead to new neffcences; n the presence of hgh margnal taxes, workers may turn down job offers wth a hgh pre-tax wage that scores low on amentes that the planner wants the workers to accept. We stll requre that taxes do not alter the orderng of the pecunary returns of jobs. The utlty flow of the current job s equal to the sum w +z. The worker wll accept a new job f and only f the characterstcs of the new job (, z ) satsfes w +z > w +z. Hence w + z s a suffcent statstcs for the npv value of the job. Defne z,k (z) = w + z w k. A worker n a job wth characterstcs (w, z) accepts a new job offer at wage level w k f and only f the level of amentes z n the new job satsfes z > z,k (z). The NPV ncome of a worker of type j n a job wth characterstcs (w, z) can thus be wrtten as (r + s)w j (z) = w + z + Ae j W j (z) + sw j c(ej ) (6) where W j (z) = n k= f j k z,k (z) (W j k ( z) W j (z)))g( z)d z (7) As above we have that c (e) = A W j (z) (8) Let ˆF j (z) denote the probablty that a worker of type j n a job at level wth amentes 4

15 z does not accept a job offer when t arrves. It follows that ˆF j (z) = n k= f j k G( z,k(z)). From the envelope t follows that W j (z) = r + s + Ae j ( ˆF j (z)) (9) Lemma 2 for a gven dstrbuton F j wth fnte support, the equlbrum exsts. Proof. Let us sketch the proof. The value functons W j (z) are real functons defned on a closed subset of R n+. Let D denote the set of contnuous functons defned on that subset and bounded above by (y n + z max )/r. We know that D s complete under the sup norm. Equaton (6)-(8) defne a mappng Γ : D D. It follows trvally that Γ s contnuous, and ncreasng. Let ē < denote the supremum of e. Snce the set of value functons s bounded t follows that ē s bounded. We want to show that Blackwell s suffcent condton holds. Let δ be a strctly postve constant. For any W D t follows that Γ(W + δ) ΓW + r r+aēδ = W + βδ wth β <. Hence the Blackwell suffcent condton s satsfed, and Γ s a contracton mappng. From the contracton mappng theorem t follows that Γ has a unque fx-pont. Let us consder the effect on W j (z) of a change n w. Wthout amentes, ths would only nfluence the NPV values n frms wth wage at or below w (n addton to the effects through W ). Wth amentes ths s no longer the case, as workers may accept jobs wth lower wages. As the envelope theorem stll apples, the effect on effort level and acceptance decsons do not have frst order effects on W j and can be gnored. Hence (r + s) dw j (z) = + n k= f j k z,k (z) ( dw j k ( z) dw j (z)) )g( z)d z + s dw j The frst term on the rhs s the drect effect of an ncrease n the wage. The second term s the ndrect effect, showng the effect on asset values after a job swtch, whch s nfluenced by a change n w as the worker may swtch back to a job at level w n the future. Note that dw j W j (z), as the latter does not take nto account that the value of stayng n a job of type wll be hgher f the worker s rehred at a job on ths level n the future. Remark 6 In the lmt, as the number of job types go to nfnty and f j k k, the probablty that a worker returns to a job of type f acceptng a better job offer 5

16 goes to zero. In ths case dw j = W j (z) gven by (9). By defnng e as a functon of w + z, takng the dervatve of e and nsertng for dw j equaton n e that can be solved, at least numercally. gves a second order dfferental The effect of an ncrease n w on workers hred n other job types read (r + s) dw j l (z) = n k= f j k z,k (z) ( dw j k ( z) dw j l ((z)) )g( z)d z + s dw j The expresson s more complcated than one may frst expect, and ths s due to the endogenous search effort, whch vares between states. As en example, suppose s a hgh-type job and w l + z s low. The search effort ntally s s hgh, and hence an ncrease n w has a relatvely hgh mpact on W j (z). If the worker transfers to a job of type k wth a hgher w k + z, the search effort falls, and the effect of an ncrease n w s smaller. Ths feeds back nto dw j l (z). Note also that n ths case, we cannot abstract from the thrd term when the number of types ncrease. True, the probablty of movng to state goes to zero, but that s also true for dw j l (z). Relatvely speakng, the thrd term does not vansh. The total ncome a worker earns gross of taxes can be wrtten as rm j (z) = y + z + Ae j (z) j M j (z + y ) c(e j (z)) (2) where j M j (z) s defned analogous wth j W (z),.e., M j (z) = n f j k k= z,k (z) (M j k ( z) M j (z)))g( z)d z Note that the threshold z,k (z) s determned so as to maxmze W j, not M j. Consder an ncrease n w. Above ths nfluenced M j through ts effect on search effort of workers at rng and below. Now t may potentally nfluence search ntensty n jobs above rng as well. In addton we get a new effect through the workers rankng of alternatves. Taxes drve a wedge between productvty and wage, and ths may dstort the worker s job acceptance decson. If taxes (n levels) are ncreasng wth the wage as optmal taxes typcally prescrbe a worker wll put less weght on gross wages (.e. productvty) and more on amentes than the planner wll. Hence a worker wll have a tendency to accept too few jobs that offer a hgher wage than hs 6

17 current wage, and too many jobs that offer a lower wage. Formally, the dervatve of (2) reads (after addng and subtractng W as above), see equaton (9) (r + s) dm j (z) = dej (z) + Ae j (z) n n A (M W ) + Ae j (z) k= f kg( z,k (z))(t t k ) k= f j k z,k (z) ( dm j k ( z) r + s + Ae j j (z)( ˆF (z)) dm j (z)) )g( z)d z + s dm j (2) The frst thrd and fourth terms are as above, the new term s the second term. It shows the effect of a wage change on the acceptance decson of job offers. An ncrease n w decreases the acceptance rate of other job offers, and ths wll gve a gan/loss that depends on the dfference n tax rate. Ths dfference can be wrtten as n k= f kg( z,k (z))(m j k ( z,k(z)) M j (z)). Note that (r+s)m j (z)) = w +t + j M j (z + y ) c(e j (z)). Snce we are lookng for the margnal swtch, the NPV ncome and search and acceptance behavour s the same before and after the swtch. It follows that the flow change n value created s the dfference n taxes, and that the flow dfference ends at rate Ae j j (z)( ˆF (z)).8 The thrd term reflects the mpact on the value of future jobs, whch agan reflects that the worker may return to a job of type. The fnal term shows the effect of an ncrease n w on the value of beng unemployed. Remark 7 In the lmt, as the number of job types go to nfnty and f j k k, the probablty that a worker returns to a job of type f acceptng a better job offer goes to zero. In ths case the thrd and fourth terms vansh. Fnally, consder a change n w on M j k (z) the value of beng n job l. The frst thng to note s that an ncrease n w wll not nfluence the choce between two jobs 8 An alternatve formulaton of the total ncome s the followng.: Let T j (z) denote the expected dscounted future tax payments of a worker of type j n a job of type wth amentes z. It follows that M j (z) W j (z) + T j (z). Wth ths formulaton, the dervatves derved above follows easly. 7

18 w l and w k, k, l. It follows that (r + s) dm j l (z) = del (z) A (M l W l ) + Ae j dw l (z) f g( z l, (z))(t t l ) r + s + Ae j j (z)( ˆF l (z)) n + f j k ( dm j k ( z) dm j l (z)) )g( z)d z k= z,k (z) + s dm j (22) Fnally, consder the planner s problem. Formally, the problem s equvalent to the planner s maxmzaton problem wthout amentes, wth the old asset values replaced wth the new ones. In partcular, the trade-off between equty and ncentves to search s stll present. However, there are new elements n the trade-off related to the workers acceptance decson of frms. Wth the possble excepton of the state of beng unemployed, taxes wll typcally be ncreasng n job types because of equty consderatons. As a result, workers wll have a tendency to accept too few hgh-wage job offers (wth low amentes) and accept too many low-wage offers (wth hgh amentes). Ths effect s partcularly prevalent at the hghest pad job, jobs for whch unambguously too few workers employed n other job types accept and too many workers qut from after successful on-the-job search. Hence the ntroducton of amentes ncreases the socal cost of redstrbuton, as there s a new margn, acceptance rate of jobs, that taxes dstort (n addton to search ntensty). 5 Smulaton of Model Wthout Amenetes In ths secton we show a smulaton of the model wthout amentes. 5. Parameterzaton The unt of tme s a year. Followng Chrstensen, Lentz, Mortensen, Neumann, and Werwatz (25) we assume that the search cost functon c(e) s gven by c(e) = c + /c e +/c We assume that y s unformly dstrbuted on the nterval [y l, y h ] wth y = y l and y n = y h. Each worker type j draw from a worker specfc offer dstrbuton, F j. Ths 8

19 s approxmated by a beta dstrbuton wth parameters α j and β j. The socal welfare functon s gven by Γ = J τ j log(w j ) f γ = j= τ j (W j ) γ γ f γ where W j s the value functon for a type j worker n unemployment. To make the llustraton easer we set the number of worker types to 3 (J = 3) and the number of frm types to (n = ). Each type s equally lkely so τ j = /3. Table shows the the parameter values. We set s = such that the average employment length s 5 years. A s set to ht an unemployment rate of around n the no-tax case. We normalze c =, snce ths cannot be separately dentfed from A. Followng the results n Chrstensen, Lentz, Mortensen, Neumann, and Werwatz (25) we set c = such that the search cost functon s quadratc. We normalze y l = and set the hghest level of productvty to be 5 tmes hgher. 9

20 Table : Parameters Parameter Descrpton Value s j = s Job destructon rate r Effectve dscount rate.5 A Scale parameter n search technology 5 κ j = κ Worker type effcency c Scale parameter n search cost functon c Elastcty of search cost functon y l Lower bound on productvty dstrbuton y h Upper bound on productvty dstrbuton 5 UI fact y = UI fact y l M Amount to rase.3 J j= τ j n = g jm j γ Constant relatve rsk averson α Worker type, Offer Dstr., Alpha Parameter.75 α 2 Worker type 2, Offer Dstr., Alpha Parameter 5 α 3 Worker type 3, Offer Dstr., Alpha Parameter.95 β Worker type, Offer Dstr., Beta Parameter 5 β 2 Worker type 2, Offer Dstr., Beta Parameter 4 β 3 Worker type 3, Offer Dstr., Beta Parameter 3 M s set such that the government set taxes to rase 3 of the output n the no-tax equlbrum. We are gong to refer to workers of type, 2, and 3 as low, medum, and hgh, respectvely, snce F 3 F 2 F. 5.2 Results In ths secton we compare fout dfferent equlbra; ) one wthout any taxaton, 2) one wth a poll-tax, 3) one wth proportonal taxaton, and 4) one wth optmal taxaton. In the frst equalbrum M =, snce there s no taxaton to fnance t, whle n the last three cases we set M > and to the same amount such that n each of the tax schemes the government collects the same revenue. Table 2 shows some key numbers for the dfferent equlbra. We can see that the unemployment rate dffers across worker types wth workers of type beng most 2

21 Table 2: Key Equlbrum Objects Worker Type Worker Type 2 Worker Type 3 No Tax ( M = ) Unemployment Rate Average Before Tax Income Average Net of Tax Income W j Ω Poll Tax ( M > ) Unemployment Rate Average Before Tax Income Average Net of Tax Income W j Ω Proportonal Tax ( M > ) Unemployment Rate Average Before Tax Income Average Net of Tax Income W j Ω Optmal Tax ( M > ) Unemployment Rate Average Before Tax Income Average Net of Tax Income W j Ω unemployed. However, the unemployment rates do not dffer that much, snce all workers have smlar job destructon rates. Imposng a poll-tax does not dstort choces, but average net of tax ncome decreases and thus the value of unemployment decreases. Proportonal taxaton clearly dstorts search choces as llustrated by the ncrease n unemployment and the decrease n average net of tax ncome. The optmal tax schedule decrease unemployment and ncrease net of tax ncome compared to the proportonal tax scheme. Fnally, Table 2 shows Ω, the welfare as evaluated by the planner. We note that the welfare rankng s as expected. The proportonal tax regme delvers the lowest welfare, wth the poll tax regme begn second. The optmal tax regme delvers the hghest welfare by balancng equty and effcency concerns at the margn. In fgure we show the output, whch s equal to the wage ncome before taxaton, of each state. Output s unformly dstrbuted across frms and we mposed a UIB level 2

22 of 5 percent of the lowest output. We also show the net of tax ncome. In the polltax case all workers pay the same and unemployed workers actually receves negatve payments. In the proportonal tax case all workers pay 26 percent of ther ncome. Fnally, n the optmal tax case workers n all states pay taxes wth unemployed worker payng y n taxes. Our smulatons are set up such that all worker types face (almost) the same unemployment rsk as shown n Tables and 2. Thus, taxng workers n ths state s a relatvely effcent way of collectng taxes whle addng more ncentves to search for hgher payng jobs that are also taxed. Ths effect my be watered down or vansh altogether dependng on the precse specfcaton of the job destructon process. 5 Before Tax Income Net of Tax Income 5 Before Tax Income Net of Tax Income Unemp Unemp Before Tax Income Net of Tax Income (a) No Tax 5 Before Tax Income Net of Tax Income (b) Poll Tax Unemp Unemp (c) Proportonal Tax (d) Optmal Tax Fgure : Gross and Net Income Fgure shows tax rate on the rght sde y-axs. Ths s just calculated as the percentage dfference between the net of tax ncome and the before tax ncome as shown n fgure. The bars show the dstrbuton of worker types condtonal on the 22

23 state of the worker. The state condtonal dstrbutons are qute stable across the dfferent tax regmes. Note, however, as we shall see further below, ths does not mply taxaton has no mpact on the allocaton of labor. However, we can conclude that the response to the dfferent tax regmes s smlar across worker-types. For the optmal tax scheme, the tax s decreasng up untl the 8th percentle of frms. After the 8th percentle the tax rate start to ncrease. The reason s that n those frms around 5-65 percent of the workers are of the hghest type. Thus, taxng workers n ths state s a very good way of drectng taxaton at the hghest type Unemp Unemp Type Type 2 Type 3 Tax rate Type Type 2 Type 3 Tax rate (a) No Tax.4 (b) Poll Tax Unemp Unemp Type Type 2 Type 3 Tax rate Type Type 2 Type 3 Tax rate (c) Proportonal Tax (d) Optmal Tax Fgure 2: State Condtonal Worker Dstrbuton In fgure 3 we show the CDF s of the offer (F ) and steady state (G) dstrbutons for each taxaton scheme. The dfference between the offer and steady state dstrbutons s largest for the hgh worker type. Ths reflects that the ncentves to search for a better job s hghest for ths type, whch we show n fgure 4. It s clear that taxaton only nfluences search ntensty n the proportonal case, whereas t does not change search ntensty that much n the optmal tax case. Ths s a clear ndcaton that desgnng the tax scheme s mportant n order to mnmze the cost. 23

24 (a) No Tax (b) Poll Tax (c) Proportonal Tax (d) Optmal Tax Fgure 3: Dstrbuton of Workers Across Frm Types Ths pont s further llustrated by Fgure 4 whch shows the search ntenstes for every possble worker-frm type combnaton (ncludng unemployment) and for each of the four tax regmes we consder. We note that the proportonal tax regme depresses search for every worker types at all rungs at the job ladder relatve to the nondstorted search choces n panels (a) and (b). Under the optmal tax regme, workers search harder at the bottom and mddle rungs, but less at the hgher rungs, reflectng the progressvty of the optmal tax system for very hgh wages that allow the planner to target hgh-type workers. 24

25 Unemp Unemp (a) No Tax (b) Poll Tax Unemp Unemp (c) Proportonal Tax (d) Optmal Tax Fgure 4: Search Intensty Fgure 5 llustrates how dfferent tax regmes mpact the allocaton of labor. Fgures 5a, 5b, and 5c shows the steady state dstrbuton of type (low), type 2 (medum) and type 3 (hgh) workers across frm types,.e. across the job ladder. We note that for every worker type, the labor allocaton under the optmal tax regme strctly and clearly domnates the allocaton under a (revenue neutral) proportonal tax. The allocaton under the non-dstorted poll-tax regme of course domnates the allocaton under the optmal tax regme. 25

26 Unemp Unemp No tax/poll-tax Proportonal tax Optmal tax (a) Worker type No tax/poll-tax Proportonal tax Optmal tax (b) Worker type Unemp No tax/poll-tax Proportonal tax Optmal tax (c) Worker type 3 Fgure 5: Labor allocaton Fnally, n fgure 6 we show how the optmal tax schedule depends on the coeffcent of relatve rsk-averson n the welfare functon. Recall that γ = was the baselne value. 26

27 Unemp = = = Fgure 6: Optmal Tax Schemes: γ 5.3 Robustness Checks In ths secton we show results for the optmal taxaton f M = and UIfact =.95. Frst, we set M = n order to see how ths affects optmal polcy. Ths corresponds to the case, where all tax-ncome s redstrbuted through transfers. The state condtonal tax rate s decreasng from unemployment as t was prevously, but the drop s much larger than n the baselne case wth M >. The reason s that redstrbutng taxes to workers n low frm types s now the most effcent way of allocatng ressources to the low worker type. 27

28 Unemp Unemp Type Type 2 Type 3 Tax rate (a) Optmal Tax, baselne, M > Type Type 2 Type 3 Tax rate (b) Optmal Tax, M = Fgure 7: State Condtonal Worker Dstrbuton We now set UI frac =.95 nstead of. Ths does not change that much, whch s only natural snce all the ncome was taxed away for the unemployed n the baselne case. Ths s stll optmal to do, snce t s a good way of taxng the hgh type workers wthout dstortng search ncentves Unemp Unemp Type Type 2 Type 3 Tax rate (a) Optmal Tax, baselne, UI frac = Type Type 2 Type 3 Tax rate (b) Optmal Tax, UI frac =.95 Fgure 8: State Condtonal Worker Dstrbuton 6 Data Our emprcal analyss uses an admnstratve matched employer-employee dataset coverng the entre Dansh populaton durng We chose ths perod, snce t has a stable ncome tax system. On the worker sde, the dataset contans ndvdual labor market hstores measured at a daly frequency, job-specfc annual average hourly wages, detaled nformaton on ndvdual tax flngs, and a vector of background characterstcs such as gender, educaton, ncome measures, and martal status. We use a tax smulator coded by Kleven and Schultz (24) to compute ndvdual margnal tax 28

29 rates. On the frm sde, the data contans nformaton on value added at an annual frequency as well as some background characterstcs, e.g. ndustry. 6. Data sources We buld our matched employer-employee panel from three sources, () labor market spell data, ncludng hourly wage nformaton, () nformaton from IDA 9, a Dansh regster-based matched employer-employee database mantaned by Statstcs Denmark contanng socoeconomc nformaton on workers and some background nformaton on frms, and () nformaton on frms sales and purchases from frm-level value added tax (VAT) accounts admnstered by the Dansh tax authortes. We lnk the data sources usng worker and frm dentfers. Labor market spell data. The labor market spell data contans ndvdual job and non-employment spells. Informaton on job spells s avalable for the perod for all legal Dansh resdents aged 5-74, and s obtaned by combnng a number of admnstratve regsters. As mentoned above, for ths paper, we restrct attenton to the perod A job spell s defned as a contnuous perod of prmary employment at a gven frm. 2 Job spells, and therefore labor market transtons, are measured at the frm-level. A frm may consst of multple workplaces. Contnuous employment at dfferent workplaces wthn a frm s consdered as a sngle job spell. Nonemployment spells are perods where no job spells are recorded. We are not able to dstngush between nonemployed who actvely search for a job and nonpartcpants. Thus, we lump the two groups together nto non-employment and mpose sample selecton rules descrbed below to select those wth a strong labor market attachment. The unt of observaton n the labor market spell data s a person-spell-year. The job spell data ncludes worker and frm dentfers, start- and end-dates of the job, the annual earnngs pertanng to the job, as well as an estmate of the annual number of hours worked n the job. 3 Hence, we can compute (an estmate of) the average annual wage 9 IDA s an acronym of Integreret Database for Arbejdsmarkedsforsknng. Worker dentfers are obtaned from the socal securty regstry (the CPR-regstry) and frm dentfers are based on the Dansh busness regstry (the CVR-regstry). Hennng Bunzel at Aarhus Unversty has been nstrumental n developng the job spell data descrbed here. The data s descrbed n more detal n Bunzel and Hejlesen (26). 2 Prmary attachment s evaluated calendar month by calendar month. For each ndvdual n each month, the prmary employer s defned as the frm at whch the ndvdual works the hghest number of hours n the current and next two calendar months. 3 Annual hours are estmated usng nformaton on mandatory penson contrbutons. Lund and Vejln (25) develop and mplement a procedure for computng annual hours n a job n the IDA data for the 29

30 rate n each job. We overwrte short non-employment spells whch occur between a job to job transton between two dfferent frms or between a recall,.e. between two consecutve job spells at the same frm. For job to job transtons we overwrte up to 4 days of non-employment and for recalls we overwrte up to 2 weeks of non-employment. IDA data. IDA conssts of several sub-panels. We use the sub-panels IDA-P and IDA-S. IDA-P contans annual background nformaton from publc regsters on all ndvduals aged 5-74 resdng legally n Denmark on the 3st of December. We retan IDA-P nformaton on age, gender, hghest completed educaton ncludng date of completon, and nformaton on any ongong educaton subsequent to the hghest completed educaton. We use educaton nformaton to dentfy number of years of schoolng for each ndvdual, whch wll allow us to account for observed educatonal heterogenety n labor market behavor. Date of completon of the hghest completed educaton s used to defne labor market entry. The unt of observaton n IDA-P s a person-year. IDA-S contans background nformaton from publc records on all physcal workplaces n Denmark. Some jobs nvolves work that s carred out at changng locatons. Statstc Denmark desgnate such jobs at takng place at fcttous workplaces, and the fcttous workplaces are excluded from IDA-S. 4 We retan nformaton on ndustry afflaton and a publc sector ndcator from IDA-S. Our analyss s carred out at the frm-level. We take a frm s ndustry afflaton and publc sector status to be the ndustry and publc sector status of ts largest workplace. 5 The unt of observaton n the (aggregated) IDA-S panel s a frm-year. VAT data. Data on frms sales and purchases are obtans from the panels MOMS and MOMM, constructed from frm s VAT accounts mantaned by the Dansh tax authortes. Dansh frms settle VAT accounts monthly, quarterly or annually dependng on the sze of ther revenue. MOMS covers the perod and contans annual sales and purchases. MOMM s a monthly panel startng n January, 2. We use MOMM data up untl December, 25, aggregatng the monthly nformaton to an perod 98-27, prmarly usng nformaton on mandatory penson contrbutons. We adapt ths procedure for the spell data wth some mnor smplfcatons. 4 Approxmately 95% of workers are workng at non-fcttous workplaces,.e. has a fxed physcal workplace. 5 The sze of the workplace s the number of workers wth prmary employment at the workplace on the 28th of November, as measured n IDA-S. 3