Kultti, Klaus

Miettunen, Antti

Virrankoski, Juha

Helsingin yliopiston digitaalinen arkisto

öMmföäflsäafaäsflassflassflasffffffffffffffffffffffffffffffffffffDiscussion PapersEquilibrium Unemployment Duration in an Urn-Ball Model of the Labour MarketKlaus KulttiUniversity of Helsinki and HECERAntti MiettunenUniversity of Helsinki and HECERandJuha VirrankoskiUniversity of Helsinki, RUESG and HECERDiscussion Paper No. 189October 2007ISSN 1795-0562HECER – Helsinki Center of Economic Research, P.O. Box 17 (Arkadiankatu 7), FI-00014University of Helsinki, FINLAND, Tel +358-9-191-28780, Fax +358-9-191-28781,E-mail info-hecer@helsinki.fi, Internet www.hecer.fiHECERDiscussion Paper No. 189Equilibrium Unemployment Duration in an Urn-Ball Model of the Labour Market*AbstractWe derive the equilibrium unemployment duration in response to a change inunemployment when matches are formed by an urn-ball process. We show that theduration is more sensitive to unemployment than conventionally thought. This result comesfrom solving the equilibrium of the model instead of just varying the unemployment-vacancy ratio.JEL Classification: J64Keywords: unemployment, searchKlaus Kultti Antti MiettunenDepartment of Economics Department of EconomicsUniversity of Helsinki University of HelsinkiP.O. Box 17 (Arkadiankatu 7) P.O. Box 17 (Arkadiankatu 7)FI-00014 University of Helsinki FI-00014 University of HelsinkiFINLAND FINLANDe-mail: klaus.kultti@helsinki.fi e-mail: antti.t.miettunen@helsinki.fiJuha VirrankoskiResearch Unit of Economic Structures and Growth (RUESG)Department of EconomicsUniversity of HelsinkiP.O. Box 17 (Arkadiankatu 7)FI-00014 University of HelsinkiFINLANDe-mail: juha.virrankoski@helsinki.fi* We thank participants at EALE 2006 conference in Prague, ESEM 2007 conference inBudapest, and HECER seminars for useful comments. Juha Virrankoski gratefullyacknowledges financial support from Yrjö Jahnsson Foundation.1 IntroductionSearch theory is the main modelling framework used in the theory of labour markets.Finding a partner is time consuming and uncertain, and consequently unemployed andvacancies meet in some kind of random manner. Often the exact meeting process isside stepped by postulating a matching function that tells the number of employmentrelationships produced given the number of unemployed and vacancies. Denoting thenumber of unemployed by U and the number of vacancies by V the number of matchesis then given byM = m(U, V ) where m is the matching function. Usually, m is assumedto be increasing in both arguments and homogenous of degree one. On average, anunemployed worker finds a job during a period with probabilitym(U, V )/U , and a vacantjob is filled with probability m(U, V )/V . In a steady state equilibrium the inverses ofthese probabilities are the mean durations of unemployment and vacancies, respectively.In a survey of matching functions Petrongolo and Pissarides (2001, p. 402) claimthat a simple urn-ball model fails to produce unemployment durations that are observedempirically: “. . . It is, however, too naïve to be empirically a good approximation tomatching in real labor markets. For example, it implies an implausible combinationof levels and durations of unemployment. If the level of unemployment and vacanciesis the same, the mean duration of unemployment is 1.58 periods, and if the level ofunemployment is three times as high as that of vacancies, mean duration is 3.16. Inactual labor markets duration would rise by more than the function£M = V¡1− e−U/V¢¤implies when the level of unemployment is higher.”There are basically two methods for assessing whether the simple urn-ball modelproduces empirically observed unemployment durations. As the duration depends onlyon ratio U/V , one can vary the ratio and compare the theoretical durations to theobserved ones. This requires reliable data on both unemployed and vacancies. If data onvacancies, for example, is missing or unreliable but there is data on unemployment andjob separation probability then one can evaluate the urn-ball model by using equilibriumanalysis. We choose this approach.We study how the duration of unemployment is aﬀected in an urn-ball model when the1absolute number of unemployed changes. We specify an equilibriummodel where changesin one variable cannot take place without changes in other variables. In particular, anincrease in the number of unemployed requires a decrease in the number of firms, or anincrease in the labour force or in the job separation rate. Keeping track of the eﬀectsof these changes requires that one solves the new equilibrium values for the stocks ofactive firms and workers as well as the stocks of unemployed and vacancies. It turns outthat when the number of unemployed is, say, tripled this necessitates a decrease in thenumber of vacancies such that the equilibrium ratio U/V more than triples. Thus, theduration is more sensitive to unemployment than one would expect by just tripling theunemployment-vacancy ratio.We derive the urn-ball model in Section 2. In Section 3 we present the equilibriumanalysis when unemployment increases because the number of firms decreases. In Section4 we consider a model where change in unemployment is driven by a change in the labourforce. Section 5 concludes.2 The Urn-Ball ModelIn the urn-ball model the vacancies present the urns, and the unemployed play the roleof the balls which are randomly placed in the urns. The number of balls ending up in anurn is thus binomially distributed. Some urns end up with one ball, some urns end upwith several balls, and some urns remain empty. The coordination failure is due to thelack of information about the other applicants’ actions. The model can be enriched buthere we focus just on the equilibrium meetings that are assumed symmetric and random.We do not say anything about how the meetings come about but for instance, a modelwith posted wages exhibits the matching function (1) in a symmetric equilibrium.The matching function isM = Vh1− (1− 1/V )Ui, (1)where (1− 1/V )Udenotes the probability that a vacancy receives no applications. Theprobability that a vacancy receives at least one application multiplied by V gives us (1),2the expected number of matches. For large U and V the magnitude (1− 1/V )Ucan beapproximated with e−U/V , i.e. the binomial distribution can be approximated with aPoisson distribution with parameter U/V , giving the following matching function:M = V¡1− e−U/V¢. (2)Denoting the mean duration of unemployment by D we haveD = (M/U)−1 =£(V/U)¡1− e−U/V¢¤−1. (3)We can easily verify the durations mentioned by Petrongolo and Pissarides. Let U = V .Then D = (1− e−1)−1 ≈ 1.58. If U = 3V , we have D = [(1/3) (1− e−3)]−1 ≈ 3.16.3 The Duration of Unemployment when the Numberof Employers ChangesVarying the number of employers, or production possibilities, is a natural way of mod-elling changes in unemployment. Employers enter or exit the economy because of shocksto productivity, demand, discount factor, recruiting cost, bargaining power etc. Analysinghow these “ultimate” factors aﬀect unemployment and its duration requires a job creationcondition (see e.g. Pissarides 2000, ch. 1), but in order to test the matching function wetake the change in the number of employers as given.The labour force consists of W workers, of whom U are unemployed, and the rest areemployed. There are E employers who either have hired one worker or are vacant. Thenumber of vacancies is denoted V . There are equally many employed workers and filledjobs:W − U = E − V. (4)We assume that these numbers are large and that the agents meet according to theurn-ball matching function (2). Jobs terminate at an exogenous separation probabilitys ∈ [0, 1], and the worker and the employer start looking for partners again. The flow intounemployment is equal to s (W − U), and it is equal to the flow s (E − V ) into vacancies.3We consider a steady-state equilibrium where the flows out of and into unemploymentare equal:V¡1− e−U/V¢= s(W − U). (5)Equation (5) is the Beveridge curve. For givenW and s it depicts the steady state valuesfor U and V . It is straightforward to show that it is downward sloping in (U, V ) planeand convex to the origin.The model consists of equations (4) and (5) where E, W , and s are exogenous vari-ables, and U and V are determined in an equilibrium. It is trivial to show that (4) and(5) have a unique solution for U and V for given values of E, W and s even thoughwe cannot solve for U and V explicitly. In the analysis of this section we let E changeand determine the equilibrium responses of U and V . Also a variation in s aﬀects thesteady state, but there is strong empirical evidence that s is practically constant (“Myfindings that the job-finding rate is strongly procyclical and the separation rate is nearlyacyclical oppose the conventional wisdom that recessions are primarily characterized bya high separation rate.” (Shimer 2005. p. 493)).We solve the unemployment duration by tracking the Beveridge curve numericallywhen E changes. To avoid computational problems we, instead of decreasing E directly,let U increase. The algorithm is the following: (i) Give initial values to E, W , and s.The values of E and W must be large in order to justify the use of matching function(2), and they must be close to each other to facilitate computation. (ii) Use equation(4) and the Beveridge curve (5) to calculate the initial values of U and V . (iii) Giveincreasing values to U , and use the Beveridge curve to compute the corresponding V .(iv) Use equation (3) to calculate the duration. Finally, one can use equation (4) to findthe value of E which gives the U, V,D calculated.A Numerical ExampleLet E = W initially, in other words, there is no structural unemployment. ThenU = V in equilibrium. We postulate that W and the initial value of E are one hundredmillion. The shape of the Beveridge curve and thereby unemployment duration dependson the value of s, and therefore we use two alternative period lengths: a week and a4month. We pick the monthly separation probability from U.S. data1: sm = 0.033. Theweekly separation probability sw is then determined by sm = 1 − (1− sw)30/7. Forsm = 0.033 this gives sw = 0.008. Initially, U = V = 1249000 using sw = 0.008, andU = V = 4961500 using sm = 0.033. We compute the unemployment duration whenunemployment changes from the initial value to eightfold.Figure 1: Unemployment duration for multiples of U and multiples of U/V .Figure 1 depicts unemployment duration in periods when U/V changes from one toeight and when U changes from initial value (1249000 or 4961500, depending on thevalue of s) to eightfold. The duration responds more sharply to an increase in absoluteunemployment than to an increase in U/V . If we are interested in how the duration1Total separations, nonfarm, seasonally adjusted, 2001 - 2007. Source: Department of Labor, Bureauof Labor Statistic. sana5changes when something happens to the absolute number of unemployed people, thenthis exercise is quite satisfying in the sense that the too small duration generated by thetoo straightforward use of the urn-ball model, namely 3.16 turns out not to be the trueduration. When the separation rate becomes larger, the duration responds more stronglyto a change in unemployment. Table 1 shows the computed durations and the associatedvalue of U/V when U increases from the initial value to eightfold. In tables 1 and 2 wedenote Un = nU1.Table 1Unemployment duration and unemployment/vacancy ratio when the number of firmschanges.U1 U2 U3 U4 U5 U6 U7 U8D when sw = 0.008 1.58 3.20 4.87 6.58 8.33 10.13 11.98 13.89U/V when sw = 0.008 1 3.05 4.83 6.57 8.33 10.13 11.98 13.89D when sm = 0.033 1.58 3.34 5.30 7.50 10.00 12.85 16.13 19.95U/V when sm = 0.033 1 3.20 5.27 7.50 10.00 12.86 16.13 19.954 The Duration of Unemployment when the LabourForce ChangesAssume next that changes in unemployment are driven by changes in the labour force(or, equivalent in this context, participation) while keeping the number of employersfixed. The labour force changes because of exogenous shocks like a change in the valueof leisure, but for our purposes we take the change in W as given. As before, each firmcan hire one worker, which implies W − U = E − V . In a steady stateV¡1− e−U/V¢= s (E − V ) (6)where s (E − V ) is the flow from the filled jobs to vacancies. Using (6) instead of (5)allows fixing the value of E. The algorithm for computing the equilibrium unemployment6duration is the same as above. For convenient computation we do not increaseW directlybut we let U increase.A Numerical ExampleLet W and E to be one hundred million initially, and we use the same separationprobabilities as above. Table 2 depicts unemployment duration and the correspondingU/v when unemployment changes from the initial value (1249000 or 4961500, dependingon the value of s) to eightfold.Table 2Unemployment duration and unemployment/vacancy ratio when the size of the labourforce changes.U1 U2 U3 U4 U5 U6 U7 U8D when sw = 0.008 1.58 3.15 4.72 6.30 7.87 9.45 11.02 12.60U/V when sw = 0.008 1 3.00 4.68 6.29 7.87 9.45 11.02 12.60D when sm = 0.033 1.58 3.11 4.66 6.21 7.77 9.32 10.87 12.43U/V when sm = 0.033 1 2.95 4.61 6.20 7.77 9.33 10.87 12.43Comparing the durations in tables 1 and 2 we see that the duration is more sensi-tive to a decrease in E than to a an increase in W , given the increase in the absoluteunemployment. This is because Beveridge curve (6) is flatter than Beveridge curve (5)for all values of U (when U is on the horizontal axis and V on the vertical axis): DenoteU/V ≡ x. On (5) we have dV/dU = − (e−x + s) / (1− e−x − xe−x), and on (6) we havedV/dU = − (e−x) / (1− e−x − xe−x + s). The diﬀerence of the absolute slopes is equalto s (1− xe−x + s) / [(1− e−x − xe−x) (1− e−x − xe−x + s)] > 0 ∀x > 0.5 ConclusionWe have shown how to compute the equilibrium duration of unemployment when un-employment increases, in a model where unemployed and vacancies meet according to7a simple urn-ball matching function. The increase in unemployment is assumed to bedriven by a decrease in the number of employers or an increase in the labour force. Thismethod can be used to evaluate the empirical validity of the urn-ball matching functionif one has data on unemployment and job separation probabilities but not on vacancies.We show that the duration is more sensitive to unemployment than one would expect byjust changing the unemployment-vacancy ratio.ReferencesDavidson, C., and Woodbury, S. (1989) “The Displacement Eﬀect of ReemploymentBonus Programs”, Upjohn Institute StaﬀWorking Paper No. 90-02, February 1989.Jackman, R., Layard, R., and Pissarides, C. (1989) “On Vacancies”, Oxford Bulletinof Economics and Statistics 51, 4.Petrongolo, B., and Pissarides, C. (2001) “Looking into the Black Box: A Survey ofthe Matching Function”, Journal of Economic Literature XXXIX, June, 390-431.Pissarides, C. (2000) “Equilibrium Unemployment Theory” The MIT PressShimer (2005) “The Cyclicality of Hires, Separations, and Job-to-Job Transitions”Federal Reserve Bank of St. Louis Review, July/August 2005, 87(4), pp. 493-507.U.S. Department of Labor, Bureau of Labor Statistic8

Equilibrium Unemployment Duration in an Urn-Ball Model of the Labour Market

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