Job search, occupational choice and learning

Büyükbaşaran, Tayyar Article Job search, occupational choice and learning Central Bank Review (CBR) Provided in Cooperation with: Central Bank of The Republic of Turkey, Ankara Suggested Citation: Büyükbaşaran, Tayyar (2020) : Job search, occupational choice and learning, Central Bank Review (CBR), ISSN 1303-0701, Elsevier, Amsterdam, Vol. 20, Iss. 3, pp. 85-97, https://doi.org/10.1016/j.cbrev.2020.03.004 This Version is available at: https://hdl.handle.net/10419/297920 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/4.0/ Job search, occupational choice and learning Tayyar Büyükbas¸aran 1 Central Bank of the Republic of Turkey, Turkey article info Article history: Received 27 February 2020 Received in revised form 8 March 2020 Accepted 30 March 2020 Available online 29 June 2020 Keywords: Search Matching Occupational choice Learning Unemployment Discouraged workers Beveridge curve abstract This paper investigates the labor market consequences of incomplete information about workers’own job searching process and best occupations fitting to them. A search and learning model is provided in order to analyze these effects. In the model, search outcomes relay information about workers’job finding abilities and appropriate occupations suited to them, and workers use this information to infer their types. Our theory explains how search outcomes during unemployment can change the beliefs of workers about their job finding ability and consequently affect their decisions including the occupational choices. Characterization of the model results in a simple value function with reservation level of prior belief property that is similar to reservation wage property. Some interesting facts about both micro and macro data are identified and our model’s explanation of these facts is discussed. Particularly, our characterization gives rational for why workers with less experience in searching have (1) longer unemployment duration and (2) higher probability of changing occupation by reemployment, and (3) why shifts in Beveridge curve may be observed. Theory can also be used to (4) explain the discouraged worker phenomenon. ©2020 Central Bank of The Republic of Turkey. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction If workers do not have complete information about their job search and matching process such as their ability to find a job and the best occupation suitable for them, search outcomes provides important guidance. Even the same ability workers may have differences in search outcomes initially caused by chance, which bring differences in their beliefs about their abilities and their best-suited occupation. Differences in beliefs further affect their future search decisions, which eventually may have a substantial effect on occupational choice and unemployment duration. Consider two workers, who are the same age and have the same skill set, same education etc. One worker has been working at the same firm and occupation for ten years and never has been separated, the other worker had separated and had to search for a job every year for ten years, possibly trying different occupations. The second worker obviously is more experienced in searching for a job, and therefore has a better understanding of her job finding ability and about which occupation she can find a job more easily. She may be called as “a bird in the air”. On the other hand, first worker is less experienced in searching for a job. She may not have a good understanding of which occupation she can find a job more easily. She may be called as “afish out of water”. Unfortunately, to the best of our knowledge there is not a data source which provides either a full search history or a belief history of workers for US economy. Nevertheless, the Current Population Survey’s (CPS) ‘Displaced Worker, Employee Tenure, and Occupational Mobility Supplement’ (to be called as Displaced worker supplement from now on) includes data of displaced workers about their previous and current occupations, job tenures and wage rates. In this paper, previous job tenure of displaced workers will be used as a proxy in order to identify these two different group of workers. Displaced workers with longer previous job tenures is assumed to be more likely in the first group, displaced workers with short previous job tenure is assumed to be more likely in the second group. Our model in this paper will be a base to state actually that displaced workers with longer previous job tenure are more likely to be inexperienced workers in searching compared to same age displaced workers with shorter previous job tenure. Displaced worker supplement have the following stark features: First, compared to displaced workers with shorter previous job tenures, the displaced workers with longer tenures in previous job are more likely to have (1) longer unemployment durations, and (2) different reemployment occupations than previous occupations. Second, during the current recession, (3) the ratio of displaced E-mail address: tayyar[email protected].tr. Peer review under responsibility of the Central Bank of the Republic of Turkey. 1 The author is thankful to VV Chari, Larry Jones and Chris Phelan and attendants of growth workshop in University of Minnesota for their comments. Contents lists available at ScienceDirect Central Bank Review journal homepage: http://www.journals.elsevier.com/central-bank-review/ https://doi.org/10.1016/j.cbrev.2020.03.004 1303-0701/©2020 Central Bank of The Republic of Turkey. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/). Central Bank Review 20 (2020) 85e97 workers with longer past tenure to displaced worker with shorter past tenure has been increased, which results in longer average unemployment duration. Aggregate data on the labor market suggests that the current recession is different from the previous periods. Firstly, there is a (4) higher number of discouraged workers during the current recession. The discrepancy between the U4 unemployment rate and the U3 unemployment rate 2 has been increased from 0.2 (historical average) to 0.6 percent (average during current recession), which amounts to around 1.2 million discouraged worker throughout US. Secondly, during the current recession (5) average unemployment duration increases significantly. Finally, the (6) Beveridge curve, an empirical relation between vacancy and unemployment rates, does not appear to hold during thecurrentrecessionwhereas it was morepronounced inpreviousperiods.After2008,unemploymentrate havebeenhigher thanthe Beveridge curve suggests,which can be consideredas a shift. This paper will attempt to give a possible explanation for these facts. In our model, workers are heterogeneous in their job finding ability and their suitability for certain occupations. Because workers have incomplete information about their types, they do not precisely know their job finding abilities and their best-suited occupations. They learn about their types from observing their search outcomes. Firms, which are subject to free entry condition, will post vacancies at a cost and commit to pay a wage rate if a successful match occurs. Characterization of an equilibrium with these features gives some qualitative explanation in the line with the facts presented beforehand. Firstly, people with worse priors about their types (in the sense that their belief is far away from their actual type) direct their search for a job in a less suitable occupation. Because job finding probabilities depend on actual type of a worker and on suitability of occupation for the worker, search outcomes will give valuable information to the workers to direct themselves to more suitable occupations. Since this learning process takes time, people with less experience in searching stay unemployed longer and are more likely to change occupation on average. This learning process through search outcomes gives a rationale for the facts (1) and (2) from the Displaced worker supplement. Secondly, if the amount of workers with worse priors in the unemployment pool increases (fact 3), the average unemployment duration also increases and the observed unemploymentvacancy relation shifts as seen in the data. This gives a rationale for some aggregate data facts (3), (5) and (6). Finally, with costly search extension to the main model, workers whose prior get worse and worse because of negative search outcomes might decide not to search any longer but stay out of the labor force becoming discouraged workers, a rationale for fact (4). Our model is a directed search model in the spirit of Acemoglu and Shimer (1999).InBurdett and Vishwanath (1988) and Bikhchandani and Sharma (1996), workers also learn during the search process, but workers learn about unknown common distribution whereas in our paper workers learn about their own types. In Jovanovic (1979), a matched worker-firm pair draws an unobservable match quality from a known distribution. Through noisy signals which is correlated by match quality, both workers and firms learn about the actual match quality and decide whether to continue the match. In all these papers, workers are homogeneous and learning is about an unobservable draw from a known distribution. In our paper, workers are heterogeneous in terms of their ability to find a job and their best suited occupation. Moreover, unlike those papers, workers learn about their own type by the search outcomes. Model in this paper is closely related to Gonzalez and Shi (2010). Our model also is a competitive search model with different types of workers with different job finding probabilities. The productive technology of search effort are also very similar. However, in Gonzalez and Shi (2010) the workers do not have any occupational choice; all the jobs are homogeneous in terms of productivity and matching probability of same type of workers. Our model will assume that the jobs are differentiated, therefore they can be interpreted as different occupations. Hence, different than Gonzalez and Shi (2010) our value function will have an endogenous reservation prior belief property. Our model is also related with Falk et al. (2006), which also uses different types of workers with different job finding probabilities and giving a motivation for discouraged worker phenomenon. However, they have homogeneous jobs also; therefore, they cannot address the occupational choices addressed in our paper. Moreover, they used a continuous time approach and random search with Nash bargaining whereas our model is a discrete time competitive search model. InPapageorgiou (2014), both workersand jobs are heterogeneous. Jobs are divided among different occupations and workers divided into different unobservable types as in our paper. Workers also learn about their types through observing outcomes as in our paper. However, in Papageorgiou (2014) workers infer about their types observing noisy signal of their productivity during employment and decide on attempting to change occupation. Our model focus on information content of search outcomes rather than information content of productivity of a worker in certain occupation. Therefore the learning processis different. Moreover, in our model search outcomes rather than the match quality convey information to workers. Model in this paper uses the block recursive structure as in Gonzalez and Shi (2010) and Menzio and Shi (2010). Our model lacks aggregate productivity shocks, whose effects are analyzed by Moscarini and Postel-Vinay (2013). Aggregate productivity or sector specific productivity shocks are other reasons for unemployment as well as occupational choice. However, this paper focus on information and learning content of search outcomes, therefore it deliberately lacks aggregate and sector specific productivity shocks. Model in this paper also lacks negative duration dependence, i.e. adverse effect of longer unemployment duration, as in Kroft et al. (2013). Negative duration dependence is not crucial for the results of this paper; nevertheless, addition of it would put a further channel such that the workers who are inexperienced in job search stay unemployed even longer. The paper is organized as follows. After this introduction, Section 2discusses data motivation. Section 3presents the model. Section 4defines the equilibrium, characterizes it and gives an extension to model in order to explain discouraged worker notion. Section 5concludes. 2. Data motivation We have used two different data sources. First is Current Population Survey’s (CPS) Displaced worker supplement. Second is aggregate data obtained from the BLS by CPS and Job Openings and Labor Turnover Survey (JOLTS). Displaced worker supplement is conducted every two years in Januaries, which asks additional questions to the displaced workers. Displaced workers in this supplement are those who involuntarily separated from their jobs during the past three years before survey date by (i) mass layoff, (ii) plant closure or (iii) 2 U3 is the convention unemployment rate whereas the U4 includes the discouraged workers to calculation. U5 unemployment rate includes marginally attached workers and U6 unemployment rate includes the workers who work parttime for economic reasons (NPTFER). Formal definitions stated by Bureau of Labor Statistics are in Appendices - Definition of Different Unemployment Conventions. T. Büyükbas¸aran / Central Bank Review 20 (2020) 85e9786 abolishment of their position 3 rather than because of individual job performance. Therefore supplement data has its own limitations: First, workers are surveyed just once, providing information on one post-displacement data, rather than about their full history of experiences over time. So it is not possible to obtain panel data from this survey. Second, it does not include all unemployment pool but only displaced workers; hence, voluntary quits and fires by case are not sampled. However, it also has one main advantage; it is a huge survey of around 150,000 individuals who are weighted to represent US workforce. We have taken eight supplements 4 to obtain data on displaced workers from 1999 to 2016, including their demographic information, total unemployment duration, previous and current occupations, previous and current job tenures, and previous and current wage rates etc. As stated beforehand, we are trying to identify the workers who are more experienced in search. If search provides some valuable information about job finding ability and suitability of their skills to find a job in certain occupations, then people with more experience in job search are more likely to realize their abilities and more likely to pick suitable occupation to search for. Because of that, experienced workers in searching would find a job more easily and their total unemployment duration would be lower compared to inexperienced ones. Unfortunately, Displaced worker supplement does not include full search experience of the workers. Nonetheless, it includes previous job tenure of displaced workers, and this information will be used as a proxy for total search experience. It will be assumed that a worker who had to search for a job e.g. one years ago and now have to search again will be regarded as more experienced in searching than the same age worker who had to search for a job ten years ago and now have to search again on average. Hence workers with shorter previous tenure is more experienced in searching than same age workers with longer previous job tenure. Validity of the data motivation depends on whether this is a good proxy for overall search experience. Our model will not include imperfectness in recalling learning experience or unobservable aggregate/idiosyncratic fluctuations or trends in labor market. However, if people cannot recall their experience from far past perfectly or if the conditions of the current labor market has changed, farther the previous job search experience lesser the information content of it. Hence a worker who had search experience one year ago will remember or will be able to ‘use’his/her experience from that search much better than a worker who had the same search experience ten years ago and never searched again. If this is actually the case, the most current previous job search process will be more informative and therefore the proxy mentioned here is actually a good one. Moreover, if anything else is constant, among the same age displaced workers, workers who have longer tenures on his previous job have a shorter time span for experimenting (i.e. off-the-job-searching) compared to displaced workers who have shorter previous job tenure on average. Table 1 5 summarizes the main findings in this analysis. We will present only relevant statistics in this section; detailed data work can be seen in Appendix-Data section. All figures inTable 1 are in terms of row percentage, e.g. in first mini-table, 42% implies that forty two percent of all workers with short previous tenure duration have changed their occupation upon reemployment and 58% of workers with short previous tenure duration have not changed occupation upon reemployment. Following observations from Table 1 worth noting. First, from first mini-table of Table 1,uponre-employment new occupation of displaced workers with longer previous job tenure is more likely to be different than their previous occupation compared to displaced worker with shorter previous job tenure. 47% of displaced workers with long previous job tenure change occupation after re-employment whereas only 42% displaced workers with short previous job tenure change occupation after re-employment. Second, from second mini-table of Table 1,displacedworkerswith longer tenure in his previous job before displacement have longer unemployment durations. 47% of displaced workers with long previous job tenure have long unemployment durations while only 38% displaced workers with short previous job tenure have long unemployment durations. These two findings appear to be a puzzle if one considers standard occupation specific human capital approach. Under that approach,it isassumed that if a person stayed longer in his previous occupation, he would obtain more occupation specifichuman capital. Therefore, one would expect a displaced worker with longer previous job tenure to find a job in his previous occupation once re-employed. Moreover, if everythingelse is the same (including reservation wage, wealth and other types of human capital), the person with higher occupation specific human capital is expected to find ajob in shorter duration. Our hypothesis gives a rationale to solve thispuzzlebetween data facts and standard human capital approach. Current aggregate data on labor market suggests that the current recession is different than the previous time periods. Firstly, there is a higher number of discouraged workers during current recession. Fig. 1 shows U4 and U3 unemployment rates (see footnote 2). The discrepancy between U4 unemployment rate and U3 unemployment rate has been increased from 0.2 percent (historical average) to 0.6 percent (average during current recession), which amounts to around 1.2 million discouraged worker throughout US. Secondly, during the current recession average unemployment duration increases significantly. Fig. 2 shows that average unemployment duration has increased from around 17 weeks of historical average to around 28 weeks during recession. Finally, the empirical relation between vacancy rate and unemployment rate, namely Beveridge curve, does not appear to hold during current recession whereas it was more pronounced in previous periods. Fig. 3 shows that there is a shift in this empirical relation after recession such that corresponding unemployment rate is higher than the Beveridge curve suggests for each vacancy rate. In the next chapter we will provide a model which will be used to account for these facts. This model also will be a candidate to explain the puzzle between these data facts and standard human capital approach. 3. Model 3.1. Environment Model environment is similar to Gonzalez and Shi (2010), but 3 It means position is abolished and no new employee will take place of him after his/her separation. 4 Displaced worker supplements 2002, 2004,2006,2008, 2010, 2012, 2014 and 2016. 5 For occupations two digit occupation codes of CPS classification has been chosen. Change in occupation means that if displaced worker is reemployed, his/ her new occupation is different than his previous occupation in two digit codes. Long (short) tenure means that the displaced worker has worked for 9 years or more (8 years or less) in his/her previous job before displacement. Long (short) unemployment duration means that it takes more (strictly less) than 18 weeks to find a new job. Arranged displacements are disregarded, i.e. workers who did not stayed unemployed after displacement but immediately found job are taken as arranged displacement and were not included in the dataset. These summary statistics are for displaced workers who are male, white, aged between 35 and 45, having educational attainment of at least high school, and having same eligibility for unemployment insurance. There are around 3,100 observations in this group. Not all the workers have all relevant data like previous and/or current tenure duration. The table includes observations whenever relevant data exists. We have also looked at the other group of employees with respect to other demographic and educational characteristics. The statistics are not changing qualitatively. We have also done robustness check on the specification of long and short durations and its cut-off points. Again, the results does not change qualitatively. We tried to control for industries and occupations but small sample sizes directs us away from that approach. T. Büyükbas¸ aran / Central Bank Review 20 (2020) 85e97 87 unlike their setup there are different occupations, and unemployed workers direct their search to different occupations according to their beliefs. There is a unit measure of infinitely-living workers which is divided to employed and unemployed (and out-of-labor force with the model extension discussed in Section 4.6). Measure of firms in each occupation will be determined endogenously by free entry. All the agents are risk neutral and discount the future at a rate r>0. Employed workers produces an amount of homogeneous goods according to their occupation until a separation or exit shock hits the worker. Unemployed worker searches for a job and receives a utility of b>0 in each period which constitutes leisure benefit of being unemployed and/or unemployment benefit. As in Gonzalez and Shi (2010), each worker has an unknown permanent ability of i, which is either high hor low l;and has an associated productivity parameter s h and s l respectively, where s h , s l 2ð0;1Þand s h  s l . Each new worker in the market has ability i with probability p i 2ð0;1Þ,wherep h ¼1p l . There are two different occupations j, either good gor bad b, and associated productivity parameters m g and m b respectively, where m g ; m b 2ð0;1Þ, m b ¼1 m g and m g > m b . We will see that productivity of employed workers will not differ according to type of workers. On the other hand occupation ghas better prospects for type hunemployedworkers and occupation bhas better prospects for type lunemployedworkers in terms of finding a job. Ability and occupation determines a worker’ssearch productivity as follows. First, a worker picks either occupation gor b and firms decide to open a vacancy position. Second, standard randomized matching occurs between unemployed workers and vacancy positions. So far, everything is very conventional as in directed search and matching model. Then following unconventional productivity assignment makes finding a job in distinct occupations Table 1 Basic facts from micro data: the displaced workers with longer tenures in previous job are more likely to have (1) longer unemployment durations, and (2) different reemployment occupations. 1 Change of Occupation 2 Unemployment Duration Prev. tenure Change No Change Prev. tenure Short ( <18 w) Long (18 w) Short (8 y) 42% 58% Short (8 y) 62% 38% Long ( >8y) 47% 53% Long ( >8y) 53% 47% Fig. 1. Different definitions of UE and Role of Discouraged Workers. T. Büyükbas¸aran / Central Bank Review 20 (2020) 85e9788 differentfordistincttypesofunemployedworkers. Nature(whichcan see the type of the worker and assigns productivities accordingly) moves and assign a productivity to a matched worker according to his type and the occupation he chooses. If a worker is applying for occupation g;productivity of worker of type hwill be y g >0with probability s h m g and y’<0withprobability1 s h m g (otherwise); productivity of worker of type lwill be y g >0withprobability s l ð1 m g Þand y’<0 otherwise. In market for occupation b, a worker with type hwill have a productivity of y b >0withprobability s h m b and otherwise; a worker of type lwill have a productivity of y b >0with probability s l ð1 m b Þand y’<0 otherwise. That is productivity of an employed worker is occupation specific and it does not depend on the type of the worker, whereas probability of matching with positive productivity depend on type of the worker as well as the occupation. Worker meets randomly drawn firm which offers a job in that occupation.Bothfirmand worker can see the productivitybut not the type of the worker. Similar to Gonzalez and Shi (2010) we will call sm components as productive units. High ability workers are more likely to be productive than a low-ability worker in occupationh.Obviously, afirm will hire the worker only if worker has positive productivity. Note that every employed worker in occupation ghas a productivity y g , and every employed worker in occupation lhas a productivity y l . We also assume different separation shocks for different occupations: an employed worker in occupation ghas a probability d g >0to separate and join unemployed pool and an employed worker in occupation bhas a probability d b >0 to separate. Hence occupations g and bare differentiated by their prospects of job finding probability fordifferent types of unemployed workers, by total product produced by each worker and by their separation probability. Note that by construction, in this setup, every employed worker, regardless of her type, produces same amount of product in a certain occupation. In this way, we focus on learning aspects of search outcomes of unemployed workers rather than the job productivity signal of employed workers. Hence, we have clearly separated informativeness of search outcomes from informativeness of on-the job productivity. This study will only focus on informativeness of search outcomes and should be seen as a complementary research rather than a substitute to the studies (e.g. Jovanovic, 1979) about informativeness of on-the-job productivity. Nevertheless, please note that we have used similar concepts in different context, therefore there remains a possibility of confusing these concepts if one is more familiar with studies about informativeness of on-the-job productivity. As mentioned in Gonzalez and Shi (2010), this formulation of worker’s ability to find a job can be interpreted in the context of worker and firm specific skill bundle of Lazear (2009). In that context, different firms or different occupations require different skill bundles and workers are heterogeneous in terms of their skill bundles. A firm reviews the worker in order to understand whether his/her skill bundle would fit the firm. In our study, we assume that high type workers have a higher probability to fit the firm which offer a job with occupation gcompared to low type workers. Likely, low types have a higher probability to fitafirm offering occupation bthan to fitafirm offering an occupation g. Learning will take place once an unemployed worker searches for a job. Since after a long history the worker would be able to learn his/her actual ability, we will assume an exit shock hits him. Therefore, in each period, there is a probability of j 2ð0;1Þthat a person dies, regardless of her employment status. We will use the search and matching approach as follows: The number of matches is given by a matching function F:R 2 /R.We will use the index 6 x, rate of a match occurs, as the argument of all the following functions. vðxÞdenotes total measure of vacancies created in the economy, whereas uðxÞdenotes total measure of unemployed workers, just the sum of the measure of unemployment workers of each type. Domain of xis. X¼½0;1=ð s h m g Þ A function FðuðxÞ;vðxÞÞ gives the number of matches in the economy. Therefore matching rate index xis x¼FðuðxÞ;vðxÞÞ uðxÞ Using matching function, ordinary definitions in competitive search model follows: the matching probability of a vacancy in economy is F=v¼x= l ðxÞ, where l ðxÞ≡vðxÞ=uðxÞis the tightness in the labor market. As in Gonzalez and Shi (2010), we will assume the following standard assumptions for the matching function: Assumption I. (Regularity conditions of matching function) Function Fis such that (i) strictly increasing, strictly concave, and twice differentiable in each argument; (ii) Fis linearly homogeneous; (iii) Fð1;0Þ¼0Fð1;∞Þ1=ð s h m g Þ, and x= l ðxÞ1 for all x1=ð s h m g Þ. Remark 1: Since Fð1; l Þ¼xthis assumption implies that v l ðxÞ vx> l ðxÞ x>0;v2 l ðxÞ vx2>0 for all x2X(1) moreover, x l ðxÞ is strictly decreasing in. x: By these specifications labor market is characterized by a wage level, Wðx;jÞ, and a tightness, l ðxÞ. Every agent in the market takes Wð$Þand l ð$Þas given, which will be determined in equilibrium. In each period, an unemployed worker chooses in which occupation j to search for a job. A firm set the wage menu fðwðx;jÞ; l ðxÞÞ :j2fg;bgg taken the equilibrium wage menu, fðWðx;jÞ; l ðxÞÞ :j2fg;bg;x2Xgas given and commit to pay wage rate wðx;jÞif a productive match on occupation joccurs. 3.2. Value function of firms and free entry Any firm can post a vacancy in economy after incurring a cost c2ð0;y b Þ. If an occupation jis filled at a wage rate w, value of that filled occupation jto the firm discounted to the end of previous period is ð1þrÞJj fðwÞ¼yjwþð1 j Þ1 d jJj fðwÞ(2) Match probability is x l ðxÞ and continuation value of the match is ð1 j ÞJ j f ðWðx;jÞÞ. Therefore solving J j f from the equation above, value of opening vacancy is Fig. 2. Average and median unemployment duration. 6 Tightness, l , could have been used as argument instead x. In fact, in literature use of l is more common than x. But, in this paper xis more convenient for derivations. T. Büyükbas¸ aran / Central Bank Review 20 (2020) 85e97 89 JvðxÞ¼ cþx l ðxÞ yjWðx;jÞ Aj (3) where A j ≡ rþ j 1 j þ d j is a constant for j2fg;bg. Note that left hand side does not depend on j, since there should not be any arbitrage for the firm to open a vacancy in occupation gor occupation bdue to free entry. Precisely, J v ðxÞand the number of vacancies,vðxÞsatisfy J v ðxÞ0 and vðxÞ0 by condition of free entry, where the two inequalities hold with complementary slackness. Thus, if vðxÞ>0, the wage rate is Wðx;jÞ¼yjcAj l ðxÞ.x(4) for j2fg;bg. By Remark 1, wage function has following properties: 1)W0ðx;jÞ<02)xWðx;jÞis strictly concave for j2fg;bg. 3.3. Learning from search outcomes Workers learn about their types by observing their search outcomes using Bayesian updating. We will denote a worker’s prior expectation of being a high type 7 as P h and call it worker’s prior belief. New born workers entering the market will have a belief of P h ¼p h , where p h 2ð0;1Þis a scalar and common knowledge to every agent in the market. Lets say that P h is the prior belief about being a high type. Workers uses Bayesian updating over search outcomes. Updating will depend on in which particular occupation they try to find a job and on his/her search outcomes. Let o¼1 indicates that he has been productive in his current search (find the job) and o¼0 indicates that he has not been productive (fails to find the job). If a productive match occurs (worker finds a job) then posterior belief of being high type conditional on worker is searching in occupation jwhere the match rate in labor market is xis Pðhjx;j;o¼1Þ¼ Pðo¼1jx;j; s hÞPh Pðo¼1jx;j;hÞPhþPðo¼1jx;j;lÞPl ¼ s h m jxPh s h m jxPhþ s l1 m jxPl ¼1 1þ s l1 m j s h m j Pl Ph (5) Posterior belief of being high type conditional on worker is searching in occupation jand has not been productive is Pðhjx;j;o¼0Þ¼ Pðo¼0jx;j; s hÞPh Pðo¼0jx;j; s hÞPhþPðo¼0jx;j; s lÞPl ¼1 s h m jxPh 1 s h m jxPhþ1 s l1 m jxPl ¼1 1þ1 s l1 m jx 1 s h m jx Pl Ph (6) It is informative to compare posterior beliefs Pðhj:Þwith prior beliefs P h . For that purpose, the multiplier at the dominator just before P l P h is useful. If that multiplier is smaller than 1 than posterior is higher than prior and vice versa. Note that if search occurs in occupation j¼hand match is successful o¼1, then posterior of being high type is higher then prior (Pðhj:Þ>P h ) since s l ð1 m h Þ s h m h <1. If search occurs in occupation j¼gwith match rate xand match is not successful o¼0, then posterior of being high type is smaller then its prior (Pðhj:Þ<P h ) since 1 s l ð1 m h Þx 1 s h m h x >1. Moreover, increase on the belief of being high type with productive match does not depend on the rate x, whereas the decrease in beliefs with a nonproductive match is higher for higher x’s. Because, xdoes not affect the likelihood ratio of a successful match between the two types, while the fail in matching with a higher rate of job offers in occupation hgive a stronger signal of being type l. If the search has occurred for occupation j¼b, then a match will decrease, increase or not-change the belief of being high type depending on whether s l ð1 m b Þ s h m b >1, s l ð1 m b Þ s h m b <1or s l ð1 m b Þ s h m b ¼1, respectively. Again, fail in finding a job will decrease, increase or not-change the belief of being high type depending on whether s l ð1 m b Þx s h m b x <1, s l ð1 m b Þx s h m b x >1or s l ð1 m b Þx s h m b x ¼1, respectively. Among this alternatives assuming either s l ð1 m b Þ s h m b >1or s l ð1 m b Þ s h m b ¼1 is meaningful in the sense that new information (i.e. being accepted by occupation b) does not take posterior away from correct type of worker on average. This observation is summarized in Assumption II below. Moreover, as a special case we will keep no information content case on the search outcome in occupation b. The reason is that it may be more enlightening to characterize equilibrium with two different occupations; one with information content and one without the information content with respect to search outcomes. In such a characterization, the workers’behavior towards information content of occupation, as well as interaction between information content of search and other labor market variables such as wage rate, unemployment rate and unemployment duration can be more informative. Assumption II. (Search outcome gives information about true type) s h  s l ; m g > m b and s l ð1 m b Þ s h m b . 3.4. Value function of workers Consider first a worker with belief P h of being high type who is employed at wage win occupation jin any period. Denote the worker’s value function, discounted to the end of the previous period, as J e ðP h ;w;jÞ:After producing and obtaining the wage w, the separation shock forces the worker into unemployment with probability d j depending on occupation and then, independently, the exit shock forces the worker out of the market with probability j . If the worker remains employed after these two shocks, the continuation value is J e ðP h ;w;jÞ. If the worker is separated from the job but remains in the market, the continuation value is denoted VðP h Þ. If the worker is out of the market, the continuation value is 0. Thus Bellman equation for J e ð1þrÞJeðPh;w;jÞ¼wþð1 j Þð1 d ÞJeðPh;wÞþ d jVðPhÞ This yields JeðPh;w;jÞ¼1 Ajhw 1 j þ d jVðPhÞi(7) where A j ≡ rþ j 1 j þ d j is a constant for j2fg;bg. Now consider an unemployed worker who enters a period with belief P h . If he chooses occupation j, expected probability of finding a job is PðP h ;j;xÞwhere 7 Prior probability of being low type P l could have been used as belief structure without loss of generality. T. Büyükbas¸aran / Central Bank Review 20 (2020) 85e9790 PðPh;j;xÞ¼xPh s h m jþð1PhÞ s l1 m j ¼xCjPhþDj(8) where C j ¼ð s h þ s l Þ m j  s l and D j ¼ s l ð1 m j Þare occupation specific positive constants for j2fg;bg. His belief will be updated by Pðhjx;j;o¼1Þvia equation (5) if he finds a job and by Pðhjx;j;o¼0Þvia equation (6) if he/she fails. If she finds a job, her value function will be maxfJ e ðPðhjx;j;o¼1Þ;Wðx;jÞÞ; VðPðhjx;j;o¼1ÞÞg; if he fails to find a job, the value function will be VðPðhjx;j;o¼0ÞÞ:His expected return to apply occupation j excluding the unemployment benefit, is ð1 j ÞRðP h ;j;xÞ, where RðPh;j;xÞ¼PðPh;j;xÞmaxfJeðPðhjx;j;o¼1Þ;Wðx;jÞÞ;VðPðhjx;j;o¼1ÞÞg þð1PðPh;x;jÞÞVðPðhjx;j;o¼0ÞÞ Since we discounted value functions to the end of previous period, then ð1þrÞVðPhÞ¼bþð1 j Þmax j2fg;bgRðPh;j;xÞ(9) As in Gonzalez and Shi (2010), we assume that the workers always accept the job offer in any occupation. In other words, JeðPðhjx;j;o¼1Þ;Wðx;jÞ;jÞ>VðPðhjx;j;o¼1ÞÞ for all Ph2½0;1and j2fg;bg (10) Using definition of J e from equation (7), this condition is equivalent to Wðx;jÞ>ðrþ j ÞVðPðhjx;j;o¼1ÞÞ for all Ph2½0;1;x2Xand j2fg;bg(11) This condition is equivalent to condition (12) below. As Xis bounded, this condition can be satisfied for sufficiently high productivity y j and sufficiently small unemployment benefitband cost of opening vacancy c. This is reasonable since equilibrium wage rate depends positively on productivity y j which in turn make continuation value of employment higher whereas unemployment benefitbincreases continuation value of staying unemployed. Assumption III. (Reservation wage always met) Assume that productivity in the market j2fg;bgsatisfy that yjb c>Ajþ s hxfor all x2X(12) There are two reasons for this assumption. First, it is already well known that the workers will prolong their unemployment duration if their reservation wage has not been met. However, in this paper, the focus is on the unemployment caused by learning process of workers. Best way to isolate this type of unemployment is closing the reservation wage channel. Second reason is a technical one: with the use of this assumption the value function will be identified further by Theorem 3. Under Assumption II and Assumption III, we can rewrite expected return function using definition of J e from equation (7) RðPh;j;xÞ¼xCjPhþDj"1 Aj Wðx;jÞ 1 j þ d j Aj VPðhjx;j;o¼1Þ# þ1CjPhxDjxVPðhjx;j;o¼0Þ(13) Finally, under Assumption II and III value function is ð1þrÞVðPhÞ¼bþð1 j Þ max j2fg;bg(xCjPhþDj"1 Aj Wðx;jÞ 1 j þ d j Aj VPðhjx;j;o¼1Þ# þ1CjPhxDjxVPðhjx;j;o¼0Þ) (14) We use equation (14) to characterize equilibrium. 4. Definition and characterization of equilibrium 4.1. Definition of equilibrium Definition 1. The stationary symmetric equilibria with learning consists of value functions (J e ;V;J f ;J v ), worker choices (j),a wage function Wðx;jÞand a sequence of beliefs such that (a) The value functions (J e ;V;J f ;J v ) satisfies ð7Þð9Þ;ð2Þð3Þ, respectively (b) Given the wage function, all workers with same belief P h use same optimal strategy (j) j ¼gðP h Þ2G j ðP h Þwhich solves. ð9Þ (c) Bayesian Update: A worker with belief P h and optimal strategy ðjÞas in (b) update his belief with Pðhjx;j;o¼1Þas in (5) if he/ she finds a job and with Pðhjx;j;o¼0Þas in (6) if he/she fails to find a job. (d) Free entry: Wage function Wðx;jÞsatisfies (4). (e) Consistency: For every labor market, the measure of all vacancies divided by the measure of unemployed workers is equal to l ðxÞ.▪ After this general definition of the equilibrium, we will characterize equilibrium under Assumption I, II and III. 4.2. Some characterization of the equilibrium Theorem 2. (Existence Of Equilibrium) Under Assumptions I and III, there exists an equilibrium where all matches are accepted. Proof. Existence of value functions J f and J v are very standard and will be omitted here. One can check e.g. Rogerson et al. (2005) for the arguments. Existence of J e depends on existence of V. For existence of V;it is almost immediate to check that the right-hand side of (9) satisfies the Blackwell sufficiency conditions. Using standard arguments in Stokey et al. (1989), one can show that a unique Vexists, which is positive, bounded and continuous on M. Moreover, the correspondence of maximizers G j is non-empty, closed and upper hemicontinuous. Sufficiency of Assumption III for all matches accepted is little detailed and will be addressed at the appendix. ▪ Although the existence result does not depend on Assumption II, it will help us to characterize the value function with a reservation prior property. Theorem 3. (Reservation Belief Property).Under Assumptions I, II and III, there exists P * 2ð0;1Þsuch that all unemployed workers with a belief P h <P * choses to search in occupation b and all unemployed workers with a belief P h >P * choses to search in occupation g. Value function VðP h Þis (weakly) convex and strictly increasing on ½0;1. Proof. Under Assumption II and III, value function is represented by (14). Since Vis unique, first argument in max operator at the right hand side is strictly decreasing on [0,1] whereas second T. Büyükbas¸ aran / Central Bank Review 20 (2020) 85e97 91 argument is strictly increasing. I will just show that second argument is strictly increasing on [0,1] (proof of the other argument is very similar): Let Vbe a weakly increasing function, P ha >P hb where P ha ;P hb 2 ½0;1and g i ¼gðP hi Þ2GðP hi Þbe particular optimum choices where i2fa;bg. Then RðPha;gaÞRðPhb;gbÞ RðPha;gbÞRðPhb;gbÞ gbDjþCjPha1 AhWðgb;hÞ 1 j þ d hVðPðPha;g;1ÞÞ þ1gbDjþCjPhaVðPðPha;g;0ÞÞ gbDjþCjPhb1 AhWðgb;hÞ 1 j þ d hVðPðPhb;g;1ÞÞ þ1gbDjþCjPhbVðPðPhb;gb;0ÞÞ ¼gbDjðPha PhbÞ1 Ah Wðgb;hÞ 1 j þgbDjþCjPha1 Ah d hVðPðPha;gÞ gbDjþCjPhb1 A d hVðPðPhb;g;1ÞÞ þ1gbDjþCjPhaVðPðPha;g;0ÞÞ 1gbDjþCjPhbVðHðPhb;gb;0ÞÞ gbDjðPha PhbÞ1 Ah Wðgb;hÞ 1 j þgbDjþCjPha1 Ah d hVðPðPhb;g;1Þ gbDjþCjPhb1 Ah d hVðPðPhb;g;1ÞÞ þ1gbDjþCjPhaVðPðPb;g;0ÞÞ 1gbDjþCjPhbVðPðPhb;gb;0ÞÞ ¼gbCðPha PhbÞWðgb;hÞ Ahð1 j Þþ d h Ah VðPðPhb;g;1Þ VðPðPhb;g;0ÞÞ >gbCðPha PhbÞ½VðPðPhb;g;1ÞÞVðPðPhb;g;0ÞÞ 0 First inequality uses the fact that g a is the maximizer for P ha : Second inequality uses VðPðP ha ;gÞÞVðPðP hb ;g;1ÞÞ and VðPðP ha ;g b ; 0ÞÞ  VðPðP hb ;g b ;*ÞÞ:Strict inequlaity uses Assumption III which is equivalent to Wðx; m m Þ>ðrþ j ÞVð4ðP; m m ÞÞ:Last inequality uses the fact that PðP hb ;g;o¼1Þ>PðP hb ;g;o¼0Þand V is weakly increasing function. Since first component of max operator is decreasing and second operator is strictly increasing we have a reservation prior property (Note that we did not restrict P * to be interior of ½0;1.For that a relevant bound on y g y b is sufficient). Finally,proof of convexity of Value function is similar to proof in Nyarko (1994).▪ 4.3. Steady state distributions Denote e j i ðP h Þas measure of type-i workers employed in occupation-j with belief P h and u i ðP h Þas measure of unemployed type-iworkers before labor market opens in a period for i2fh;lg and occupation j2fg;bg. Probability of new borns to be type iis p i . New workers enter with belief p 0 ¼p h . Let Tðp 0 Þbe tree of equilibrium beliefs generated from p 0 . Then stationary distributions of workers over beliefs is fðe j i ðpÞ;u i ðpÞÞ j2fg;bg i2fh;lg :p2Tðp 0 Þg Unemployed workers are in 3 groups: Newborns, UE was E in previous period, UE was UE in previous period: 1. Newborns: Outflow and inflow from this group is j p i . Therefore this group is always stationary uh iðp0Þ¼ j pip2Tðp0Þi2fh;lg(15) 2. Unemployed - was employed in previous period. Outflow: All workers move out from this group. Inflow: Separated from jobs and survive Occupation j: Belief is 4ðpÞfor some p2Tðp 0 Þ uið4ðpÞÞ¼X j ð1 j Þ d jej ið4ðpÞÞ;p2Tðp0Þi2fh;lg(16) 3. Unemployed - was unemployed in previous period Occupation j: Belief is nðpÞ¼Pðp;x;j;o¼0Þfor some p2Tðp 0 Þ. Everybody outflow. Inflow is who survives and fails to find a job uj HðnðpÞÞ¼ð1 j Þ1 s h m hxJðpÞuHðpÞ;p2Tðp0Þ(17) uj LðnðpÞÞ¼ð1 j Þ1 s Lð1 m hÞxJðpÞuLðpÞ;p2Tðp0Þ(18) Employed workers are just in one group. For Occupation j, belief is p2Tðp 0 Þ. Outflow is worker who dies or separated. Inflow is worker who find a job among searchers:  j þð1 j Þ d jej Hð4ðpÞÞ¼ð1 j Þ s H m jxJðpÞuHðpÞ;p2Tðp0Þ (19)  j þð1 j Þ d jej Lð4ðpÞÞ¼ð1 j Þ s L1 m jxJðpÞuLðpÞ;p2Tðp0Þ (20) The stationary distribution is determined by (15)e(20) and with the requirement that the total measure of workers is one. Because the equilibrium is block recursive, optimal choices are independent of the distribution, and so (15)e(20) are linear equations of the Fig. 3. Shift in Beveridge curve. T. Büyükbas¸aran / Central Bank Review 20 (2020) 85e9792