UW-Madison ECON 312 - Lecture 24 - Search and Matching

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lect24-front.pdfPages from MortensonPissarides_SL.pdfLecture 24Search and MatchingNoah WilliamsUniversity of Wisconsin - MadisonEconomics 312Williams Economics 312Search and MatchingSo far we have considered only the worker’s problem,taking the wage distribution as given.But firms also need to search to hire workers.This gives rise to a matching problem, which was studiedby Pissarides (1985) and Mortensen and Pissarides (1994).This now serves as the benchmark model for studyingunemployment.This discussion and notation follows Romer, Chapter10.6-10.7.Williams Economics 312ModelTime is continuous.Demographics:There are¯L identical workers.They live forever (or they could die stochastically).Preferences:Utility = consumption (one good).Discount rate r.3 / 45TechnologyOutput is produced from labor only.Production can take place only in a worker-job match.Each match consists of exactly one job / one worker.When matched, a match produces a flow output of A.4 / 45Model: The logicEnter the "period" withU unemployed workersF =¯L − U job matches.E = F employed workersbE matches break up (exogenously)Firms post V vacancies, paying a cost.5 / 45Model: The logicUnemployed workers and vacancies meet at random.Workers who don’t meet a firm stay unemployed, consume 0.In a match:Firm and worker bargain over the wage (no contracts!).If no agreement is reached, the job becomes vacant and the workerbecomes unemployed.If agreement is reached, the pair produces until exogenous breakupoccurs.6 / 45WorkersWorkers live forever and maximize the expected present value ofearnings.The discount rate is r (exogenous).The only decisions: in wage negotiation.7 / 45FirmsFirms can create jobs (vacancies) at a flow cost of C per unit of time.A filled job produces A and pays w (endogenous) to the worker.The firm keeps the profit: A − w − C.8 / 45MatchingA matching function describes how workers are matched tovacancies.The number of matches per period isM (U,V ) = K UβVγ(1)We take M (U, V ) as given.Matching functions can be derived from micro-foundations.More vacancies or more unemployed workers result in more matches.9 / 45Steady state restrictionsFocus on situations where E,U, V are constant.The number of employed workers changes according to˙E = M (U,V )− bE (2)where b is the exogenous rate of match dissolution.In steady state˙E = 0:M (U,V ) = bE (3)10 / 45Steady state restrictionsThe number of unemployed follows˙U = bE − M (U,V ) (4)= −˙E (5)˙U = 0 is implied by˙E = 0.11 / 45DefinitionsDefine the rate of exit from unemploymenta =M (U,V )U(6)Define the rate at which vacancies are filled:α =M (U,V )V(7)12 / 45Solution methodAssume that all workers receive the same wage w when matched (verifythis later).For a given wage, there is only one decision to be made: how manyvacancies to create.Assume that vacancies are created until they yield zero profit (freeentry).We need to find the value of an open vacancy (VV).Then we need to find the bargained wage.For this we need to know the valuesof being employed (VE) or unemployed (VU).of a filled vacancy (VF).13 / 45Workers: EmployedThe value of being employed isrVE= w + b (VU− VE) (8)Or:VE=w + bVU+ (1 − b)VE1 + rIntuition:Receive a flow benefit w.With probability b switch to unemployment and lose VU− VE.14 / 45Employed worker: DerivationConsider the value of being employed for a short period ∆t.Receive flow benefit w, discounted at r.Probability of remaining in the match: e−bt.Value:R∆t0e−(r+b)tw dt =1−e−(r+b)∆tr+bw.15 / 45Employed worker: DerivationAt the end, at t + ∆t:continue as unemployed with probability 1 − e−b∆t.continue in match with probability e−b∆t.Value: e−r∆te−b∆tVE(∆t) +1 − e−b∆tVU(∆t) .16 / 45Employed worker: DerivationValue of being employed is then:VE(∆t) =1−e−(r+b)∆tr+bw + e−r∆te−b∆tVE(∆t) +1 − e−b∆tVU(∆t) =wr+b+(1−e−b∆t)e−r∆t1−e−(r+b)∆tVU(∆t).Take the limit as ∆t → 0.Use l’Hopital’s rule to evaluate the ratio in front of VU. It becomesbr+b. ThereforeVE=wr + b+br + bVURearrange. Done.17 / 45Unemployed WorkerrVU= 0 + a (VE− VU)OrVU=0 + aVE+ (1 − a)VU1 + rReceive nothing right now.With probability a switch to "employed."18 / 45Unfilled VacanciesrVV= −C + α (VF− VV)OrVV=−C + αVF+ (1 − α)VV1 + rPay the vacancy cost C.With probability α fill it and receive VF.19 / 45Filled vacanciesrVF= A − w − C + b (VV− VF)OrVF=A − w − C + bVV+ (1 − b)VF1 + rReceive the profit A − w − C.With probability b lose the match and receive VV.20 / 45Stationary equilibriumA stationary equilibrium determines (VU,VE,VV,VF,E,U,V,w) such that:the values Vxare determined as above.the labor market "clears:"¯L = E + U.the number of employed is constant: M (U,V ) = bE.creating new vacancies yields zero profit: VV= 0wages are somehow determined (this is where VU,VEcome in).In addition: a,α are defined above as functions of U,V.21 / 45Wage determinationWhat happens when firms and workers meet?The worker accepts any wage such that VE≥ VU.The firm accepts any wage such that VF≥ VV.Bargaining pins down the exact distribution of the surplus.We make an assumption: the surplus is evenly divided:VE− VU= VF− VV(9)Note: there is no good theory that would pin down how the surplus issplit.22 / 45Model summary IObjects: (VU,VE,VV,VF,E,U,V ,w).Flow equations:¯L = E + U (10)M (U,V ) = bE (11)Values:rVE= w + b (VU− VE) (12)rVU= a (VE− VU) (13)rVV= −C + α(VF− VV) = 0 (14)rVF= A − w − C − b(VF− VV) (15)23 / 45Model summary IIBargaining:VE− VU= VF− VV(16)Definitions:a =M (U,V )U(17)α =M (U,V )V(18)24 / 45Solving the modelThis is just algebra: solve the 8 equations for the 8 unknowns.Step 1: substitute out the value functions.Start from bargaining:VE− VU= VF− VV(19)From the definitions:VE− VU=wa + b + r(20)VF− VV=A − wα + b + r(21)25 / 45Solving the modelSolve for the wage:w =(a + b + r)Aa + α + 2b + 2r(22)Intuition:the surplus (A) is equally divided when α = a.if workers have a harder time finding jobs (low a), their surplus shareshrinks.The next step: express everything in terms of E.26 / 45Job Finding RateFind a in terms of E.a (E) =M(U,V )U=bE¯L − Ea is increasing in E.Higher employment → faster exit from unemployment.27 / 45Vacancy Filling RateFind α in terms of E.α =M(U,V )V=bEVα is increasing in E, but only for given V .28 / 45Vacancy Filling RateSolve the matching function


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