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1 Dopouts in a Nonstationay Seach Model: Unemployment vs. Nonpaticipation a Tade-off? ALISHER ALDASHEV 1 Univesity of Regensbug Institut fü Volksitschaftslehe, Univesitätstaße 31, Regensbug, Gemany alishe.aldashev@ii.uni-egensbug.de JEL code: J64, J22 Keyods: seach model, dopouts, nonstationaity, median-peseving-spead, potential seach time Abstact The pape pesents a seach model ith time-dependent offe aival ate, hich leads to nonstationay esevation ages. The pape shos that thee exists some citical point in time afte hich it is not optimal to seach and okes ithda fom the labou maket. Thus, the potential time an unemployed may spend looking fo a job is the solution to a maximisation poblem. Moeove, it is shon that the policy-make should be aae that measues aimed at educing unemployment ould function at the expense of paticipation. The effect on total employment is ambiguous in the model. 1 Financial suppot of the Deutsche Foschungsgemeinschaft (DFG) though the eseach poject MO523/41 Flexibilität de Lohnstuktu, Ungleichheit und Beschäftigung - Eine vegleichende Mikodatenuntesuchung fü die USA und Deutschland is gatefully acknoledged.

2 I. Intoduction Seach models have become an inceasingly popula fameok in labou maket analysis. Most of seach models addess the poblem of unemployment. Paticipation in the labou maket so fa has eceived unfaily little attention. The pape is aimed at filling this gap by incopoating ithdaals fom the labou maket into a nonstationay job seach model. The model shos that a citical time exists afte hich it is not optimal fo the unemployed to seach anymoe and they ithda fom the labou maket. Results concening compaative dynamics of the esevation age ae then deived. Folloing Aldashev and Mölle (2005) the model also accounts fo asymmety in age offe distibution and diffeentiates beteen dispesion belo and above the median. It is also shon that esevation ages positively affect both potential seach time and unemployment duation. A eade may then deduce policy ecommendations concening the tadeoff beteen unemployment and paticipation. Since the pioneeing ok of Stigle (1962), seach models have been idely used in labou maket theoy. Applications of seach theoy give pedictions about individuals esevation ages, unemployment duations and eemployment oppotunities. Most of the seach model fameoks imply time-invaiant esevation ages. Economic eality suggests, hoeve, that they ae not. As ealy as in 1967, Kaspe povided empiical evidence of declining esevation ages ove the seach span. Attempts have been undetaken to explain this phenomenon theoetically. Gonau (1971) claimed that constant esevation age hypothesis does not hold if the infinite life hoizon assumption is elaxed. Hoeve, as suggested by Motensen (1986) this is athe an aging effect hich cannot explain elatively lage ates of decline in esevation ages fo elatively young

3 okes epoted in seveal studies. Hence, ith the exception of eldely okes close to the etiement age, infinite time hoizon is not a stumbling point. Motensen (1986) povides an elegant explanation of declining esevation ages by imposing a cedit maket constaint. The geneal nonstationay job seach model can be found in van den Beg (1990) hee nonstationaity of esevation ages may aise due to time-dependence of any exogenous vaiable. Modelling ithdaals fom the labou maket eceived vey little attention in the seach liteatue. The dopouts ee aleady mentioned in the seach model of McCall (1970). Hoeve, the McCall s model is static and the dopout condition in his model is the same as the paticipation constaint as defined in Motensen (1986). This can ell suit the decision making pocess hethe to ente the labou maket o not but fails to explain the exits fom the labou maket of the paticipating okes. In this pape I m concened ith dynamic aspects of seach; and as I ill sho, ithdaals fom labou foce ae a logical outcome of the nonstationay job seach. In the model pesented hee esevation ages decline due to discimination of long-tem unemployed by fims. Fims could use the length of the unemployment spell as an indicato fo a possible depeciation of skills. As a consequence, they ould favou shot-tem unemployed if they had the choice. This discimination mechanism oks though the job-offe aival ate only (eveything else being equal). As a esult, the model pedicts a decline in esevation ages ove the seach span. The pape is oganised as follos: section II povides the theoetical explanation of declining esevation ages and dopouts fom the labou maket, section III discusses policy implications, and section IV concludes.

4 II. Theoetical Model Conside a standad seach model in the tadition of McCall (1970) and Motensen (1986). Unemployed okes ae identical and live foeve. They possess the knoledge about the paametes of the age offe distibution F, hich is assumed to be timeinvaiant, but they have no infomation hen job offes aive and hat ages ae associated ith them. Once accepted by a fim, okes must immediately eply (accept the job o decline), so no aiting is alloed. Once the job is ejected it cannot be ecalled. Seaching involves a diect cost c pe peiod. Hence, by not paticipating agents can alays enjoy the pue leisue b (ithout incuing the cost c). In a standad seach model jobs aive to the unemployed ith a time-invaiant Poisson aival ate λ. In this model I intoduce a time dependent aival ate λ () t, ith d λ( t) < 0 dt and lim λ( t) = 0. This specification means that the longe the seach, the less t likely is an offe to aive. The highest possible aival ate happens at t = 0. The Bellman equation fo the optimal value of seach can be given as Ω () t = ( b c) τ + β() τ q( m, τ,) t max[ Ω () t, W]d G( m) + q(0, τ,) t Ω ( t+ τ) = m= 1 0 = ( b c) τ + βτ q( m, τ) max[0, W Ω ( t)]d G( m) + q( m, τ) Ω ( t) + q(0, τ, t) Ω ( t+ τ) m= 1 0 m= 1 (0.1) In continuous time afte some manipulations 2 the equation simplifies to: λ t d t () t = b c+ ( () t ) d F + dt () t, (0.2) 2 Deivations ae given in the Appendix 1.1

5 hich gives the esevations age as a function of time. The aival ate is declining ove time and appoaches zeo in the limit. Folloing the assumption that at t = d λ( t) = 0, one obtains: dt lim t = b c. (0.3) t t * b-c b The time dependent aival ate specification yields a esevation age as a deceasing function ove time, appoaching in the limit its loest value given in (0.3). One must be aae, hoeve, that the esevation age cannot fall belo b because a oke has alays an option to dop out of the labou foce and ean pue leisue hich is oth b pe peiod. The implication of that is that thee is a citical time, denote it as t *, such that * t b = and * * * t t t t t > < < <. * Since t > t < b it is not optimal Figue 1: The potential seach time anymoe fo a oke to seach afte * t and t * is the time at hich the unemployed oke dops out of the labou maket and joins the pool of discouaged okes. The potential seach time is then the maximum amount of time an agent is illing to allocate to his/he seach and is equal to t *.

6 Figue 1 shos the esevation age as a function declining ith time (ed cuve) ith an asymptote b c. The geen line indicates the value of leisue b, hich intesects the esevation age function at t *. At t * the oke dops out and eceives b. The esevation age equation can be eitten as a diffeential equation: () λ t d t () t = max b, b c+ ( () t ) d F + d t () t λ Conside a special case: λ () t = 2 t 1 +, (0) max [ ( t )] Solving (0.4) fo (0) fo this special case yields 3 : (). (0.4) λ = λ = λ ith d λ ( t ) = 0 at t = 0. dt λ ( 0) = b c+ ( () t ) d F. (0.5) () t With this specification, the stating value of the esevation age is identical to the esevation age in a stationay job seach model. An impotant implication of the nonstationay aival ate is that it intoduces dopouts into the seach model fameok. Moeove, it gives altenative explanation to the declining esevation ages. Potential seach time The model shos that at time t *, hen an agent dops out of the labou foce, his/he es- * evation age equals b. Solving the diffeential equation in (0.4) fo t and substi- * tuting b fo t gives the solution fo t *. The potential seach time implies the maximum time an unemployed oke is illing to allocate to his/he seach. This means that afte t * the oke ithdas fom the labou maket unde condition that 3 See Appendix 1.1

7 he/she has not accepted a job ithin this peiod. An impotant esult is that the dopout time is not a andom vaiable, it is the choice vaiable fo seaches, and is independent of the pobability of tansition fom unemployment into employment. Looking at the figue 1 one can see that potential seach time is inceasing ith esevation ages. Hence, exogenous factos hich push esevation age up incease the potential seach time. The compaative dynamics fo the esevation age ae time-dependent in tems of magnitude. Hoeve, since the aival ate is monotonically declining ove time and the age distibution is time-invaiant, the compaative dynamics fo the esevation age in tems of the sign of the effect ae the same acoss all time peiods. The standad esults of the seach models ae that the esevation ages go up ith the mean of the age offe distibution and ith the mean-peseving spead of the age distibution. This is tue fo the symmetic distibution. But hat happens if the spead paametes fo the left tail of the age distibution and fo the ight tail may vay sepaately? The complication aising hee is that changing the speads in the left and ight tail unpopotionately ill affect the mean. The poblem can be handled by intoducing the median-peseving spead. This implies that: F, σ, σ = 1/2, (0.6) L hee is the median age and σ L and σ R ae the median-peseving speads in the left and ight tail espectively hich can vay feely and independently of one anothe. Suppose σl < σ L and σr < σ R, then fo any abitay constants a< and b>, the folloing inequalities hold: R

8 a b F (, σ )d < F (, σ )d L F (, σ )d > F (, σ )d. R a b L R (0.7) To calculate the compaative dynamics fo median-peseving speads let fo simplicity b c= 0 and λ / = 1. This ill not change the qualitative esults as these vaiables ae exogenous and ae independent of the paametes of the age distibution. Reite the equation fo the esevation age:. (0.8) 4 () t = () t d F = () t d F + () t d F () t () t Integating (0.8) by pats yields: 1 1 () t = ( () t ) Fd+ d F () t 2 2 () t. (0.9) Poposition 1. Resevation age inceases ith the median age and the spead above the median and deceases ith the spead belo the median. Poof. See Appendix 1.2. These esults sho that highe inequality in the left tail of the distibution leads to eduction in esevation ages, hich ould make the agents leave the labou maket ealie. III. Policy Issues The policy-make s pimay concen is loeing the duation of unemployment. The instantaneous pobability o hazad of tansition fom unemployment into employment can be itten as: ( ) φ() t = λ ()1 t F () t. (1.1) 4 If the esevation age is belo the median. Fo esevation ages above the median see Appendix 1.2.

9 Measues hich educe esevation ages of the unemployed okes incease the hazad of exit fom unemployment into employment. Hoeve, as it as aleady shon in the pevious section loe esevation age imply shote potential seach. Conside S identical okes having identical esevation ages, () 1 t. The length of potential seach is t * 1. The numbe of exits fom unemployment into employment is t 1 * S 1 exp ( t) dt φ. Suppose no that due to a change in some facto esevation age dops to () 2 t and the length of potential seach becomes t * 2, so that t * * 2 < t1 and <. The hazad has inceased as 2 < 1. Hoeve, this does not necessaily mean 2 1 that the numbe of exits fom unemployment into employment has isen. Wokes ith esevation ages () 2 t and ho did not accept any job until t * 2 ithda fom the la- bou maket. Wokes ith esevation ages () 1 t do not dop out at t * 2 but continue thei seach until * * * t 1. Whethe the numbe of tansitions into employment duing t1 t2 may compensate the diffeence in the numbe of exits fom unemployment into employment duing t * 2 is ambiguous. Figues 2 and 3 sho possible scenaios of policies hich change esevation ages.

10 Figue 2. Changes in esevation ages, incease in the oveall employment Figue 2 shos the fist possible scenaio. When the esevation age as () 1 t and the potential seach time t *, the shae of exits into employment as 1 P 1. When esevation ages deceased to () 2 t, the potential seach time dopped to t * 2 but the shae of exits into employment has inceased to P 2.

11 Figue 3. Changes in esevation ages, decease in the oveall employment Figue 3 shos the second possible scenaio. Hee esevation ages deceased to () 2 t, the potential seach time dopped to t * 2 such that the shae of exits into employment has deceased to P 2. The tade-off beteen loeing unemployment and loeing paticipation poses a cetain dilemma fo a policy-make. If the ultimate goal is to incease oveall employment then the pictue is mixed as the theoy alone does not say hich scenaio ould

12 pevail (this depends on the magnitudes of exogenous vaiables in equation (0.4) and emains in geneal ambiguous). If the pimay concen is the numbe of egisteed unemployed then the measues aimed at loeing esevation ages ae adequate. IV. Conclusions The seach theoetical model ith declining aival ates ove time povides an explanation fo exits of the unemployed okes out of the labou foce. It is also shon that the implications of the standad seach theoy need to be econsideed once the age distibution is non-symmetically and unpopotionately dispesed in tails. Namely, a highe median-peseving spead in the left tail leads to loe esevation ages and, hence, loes unemployment duation. Hoeve, as the pape establishes, one should keep in mind that this involves a tade-off. Loe esevation ages also mean highe exit ate fom unemployment into nonpaticipation. Anseing the question hethe loe unemployment duation offsets highe dopout ate equies empiical testing.

13 Appendix 1.1 Collecting tems in equation (0.1) yields: () Ω t Ω t+ τ = ( b c) τ + β( τ) q( m, τ) max[0, W Ω ( t)] g( ) d+ m= 1 m= βτ qm (, τ) Ω ( t) + βτ q(0, τ, t) Ω t+ τ Ω t+ τ m (2.1) Note that βτ = e τ and ( 0, τλ, ) ( t) q = e λ τ, hence one could simplify equation (2.1): () Ω t Ω t+ τ = ( b c) τ + β( τ) q( m, τ) max[0, W Ω ( t)] g( ) d+ m= 1 m= 1 0 () t τ + βτ qm (, τ) Ω( t) Ω t+ τ 1 e e λ τ m (2.2) Moeove, in continuous time: τ 0 ( t τ ) ( t) d ( t) Ω + Ω Ω lim = ; τ d t q(1, τλ, ) q( m, τλ, ) lim = λ () t ; lim = 0, fo m>1 τ τ τ 0 τ 0 τ 0 τ λ() t τ ( 1 e e ) () t lim = + λ ; τ (2.3) Hence, dividing (2.2) by τ and collecting tems yields: () t d Ω = b c+ λ () t max[0, W Ω ( t)]d F + d t () t () t () t ( λ() t ) + λ Ω Ω + = 0 () max[0, ]d () = b c+ λ t W Ω t F Ω t 0 (2.4) Remembeing that () t () t Ω = e can eite (2.4) as: λ t d t () t = b c+ ( () t ) d F + (2.5) d t

14 o: d d t () t λ d () = b c t t F + t (2.6) At t = 0 λ () t assumes its maximum value and at 0 t = and ( 0) simplifies to: d λ ( t) dt = 0 at t = 0. Hence, d dt () t = 0 λ = + 0 b c 0 d F (2.7) Appendix 1.2 Define an implicit function ( (), t L, R) ( L R) Ψ σ σ, so that: Ψ (), t σ, σ = 2 () t + F d d F = 0 (3.1) 2 () t By the means of the implicit function theoem Ψ( t σl σr) ( L R) d ( t) (),, / x =. d x Ψ (), t σ, σ / () t ( (), t, ) Ψ σ σ () t L R = 2 F ( t) > 0 (3.2) Hence, ( (), t L, R) d ( t) Ψ σ σ sign = sign d x x The folloing esult ould necessay: d F > 0 and F < 0 (3.3) This can be explained by the fact that shifting the median does not change the pobability mass above the median (by the definition of the median) but this eassigns moe pobability mass to highe ages. In some ay this is analogous to the tuncated mean.

15 Diffeentiating the implicit function ith espect to the median yields: ( (), t L, R) Ψ σ σ = 1/2+ 1/2 + F d d F = () t () t = F d d F < 0 (3.4) By the means of (3.3). And diffeentiating the implicit function ith espect to the spead in the left tail yields, assuming that the esevation lies belo the median: By the means of (0.7). ( (), t L, R) Ψ σ σ = F d> 0 σ σ, (3.5) L L () t When the esevation age is above the median then the effect is zeo. One must note hoeve, even if () t >, at some point in time (since esevation ages decline ove the seach span), call it t, ( t ) =. Hence, at t > t the spead in the left tail ould still play a ole o fomally: ( ( t t ), L, R) Ψ σ σ = 0 σ L Ψ( ( t > t ), σl, σr) > 0 σ L (3.6) Diffeentiating the implicit function ith espect to the spead in the ight tail yields: ( (), t L, R) Ψ σ σ = d F > 0 σ σ, (3.7) R R

16 The esult in (3.7) is staightfoad as the highe spead in the ight tail ould mean taking aay pobability mass fom loe ages and edistibuting this pobability mass to the highe ages. 5 As a esult d ( t ) < 0, d ( t ) > 0, and d ( t ) 0 d σ d σ d >. L R 5 Think of the tuncated mean

17 Refeences ALBRECHT, J. W. and AXELL B. (1984), An equilibium model of seach unemployment, Jounal of Political Economy 92(51), ALDASHEV, A. and MÖLLER, J. (2005), Wage inequality, esevation ages, and labou maket paticipation. Testing the implications of a seach theoetical model ith egional data (Woking Pape). BERG, G. J. van den (1990), Nonstationaity in job seach theoy, Revie of Economic Studies 57, BLANCHARD, O. (1996), The plight of the long-tem unemployed (Woking Pape). GRONAU, R. (1971), Infomation and fictional unemployment, Ameican Economic Revie, 61, KIEFER, N. M. (1988), Economic duation data and hazad functions, Jounal of Economic Liteatue 26, KIEFER, N. M. and NEUMANN, G. R. (1979), An empiical job-seach model, ith a test of the constant esevation-age hypothesis, Jounal of Political Economy 87(1), LANCASTER, T. (1979), Econometic methods fo the duation of unemployment, Econometica, 47, MCCALL, J. J. (1970), Economics of infomation and job seach, Quately Jounal of Economics 84, MORTENSEN, D. T. (1986), Job seach and labou maket analysis, in Ashenfelte, O. and Layad, R. (eds), Handbook of Labou Economics (Amstedam: Noth-Holland). MORTENSEN, D. T. (2003), Wage Dispesion: Why Ae Simila Wokes Paid Diffeently? (Cambidge: Massachusetts). MORTENSEN, D. T. and PISSARIDES, C. A. (1994), Job ceation and job destuction in the theoy of unemployment, Revie of Economic Studies 61(3), PISSARIDES, C. A. (1979), Job matchings ith state employment agencies and andom seach, Economic Jounal 89, PISSARIDES, C. (1990), Equilibium Unemployment Theoy (Basil Blackell: Oxfod). STIGLER, G. J. (1962), Infomation in the labo maket, Jounal of Political Economy 70,

Notes on McCall s Model of Job Search. Timothy J. Kehoe March if job offer has been accepted. b if searching

Notes on McCall s Model of Job Seach Timothy J Kehoe Mach Fv ( ) pob( v), [, ] Choice: accept age offe o eceive b and seach again next peiod An unemployed oke solves hee max E t t y t y t if job offe has