Lecture 5 Search and matching theory

Transcription

1 Lecture 5 Search and matching theory Leszek Wincenciak, Ph.D. Warsaw University December 16th, 2009

2 2/48 Lecture outline: Introduction Search and matching theory Search and matching theory The dynamics of unemployment Job creation by firms Wage determination and the steady state Comparative statics

3 Introduction 3/48 Introduction Even in the absence of marked changes in overall employment, there are simultaneous processes of job creation and destruction, reaching 20% of total employment in manufacturing during a year Workersaresearchingforthebestjobs Firmsarelookingforthebestworkers Searchingforjoboraworkerandmatchingtakestimeandis costly This leads to frictional unemployment

4 Introduction 4/ q1 1996q1 1997q1 1998q1 1999q1 2000q1 2001q1 2002q1 2003q1 2004q1 JC3 JD3 Figure 1. Job creation and destruction rates in Polish manufacturing, LFS data

5 Search and matching theory 5/48 Search and matching theory Search and matching theory

6 Search and matching theory 6/48 Search and matching theory Frictional unemployment Firms create job openings(vacancies) Workers search for jobs Matchofaworkerandavacancyresultsinaproductivejob Matching is not coordinated(workers and firms dedicate time and resources to find a suitable match) Probabilitythatafirmoraworkerfindthepartnerdepends on a relative number of vacant jobs and unemployed workers Laborsupply(L)=unemployed+employed Labordemand=filledjobs+vacancies

7 Search and matching theory 7/48 Search and matching theory Totalnumberofunemployedworkers ul Totalnumberofvacancies vl Total number of matches between unemployed workers and vacantfirmsineachunitoftime ml Theprocessofmatchingissummarizedbyamatching function, which expresses the number of newly created jobs (ml)asafunctionofthenumberofunemployedworkers (ul) and vacancies(vl): ml = m(ul,vl) (1)

8 Search and matching theory 8/48 Search and matching theory The matching function, assumed to be increasing in both arguments, can be thought of as similar to aggregate production function. Workers and vacant jobs can be viewed as productive inputs which produce a match, which results in a productive job. Creation of employment requires presence of both unemployed workersandvacantjobs m(0,0) = m(0,vl) = m(ul,0) = 0. Inordertohaveaconstantunemploymentrateinagrowing economy, we need the matching function to exhibit a constant returns to scale(empirical evidence seems to support this assumption).

9 Search and matching theory 9/48 Search and matching theory IncaseofCRSmatchingfunction,wecanwrite: m = m(ul,vl) = m(u,v). (2) L Thefunction m( )determinestheflowofworkerswhofindajob and who exit the unemployment pool within each time interval. Considerthecaseofanunemployedworker:ateachmomentin time,theworkerwillfindajobwithprobability p = m( )/u.with constant returns to scale for m( ) we may thus write: m(u, v) u ( = m 1, v ) p(θ), (3) u where θ = v/u is called the labor market tightness.

10 Search and matching theory 10/48 Search and matching theory The instantaneous probability p that a worker finds a job is positively related to the tightness of the labor market which is measuredby θ,theratiobetweenthenumberofvacanciesand unemployed workers. An increase in θ, reflecting a relative abundance of vacant jobs relative to unemployed workers, leads to anincreasein p.

11 Search and matching theory 11/48 Search and matching theory The average length of an unemployment spell is given by 1/p(θ), andisthusinverselyrelatedto θ.similarly,therateatwhich avacantjobismatchedtoaworkermaybeexpressedas: m(u, v) v ( = m 1, v ) u u v = p(θ) q(θ), (4) θ a decreasing function of the vacancy/unemployment ratio. An increasein θreducestheprobabilitythatavacancyisfilledand 1/q(θ) measures the average time that elapses before a vacancy is filled. Thedependenceof pand qon θcapturesthedualexternality betweenagentsinthelabormarket:anincreaseinthenumberof vacancies relative to unemployed workers increases the probability thataworkerfindsajob(dp( )/dv > 0),butatthesametimeit reduces the probability that a vacancy is filled(dq( )/dv < 0).

12 Search and matching theory 12/48 The dynamics of unemployment The dynamics of unemployment

13 Search and matching theory 13/48 The dynamics of unemployment The dynamics of unemployment Changes in unemployment result from a difference between the flowofworkerswholosetheirjobandbecomeunemployed,and theflowofworkerswhofindajob.theinflowintounemployment isdeterminedbythe separationrate whichwetakeasgivenfor themoment:ateachmomentintimeafraction sofjobs (correspondingtoafraction 1 uofthelaborforce)ishitby ashockthatreducestheproductivityofthematchtozero:inthis casetheworkerlosesherjobandreturnstothepoolof unemployed,whilethefirmisfreetoopenupavacancyinorderto bring employment back to its original level. Given match destruction rate s, jobs therefore remain productive for an average period of 1/s.

14 Search and matching theory 14/48 The dynamics of unemployment Given these assumptions we can now describe the dynamics of the number of unemployed workers. Since L is constant, d(ul)/dt = ul and hence: ul = s(1 u)l p(θ)ul u = s(1 u) p(θ)u. (5) The dynamics of the unemployment rate depend on the tightness ofthelabormarket θ:atahighvaluefortheratioofvacanciesto unemployed workers, workers easily find a job leading to a large flow out of unemployment.

15 Search and matching theory 15/48 The dynamics of unemployment From equation(5) we can immediately derive the steady state relationship between the unemployment rate and θ: u = s s + p(θ). (6) Since p ( ) > 0,thepropertiesofthematchingfunctiondetermine anegativerelationbetween θand u:ahighervalueof θ correspondstoalargerflowofnewlycreatedjobs.inordertokeep unemployment constant, the unemployment rate must therefore increase to generate an offsetting increase in the flow of destroyed jobs. To obtain job creation and destruction rates, we may divide the flowsintoandoutofemploymentbythetotalnumberofemployed workers (1 u)l.therateofdestructionissimplyequalto s, whiletherateofjobcreationisgivenby p(θ)[u/(1 u)].

16 Search and matching theory 16/48 The dynamics of unemployment θ v v = θ 0 u θ 0 v 0 u 0 u = 0 u θ 0 u 0 u = 0 u Figure 2. Dynamics of the unemployment rate

17 Search and matching theory 17/48 The dynamics of unemployment The steady-state relationship(6) is illustrated graphically in the leftpaneloffigure5:toeachvalueof θcorrespondsaunique value for the unemployment rate. Moreover, the same properties of m( ) ensure that this curve is convex. For points above or below u = 0,theunemploymentratetendstomovetowardsthe stationaryrelationship:keeping θconstantat θ 0,avalue u > u 0 causesanincreaseintheflowoutofunemploymentandadecrease oftheflowintounemployment,bringing ubackto u 0.

18 Search and matching theory 18/48 The dynamics of unemployment Moreover, given u and θ, the number of vacancies is uniquely determinedby v = θu,where vdenotesthenumberofvacancies asaproportionofthelaborforce.thepictureontherighthand sideofthefigureshowsthecurve u = 0in (v,u)space.thislocus isknownasthebeveridgecurve,andidentifiesthelevelof vacancies v 0 thatcorrespondstothepair (θ 0,u 0 )ofthelefthand panelofthefigure5. It is important to note that variations in the labor market tightness areassociatedwithamovementalongthecurve u = 0,while changes in the separation rate s or the efficiency of the matching process(captured by the properties of the matching function) correspondtomovementsofthecurve u = 0.

19 Search and matching theory 19/48 The dynamics of unemployment Forexample,anincreasein soradecreaseinthematching efficiencycauseanupwardshiftof u = 0.Equation(6)givesafirst steady state relationship between u and θ. To find the actual equilibrium values, we need to specify a second relationship between these variables. This second relationship can be derived fromthebehavioroffirmsandworkersonthelabormarket.

20 Search and matching theory 20/48 Job creation by firms Job creation by firms

21 Search and matching theory 21/48 Job creation by firms Job creation by firms Thecrucialdecisionoffirmsconcernsthesupplyofjobsonthe labormarket.thedecisionofafirmwhethertocreateavacancy depends on the expected future profits over the entire time horizon ofthefirm,whichweassumetobeinfinite.formally,each individual firm solves an intertemporal optimization problem taking as given the aggregate labor market conditions which are summarized by θ, the labor market tightness. Individual firms therefore disregard the effect of their decisions on θ, and consequently on the matching rates p(θ) and q(θ). To simplify the analysis,weassumethateachfirmcanofferatmostonejob.if thejobisfilled,thefirmreceivesaconstantflowofoutputequalto y.moreover,itpaysawage wtotheworkerandittakesthiswage as given. The determination of this wage will be described later on.

22 Search and matching theory 22/48 Job creation by firms Onthecontrary,ifthejobisnotfilled,thefirmincursaflowcost c,whichreflectsthetimeandresourcesinvestedinthesearchfor suitable workers. Firms therefore find it attractive to create avacancyaslongasitsvalue,measuredintermsofexpected profits,ispositive;intheoppositecase,thefirmwillnotfindit attractivetoofferavacancyandwillexitthelabormarket.the valuethatafirmattributestoavacancy(denotedby V)andto afilledjob(j)canbeexpressedusingtheassetequations.given aconstantrealinterestrate r,wecanexpressthesevaluesas: rv (t) = c + q(θ(t))(j(t) V (t)) + V (t), (7) rj(t) = (y w(t)) + s(v (t) J(t)) + J(t). (8)

23 Search and matching theory 23/48 Job creation by firms (7)and(8)areexplicitfunctionsoftime.Theflowreturnof avacancyisequaltoanegativecostcomponent ( c),plusthe capitalgainincasethejobisfilledwithaworker (J V ),which occurswithprobability q(θ),plusthechangeinthevalueofthe vacancyitself ( V ).Similarly,(8)definestheflowreturnofafilled jobasthevalueoftheflowoutputminusthewage (y w),plus thecapitalloss (V J)incasethejobisdestroyed,whichoccurs withprobability s,plusthechangeinthevalueofthejob ( J).

24 Search and matching theory 24/48 Job creation by firms Subtracting(7) from(8) yields the following expression for the differenceinvaluebetweenafilledjobandavacancy: r(j(t) V (t)) =(y w(t) + c) [s + q(θ(t))](j(t) V (t)) + ( J(t) V (t)). (9)

25 Search and matching theory 25/48 Job creation by firms Now,ifwefocusonsteadystateequilibriawecanimpose V = J = 0inequations(7)and(8).Moreover,weassumefree entryoffirmsandasaresult V = 0:newfirmscontinuetooffer vacantjobsuntilthevalueofthemarginalvacancyisreducedto zero.substituting V = 0in(7)and(8)andcombiningthe resulting expressions for J, we get: J = c/q(θ) J = (y w)/(r + s) } y w = (r + s) c q(θ). (10) Equation(7)givesusthefirstexpressionfor J.Accordingtothis conditiontheequilibriumvalueofafilledjobisequaltothe expectedcostsofavacancy,thatistheflowcostofavacancy c times the average duration of a vacancy 1/q(θ).

26 Search and matching theory 26/48 Job creation by firms Thesecondconditionfor Jcanbederivedfrom(8):thevalueof afilledjobisequaltothevalueoftheconstantprofitflow y w. Theseflowreturnsarediscountedatrate r + stoaccountforboth impatience and the risk that the match breaks down. Equating these two expressions yields the final solution(10), which gives the marginal condition for employment in a steady state equilibrium: the marginal productivity of the worker(y) needs to compensate thefirmforthewage wpaidtotheworkerandfortheflowcostof opening a vacancy. Thelatterisequaltotheproductofthediscountrate r + sand the expected costs of a vacancy c/q(θ).

27 Search and matching theory 27/48 Job creation by firms Thislasttermisjustlikeanadjustmentcostforthefirm s employment level. It introduces a wedge between the marginal productivityoflaborandthewagerate,whichissimilartothe effectofthehiringcosts.however,inthismodelthesizeofthe adjustment cost is endogenous and depends on the aggregate conditions on the labor market. In equilibrium, the size of the adjustment costs depend on the unemployment rate and on the number of vacancies, which are summarized at the aggregate level bythevalueof θ.if,forexample,thevalueofoutputminuswages (y w) increases, then vacancy creation will become profitable (V > 0)andmorefirmswillofferjobs.Asaresult, θwillincrease, leadingtoareductioninthematchingrateforfirmsandan increaseintheaveragecostofavacancyandboththeseeffects tendtobringthevalueofavacancybacktozero.

28 Search and matching theory 28/48 Job creation by firms Finally, notice that equation(10) still contains the wage rate w. This is an endogenous variable. Hence the job creation condition (10)isnotyetthesteadystateconditionwhichtogetherwith(6) wouldallowustosolvefortheequilibriumvaluesof uand θ.to completethemodelweneedtoanalyzetheprocessofwage determination, to which we now turn.

29 Search and matching theory 29/48 Wage determination and the steady state Wage determination and the steady state

30 Search and matching theory 30/48 Wage determination and the steady state Wage determination Theprocessofwagedeterminationthatweadopthereisbasedon the fact that the successful creation of a match generates asurplus.thatis,thevalueofapairofagentsthathaveagreed tomatch(thevalueofafilledjobandanemployedworker)is largerthanthevalueoftheseagentsbeforethematch(thevalue ofavacancyandanunemployedworker).thissurplushasthe natureofamonopolisticrentandneedstobesharedbetweenthe firm and the worker during the wage negotiations. Here we shall assume that wages are negotiated at a decentralized level between each individual worker and her employer. Since workers and firms are identical, all jobs will therefore pay the same wage.

31 Search and matching theory 31/48 Wage determination and the steady state Wage determination Let Eand Udenotethevaluethataworkerattributesto employment and unemployment, respectively. The joint value of amatch(givenbythevalueofafilledjobforthefirmandthe valueofemploymentfortheworker)canthenbeexpressedas J + E,whilethejointvalueincasethematchopportunityisnot exploited(givenbythevalueofavacancyforafirmandthevalue ofunemploymentforaworker)isequalto V + U.Thetotal surplusofthematchisthusequaltothesumofthefirm ssurplus, J V,andtheworker ssurplus, E U: (J + E) (V + U) (J V ) + (E U). (11)

32 Search and matching theory 32/48 Wage determination and the steady state Wage determination Thematchsurplusisdividedbetweenthefirmandtheworker through a wage bargaining process. We take their relative bargaining strength to be exogenously given. Formally, we adopt the assumption of Nash bargaining. This assumption is common in models of bilateral negotiations. It implies that the bargained wage maximizesageometricaverageofthesurplusofthefirmandthe worker, each weighted by a measure of their relative bargaining strength(nash maximand). In our case the assumption of Nash bargaining gives rise to the following optimization problem: max w (J V )1 β (E U) β, (12) where 0 β 1 denotes the relative bargaining strength of the worker.

33 Search and matching theory 33/48 Wage determination and the steady state Wage determination Given that the objective function is a Cobb-Douglas one, we can immediately express the solution(the first order conditions) of the problem as: E U = β (J V ) E U = β[(j V ) + (E U)].(13) 1 β The surplus that the worker appropriates in the wage negotiations (E U)isthusequaltoafraction βofthetotalsurplusofthejob.

34 Search and matching theory 34/48 Wage determination and the steady state Similartowhatisdonefor Vand Jin(7)and(8),wecanexpressthe values E and U using the relevant asset equations(reintroducing the dependence on time t): re(t) = w(t) + s(u(t) E(t)) + Ė(t) (14) ru(t) = z + p(θ)(e(t) U(t)) + U(t). (15) Fortheworkertheflowreturnonemploymentisequaltothewageplus thelossinvalueiftheworkerandthefirmseparate,whichoccurswith probability s,plusanychangeinthevalueof Eitself;thereturnon unemploymentisgivenbytheimputedvalueofthetimethataworker doesnotspendworking,denotedby z,plusthegainifshefindsajoband thechangeinthevalueof U.Parameter zincludesthevalueofleisure and/or the value of alternative sources of income including possible unemployment benefits. It is assumed to be exogenous and fixed.

35 Search and matching theory 35/48 Wage determination and the steady state Restrictingattentiontosteadystateequilibria,sothat Ė = U = 0, wecanderivethesurplusoftheworker E Udirectlyfrom(14) and(15). E U = w z r + s + p(θ). (16) According to(16) the surplus of a worker depends positively on the difference between the flow return during employment and unemployment (w z) and negatively on the separation rate s and on θ:anincreaseintheratioofvacanciestounemployedworkers increases the exit rate out of unemployment and reduces the average length of an unemployment spell.

36 Search and matching theory 36/48 Wage determination and the steady state Using(16) and noting that in steady state equilibrium J V = J = y w r + s, wecansolvetheexpressionfortheoutcomeofthewage negotiations given by(13) as: w z r + s + p(θ) = β y w 1 β r + s. Rearranging terms, and using(10), we obtain the following equivalent expressions for the wage: w z = β[(y + cθ w) + (w z)] (17) w = z + β(y + cθ z). (18)

37 Search and matching theory 37/48 Wage determination and the steady state Equation(17)istheversionintermsoftheflowsofequation(13): the flow value of the worker s surplus, i.e. the difference between thewageandalternativeincome z,isafraction βofthetotalflow surplus.theterm y w + cθrepresentstheflowsurplusofthe firm,where cθdenotestheexpectedcostsavingsifthefirmfills ajob.moreover,thewageisapureredistributionfromthefirmto theworker.ifweeliminatethewagepaymentsin(17)weobtain theflowvalueofthetotalsurplusofafilledjob y + cθ z,which isequaltothesumofthevalueofoutputandthecostsavingof the firm minus the alternative costs of the worker. Finally, equation(18) expresses the wage as the sum of the alternative incomeandthefractionofthesurplusthataccruestotheworker.

38 Search and matching theory 38/48 Wage determination and the steady state Itcaneasilybeverifiedthattheonlyinfluenceofaggregatelabor marketconditionsonthewageoccurvia θ,theratioofvacancies to unemployed workers. The unemployment rate u does not have any independent effect on wages. The explanation is that wages arenegotiatedafterafirmandaworkermeet.inthissituationthe matchsurplusdependson θ,aswesawabove.thisvariable determines the average duration of a vacancy, and hence the expectedcostsforthefirmifitwouldcontinuetosearch.the determination of the equilibrium wage completes the description of the steady state equilibrium.

39 Search and matching theory 39/48 Wage determination and the steady state The equilibrium can be summarized by equations(6),(10) and (18)whichweshallrefertoas BC(Beveridgecurve), JC(job creation condition) and W(wage equation): s u = s + p(θ) w = y (r + s) c q(θ) (BC) (19) (JC) (20) w = (1 β)z + β(y + cθ) (W) (21)

40 Search and matching theory 40/48 Wage determination and the steady state Foragivenvalueof θ,thewageisindependentofthe unemployment rate. The system can therefore be solved recursively fortheendogenousvariables u, θand w.usingthedefinitionfor θ wecanthensolvefor v.thelasttwoequationsjointlydetermine the equilibrium wage w and the ratio of vacancies/unemployed θ, asisshownintheleftpaneloffigure6.given θ,wecanthen determine the unemployment rate u, and consequently also v, which equate the flows into and out of unemployment(the right hand panel of the figure).

41 Search and matching theory 41/48 Wage determination and the steady state w v JC + W W w 0 v 0 θ 0 JC θ θ 0 u 0 BC u Figure 3. Equilibrium of the labor market with frictional unemployment

42 Search and matching theory 42/48 Comparative statics Comparative statics

43 Search and matching theory 43/48 Comparative statics Comparative statics This dual representation facilitates the static comparative analysis, whichisintendedtoanalyzetheeffectofchangesinthe parameters on the steady state equilibrium. Scenario 1: Assume, that we observe an increase in unemployment benefits,acomponentof z,oranincreaseintherelative bargaining strength of workers β.

44 Search and matching theory 44/48 Comparative statics Comparative statics, z or β w W v JC + W w 1 w 0 θ 1 θ 0 W JC θ v 0 v 1 θ 0 θ 1 u 0 u 1 JC + W BC u Figure4.Theeffectsofanincreasein zor β

45 Search and matching theory 45/48 Comparative statics Interpretation As a result of an increase in unemployment benefits(captured by z)oranincreaseinthebargainingpoweroftheworkers(captured by β)thewagecurvedefinedby(21)shiftsupwards.thiscauses anincreaseinthewageandareductioninthelabormarket tightness, θ. This reduction, along the Beveridge Curve(BC), is accompaniedbyanincreasein uandareductionin v. Scenario 2: Consider now an adverse aggregate shock, resulting in adecreasein y.

46 Search and matching theory 46/48 Comparative statics Comparative statics, y w W v JC + W w 0 W JC + W w 1 θ 1 θ 0 JC JC θ v 0 v 1 θ 0 θ 1 u 0 u 1 BC u Figure 5. The effects of an adverse aggregate shock

47 Search and matching theory 47/48 Comparative statics Areductionin ymovesboth JCand Wschedulesdownwards. Thisresultsinreductionofthewagebuthasanambiguouseffect on θ. However, formal analysis shows that in a stationary equilibrium θalsodecreases(providedthat 0 β 1).Atthe sametimethecurve BCdoesnotshift,sothattheunemployment rate must increase while the number of vacancies v is reduced. Scenario 3: Consider now a reallocative shock, i.e. the increase in the separation rate s.

48 Search and matching theory 48/48 Comparative statics Comparative statics, s w W v JC + W w 0 w 1 JC + W θ 1 θ 0 JC JC θ? θ 0 θ 1 u 0 u 1 BC BC u Figure 6. The effects of an adverse reallocative shock